University of California • Berkeley

Gift of

MRS. GRIFFITH C. EVANS

^1-^^ l^i cLu^o^ !'^'

HISTORY OF

THE THEORY OF PROBABILITY.

A HISTOEY

OF THE

MATHEMATICAL THEORY OF PROBABILITY

T

FROM THE TIME OF PASCAL TO TEAT

OF LAPLACE.

BY

I. TODHUNTEH, M.A., F.RS.

1

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\XYr O' C P . UlMOAy^^ ^

(Q^aml)til5ge nnti Hoution: MACMILLAN AND CO.

I86^

CambritJcjc:

PRINTED BY C. J. CLAY. M.A. AT THE UNIVERSITY PRESS.

PREFACE.

The favourable reception which has been granted to my History of the Calculus of Variations daring the Xineteenth Century has encouraged me to undertake another work of the same kind. The subject to which I now invite attention has high claims to consideration on account of the subtle problems which it involves, the valuable contributions to analysis which it has produced, its important practical applications, and the eminence of those who have cultivated it.

The nature of the problems which the Tlieory of Probability contemplates, and the influence which this Theory has exercised on the progress of mathematical science and also on the concerns of practical life, cannot be discussed within the limits of a Preface ; we may however claim for our subject all the interest wdiicli illus- trious names can confer, by the simple statement that nearly every gi-eat mathematician within the range of a century and a half will come before us in the course of the history. To mention only the most distinguished in this distinguished roll — we sliall find here — Pascal and Format, worthy to be associated by kindred genius and character— De Moivre with his rare powers of analysis, which seem to belong only to a later epoch, and which justify the honour in which he was held by Newton — Leibnitz and the emi- nent school of which he may be considered the founder, a school including the Bernoullis and Euler — D'Alembert, one of the most conspicuous of those who brought on the French revolution, and Condorcet, one of the most illustrious of its victims — Lagrange and Laplace who survived until the present century, and may be regarded as rivals at that time for the suj^remacy of the mathe- matical world.

I will now give an outline of the contents of the book.

The first Chapter contains an account of some anticipations of the subject which are contained in the writings of Cardan, Kepler and Galileo.

The second Chapter introduces the Chevalier de Mere' who having puzzled himself in vain over a problem in chances, fortunately turned for help to Pascal : the Problem of Points is discussed in the correspondence between Pascal and Format, and thus the Theory of Probability begins its career.

Vi PREFACE.

The third Chapter analyses the treatise in which Huygens in 1659 exhibited what was then known of the subject. Works such as this, which present to students the opportunity of becoming acquainted with the speculations of the foremost men of the time, cannot be too highly commended ; in this respect our sub- ject has been fortunate, for the example which was afforded by Huygens has been imitated by James Bernoulli, De Moivre and Laplace — and the same course might with great advantage be pursued in connexion with other subjects by mathematicians in the present day.

The fourth Chapter contains a sketch of the early history of the theory of Permutations and Combinations ; and the fifth Chap- ter a sketch of the early history of the researches on Mortality and Life Insurance. Neither of these Chapters claims to be ex- haustive ; but they contain so much as may suffice to trace the connexion of the branches to which they relate with the main sub- ject of our history.

The sixth Chapter gives an account of some miscellaneous in- vestigations between the years 1670 and 1700. Our attention is directed in succession to Caramuel, Sauveur, James Bernoulli, Leibnitz, a translator of Huygens's treatise whom I take to be Arbuthnot, Roberts, and Craig — the last of whom is notorious for an absurd abuse of mathematics in connexion with the probability of testimony.

The seventh Chapter analyses the Ars Conjectandi of James Bernoulli. This is an elaborate treatise by one of the greatest mathematicians of the age, and although it was unfortunately left incomplete, it affords abundant evidence of its author's ability and of his interest in the subject. Especially we may notice the famous theorem which justly bears the name of James Bernoulli, and which places the Theory of Probability in a more commanding position than it had hitherto occupied.

The eighth Chapter is devoted to Montmort. He is not to be compared for mathematical power with James Bernoulli or De Moivre; nor does he seem to have formed a very exalted idea of the true dignity and importance of the subject. But he was en- thusiastically devoted to it; he spai^ed no labour himself, and his influence direct or indirect stimulated the exertions of Nicolas Bernoulli and of De Moivre.

The ninth Chapter relates to De Moivre, containing a full analysis of his Doctrine of Chances, De Moivre brought to bear on the subject mathematical powers of the highest order ; these powers are especially manifested in the results which he enun- ciated respecting the great problem of the Duration of Play. Unfortunately he did not publish demonstrations, and Lagrange

PREFACE. Vii

himself more than fifty years later found a good exercise for his analytical skill in supplying the investigations ; this circumstance compels us to admire De Moivre's powers, and to regret the loss which his concealment of his methods has occasioned to mathe- matics, or at least to mathematical history.

De Moivre's Doctrine of Chances formed a treatise on the subject, full, clear and accurate ; and it maintained its place as a standard work, at least in England, almost down to our own day.

The tenth Chapter gives an account of some miscellaneous investigations between the years 1700 and 17-30. These inves- tio-ations are due to Nicolas Bernoulli, Arbuthnot, Browne, Mairan, Nicole, Buffon, Ham, Thomas Simpson and John Bernoulli.

The eleventh Chapter relates to Daniel Bernoulli, containing an account of a series of memoirs published chiefly in the volumes of the Academy of Petersburg ; the memoirs are remarkable for boldness and originality, the first of them contains the celebrated theory of Moral Expectation.

The twelfth Chapter relates to Euler ; it gives an account of his memoirs, which relate j^rincipally to certain games of chance.

The thirteenth Chapter relates to D'Alembert ; it gives a full account of the objections which he urged against some of the fundamental principles of the subject, and of his controversy with Daniel Bernoulli on the mathematical investisj-ation of the ^ain to human life which would arise from the extirpation of one of the most fatal diseases to which the human race is liable.

The fourteenth Chapter relates to Bayes ; it explains the me- thod by which he demonstrated his famous theorem, which may be said to have been the origin of that part of the subject which relates to the probabilities of causes as inferred from observed effects.

The fifteenth Chapter is devoted to Lagrange ; he contributed to the subject a valuable memoir on the theory of the errors of observations, and demonstrations of the results enunciated by De Moivre respecting the Duration of Play.

The sixteenth Chapter contains notices of miscellaneous inves- tigations between the years 1750 and 17^0. This Chapter brings before us Kaestner, Clark, Mallet, John Bernoulli, Beguelin, Michell, Lambert, Buffon, Fuss, and some others. The memoir of Michell is remarkable ; it contains the famous argument for the existence of design drawn from the fact of the closeness of certain stars, like the Pleiades.

The seventeenth Chapter relates to Cordorcet, who published a large book and a long memoir upon the Theory of Probability. He chiefly discussed the probability of the correctness of judg- ments determined by a majority of votes ; he has the merit of first

vlii PREFACE.

submitting this question to mathematical investigation, but his own results are not of great practical importance.

The eighteenth Chapter relates to Trembley. He wrote several memoirs with the main design of establishing by elementary methods results which had been originally obtained by the aid of the higher branches of mathematics ; but he does not seem to have been very successful in carrying out his design.

The nineteenth Chapter contains an account of miscellaneous investigations between the years 1780 and 1800. It includes- the following names ; Borda, Malfatti, Bicquilley, the writers in the mathematical portion of the Encydopedie Methodique, D'Anieres, Waring, Prevost and Lhuilier, and Young.

The twentieth Chapter is devoted to Laplace ; this contains a full account of all his writings on the subject of Probability. First his memoirs in chronological order, are analysed, and then the great work in which he embodied all his own investigations and much derived from other writers. 1 hope it will be found that all the parts of Laplace's memoirs and work have been carefully and clearly expounded ; I would venture to refer for examples to Laplace's method of approximation to integrals, to the Problem of Points, to James Bernoulli's theorem, to the problem taken from Buffon, and above all to the famous method of Least Squares. With respect to the last subject I have availed myself of the guidance of Poisson's luminous analysis, and have given a general investigation, applying to the case of more than one unknown element. I hope I have thus accomplished something towards ren- dering the theory of this important method accessible to students.

In an Appendix I have noticed some writings which came under my attention during the printing of the work too late to be referred to their proper places.

I have endeavoured to be quite accurate in my statements, and to reproduce the essential elements of the original works which I have analysed. I have however not thought it indispen- sable to preserve the exact notation in which any investigation w^as first presented. It did not appear to me of any importance to retain the specific letters for denoting the known and unknown quantities of an algebraical problem which any writer may have chosen to use. Very often the same problem has been dis- cussed by various writers, and in order to compare their methods with any facility it is necessary to use one set of symbols through- out, although each writer may have preferred his peculiar set. In fact by exercising care in the choice of notation I believe that my exposition of contrasted methods has gained much in brevity and clearness without any sacrifice of real fidelity.

I have used no symbols which are not common to all mathc-

PREFACE. IX

matical literature, except \n wliicli is an abbreviation for the pro- duct 1 . 2, ...'?i, frequently but not universally employed : some such symbol is much required, and I do not know of any which is pre- ferable to this, and I have accordingly introduced it in all my publications.

There are three important authors whom I have frequently cited whose works on Probability have passed through more than one edition, Montmort, De Moivre, and Laplace : it may save trouble to a person who may happen to consult the present volume if I here refer to pages 79, 13G, and 495 where I have stated which editions I have cited.

Perhaps it may appear that I have allotted too much space to some of the authors whose works I examine, especially the more ancient ; but it is difficult to be accurate or interesting if the nar- rative is confined to a mere catalogue of titles : and as experience shews that mathematical histories are but rarely undertaken, it seems desirable that they should not be executed on a meagre and inadequate scale.

I will here advert to some of my predecessors in this depart- ment of mathematical history ; and thus it will appear that I have not obtained much assistance from them.

In the third volume of Montucla's Histoire des Mathematiqiies pages 380—426 are devoted to the Theory of Probability and the kindred subjects. I have always cited this volume simply by the name Montucla, but it is of course well known that the third and fourth volumes were edited from the author's manuscripts after his death by La Landc. I should be sorry to apj^ear ungrateful to Montucla; his work is indispensable to the student of mathema- tical history, for whatever may be its defects it remains without any rival. But I have been much disappointed in what he says respecting the Theory of Probability ; he is not copious, nor accu- rate, nor critical. Hallaui has characterised him with some severity, by saying in reference to a point of mathematical history, " Mon- tucla is as superficial as usual :" see a note in the second Chapter of the first volume of the History of the Literature of Europe.

There are brief outlines of the history involved or formally incorporated in some of the elementary treatises on the Theory of Probability : I need notice only the best, which occurs in the Treatise on Probability published in the Library of L^seful Know- ledge. This little work is anonymous, but is known to have been written by Lubbock and Drinkwater ; the former is now Sir John Lubbock, aud the latter changed his name to Drinkwater-Bethune : see Professor De Morgan's Arithmetical Books... page 106, a letter by him in the Assurance Magazine, Yol. TX. page 238, and another letter by him in the Times, Dec. 16, 1862. The treatise is inter-

X PREFACE.

esting and valuable, but I have not been able to agree uniformly with the historical statements which it makes or implies.

A more ambitious work bears the title Histoire dii Calcul des Prohabilites depuis ses origines jusqud nos jours par Charles Gouraud... Paris, 184^8. This consists of 148 widely printed octavo pages ; it is a popular narrative entirely free from mathematical symbols, containing however some important specific references. Exact truth occasionally suffers for the sake of a rhetorical style unsuitable alike to history and to science; nevertheless the general reader will be gratified by a lively and vigorous exhibition of the whole course of the subject. M. Gouraud recognises the value of the purely mathematical part of the Theory of Probability, but will not allow the soundness of the applications which have been made of these mathematical formulse to questions involving moral or political considerations. His history seems to be a portion of a very extensive essay in three folio volumes containing 1929 pages written when he was very young in competition for a prize pro- posed by the French Academy on a subject entitled Theorie de la Certitude; see the Rapport by M. Franck in the Seances et Tra- vaux de V Academie des Sciences morales et politiques, Vol. x. pages 372, 382, and Vol. XI. page 139. It is scarcely necessary to remark that M. Gouraud has gained distinction in other branches of literature since the publication of his work which we have here noticed.

There is one history of our subject which is indeed only a sketch but traced in lines of light by the hand of the great master himself: Laplace devoted a few pages of the introduction to his celebrated work to recording the names of his predecessors and their contributions to the Theory of Probability. It is much to be regretted that he did not supply specific references through- out his treatise, in order to distinguish carefully between that which he merely transmitted from preceding mathematicians and that which he originated himself.

It is necessary to observe that in cases where I point out a similarity between the investigations of two or more writers I do not mean to imply that these investigations could not have been made independently. Such coincidences may occur easily and naturally without any reason for imputing unworthy conduct to those who succeed the author who had the priority in publication. I draw attention to this circumstance because I find with regret that from a passage in my former historical work an inference has been drawn of the kind which I here disclaim. In the case of a writer Uke Laplace who agrees with his predecessors, not in one or two points but in very many, it is of course obvious that he must have borrowed largely, and we conclude that he supposed the

PREFACE. XI

erudition of his contemporaries would be sufficient to prevent them from ascribing to himself more than was justly due.

It will be seen that I have ventured to survey a very extensive field of mathematical research. It has been mv aim to estimate carefully and impartially the character and the merit of tlie numerous memoirs and works which I have examined; my criti- cism has been intentionally close and searching, but I trust never irreverent nor unjust. I have sometimes explained fully the errors which I detected; sometimes, when the detailed exposition of the error would have recpiired more space than the matter deserved, I have given only a brief indication which may be serviceable to a student of the original production itself I have not hesitated to introduce remarks and developments of my own whenever the subject seemed to require them. In an elaborate German review of my former puljlication on mathe- matical history it was suggested that my own contributions were too prominent, and that the purely historical character of the work was thereby impaired; but I have not been induced to change my plan, for I continue to think that such additions as I have been able to make tend to render the subject more in- telligible and more complete, without disturbing in any serious degree the continuity of the history. I cannot venture to expect that in such a difficult subject I shall be quite free from error either in my exposition of the labours of others, or in my own contributions; but I hope that such failures will not be numerous nor important. I shall receive most gratefully intimations of any errors or omissions whicli may be detected in the work.

I have been careful to corroborate mv statements bv exact quotations from the originals, and these I have given in the lan- guages in which they were published, instead of translating them ; the course which I have here adopted is I understand more agree- able to foreign students into whose hands the book may fall. I have been careful to preserve the historical notices and references which occurred in the works I studied ; and by the aid of the Table of Contents, the Chronological List, and the Index, which accompany the present volume, it will be easy to ascertain with regard to any proposed mathematician down to the close of the eighteenth century, whether he has written au}'thing upon the Theory of Probability.

I have carried the history down to the close of the eighteenth century ; in the case of Laplace, however, I have passed beyond this limit: but by far the larger part of his labours on the Theory of Probability were accomplished during tlie eighteenth century, though collected and republished by him in his celebrated work in the early part of the present century, and it was therefore conve-

Xll PREFACE.

nient to include a full account of all his researches in the present volume. There is ample scope for a continuation of the work which should conduct the history through the period which has elapsed since the close of the eighteenth century ; and I have already made some progress in the analysis of the rich materials. But when I consider the time and labour expended on the present volume, although reluctant to abandon a long cherished design, I feel far less sanguine than once I did that I shall have the leisure to arrive at the termination I originally ventured to pro- pose to myself

Although I wish the present work to be regarded princijDally as a history, yet there are two other aspects under which it may solicit the attention of students. It may claim the title of a com- prehensive treatise on the Theory of Probability, for it assumes in the reader only so much knowledge as can be gained from an elementary book on Algebra, and introduces him to almost every process and every species of problem which the literature of the subject can furnish; or the work may be considered more spe- cially as a commentary on the celebrated treatise of Laplace, — and perhaps no mathematical treatise ever more required or more deserved such an accompaniment.

My sincere thanks are due to Professor De Morgan, himself conspicuous among cultivators of the Theory of Probability, for the kind interest which he has taken in my work, for the loan of scarce books, and for the suggestion of valuable references. A similar interest was manifested by one prematurely lost to science, whose mathematical and metaphysical genius, attested by his marvellous work on the Laws of Thought, led him naturally and rightfully in that direction which Pascal and Leibnitz had marked with the unfading lustre of their approbation; and who by his rare ability, his wide attainments, and his attractive character, gained the affection and the reverence of all who knew him.

I. TODHUNTER.

Cambridge, May, 1865.

CONTENTS.

PAGE

Chapter I. Cardan. Kepler. Galileo . . l

Commentary on Dante, i. Cardan, Be Ludo Alece, i, Kepler, De Stella Nova, 4. Galileo, Considerazione sopra il Giuco del Dadi, 4 ; Lettcre, 5.

Chapter II. Pascal and Fermat .... 7

Quotations from Laplace, Poisson, and Boole, 7. De Mare's Problems, 7, Problem of Points, 9. De Merc's dissatisfaction, 11. Opinion of Leib- nitz, 12. Fermat's solution of the Problem of Points, 13. Roberval, 13. Pascal's error, 14. The Arithmetical Triangle, 17. Pascal's design, 20. Contemporary mathematicians, 21.

Chapter III. Huygens 22

De liatiodniis in Ludo Alece, 22. English translations, 23. Huygens's solu- tion of a problem, 24 ; Problems proposed for solution, 25.

Chapter IV. On Combinations . . . .20

"William Buckley, 26. Bernardus Bauhusius and Erycius Puteanus, 27. Quo- tation from James Bernoulli, 28. Pascal, 29. Schooten, 30. Leibnitz, Dissertaiio de Arte Comhinatoria, 31 ; his fruitless attempts, 33. Wallis'3 Algebra, 34; his errors, 35.

Chapter V. Mortality and Life Insurance . 37

John Graunt, 37. Van Hudden and John de Witt, 38, Sir William Petty, 39. Correspondence between Leibnitz and James Bernoulli, 40. Halley, 4 1 ; his table, 42 ; geometrical illustration, 43.

Chapter YI. Miscellaneous Investigations between

THE YEARS 1670 AND 1700 -ii

Caramuel's Matliesis Biceps, 44 ; his errors, 45, 46. Sauveur on Bassette, 46. James Bernoulli's two problems, 47. Leibnitz, 47; his error, 48. Of the Laws of Chance, ascribed to Motte, 48 ; really by Arbuthnot, 49 ; quotation from the preface, 50 ; error, 52 ; problem proposed, 53. Francis Roberts, An Arithmetical Paradox, 53, Craig's Theologies Chris- tiance Principia Maihematica, 54. Credihility of Human Testimony, 55.

Chapter YII. Jaihes Bernoulli . . . .56

Correspondence with Leibnitz, 56 ; Ars Conjectandi, 57. Error of Montucia, ^S. Contents of the Ars Conjectandi, 58. Problem of Points, 59. James Bernoulli's own method for problems on chances, 60; his solution of a

XIV CONTENTS.

PAGE

problem on Duration of Play, 6i ; he points out a plausible mistake, 63; treats of Permutations and Combinations, 64 ; his Numbers, 65 ; Pro- blem of Points, 66 ; his problem with a false but plausible solution, 67 ; his famous Theorem, 71 ; memoir on infinite series, 73; letter on the game of Tennis, 75. Gouraud's opinion, 77.

Chapter VIII. Montmoet 78

Fontenelle's Eloge, 78. Two editions of Montmort's book, 79 ; contents of the book, 80; De Moivre's reference to Montmort, 81; Montmort treats of Combinations and the Binomial Theorem, 82 ; demonstrates a formula given by De Moivre, 84 ; sums certain Series, 86 ; his researches on Pha- raon, 87; Treize, 91; Bassette, 93. Problem sob ed by a lady, 95. Pro- blem of Points, 96; Bowls, 100; Duration of 1 lay, loi ; Her, 106; Tas, no. Letter from John Bernoulli, 113. Nicolas Bernoulli's game of chance, 116. Treize, 120. Summation of Series, 121. Waldegrave's problem, 122, Summation of Series, 125. Malebranche, 126. Pascal, 128. Sum of a series, 129. Argument by Arbuthnot and 's Gravesande on Divine Providence, 130. James Bernoulli's Theorem, 131. Montmort's views on a History of Mathematics, 132. Problems by Nicolas Ber- noulli, 133. Petersburg Problem, 134.

Chapter IX. De Moivre 135

Testimony of John Bernoulli and of Newton, 135. Editions of the Doc- trine of Chances, 136. De Mensura Sortis, 137. De Moivre's approximate formula, 138; his Lemma, 138; Waldegrave's problem, 139; Duration of Play, 140; Doctrine of Chances, 141; Litroduction to it, 142; con- tinued fractions, 143; De Moivre's approximate formula, 144; Duration of Play, 147; Woodcock's problem, 147; Bassette and Pharaon, 150; Numbers of Bernoulli, 151; Pharaon, 152; Treize or Rencontre, 153; Bowls, 159; Problem on Dice, 160; Waldegrave's problem, 162; Hazard, 163; Whist, 164; Piquet, 166; Dirration of Play, 167; Recur- ring Series, 178; Cuming's problem, 182 ; James Bernoulli's Theorem, 183 ; problem on a Run of Events, 184; Miscellanea Analytica, 187; contro- versy with Montmort, 188; Stirling's theorem, 189; Arbuthnot's argu- ment, 193.

Chapter X. Miscellaneous Investigations BET^yEEN

THE YEARS 1700 AND 1750 191?

Nicolas Bernoulli, 194. Barbeyrac, 196. Arbuthnot's argument on Divine Providence, 197. Waldegrave's problem, 199. Browne's translation of Huygens's treatise, 199. Mairan on Odd and Even, 200. Nicole, 201. BufFon, 203. Ham, 203. Trente-et-quarante, 205. Simpson's Nature and Laws of Chance, 206; he adds something to De Moivre's results, 207; sums certain Series, 210; his Miscellaneous Tracts, ■21 1. Problem by John Bernoulli, 212.

CONTENTS. XV

PAGB

Chapter XL Daniel Bernoulli . . . .213

Theory of Moral Expectation, 213; Petersburg Problem, 220; Inclination of planes of Planetary Orbits, 122 ; Small-pox, 224; mean dm-ation of mar- riages, 229; Daniel Bernoulli's problem, 231 ; Births of boys and girls, 235; Errors of observations, 236,

Chapter XIL Euler 239

Treize, 239; Mortality, 240; Annuities, 242; Pharaon, 243; Lottery, 2^5; Lottery, 247; notes on Lagrange, 249; Lottery, 250; Life Assurance, 256.

Chapter XIII. D'Alembert 258

Croix ou Pile, 258; Petersburg Problem, 259; Small-pox, 265; Petersburg Problem, 275; Mathematical Expectation, 276; Inoculation, 277; Croix ou Pile, 279; Petersburg Problem, 280; Inoculation, 282; refers to Laplace, 287; Petersburg Problem, 288; error in a problem, 290.

Chapter XI Y. Bates 2.94

Bayes's theorem, 295; his mode of investigation, 296; area of a curve, 298. Price's example, 299. Approximations to an area, 300,

Chapter XV. Lagrange 301

Theory of errors, 301; Recurring Series, 313; Problem of Points, 315; Dura- tion of Play, 316; Annuities, 320.

Chapter XVI. Miscellaneous Investigations be- tween the years iToO AND 1780 . . .321

Kaestner, 321. Dodson, 322. Hoyle, 322. Clark's Laics of CJiauce, 323. Mallet, 325, John Bernoulli, 325. Beguelin, on a Lottery problem, 3 28 ; on the Petersburg Problem, 332. Michell, 332. John Bernoulli, 335. Lambert, 335. Mallet, 337. Emerson, 343. Buffon, on gambling, 344 ; ou the Petersburg Problem, 345 ; his own problem, 347. Fuss, 349.

Chapter XVIL Condorcet 351

Dlscours Preliminaire, 351; Essai, 353; first Hj-pothesis, 353; second Hypo- thesis, 357; problem on a Run of Events, 361 ; election of candidates for an ofl&ce, 370; problems on inverse probability, 37S; Risk which may be neglected, 3S6 ; Trial by Jury, 388; advantageous Tribunals, 391; ex- pectation, 392 ; Petersburg Problem, 393 ; evaluation of feudal rights, 395 ; probability of future events, 398; extraordinary facts, 400; credibility of Roman History, 406. Opinions on Condorcet's merits, 409.

XVI CONTENTS.

PAGR

Chapter XVIII, Teembley . . . . • . 411

Problem of Points, 412; probability of causes, 413; problem of births, 415; lottery problem, 421; small-pox, 423; duration of marriages, 426; theory of errorS; 428 ; Her, 429.

Chapter XIX. Miscellaneous Investigations be- tween the years 1780 AND 1800 . . . 432

Prevost, 432. BorJa, 432. Malfatti, 434. Bicquilley, 438. Encyclopedie Me- tkodique, 441. D'Anieres, 445, Waring, 446. Ancillon, 453. Prevost and Lhuilier, 45 3. Young, 463.

Chapter XX. Laplace 464

Memoirs of 1774, 464; recurring series, 464; Duration of Play, 465; Odd and Even, 465; probability of causes, 465 ; theory of errors, 468; Peters- burg Problem, 470; Memoir of 1773, 473; Odd and Even, 473; Problem of Points, 474 ; Duration of Play, 474 ; Inclination of Orbits of Comets, 475 ; Memoir of 1781, 476 ; Duration of Play, 476; approximation to integrals, 478; problem of births, 482; theory of errors, 484; Memoir of 1779, 484 ;. Generating Functions, 484; Memoir of 1782, 485; Memoirs of 1783, 485; Memoir of 1809, 487; Memoir of 18 10, 489; Connaissance des Terns, 490; Problem on Comets, 491; Theorie...des Probalilites, 495; editions of it, 495; dedication to Napoleon, 496; Laplace's researches in Physical Astronomy, 499^ Pascal's argument, 500; illusions, 501; Bacon, 503; Livre I. 505 ; Generating Functions, 505 ; Method of approximation, 512 ; examples, 516; Livre II. first Chapter, 527; second Chapter 527; Odd and Even, 527; Problem of Points, 528; Fourth Supplement, 532; Walde- grave's Problem, 535; Run of Events, 539; Inclination of the Orbits of Planets, 542; election of candidates, 547; third Chapter, 548; James Bernoulli's Theorem, 548; Daniel Bernoulli's problem, 558; fourth Chap- ter, 560; Poisson's problem, 561; Least Squares, 571; history of this subject, 588; fifth Chapter, 589. BufFon's problem, 590; sixth Chapter, 592; a Definite Integral, 594; seventh Chapter, 598; eighth Chapter, 601 ; Small-pox, 60 r; duration of marriages, 602; ninth Chapter, 605 ; exten- sion of James Bernoulli's Theorem, 607 ; tenth Chapter, 609 ; inequal- ity, 609; eleventh Chapter, 609; first Supplement, 610; second Supple- ment, 611; third Supplement, 612; quotation fi-om Poisson, 613.

Appendix . 614

John de Witt, 614. Rizzetti, 614. Kahle, 615. 's Gravesande, 616. Quotation from John Bernoulli, 616. Mendelsohn, 616. Lhuiher, 618. Waring, 618.

r

CHAPTER I.

CARDAN. KEPLER. GALILEO.

1. The practice of games of chance must at all times have directed attention to some of the elementary considerations of the Theory of Probability. Libri finds in a commentary on the Divina Commedia of Dante the earliest indication of the different proba- bility of the various throws which can be made with three dice. The passage from the commentary is quoted by Libri ; it relates to the first line of the sixth canto of the Purgatorio. The com- mentary was published at Venice in 1477. See Libri, Histoire des Sciences Mathematiques en Italie, Vol. ii. p. 188.

2. Some other intimations of traces of our subject in older writers are given by Gouraud in the following passage, unfor- tunately without any precise reference.

Les anciens paraissent avoir eutierement ignore cette sorte de calcul. L'eruditioii moderne en a, il est vrai, trouve quelques traces dans un poeme en latin barbare intitule : De Vetidq, oeuvre d'un nioine du Bas- Empire, dans un commentaire de Dante de la fin du XY^ siecle, et dans les ecrits de plusieurs matliematiciens italiens du moyeu age et

de la renaissance, Pacioli, Tartaglia, Peverone ; Go\irsi\\d,IIisto{re

du Calcul des Frohahilites, page 3.

3. A treatise by Cardan entitled De Ludo Alece next claims our attention. This treatise was published in 1663, in the first volume of the edition of Cardan's collected works, long after Cardan's death, which took -place in 1576.

1

2 CARDAN.

Montmort says, " Jerome Cardan a donne un Traits De Ludo Alese ; mais on n'y trouve que de I'erudition et des reflexions morales." Essai d'Analyse.-.ip. XL. Libri says, "Cardan a ecrit un traite special de Ludo Alece, ou se trouvent resolues plusieurs questions d'analyse combinatoire." Histoire, Vol. ill. p. 176. The former notice ascribes too little and the latter too much to Cardan.

4. Cardan's treatise occupies fifteen folio pages, each containing two columns; it is so badly printed as to be scarcely intelligible. Cardan himself was an inveterate gambler ; and his treatise may be best described as a gambler's manual. It contains much mis- cellaneous matter connected with gambling, such as descriptions of games and an account of the precautions necessary to be employed in order to guard against adversaries disposed to cheat : the discussions relating to chances form but a small portion of the treatise.

5, As a specimen of Cardan's treatise we will indicate the contents of his thirteenth Chapter. He shews the number of cases which are favourable for each throw that can be made with two dice. Thus two and twelve can each be thrown in only one way. Eleven can be thrown in two ways, namely, by six appear- ing on either of the two dice and five on the other. Ten can be thrown in three ways, namely, by five a23pearing on each of the dice, or by six appearing on either and four on the other. And so on.

Cardan proceeds, *'Sed in Ludo fritilli undecim puncta adjicere decet, quia una Alea potest ostendi."...The meaning apparently is, that the person who throws the two dice is to be considered to have thrown a given number when one of the dice alone exhibits that number, as well as when the number is made up by the sum of the numbers on the two dice. Hence, for six or any smaller number eleven more ftivourablc cases arise besides those already considered.

Cardan next exhibits correctly the number of cases which are favourable for each throw that can be made with three dice. Thus three and eighteen can each be thrown in only one way ; four and

CARDAN, 3

seventeen can each be thrown in three ways ; and so on. Cardan also gives the following list of the number of cases in Fritillo :

12 34 5 6789 10 11 12

108 111 115 120 12G 133 33 36 37 36 33 26

Here we have corrected two misprints by the aid of Cardan's verbal statements. It is not obvious what the table means. It might be supposed, in analogy with what has already been said, that if a person throws three dice he is to be considered to have thrown a given number when one of the dice alone exhibits that number, or when two dice together exhibit it as their sum, as well as when all the three dice exhibit it as their sum : and this would agree wdth Cardans remark, that for numbers higher than twelve the favourable cases are the same as those already given by him for three dice. But this meaning does not agree with Cardan's table ; for with this meaning we should proceed thus to find the cases favourable for an ace : there are 5^ cases in which no ace appears, and there are 6' cases in all, hence there are 6^ — 5^ cases in which we have an ace or aces, that is 91 cases, and not 108 as Cardan gives.

The connexion between the numbers in the ordinary mode of using dice and the numbers which Cardan gives appears to be the following. Let n be the number of cases which are favour- able to a given throw in the ordinary mode of using three dice, and N the number of cases favourable to the same throw in Cardan's mode ; let m be the number of cases favourable to the given throw in the ordinary mode of using two dice. Then for any throw not less than thirteen, N=n ; for any throw between seven and twelve, both inclusive, N = Sni + n ; for any throw not greater than six, i\^= 108 + 3?/i + n. There is only one deviation from this law ; Cardan gives 26 favourable cases for the throw twelve, and our proposed law would give 3 + 25, that is 28.

We do not, however, see what simple mode of playing with three dice can be suo'o-ested which shall oive favourable cases agreeing in number with those determined by the above law.

6. Some further account of Cardan's treatise will be found

1—2

^ KEPLER.

in the Life of Cardan, by Henry Moiiey, Vol. I. pages 92 — 95. Mr Morley seems to misunderstand the words of Cardan which he quotes on his page 92, in consequence of which he says that Cardan " lays it down coolly and philosophically, as one of his first axioms, that dice and cards ought to be played for money." In the passage quoted by Mr Morley, Cardan seems rather to admit the propriety of moderation in the stake, than to assert that there must be a stake; this moderation Cardan recommends elsewhere, as for example in his second Chapter. Cardan's treatise is briefly noticed in the article Prohability of the English Cyclopcedia.

7. Some remarks on the subject of chance were made by Kepler in his work De Stella Kova in pede Serjjentarii, which was published in 1606. Kepler examines the different opinions on the cause of the appearance of a new star which shone with great splendour in 1604, and among these opinions the Epicurean notion that the star had been produced by the fortuitous concurrence of atoms. The whole passage is curious, but we need not repro- duce it, for it is easily accessible in the reprint of Kepler's works now in the course of publication ; see Joannis Kepleri Astronomi Opera Omnia edidit Dr Ch. Frisch, Vol. ii. pp. 714 — 716. See also the Life of Kepler in the Library of Useful Knoiuledge, p. 13. The passage attracted the attention of Dugald Stewart ; see his Works edited by Hamilton, Vol. I. p. 617.

A few words of Kepler may be quoted as evidence of the soundness of his opinions ; he shows that even such events as throws of dice do not happen without a cause. He says,

Quare hoc jactu Venus cecidit, illo canis 1 Nimh'um lusor liac vice tessellam alio latere arripuit, aliter marm condidit, aliter intus agitavit, alio impetii animi maniisve projecit, aliter interflavit aura, alio loco alvei imj)egit. JSTihil hie est, quod sua causa sic caruerit, si quis ista subtilia posset coiisectavi.

8. The next investigation which we have to notice is that by Galileo, entitled Consider azione sopyu il Giuco dei Dadi. The date of this piece is unknown; Galileo died in 1642. It appears that a friend had consulted Galileo on the following dilBculty : with three dice the number 9 and the number 10 can each be produced by six different combinations, and yet experience shows that the

GALILEO. b

number 10 is oftener thrown than the number 9. Galileo makes a careful and accurate analysis of all the cases which can occur, and he shows that out of 216 possible cases 27 are favourable to the appearance of the number 10, and 25 are favourable to the appearance of the number 9.

The piece will be found in Vol. xiv. pages 293 — 290, of Le

Opere cU Galileo Galilei, Firenze, 1855. From the Biblio-

grafia Galileiana given in Vol. XV. of this edition of Galileo's works we learn that the piece first aj^peared in the edition of the works published at Florence in 1718 : here it occurs in Vol. III. pages 119 — 121.

9. Libri in his Histoire des Sciences Mathematiques en Italie, Vol. IV. page 288, has the following remark relating to Galileo : ..."Ton voit, par ses lettres, qu'il s'etait longtemps occupe d'une question delicate et non encore resolue, relative h, la maniere de compter les erreurs en raison geometrique ou en proportion arithm^tique, question qui touche ^galement au calcul des pro- babilites et a Tarithmetique politique." Libri refers to Vol. ii. page 00, of the edition of Galileo's works published at Florence in 1718 ; there can, however, be no doubt, that he means Vol. iii. The letters will be found in Vol. xiv. pages 231 — 284' of Le Opere... di Galileo Galilei, Firenze, 1855 ; they are entitled Lettere intorno la stwia di un cavallo. We are informed that in those days the Florentine gentlemen, instead of wasting their time in attention to ladies, or in the stables, or in excessive eraminfr. 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 hundred crowns, one person estimated it at ten crowns and another at a thousand ; which of the two made the more extra vagrant estimate ? Amoncr the persons who were consulted was Galileo ; he pronounced the two estimates to be equally extravagant, because the ratio of a thousand to a hundred is the same as the ratio of a hundred to ten. 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 thousand above a hundred is gi'eater than that of a hundred above ten. Various letters of

6 GALILEO.

Galileo and Nozzolini are printed, and also a letter of Benedetto Castelli, who took the same side as Galileo ; it appears that Galileo had the same notion as Nozzolini when the question was first 23roposed to him, but afterwards changed his mind. The matter is discussed by the disputants in a very lively manner, and some amusing illustrations are introduced. It does not appear, however, that the discussion is of any scientific interest or value, and the terms in which Libri refers to it attribute much more importance to Galileo's letters than they deserve. The Florentine gentlemen when they renounced the frivolities already mentioned might have investigated questions of greater moment than that which is here brought under our notice.

CHAPTER II.

PASCAL AND FERMAT.

10. The indications which we have given in tlie preceding Chapter of the subsequent Theory of Probability are extremely slight; and we find that \vriters on the subject have shewn a jus- tifiable pride in connecting the true origin of their science with the great name of Pascal. Thus,

EUe doit la naissance h deux Georaetres frangais du dix-septieme si^cle, si fecond en grands hommes et en grandes decouvertes, et peut- ^tre de tons les siecles celiii qui fait le plus d'honneur a I'esprit humain. Pascal et Fermat se proposerent et resolurent quelqucs pro- blemes sur les probabilites... Laplace, Tlieorie . . .des Prob. 1st edition, page 3.

XJn probleme relatif aux jeux de liasard, propose a un austere jan- seniste par un homme du monde a ete I'origine du calcul des probabilites. Poisson, Recherches sur la Prob. page 1.

The problem which the Chevalier de Mere (a reputed gamester) proposed to the recluse of Port Royal (not yet witlidi-awn from the in- terests of science by the more distracting contemplation of the "great- ness and the misery of man''), was the first of a long series of problems, destined to call into existence new methods in matliematical analvsis, and to render valuable service in the practical concerns of life." Boole, Laws of Thought, page 243.

11. It appears then that the Chevalier de Mere proposed certain questions to Pascal ; and Pascal con^esponded with Fer- mat on the subject of these questions. Unfortunately only a portion of the correspondence is now accessible. Three letters

8 PASCAL AND FERMAT.

of Pascal to Format on this subject, which were all written in 165-i, were published in the Varia Opera Mathematica D. Petri de Fer7nat... Tolosse, 1679, pages 179 — 188. These letters are reprinted in Pascal's works ; in the edition of Paris, 1819, they occur in Yol. iv. pages 360 — 888. This volume of Pascal's works also contains some letters written by Format to Pascal, which are not given in Format's works ; two of these relate to Probabilities, one of them is in reply to the second of Pascal's three letters, and the other apparently is in reply to a letter from Pascal which has not been preserved ; see pages 385 — 388 of the volume.

We will quote from the edition of Pascal's works just named. Pascal's first letter indicates that some previous correspondence had occurred which we do not possess ; the letter is dated July 29, 1654. He begins.

Monsieur, L'impatience me prend aussi-bieii qu a vous ; et quoique je sois encore au lit, je ne puis m'empeclier de vous dire que je re9us hier au soir, de la part de M. de Carcavi, votre lettre sur les partis, que j'admire si fort, que je ne puis vous le dire. Je n'ai pas le loisir de m'etendre ; mais en un mot vous avez trouve les deux partis des des et des parties dans la parfaite justesse : j'en suis tout satisfait ; car je ne doute plus maintenant que je ne sois dans la verite, apres la rencontre admirable oil je me trouve avec vous. J'admire bien da vantage la metliode des parties que celle des des ; j'avois vu plusieurs personnes trouver celle des des, comme M. le chevalier de Mere, qui est celui qui m'a propose ces questions, et aussi M. de Roberval ; mais M. de Mere n'avoit jamais pu trouver la juste valeur des parties, ni de biais pour y arriver : de sorte que je me trouvois seul qui eusse connu cette proportion.

Pascal's letter then proceeds to discuss the problem to which it appears from the above extract he attached the greatest importance. It is called in English the Problem of Points, and is thus enun- ciated : two players want each a given number of points in order to win ; if they separate without playing out the game, how should the stakes be divided between them ?

The question amounts to asking what is the probability which each player has, at any given stage of the game, of winning the game. In the discussion between Pascal and Fermat it is sup-

PASCAL AND FERMAT. 9

posed that the players have equal chances of whining a single point.

12. We will now give an account of Pascal's investigations on the Problem of Points ; in substance we translate his words.

The following is my method for determining the share of each player, when, for example, two players play a game of three points and each player has staked 32 pistoles.

Suppose that the first player has gained two points and the second player one point ; they have now to play for a point on this condition, that if the first player gains he takes all the money which is at stake, namely 6^ pistoles, and if the second player gains each player has two points, so that they are on terms of equality, and if they leave off playing each ought to take 32 pistoles. Thus, if the first player gains, 64 pistoles belong to him, and if he loses, 32 pistoles belong to him. If, then, the players do not wish to play this game, but to separate without playing it, the first player w^ould say to the second " I am certain of 32 pistoles even if I lose this game, and as for the other 32 pistoles perhaps I shall have them and perhaps you will have them ; the chances are equal. Let us then divide these 32 pistoles equally and give me also the 32 pistoles of which I am certain." Thus the first player wdll have 48 pistoles and the second 16 pistoles.

Next, suppose that the first player has gained two points and the second player none, and that they are about to play for a point ; the condition then is that if the first player gains this point he secures the game and takes the 64 pistoles, and if the second player gains this point the players will then be in the situation already examined, in which the first player is entitled to 48 pistoles, and the second to 16 pistoles. Thus if they do not wish to play, the first player would say to the second " If I gain the point I gain 64 pistoles ; if I lose it I am entitled to 48 pistoles. Give me then the 48 pistoles of which I am certain, and divide the other 16 equally, since our chances of gaining the point are equal." Thus the first player will have 56 pistoles and the second player 8 pistoles.

Finally, suppose that the first player has gained one point and

10 PASCAL AND FERMAT.

the second player none. If they proceed to play for a point the condition is that if the first player gains it the players will be in the situation first examined, in which the first player is entitled to 5Q pistoles ; if the first player loses the point each player has then a point, and each is entitled to 32 pistoles. Thus if they do not wish to play, the first player would say to the second " Give me the 82 pistoles of which I am certain and divide the remainder of the 56 pistoles equally, that is, divide 24 pistoles equally." Thus the first player will have the sum of 32 and 12 pistoles, that is 44 pistoles, and consequently the second will have 20 pistoles.

13. Pascal then proceeds to enunciate two general results without demonstrations. We will give them in modern notation.

(1) Suppose each player to have staked a sum of money denoted by A ; let the number of points in the game be n+ 1, and suppose the first player to have gained n points and the second player none. If the players agree to separate without playing

A

any more the first player is entitled to 2 A — ~ .

(2) Suppose the stakes and the number of points in the game as before, and suppose that the first player has gained one point and the second player none. If the players agree to separate without playing any more, the first player is entitled to

, 1 . 3 . 5 . . . (2n - 1)

■^2.4.6... 2/1 •

Pascal intimates that the second theorem is difficult to prove. He says it depends on two propositions, the first of which is purely arithmetical and the second of which relates to chances. The first amounts in fact to the proposition in modern works on Algebra which gives the sum of the co-efficients of the terms in the Binomial Theorem. The second consists of a statement of the value of the first player's chance by means of combinations, from which by the aid of the arithmetical proposition the value above given is deduced. The demonstrations of these two results may be obtained from a general theorem which will be given later in the present Chapter ; see Art. 23. Pascal adds a table which

PASCAL AND FERMAT. 11

exhibits a complete statement of all the cases which can occur in a game of six points.

14. Pascal then proceeds to another topic. He says

Je n'a pas le temps de vous envoyer la demonstration d'une difficulte qui etonnoit fort M. de Mere : car il a tres-bon esprit, mais il n'est pas geometre ; c'est, comme vous savez, un grand defaut; etmeme ilne com- prend pas qu'une ligne mathematique soit divisible a I'infini, et croit fort bien entendre qu'elle est composee de points en nombre fini, et jamais je n*ai pu Ten tirer ; si vous pouviez le faire, on le rendroit parfait. II me disoit done qu'il avoit trouve faussete dans les nombres par cette raison.

The difficulty is the following. If we undertake to throw a six with one die the odds are in favour of doing it in four throws, being as 671 to 625 ; if we undertake to throw two sixes with two dice the odds are not in favour of doing it in twenty-four throws. Nevertheless 24 is to 86, which is the number of cases with two dice, as 4 is to 6, which is the number of cases with one die. Pascal proceeds

"Voilk quel etoit son grand scandale, qui lui faisoit dire hautement que les propositions n'etoient pas constantes, et que I'arithmetique se d^mentoit. Mais vous en verrez bien aisement la raison, par les prin- cipes o^ vous etes.

15. In Pascal's letter, as it is printed in Fermat's works, the name de Mere is not given in the passage we have quoted in the preceding article ; a blank occurs after the 21. It seems, however, to be generally allowed that the blank has been filled up correctly by the publishers of Pascal's works : Montmort has no doubt on the matter ; see his p. XXXII. See also Gouraud, p. 1 ; Lubbock and Drinkwater, p. 41. But there is certainly some difficulty. For in the extract which we have given in Art. 11, Pascal states that M. de Mere could solve one problem, celle des des, and seems to imply that he failed only in the Problem of Points. Montucla says that the Problem of Points w^as proposed to Pascal by the Chevalier de Mer^, " qui lui en proposa aussi quelques autres sur le jeu de des, comme de detemiiner en combien de coups on pent parier d'amener une rafle, &c. Ce chevalier, plus bel esprit que

12 PASCAL AND FERMAT.

geom^tre ou analyste, rdsolut a la verite ces derni^res, qui ne sont pas bien difficiles ; mais il echoua pour le precedent, ainsi que Roberval, a qui Pascal le proposa." p. 384. These words would seem to imply that, in Montucla's opinion, M. de Mere was not the person alluded to by Pascal in the passage we have quoted in Article 14. We may remark that Montucla was not justified in suofsrestinof that M. de Mere must have been an indifferent mathe- matician, because he could not solve the Problem of Points ; for the case of Roberval shews that an eminent mathematician at that time might find the problem too difficult.

Leibnitz says of M. de Mere, " II est vrai cependant que le Che- valier avoit quelque genie extraordinaire, meme pour les Mathe- matiques ;" and these words seem intended seriously, although in the context of this passage Leibnitz is depreciating M. de Merd. Leibnitii, Opera Omnia, ed. Dutens, Vol. ii. part 1. p. 92.

In the Nouveaiix Essais, Li v. IV. Chap. 16, Leibnitz says, *' Le Chevalier de Mere dont les Agrements et les autres ouvrages ont ete imprimes, homme d'un esprit jDenetrant et qui etoit joueur et philosophe."

It must be confessed that Leibnitz speaks far less favourably of M. de Mere in another place. Opera, Vol. V. p. 203. From this pas- sage, and from a note in the article on Zeno in Bayle's Dictionary, to which Leibnitz refers, it appears that M. de Mere maintained that a magnitude was not infinitely divisible : this assists in identi- fying him with Pascal's friend who would have been jDerfect had it not been for this single error.

On the whole, in spite of the difficulty which we have pointed out, we conclude that M. de Mer^ really was the person who so strenuously asserted that the propositions of Arithmetic were in- consistent with themselves ; and although it may be unfortunate for him that he is now known principally for his error, it is some compensation that his name is indissolubly associated with those of Pascal and Fermat in the history of the Theory of Probability.

16. The remainder of Pascal's letter relates to other mathe- matical topics. Fermat's reply is not extant ; but the nature of it may be inferred from Pascal's next letter. It appears that Fermat

PASCAL AND FERMAT. 13

sent to Pascal a solution of the Problem of Points depending on combinations.

Pascal's second letter is dated August 24th, 1654. He says that Fermat's method is satisfactory when there are only two players, but unsatisfactory when there are more than two. Here Pascal was wrong as we shall see. Pascal then gives an example of Fermat's method, as follows. Suppose there are two players, and that the first wants two points to win and the second three points. The game will then certainly be decided in the course of four trials. Take the letters a and h and write down all the combina- tions that can be formed of four letters. These combinations are the following, 16 in number :

a

a

a

a

a

h

a

a

h

a

a

a

h

h

a

a

a

a

a

h

a

h

a

h

h

a

a

h

h

h

a

b

a

a

h

a

a

h

h

a

h

a

h'

a

h

h

h

a

a

a

h

h

a

h

h

h

h

a

h

h

h

h

h

h

Now let A denote the player who wants two points, and B the player who wants three points. Then in these 16 combinations every combination in which a occurs twice or oftener represents a case favourable to A, and every combination in which h occurs three times or oftener represents a case favourable to B. Thus on counting them it will be found that there are 11 cases favourable to A, and 5 cases favourable to B ; and as these cases are all equally likely, -4's chance of winning the game is to -S's chance as 11 is to 5.

17. Pascal says that he communicated Fermat's method to Roberval, who objected to it on the following ground. In the example just considered it is supposed that four trials will be made ; but this is not necessarily the case ; for it is quite possible that the first player may win in the next two trials, and so the game be finished in two trials. Pascal answers this objection by stating, that although it is quite possible that the game may be finished in two trials or in three trials, yet we are at liberty to conceive that the players agree to have four trials, because, even if the game be decided in fewer than four trials, no difference will be

14j pascal and fermat.

made in the decision by the superfluous trial or trials. Pascal j)uts this point very clearly.

In the context of the first passage quoted from Leibnitz in Art. 15, he refers to " les belles pensees de Alea, de Messieurs Fermat, Pascal et Huygens, oil Mr. Roberval ne pouvoit ou ne vouloit rien comprendre."

The difficulty raised by Roberval was in effect reproduced by D'Alembert, as we shall see hereafter.

18. Pascal then proceeds to apply Format's method to an example in which there are three players. Suppose that the first player wants one point, and each of the other players two points. The game will then be certainly decided in the course of three trials. Take the letters a, h, c and write down all the combinations which can be formed of three letters. These combinations are the following, 27 in number:

a	a	a	h	a	a	C	a	a
a	a	h	h	a	b	c	a	b
a	a	c	h	a	c	c	a	c
a	h	a	b	b	a	c	b	a
a	h	h	b	b	b	c	b	b
a	h	c	b	b	c	c	b	c
a	c	a	b	c	a	c	c	a
a	c	b	b	c	b	c	c	b
a	c	c	b	c	G	c	c	c

Let A denote the player who wants one point, and B and C the other two players. By examining the 27 cases, Pascal finds 13 Avhich are exclusively favourable to A, namely, those in which a occurs twice or oftener, and those in which a, b, and c each occur once. He finds 3 cases which he considers equally favourable to A and B, namely, those in which a occurs once and b twice ; and similarly he finds 3 cases equally favourable to A and C. On the whole then the number of cases favourable to A may be considered to be 13 + f + f, that is 16. Then Pascal finds 4 cases which are exclusively favourable to B, namely those represented by bbb, ebb, bcb, and bbc ; and thus on the whole the number of cases

PASCAL AND FERMAT. 15

favourable to B may be considered to be 4 + |, that is 5^. Simi- larly the number of cases favourable to C may be considered to be 5^. Thus it would appear that the chances oi A, B, and C are respectively as 16, 5i, and 51

Pascal, however, says that by his own method he had found that the chances are as 17, 5, and 5. He infers that the differ- ence arises from the circumstance that in Fermat's method it is assumed that three trials will necessarily be made, which is not assumed in his own method. Pascal was wrong in supposing that the true result could be affected by assuming that three trials w^ould necessarily be made ; and indeed, as we have seen, in the case of two players, Pascal himself had correctly maintained against Roberval that a similar assumption was legitimate.

19. A letter from Pascal to Format is dated August 29th, 1654. Format refers to the Problem of Points for the case of three players; he says that the proportions 17, 5, and 5 are correct for the example which we have just considered. This letter, how- ever, does not seem to be the reply to Pascal's of August 24th, but to an earlier letter which has not been preserved.

On the 25th of September Format writes a letter to Pascal, in which Pascal's error is pointed out. Pascal had supposed that such a combination as ace represented a case equally favour- able to A and C\ but, as Format says, this case is exclusively favourable to A, because here A gains one point before C gains one ; and as A only wanted one point the game is thus decided in his favour. When the necessary correction is made, the result is, that the chances of A, B, and C are as 17, 5, and 5, as Pascal had found by his own method.

Fermat then gives another solution, for the sake of Roberval, in which he does not assume that three trials will necessarilv be made; and he arrives at the same result as before.

In the remainder of his letter Fermat enunciates some of his memorable propositions relating to the Theory of Numbers.

Pascal replied on October 27th, 1654, to Fermat's letter, and said that he was entirelv satisfied.

16 PASCAL AND FERMAT.

20. There is another letter £i'oni Fermat to Pascal which is not dated. It relates to a simple question which Pascal had pro- posed to Fermat. A person undertakes to throw a six with a die in eight throws ; supposing him to have made three throws with- out success, what portion of the stake should he be allowed to take on condition of giving up his fourth throw ? The chance of success is J, so that he should be allowed to take J of the stake on con- dition of giving up his throw. But suppose that we wish to esti- mate the value of the fourth throw before any throw is made. The first throw is worth J of the stake ; the second is worth J of what remains, that is -^ of the stake ; the third throw is worth i of w^hat now remains, that is -ff^ of the stake ; the fourth throw is worth J of what now remains, that is -^-ff-Q of the stake.

It seems possible from Format's letter that Pascal had not dis- tinguished between the two cases ; but Pascal's letter, to which Format's is a reply, has not been preserved, so that we cannot be certain on the point.

21. We see then that the Problem of Points was the prin- cipal question discussed by Pascal and Fermat, and it was certainly not exhausted by them. For they confined themselves to the case in which the players are supposed to possess equal skill; and their methods would have been extremely laborious if applied to any examples except those of the most simple kind. Pascal's method seems the more refined ; the student will perceive that it depends on the same principles as the modern solution of the problem by the aid of the Calculus of Finite Differences ; see Laplace, Theorie...cles Proh. page 210.

Gouraud awards to Format's treatment of the problem an amount of praise which seems excessive, whether we consider that treatment absolutely or relatively in comparison with Pascal's ; see his page 9,

22. We have next to consider Pascal's Traite du triangle arithmetique. This treatise was printed about 1G5-4, but not pubhshed until 1665 ; see Montucla, p. 387. The treatise will be found in the fifth volume of the edition of Pascal's works to which we have already referred.

PASCAL AND FERMAT. 17

The Arithmetical Triangle in its simplest form consists of the

following	table :
		1 1	1	1	1	1	1	1
		2 3	4	0	6	7	8	9
		3 G	10	15	21	28	30 ...
		4 10	20	35	dQ	8-i..	t
		5 15	35	70	120..	•
		6 21	50	120.	■ •
		7 28	81^.	• •
		8 3G..	•
		%J a • •
J. •	• •

In the successive horizontal rows we have what are now called the figurate numbers. Pascal distinguishes them into orders. He calls the simple units 1, 1, 1, 1,... which form the first row, num- bers of the first order; he calls the numbers 1, 2, 3, 4,... which form the second row, numbers of the second order; and so on. The numbers of the third order 1, 3, 0, 10,... had already received the name oi triangular numbers; and the numbers of the fourth order 1, 4, 10, 20,... the name oi pyr^amidal numbers. Pascal says that the numbers of the fifth order 1, 5, 15, 35,... had not yet received an express name, and he proposes to call them triangulo- triangulaires.

In modern notation the if^ term of the r*^ order is

n(ii + l) ... {n + r - 2)

r-1

Pascal constructs the Arithmetical Triangle by the foUowdng definition ; each number is the sum of that immediately above it and that immediately to the left of it. Thus

10 = 4 + 0, 35 = 20 + 15, 126 = 70 + 50,...

The properties of the numbers are developed by Pascal with great skill and distinctness. For example, suppose we require the sum of the first n terms of the r^^ order : the sum is equal to the number of the combinations of n + r — 1 things taken r at a time, and Pascal establishes this by an inductive proof

2

18 PASCAL AND FERMAT.

23. Pascal applies liis Arithmetical Triangle to various subjects ; among tliese we have the Problem of Points, the Theory of Com- binations, and the Powers of Binomial Quantities. We are here only concerned with the application to the first subject.

In the Arithmetical T^^iangle a line drawn so as to cut off an equal number of units from the top horizontal row and the extreme left-hand vertical column is called a base.

The bases are numbered, beginning from the top left-hand corner. Thus the tenth base is a line drawn through the num- bers 1, 9, 36, 84, 12G, 12G, 84, 36, 9, 1. It will be perceived that the r*^ base contains r numbers.

Suppose then that A wants m points and that B wants n points. Take the {m + ii)^^ base; the chance oi A is to the chance of B as the sum of the first n numbers of the base, beginning at the highest row, is to the sum of the last m numbers. Pascal establishes this by induction.

Pascal's result may be easily she^vn to coincide with that obtained by other methods. For the terms in the (m + ti)"^ base are the coefficients in the expansion of (1 -f xY'^''~^ by the Binomial Theorem. Let m + n — l=r\ then Pascal's result amounts to saying that the chance of A is proportional to

- r (r — 1) r (r — l) ... (r — n-\-2)

I . z n — 1

and the chance of B proportional to

Ij^yj^ r (r-1) ^ ^^^^ ^ r{r-l)...{r-m + 2)

1.2 ^^1-1

This agrees with the result now usually given in elementary treatises; see Algebra, Chapter Liii.

24. Pascal then notices some particular examples. (1) Sup- pose that A wants one point and B wants n points. (2) Suppose that A wants n — 1 points and B wants n points. (3) Suppose that A wants n— 2 points and B wants n points. An interesting relation holds between the second and third examples, which we will exhibit.

PASCAL AXD FERMAT. 19

Let M denote the number of cases which are favourable to A , and N the number of cases which are favourable to B, Let r = 2/1 - 2.

In the second example we have

M — N.= . -^ = X say.

\n— 1 I ;2 — 1 "^

Then if 2 aS' denote the whole sum at stake, A is entitled to -^ . — ^— , that is to — (2*' +X)\ so that he may be considered to have recovered his own stake and to have won the fraction ^7 of his adversary's stake.

In the third example we have il/ + lY = T-\

2 r - 1 2 (?2 - 1) 1 r - 1 1\{n-\\

n — \ ?i— 2 \n — 1 In — 1

Thus we shall find that A may be considered to have recovered his own stake, and to have won the fraction ■— j of his adversary's

stake.

Hence, comparing the second and third examples, we see that if the player who wins the first point also wins the second point, his advantage when he has gained the second point is double what it was when he had gained the first point, whatever may be the number of points in the game,

25. We have now analysed all that has been preser\'ed of Pascal's researches on our subject. It seems however that he had intended to collect these researches into a complete treatise. A letter is extant addressed by him Celeberrimce Matheseos Academice Parisiensi ; this Academy was one of those voluntary associations which preceded the formation of formal scientific societies : see Pascal's Works, Vol. iv. p. 356. In the letter Pascal enumerates various treatises which he had prepared and which he hoped to

20 PASCAL AND FERMAT.

publish, among wliicli was to be one on chances. His language shews that he had a high opinion of the novelty and importance of the matter he proposed to discuss ; he says,

Novissima autem ac penitus intentatse materise tractatio, scilicet de compositione alece in hid is ijysi subjeclis, qnod gallico nostro idiomate dicitur (/aire les ^;ar^is cles jeux) : ubi ancej)s fortuna sequitate rationis ita reprimitur ut utrique lusorum quod jure competit exacte semper assignetur. Quod quidem eo fortius ratiocinando quserendnm, quo minus tentando investigari possit : ambigiii enim sortis eventus fortiiitse contingentise potius quam nattirali necessitati meritb tribuuntur. Ideo res hactenus erravit incerta ; nunc autem qu?e experimento rebellis fuerat, rationis dominium effugere non potuit : eam quippe tanta se- curitate in artem per geometriam reduximus, ut certitudinis ejus j^articeps facta, jam audacter prodeat ; et sic matheseos demonstrationes cum alese incertitudine jungendo, et qu?e contraria videntur conciliando, ab utraque nominationem suam accipiens stupendum hunc titulum jure sibi arrogat : alece geometria.

But the design was probably never accomplished. The letter is dated 1651; Pascal died in 1662, at the early age of 39.

26. Neglecting the trifling hints which may be found in pre- ceding writers we may say that the Theory of Probability really commenced with Pascal and Format ; and it would be difficult to find two names which could confer higher honour on the subject.

The fame of Pascal rests on an extensive basis, of which mathematical and physical science form only a part ; and the regret which we may feel at his renunciation of the studies in which he gained his earliest renown may be diminished by reflect- ing on his memorable Letters, or may be lost in deeper sorrow wdien we contemplate the fragments which alone remain of the great work on the evidences of religion that was to have engaged the efforts of his maturest powers.

The fame of Format is confined to a narrower range ; but it is of a special kind which is without a parallel in the history of science. Format enunciated various remarkable propositions in the theory of numbers. Two of these are more important than the rest; one of them after bafiling the powers of Euler and La- grange finally yielded to Cauchy, and tlie other remains still un-

PASCAL AND FEEMAT. 21

conquered. The interest which attaches to the propositions is increased by the uncertainty which subsists as to whether Fermat himself had succeeded in demonstrating them.

The French government in the time of Louis Philippe assigned a grant of money for publishing a new edition of Format's works ; but unfortunately the design has never been accomplished. The edition which we have quoted in Art. 11 has been reprinted in facsimile by Friedlander at Berlin in 1861.

27. At the time when the Theory of Probability started from the hands of Pascal and Fermat, they were the most distinguished mathematicians of Europe. Descartes died in 1650, and Newton and Leibnitz were as yet unknown ; Newton was born in 1642, and Leibnitz in 1646. Huygens was born in 1629, and had already given specimens of his powers and tokens of his future eminence; but at this epoch he could not have been placed on the level of Pascal and Fermat. In England Wall is, born in 1616, and appointed Savilian j^rofessor of geometry at Oxford in 1649, was steadily rising in reputation, while Barrow, born in 1630, was not appointed Lucasian professor of mathematics at Cambridge until 1663.

It might have been anticipated that a subject interesting in. itself and discussed by the two most distinguished mathematicians of the time would have attracted rapid and general attention ; but such does not appear to have been the case. The two great men themselves seem to have been indifferent to any extensive publi- cation of their investigations; it was sufficient for each to gain the approbation of the other. Pascal finally withdrew from science and the world ; Fermat devoted to mathematics only the leisure of a laborious life, and died in 1665.

The invention of the Differential Calculus by Newton and Leibnitz soon offered to mathematicians a subject of absorbing interest ; and we shall find that the Theory of Probability advanced but little during the half century which followed the date of the correspondence between Pascal and Fermat.

CHAPTER III.

HUYGENS.

28. We have now to speak of a treatise by Hu3^gens entitled Be Ratiociniis in Ludo Alece. This treatise was first printed by Schooten at the end of his work entitled Francisci a Bcliooten Exercitationum Mathematicarum Lihri quinque ; it occupies pages 519... 534 of the volume. The date 1658 is assigned to Schooten's work by Montucla, but the only copy which I have seen is dated 1657.

Schooten had been the instructor of Huygens in mathematics ; and the treatise which we have to examine was communicated by Huygens to Schooten w^ritten in their vernacular tongue, and Schooten translated it into Latin.

It appears from a letter written by Schooten to Wallis, that Wallis had seen and commended Huygens's treatise ; see Wallis's Algebra, 1693, p. 833.

Leibnitz commends it. Leibnitii Opera Omnia, ed. Dutens, Vol. VI. part 1, p. 318.

29. In his letter to Schooten which is printed at the beginning of the treatise Huygens refers to his predecessors in these words : Sciendum verb, quod jam pridem inter prsestantissimos totd Gallia Geometras calculus hie agitatus fuerit, ne quis indebitam mihi primse inventionis gloriam hac in re tribuat. Huygens ex- presses a very high opinion of the importance and interest of the subject he was bringing under the notice of mathematicians.

30. The treatise is reprinted with a commentary in James Bernoulli's Ars Conjectandi, and forms the first of the four parts

huvgp:ns. 2.3

of which that work is composed. Two English translations of the treatise have been published ; one which has been attributed to Motte, but which was probably by Arbuthnot, and the other by W. Browne.

31. The treatise contains fourteen propositions. The first pro- position asserts that if a player has equal chances of gaining a sum represented by a or a sum represented by b, his expectation is ^ (a + b). The second proposition asserts that if a player has equal chances of gaining a or 6 or c, his expectation is J (a + 6 + c). The third proposition asserts that if a player has 2^ chances of gaining a

and q chances of gaining b, his expectation is — .

i^ + 2'

It has been stated with reference to the last proposition :

*' Elementary as this truth may now appear, it was not received

altogether without opposition." Lubbock and Drinhwater, p. 42.

It is not obvious to what these words refer; for there does not

appear to have been any opposition to the elementary principle,

except at a much later period by D'Alembert.

82. The fourth, fifth, sixth, and seventh propositions discuss simple cases of the Problem of Points, when there are two players ; the method is similar to Pascal's, see Art. 12. The eiirhth and ninth propositions discuss simple cases of the Problem of Points when there are ^/i?'e^ players ; the method is similar to that for two players.

83. Huygens now proceeds to some questions relating to dice. In his tenth proposition he investigates in how many throws a player may undertake to throw a six with a single die. In his eleventh proposition he investigates in how many throws a player may undertake to throw twelve with a pair of dice. In his twelfth proposition he investigates how many dice a player must have in order to undertake that in one throw two sixes at least may appear. The thirteenth proposition consists of the following problem. A and B play with two dice ; if a seven is thrown, ^1 wins; if a ten is thrown, B Avins; if any other number is thrown, the stakes are divided : compare the chances of A and B. They are shewn to be as 13 is to 11.

24 ^ HUYGENS.

84. The fourteenth proposition consists of the following problem. A and B play with two dice on the condition that A is to have the stake if he throws six before B throws seven, and that B is to have the stake if he throws seven before A throws six ; ^ is to begin, and they are to throw alternately ; compare the chances of A and B.

We will give the solution of Huygens. Let B's chance be worth X, and the stake a, so that a — a? is the worth of ^'s chance ; then whenever it is ^.'s turn to throw x will express the value of B's chance, but when it is i>'s own turn to throw his chance will have a different value, say ?/. Suppose then A is about to throw ; there are 36 equally likely cases ; in 5 cases A wins and B takes nothing, in the other 81 cases A loses and B's turn comes on, which is worth y by supposition. So that by the third propo- sition of the treatise the expectation of B is ^ — - , that is,

^2l, Thus So 81?/

Now suppose B about to throw, and let us estimate ^'s chance. There are S6 equally likely cases ; in 6 cases B wins and A takes nothing ; in the other 80 cases B loses and ^'s turn comes on again, in which case B's chance is worth x by supposition. So

that the expectation of B is — ^j^ — . Thus

(ja-^SOx

81«

From these equations it will be found that x = -^ , and thus

80cj

a — x=^

61

, so that ^'s chance is to ^'s chance as 80 is to 81.

85. At the end of his treatise Huygens gives five problems without analysis or demonstration, which he leaves to the reader. Solutions are given by Bernoulli in the Ars Conjectandi. The following are the problems.

(1) A and B play with two dice on this condition, that A gains if he throws six, and B gains if he throws seven. A first has one

HUYGENS. 25

throw, then B has two throAvs, then A two throws, and so on until one or the other gains. Shew that ^'s chance is to J5's as 10355 to 12276.

(2) Three players A, B, C take twelve balls, eight of which are black and four white. They play on the following condition ; they are to draw blindfold, and the first who draws a white ball wins. A is to have the first turn, B the next, G the next, then A again, and so on. Determine the chances of the players.

Bernoulli solves this on three suppositions as to the meaning ; first he supposes that each ball is replaced after it is drawn ; secondly he supposes that there is only one set of twelve balls, and that the balls are not replaced after being drawn ; thirdly he supposes that each player has his own set of twelve balls, and that the balls are not replaced after being drawn.

(3) There are forty cards forming four sets each of ten cards ; A plays with B and undertakes in drawing four cards to obtain one of each set. Shew that ^'s chance is to -S's as 1000 is to 8139.

(4) Twelve balls are taken, eight of which are black and four are white. A pla3^s with B and undertakes in drawing seven balls blindfold to obtain three white balls. Compare the chances of A and B.

(5) A and B take each twelve counters and play with three dice on this condition, that if eleven is throAA-n A gives a counter to B, and if fourteen is thrown B gives a counter to A ; and he wins the game who first obtains all the counters. Shew that A 's chance is to ^'s as 244140625 is to 282429536481.

oQ>. The treatise by Huygens continued to form the best account of the subject until it was superseded by the more elabo- rate works of James Bernoulli, Montmort, and De Moivre. Before we speak of these we shall give some account of the history of the theory of combinations, and of the inquiries into the laws of mortality and the principles of life insurance, and notices of various miscellaneous investigations.

CHAPTER IV.

ON COMBIjSTATIONS.

87. The theory of combinations is closely connected witli the theory of probability ; so that we shall find it convenient to imi- tate Montucla in giving some account of the writings on the former subject up to the close of the seventeenth century.

88. The earliest notice we have found respecting combinations is contained in Wallis's Algebra as quoted by him from a work by William Buckley; see Wallis's Algebra 1693, page 489. Buckley was a member of King's College, Cambridge, and lived in the time of Edward the Sixth. He wrote a small tract in Latin verse con- taining the rules of Arithmetic. In . Sir John Leslie's Pliilosophj of Arithmetic full citations are given from Buckley's work; in Dr. Peacock's History of A rithmetic a citation is given ; see also De Morgan's Arithmetical Books from the invention of Printing .. .

Wallis quotes twelve lines which form a Regula Comhinationis, and then explains them. We may say briefly that the rule amounts to assigning the whole number of combinations which can be formed of a given number of things, when taken one at a time, or two at a time, or three at a time,. . . and so on until they are taken all together. The rule shews that the mode of proceeding was the same as that which we shall indicate hereafter in speaking of Schooten ; thus for four things Buckley's rule gives, like Schoo- ten's, 1 + 2 + 4 + 8, that is 15 combinations in all.

By some mistake or misprint Wallis apparently overestimates the age of Buckley's work, when he says *' . . . in Arithmetica sua,

lUUHUsius. 27

versibus scripta ante annos plus minus 190;" in the ninth Chapter of the Algebra the date of about 1550 is assigned to Buckley's death.

89. We must now notice an example of combinations which is of historical notoriety although it is very slightly connected with the theory.

A book was published at Antwerp in 1617 by Erycius Pu- teanus under the title, Erycii Puteani Fietatis TJiaumata in Bernardi Bauhusii ^ Societate Jesu Proteum Parthenium. The book consists of IIG quarto pages, exclusive of seven pages, not numbered, which contain an Index, Censura, Summa Privilegii, and a typographical ornament.

It appears that Bernardus Bauhusius composed the following line in honour of the Virgin Mary :

Tot tibi sunt dotes, Virgo, quot sidera copIo.

This verse is arranged in 1022 different ways, occupying 48 pages of the work. First we have 54 arrangements commencing Tot tibi; then 25 arrangements commencing Tot sunt; and so on. Although these arrangements are sometimes ascribed to Puteanus, they ajD- pear from the dedication of the book to be the work of Bauhusius himself; Puteanus supplies verses of his own and a series of chap- ters in prose which he calls Thaumata, and which are distinguished by the Greek letters from A to O inclusive. The number 1022 is the same as the number of the stars accordino- to Ptolemy's Cata- logue, wdiich coincidence Puteanus seems to consider the great merit of the labours of Bauhusius ; see his page 82.

It is to be observed that Bauhusius did not profess to include all the possible arrangements of his line; he expressly rejected those which would have conveyed a sense inconsistent with the glory of the Virgin Mary. As Puteanus sa3\s, page 103, Dicere horruit Vates :

Sidera tot ca?lo, Virgo, quot sunt tibi Dotes,

imb in hunc sensum producere Proteum recusavit, ne laudem immi- nueret. Sic igitur contraxit versuum numerum ; ut Dotium augeret.

40. The line due to Bauhusius on account of its numerous an-angements seems to have attracted gi'eat attention during the following century ; the discussion on the subject was finally settled

28 PASCAL.

by James Bernoulli in his Ars Coiijectandi, where he thus details the history of the problem.

. , . Quemadmodum cernere est in hexametro a Bernli. Bauhusio Jesuita Lovaniensi in laudem Virginis Deiparse constructo :

Tot tihi sunt Dotes, Virgo, quot sidera ccdo ; qiiem dignnm peculiari opera duxerunt plures Viri celebres. Erycius Puteanus in libello, quern. Tliaumata Pietatis inscripsit, variationes ejus utiles integris 48 paginis enumerat, easque numero stellarum, quarum vulgb 1022 recensentur, accommodat, omissis scrupulosius illis, quse di- cere videntur, tot sidera cselo esse, quot Marine dotes; nam Mariae dotes esse multo plures. Eundem numerum 1022 ex Puteano repetifc Gerh. Yossius, cap. 7, de Scient. Matliemat. Prestetus Gallus in prima editione Element. Matliemat. pag. 358. Proteo huic 2196 variationes attribuit, sed facta revisione in altera edit. torn. pr. pag. 133. numerum earum dimidio fere auctum ad 3276 extendit. Industrii Actorum Lips. Collectores m. Jun. 1686, in recensione Tractatus Wallisiani de Algebra, numerum in qusestione (quem Auctor ipse definire non fuit ausus) ad 2580 determinant. Et ipse postmodum Wallisius in edit, latina operis sui Oxon. anno 1693. impressa, pagin. 494. eundem ad 3096 profert. Sed omnes adliuc a vero deficientes, ut delusam tot Yirorum post adhibitas quoque secundas curas in re levi perspicaciam meritb mireris. Ars Conjectandi, page 78.

James Bernoulli seems to imply that the two editions of Wallis's Algebra differ in their enumeration of the arrangements of the line due to Bauhusius ; but this is not the case : the two editions agree in investigation and in result.

James Bernoulli proceeds to say that he had found that there could be 3312 arrangements without breaking the law of metre; this excludes spondaic lines but includes those which have no caesura. The analysis which produces this number is given.

41. The earliest treatise on combinations which we have ob- served is due to Pascal. It is contained in the work on the Arithmetical Triangle which we have noticed in Art. 22; it will also be found in the fifth volume of Pascal's works, Paris 1819, pages 86—107.

The investigations of Pascal on combinations depend on his Arithmetical Triangle. The following is his principal result; we express it in modern notation.

PASCAL. 29

Take an Arithmetical Triangle with r numbers in its base; then the sum of the numbers in the _29"' horizontal row is equal to the multitude of the combinations of r things taken p at a time. For example, in Art 22 we have a triangle with 10 numbers in its base ; now take the numbers in the 8th horizontal column ; their sum is 1 4-8 + 36, that is 45; and there are 45 combinations of 10 things taken 8 at a time. Pascal's proof is inductive. It may be observed that multitudo is Pascal's word in tlie Latin of his treatise, and multitude in the French version of a part of the treatise which is given in pages 22 — 30 of the volume.

From this he deduces various inferences such as the followino-. Let there be n things ; the sum of the multitude of the combinations which can be formed, one at a time, two at a time,... , up to n at a time, is 2''— 1.

At the end Pascal considers this problem. Datis duobus numeris inaequalibus, invenire quot modis minor in majore combinetur. And from his Arithmetical THangle he deduces in effect the follow- ing result ; the number of combinations of r things taken p at a time is

(^+1) (p + 2) (;; + 3)...r

■P

After this problem Pascal adds.

Hoc problemate tractatum liiiuc absolvere constitiieram, non tamen omniiio sine molestia, cum niulta alia parata liabeam ; sed ubi tanta ubertas, vi moderanda eat fames : his ergo pauca hsec subjiciam.

Eruditissimus ac milii charisimus, D.D. de Ganieres, circa combina- tiones, assiduo ac peiiitili labore, more suo, incumbens, ac indigens facili constructione ad inveniendum quoties numerus datus in alio dato combinetur, hanc ipse sibi praxim instituit.

Pascal then gives the rule ; it amounts to this ; the num- ber of combinations of r things taken |) at a time is

r (>'- 1)... {r-p+ 1)

■ {p ■

This is the form with which we are now most familiar. It may be immediately shewn to agree with the form given before by Pascal, by cancelling or introducing factors into both numerator and denominator. Pascal however savs, Excellentem hanc solu-

.so SniOOTEN.

tionem ipse mihi ostendit, ac etiam demonstranJam proposiiit, ipsam ego san^ miratus sum, sed difficultate territus vix opus suscepi, et ipsi authori relinquendum existimavi; attamen trianguli arith- metici auxilio, sic proclivis facta est via. Pascal then establishes the correctness of the rule by the aid of his Arithmetical Triangle; after which he concludes thus, Hac demonstratione assecuta, jam reliqua quae invitus supprimebam libenter omitto, adeo dulce est amicorum memorari.

42. In the work of Schooten to which w^e have already re- ferred in Art. 28 we find some very slight remarks on combinations and their applications; see pages 873 — 403. Schooten's first sec- tion is entitled, Ratio inveniendi electiones omnes, qu^ fieri pos- sunt, data multitudine rerum. He takes four letters a, h, c, d, and arranges them thus,

a.

h. ah.

c. ac. he. ahc.

d. ad. hd. abd. cd. acd. bed. abed.

Thus he finds that 15 elections can be made out of these four letters. So he adds, Hinc si per a designatur unum malum, jDer b unum pirum, per c unum prunum, et per d unum cerasum, et ipsa alitor atque alitor, ut supra, eligantur, electio eorum fieri poterit 15 diversis modis, ut sequitur

Schooten next takes five letters ; and thus he infers the result which we should now express by saying that, if there are n letters the whole number of elections is 2"— 1.

Hence if a, b, c, d are prime factors of a number, and all dif- ferent, Schooten infers that the number has 15 divisors excludinsf unity but including the number itself, or 1 6 including also unity.

Next suppose some of the letters are repeated; as for example suppose we have a, a, b, and c ; it is required to determine how many elections can be made. Schooten arranges the letters thus,

a.

a. aa.

h. ah. aab.

c. ac. aac. be. ahc. aabc. We have thus 2 + 3 + 6 elections.

LEIBNITZ. 3 1

Similarly if the proposed letters are a, a, a, b, h, it is found that 11 elections can be made.

In his following sections Schooten proceeds to apply these results to questions relating to the number of divisors in a number. Thus, for example, supposing a, h, c, d, to be different prime factors, numbers of the following forms all have 16 divisors, ahcd, a^hc, a^b^, a^b, a)^. Hence the question may be asked, what is the least number which has 10 divisors? This question must be answered by trial ; we must take the smallest prime numbers 2, 8,. . . and substitute them in the above forms and pick out the least number. It will be found on trial that the least number is 2^. 3. 5, that is 120. Similarly, suppose we require the least number which has 24 divisors. The suitable forms of numbers for 24 divisors are ci^bcd, a^¥c, oJ'bc, a^¥, a'b'^, o}^h and a^^. It will be found on trial that the least number is 2^ 3^. 5, that is 360.

Schooten has given two tables connected with this kind of question. (1) A table of the algebraical forms of numbers which have any given number of divisors not exceeding a hundred ; and in this table, when more than one form is given in any case, the first form is that which he has found by trial will give the least number with the corresponding number of divisors. (2) A table of the least numbers which have any assigned number of divisors not exceeding a hundred. Schooten devotes ten pages to a list of all the prime numbers under 10,000.

43. A dissertation was pubHshed by Leibnitz in 1666, entitled Dissertatio de Arte Combinatoma; part of it had been previously published in the same year under the title of Disputatio arith- metica de comjilexionihus. The dissertation is interesting as the earliest work of Leibnitz connected with mathematics ; the con- nexion however is very slight. The dissertation is contained in the second volume of the edition of the works of Leibnitz by Dutens ; and in the first volume of the second section of the mathematical works of Leibnitz edited by Gerhardt, Halle, 1858. The dissertation is also included in the collection of the philoso- phical writings of Leibnitz edited by Erdmann, Berlin, 1840.

44. Leibnitz constructs a table at the beginning of his dis-

32 LEIBNITZ.

sertation similar to Pascal's Arithmetical Triangle, and applies it to find the number of the combinations of an assigned set of things taken two, three, four,... together. In the latter part of his disser- tation Leibnitz shews how to obtain the number of permutations of a set of things taken all together ; and he forms the product of the first 24* natural numbers. He brings forward several Latin lines, including that which we have already quoted in Art. 39, and notices the great number of arrangements which can be formed of them.

The greater part of the dissertation however is of such a character as to confirm the correctness of Erdmann's judgment in including it among the philosophical works of Leibnitz. Thus, for example, there is a long discussion as to the number of moods in a syllogism. There is also a demonstration of the existence of the Deity, which is founded on three definitions, one postulate, four axioms, and one result of observation, namely, aliquod corpus movetur.

4iD. We will notice some points of interest in the dissertation.

(1) Leibnitz proposes a curious mode of expression. When a set of things is to be taken two at a time he uses the S3rmbol com2natio (combinatio) ; when three at a time he uses conSnatio (conternatio) ; when four at a time, con4natio, and so on.

(2) The mathematical treatment of the subject of combina- tions is far inferior to that given by Pascal ; probably Leibnitz had not seen the work of Pascal. Leibnitz seems to intimate that his predecessors had confined themselves to the combina- tions of things two at a time, and that he had himself extended the subject so far as to shew how to obtain from his table the combinations of things taken together more than two at a time ; generaliorem modum nos deteximus, specialis est vidgatus. He gives the rule for the combination of things two at a time, namely,

that which we now express by the formula ^ — -^ ; but he does

not give the similar rule for combinations three, four,... at a time, which is contained in Pascal's work.

(3) After giving his table, which is analogous to the Arith-

LEIBNITZ. S3

metical Triangle, he adds, "Adjiciemus hie Theoremata quorum TO on ex ipsa tabula manifestum est, to Slotl ex tabulae funda- niento." The only theorem here that is of any importance is that which we should now express thus : if n be prime the number of combinations of n things taken r at a time is divisible by n.

(4) A passage in which Leibnitz names his predecessors may be quoted. After saying that he had partly furnished the matter himself and partly obtained it from others, he adds,

Quis ilia primus detexerit ignoramus. Scliwentenis Belie. 1. 1, Sect. 1, prop. 32, apud Hieronymum Cardanum, Johannem Buteonem et Nicolaum Tartaleam, extare dicit. In Cardani tameu Practica Arith- metica quae prodiit Mediolani anno 1539, nihil reperimus. Inprimis dilucide, quicquid dudum habetur, proposuit Christoph. Clavius in Com. supra Joh. de Sacro Bosco Spliaer. edit. Bomte forma 4ta anno 1785. p. 33. seqq.

With respect to Schwenter it has been observ^ed,

Schwenter probably alluded to Cardan s book, " De Proportionibus," in which the figurate numbers are mentioned, and their use shown in the extraction of roots, as employed by Stifel, a German algebraist, who wrote in the early part of the sixteenth century. Lubbock and Drinkwater, page 45.

(5) Leibnitz uses the symbols -1 = in their present sense ;

he uses -— ^ for multiplication and --^ for division. He uses the word productiun in the sense of a sum : thus he calls 4 the pro- ductum of 3 + 1.

46. The dissertation shews that at the age of twenty years the distinguishing characteristics of Leibnitz were strongly de- veloped. The extent of his reading is indicated by the numerous references to authors on various subjects. We see evidence too that he had already indulged in those dreams of impossible achieve- ments in which his vast powers were uselessly squandered. He vainly hoped to produce substantial realities by combining the precarious definitions of metaphysics with the elementary tniisms of logic, and to these fruitless attempts he gave the aspiring titles of universal science, general science, and philosophical calculus. See Erdmann, pages 82 — 91, especially page 84.

3

34 ^yALLIS.

47. A discourse of coinhinations, alternations, and aliquot parts is attached to the English edition of Wallis's Algebra pub- lished in 1685. In the Latin edition of the Algebra, published in 1693, this j^art of the work occupies pages 485 — 529.

In referring to Wallis's Algebra we shall give the pages of the Latin edition ; but in quoting from him we shall adopt his own English version. The English version was reprinted by Maseres in a volume of reprints which was published at London in 1795 under the title of The Doctrine of Permutations and Gomhinations, being an essential and fundamental part of the Doctrine of Chances.

48. "Wallis's first Chapter is Of the variety of Elections, or Choise, in taking or leaving One or more, out of a certain Num- her of things proposed. He draws up a Table which agrees with Pascal's Arithmetical Triangle, and shews how it may be used in finding the number of combinations of an assigned set of things taken two, three, four, five,... at a time. Wallis does not add any thing to what Pascal had given, to whom however he does not refer ; and Wallis's clumsy parenthetical style con- trasts very unfavourably with the clear bright stream of thought and language which flowed from the genius of Pascal. The chapter closes with an extract from the Arithmetic of Buckley and an explanation of it ; to this we have aU'eady referred in Art. 38.

49. Wallis's second Chapter is Of Alternations, or the different change of Order, in any Number of things ptroposed. Here he gives some examples of what are now usually called permutations ; thus if there are four letters a, h, c, d, the number of permutations when they are taken all together is 4 x 3 x 2 x 1. Wallis accord- ingly exhibits the 24 permutations of these four letters. He forms the product of the first twenty-four natural numbers, which is the number of the permutations of twenty-four things taken all toge- ther.

Wallis exhibits the 24 permutations of the letters in the word Roma taken all together ; and then he subjoins, *' Of which (in Latin) these seven are only useful; Roma, ramo,oram,mora, maro, armo, amor. The other forms are useless ; as affording no (Latin) word of known signification."

WALLIS. 35

Wallis then considers the case in which there is some repetition among the quantities of which we require the permutations. He takes the letters which compose the word Messes. Here if there were no repetition of letters the number of permutations of the letters taken all together would^ be 1x2x3x4x5x0, that is 720 ; but as Wallis explains, owing to the occurrence of the letter e twice, and of the letter s thrice, the number 720 must be divided by 2 X 2 X 3, that is by 12. Thus the number of permutations is reduced to 60. Wallis exhibits these permutations and then sub- joins, " Of all which varieties, there is none beside messes itself, that affords an useful AnagTam." The chapter closes with Wallis's attempt at determining the number of arrangements of the verse

Tot tibi sunt dotes, virgo, quot sidera caelo.

The attempt is followed by these w^ords, " I will not be posi- tive, that there may not be some other Changes : (and then, those may be added to these :) Or, that most of these be twice repeated, (and if so, those are to be abated out of the Number :) But I do not, at present, discern either the one and other."

Wallis's attempt is a very bad specimen of analysis ; it involves both the errors he himself anticipates, for some cases are omitted and some counted more than once. It seems strange that he should have failed in such a problem considering the extraordinary powers of abstraction and memory which he possessed ; so that as he states, he extracted the square root of a number taken at random wdth 53 figures, in tenebris decumbens, sola fretus memoria. See his Algebra, page 150.

50. Wallis's third Chapter is Of the Divisors and Aliquot paints, of a Number i^roposed. This Chapter treats of the resolu- tion of a number into its prime factors, and of the number of divisors Avhich a given number has, and of the least numbers which have an assigned number of divisors.

51. Wallis's fourth Chapter is Monsieur Fermafs Problems con- cerning Divisors and Aliquot Parts. It contains solutions of two problems which Fermat had proposed as a challenge to Wallis and the English mathematicians. The problems relate to what is now called the Theory of Numbers.

o

o-

8G PRESTET.

52. Thus the theory of combinations is not applied by Wallis in any manner that materially bears upon our subject. In fact the influence of Format seems to have been more powerful than that of Pascal ; and the Theory of Numbers more cultivated than the Theory of Probability.

The judgment of Montmort seems correct that nothing of any importance in the Theory of Combinations previous to his own Avork had been added to the results of Pascal. Montmort, on his page XXXV, names as writers on the subject Prestet, Tacquet, and Wallis. I have not seen the works of Prestet and Tacquet ; Gouraud refers to Prestet's Nouveaux elements de mathematiqiies, 2® ed., in the following terms, Le pere Prestet, enfin, fort habile geom^tre, avait explique avec infiniment de clart^, en 1689, les principaux artifices de cet art ingenieux de composer et de varier les grandeurs. Gouraud, page 23.

CHAPTER V,

MORTALITY AND LIFE INSURANCE.

53. The history of the investigations on the laws of mortality and of the calculations of life insurances is sufficiently important and extensive to demand a separate work ; these subjects were originally connected with the Theory of Probability but may now be considered to form an independent kingdom in mathematical science : we shall therefore confine ourselves to tracing their origin.

54. According to Gouraud the use of tables of mortality was not quite unknown to the ancients: after speaking of such a table as unkno'svn until the time of John de Witt he subjoins in a note,

Inconnue du moins des modernes. Car il paraitrait par un passage du Digeste, ad legem Falcidlam, xxxv. 2, 68, que les Romains n'en ignoraieut pas absolument I'usage. Voyez "k ce sujet M. Y. Leclerc, Des Journaux chez les Romains, p. 198, et une curieuse dissertation: De prohabilitate vitce ejusqite usu forensi, etc., d'un certain Schmelzer (Goettingue, 1787, in-8). Gouraud, page 14.

55. The first name which is usually mentioned in connexion with our present subject is that of John Graunt : I borrow a notice of him from Lubbock and Drinkwater, page 4-i. After referrino: to the reoisters of the annual numbers of deaths in London which began to be kept in 159:^, and which with some

38 GRAUNT.

intermissions between 1d94< and 1603 have since been regularly continued, they proceed thus.

They were first intended to make known the progress of the plague ; and it was not till 1662 that Captain Graunt, a most acute and intel- ligent man, conceived the idea of rendering them subservient to the ulterior objects of determining the population and growth of the me- tropolis ; as before his time, to use his own words, " most of them who constantly took in the weekly bills of mortality, made little or no use of them than so as they might take the same as a text to talk upon in the next company; and withal, in the plague time, how the sickness increased or decreased, that so the rich might guess of the necessity of their removal, and tradesmen might conjecture what doings they were like to have in their respective dealings." Graunt was careful to pub- lish with his deductions the actual returns from which they were obtained, comparing himself, when so doing, to "a silly schoolboy, coming to say his lesson to the world (that peevish and tetchie master,) who brings a bundle of rods, wherewith to be whipped for every mistake he has committed." Many subsequent writers have betrayed more fear of the punishment they might be liable to on making similar disclosures, and have kept entirely out of sight the sources of their conclusions. The immunity they have thus purchased from contradiction could not be obtained but at the expense of confidence in their results.

These researches procured for Graunt the honour of being chosen a fellow of the Koyal Society, . . .

Gouraud says in a note on his page 16,

...John Graunt, homme sans geometric, mais qui ne manquait ni de sagacite ni de bon sens, avait, dans une sorte de traite d'Arithme- tique politique intitule: Natural and 'political observations .. .made itpon the hills of mortality^ etc., rassemble ces difierentes listes, et donne meme i^ihid. chap, xi.) un calcul, a la verite fort grossier, mais du moins fort original, de la mortalite probable \ chaque age d'un certain nombre d'individus supposes n6s viables tons au meme instant.

See also the AtJienceum for October 31st, 1863, page 537.

56. The names of two Dutchmen next present themselves,

Van Hudden and John de Witt. Montucla says, page 407,

Le probleme des rentes viageres fut traite par Van Hudden, qui quoique geometre, ne laissa pas que d'etre bourguemestre d' Amsterdam,

JOHN DE WITT. 30

et par le c61ebre pensionnaire d'Hollande, Jean de Witfc, iin dea pre^ miers promoteurs de la geometrie de Descartes. Jlgnore le titre de I'ecrit de Hudden, mais celui de Jean de Witt etoit intitule : De vardye van de lif-renten na j^^oportie van de los-renten, ou la Valeur des rentes viageres en raison des ventes lihres ou remboursahles (La Haye, 1C71). lis etoient I'un et I'autre plus a portee que personne d'en sentir I'impor- tance et de se procurer les depouillemens necessaires de registres de inor- talitc; aussi Leibnitz, passant en Hollande quelques annees apres, fit tout son possible pour se procurer I'ecrit de Jean de Witt, mais il ne pent y parvenir; il n'etoit cependant pas absolument perdu, car M. Ni- colas Struyck {Inleiding tot het algemeine geography, &c. Amst. 1740, in 4o. p. 345) nous apprend qu'il en a eu un exemj)laire entre les mains; il nous en donne un precis, par lequel on voit combien Jean de Witt raisonnoit juste sur cette matiere.

Le chevalier Petty, Anglois, qui s'occupa beaucoup de calculs poli- tiques, entrevit le probleme, mais il n'etoit pas assez geometre pour le traiter fructueusement, en sorte que, jusqu'a Halley, I'Angleterre et la France qui emprunterent tant et ont tant empruntc de2:)uis, le firent comme des aveugles ou comme de jeunes debauclics.

57. Witli respect to Sir William Petty, to whom Montucla refers, we may remark that his writings do not seem to Iiave been very important in connexion with our present subject. Some account of them is given in the article A rithmetique Politique of the original French Encyclopedie ; the article is reproduced in the Encyclopedie Methodique. Gouraud speaks of Petty thus in a note on his page 1 6,

Apres Graunt, le chevalier W. Petty, dans differents essais d'eco- nomie politique, oi\ il y avait, il est vrai, plus d 'imagination que de jugement, s'etait, de 1682 a 1687, occupe de semblables recherclies.

58. W^ith respect to Van Hudden to whom Montucla also refers we can only add that his name is mentioned with appro- bation by Leibnitz, in conjunction with that of John de Witt, for his researches on annuities. See Leihnitii Opera Omnia, ed. Dutens, Vol. II. part 1, page 93 ; Vol. Yl. part 1, page 217.

69. With respect to the work of John de Witt we have some notices in the correspondence between Leibnitz and James Bernoulli; but these notices do not literallv confirm Montucla's

40 JOHN DE WITT.

statement respecting Leibnitz : see Leihnizens Matliematische Schriften herausgegehen von C. I. Gerhardt, Erste Abtheilung. Band ill. Halle 1855. James Bernoulli says, page 78,

Nuper in Menstruis Excerptis Hanoverae imjoressis citatum, inveni Tractatum quendam mihi ignotum Pensionarii de Wit von Subtiler Ausreclinung des valoris der Leib-Renten. Fortasse is quaedam hue facientia liabet; quod si sit, copiam ejus mihi alieunde fieri percuperem.

In liis reply Leibnitz says, page 84,

Pensionarii de Wit libellus exiguus est, ubi aestimatione ilia nota utitur a possibilitate casuum aequalium aequali et liinc ostendit re- ditus ad vitam sufiicientes pro sorte a Batavis solvi. Ideo Belgice scripserat, ut aequitas in vulgus apjDareret.

In his next letter, page 89, James Bernoulli says that De Witt's book will be useful to him; and as he had in vain tried to obtain it from Amsterdam he asks for the loan of the copy which Leibnitz possessed. Leibnitz replies, page 93,

Pensionarii Wittii dissertatio, vel potius Scheda impressa de re- ditibus ad vitam, sane brevis, extat quidem inter chartas meas, sed cum ad Te mittere vellem, reperire nondum potui. Dabo tamen operam ut nanciscare, ubi primum domi eruere licebit alicubi latitantem.

James Bernoulli again asked for the book, page 95. Leibnitz replies, page 99,

Pensionarii Wittii scriptum nondum satis quaerere licuit inter char- tas; non dubito tamen, quin sim tandem reperturus, ubi vacaverit. Sed vix aliquid in eo novum Tibi occurret, cum fundamentis iisdem ubique insistat, quibus cum alii viri docti jam erant usi, tum Paschalius in Triangulo Aritlimetico, et Hugenius in diss, de Alea, nempe ut medium Arithmeticum inter aeque incerta sumatur; quo fundamento etiam rustic! utuntur, cum praediorum pretia aestimant, et rerum fis- calium curatores, cum reditus praefecturarum Principis medios consti- tuunt, quando se offert conductor.

In the last of his letters to James Bernoulli which is given, Leib- nitz implies that he has not yet found the book ; see page 103.

We find from pages 767, 769 of the volume that Leibnitz attempted to procure a copy of De Witt's dissertation by the aid of John Bernoulli, but without success.

These letters were written in the years 1703, 1704, 1705.

HALLEY. 41

60. The political fame of John de Witt has overpowered that which he might have gained from science, and thus his mathe- matical attainments are rarely noticed. We may therefore add that he is said to have published a work entitled Elementa linea- rum curvarum, Leyden 1650, which is commended by Condorcet ; see Condorcet's Essai...d'Analyse... i>age CLXXXiv.

CI. We have now to notice a memoir by Halley, entitled An estimate of the Degrees of the Mortality of Mankind, dravm from carious Tables of the Births and Funerals at the City of Breslaiv; with an Attempt to ascertain the Price of Annuities upon Lives.

This memoir is published in Vol. xvil. of the Philosophical Transactions, 1693 ; it occupies pages 596 — 610.

This memoir is justly celebrated as having laid the foundations of a correct theory of the value of life annuities.

62. Halley refers to the bills of mortality which had been published for London and Dublin ; but these bills were not suit- able for drawing accurate deductions.

First, In that the Number of the People was wanting. Secondly, That the Ages of the People d}dng was not to be had. And Lastly, That both London and Dublin by reason of the great and casual Accession of Strangers who die therein, (as appeared in both, by the great Excess of the Funerals above the Births) rendered them incapable of being Standards for this purpose; which requires, if it were possible, that the People we treat of should not at all be changed, but die where they were born, without any Adventitious Increase from Abroad, or Decay by Migration elsewhere.

63. Halley then intimates that he had found satisfactory data in the Bills of Mortality for the city of Breslau for the years 1687, 88, 89, 90, 91 ; which *'had then been recently communi- cated by Neumann (probably at Halley's request) through Justell, to the Royal Society, in whose archives it is supposed that copies of the original registers are still preserved." Lubbock and Drink- luater, page 45.

64. The Breslau registers do not appear to have been pub- lished themselves, and Halley gives only a very brief introduction

42 HALLEY.

to the table which he deduced from them. Halley's table is in the following form:

1

2 3 4

1000 855

798 760

The left-hand number indicates ages and the right-hand num- ber the corresponding number of persons alive. We do not feel confident of the meaning of the table. Montucla, page 408, under- stood that out of 1000 persons born, 855 attain to the age of one year, then 798 out of these attain to the age of two years, and so on.

Daniel Bernoulli understood that the number of infants born is not named, but that 1000 are supposed to reach one year, then 855 out of these reach two years, and so on. Hist de VAcad. ... Paris, 1760.

^D. Halley proceeds to shew the use of his table in the calcu- lation of annuities. To find the value of an annuity on the life of a given person we must take from the table the chance that he will be alive after the lapse of n years, and multiply this chance by the present value of the annual payment due at the end of n years ; we must then sum the results thus obtained for all values of n from 1 to the extreme possible age for the life of the given person. Halley says that " This will without doubt appear to be a most laborious Calculation." He gives a table of the value of an annuity for every fifth year of age up to the seventieth.

^Q. He considers also the case of annuities on joint lives, or on one of two or more lives. Suppose that we have two persons, an elder and a younger, and we wish to know the probability of one or both being alive at the end of a given number of years. Let N be the number in the table opposite to the present age of the younger person, and R the number opposite to that age in- creased by the given number of years ; and let N=R-\- Y, so that Y represents the number who have died out of N in the given number of years. Let n, r, y denote similar quantities for the elder age. Then the chance that both will be dead at the end

HALLEY

43

of the given number of years is —■ ; the chance that the younger

Till

will be alive and the elder dead is -r^ ; and so on.

Halley gives according to the fashion of the time a geometri- cal illustration.

D 1

B

E _C

G

H

Let AB or CD represent N, and DE or BH represent R, so that EC or HA represents F. Similarly AC, AF, CF may represent n, r, y. Then of course the rectangle ECFG represents Ty, and so on.

In like manner, Halley first gives the proposition relating to three lives in an algebraical form, and then a geometrical illus- tration by means of a parallelepiped. We find it difficult in the present day to understand how such simple algebraical pro- positions could be rendered more intelligible by the aid of areas and solids.

67. On pages 654^ — 6oQ of the same volume of the Pliiloso- pMcal Transactions we have Some further Considerations on the Breslaiu Bills of Mortality. By the same Hand, d'C.

68. De Moivre refers to Halley's memoir, and republishes the table; see Be Moivre's Doctrine of Chances, pages 261, ^^o.

CHAPTER VI.

MISCELLANEOUS INVESTIGATIONS Between the yeaes 1670 and 1700.

69. The present chapter will contain notices of various con- tributions to our subject, which were made between the publi- cation of the treatise by Huygens and of the more elaborate works by James Bernoulli, Montmort, and De Moivre.

70. A Jesuit named John Caramuel published in 1670, under the title of Mathesis Bicej^s, two folio volumes of a course of Mathematics ; it appears from the list of the author's works at the beginning of the first volume that the entire course was to have comprised four volumes.

There is a section called Gomhinatoria which occupies pages 921 — 1036, and part of this is devoted to our subject.

Caramuel gives first an account of combinations in the modern sense of the word; there is nothing requiring special attention here : the work contains the ordinary results, not proved by general symbols but exhibited by means of examples. Caramuel refers often to Clavius and Izquierdus as his guides.

After this account of combinations in the modern sense Cara- muel proceeds to explain the Ars Lidliana, that is the method of affording assistance in reasoning, or rather in disputation, proposed by Raymond Lully.

71. Afterwards we have a treatise on chances under the title of Kyheia, quce Combinatorioe genus est, de Alea, et Ludis FortuncB

CARAMUEL. 45

serio disputans. This treatise includes a reprint of tlie treatise of Huygens, which however is attributed to another person. Cara- muel says, page 984,

Dum hoc Syntagma Perilhistri Domino N. Viro eruditissimo com- municarem, ostendit etiam mihi ingeniosam quamdam de eodem argu- ment© Diatribam, quam ^ Christiano Severino Longomontano fuisse scriptam putabat, et, quia est curiosa, et brevis, debuit huic Qusestioni subjungi...

In the table of contents to his work, page xxviii, Caramuel speaks of the tract of Huygens as

Diatribe ingeniose a Longomontano, ut putatur, de hoc eodem argu- mento scripta : nescio an evulgata.

Longomontanus was a Danish astronomer who lived from 15G2 to 1647.

72. Nicolas Bernoulli speaks very severely of Caramuel. He says XJn Jesuite nomme Caramuel, que j'ai citd dans ma These... mais comme tout ce qu'il donne n'est qu'un amas de paralogismes, je ne le compte pour rien. Montmort, p. 387.

By his T}ie$e Nicolas Bernoulli probably means his Specimina Artis conjectandi..., which will be noticed in a subsequent Chapter, but Caramuel's name is not mentioned in that essay as reprinted in the A da Erud. . . . Suppl.

John Bernoulli in a letter to Leibnitz speaks more favourably of Caramuel ; see page 715 of the volume cited in Art. 59.

73. Nicolas Bernoulli has exaggerated the Jesuit's blunders. Caramuel touches on the following points, and correctly : the chances of the throws with two dice ; simple cases of the Problem of Points for two players ; the chance of throwing an ace once at least in two throws, or in three throws ; the game of Passe-dix.

He goes Avrong in trying the Problem of Points for three players, which he does for two simple cases ; and also in two other problems, one of which is the fourteenth of Huygens's treatise, and the otlier is of exactly the same kind.

Caramuel's method with the fourteenth problem of Huygens's treatise is as follows. Suppose the stake to be 36 ; then A's chance

46 SAUVEUR.

5 5

at his first throw is ^ , and ^ x 86 = 5 ; thus taking 5 from 86 we

may consider 81 as left for B. Now B's chance of success in a single

throw is ^ ; thus — x 81, that is 5 J, may be considered the value oO oO

of his first throw.

Thus Caramuel assigns 5 to J. and 5 J to B, as the value of

their first throws respectively ; then the remaining 25f he proposes

to divide equally between A and B. This is wrong : he ought to

have continued his process, and have assigned to A for his second

5 6

throw ^ of the 25f , and then to B for his second throw -^ of the

remainder ; and so on. Thus he Avould have had for the shares of each player an infinite geometrical progression, and the result would have been correct.

It is strange that Caramuel went wrong when he had the treatise of Huygens to guide him ; it seems clear that he followed this oruidance in the discussion of the Problem of Points for Uvo players, and then deserted it.

74. In the Journal des Scavans for Feb. 1679, Sauveur gave some formulae without demonstration relating to the advantage of the Banker at the game of Bassette. Demonstrations of the for- mulae will be found in the Ars Conjectandi of James Bernoulli, pages 191 — 199. I have examined Sauveur's formulae as given in the Amsterdam edition of the Journal. There are six series of formulae ; in the first five, which alone involve any difficulty, Sauveur and Bernoulli agree : the last series is obtained by simply subtracting the second from the fifth, and in this case by mistake or misprint Sauveur is wrong. Bernoulli seems to exaggerate the discrepancy when he says, Qu5d si quis D.ni Salvatoris Tabellas cum hisce nostris contulerit, deprehendet illas in quibusdam locis, praesertim ultimis, nonnihil emendationis indigere. Montucla, page 390, and Gouraud, page 17, seem also to think Sauveur more inaccurate than he really is.

An eloge of Sauveur by Fontenelle is given in the volume for 1716 of the Hist, de F Acad.... Paris. Fontenelle says that Bassette was more beneficial to Sauveur than to most of those who

LEIBXITZ. 47

played at it with so much fury ; it was at the request of the Marquis of Dangeau that Sauveur undertook the investigation of the chances of the game. Sauveur was in consequence introduced at court, and had the honour of explaining his calculations to the King and Queen. See also Montmor^t, page xxxix.

75. James Bernoulli proposed for solution two problems in chances in the Journal des Sgavans for 1685. They are as follows :

1. A and B play with a die, on condition that he who first throws an ace wins. First A throws once, then B throws once, then A throws twice, then B throws twice, then A throws three times, then B throws three times, and so on until ace is thrown.

2. Or first A throws once, then B twice, then A three times, then B four times, and so on.

The problems remained unsolved until James Bernoulli himself gave the results in the Acta Eruditorum for 1690. Afterwards in the same volume Leibnitz gave the rcsidts. The chances involve infinite series which are not summed.

James Bernoulli's solutions are reprinted in the collected edition of his works, Geneva, 17^4 ; see pages 207 and 430. The problems are also solved in the Ars Conjectandi, pages 52 — oG.

76. Leibnitz took great interest in the Theory of Probability and shewed that he was fully alive to its importance, although he cannot be said himself to have contributed to its advance. There was one subject which especially attracted his attention, namely that of games of all kinds ; he himself here found an exercise for his inventive powers. He believed that men had noAvhere shewn more ingenuity than in their amusements, and that even those of children might usefully engage the attention of the greatest mathe- maticians. He wished to have a systematic treatise on games, comprising first those which depended on numbers alone, secondly those which depended on position, like chess, and lastly those which depended on motion, like billiards. This he considered would be useful in bringing to perfection the art of invention, or

48 ARBUTHXOT.

as he expresses it in another place, in bringing to perfection the art of arts, which is the art of thinking.

See Leihnitii Opera Omnia, ed. Dutens, Vol. v. pages 17, 22, 28, 29, 203, 206. Vol. Vi. part 1, 271, 304. Erdmann, page 175.

See also Opera Omnia, ed. Dutens, Vol. vi. part 1, page 36, for the design which Leibnitz entertained of writing a work on estimating the probability of conclusions obtained by arguments.

77. Leibnitz however furnishes an example of the liability to error which seems peculiarly characteristic of our subject. He says. Opera Omnia, ed. Dutens, Vol. vi. part 1, page 217,

...par exemple, avec deux des, il est aussi faisable de jetter douze points, que d'en jetter onze ; car Tun et I'autre no se peut faire que d'une seule manierej mais il est trois fois plus faisable d'en jetter sept; car cela se peut faire en jettant six et un, cinq et deux, quatre et trois; et une combinaison ici est aussi faisable que I'autre.

It is true that eleven can only be made up of six and five ; but the six may be on either of the dice and the five on the other, so that the chance of throwing eleven with two dice is twice as great as the chance of throwing twelve : and similarly the chance of throwing seven is six times as great as the chance of throwing twelve.

78. A work entitled Of the Laws of Chance is said by Montu- cla to have appeared at London in 1692; he adds mais n'ayant jamais rencontr^ ce livre, je ne puis en dire davantage. Je le soupconne n^anmoins de Benjamin Motte, depuis secretaire de la society royale. Montucla, page 391.

Lubbock and Drink water say respecting it, page 43, This essay, which was edited, and is generally supposed to have been written by Motte, the secretary of the Koyal Society, contains a translation of Huyghens's treatise, and an ajDplication of his princi- ples to the determination of the advantage of the banker at pharaon, hazard, and other games, and to some questions relating to lotteries.

A similar statement is made by Galloway in his Treatise on Prohahility, page 5.

79. It does not appear however that there was any fellow of the Royal Society named Motte; for the name does not occur

ARBUTHNOT. 49

in the list of fellows given in Thomson's History of the Royal Society.

I have no doubt that the work is due to Arbuthnot. For there is an English translation of Huygens's treatise by W. Browne, published in 1714 ; in his Advertisement to the Reader Browne says, speaking of Huygens's treatise,

Besides the Latin Editions it has pass'd thro', the learned Dr Arbuthnott publish'd an English one, together with an Application of the General Doctrine to some pai-ticular Games then most in use; which is so intirely dispers'd Abroad, that an Account of it is all we can now meet with.

This seems to imply that there had been no other transla- tion except Arbuthnot's; and the words ''an Application of the General Doctrine to some particular Games then most in use" agree very well with some which occur in the work itself: ''It is easy to apply this method to the Games that are in use amongst us." See page 28 of the fourth edition.

Watt's Bihliotheca Britannica, under the head Arbuthnot, places the work with the date 1G92.

80. I have seen only one copy of this book, which was lent to me by Professor De Morgan. The title page is as follows:

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 fourth edition, ro^is'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, jt.dcc.xxxviii.

81. I proceed to describe the work as it appears in the fourth edition.

The book is of small octavo size; it may be said to consist of two parts. The first part extends to page 49 ; it contains a trans- lation of Huygens's treatise with some additional matter. Page 50 is blank ; page 51 is in fact a title page containing a reprint

4.

50 ARBUTHNOT.

of part of the title we have already given, namely from "a de- monstration" down to "Backgammon."

The words which have been quoted from Lubbock and Drink- water in Art. 78, seem not to distinguish between these two parts. There is nothing about the " advantage of the banker at Pharaon" in the first part; and the investigations which are given in the second part could not, I believe, have appeared so early as 1692: they seem evidently taken from De Moivre. De Moivre says in the second paragraph of his preface,

I had not at that time read anything concerning this Subject, hut Mr. Huygens's Book, de Eatiociniis in Ludo Alese, and a little Eng- lish Piece (which was properly a Translation of it) done by a very in- genious Gentleman, who, tho' capable of carrying the matter a great deal farther, was contented to follow his Original; adding only to it the computation of the Advantage of the Setter in the Play called Hazard, and some few things more.

82. The work is preceded by a Preface written with vigour but not free from coarseness. We will give some extracts, which show that the writer was sound in his views and sagacious in his expectations.

It is thought as necessary to write a Preface before a Book, as it is judg'd civil, when you invite a Friend to Dinner to proffer him a Glass of Hock beforehand for a Whet: And this being maim'd enough for want of a Dedication, I am resolv'd it shall not want an' Epistle to the Beader too. I shall not take upon me to determine, whether it is lawful to play at Dice or not, leaving that to be disputed betwixt the Fanatick Parsons and the Sharpers ; I am sure it is lawful to deal with Dice as with other Epidemic Distempers;

A great part of this Discourse is a Translation from Mons. Huy- gens's Treatise, De ratiociniis in ludo Alese; one, who in his Improve- ments of Philosophy, has but one Superior, and I think few or no equals. The whole I undertook for my own Divertisement, next to the Satisfaction of some Friends, who would now and then be wran- gling about the Proportions of Hazards in some Cases that are here decided. All it requir'd was a few spare Hours, and but little Work for the Brain; my Design in publishing it, was to make it of more general Dse, and perhaps persuade a raw Squire, by it, to keep his Money in his Pocket; and if, upon this account, I should incur the

ARBUTHNOT. 51

Clamours of the Sharpers, I do not m^^ch regard it, since they are a sort of People the World is not bound to provide for

...It is impossible for a Die, with snch determin'd force and di- rection, not to fall on such a determin'd side, and therefore I call that Chance which is nothing but want of Art ;

The Reader may here observe the Force of Numbers, which can be successfully applied, even to those things, which one would imagine are subject to no Rules. There are very few things which we know, which are not capable of Ijeing reduc'd to a Mathematical Reasoning; and when they cannot, it's a sign our Knowledge of them is very small and confus'd; and where a mathematical reasoning can be had, it's as great folly to make use of any other, as to grope for a thing in the dark, when you have a Candle standing by you. I believe the Cal- culation of the Quantity of Probability might be improved to a very useful and pleasant Speculation, and applied to a great many Events which are accidental, besides those of Games ;

...There is likewise a Calculation of the Quantity of Probability founded on Experience, to be made use of in Wagers about any thing; it is odds, if a Woman is with Child, but it shall be a Boy; and if you would know the just odds, you must consider the Proportion in the Bills that the Males bear to the Females: The Yearlv Bills of Mortality are observed to bear such Proportion to the live People as 1 to 30, or 2Q; therefore it is an even Wager, that one out of thir- teen dies within a Year (which may be a good reason, tho' not the true, of that foolish piece of Superstition), because, at this rate, if 1 out of 26 dies, you are no loser. It is but 1 to 18 if you meet a Parson in the Street, that he proves to be a Non-Juror, because there is but 1 of 36 that are such.

83. Pages 1 to 25 contain a translation of Huygens's treatise including the five problems which he left unsolved. Respecting these our author says

The Calculus of the preceding Problems is left out by Mons. Huy- gens, on purpose that the ingenious Reader may have the satisfiiction of applying the former method himself; it is in most of them more labo- rious than difficult : for Example, I have pitch'd upon the second and third, because the rest can be solv'd after the same Method.

Our author solves the second problem in the first of the three senses which it may bear according to the Ars Conjectandi,

4—2

52 ARBUTHNOT.

and he arrives at the same result as James Bernoulli on page 58 of the Ars Conjectandi. Our author adds,

I have suppos'd here the Sense of the Problem to be, that when any- one chus'd a Counter, he did not diminish their number; but if he miss'd of a white one, put it in again, and left an equal hazard to him

who had the following choice; for if it be otherwise suppos'd, ^'s share

55 9

will be Y9^ » which is less than Yq •

55

This result ^-^ however is wrong in either of the other two

senses which James Bernoulli ascribes to the problem, for which he

77 101 obtains j^ and z-^ respectively as the results ; see Art. 35.

84. Then follow some other calculations about games. We have some remarks about the Boyal-Oak Lottery which are analo- gous to those made on the Play of the Royal Oak by De Moivre in the Preface to his Doctrine of Chances.

A table is g^iven of the number of various throws which can be made with three dice. Pages 84 — 39 are taken from Pascal ; they seem introduced abruptly, and they give very little that had not already occurred in the translation of Huygens's treatise.

85. Our author touches on Whist ; and he solves two problems about the situation of honours. These solutions are only approxi- mate, as he does not distinguish between the dealers and their adversaries. And he also solves the problem of comparing the chances of two sides, one of which is at eight and the other at nine; the same remark applies to this solution. He makes the chances as 9 to 7; De Moivre by a stricter investigation makes them nearly as 25 to 18. See Doctrine of Chances, page 176.

86. Our author says on page 43,

All the former Cases can be calculated by the Theorems laid down by Monsieur Huygens; but Cases more compos'd require other Prin- ciples; for the easy and ready Computation of which, I shall add one Theorem more, demonstrated after Monsieur Huygens's method.

The theorem is : " if I have p Cliances for a, q Chances for h,

ROBERTS. 53

and r Chances for c, then my hazard is worth ^J- — — " Our

]_)^- q + r

author demonstrates this, and intimates that it may be extended

to the case when there are also s Chances for d, &c.

Our author then considers the game of Hazard. He gives an investigation similar to that in De Moivre, and leading to the same results; see Doctrine of Chances, page IGO.

87. The first part of the book concludes thus :

All those Problems suppose Chances, which are in an equal proba- bility to happen; if it should be suppos'd otherwise, there will arise variety of Cases of a quite different nature, which, perhaps, 'twere not unpleasant to consider : I sliall add one Problem of that kind, leaving the Solution to those who think it merits their pains.

In Parallel ipipedo cujus latera sunt ad iuvicem in ratione a,b,c: Invenire quota vice quivis suscipere potest, ut datum quodvis planum, v.g. aSjaciat.

The problem was aftersvards discussed by Thomas Simpson ; it is Problem xxvil, of his Nature and Laius of CJiance.

88. It will be convenient to postpone an account of the second part of the book until after we have examined the works of De Moivre.

89. We next notice An Arithmetical Paradox, concerning the Chances of Lotteries, by the Honourable Francis Roberts, Esq. ; Fellow of the R S.

This is published in Vol. xvii. of the Philosophical Trans- actions, 1693 ; it occupies pages 677 — 681.

Suppose in one lottery that there are three blanks, and three prizes each of 16 pence ; suppose in another lottery that there are four blanks, and two prizes each of 2 shilliugs. Now for one drawing, in the first lottery the expectation is ^ of 16 pence, and in the second it is J of 2 shillings ; so that it is 8 pence in each case. The paradox which Roberts finds is this ; suppose that a gamester pays a shilling for the chance in one of these lotteries ; then although, as we have just seen, the expectations are equal, yet the odds against him are 3 to 1 in the first lottery, and only 2 to 1 in the second.

Oi CRAIG.

The paradox is made by Roberts himself, by his own arbitrary definition of odds.

Supposing a lottery has a blanks and h prizes, and let each prize be r shillings ; and suppose a gamester gives a shilling for one drawing in the lottery; then Roberts says the odds against

a 1

him are formed by the product of j ^^^ T > ^^^^ '^^) "tl^® ^^^^

are as a to Z> (r — 1). This is entirely arbitrary.

The mere algebra of the paper is quite correct, and is a curious specimen of the mode of work of the day.

The author is doubtless the same whose name is spelt Robartes in De Moivre's Preface.

90. I borrow from Lubbock and Drinkwater an account of a work which I have not seen ; it is given on their page 45.

It is not necessary to do more than mention an essay, by Craig, on the probability of testimony, which appeared in 1699, under the title of "Theologi£e Cliristianse Principia Mathematica." This attempt to introduce mathematical language and reasoning into moral subjects can scarcely be read with seriousness ; it has the appearance of an insane parody of Newton's Principia, which then engrossed the attention of the mathematical world. The author begins by stating that he considers the mind as a movable, and arguments as so many moving forces, by which a certain velocity of suspicion is produced, &c. He proves gravely, that suspicions of any history, transmitted through the given time (cceteris ^9aH62^s), vary in the duplicate ratio of the times taken from the beginning of the history, with much more of the same kind with respect to the estimation of equable pleasure, uniformly accele- rated pleasure, pleasure varying as any power of the time, &c. &c.

It is stated in biographical dictionaries that Craig's work was reprinted at Leipsic in 1755, with a refutation by J. Daniel Titius ; and that some Anwiadversiones on it were published by Peterson in 1701.

Prevost and Lhuilier notice Craig's work in a memoir published in the Memoires de VAcad... .Beiiin, 1797. It seems that Craig con- cluded that faith in the Gospel so far as it depended on oral tra- dition expired about the year 800, and that so far as it depended on written tradition it would expire in the year 3150. Peterson

CEAIG. 55

by adopting a different law of diminution concluded that faith would expire in 1789.

See Montmort, page XXXVIII. ; also the Athenceum for Nov, 7th, 1863, page Gil.

91. A Calctdation of the C^'edihility of Human Testimony is contained in Vol. xxi. of the Philosophical Transactions; it is the volume for 1699 : the essay occupies pages 359 — 365. The essay is anonymous ; Lubbock and Drinkwater suggest that it may be by Craig.

The views do not agree with those now received.

First suppose we have successive witnesses. Let a report be transmitted through a series of n witnesses, whose credibilities are Pi' P^y-'Pn' the essay takes the jDroduct j^^j^g '"Pn ^s representing the resulting probability.

Next, suppose we have concurrent witnesses. Let there be two witnesses ; the first witness is supposed to leave an amount of un- certainty represented by 1 —p{, of this the second witness removes the fraction p^, and therefore leaves the fraction (1 —p^ (1 — p^ : thus the resulting probability is ^ — 0- — 2\) 0- ~2^2)- Sii^^iiarly if tliere are three concurrent testimonies the resulting probability is 1 — (1 —2\) (1 — i^a). 0- —P^) '} ^^^^ s^ 0^ ^^^' '^ greater number.

The theory of this essay is adopted in the article Prohahilite of the original French Encyclopedie, which is reproduced in the Encyclopedie Methodique: the article is unsigned, so that we must apparently ascribe it to Diderot. The same theory is adopted by Bicquillcy in his work Bu Calcul des Frohahilites.

CHAPTER VII.

JAMES BERNOULLI.

92. We now propose to give an account of the Ars Conjec- tandi of James Bernoulli.

James Bernoulli is the first member of the celebrated family of this name who is associated with the history of Mathematics. He was born 27th December, 1654, and died 16th August, 1705. For a most interesting and valuable account of the whole family we may refer to the essay entitled Die Mathematiker Bernoulli. . . von Frof. Dr. Peter Merian, Basel, 1860.

93. Leibnitz states that at his request James Bernoulli studied the subject. Feu Mr. Bernoulli a cultive cette mati^re sur mes exhortations; Leibnitii Opera Omnia, ed. Dutens, Vol. Vl. part 1, page 217. But this statement is not confirmed by the correspond- ence between Leibnitz and James Bernoulli, to which we have already referred in Art. 59. It appears from this correspondence that James Bernoulli had nearly completed his work before he was aware that Leibnitz had heard any thing about it. Leibnitz says, page 71,

Audio a Te doctrinam de aestimandis probabilitatibus (quam ego magni facio) non parum esse excultam. Vellem aliqiiis varia ludendi genera (in quibus pulchra hujus doctrinae specimina) mathematice trac- taret. Id simul amoenum et utile foret nee Te aut quocunque gra- ^issimo Mathematico indignum.

James Bernoulli in reply says, page 77,

Scire libenter velim, Amplissime Vir, a quo habeas, quod Doctrina de probabilitatibus aestimandis a me excolatur. Yerum est me a plu-

JAMES BERNOULLI. 57

ribus retro annis hujusmodi speciilatlonibus magnopere delectari, ut vix piitem, quemquani plura super his meclitatum esse. Animus etiam erat, Tractatum quendam conscribendi de hac materia ; sed saepe per integros annos seposui, quia naturalis meus torjoor, quem accessoria vale- tiidinis meae infirmitas immane quantum auxit, facit ut aegerrime ad Bcribendum accedam ; et saepe mihi optarem amanuensem, qui cogitata mea leviter sibi indicata plene divinare, scriptisque consignare posset. Absolvi tamen jam maximam Libri partem, sed deest adliuc praecipua, qua artis conjee tandi principia etiam ad civilia, moralia et oeconomia applicare doceo...

James Bernoulli then proceeds to speak of the celebrated theorem which is now called by his name.

Leibnitz in his next letter brings some objections against the theorem ; see page 83 : and Bernoulli replies ; see page 87. Leib- nitz returns to the subject; see page 9-i: and Bernoulli briefly replies, page 97,

Quod Yerisimilitudines spectat, et earum augmentum pro aucto soil, observationum numero, res omnino se habet ut scripsi, et certus sum Tibi placituram demonstration em, cum publicavero.

94. The last letter from James Bernoulli to Leibnitz is dated 3rd June, 1705. It closes in a most painful manner. We here see him, who was perhaps the most famous of all who have borne his famous name, suffering under the combined sorrow arising from illness, from the ingratitude of his brother John who had been his pupil, and from the unjust suspicions of Leibnitz who may be considered to have been his master :

Si inimor vere narrat, redibit cei'te frater meus Basileam, non tamen Graecam (cum ipse sit dva\<jidf3-r]Tos) sed meam potius stationem (quara brevi cum vita me derelicturum, forte non vane, existimat) occupatunis. De iniquis suspicionibus, quibus me immerentem onerasti in Tuis pe- nultimis, alias, ubi plus otii nactus fuero. Nimc vale et fave etc.

95. Tlie Ars Conjectandi was not published until eight years after the death of its author. The volume of the Hist, de r A cad.... Pains for 1705, published in 1706, contains Fontenelle's Eloge of James Bernoulli. Fontenelle here gave a brief notice, derived from Hermann, of the contents of the Ars Conjectandi then unpublished. A brief notice is also give in another Eloge of

58 JAMES BERNOULLI.

James Bernoulli wliicli appeared in the Journal des Bgavans for 1706: this notice is attributed to Saurin by Montmort; see his page IV.

References to the work of James Bernoulli frequently occur in the correspondence between Leibnitz and John Bernoulli ; see the work cited in Art. 59, pages 367, 377, 836, 8i5, 847, 922, 923, 925, 931.

96. The A^^s Conjectandi was published in 1713. A preface of two pages was supplied by Nicolas Bernoulli, the son of a brother of James and John. It appears from the preface that the fourth part of the work was left unfinished by its author ; the publishers had desired that the work should be finished by John Bernoulli, but the numerous engagements of this mathematician had been an obstacle. It was then proposed to devolve the task on Nicolas Bernoulli, who had already turned his attention to the Theory of Probability. But Nicolas Bernoulli did not con- sider himself adequate to the task; and by his advice the work was finally published in the state in which its author had left it; the words of Nicolas Bernoulli are, Suasor itaque fui, ut Tractatus iste qui maxima ex parte jam impressus erat, in eodem quo eum Auctor reliquit statu cum publico communicaretur.

The Ars Conjectandi is not contained in the collected edition of James Bernoulli's works.

97. TYvqAvs Conjectandi, including a treatise on infinite series, consists of 306 small quarto pages besides the title leaf and the preface. At the end there is a dissertation in French, entitled Lettre d un Amy, sur les Parties du Jeu de Paume which occu- pies 35 additional pages. Montucla speaks of this letter as the work of an anonymous author ; see his page 391 : but there can be no doubt that it is due to James Bernoulli, for to him Nicolas Bernoulli assigns it in the preface to the J.rs Conjectandi, and in his correspondence with Montmort. See Montmort, page 333.

98. The Ars Conjectandi is divided into four parts. The first part consists of a reprint of the treatise of Huygens De Ra- tiociniis in Ludo Alece, accompanied with a commentary by James Bernoulli. The second part is devoted to the theory of permu- tations and combinations. The third part consists of the solution

JAMES BEENOULLT. 59

of various problems relating to games of chance. The fourth part proposed to apply the Theory of Probability to questions of interest in morals and economical science.

We may observe that instead of the ordinary symbol of equality, = James Bernoulli uses x, which Wallis ascribes to Des Cartes; see Walliss Algebra, 1693, page 138.

99. A French translation of the first part of the Ars Con- jectandi was published in 1801, under the title of LArt de

Conjecturer, Tradidt du Latin de Jacques Bernoulli; Avec des Observations, Eclair cissemens et Additions. Far L. G. F. Vastel,... Caen. 1801.

The second part of the Ars Conjectandi is included in the volume of reprints which we have cited in Art. 47; Maseres in the same volume gave an English translation of this part.

100. The first part of the Ars Conjectandi occupies pages 1 — 71 ; with respect to this part we may observe that the com- mentary by James Bernoulli is of more value than the original treatise by Huygens. The commentary supplies other proofs of the fundamental propositions and other investigations of the pro- blems; also in some cases it extends them. We will notice the most important additions made by James Bernoulli.

101. In the Problem of Points with two players, James Bernoulli gives a table which furnishes the chances of the two players when one of them wants any number of points not exceeding nine, and the other wants any number of points not exceeding seven ; and, as he remarks, this table may be j^rolonged to any extent; see his page 16.

102. James Bernoulli gives a long note on the subject of the various throws which can be made with two or more dice, and the number of cases favourable to each throw. And we may especially remark that he constructs a large table which is equi- valent to the theorem we now express thus : the number of ways in which ni can be obtained by throwing n dice is equal to the co-efficient of ^'" in the development of {x + x^ -{- x^ -\- x^ ^ x° -\- x^ in a series of powers of x. See his page 21;.

60 JAMES BERNOULLI.

103. The tenth problem is to find in how many trials one may undertake to throw a six with a common die. James Bernoulli gives a note in reply to an objection which he suggests might be urged against the result; the reply is perhaps only intended as a popular illustration : it has been criticized by Prevost in the NoiLveaux Memoir es de FA cad.... Berlin for 1781.

104. James Bernoulli gives the general expression for the

chance of succeeding m times at least in n trials, when the chance

of success in a single trial is known. Let the chances of success

b c

and failure in a single trial be - and - respectively: then the

required chance consists of the terms of the expansion of - + — )

from ( - j to the term which involves - j [ - J , both inclusive.

This formula involves a solution of the Problem of Points for two players of unequal skill; but James Bernoulli does not ex- plicitly make the application.

105. James Bernoulli solves four of the five problems which Huygens had placed at the end of his treatise ; the solution of the fourth problem he postpones to the third part of his book as it depends on combinations.

106. Perhaps however the most valuable contribution to the subject which this part of the work contains is a method of solving problems in chances which James Bernoulli speaks of as his own, and which he frequently uses. We will give his solution of the problem which forms the fourteenth proposition of the treatise of Huygens : we have already given the solution of Huygens him- self; see Art. 34.

Instead of two players conceive an infinite number of players each of whom is to have one throw in turn. The game is to end as soon as a player whose turn is denoted by an odd number throws a six, or a player whose turn is denoted by an even number throws a seven, and such player is to receive the whole sum at stake. Let h denote the number of ways in which six can be thrown, c the number of ways in which six can fail; so that 6 = 5,

JAMES BERNOULLI. 61

and c = 31 ; let e denote the number of ways in which seven can be thrown, and /the number of ways in which seven can fail, so that e = 6, and /= 30 ; and let a = 6 4- c = e +/

Now consider the expectations of the different players ; they are as follows:

I.	II.	III.	IV.	V.	YI.	YIL	YIIL...
h a'	ce 2 > a	hcf a''			e'er

For it is obvious that - expresses the expectation of the first

player. In order that the second player may win, the first throw

must fail and the second throw must succeed ; that is there are ce

ce favourable cases out of o^ cases, so the expectation is -2 . In

order that the third player may win, the first throw must fail,

the second throw must fail, and the third throw must succeed;

that is there are cfh favourable cases out of a^ cases, so the ex-

Icf pectation is — . And so on for the other players. Now let a a

single player. A, be substituted in our mind in the place of the

first, third, fifth,...; and a single player, B, in the place of the

second, fourth, sixth.... We thus arrive at the problem proposed

by Huygens, and the expectations of A and B are given by two

infinite geometrical progressions. By summing these progressions

we find that ^'s expectation is -3 — -, and 5's expectation is

CB

; the proportion is that of 30 to 81, which agrees with

the result in Art. 31.

107. The last of the five problems which Huygens left to be solved is the most remarkable of all ; see Art. 35. It is the first example on the Duration of Play, a subject which afterwards exercised the highest powers of De Moi\Te, Lagrange, and Laplace. James Bernoulli solved the problem, and added, without a demon- stration, the result for a more general problem of which that of Huygens was a particular case; see Ars Conjectandi page 71.

62 JAMES BERNOULLI.

Suppose A to have m counters, and B to have n counters ; let their chances of winning in a single game be as a to 6 ; the loser in each game is to give a counter to his adversary : required the chance of each player for winning all the counters of his adversary. In the case taken by Huygens m and n were equal.

It will be convenient to give the modern form of solution of the problem.

Let u^ denote J.'s chance of winning all his adversary's count- ers when he has himself w counters. In the next game A must either win or lose a counter; his chances for these two contin- gencies are r and t- respectively: and then his chances

of winning all his adversary's counters are u^_^_^ and u^_^ respectively.

Hence

_ a h

This equation is thus obtained in the manner exemplified by Huygens in his fourteenth proposition; see Art. 34.

The equation in Finite Differences may be solved in the or- dinary way; thus we shall obtain

where C^ and C^ are arbitrary constants. To determine these constants we observe " that ^'s chance is zero when he has no counters, and that it is unity when he has all the counters. Thus u^ is equal to 0 when x is 0, and is equal to 1 when x is m + n. Hence we have

0=0.+ a„ 1 = 0.+ c,g)

«x

«!+n

therefore ^i — ~ ^2~

m+n 1 m+n '

Hence u^ =

^m+n _ ^m+n-:c J^,

X rn-^n J vi+n

To determine ^'s chance at the beginning of the game we must put x = m; thus we obtain

7/ =

JAMES BERXOULLI. 63

In precisely tlie same manner we may find jS's chance at any

stage of the game ; and his chance at the beginning of the game

will be

h"" (g^ - If)

It will be observed that the sum of the chances of A and B at the beginning of the game is unitif. The interpretation of this result is that one or other of the players must eventually win all the counters; that is, the play must terminate. This might have been expected, but was not assumed in the investigation.

The formula which James Bernoulli here gives will next come before us in the correspondence between Nicolas Bernoulli and Montmort; it was however first published by De Moi\Te in his De Mensiira Soiiis, Problem ix., where it is also demonstrated.

108. We may observe that Bernoulli 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. Thus, on his page 15, he points out a mistake into which it would have been easy to fall, nisi nos calculus aliud clocuisset He adds,

Qao ipso proin monemiir, ut cauti siraiis in jiidicando, 'nee ratio- cinia nostra super qiiacunque statim aiialogia in rebus deprehensji fun- dare suescamus; quod ipsum tamen etiam ab iis, qui vel maxinie sapere videntur, nimis frequenter fieri solet.

Again, on his page 27,

Quae quideiu eum in finem hie adduce, ut palam fiat, quam parum fideudum sit ejusmodi ratiociniis, qu?e corticem tantuiu attingunt, nee in ipsam rei naturam altius penetrant; tametsi in toto vitse usu etiam. apud sapientissimos quosque nihil sit frequentius.

Again, on his page 29, he refers to the difficulty which Pascal says had been felt by M. de * * * *, whom James Bernoulli calls Anonymus quidam coetera subacti judicii Yir, sed Geometriae expers. . James Bernoulli adds,

Hac enim qui imbuti sunt, ejusmodi erai'Tto^avetai minime moran- tur, probe conscii dari innumera, qua3 admoto calculo aliter se habere comperiuntur, quam initio apparebaut; ideoque sedulb cavent, juxta id quod semel iterumque monui, ne quicquam analogiis temere tribuant.

64 JA^IES BERNOULLI.

109. The second part of the Ars Conjectandi occupies pages 72 — ] 87 : it contains the doctrine of Permutations and Combina- tions. James Bernoulli says that others have treated this subject before him, and especially Schooten, Leibnitz, Wallis and Prestet ; and so he intimates that his matter is not entirely new. He con- tinues thus, page 73,

...tametsi qusedam non contemnenda de nostro adjecimus, inprimis demonstrationem generalem et facilem proprietatis numerorum figura- torum, cui csetera pleraque innituntur, et quam nemo quod sciam ante nos dedit eruitve.

110. James Bernoulli begins by treating on permutations; he proves the ordinary rule for finding the number of permuta- tions of a set of things taken all together, when there are no repetitions among the set of things and also when there are. He gives a full analysis of the number of arrangements of the verse Tot tibi sunt dotes, Virgo, quot sidera coeli ; see Art. 40. He then considers combinations ; and first he finds the total number of ways in which a set of things can be taken, by taking them one at a time, two at a time, three at a time, ...He then proceeds to find what we should call the number of combinations of n things taken r at a time ; and here is the part of the subject in which he added most to the results obtained by his predecessors. He gives a figure which is substantially the same as Pascal's Arith- metical Triangle; and he arrives at two results, one of which is the well-known form for the nth. term of the rth order of figurate numbers, and the other is the formula for the sum of a given number of terms of the series of figurate numbers of a given order ; these results are expressed definitely in the modern notation as we now have them in works on Algebra. The mode of proof is more laborious, as might be expected. Pascal as we have seen in Arts. 22 and 41, employed without any scruple, and indeed rather with approbation, the method of induction : James Bernoulli however says, page 95,... modus demonstrandi per inductionem parum scientificus est.

James Bernoulli names his predecessors in investigations on figurate numbers in the following terms on his page 95 :

Multi, ut hoc in transitu notemus, numerorum figuratorum contem-

JAMES BERNOULLI. 65

plafcionibua vacarunt (quos inter Faulliaberus et Remmelini TJlmenEes, Wallisius, Mercator in Logarithmotechnia, Prestetus, aliique)...

111. We may notice that James Bernoulli gives incidentally on his page 89 a demonstration of the Binomial Theorem for the case of a positive integral exponent. Maseres considers this to be the first demonstration that appeared ; see page 283 of the work cited in Ai't. 47.

112. From the summation of a series of figurate numbers James Bernoulli proceeds to derive the summation of the powers of the natural numbers. He exhibits definitely 2?i, Sn^ 2n^... up to Xw^" ; he uses the sj^mbol / where we in modern books use S. He then extends his results by induction without demonstration, and introduces for the first time into Analysis the coefficients since so famous as the numbers of Bernoulli. His general formula is that

^ , n'"-' n' c . ^_, c(c-l)(c-2) J, ,_^

c(c-l)(c-2)(c-3)(o-4) _, ^ 2.3.4.5.6

c(c-.l)(o-2)(c-3)(c-4)(c-5)(c~6) "^ 2.3.4.5.6.7.8 '^'"

where ^ = 6 ' ^ = " SO ' ^ = A' ^ = - i' -

He gives the numerical value of the sum of the tenth powers of the first thousand natural numbers ; the result is a number with thirty-two figures. He adds, on his page 98,

E quibus apparet, quam inutilis censenda sit opera Jsmaelis Bul- lialdi, quam conscribendo tarn spisso volumini Arithmeticae sufe Infijii- torum impendit, ubi niliil prgestitit aliud, quam ut primarum tantum sex potestatum summas (partem ejus quod unica nos consecuti sumus pagina) immense labore demonstratas exhiberet.

For some account of Bulliald's sjnssum volumen, see Wallis's Algebra, Chap. LXXX.

113. James Bernoulli gives in his fourth Chapter the rule now well known for the number of the combinations of ti thiners

66 JAMES BERNOULLI.

taken c at a time. He also draws various simple inferences from the rule. He digresses from the subject of this part of his book to resume the discussion of the Problem of Points ; see his page 107. He gives two methods of treating the problem by the aid of the theory of combinations. The first method shews how the table which he had exhibited in the first part of the A7'S Con- jectandi might be continued and the law of its terms expressed; the table is a statement of the chances of A and B for winning the game when each of them wants an assigned number of points. Pascal had himself given such a table for a game of six points ; an extension of the table is given on page 16 of the Ars Con- jectandi, and now James Bernoulli investigates general expressions for the component numbers of the table. From his investigation he derives the result which Pascal gave for the case in which one player wants one point more than the other player. James Ber- noulli concludes this investigation thus ; Ipsa solutio Pascaliana, quae Auctori suo tantopere arrisit.

James Bernoulli's other solution of the Problem of Points is much more simple and direct, for here he does make the application to which we alluded in Art. 101^ Suppose that A wants m points and B wants 7i points ; then the game will certainly be decided in m + n — 1 trials. As in each trial A and B have equal chances of success the whole number of possible cases is 2"'"^""\ And A wins the game if B gains no point, or if B gains just one point, or just two points,... or any number up to w — 1 inclusive. Thus the number of cases favourable to A is

! + ;. + _-_ + ^ + ... + ^^^ ^

where //< = m -f w — 1 .

Pascal had in effect advanced as far as this; see Art. 23: but the formula is more convenient than the Arithmetical Triangle.

114. In his fifth Chapter James Bernoulli considers another question of combinations, namely that which in modern treatises is enunciated thus : to find the number of homogeneous products of the r^^ degree which can be formed of n symbols. In his sixth Chapter he continues this subject, and makes a slight reference to

JAMES BERNOULLI. 67

the doctrine of the number of divisors of a given number; for more information he refers to the works of Schooten and WaUis, which we have already examined ; see Arts. 42, 47.

115. In his seventh Chapter James Bernoulli gives the for- mula for what we now call the number of permutations of n things taken c at a time. In the remainder of this part of his book he discusses some other questions relating to permutations and com- binations, and illustrates his theory by examples.

116. The third part of the Ars Conjectandi occupies pages 138 — 209; it consists of twenty-four problems which are to illus- trate the theory that has gone before in the book. James Ber- noulli gives only a few lines of introduction, and then proceeds to the problems, which he says,

...nullo fere habito selectu, prout in adversariis reperi, proponam, prre- niissis etiam vel intersj)ersis nonnuUis facilioribus, et in quibua nidlus combiiiationum usus apparet.

117. The fourteenth problem deserves some notice. There are two cases in it, but it will be sufficient to consider one of them. A is to throw a die, and then to repeat his throw as many times as the number thrown the first time. A is to have the whole stake if the sum of the numbers given by the latter set of throws exceeds 12; he is to have half the stake if the sum is equal to 12; and he is to have nothing if the sum is less than 12. Required the value of his expectation. It is found to be

^Y^^rr , Avliich is rather less than ^ . After giving the connect

solution James Bernoulli gives another which is plausible but false, in order, as he says, to impress on his readers the necessity of caution in these discussions. The following is the false solution.

A has a chance equal to -x of throwing an ace at his first trial;

in this case he has only one throw for the stake, and that throw may give him with equal probabihty any number between 1 and 6

inclusive, so that we may take ^ (1 + 2 + 34-44-5+6), that is

31, for his mean throw. We may observe that 3^ is the Arith-

5—2

68 JAiyiES BERNOULLI.

metical mean between 1 and 6. Again A has a chance equal to -

of throwing a two at his first trial ; in this case he has two throws for the stake, and these two throws may give him any number between 2 and 12 inclusive; and the probability of the number 2 is the same as that of 12, the probability of 3 is the same as

that of 11, and so on; hence as before we may take ^ (2 + 12),

that is 7, for his mean throw. In a similar way if three, four, five, or six be thrown at the first trial, the corresponding means of the numbers in the throws for the stake will be respectively lOi, 14i, 17^, and 21. Hence the mean of all the numbers is

^ m + 7 + lOi + 1-i + I7i + 21], that is 121;

and as this number is greater than 12 it might appear that the odds are in favour of A.

A false solution of a problem will generally appear more plau- sible to a person who has originally been deceived by it than to another person who has not seen it until after he has studied the accurate solution. To some persons James Bernoulli's false solu- tion 'would appear simply false and not plausible; it leaves the problem proposed and substitutes another which is entirely differ- ent. This may be easily seen by taking a simple example. Suppose that A instead of an equal chance for any number of throws between one and six inclusive, is restricted to one or six throws, and that each of these two cases is equally" likely. Then,

as before, we may take -^ (8 J + 21], that is 12J as the mean

throw. But it is obvious that the odds are against him; for if he has only one throw he cannot obtain 12, and if he has six throws he will not necessarily obtain 12. The question is not what is the mean number he will obtain, but how many throws will give him 12 or more, and how many will give him less than 12. James Bernoulli seems not to have been able to make out more than that the second solution must be false because the first is unassailable; for after saying that from the second solution we might suppose the odds to be in fiiv^our of A, he adds, Hujus

JAMES BERNOULLI. G9

aiitem contrarium ex priore solutione, quae sua luce radiat, ap- paret; ...

The problem has been since considered by Mallet and by Fuss, who agree with James Bernoulli in admitting the plausibility of the false solution.

118. James Bernoulli examines in detail some of the games of chance which were popular in his day. Thus on pages 167 and 168 he takes the game called Cinq et neuf. He takes on pages 16.0 — 174* a game which had been brought to his notice by a stroller at fairs. According to James Bernoulli the chances were against the stroller, and so as he says, istumque proin hoc alese genere, ni praemia minuat, non multum lucrari posse. We might desire to know more of the stroller who thus supplied the occasion of an elaborate discussion to James Bernoulli, and who offered to the public the amusement of gambling on terms unfavourable to himself.

James Bernoulli then proceeds to a game called Trijaques. He considers that, it is of great importance for a placer to main- tain a serene composure even if the cards are unfavourable, and that a previous calculation of the chances of the game will assist in securing the requisite command of countenance and temper. As James Bernoulli speaks immediately afterwards of what he had himself formerl}^ often observed in the game, we may perhaps infer that Trijaques had once been a favourite amusement with him.

119. The nineteenth problem is thus enunciated,

In quolibet Alese genere, si ludi Oeconomus sen Dispensator {le Banquier du Jeu) nonnihil habeat praerogativse in eo consistentis, ut paulo major sit casuiim nnmeriis quibus vincit quam quibus perdit; et major simul casuum numerus, quibus in officio Oeconomi ])ro ludo sequenti confirmutur, quam quibus ceconomia in collusorem transfertur. Quanitur, quanti privilegium hoc Oeconomi sit lestimandum ?

The problem is chiefly remarkable from the fact that James Bernoulli candidly records two false solutions which occuiTed to him before he obtained the true solution.

120. The twenty-first problem relates to the game of Bassette;

70 JAMES BERNOULLI.

James Bernoulli devotes eiglit pages to it, his object being to estimate the advantage of the banker at the game. See Art. 74>.

The last three problems which James Bernoulli discusses arose from his observing that a certain stroller, in order to entice persons to play with him, offered them among the conditions of the game one which was apparently to their advantage, but which on investigation was shewn to be really pernicious ; see his pages 208, 209.

121. The fourth part of the Ay^s Conjectandi occupies pages 210 — 239 ; it is entitled Pars Quai'ta, tradens usum et apj^licatio- nem prwcedentis Doctrince in Civilibus, Moralihus et Oeconomicis. It was unfortunately left incomplete by the author; but nevertheless it may be considered the most important part of the whole work. It is divided into five Chapters, of which we will give the titles.

I. Prceliminaria qucedam de Certitudine, Prohahilitate, Neces- sitate, et Contingentia Rerum.

II. De Scieniia et Conjectura. De Arte Conjectandi. De Argumentis Conjecturanmi, Axiomata quwdam generalia hue pertinentia.

III. De variis argiimentorum generihus, et quomodo eorum pondera wstimentur ad supputandas rerum prohahilitates.

lY. De duplici Modo investigandi mimeros casiium. Quid sentiendum de illo, qui instituitur per experimenta. Prohlenia singulare eani in rem propositum, &c.

V. Solutio Prohlematis prcecedentis.

122. We will briefly notice the results of James Bernoulli as to the probability of arguments. He distinguishes arguments into two kinds, pure and mixed. He says, Pura voco, quoe in qui- busdam casibus ita rem probant, ut in aliis nihil positive probent : Mixta, quae ita rem probant in casibus nonnullis, ut in cieteris probent contrarium rei.

Suppose now we have three arguments of the pure kind lead- ing to the same conclusion; let their respective probabilities be

JAMES BERNOULLI. 71

c f %

1 — , 1 — ^, 1 — • Then the resulting probability of the con-

elusion is 1 — ~- . This is obvious from the consideration that adg

any one of the arguments would establish the conclusion, so that

the conclusion fails only when all the arguments fail.

Supj)ose now that we have in addition two arguments of the

mixed kind : let their respective probabilities be — ^^ , .

Then James Bernoulli gives for the resulting probability

, cfiru

1 — -^

adg (ru + qt) '

But this formula is inaccurate. For the supposition q = 0 amounts to having one argument absolutehj decisive against the conclusion, while yet the formula leaves still a certain probability for the conclusion. The error was pointed out by Lambert; see Pre vest and Lhuilier, Memoir es de F Acad.... Berliii iov 1797.

123. The most remarkable subject contained in the fourth part of the Ars Conjectandi is the enunciation and investigation of what we now call Bernoulli s Theorem. It is introduced in terms which shew a high opinion of its importance :

Hoc igitur est illud Problema, quod evulgauduni hoc loco proposui, postquam jam per vicenniiini pressi, et cujus turn novitas, turn summa utilitas cum pari conjuucta difficultate omnibus reliquis hujus doc- triiiae capitibus pondus et pretium superaddere potest. Ars Conjectandij page 227. See also De Moivre's Doctrine of Chances , page 2d^.

We will now state the purely algebraical part of the theorem. Suppose that (r + s)**' is exj)anded by the Binomial Theorem, the letters all denoting integral numbers and t being equal to r + s. Let u denote the sum of the greatest term and the n preceding terms and the n following terms. Then by taking n large enough the ratio of u to the sum of all the remaining terms of the expan- sion may be made as gi-eat as we please.

If we wish that this ratio should not be less than c it will be sufficient to take n equal to the greater of the two following ex- pressions,

72 JAMES BERNOULLI.

log c + log {s - 1) /^ ^ s \ s__

log (r + 1) - log r V r + 1/ r + 1'

and logc + log(r-l) A^

loor(s+ l)-log5 V

(S + 1) - log 5 V 5 + 1/ 5+1

James Bernoulli's demonstration of this result is long but perfectly satisfactory ; it rests mainly on the fact that the terms in the Binomial series increase continuously up to the greatest term, and then decrease continuously. We shall see as we proceed with the history of our subject that James Bernoulli's demonstra- tion is now superseded by the use of Stirling's Theorem.

124. Let us now take the application of the algebraical result to the Theory of Probability. The greatest term of (r + 5)"', where t=r-\-s is the term involving r"''^"'. Let r and s be proportional to the probability of the happening and failing of an event in a single trial. Then the sum of the 2?i + 1 terms of (r + s)"^ which have the greatest term for their middle term corresponds to the probability that in nt trials the number of times the event happens will lie between n{r—l) and n (r+ 1), both inclusive ; so that the ratio of the number of times the event happens to the whole number of

7* + 1 T ~— 1.

trials lies between and . Then, by taking for n the

t f

greater of the two expressions in the preceding article, we have

the odds of c to 1, that the ratio of the number of times the event

7* + 1

happens to the whole number of trials lies between and

r-1

t ' As an example James Bernoulli takes

r = 30, 5=20, t=50.

He finds for the odds to be 1000 to 1 that the ratio of the number of times the event happens to the whole number of trials

31 29 . .

shall lie between —r and ~r, it will be sufficient to make 25550

t)0 50

trials ; for the odds to be 10000 to 1, it will be sufficient to make

31258 trials ; for the odds to be 100000 to 1, it will be sufficient

to make 36966 trials; and so on.

JA3IES BERNOULLI. 73

125. Suppose then that we have an urn containing white balls

and black balls, and that the ratio of the number of the former

to the latter is known to he that of 3 to 2. We learn from the

preceding result that if we make 25550 drawings of a single ball,

replacing each ball after it is drawn, the odds are 1000 to 1 that

31 29

the white balls drawn lie between —- and — : of the whole num-

50 oO

ber drawn. This is the direct use of James Bernoulli's theorem.

But he himself proposed to employ it inversely in a far more

important way. Suppose that in the preceding illustration we

do not know anything beforehand of the ratio of the white balls

to the black ; but that we have made a larg-e number of drawings,

and have obtained a white ball B times, and a black ball S times :

then according to James Bernoulli we are to infer that the

ratio of the white balls to the black balls in the urn is approxi-

r)

mately — . To determine the precise numerical estimate of the

probability of this inference requires further investigation : we shall find as we proceed that this has been done in two ways, by an inversion of James Bernoulli's theorem, or by the aid of another theorem called Bayes's theorem ; the results apj^roximately agree. See Laplace, Theorie.,.des Proh.... pages 282 and 3CG.

126. We have spoken of the inverse use of James Bernoulli's theorem as the most important; and it is clear that he himself was fully aware of this. This use of the theorem was that which Leibnitz found it difficult to admit, and which James Bernoulli maintained against him; seethe correspondence quoted in Art. 59, pages 77, 83, 87, 94, 97.

127. A memoir on infinite series follows the Ars Conjectandi, and occupies pages 24)1 — 306 of the volume ; this is contained in the collected edition of James Bernoulli's works, Geneva, 1744 : it is there broken up into parts and distributed through the two volumes of which the edition consists.

This memoir is unconnected with our subject, and we will therefore only briefly notice some points of interest which it presents.

74 JAJVIES BERNOULLI.

128, James Bernoulli enforces tlie importance of the subject in the following terms, page 243,

Cseterum quantse sit necessitatis pariter et utilitatis hasc serierum contemplatio, ei sane ignotum esse non poterit, qui perspectum habuerit, ejusmodi series sacram quasi esse anchoram, ad quam in maxime arduis et desperatse solutionis Problematibus, ubi omnes alias humani ingenii vires naufragium passae, velut ultimi remedii loco confugiendum est.

129. The principal artifice employed by James Bernoulli in this memoir is that of subtracting one series from another, thus obtaining a third series. For example,

let /S'=l + R+iT+ ... +

2 ' 3 n + l '

a ..11 11

then b= l + -^ + o+"- + ~-^ TT 5

z 3 n 71 + 1

1 r ^ -, 111 11

therefore 0 = — 1 + ^ — ^ + ^ — ^ + - — - + . . . + -7 — — rr +

1 . 2 ' 2 . 3 ' 3 . 4 ' •" ' 7i(?i + l) n + 1 '

, . Ill 1,1

therelore -z — ^ + - — ^ + ^ — r + • • . H — 7 — — ty = 1 —

1.2' 2. 33. 4' ' n{n+l) n+1'

Thus the sum of n terms of the series, of which the r^^ term is 1 . n

IS

r (r + 1) ' n + 1 '

ISO. James Bernoulli says that his brother first observed

1111

that the sum of the infinite series -+ — +- + y + ...is infinite ;

i. jLi O ^

and he gives his brother's demonstration and his own ; see his page 250.

131. James Bernoulli shews that the sum of the infinite series _ _|_ — ^ + -j- . . . is finite, but confesses himself unable to give

the sum. He says, page 254, Si quis inveniat nobisque commu- nicet, quod industriam nostram elusit hactenus, magnas de nobis

crratias feret. The sum is now known to be 7r ; this result is due

to Euler : it is given in his Introductio in Analysin Infinitorum, 1748, Vol. L page 130.

JAMES BERNOULLI. 75

132. James Bernoulli seems to be on more familiar terms with infinity than mathematicians of the present day. On his page 262 we find him stating, correctly, that the sum of the infinite

series —-r + —p^+ -77, + -77 + . . . is infinite, for the series is greater \/i v^ V^ V"*

1111

than 7 + Q + Q + 7 + ... He adds that the sum of all the odd

terms of the first series is to the sum of all the even terms as \/2 — 1 is to 1 ; so that the sum of the odd terms would appear to be less than the sum of the even terms, which is impossible. But the paradox does not disturb James Bernoulli, for he adds,

...cujus evavTLO(fiaveLas rationem, etsi ex infiniti natiira finito intel- lectui comprehendi non posse videatur, nos tamen satis perspectam habemus.

183. At the end of the volume containing the Ars Conjectandi we have the Lettre a un Amy, sur les Parties da Jen de Faume, to which we have alluded in Art. 97.

The nature of the problem discussed may be thus stated. Suppose A and B two players ; let them play a set of games, say five, that is to say, the player gains the set who first wins five games. Then a certain number of sets, say four, make a match. It is required to estimate the chances of A and B in various states of the contest. Suppose for example that A has won two sets, and B has won one set ; and that in the set now current A has won two games and B has won one game. The problem is thus somewhat similar in character to the Problem of Points, but more complicated. James Bernoulli discusses it very fully, and presents his result in the form of tables. He considers the case in which the players are of unequal skill ; and he solves various problems arising from particular circumstances connected with the game of tennis to which the letter is specially devoted.

On the second page of the letter is a very distinct statement of the use of the celebrated theorem known by the name of Ber- noulli ; see Art. 123.

134. One problem occurs in ihi^ Lettre a un Amy... which it may be interesting to notice.

Suppose that A and B engage in play, and that each in turn

76 JAMES BERNOULLI.

by the laws of tlie game has an advantage over his antagonist. Thus suppose that ^'s chance of winning in the 1st, 3rd, 5th... games is always p, and his chance of losing q) and in the 2nd, 4th, 6th... games suppose that ^'s chance of winning is q and his chance of losing/?. The chance of B is found by taking that of A from unity ; so that B's chance is p or 5' according as ^'s is q or p.

Now let A and B play, and suppose that the stake is to be assigned to the player who first wins n games. There is however to be this peculiarity in their contest : If each of them obtains n — 1 games it will be necessary for one of them to win two games in succession to decide the contest in his favour; if each of them wins one of the next two games, so that each has scored n games, the same law is to hold, namely, that one must win two games in succession to decide the contest in his favour ; and so on.

Let us now suppose that n = 2, and estimate the advantage of A. Let X denote this advantage, >S^ the whole sum to be gained.

Now A may win the first and second games ; his chance for this \^ pq, and then he receives S. He may win the first game, and lose the second ; his chance for this is p^. He may lose the first game and win the second; his chance for this is ^. In the last two cases his position is neither better nor worse than at first ; that is he may be said to receive x.

Thus X = pq S -{■ {p"^ -{- q^) X \

r pq S pq S S

therefore a?=., ^ ., 2= ^ =7T •

1 —p — q zpq A

Hence of course J5's advantage is also - . Thus the players

are on an equal footing.

James Bernoulli in his way obtains this result. He says that whatever may be the value of n, the players are on an equal foot- ing ; he verifies the statement by calculating numerically the chances for n = 2, 8, 4 or 5, taking^ = 2q. See his pages 18, 19.

Perhaps the following remarks may be sufficient to shew that whatever n may be, the players must be on an equal footing. By the peculiar law of the game which we have explained, it follows that the contest is not decided until one player has gained at least n games, and is at least two games in advance of his adversary.

JAMES BERNOULLI. 77

Thus the contest is either decided in an even number of games, or else in an odd number of games in which the victor is at least three games in advance of his adversary : in the last case no ad- vantage or disadvantage will accrue to either player if they play one more game and count it in. Thus the contest may be con- ducted without any change of probabilities under the following laws: the number of games shall be even, and the victor gain not less than n and be at least two in advance of his adversary. But since the number of games is to be even we see that the two players are on an equal footing.

135. Gouraud has given the following summary of the merits of the A7^s Conjectandi ; see his page 28 :

Tel est ce livre de YArs conjectandi, livre qui, si Ton considere le temps ou il fut compose, I'origiualite, Fetendue et la penetration d'esprit qu'y montra son autenr, la fecondite etonnante de la constitution scientifique qu'il donna au Calcul des probabilites, I'influence enfin qu'il devait exercer sur deux siecles d'analyse, pourra sans exageration etre regarde comme un des monuments les plus importants de I'histoire des matliematiques. II a place a jamais le nom de Jacques Bernoulli parmi les noms de ces inventeurs, a qui la posterite reconnaissante rejiorte tou- jours et a bon droit, le plus pur merite des decouvertes, que sans leur premier effort, elle n'aurait jamais su faire.

Tliis 2^aneg}Tic, however, seems to neglect the simple fact r.f the date of inihlication of the Ars Conjectandi, which was really subsequent to the first appearance of Montmort and De Moivre in this field of mathematical investigation. The researches of James Bernoulli were doubtless the earlier in existence, but they were the later in appearance before the world ; and thus the influence which they might have exercised had been already produced. The problems in the first three parts of the Ars Conjectandi cannot be considered equal in importance or difliculty to those which we find investigated by Montmort and De Moivre ; but the memorable theorem in the fourth part, which justly bears its author's name, will ensure him a permanent \)\'d.cQ in the history of the Theory of Probability.

CHAPTER VIII.

MONTMORT.

186. The work which next claims attention is that of Mont- mort; it is entitled Essai d! Analyse stir les Jeux de Hazards.

Fontenelle's Hloge de M. de Montmort is contained in the volume for 1719 of the Hist, de V Acad... Paris, which was pub- lished in 1721 ; from this we take a few particulars.

Pierre Eemond de Montmort was born in 1678. Under the influence of his guide, master, and friend, Malebranche, he devoted himself to religion, philosophy, and mathematics. He accepted with reluctance a canonry of Notre-Dame at Paris, which he re- linquished in order to marry. He continued his simple and retired life, and we are told that, j^ar un honheur assez singidier le mariage lui rendit sa maison plus agreahle. In 1708 he pub- lished his work on Chances, where with the courage of Columbus he revealed a new world to mathematicians.

After Montmort's work appeared De Moivre published his essay De Mensura Sortis. Fontenelle says,

Je ne dissimulerai point qui M. de Montmort fut vivement pique de cet ouvrage, qui lui parut avoir ete entierement fait sur le sien, et d'apres le sien. II est vrai, qu'il y 6toit loue, et n'etoit-ce pas assez, dira-t-on 1 mais un Seigneur de fief n'en quittera pas pour des louanges celui qu'il pretend lui devoir foi et liommage des terres qu'il tient de lui. Je parle selon sa pretention, et ne decide nulloinent s'il etoit en efi'et le Seigneur.

Montmort died of small pox at Paris in 1719. He had been engaged on a work entitled Histoire de la Geometrie, but -had not

MONTMORT. 79

proceeded far with it; on this subject Fontenelle has some inter- esting remarks. See also Montucla's Histoire des Mathematiques, first edition, Preface, page vii.

137. There are two editions of Montmort's work; the first appeared in 1708; the second is sometimes said to have appeared in 1713, but the date 1714 is on the title page of my copy, which appears to have been a present to 'sGravesande from the author. Both editions are in quarto; the first contains 189 pages with a preface of xxiv pages, and the second contains 414 pages with a preface and advertisement of XLII pages. The increased bulk of the second edition arises, partly from the introduction of a treatise on combinations which occupies pages 1 — 72, and partly from the addition of a series of letters which passed between Montmort and Nicholas Bernoulli with one letter from John Bernoulli. The name of Montmort does not appear on the title page or in the work, except once on page 338, where it is used with respect to a place.

Any reference which we make to Montmort's work must be taken to apply to the second edition unless the contrary is stated.

Montucla says, page 394, speaking of the second edition of Montmort's work, Cette edition, independamment de ses aug- mentations et corrections faites a la premiere, est remarquable par de belles gravures a la tete de chaque partie. These engravings are four in number, and they occur also in the first edition, and of course the impressions will naturally be finer in the earlier edition. It is desirable to correct the eiTor implied in Montucla's state- ment, because the work is scarce, and thus those who merely wish for the engravings may direct their attention to the first edition, leaving the second for mathematicians,

138. Leibnitz corresponded with Montmort and his brother; and he records a very favourable opinion of the work we are now about to examine. He says, however, J'aurois souhaite les loix des Jeux un peu mieux decrites, et les termes expliques en favour des dtrangers et de la posterite. Leibnitii Opera Omnia, ed. Dutens, Vol. v. pages 17 and 28.

Reference is also made to Montmort and his book in the cor- respondence between Leibnitz and John and Nicholas Bernoulli ;

80 MONTMORT.

see the work cited in Art. 59, pages 827, 836, 837, 8-i2, 846, 903, 985, 987, 989.

139. We will now give a detailed account of Montmort's work ; we will take the second edition as our standard, and point out as occasion may require when our remarks do not apply to the first edition also.

140. The preface occupies XXIV pages. Montmort refers to the fact that James Bernoulli had been engaged on a work entitled De arte conjectandi, which his premature death had prevented him from completing. Montmort's introduction to these studies had arisen from the request of some friends that he would determine the advantage of the banker at the game of Pharaon; and he had been led on to compose a work which might compensate for the loss of Bernoulli's.

Montmort makes some judicious observations on the foolish and superstitious notions which were prevalent among persons devoted to games of chance, and proposes to check these by shew- ing, 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 to their own ignorance. Per- haps however he speaks rather as a philosopher than as a gambler when he says positively on his page vili,

On joueroit sans donte avec plus d'agrement si Ton pouvoit sgavoir a chaqne coup I'esperance qu'on a de gagner, ou le risque que I'on court de perdre. On seroit plus tranquile sur les evenemens du jeu, et on sentiroit mieux le ridicule de ces plaintes continuelles ausquelles se laissent aller la plupart des Joueurs dans les rencontres les plus com- munes, lorsqu'elles leur sout conti'aires.

141. Montmort divides his work into four parts. The first part contains the theory of combinations ; the second part discusses certain games of chance depending on cards ; the third part dis- cusses certain games of chance depending on dice; the fourth part contains the solution of various problems in chances, including the five problems proposed by Huygens. To these four parts must be added the letters to which we have alluded in Art. 137.

MONTMORT. 81

Montmort gives his reasons for not devoting a part to the appli- cation of his subject to political, economical, and moral questions, in conformity with the known design of James Bernoulli; see his pages XIII — XX. His reasons contain a good appreciation of the difficulty that must attend all such applications, and he thus states the conditions under which we may attempt them with advantage: 1^. borner la question que Ton se propose h un petit nombre de suppositions, etablies sur des faits certains; 2". faire abstraction do toutes les circonstances ausquelles la liberte de I'homme, cet ocueil perpetuel de nos connoissances, pourroit avoir quelque part. Montmort praises highly the memoir by Halley, which we have already noticed ; and also commends Petty's Political A rithmetic ; see Arts. 57, 01.

Montmort refers briefly to his predecessors, Huygens, Pascal, and Format. He says that his work is intended principally for mathematicians, and that he has fully explained the various games which he discusses because, pour I'ordinaire les S^avans ne sont pas Joueurs; see his page xxiii.

142. After the preface follows an Avertissement which was not in the first edition. Montmort sa3^s that two small treatises on the subject had appeared since his first edition; namely a thesis by Nicolas Bernoulli De arte conjectandi in Jure, and a memoir by De Moivre, De meiisura sortis.

Montmort seems to have been much displeased with the terms in which reference was made to him by De Moivre. De Moivre had said,

Ilugenius, primus quod sciani regulas tradidit ad istius generis Pro- blematum Solutionem, quas nuperrimus autor Gallus variis exemplis pulclire illustravit ; sed non videntur viri clarissimi ea simplicitate ac generalitate usi fuisse quam natura rei postulabat : etenirn dum p] ures quantitates incognitas usurpant, ut varias Collusorum conditiones re- praesentent, calculum siumi nimis perplexum redduut ; diimque Colhi- sorum dexteritatem semper aequalem pomint, doctriuam hanc ludorum intra limites nimis arctos continent.

Montmort seems to have taken needless offence at these words ; he thought his own performances were undervalued, and accord- ingly he defends his own claims : this leads him to give a sketch

6

82 MOXTMORT.

of the history of the Theory of Probability from its origin. He attributes to himself the merit of having explored a subject which had been only slightly noticed and then entirely forgotten for sixty years ; see his page xxx.

143. The first part of Montmort's work is entitled TraiU des Combinaisons ; it occupies pages 1 — 72. Montmort says, on his page XXV, that he has here collected the theorems on Combina- tions which were scattered over the work in the first edition, and that he has added some theorems.

Montmort begins by explaining the properties of Pascal's Arith- metical Triangle. He gives the general expression for the term which occupies an assigned place in the Arithmetical Tiiangle. He shews how to find the sum of the squares, cubes, fourth powers, . . . of the first n natural numbers. He refers, on his page 20, to a book called the New introduction to the Mathematics written by M. Johnes, scavant Geometre Anglois. The author here meant is one who is usually described as the father of Sir William Jones. Montmort then investigates the number of permutations of an assigned set of things taken in an assigned number together.

14-i. Much of this part of Montmort's work would however be now considered to belong rather to the chapter on Chances than to the chapter on Combinations in a treatise on Algebra. We have in fact numerous examples about drawing cards and throwing dice.

We will notice some of the more interesting points in this part. We may remark that in order to denote the number of combinations of n things taken r at a time, Montmort uses the symbol of a small rectangle with n above it and r below it.

145. Montmort proposes to establish the Binomial Theorem; see his page 32. He says that this theorem may be demonstrated in various ways. His own method will be seen from an example. Suppose we require (a + 6)^ Conceive that we have four counters each having two faces, one black and one white. Then Montmort has already shewn by the aid of the Arithmetical Triangle that if the four counters are thrown promiscuously there is one way ia which all the faces presented will be black, four ways in which

MONTMORT. 83

three faces will be black and one white, six ways in which two faces will be black and two white; and so on. Then he reasons thus: we know by the rules for multiplication that in order to raise a + h to the fourth power (1) we must take the fourth power of a and the fourth power of h, which is the same thing as taking the four black faces and the four white faces, (2) we must take the cube of a with b, and the cube of b with a in as many ways as possible, which is the same thing as taking the three black faces with one white face, and the three white faces with one black face, (3) we must take the square of a with the square of b in as many ways as possible, which is the same thing as taking the two black faces with the two white faces. Hence the coefficients in the Binomial Theorem must be the numbers 1, 4, 6, which we have already obtained in considering the cases which can arise with the four counters.

l-iG. Thus in fact Montmort argues a priori that the coeffi- cients in the expansion of {a + hy must be equal to the numbers of cases corresponding to the different ways in which the white and black faces may appear if n counters are thrown 2)romiscuously, each counter having one black face and one white face.

Montmort gives on his page 3i a similar interpretation to the coefficients of the multinomial theorem. Hence we see that he in some cases passed from theorems in Chances to theorems in pure Algebra, while we now pass more readily from theorems in pure Algebra to their application to the doctrine of Chances.

147. On his page 42 Montmort has the following problem: There are jj dice each having the same number of faces; find the number of ways in which when they are thrown at random we can have a aces, b twos, c threes, . . .

The result will be in modern notation

\a \b[G...

He then proceeds to a case a little more complex, namely where we are to have a of one sort of faces, h of another sort, c of a third sort, and so on, without specifying whether the a faces

G— 2

8^ MONTMOET.

are to be aces, or twos, or threes, ,.., and similarly without specify- ing for the h faces, or the c faces, . . .

He had given the result for this problem in his first edition, page 137, where the factors B, C, JD, E, F,... must however be omitted from his denominator ; he suppressed the demonstration in his first edition because he said it would be long and abstruse, and only intelligible to such persons as were capable of discovering it for themselves.

148. On his page 46 Montmort gives the following problem, which is new in the second edition : There are n dice each having /faces, marked with the numbers from 1 to/; they are thrown at random : determine the number of ways in which the sum of the numbers exhibited by the dice will be equal to a given number p.

"We should now solve the problem by finding the coefficient of x^ in the expansion of

(a; + 03^ + 03'+ ...+x^Y,

/I — x^y^

that is the coefficient of x^'"' in the expansion of I = J , that is

in the expansion of (1 — x)'"" (1 — x^y. Let p — n = s; then the required number is

n (ii+l) ... (n-h s —1) 71 (72 + 1) ... (n+s —f— 1)

«-/

n(n-l) n(n + V) ... (n+ s —2f- 1) 1.2 l.s-2/

The series is to be continued so long as all the factors which occur are positive. Montmort demonstrates the formula, but in a much more laborious way than the above.

149. The preceding formula is one of the standard results of the subject, and we must now trace its history. The formula was first published by De Moivre without demonstration in the Be Mensura Sortis. Montmort says, on his page 364, that it was derived from page 141 of his first edition; but this assertion is quite un- founded, for all that we have in Montmort's first edition, at the place cited, is a table of the various throws which can be made with any number of dice up to nine in number. Montmort how-

MONTMORT. 85

ever shews by tlie evidence of a letter addressed to John Bernoulli, dated 15th November, 1710, that he was himself acquainted with the formula before it was published by De Moivi-e ; see Montmort, page 307. De Moivre first published his demonstration in his Miscellanea Analytica, 1730, where he ably replied to the asser- tion that the formula had been derived from the first edition of Montmort's work ; see Miscellama Analytica, pages 191 — 197. De Moivre's demonstration is the same as that which we have given.

150. Montmort then proceeds to a more difficult question. Suppose we have three sets of cards, each set containing ten cards marked with the numbers 1, 2, . . . 10. If three cards are taken out of the thirty, it is required to find in how many ways the sum of the numbers on the cards will amount to an assigned number.

In this problem the assigned number may arise (1) from three cards no two of which are of the same set, (2) from three cards two of which are of one set and the third of another set, (3) from three cards all of the same set. The first case is treated in the problem, Article 148; the other two cases are new.

Montmort here gives no general solution; he only shews how a table may be made registering all the required results.

He sums up thus, page 62 : Cette methode est un peu longue, mais j'ai de la peine a croire qu'on puisse en trouver une plus courte.

The problem discussed here by Montmort may be stated thus : We require the number of solutions of the equation x -\- y + z = p, under the restriction that x, y, z shall be positive integers lying between 1 and 10 inclusive, and p a positive integer wdiich has an assigned value lying between 3 and 30 inclusive.

151. In his pages 63 — 72 Montmort discusses a problem in the summation of series. We should now enunciate it as a general question of Finite Differences : to find the sum of any assigned number of terms of a series in which the Finite Differences of a certain order are zero.

In modern notation, let iin denote the n^^ term and suppose that the {in + 1)*^ Finite Difference is zero.

86 MONTMOET.

Then it is shewn in works on Finite Differences, that

i(n = % + 'i^^Uo 4- -J — 2~ -^'^^^ + • • •

, yi(?i--l) ...(??-m+l) .,„

i -» j 11 Uq .

[m

This formula Montmort gives, using A, B, C,... for Aw^j AV^,

By the aid of this formula the summation of an assigned

number of terms of the proposed series is reduced to depend on the

,. ^ . ^ ,., n (n—1) ... (n — r+1) .

summation of series of which — ^ — j — ^ ^ may be

taken as the type of the general term ; and such summations have been already effected by means of the Arithmetical Triangle and its properties.

152. Montmort naturally attaches great importance to this general investigation, which is new in the second edition. He says, page ^5^

Ce Problerae a, comme Ton voit, toute I'etendue et toute I'universa- lite possible, et semble ne rien laisser a desirer sur cette matiere, qui n'a encore et6 traitee par personne, que je s^ache : j'en avois obmis la de- monstration dans le Journal des Sgavans du mois de Mars 1711.

De Moivi'e in his Doctrine of Chances uses the rule which Montmort here demonstrates. In the first edition of the Doctrine of Chances, page 29, we are told that the "Demonstration may be had from the Methodus Differentialis of Sir Isaac Xewton, printed in his Analysis!' In the second edition of the Doctrine of Chances, page 52, and in the third edition, page 59, the origin of the rule is carried further back, namely, to the fifth Lemma of the Princijna, Book iii. See also Miscellanea Analytica, page 152.

De Moivre seems here hardly to do full justice to Montmort ; for the latter is fairly entitled to the credit of the first explicit enunciation of the rule, even though it may be implicitl}^ contained in Newton's Princijna and Methodus Differentialis.

153. Montmort's second part occupies pages 73 — 172 ; it