Clotnell Univetaity lUibtwcs Jtlfara, Nrw fmrlt Cornell University Library QA 173.T63 History of the mathematical theory of pr 3 1924 005 730 282 B Cornell University B Library The original of tiiis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924005730282 HISTORY OF THE THEOBY OF PROBABILITY. A HISTOKY OF THE MATHEMATICAL THEORY OF PROBABILITY FEOM THE TIME OF PA 8 GAL TO THAT OF LAPLACE. BY I?*TODHUNTER, M.A., F.RS. ©ambrtlrge ani Hotttion: MACMILLAN AND CO. 1865. (Eamfirage: FKtNTED BY C. J. CLAY, M.J AT THE UNIVERSITY PRESS. PREFACE. The favourable reception which has been granted to my History of the CoIcuItxs of Variations during the Ifineteenth 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 Theory of Probabihty 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 which illus- trious names can confer, by the simple statement that nearly every great mathematician within the range of a century and a half wiU come before us in the course of the history. To mention only the most distinguished in this distinguished roll — we shall find here — Pascal and Fermat, 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 BernouUis 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 supremacy 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 M^rd 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 Fermat, 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 Lisurance. 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 la,st 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 spared 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 fuU 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 gi-eat problem of the Duration of Play. Unfortunately he did not pubhsh demonstrations, and Lagrange PREFACE. VU 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 1750. These inves- tigations 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 principally 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 investigation of the gain 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 1780. This Chapter brings before us Kaestner, Clark, Mallet, John Bernoulli, Beguehn, Michell, Lambert, Buffon, Fuss, and some others. The memoir of MicheU 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 Viu PREFACE. STlbmitting 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 JEncyclopedie Metkodique, 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. I 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 unknoTVTi element. I hope I have thus accompHshed 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 was 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 facihty 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 mathe- PREFACE. IX matical literature, except \n which is an abbreviation for the pro- duct 1.2, ...n, 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, 136, 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 Mathematiques 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 Lande. I should be sorry to appear 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. HaUam 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 Useful 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, and 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, Vol. ix. page 238, and another letter by him in the Times, Dec. 16, 1862. The treatise is inter- 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 du Galcul des Probahilites depuis ses origines jusqu'cb. nos jours par Charles Gouraud... Paris, 1848. 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 stibject. 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 formulae 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 I'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 like 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 my aim to estimate carefully and impartially the character and the merit of the 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 required 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 publication 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 en-or 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 which may be detected in the work. I have been careful to corroborate my statements by 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 anything 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 Theoiy of Probability were accomplished during the 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- XU 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 principally 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 accoinpaniment. 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 Cambeidgb, May, 1865. CONTENTS. Chapter I. Cabdan. Kepler. Galileo . . l Commentary pn Dante, i. Cnrdan, J9e Imdo Alece, i. Kepler, De Stella Nma, 4. Galileo, Considerazione sopra il Giuco dei Dadi, 4 ; Lettere, 5. Chapter II. Pascal and Fermat .... 7 Quotations from Laplace, Poisson, and Boole, 7. De M^r^'s Problems, 7. Problem of Points, 9. De M^r^'s dissatisfaction, 11. Opinion of Leib- nitz, 12. Termat's solution of the Problem of Points, 13. Eoberval, 13. Pascal's error, 14. The Arithmetical Triangle, 17. Pascal's design, 20. Contemporary mathematicians, 21. Chapter III. Huygens 22 De Saiiodniis in Ludo Alex, 22. English translations, 23. Huygens's solu- tion of a problem, 24; Problems proposed for solution, 25. CBU.PTER IV. On Combinations .... 26 William Buckley, 26. Bernardus Bauhusius and Eryciua Puteanus, 27. Quo- tation from James Bernoulli, 28. Pascal, 29. Schooten, 30. Leibnitz, Dissertatio de Arte Conibinatoria, 31 ; his fruitless attempts, 33. Wallis's 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, 41 ; his table, 42 ; geometrical illustration, 43. Chapter VI. Miscellaneous Investigations between the tears 1670 AND 1700 44 Caramuel's Maihesis 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 Theologioe Chris- tiance Principia MathemaUca, 54. Credibility of Buman Tealimomy, 55. Chapter VII. James Bernoulli . . . .56 Correspondence with Leibnitz, 56 ; Art Ccmjectandi, 57. Error of Montucla, 58. Contents of the Ars Conjeciandi, 58. Problem of Points, 59. James Bernoulli's own method for problems on chances, 60; his solution of a XIV CONTENTS. TAOS problem on Duration of Play, 6i ; he points out a plausible mistake, 63 ; treats of Permutations and Combinations, 64 ; bie 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 "^^ Fontenelle's Moge, 78. Two editions of Montmort's boot, 79; contents of the ' book, 80; De Moiyre'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 sol- ed by a lady, 95. Pro- blem of Points, 96; Bowls, 100; Duration of I lay, loi; Her, 106; Tas, no. Letter from John Bernoulli, 113. Nicolas BernoulU's game of chance, 116. Treize, 120. Summation of Series, 121. Waldegrave's problem, 122. Summation of Series, 125. Malebranohe, 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; Introduction 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, i66; Duration 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; Aibuthnot's argu- ment, 193. Chapter X. Miscellaneous Investigations between the years 1700 AND 1750 194 Nicolas Bernoulli, 194. Barbeyrac, 196. Arbuthnot's argument on Divine Providence, 197. Waldegrave's problem, 199. Brovpne'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, 211. Pro'jlem by John Bernoulli, 212. CONTENTS. XV Chapter XL Daniel Bernoulli . . . .213 Theory of Moral Expectation, 213; Petersburg Problem, 220; Inclination of planes of Planetary Orbits, 222; Smallpox, 224; mean duration' of mar- riages, 229; Daniel Bernoulli's problem, 231 ; Births of boys and girls, 235; Errors of observations, 236. Chapter XII. Euler 239 Treize, 239; Mortality, 240; Annuities, 242; Pharaon, 243; Lottery, 245; Lottery, 247; notes on Lagrange, 249; Lottery, 250; Life Assurance, 256. Chapter XIII. D'Alembert 258 Croix Ott Pile, 258; Petersburg Problem, 259; Small-pox, 265; Petersburg Problem, 275; Mathematical Expectation, 276; Inoculation, 277 i Croix ou Pile, 279; Petersburg Problem, 280; Inoculation, 282; refers to Laplace, 287; Petersburg Problem, 288; error in a problem, 290. Chapter XIV. Bates 294 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 ; Eeourring Series, 313; Problem of Points, 315; Dura- tion of Play, 316; Ajinuities, 320. Chapter XVI. Miscellaneous Investigations be- tween THE TEARS 1750 AND 1780 . . . 321 Kaestner, 321. Dodson, 322. Hoyle, 322. Clark's Laws of Chance, 323. Mallet, 325. John Bernoulli, 325. Begueliu, on a Lottery problem, 328 ; on the Petersburg Problem, 332. Michell, 332. John BemouUi, 335. Lambert, 335. Mallet, 337. Emerson, 343. Buffon, on gambling, 344 ; on the Petersburg Problem, 345 ; his own problem, 347. Fuss, 349. Chapter XVII. Condorcet 351 IHscours PrSUminaire, 351; ^isai, 353; first Hypothesis, 353; second Hypo- thesis, 357; problem on a Run of Events, 361; election of candidates for anofSce, 370; problems on inverse probability, 378; Risk which maybe neglected, 386; 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, PAGE Chapter XVIIL Trembley 411 Problem of Points, 412 ; probability of causes, 413; problem of birtbs, 415; lottery problem, 421; small-pox, 423; duration of marriages, 426; theory of errors, 428 ; Her, 429. Chapter XIX. Miscellaneous Investigations be- tween the tears 1780 AND 1800 . . ■ 432 Prevost, 432. Borda, 432. Malfatti, 434. Bioquilley, 438. EncyclopMie MS- thodique, 441. D'Anieres, 445. Waring, 446. AnciUon, 453. Prevost and Lhuilier, 453. Young, 463. Chapter XX. Laplace 464 Memoirs of 1774, 464; reoiorrlng 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 birtbs, 482 ; theory of errors, 484; Memoir of 1779, 484 ; Generating Functions, 484; Memoirof 1782, 485; Memoirs of 1783, 485; Memoir of 1809, 487 ; Memoir of 1810, 489 ; Connaissance des Terns, 490; Problem on Comets, 491; TMoric.des ProldbiliUs, 495; editions of it, 495; dedication to Napoleon, 496; Laplace's researches in Physical Astronomy, 499,; Pascal's argument, 500 ; illusions, 501 ; Bacon, 503 ; lAvre I. 505 ; Generating Functions, 503 ; 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 Theoi:em, S48; Daniel Bernoulli's problem, 558; fourth Chap- ter, 560; Poisson's problem, 561; Least Squares, 571; history of this subject, s88 ; fifth Chapter, 589. Buffon's problem, 590; sixth Chapter, 592; a Definite Integral, 594; seventh Chapter, 598; eighth Chapter, 601; Small-pox, 601 ; 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, 6io; second Supple- ment, 611; third Supplement, 612; quotation from Poisson, 613. Appendix ' 614 John de Witt, 614. Eizzetti, 614. Kahle, 615. 's Gravesande, 616. Quotation from John Bernoulli, 616. Mendelsohn, 616. Lhuilier, 618. Waring, 618. CHAPTER I. CAEDAJSf. 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 Bivina 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 MathSmatiques en Italic, 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 entierement ignorl cette sorte de calcul. L'6rudition modeme en a, il est vrai, trouv6 quelques traces dans un poeme en latin barbare intitulS : De Tetula, oeuvre d'un moine du Bas- Bmpire, dans un commentaire de Dante de la fin du XV * siScle, et dans les 6crits de plusieurs mathlmaticiens Italians du moyen ^ge et de la renaissance, Pacioli, Tartaglia, Peverone; Gout&uA, Histoire du CalcvZ des ProbabUites, 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, " JerSme Cardan a donne un Traits De Ludo Alese; mais on n'y trouve que de I'&udition et des reflexions morales." Essai d' Analyse. . . p. XL. Libri says, " Cardan a 6crit un traits special de Ludo Alece, oh se trouvent rdsolues plusieurs questions d'analyse combinatoire." Histoire, Vol. iii. 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 appearing on each of the dice, or by six appearing on either and four on the other. And so on. Cardan proceeds, "Sed in Ludo fritiUi 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 favourable 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 : 12 345 6789 10 11 12 108 111 115 120 126 133 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 with Cardan's 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 Jff 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, iV=TC ; for any throw between seven and twelve, both inclusive, iV"= 3m + n; for any throw not greater than six, JV= 108 + 3m + 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 suggested which shall give 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 Morley, 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 Probability of the English Cyclopcedia. 7. Some remarks on the subject of chance were made by Kepler in his work Be Stella Nova in pede Serpentarii, 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 160i, 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 Br Ch. Frisch, Vol. ii. pp. 714; — 716. See also the Life of Kepler in the Library of Useful Knowledge, 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 Nimirum lusor hac vice tessellam alio latere arripuit, aliter manu condidit, aliter intus agitavit, alio impetu animi manusve projecit, aliter intei-flavit aura, alio loco alvei impegit. Nihil hie est, quod sua causa sic caruerit, si quis ista subtiHa posset consectari. 8. The next investigation which we have to notice is .that by Galileo, entitled Considerazione sopra it Giuco dei Badi. The date of this piece is unknown; Galileo died in 1642. It appears that a friend had consulted Galileo on the following difficulty : 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. number 10 is ofteiier 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 — 296, of Le Opere .... di Galileo Galilei, Firenze, 1855. From the Bihlio- grafia Galileiana given in Vol. xv. of this edition of Galileo's works we learn that the piece first appeared 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 MathSmatiques en Ttalie, Vol. IV. page 288, has the following remark relating to Galileo : ..."Ton voit, par ses lettres, qu'il s'(^tait longtemps occupe d'une question delicate et non encore r^solue, relative k la manifere de compter les erreurs en raison geom^trique ou en proportion arithm^tique, question qui touche ^galement au calcul des pro- babilit^s et k I'aritbm^tique politique." Libri refers to Vol. II. page 55, of the edition of GaKleo'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 intomo la stiTna 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 gaming, were accustomed to improve themselves by learned conversation in cultivated society. In one of their meetings the following question was proposed ; a horse is really worth a hundred crowns, one person estimated it at ten crowns and another at a thousand ; which of the two made the more extravagant estimate ? Among the persons who were consulted was Galileo ; he pronounced the two estimates to be equally extravagant, because the ratio of a 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 greater than that of a hundred above ten. Various letters of 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 proposed 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 imder our notice. CHAPTER II. PASCAL AND FERMAT. 10. The indications which we have given in the preceding Chapter of the subsequent Theory of Probability are extreffiely slight ; and we find that writers on the subject have shewn a jus- tifiable pride in connecting the true origin of their science with the great name of Pascal. Thus," Elle doit la naissance k deux GeomStres fran9ais du dix-septi§me siScle, si f^cond en. grands hommes et en grandes decouvei"tes, et peut- Stre de tous les sieoles celui qui fait le plus d'honneur k I'esprit humain. Pascal et Fermat se proposlrent et risolurent quelques pro- bllmes sur las probabilit6s... Laplace, Theorie...des Proh. 1st edition, page 3. TJn probl^me relatif aux jeux de basai'd, propos6 5. un axistlre jan- sgniste par un homme du monde a 6t§ rorigine du calcul des probabilitfis. Poisson, Eecherches sur la Prob. page 1. The problem whicli the Chevalier de Mlr6 (a reputed gamester) proposed to the recluse of Port Royal (not yet withdrawn 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 mathematical analysis, and to render valuable service in the practical concerns of life." Boole, Lcms of ThougM, page 243. 11. It appears then that the Chevalier de M&d proposed certain questions to Pascal ; and Pascal corresponded with Fer- mat on the subject of these questions. Unfortunately only a portion of the correspondence is now accessible. Three letters 8 PASCAL AND FEEIIAT. of Pascal to Format on this subject, -which were all written in 1654, were published in the Varia Opera Mathematica D. Petri de Fermat... To\o&?e., 1679, pages 179 — 188. These letters are reprinted in Pascal's works ; in the edition of Paris, 1819, they occur in Vol. rv. pages 360 — 388. This volume of Pascal's works also contains some letters written by Fermat 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-bien qu'S, vous ; et quoique je sois encore au lit, je ne puis m'emp^clier de vous dire que je regus 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'gtendre ; mais en un mot vous avez trouvg les deux partis des d6s et des parties dans la parfaite justesse : j'en suis tout satisfait ; car je ne doute plus maintenant que je ne sois dans la verity, apr§s la rencontre admirable o^ je me trouve avec vous. J'admire bien davantage la methode des parties que celle des dgs; j'avois vu plusieurs personnea trouver celle des d6s, comme M. le chevalier de Mer6, qui est celui qui m'a proposi ces questions, et aussi M. de Eoberval ; 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 FEBMAT. 9 posed that the players have equal chances of winning 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 64 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 would 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 will 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 th6 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. i'inally, 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 wiU be in the situation first examined, in which the first player is entitled to 56 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 32 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 wiU give them in modem notation. (1) Suppose each player to have staked a sum of money denoted by A ; let the number of points in the game he 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...(2>.-1) ^"^^2.4.6... 2^- 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 modeito 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 FEEMAT. 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 difficult^ qui 6tonnoit fort M. de Mer6 : car il a trls-bon esprit, mais il n'est pas giomltre ; tfest, comme vous savez, un grand d6faut; etm^me ilne com- prend pas qu'une ligne matlilmatique soit divisible I. I'iafini, et croit fort bien entendre qu'eUe est compos6e de points en nombre fini, et jamais je n'ai pu Ten tirer j si vous pouviez le faire, on le rendroit parfait. II me disoit done qu'il avoit trouv6 faussetl 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 36, 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 VoUS, quel 6toit son grand scandale, qui lui faisoit dire hautement que les propositions n'6toient pas constantes, et que I'arithmgtique se d6mentoit. Mais vous en verrez bien ais^ment la raison, par les prin- cipes ofi vous ^tes. 15. In Pascal's letter, as it is printed in Format's works, the name de MSrS is not given in the passage we have quoted in the preceding article ; a blank occurs after the M. It seems, however, to be generally allowed that the blank has been filled up correctly by the pubHshers 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 M^r^ could solve one problem, celle des dSs, and seems to imply that he failed only in the Problem of Points. Montucla says that the Problem of Points was proposed to Pascal by the Chevalier de M^r^ "qui lui en proposa aii_ssi quelques autres sur le jeu de dds, comme de determiner en eombien de coups on pent parier d'amener une rafle, &c. Ce chevalier, plus bel esprit que 12 PASCAL AND FERMAT. gdomfetre ou analysts, rdsolut h la v&it^ ces derniferes, qui ne sont pas Men difficiles ; mais il ^choua pour le pr^c^dent, ainsi que Roberval, S, qui Pascal le proposa." p. 384. These words would seem to imply that, in Montucla's opinion, M. de M^r^ 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 suggesting that M. de Mdrd 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 'M.4r4, " II est vi-ai cependant que le Che- valier avoit quelque g&ie extraordinaire, mSme pour les Math^- matiques ;" and these words seem intended seriously, although in the context of this passage Leibnitz is depreciating M. de Merd Leibnitii, Opera Omnia, ed. Butens, Vol. ii. part 1. p. 92. In the Nowoeaux Essais, Liv. iv. Chap. 16, Leibnitz says, " Le Chevalier de M^r^ dont les Agrements et les autres ouvrages ont ^t^ imprimis, homme d'un esprit p^n^trant et qui ^toit joueur et philosophe." It must be confessed that Leibnitz speaks far less favourably of M. de M^r^ 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 M^re maintained that a magnitude was not infinitely divisible : this assists in identi- fying him with Pascal's friend who would have been perfect 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 Fermai 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 Format PASCAL AND FERMAT, 13 sent to Pascal a solution of the Problem of Points depending on combinations. Pascal's second letter is dated August 24tlx, 1654. He says that Format'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 b b a a a a a b a b a b b a a b b b a b a a b a abba b a b a b b b a a a b b a b b b b a b b b b b b 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 b 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, A'a chance of winning the game is to Bs chance as 11 is to 5. 17. Pascal says that he communicated Fermat's method to Koberval, 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 14 PASCAL AND FERMAT. made in the decision by the superfluous trial or trials. Pascal puts 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, oti 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 Fermat'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, b, 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 b a a c a a a a b b a h c a b a a c b a c c a c a b a b b a c b a a b b b b b c b b a b c b b c c b G a c a b c a c c a a c b b c b c c b a c c b c c c c c Let A denote the player who wants one point, and B and G the other two players. By examining the 27 cases, Pascal finds 13 which 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 G. 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 FEEMAT. 13 favourable to B may be considered to be 4 + f , that is 5^. Simi- larly the number of cases favourable to G may be considered to be 55. Thus it would appear that the chances of A, B, and G are respectively as 16, 5i and 5^. 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 Format's method it is assumed that three trials will necessarily be made, which is not assumed in his own method. Pascal was wrong in stipposing that the true result could be affected by assuming that three trials would 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 G 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 G are as 17, 5, and 5, as Pascal had found by his own method. Format then gives another solution, for the sake of Koberval, in which he does not assume that three trials will necessarily be made ; and he arrives at the same result as before. In the remainder of his letter Format enunciates some of his memorable propositions relating to the Theory of Numbers. Pascal replied on October 27th, 1654, to Format's letter, and said that he was entirely satisfied. 16 PASCAL AND FEEMAT. 20. There is another letter from Fennat 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 ^, 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 thi-ow before any throw is made. The first throw is worth ^ of the stake ; the second is worth ^ of what remains, that is ^ of the stake ; the third throw is worth ^ of what now remains, that is ^^ of the stake ; the fourth throw is worth ^ of what now remains, that is -^^^ 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 modem solution of the problem by the aid of the Calculus of Finite Differences ; see Laplace, Thiorie...des Prob. page 210. Gouraud awards to Feimat'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 TraiU du triangle arithTnStique. This treatise was printed about 1654, but not published 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 5 6 7 8 9 3 6 10 15 21 28 36.. . 4 10 20 35 56 84.. , 5 15 35 70 126. , 6 21 56 126. ^, 7 28 84. . , 8 36. , 9. , 1 .. . In the successive horizontal rows we have what are now called the figurate numbers. Pascal distinguishes them into orders. He caUs 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, 6, 10,... had already received the name of triangular numbers ; and the numbers of the fomrth order 1, 4, 10, 20,... the name oi pyramvidal 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 modem notation the w* term of the r^ order is n(n + l) ... (w + r-2) \r-l Pascal constructs the Arithmetical Triangle by the following definition ; each number is the sum of that immediately above it and that immediately to the left of it. Thus 10 = 4 + 6, 35 = 20 + 15, 126 = 70 + 56,... 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 18 PASCAL AND FEEMAT. 23. Pascal applies his Arithmetical Triangle to various subjects ; among these 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 Triangle 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 comer. Thus the tenth base is a line drawn through the num- bers 1, 9, 36, 84, 126, 126, 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 + w)* base ; the chance of A is to the chance of 5 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 estabUshes this by induction. Pascal's result may be easily shewn to coincide with that obtained by other methods. For the terms in the (m + «)"' base are the coefiicients in the expansion of (1 + x)'^^'^ by the Binomial Theorem. Let m + n — l=r; then Pascal's result amounts to saying that the chance of A is proportional to 1 I ,. I r{r-l) I r{r-l)...(r-n + 2) and the chance of B proportional to i + ,+ 4i:l1)+.... + ^(>-i)-(^-^_+2) 1.2 m— 1 This agrees with the result now usually given in elementary treatises ; see Algebra, Chapter Llli. 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 AND FERMAT. 19 Let M denote the number of cases which are favourable to A, and iV the number of cases which are favourable to B. Let r = 2n—2. Li the second example we have M + ]Sr=2', M-N= ^ \n-l \n-l = '^ "^y- Then \i 2 8 denote the whole sum at stake, A is entitled to gf- . —^ — , that is to ^ (2'' + X) ; so that he may be considered to have recovered his own stake and to have won the fraction of of his adversary's stake. In the third example we have llw— 2 \n — l\n — \ r Thus we shall find that A may be considered to have recovered \ 2"" his own stake, and to have won the fraction -^^^ of his adversary's 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 preserved 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 Celeherrimce Matheseos Academiw Parisiensi; this Academy was one of those voluntaiy associations which preceded the formation of formal scientific societies : see Pascal's Works, Tol. IV. p. 356. In the letter Pascal enumerates various treatises which he had prepared and which he hoped to 2—2 26 PASCAL AND FERMAT. publish, among which 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 penitiis intentatse materise tractatio, scilicet de composUione' alecB in ludis ipsi suhjeclis, quod gallico nostro idiomate dicitur {/aire les pa/rtis des jeux) : ubi anceps fortuna sequitate rationis ita reprimitur ut utrique lusorum quod jure competit exacts semper assignetur. Quod quidem eo fortius ratiocinaiido quserendum, quo miniis tentando investigari possit : ambigui enim sortis eventus fortuitsa contingentise potiiis quam naturali necessitati meritb tribuuntur. Ideb res hactenus erravit incerta ; nunc autem quae experimento rebellis fuerat, rationis dominium effugere non potuit : earn quippe tanta se- curitate in artem per geometriam • reduximus, ut certitudinis ejus particeps facta, jam audacter prodeat ; et sic matheseos demonstrationes cum alese incertitudine jungendo, et quae contraria videntur conciliando, ab utrique nominationem suam aocipiens stupendum hunc titulum jure sibi arrogat : alece geometria. But the design was probably never accomplished. The letter is dated 1654; 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 ProbabiHty really commenced with Pascal and Fermat ; 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 when 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 Fermat is confined to a narrower range ; but it is of a special kind which is without a parallel in the history of science. Feimat enunciated various remarkable propositions in the theory of numbers. Two of these are more important than the rest; one of them after baffling the powers of Euler and La- grange finally yielded to Cauchy, and the other remains still un- PASCAL AND FERMAT. 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 Theoiy of Probability started from the hands of Pascal and Fermat, they were the most distiuguished mathematicians of Europe. Descartes died in 1650, and Newton and Leibnitz were as yet unknown ; Newton was bom in 1642, and Leibnitz in 1646. Huygens was bom 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 Wallis, bom in 1616, and appointed Savilian professor of geometry at Oxford in 1649, was steadily rising in reputation, while Barrow, bom 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 Huygens entitled Be Ratiociniis in Ludo Alece. This treatise was first printed by Schooten at the end of his work entitled Francisci ci Schooten Exercitationum Mathematicarum Libri 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 written 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 tot^ 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 Conjecta/ndi, and forms the first of the four parts HUYGENS. 23 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 gainiag a or 6 or c, his expectation is J (a + 6 + c). The third proposition asserts that if a player has p chances of gaining a arid q chances of gaining b, his expectation is — . 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 Drinlcwater, 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. 32. 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 eighth and ninth propositions discuss sitnple cases of the Problem of Points when there are iAree players ; the method is similar to that for two players. S3. 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 inay appear. The thirteenth proposition consists of the following problem. A and B play with two dice ; if a seven is thrown, A wins; if a ten is thrown, B wins; 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 HUYGKNS. 34. 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 jB'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^ chance, but when it is B& own turn to throw his chance will have a different value, say y. 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 31 cases A loses and ^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, S- Thus So _ 2,\y """ 36 • Now suppose B about to throw, and let us estimate B 's chance. There are 36 equally likely cases ; in 6 cases B wins and A takes nothing; in the other 30 cases B loses and ^'s turn comes on again, in which case jB's chance is worth x by supposition. So that the expectation of B is — ^r^ — . Thus y = 36 6a + 30a; "36~ ' From these equations it will be found that x — -^— , and thus a-x= -^ , so that J.'s chance is to 5's chance as 30 is to 31. 35. 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) ^ and 5 play with two dice on this condition, that^ gains if he throws six, and B gains if he throws seven. A first has one HUYGENS. 25 throw, then B has two throws, then A two throws, and so on until one or the other gains. Shew that ^'s chance is to B's as 10355 to 12276. (2) Three players A, B, G 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, C 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 A's chance is to B's as 1000 is to 8139. (4) Twelve balls are taken, eight of which are black and four are white. A plays 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 thrown 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 aU the counters. Shew that ^'s chance is to Fs as 244140625 is to 282429536481. 86. 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 shaU 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 COMBINATIONS. S7. 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. 38. 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 Philosophy of Arithmetic full citations are given from Buckley's work ; in Dr. Peacock's History of Arithmetic 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 WaUis apparently overestimates the age of Buckley's work, when he says "...in Arithmetica sua. BAUHUSIUS. 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. 39. 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 Futea/ni Pietatis Thaumata in Berna/rdi Bauhusii h Societate Jesu Froteum Farthenium. The book consists of 116 quarto pages, exclusive of seven pages, not numbered, which contain an Index, Censura, Summa Privilegii, and a typographical ornament. It appears that Bemardus Bauhusius composed the following line in honour of the Virgin Mary: Tot tibi sunt dotes, Virgo, quot sidera cselo. 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 ap- 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 fi inclusive. The number 1022 is the same as the number of the stars according to Ptolemy's Cata- logue, which 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 aUthe 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 says, page 103, Dicere horruit Vates : Sidera tot cselo, 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 arrangements seems to have attracted great attention during the following century ; the discussion on the subject was finally settled 28 PASCAL. by James Bernoulli in his Ars Gortjectandi, where he thus details the history of the problem. ...Qiiemadmodiim cernere est in hexametro 3, Bernh. Bauhusio Jesuitl Lovaniensi in laudem Virginis Deiparse constructo : Tot tihi sunt Dotes, Virgo, quot sidera ccdo; quern dignum peculiar! operA duxerunt plures Viri celebres. Erycius Puteanus in libeilo, quern Thaumata Pietatis inscripait, variationes ejus utiles integris 48 paginis enumerat, easque numero stellarum, quarum vulgb 1022 recensentur, accommodat, omissis scrupulosiiis illis, quse di- cere videntur, tot sidera cselo esse, quot Marise dotes; nam Marise dotes esse multo plures. Eundem numerum 1022 ex Puteano repetit Gerh. Vossius, cap. 7, de Scient. MatLemat. Prestetus Gallus in primd editione Element. Mathemat. pag. 358. Proteo huic 2196 Tariationes attribuit, sed factd reyisione in alter4 edit. torn. pr. pag. 133. numerum ean.im dimidio fere auctum ad 3276 extendit. Industrii Actorum Lips, Collectores m. Jua. 1686, in recensione Tractatua Wallisiani de Algebr4, numerum in quaestione (quem Auctor ipse definire non fuit ausus) ad 2580 determinant. Et ipse postmodiim "WaUisius in edit, latinei, opens sui Oxon. anno 1693. impress^, pagin. 494. eundem ad 3096 profert. Sed omnea adhuc 'k vero deficientes, ut delusam tot Virorum post adhibitas quoque secundas curas in re levi perspicaciam meritb mireris. Ars Conjectamdi, 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 csesura. 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 p^ horizontal row is equal to the multitude of the combinations of r things taken j) 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+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 the 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 following. 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 m at a time, is 2" — 1. At the end Pascal considers this problem. Datis duobus numeris insequalibus, invenire quot modis minor in majore combinetur. And from his Arithmetical Triangle he deduces in effect the follow- ing result; the number of combinations of r things taken p at a time is (j)+l) (p + 2) (jj + 3)...r ^ \i — p After this problem Pascal adds. Hoc problemate tractatum tninc absolvere constitueram, non tamen omninb sine molestia, ciim multa alia parata habeam ; sed ubi tanta ubertas, vi moderanda est fames : liis ergo pauca haec subjiciam. Eruditissimus ac miM charisimus, D.D. de GaniSres, circa combina- tiones, assiduo ac pemtili labore, more suo, incumbens, ac indigens facili constructione ad inveniendum quoties numerus datus in alio dato combinetur, banc 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 (r-1)... {r-p+\) \E 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 says, Excellentem banc solu- so SCHOOTEN. tionem ipse mihi ostendit, ac etiam demonstrandam proposuit, 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 asseeutS., jam reliqua quae invitus supprimebam Ubenter omitto, adeo dulce est amicorum memorari. 42. la the work of Schooten to which we have already re- ferred in Art. 28 we find some very slight remarks on combinations and their applications; see pages 373 — 403. Schooten's first sec- tion is entitled, Eatio inveniendi electiones omnes, qus fieri pos- sunt, data multitudine rerum. He takes four letters a, b, c, d, and arranges them thus, a. h. ah. c. ac. he. ahc. d. ad. hd. ahd. 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, per b unum pirum, per c unum prunum, et per d unum cerasum, et ipsa alitor atque alitor, ut supra, eHgantur, electio eorum fieri poterit 15 diversis modis, utsequitur.... 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 excluding unity but including the number itself, or 16 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. b. ab. aab. c. ac. aac. be. abc. aabc. We have thus 2 + 3 + 6 elections. LEIBNITZ. 31 Similarly if the pi'oposed letters are a, a, a, h, b, 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, abed, a'bo, a^b^, a^b, a^. Hence the question may be asked, what is the least number which has 16 divisors? This question must be answered by trial ; we must take the smallest prime numbers 2, 3,. . . 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 a'bcd, a'b% a^bc, a%^, a'b', a^^b 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 published by Leibnitz in 1666, entitled Dissertatio de Arte Cqmbinatoria ; part of it had been previously published in the same year under the title of Disputatio ckrith- metica de compleadonibus. 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, HaUe, 1858. The dissertation is also included in the collection of the philosq- 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 aiTangements 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. 45. 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 symbol com2natio (combinatio) ; when three at a time he uses con3natio (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 vulgatus. 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. 33 metical Triangle, he adds, "Adjiciemus hie Theoremata quoram TO 3x4 ex ipsa tabula manifestum est, to Sioti ex tabulae funda- mento." The only theorem here that is of any importance is that which we should now express thus : if w 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 detexerifc ignoramus. Schwenterus Delic. 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 Sphaer. edit. Eomse forma 4ta anno 1785. p. 33. seqq. With respect to Schwenter it has been observed, 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 Drmkwater, page 45. (5) Leibnitz uses the symbols H — = in their present sense ; he uses ^^ for multiplication and ^^ for division. He uses the word productum 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 truisms 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. O M WALLIS. 47. A discourse of combinations, 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 part 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 Combinations, 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- ber 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 already referred ia Art. 38. 49. Wallis's second Chapter is Of Alternations, or the different change of Order, in any Number of things proposed. Here he gives some examples of what are now usually called permutations ; thus if there are four letters a, b, 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 1x2x3x4x5x6, 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 Anagram." The chapter closes with Wallis's attempt at determining the number of arrangements of the verse Tot tibi sunt dotes, virgo, quot sidera cselo. The attempt is followed by these words, " 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 jjhese 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." WaUis'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 with 53 figures, in tenebris decumbens, sola fretus memoria. See his AJgebra, page 450. 50. Wallis's third Chapter is Of the Divisors and Aliquot parts, of a Number proposed. This Chapter treats of the resolu- tion of a number into its prime factors, and of the number of divisors which a given number has, and of the least numbers which have an assigned number of divisors. 51. Wallis's fourth Chapter is Monsieur Fermat's 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. 3—2 36 PRESTET. 62. Thus the theory of combinations is not applied by Wallis in any manner that materially bears upon our subject. In fact the influence of Fermat 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 work 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 SUments de mathimatiqms, 2° ^d., in the following terms, Le pere Prestet, enfin, fort habile geomfetre, avait explique avec infiniment de clart^, en 1689, les principaux artifices de cet art ing^nieux de composer et de varier les grandeurs. Qouraud, page 23. CHAPTER V. MOETALITY 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 unknown until the time of John de "Witt he subjoins in a note, Inconnue du moins des modernes. Oar 11 paraitrait par un passage du Digeste, ad legem Fcdcidiam, xxxv. 2, 68, que les Komains n'en ignoraient pas absolument I'usage. Voyez 'k ce si:yet M. V. Leclerc, Des Journaux chez les Bommns, p. 198, et une curieuse dissertation: De prohoMlitate vitce ejusqiie usu /orensi, 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 44. After teferring to the registers of the annual numbers of deaths in London which began to be kept in 1592, and which with some 38 GEATJNT. intennissions between 159-4 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 Royal Society, ... Gouraud says in a note on his page 16, ...John Graunt, homme sans g6om6trie, mais qui ne manquait ni de sagacite ni de bon sens, avait, dans une sorte de trait! d'Arithme- tique politique intituM: Natural cmd political observations... made upon the hills of mortality, etc., rassembl6 ces differentes listes, et donne mime {Hid. chap, xi.) im calcul, I. la v6rite fort grossier, mais du moins fort original, de la mortality probable a chaque 4ge d'un certain nombre d'individus supposes nis viables tous au meme instant. See also the Athencmm 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 probl^me des rentes viagSres fut trait€ par Van Hudden qui quoique ggomStre, ne laissa pas que d'etre bourguemestre d' Amsterdam, JOHN DE WITT. 3D et par le c61Sbre pensionnaire d'HoUande, Jean, de "Witt, un des pre- miers promoteurs de la geomltrie de Descartes. J'ignore le titre de I'ecrit de Hudden, mais celui de Jean de Witt 6toit intitule : De vardye van de lif-renten na proportie van de los-renten, ou la Valeur des rentes viag^res en rodson des ventes libres ou remhoursables (La Haye, 1671). lis 6toient I'tin et 1' autre plus h, port€e que personne d'en sentir I'impor- tance et de se procurer les d^pouillemens nScessaires de registres de mor- talite; aussi Leibnitz, passant en Hollande qu.elques annSes aprSs, fit tout son possible pour se procurer I'Scrit de Jean de "Witt, mais il ne pent y parvenir ; il n'6toit 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 exemplaire entre les mains; il nous en donne un precis, par lequel on voit combien Jean de "Witt raisonnoit juste sur cette mati§re. Le chevalier Petty, Anglois, qui s'occupa beaucoup de calculs poli- tiques, entreyit le probleme, mais il n'gtoit pas assez geomltre pour le traiter fructueusement, en sorte que, jusqu'a, Halley, I'Angleterre et la France qui empruntSrent tant et ont tant empruntS depuis, le firent comme des aveugles ou comme de jeunes d6baucli6s. 67. With respect to Sir William Petty, to whom. Montucla refers, we may remark that his writings do not seem to have been very important in connexion with our present subject. Some account of them is given in the article ArithmStique Politique of the original French Uncyclopedie ; the article is reproduced in the Encyclopidie Mithodique. Gouraud speaks of Petty thus in a note on his page 16, AprSs Graunt, le chevalier W. Petty, dans diff&ents essais d'eco- nomie politique, ou il y avait, il est vrai, plus d 'imagination que de jugement, s'ltait, de 1682 I. 1687, occupe de semblables recherches. 58. With 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. Butens, Vol. II. part 1, page 93 ; Vol. VI. part 1, page 217. 59. 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 literally confirm Montucla's 40 JOHN DE WITT. statement respecting Leibnitz: see Leibnizens Mathematische Schriften herausgegeben von G. I. Gerhardt, Erste Abtheilung. Band iii. Halle 1855. James Bernoulli says, page 78, Nuper in Menstruis Excerptis Hanoverae impressis citatum inveni Tractatum quendam mihi ignotum Pensionarii de Wit von Subtiler Ausrechnung des valoris der Leib-Renten. Fortasse is quaedam hue facientia habet; quod si sit, copiam ejus miii alicunde fieri percuperem. In his reply Leibnitz says, page 84, Pensionarii de Wit libellus exiguus est, ubi aestimatione ilia nota utitur a possibilitate casuum aequalium aequali et hinc ostendit re- ditus ad vitam sufficientes pro sorte a Batavis solvi. Ideo Belgice scripserat, ut aequitas in vulgus appareret. In his next letter, page 89, James Bernoulli says that De Witt's book wiU 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 Sckeda 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 nanoiscare, 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, nbi vacaverit. Sed vix aliquid in eo novum Tibi occurret, cum fundamentis iisdem ubique insistat, quibus cum alii viri docti jam erant nsi, turn Paschalius in Triangiilo Arithmetico, et Hugenius in diss, de Alea, nempe ut medium Arithmeticum inter aeque incerta sumatur; quo fundamento etiam rustici utuntur, cum praediorum pretia aestimant, et rerum fis- calium curatores, cum reditus praefecturarum Principis medios consti- tuunt, quando se ofiert 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 curvarwm, Leyden 1650, which is commended by Condorcet ; see Condorcet's I!ssai.,.d^Analyse..i -page CLXXxiv. 61. We have now to notice a memoir by Halley, entitled An estimate of the Degrees of the Mortality of Mankind, draiun from curious Tables of the Births and Funerals at the City of Breslaw; with an Attempt to ascertain the Price of Annuities upon Lives. This memoir is published in Vol. xvii. 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 biUs were not suit- able for drawing accurate deductions. First, In that the If timber of the People was wanting. Secondly, That the Ages of the People dying was not to be had. And Lastly, That both London and JDuUin 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, Lf it were possible, that the People we treat of should not at all be changed, but die where they were bom, 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 Koyal Society, in whose archives it is supposed that copies of the original registers are still preserved." Lubbock and Brink- water, page 45. 64 The Breslau registers do not appear to have been pub- lished themselves, and Halley gives only a very brief introduction 4Z HALLEY. to the table which he deduced from them. following form: 1 1000 2 855 3 798 4 760 Halley's table is in the 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 bom, 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 bom 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 I'Acad. ... Paris, 1760. 65. HaUey 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. 66. 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 if 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 W=B+T, so that Y represents the number who have died out of iV" 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 will be alive and the elder dead is -^rf- ; and so on. HaUey gives according to the fashion of the time a geometri- cal illustration. J C F G H Let AB or CD represent N, and BE or BH represent R, so IhsX EC or HA represents Y. Similarly AG, AF, GF may represent n, r, y. Then of course the rectangle ECFQ 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 — 656 of the same volume of the Philoso- phical Trcmsactions we have Some further Considerations on the Breslaw Bills of Mortality. By the same HoMd, Sc.. 68. De Moivre refers to HaUey's memoir, and republishes the table; see De Moivre's Doctrine of Chances, pages 261, 345. CHAPTER VI. MISCELLANEOUS INVESTIGATIONS Between the tears 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 Biceps, 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 Combinatoria which occupies pages 921 — 1036, and part of this is devoted to our subject. Caramuel gives fii-st an account of combinations in the modem 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 modem sense Cara- muel proceeds to explain the Ars Lulliana, 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, quae Oombinatoriw genus est, de Alea, et Ludis Fortunm CARAMUEL. 45 serio disputa/ns. This treatise includes a reprint of the treatise of Huygens, which however is attributed to another person. Cara- muel says, page 984, Dum hoc Syntagma Perillustri Domino K. Viro eruditissimo com- municarem, ostendit etiam mibi iugeniosam quamdam de eodem argu- mento Diatribam, quam ^ Christiano Seyeriuo Longomontano fuisse scriptam putabat, et, quia est curiosa, et brevis, debuit huic Queestioni subjungi... In the table of contents to his work, page xxviii, Caramuel speaks of the tract of Huygens as Diatribe ingeniose h, Longomontano, ut putatur, de hoc eodem argu- mento scripta : nescio an evulgata. Longomontanus was a Danish astronomer who lived from 1562 to 1647. 72. Nicolas Bernoulli speaks very severely of Caramuel. He says Un Jesuite nomm^ Caramuel, que j'ai cit^ 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 These 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 Acta I!rud....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 wrong 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 other 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'a chance 46 SAUVEUR. 5 5 at his first throw is ^ , and ^ x 36 = 5 ; thus taking 5 from 36 we may consider 31 as left for B. Now B's chance of success in a single throw is ^ ; thus ^ x 31, that is 5 J, may be considered the value of his first throw. Thus Caramuel assigns 5 to A and 5^ to 5, as the value of their first throws respectively ; then the remaining 25| 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 25|, and then to B for his second throw ^ of the remainder ; and so on. Thus he would 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 guidance in the discussion of the Problem of PoLuts for two players, and then deserted it. 74. In the Journal des Sgavans 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, Qubd si quis D.ni Salvatoris Tabellas cum hisce nostris contulerit, deprehendet illas in quibusdam locis, prsesertim ultimis, nonnihU emendationis iadigere. Montucla, page 390, and Gouraud, page 17, seem also to think Sauveur more inaccurate than he really is. An Sloge of Sauveur by Fontenelle is given in the volume for 1716 of the Hist, de V Acad.... Paris. Fontenelle says that Bassette was more beneficial to Sauveur than to most of those who LEIBNITZ. 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 Montmort, 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 throAvn. 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 results. The chances involve infinite series which are not summed. James Bernoulli's solutions are reprinted in the collected edition of his works, Geneva, 1744 ; see pages 207 and 430. The problems are also solved in the Ars Conjectandi, pages 52 — 56. 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 nowhere 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 ARBUTHNOT. 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 dis, il est aussi faisable de jetter douze points, que d'ea jetter onze ■ car I'un et I'autre ne se peut faire que d'une seule maniSrej 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 soci^t^ royale. Montucla, page 391. Lubbock and Drinkwater say respecting it, page 43, This essay, whicb. was edited, and is generally supposed to have been written by Motte, the secretary of the Eoyal Society, contains a translation of Huyghens's treatise, and an application of his princi- ples to the determiuation 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 Probability, page 5. 79. It does not appear however that there was any fellow of the Koyal 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 particular 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 Bibliotheca Britannica, under the head Arbuthnot, places the work with the date 1692. 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, revis'd by John Ham. By whom is added, a demonstration of the gain of the banker in any circumstance of the game caU'd Pharaon; and how to determine the odds at the Ace of Hearts or Fair Chance; with the arithmetical solution of some questions relating to lotteries; and a few remarks upon Hazard and Backgammon. London. Printed for B. Motte and C. Bathurst, at the Middle-Temple Gate in Fleet-street, M.DCC.xxxvni. 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 50 AEBUTHNOT. of part of the title we have already giveB, 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, kit Mr. Huygens's Book, de Eatiociniis in Ludo Aleae, 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 wiU 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 resolVd it shall not want an Epistle to the Reader 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 Use, 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. 61 Clamours of the Sharpers, I do not much, regard it, since they are a sort of People the World is not bound to provide for ...It is impossible for a Die, with such 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 Artj The Reader may here observe the Force of Numbers, which can be suocessfiiUy 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 being 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 ChUd, 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 Pemales: The Yearly Bills of Mortality are observ'd to bear such Proportion to the live People as 1 to 30, or 26; 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 iugenious Reader may have the satisfaction 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, A'a share 55 9 will be y^Q ' ■w^hich is less than jg- . 55 This result ^Hq lio'w^ever is wrong in either of the other two senses which James Bernoulli ascribes to the problem, for which he 77 101 obtains tt-^ and r-^^ respectively as the results ; see Art. 35. 84. Then follow some other calculations about games. We have some remarks about the Royal-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 given of the number of various throws which can be made with three dice. Pages 34 — 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 Chances for a, q Chances for h, ROBERTS. 53 and r Chances for c, then my hazard is worth ^^ + ^? + c^ » q^^. p + 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 Chcmces, page 160. 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 shall add one Problem of that kind, leaving the Solution to those who think it merits their pains. In Parallelipipedo cujus latera sunt ad invicem in ratione a,b,c: Invenire quota vice quivis suscipere potest, ut datum quodvis planum, v.g. aSjaciat. The problem was afterwards discussed by Thomas Simpson ; it is Problem xxvii. of his Nature and Laws of Chance. 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 shillings. 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 IcJttery, and only 2 to 1 in the second. 54 CEAIG. The paradox is made by Roberts himself, by his own arbitrary definition of odds. Supposing a lottery has a blanks and 6 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 I him are formed by the product of t and -^y , that is, the odds are as a to 6 (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 tte probability of testimony, which appeared in 1699, under the title of "Theologiae Christianfe 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 paribus), 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 Animadversiones on it were published by Peterson in 1701. Prevost and Lhuilier notice Craig's work in a memoir published in the Memoires de VAcad. . ..Berlin, 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 611. 91. A Calculation of the Credibility 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 p^,p^,...p„: the essay takes the product ^j^^ ...^„ as representing the resulting probability. Next, suppose we have concv/rrent witnesses. Let there be two witnesses ; the first witness is supposed to leave an amount of un- certainty represented by 1 —Pi, of this the second witness removes the fraction p^, and therefore leaves the fraction (1 —p^ (1 — p^ : thus the resulting probabihty is 1 — (1 — pj (1 — pj. Similarly if there are three concurrent testimonies the resulting probability is 1 — (1 — Pi) (1 —p^ (1 — i's) J ^^^ ^° °^ ^°^ ^ greater number. The theory of this essay is adopted in the article Probabiliti of the original French Encyclopidie, which is reproduced in the Encyclopidie MSthodique: the article is unsigned, so that we must apparently ascribe it to Diderot. The same theory is adopted by BicquiUey in his work Du Galcul des ProbabilitSs. CHAPTER VIL JAMES BERNOULLI. 92. We now propose to give an account of the Ars Conjeo- 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 Prof. Br. Peter Merian, Basel, 1860. 93. Leibnitz states that at his request James Bernoulli studied the subject. Feu Mr. Bernoulli a cultiv^ cette matifere sur mes exhortations; Leibnitii Opera Omnia, ed. Butens, Vol. VI. 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 dootrinam de aestimandis probabilitatibus (quam ego magni facio) non parum esse excultam. Vellem aliquis varia ludendi genera (in quibus pulchra tujus doctrinae specimina) mattematice trac- taret. Id simul amoenum et utile foret nee Te aut quocimque gra- vissimo 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. Verum est me a plu- JAMES BERNOULLI. 57 ribus retro annis hujusmodi speculationibus magnopere delectari, ut vix putem, quemquam plura super his meditatum esse. Aninms etiam erat, Tractatum quendam conscribendi de hac materia; sed saepe per integros annos seposui, quia naturalis mens torpor, quern accessoria vale- tudinis meae infirmitas immane quantum auxit, facit ut aegerrime ad sciibendum accedam ; et saepe mihi optarem amanuensem, qui cogitata mea leviter sibi iudicata plene divinare, scriptisque consignare posset. Absolvi tamen jam maximam Libri partem, sed deest adhuc praecipua, qua artis coDJectandi 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 94: and Bernoulli briefly repHes, page 97, Quod Verisimilitudines spectat, et earum augmentum pro aucto soil, observationum numero, res omnino se habet ut scripsi, et certus sum Tibi placituram demonstrationem, 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 rumor vere narrat, redibit cei-te frater meus Basileam, non tameu Graecam (cum ipse sit avaA^a^Sijros) sed meam potius stationem (quam brevi cum vita me derelicturum, forte non. vane, existimat) occupaturus. De iniquis suspicionibus, quibus me immerentem onerasti in Tuis pe- nultimis, alias, ubi plus otii nactus fuero. Nunc vale et fave etc. 95. The Ars Conjectandi was not published until eight years after the death of its author. The volume of the Hist, de r Acad.. ..Paris for 1705, published in 1706, contains FonteneUe's Uloge 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 ■which appeared in the Journal des Sgavans for 1706: this notice is attributed to Saurin by Montmortj see his page IV. Eeferences 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, 845, 847, 922, 923, 925, 931. 96. The Ars 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. Theirs 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 Ars 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 AlecB, 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 BEENOULLI. 59 of various problems relating to games of cbance. 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 WaUis ascribes to Des Cartes; see WaRis's 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 L'Art de Gonjectv/rer, Traduit du Latin de Jacques Bernoulli; Avec des Observations, Eclairdssemens et Additions. Fa/r L. 0. 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 prolonged 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 m can be obtained by throwing n dice is equal to the co-eflScient of «"" in the development of (x + a^ + x^ + x* + af + af)" in a series of powers of x. See his page 24. 60 JAJIES 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 Nouveaux Mimoires de I' Acad.... 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 h c and failure in a single trial be - and - respectively: then the required chance consists of the terms of the expansion of ( - H — I from I - J to the term which involves ( - 1 I - 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 & = 5, JAMES BEENOULLI. 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+c = e+y! Now consider the expectations of the different players ; they are as foUows: I. II. III IV. V. VI. VII. VIII. . h a' ce a'' bcf a" c'ef e'er hcT a' •• 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 a" cases, so the expectation is — ^ . 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 cfb favourable cases out of a° cases, so the ex- hcf pectation is —4 ■ And so on for the other players. Now let 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 A's expectation is -^ > , and B's expectation is a — cj the proportion is that of 30 to 31, which agrees with ce a' -of the result in Art. 34. 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 Flay, a subject which afterwards exercised the highest powers of De Moivre, 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 pai-ticular 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 modem form of solution of the problem. Let Mj, denote A's chance of winning aU his adversary's count- ers when he has himself a; counters. In the next game A must either win or lose a counter; his chances for these two contin- gencies are r and , respectively: and then his chances of winning all his adversary's counters are u^^ and %_, respectively. Hence a b a+6 "^' ' a+b '^'^ 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 (7^ and C^ are arbitrary constants. To determine these constants we observe that A'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 when x is 0, and is equal to 1 when x is m + «. Hence we have o=c,+ o, i = c,+ c,Q" therefore ^i = ~ C Hence u^ = 1 2 ^^'^ ^m+n • m+n-a; La: To determine A's chance at the beginning of the game we must put x = m; thus we obtain oT (cT - b") JAMES BERNOITDLI. 63 In precisely tlie same manner we may find JB's chance at any stage of the game; and his chance at the beginning of the game wUl be &'»(a"-;,") It wiU be observed that the sum of the chances of A and B at the beginning of the game is unity. 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 Moivi'e in his De Mensura Sortis, 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 docuisset. He adds. Quo ipso proin monenmr, ut cauti simus in jiidicando, nee ratio- cima nostra stiper qTiS,cunque statim analogic in rebus depretens^ fun- dare suescamus; quod ipsum tamen etiam ab iis, qtd vel maximS sapere videntur, nimis frequenter fieri solet. Again, on his page 27, Quse quidem eum in finem hie adduce, ut pal^m fiat, quS.in pariim fidendum sit ejusmodi ratiociniis, quse corticem tantiini attingunt, nee in ipsam rei naturam altiiis penetrant; tametsi in toto vitse usu etiam apud sapientissimos quosque nihil sit frequentius. Again, on his page 29, he refers to the difiiculty which Pascal says had been felt by M. de * * * *, whom James Bernoulli calls Anonymus quidam eeeterk subacti judicii Vir, sed Geometrise expers. James Bernoulli adds, Hac enim qui imbuti sunt, ejusmodi evavrio^amai minim§ moran- tur, probS conscii dari innumera, quse admoto calculo aliter se habere comperiuntur, qu^m initio apparebant; ideoque sedulb oavent, juxtS, id quod semel iterumque monui, ne quicquam analogiis temerg tribuant. 61 JAMES BERNOULLI. 109. The second part of the Ars Conjectandi occupies pages 72—137: it contains the doctrine of Permutations and Combina- tions. James Bernoulli says that others have treated this subject before him, and especially Schooten, Leibnitz, WaUis 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 mimerorum 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 fuU analysis of the number of arrangements of the verse Tot tibi sunt dotes, Virgo, quot sidera cceli ; 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 J" 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 modem 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 Ai-ts. 22 and 41, employed without any scruple, and indeed rather with approbation, the method of induction: James Bernoulh however says, page 95,... modus demonstrandi per inductionem partim 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^ plationibus vacarant (quos inter Faulhaberus et Remmelini Ulmenses, Wallisius, Mercator in Logarithmoteohni^, 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 233 of the work cited in Art. 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 Sm, %n^, Sm',... up to 2w" ; he uses the symbol /where we in modem books use 2. He then extends his resillts by induction without demonstration, and introduces for the first time into Analysis the coefficients since so famous as the nvmbers of Bernoulli. His general formula is that ^'^ "0 + 1+2+2^** + 2.3.4 ^" c(c-l)(c-2)(c-3)(c-4) + 2.3.4.5.6 ^" c(c-l)(c-2)(c-8)(c - i) ( c-5)(c-6-) + 2.3.4.5.6 . 7~"8 ^"^ + - where ^ = q-^ 30 ' ^ = 42"^ = ~ 30' - 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, quim inutilis censenda sit opera Ismaelis Bul- lialdi, quam conscribendo tarn spisso volumini Arithmeticse suae Infini- torum impendit, ubi nihil prsestitit aliud, qukm ut primarum tantum sex potestatum summas (partem ejus quod unicd nos consecuti sumus pagind) immense labore demonstratas exhiberefc. For some account of BuUiald's spisswm volwimn, 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 n things 5 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 Ars 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, quBe 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. 104. Suppose that A wants m points and B wants n 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 1-1 xc I ^M-1) , /^(/^- l )(/^-2) ,1(^-1 ). ..{^ -n + 2) ^ 2 "^ J +•••+ \^\ ' where fi = m-^n — l. 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 Wallis, 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, prse- missis etiam vel interspersis nonnuUis facilioribus, et in quibus nullus combinationum 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 15295 1 „.,,,, , which is rather less than 5. Aftfer giving the correct iJllO'l 4 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 ^ 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 probability any number between 1 and 6 inclusive, so that we may take g (1 + 2-1-3 + 4-1-5-1-6), that is 3|-, for his mean throw. We may observe that 3^ is the Arith- 5—2 68 JAMES BEKNOULLI. 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, 14, 174, and 21. Hence the mean of all the numbers is ^ {%\ + 7 + lOH 14 + I7i + 21}, that is 12i ; 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^-^ {3J -1- 21j, that is 12^ 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 BernouUi 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 favour of A, he adds, Hujus JAMES BERNOULLI. 69 autem contrarium ex priore solutione, quss 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 169 — 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 prsemia 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 player 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 aftenvards of what he had himself formerly 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 qtiolibet Alese genere, si ludi Oeconomus seu Dispensator (Je Bcmquier du Jeu) nonnihil habeat prserogativse in eo consistentis, ut pauIo major sit casuum numerus quibus vincit qakra. quibus perdit; et major simul casuum mimerus, quibus in officio Oeconomi pro ludo sequenti confirmatur, qulm quibus cecouomia in coUusorem transfertur. Quseritur, quanti privilegium hoc Oeconomi sit sestimandum ? The problem is chiefly remarkable from the fact that James Bernoulli candidly records two false solutions which occurred 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 eight 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 Ars Conjectandi occupies pages 210 — 239 ; it is entitled Pars Quarta, tradens usum et appUcatio- nem prcecedentis Doctrince in Givilibus, Moralibus et Oeconondcis. 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 wMch we will give the titles. I. Prceliminaria qucedam de Gertitudine, Probabilitate, Neces- sitate, et Contingentia Merum. II. De Scientia et Conjectura. De Arte Conjectandi. De Argumentis Conjectwarum. Axiomata qucedam generalia hue pertinentia. III. De variis argumentorum generibus, et quomodo eorum pondera cestimentur ad supputandas rerum probabilitates. IV. De duplici Modo investigandi numeros casuu/m. Quid sentiendum de illo, qui instituitur per experimenta. Problema singulare earn in rem propositum, &c. V. Solutio Problematis prcecedentis. 122. We wiU briefly notice the results of James Bernoulli as to the probability of arguments. He distinguishes arguments into two kinds, pwre and mixed. He says, Pura voco, quse in qui- busdam casibus ita rem probant, ut in aliis nihil positive probent: Mixta, quEe ita rem probant in casibus nonnulUs, ut in cseteris 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 BEENOTJLLI. 71 1-5 1-^ 2> ^~Q- Then the resulting probability of the con- eft elusion is 1-.^. This is obvious from the consideration that any one of the arguments would establish the conclusion, so that the conclusion fails only -svhen all the arguments fail. Suppose now that we have in addition two arguments of the mixed kind: let their respective probabilities be — ^ , — — , ^ q+r' t + u Then James Bernoulli gives for the resulting probability 1- cfiru adg (ru + qt) But this formula is inaccurate. For the supposition q=0 amounts to having one argument absolutely decisive against the conclusion, while yet the formula leaves still a certain probability for the conclusion. The error was pointed out by Lambert; see Prevost and Lhuilier, Memoires de H Acad.... Berlin for 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 evulgandum hoc loco proposui, postquam jam per vicennmm pressi, et cnjus turn novitas, turn summa utilitas cum pari conjuncta diffioultate omnibus reliquis hujua doc- trinse capitibus pondus et pretium auperaddere potest. Ars Conjectandi, page 227. See also De Moivre's Doctrine of Chances, page 254. We will now state the purely algebraical part of the theorem. Suppose that {r + s)"' is expanded 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 great 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. IogC + log(g-l) /, « \ « log(>- + l)-logr V r + lj r + 1' and logo + log(.-l) ^^^^ - . log (s + 1) - log s V s + 1/ s + 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 + s)"*, where t=r+s is the term involving r'^s"'. Let r and sbe proportional to the probability of the happening and failing of an event in a single trial. Then the sum of the 2n+l 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— 1), and n(r+ 1), both inclusive ; so that the ratio of the number of times the event happens to the whole number of r+1 r — 1 trials lies between — - — and — -— . Then, by taking for n the 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 happens to the whole number of trials lies between —— and r-1 t ' As an example James Bernoulli takes r = aO, s=20, «=5a He finds for the odds to be lOOO to 1 that the ratio of the number of times the event happens to the whole number of trials 3-1 29 shall lie between — and — , it will be sufficient to make 25550 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. JAMES 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 — r of the whole num- 50 50 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 large number of drawings, and have obtained a white ball B times, and a black ball 8 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- 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 approximately agree. See Laplace, Thiorie...de8 Prdb.... pages 282 and 366. 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 241 — 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 JAMES BERNOULLI. 128. James Bernoulli enforces the importance of the subject in the following terms, page 243, Cseterum quantse sit necessitatis pariter at utilitatis hseo 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 Tires naufragium passse, 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, 11 ] let 8=l + l,+ t;+...+- 2 3^ ""^M + l ' then 8= 1 + 1 + 1 + ... + ! + . 1 2 ' 3 ' "-^ n^n + 1 ' therefore = - 1 + .=—= + ^r— j. + -1— + ... + — -! + _1_ ^1.2^2.3^3.4^ ^w (re +1)^M + 1' therefore .; — ^+ -—5 + — —+...+ -^-L_ = l _. ^ 1 .2 ' 2.3 ' 3.4 ' ■••^«(w + l)~" n+1" Thus the sum of n terms of the series, of which the r* term is 1 . « IS r (r + 1) ' « + 1 ■ 130. James Bernoulli saj^s that his brother first observed 1111 that the sum of the infinite series t+h + k + t+ •••is infinite ; 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 T "*" 2'"^ 3^"'"4^ "^ '■■ ^^ fiiii*6, 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 2 gratias feret. The sum is now known to be — ; this result is due to Euler : it is given in his Introductio in Analysin Infinitorum, 1748, Vol. L page 130. JAMES BERNOrLLl. 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 1111 series —rr + —j^ + -77; + -tt + • • • is infinite, for the series is greater yl v^ V" V* 1111 than T+0 + Q + 7 + '" He adds that the sum of all the odd X .^ o 4 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 EvavTto^avetas rationem, etsi ex infiniti iiatura finito intel- lectui comprehendi non posse videatur, nos tamen satis perspectam habemus. 133. At the end of the volume containing the Ars Conjectandi we have the Lettre d un Amy, sur les Parties du Jeu de Pawme, 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 oiA 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 Ai^. 123. 134. One problem occurs in this ieWre d, un Amy... which it may be interesting to notice. Suppose that A and B engage in play, and that each in turn 76 JAMES BEKNOULLI. by the laws of the 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 As, chance of winning is g- and his chance of losing p. The chance of B is found by taking that of A from imity ; so that 5*8 chance is p or g- according as ^'s is q ov 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 re — 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 w = 2, and estimate the advantage of A. Let X denote this advantage, 8 the whole sum to be gained. Now A may win the first and second games ; his chance for this v&pq, and then he receives S. He may win the first game, and lose the second ; his chance for this is jo". 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=pq8+{f-\-q')x; therefore x = ., -^ , -„ = V^ = "s • ^—p-q ^pq 2 Q Hence of course B'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 » = 2, 3, 4 or 6, takings = 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 Ars Conjectandi; see his page 28 : Tel est ce livre de YArs conjectandi, livre qui, si Ton considSre le temps oil il fut composg, I'origiualitfi, I'ltendue et la p6n6tration d'esprit qu'y montra son auteiir, la f6condit6 etonnante de la constitution scientifique qu'U donna au Calcul des probabilitfe, I'influence enfin qu'il devait exercer sur deux siSoles d'analyse, pourra sans exag6ration fetre regard^ comme un des monuments les plus importants de Fhistoire des matli6matiques. II a plac€ k jamais le nom de Jacques Bernoulli parmi les noms de ces inventeurs, h, qui la post6rit6 reconnaissante reporte tou- jours et I, bon droit, le plus pur m6rite des d6couvertes, que sans leur premier effort, elle n'aurait jamais su faire. This panegyric, however, seems to neglect the simple fact of the date of publication 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 difficulty 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 place in the history of the Theory of Probability. CHAPTER VIII. MONTMOKT. 136. The work which next claims attention is that of Mont- mort; it is entitled Essai d' Analyse sur les Jeux de Hazards. Fontenelle's Moge 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 Remond de Montmort was bom 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 N6tre-Dame at Paris, which he re- linquished in order to marry. He continued his simple and Retired Hfe, and we are told that, par un bonheur assez singulier le manage lui rendit sa maison plus agriable. 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 piquS de cet ouvrage, qui lui parut avoir &U entigrement fait sur le sien, et d'aprSs le sien. II est vrai, qu'il y gtoit loug, et n'6toit-ce pas assez, dira-t-on ? mais un Seigneur de fief n'en quittera pas pour des louanges belui qu'il pretend lui devoir foi et hommage des terres qu'il tient de lui. Je parle selon sa pretention, et ne decide nuUement s'il 6toit en effet le Seigneur. Montmort died of small pox at Paris in 1719. He had been engaged on a work entitled Histoire de la Oiometrie, but had not MONTMOET. 79 proceeded far with it; on this subject Fontenelle has some inter.-r 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 's Gravesande 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 xlji 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, ind^pendamment de ses aug- mentations et corrections faites a la premifere, est remarquable par de beUes gravures a la tSte 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 error implied in Montucla's state- ment, because the work is scarce, and thus those who merely wish for the engrq^ings may direct their attention to the first edition, leaving the second for mathematicians. 138. Leibnitz corresponded with Montmort and his brother J and he records a very favourable opinion of the work we are now about to examine. He says, however, J'aurois souhait^ les loix des Jeux un peu mieux decrites, et les termes expliquds en faveui^ des strangers et de la post^rite. Leibnitii Opera Omnia, ed: Butens, Vol. V. pages 17 and 28. Eeference is also made to Montmort and his book in the cor- respondence between Leibnitz and John and Nicholas Bernoulli j 80 MONTMORT. see the work cited in Art. 59, pages 827, 836, 837, 842, 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 viii. On joueroit sans doute avec plus d'agrlment si Ton pouvoit SQavoir 5, chaque coup Tesperance qu'on a de gagner, ou le risque que Ton court de perdre. On seroit plus tranquile sur les 6venemens 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- nnines, lorsqu'elles leur sout contraires. 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. MONTMOET. 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 k un petit nombre de suppositions, ^tabhes sur des faits certains; 2°. faire abstraction de toutes les circonstances ausquelles la liberty de I'homme, cet &ueil 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 Arithmetic ; see Arts. 57, 61. Montmort refers briefly to his predecessors, Huygens, Pascal, and Fermat. 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 xxiil. 142. After the preface follows an Avertissement which was not in the first edition. Montmort says 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 Moivi-e, De mensura 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, Hugermm, primus quod sciam regulaa tradidit ad istius generis Pro- blematum Solutionem, quas nuperrimus autor Gallus variis exemplis pulchre illustravitj sed non videntur viri clarissimi ea simplicitate ac generalitate usi ftiisse quam natura rei postulabat : etenim dum plures quantitates incognitas usurpant, ut varias CoUusorum conditiones re- praesentent, calculum smim nimis perplexum reddunt ; dumque CoUu- sorum dexteritatem semper aequalem ponunt, doctrinam 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 MONTMOET. 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 Comhinaisons ; 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 Triangle. 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, S9avant 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. 1-14. 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 + b)*. Conceive that we have four counters each having two faces, one black and one white. Then Montmort has akeady shewn by the aid of the Arithmetical Triangle that if the four counters are thrown promiscuously there is one way in which all the faces presented wiU be black, four ways in which MONTMOET. 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 + 6 to the fourth power (1) we must take the fourth power of a and the fourth power of b, 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. 146. Thus in fact Montmort argues & priori that the coeffi- cients in the expansion of (a + b)". 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 promiscuously, each counter having one black face and one white face. Montmort gives on his page 34 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 arejp 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[c.. 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 6—2 84 MONTMOKT. are to be aces, or twos, or threes, ..., and similarly without specify- ing for the b faces, or the c faces, . . . He had given the result for this problem in his first edition, page 137, where the factors B, C, D, 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 haimig /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 of in the expansion of (a; + a;' + ar'+...-l-a;/)", that is the coefficient of a;'"" in the expansion of [ = ) , that is in the expansion of (1 - a;)-" (1 - x')". Let p — n = s; then the required number is n{n+\) ...(n + s-1) n{n + l) ... (w+g-/-l) i2 ^2; ■ w (w - 1) n{n + V) ... (W + S-2/-1) 1.2 1^-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 De Mmsura 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 the 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 Moivre; 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 Miscellanea Analytica, pages 191 — 197. De Moivre's demonstration is the same as that which we have given. 150. Montmort then proceeds to a more diflScult 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 which 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 modem notation, let m„ denote the m"' term and suppose that the (m + 1)'" Finite Difference is zero. 86 MONTMOKT. Then it is shewn in works on Finite Differences, that . n(n — 1) .J , m„ = m„ + wAm, + - ^ g AX+ •■• ^ nin-l)...{n-m+l) ^„^^ _ \m This formula Montmort gives, using A, B, C,... for Aw,,, A\, AX,... By the aid of this formula the summation of an assigned number of terms of the proposed series is reduced to depend on the , . -. I- 1 n(n-V) ... (n-r + 1) summation of series of which — ^^ — , — ^^ may be [r 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 65, Ce ProblSme a, comme I'on voit, toute I'^tendue et toute I'universa- lit6 possible, et semble ne rien laisser k disirer sur cette matiere, qui n'a encore 6tg traitle par personne, que je sqacte : j'en a vols obmis la d6- 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 Newton, 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 Principia, Book ill. 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 implicitly contained in Newton's Principia and Methodus Differentialis. 153. Montmort's second part occupies pages 73 — 172 ; it re- MONTMORT. 87 lates to games of chance involving cards. The first game is that called Pharaon. This game is described by De Moivre, and some investigations given by him relating to it. De Moivre restricts himself to the case of a common pack of cards with four suits ; Montmort sup- poses the number of suits to be any number whatever. On the other hand De Moivre calculates the percentage of gain of the banker, which he justly considers the most important and difficult part of the problem ; see Doctrine of Chances, pages ix, 77, 105. Montmort's second edition gives the general results more compactly than the first. 154. We shall make some remarks in connection with Mont- mort's investigations on Pharaon, for the sake of the summation of certain series which present themselves. 155. Suppose that there are p cards in the pack, which the Banker has, and that his adversary's card occurs q times in the pack. Let u^ denote the Banker's advantage, A the sum of money which his adversary stakes. Montmort shews that supposiag that^ — 2 is greater than q. That is Montmort should 3 have this ; but he puts {pq -f)2A-\- (g' -c[)-^A, on his page 89, by mistake for g (g - 1) h ^ ; lie gets right on his page 90. Mont- mort is not quite full enough in the details of the treatment of this equation. The following results will however be found on examination. If q is even we can by successive use of the formula make u^ depend on u^ ; and then it follows from the laws of the game that M, is equal to ^ if g is equal to 2, and to ^ .4 if g is greater than 2. Thus we shall have, if q is an even number greater than 2, 88 MONTMORT. + (^-2)(iJ-3)(p-4)(p-5) , (iP-«7)(p-'y-i)---i .. + + (p - 2) (i,-3). ..(?-!)) If 2 = 2 the last term within the brackets should be doubled. Again if q is odd we can by successive use of the fundamental formula make u, depend on m,^,, and if g is greater than unity it can be shewn that m, , , = 2-=4 ir ■ Thus we shall have, if g is an odd number greater than unity, ■g-1) '-p(p-l)2^|^+ (i'-2)(i'-3) ^(p-2)(i'-2-i)(i'-?-2)(;'-?-3) (y-g)(^-g-l)...2 [ "^ "^(i'-2)(i>-3) J- If 2 = 1 we have by a special investigation u^ = '- (i'-2)(i>-3) 2 P ' If we suppose q even and ^ — g* not less than j — 1, or g' odd and p —q not less than q, some of the terms within the brackets may be simplified. Montmort makes these suppositions, and con- sequently he finds that the series within the brackets may be expressed as a fraction, of which the common denominator is the numerator consists of a series, the first term of which is the same as the denominator, and the last term is (2-2)(g-3)...2.1, or (^-1) (2-2) ... 3.2, according as g is even or odd. The matter contained ia the present article was not given by Montmort in his first edition; it is due to John Bernoulli: see Montmort's, page 287. MONTMOET. 89 156. We are thus naturally led to consider the summation of certain series.^ I,t^(,^,)^ »("+l)("H-2V..(n + .-l) so that (n, r) is the n^ number of the {r + 1)* order of figurate numbers. Let 8^(n,r) stand for ^(m, r) + ^ {n -2,r) + ^ {n - i, r) + ... , so that S{n-l,r)+j>{n-'2,r) + ^{n-^,r)+ ... = («,»• + 1); and by taking the terms in pairs it is easy to see that ^{n,r)-4,{n-l,r)+^{n-'2,,r)-^{n-Z,r) + ... = 8^{n,r-l); therefore, by addition, 8^ {n, r)=^<^{n,r-{-l)+^8^{n,r- 1). Hence, continuing the process, we shall have 8^{n,r)=\,^{n,r + \)+\<^{n.r) + \j>{n,r-l) + ... and we must consider 8j> (w, 0) = 5 "' ^ ^ ^® even, and = ^ (re+1), if n be odd. We may also obtain another expression for 8^ (m, r). For change n into n + 1 in the two fundamental relations, and subtract, instead of adding as before ; thus 8^{n,r) = \4>{n + l,r+\)-\8{n + l,r-\). Hence, continuing the process, we shall have £f<^(«,r) =i,^(« + 1, r + l) -^^ (w+ 2, r) +1^ (n + 3, r- 1) - - ^' ^ (« + r, 2) + ^^ 8 {n + r, 0). 90 MONTMORT. 157. Montmort's own solution of the problem respecting Pharaon depends on the first mode of summation explained in Art. 156, which coincides with Montmort's own process. The fact that in Montmoi-t's result when q is odd, g* — 1 terms are to be taken, and when q is even, q terms are to be taken and the last doubled, depends on the different values we have to ascribe to 8^ (n, 0) ac- cording as n is even or odd ; see Montmort's page 98. Montmort gives another form to his result on his page 99 ; this he obtained, after the publication of his first edition, from Nicolas Bernoulli. It appears however that a wrong date is here assigned to the communication of Nicolas Bernoulli ; see Mont- mort's page 299. This form depends on the second mode of sum- mation explained in Art. 156. It happens that in applying this second mode of summation to the problem of Pharaon n + r is always odd; so that in Nicolas Bernoulli's form for the result we have only one case, and not two cases according as q is even or odd. There is a memoir by Euler on the game of Pharaon in the Hist, de VAcad Berlin for 1764, in which he expresses the ad- vantage of the Banker in the same manner as Nicolas Bernoulli. 158. Montmort gives two tables of numerical results respect- ing Pharaon. One of these tables purports to be an exact exhibi- tion of the Banker's advantage at any stage of the game, supposing it played with an ordinary pack of 52 cards ; the other table is an approximate exhibition of the Banker's advantage. A remark may be made with respect to the former table. The table consists of four columns ; the first and third are correct. The second column should be calculated from the formula ^. — - , by putting for n in succession 50, 48, 46, . . . 4. But in the two copies of the second edition of Montmort's book which I have seen the column is given o-j 1 y o/> incorrectly; it begins with gg^g^ instead of ^rgri, and of the re- maining entries some are correct, but not in their simplest forms, and others are incorrect. The fourth column should be calculated 2n — 5 from the formula £ fa - 1) C -S) ' ^^ P^ttu^g for w in succession 50, 48, 46 ... 4 ; but there are errors and unreduced results in it ; MONTMORT. 91 it begins -with, a fraction having twelve figures in its denominator, whicli in its simplest form would only have four figures. ' In the only copy of the first edition which I have seen these columns are given correctly ; in both editions the description given in the text corresponds not to the incorrect forms but to the cor- rect forms. 159. Montmort next discusses the game' of Lansquenet ; this discussion occupies pages 105 — 129. It does not appear to present any point of interest, and it would be useless labour to verify the complex arithmetical calculations which it involves. A few lines which occurred on pages 40 and 41 of Montmort's first edition are omitted in the second ; while the Articles 84 and 95 of the second edition are new. Article 84 seems to have been suggested to Montmort by John Bernoulli ; see Montmort's page 288 : it relates to a point which James Bernoulli had found difficult, as we have already stated in Art. 119. 160. Montmort next discusses the game of Treize; this dis- cussion occupies pages 130 — 143. The problem involved is one of considerable interest, which has maintained a permanent place in works on the Theory of Probability. The following is the problem considered by Montmort, Suppose that we have thirteen cards numbered 1, 2, 3 ... up to 13 ; and that these cards are thrown promiscuously into a bag. The cards are then drawn out singly ; required the chance that, once at least, the number on a card shall coincide with the number expressing the order in which it is drawn. 161. In his first edition Montmort did not give any demon- strations of his results ; but in his second edition he gives two demonstrations which he had received from Nicolas Bernoulli ; see his pages 301, 302. We will take the first of these demon- strations. Let a, h, c,d,e,,.. denote the cards, n in number. Then the num- ber of possible cases is [n. The number of cases in which a is first is I w — 1. The number of cases in which b is second, but a not first, is |w — 1 — [ w — 2. The number of cases in which o is third, but a not first nor b second, is |ro — 1 — |w — 2 — / |w — 2 — |w — 3 1 92 MONTMOET. that is \n-l ~2 \n-2 + \n-3. The number of cases in -which d is fourth, but neither a, h, nor c in its proper place is ]w-l - 2 |w-2 + |w-3 - J |w-2 -2 |w-3 + 1 w - 4 |, that is |w — 1 — 3 I w — 2 + 3 1 w — 3 - I w — 4 . And generally the number of cases in which the jw* card is in its proper place, while none of its predecessors is in its proper place, is , ,M n (»n — 1) (m — 2) . „ |w-l -(m-l) |w-2 + !^ ±^ 1 |w^3 (»i — 1) (w - 2) (»i - 3) , . - i\Bi-ii We may supply a step here in the process of Nicolas Bernoulli, by shewing the truth of this result by induction. Let i^ {m, n) denote the number of cases in which the jw"" card is the first that occurs in its right place ; we have to trace the connexion between yjr (m, n) and i/r (m + 1, n). The number of cases in which the (jM + 1)* card is in its right place while none of the cards between b and the tm* card, both inclusive, is in its right place, is -^ (m, n). From this number we must reject all those cases in which a is in its right place, and thus we shall obtain •^ (w 4- 1, «). The cases to be rejected are in number ■<^(jn, n — 1). Thus •^ (m + 1, w) = •^ (m, m) — ■^ (jn, n — 1). Hence we can shew that the form assigned by Nicolas Bernoulli to -^ (»ra, «) is universally true. Thus if a person undertakes that the w* card shall be the first that is ia its right place, the number of cases favourable to him is -Jr (m, n), and therefore his chance is ■ ^ ' — ^ . \n If he undertakes that at least one card shall be in its right place, we obtain the number of favourable cases by summing •\/r (m, n) for all values of m from 1 to « both inclusive : the chance is found by dividing this sum by [n. Hence we shall obtain for the chance that at least one card is in its right place, 2^[3 [i+-+ \n • MONTMOBT. 93 We may observe that if we subtract the last expression from unity we obtain the chance that no card is in its right place. Hence if {l+"^2+-r:2 2+ + \m-l\n-l ^ J' and Bs chance of winning the game is 9' Montmort demonstrates the truth of these formulae, but we need not give the demonstrations here as they will be found in elementary works ; see Algebra, Chapter LIII. 173. In Montmort's first edition he had confined himself to the case of equal skill and had given only the first formula, 7 98 MONTMOET. SO that he had not really advanced beyond Pascal, although the formula would be more convenient than the use of the Arith- metical Triangle; see Art. 23. The first formula for the case of unequal skill was communicated to Montmort by John Ber- noulli in a letter dated March 17th, 1710 ; see Montmort's page 295. As we have already stated the formula was known to James BemoulU ; see Art. 113. The second formula for the Problem of Points must be assigned to Montmort himself, for it now appears before us for the first time. 174. It will be interesting to make some comparison between the two formulae given in Art. 172. It may be shewn that we have identically •••^ \m-\ \n-\ ^ This may be shewn by picking out the coefficients of the various powers of q in the expression on the right-hand side, making use of the relations presented by the identity (i-^r-'a-^r =(!-#'. Thus we see that \i p + q be equal to unity the two expres- sions given in Art. 172 for A's, chance are numerically equal. 175. If however ^ + g be not equal to unity the two expres- sions given in Art. 172 for ^'s chance are not numerically equal. If we suppose p-\-q less than unity, we can give the following in- terpretation to the formulae. Suppose that A 's chance of winning in a single trial is p, and B'& chance is q, and that there is the chance 1—p — q that it is a drawn contest. Then the formula i?'"jl+?ngr + 1.2 |m— 1 n — \^ ]. y + .y-^ + _^^y-2= + ... + _J^^n.^n MONTMORT. 99 expresses the chance that A shall win m points before either a single drawn contest occurs, or B wins n points. This is easily seen by examining the reasoning by which the formula is established in the case when p + qis equal to unity. But the formula rjr-l) ^^^^^ , , \z Un\n — 1^ expresses the chance that A shall win m points out of r, on the condition that r trials are to be made, and that A is not to be con- sidered to have won if a drawn contest should occur even after he has won his m points. This follows from the fact that if we expand (p + q + 1 —p — qf in powers of ^, g, \—p — q^ a, term such as Gp'q'Qt. —p — qY ex- presses the chance that A wins p points, B wins o- points, and t contests are drawn. Or we may treat this second case by using the transformation in Art. 174. Then we see that {p + q)'"'^ expresses the chance that there shall be no drawn contest after the m points which A is supposed to have won ; {p + §■)''"""' expresses the chance that there shall be no drawn contest after the m points which A is supposed to have won, and the single point which B is supposed to have won ; and so on. 176. Montmort thinks it might be easily imagined that the chances of A and B, if they respectivel}' want km and Jen points, would be the same as if they respectively wanted m and n points ; but this he says is not the case ; see his page 247. He seems to assert that as k increases the chance of the player of greater skill necessarily increases with it. He does not however demonstrate this. We know by Bernoulli's theorem that if the number of trials be made large enough, there is a very high probability that the number of points won by each player respectively will be neaiiy in the ratio of his skill ; so that if the ratio of m ton be less than that of the skill of A to the skill ofB, we can, by increasing k, obtain as great a probability as we please that A will win km points before B wins kn points. Montmort probably implies, though he does not state, the con- dition which we have put in Italics, 7—2 100 MONTMORT. 177. Montmort devotes his pages 248—257 to the discussion of a game of Bowls, which leads to a problem resembling the Pro- blem of Points. The problem was started by De Moivre in his De Mensura Sortis; see Montmort, page 366, and the Doctrine of Chances, page 121. De Moivre had supposed the players to be of equal skill, and each to have the same number of balls ; Montmort generalised the problem by supposing players of unequal skill and having unequal numbers of balls. Thus the problem was not in Montmort's first edition. Montmort gives on his page 256 a simple example of a solution of a problem which appears very plausible, but which is incorrect. Suppose A plays with one bowl and B with two bowls ; required their respective chances in one trial, assuming equal skill. Considering that any one of the three bowls is as Ukely as the 2 . 1 others to be first, the chance of £ is k and that of -4 is ^ • But by the incorrect solution Montmort arrives at a different result. For suppose A to have delivered his bowl. Then B has the chance 5 with his first bowl of beating A ; and the chance ^ x ^ of failing with his first bowl and being successful with his second. Thus £'s g chance appears to be j . Montmort considers the eiTor of this so- lution to lie in the assumption that when B has failed to beat A with his first bowl it is still an even chance that he wiU beat A with his second bowl : for the fact that B failed with his first bowl suggests that A 's bowl has a position better than the average, so that B\ chance of success with his second bowl becomes less than an even chance. 178. Montmort then takes four problems in succession of trifling importance. The first relates to a lottery which was started in Paris in 1710, in which the projector had offered to the public terms which were very disadvantageous to himself The second is an easy exercise in combinations. The third relates to a game called Le Jeu des Ouhlieux. The fourth is an extension of Huygens's eleventh problem, and is also given in the Ars Conjec- tandi, page 34 These four problems are new in the second edition. MONTMOBT. 101 179. Montmort now passes to a problem of a more important character- which occupies his pages 268 — 277, and which is also new in the second edition ; it relates to the Duration of Play ; see Art. 107. Suppose A. to have m counters and B to have n counters ; let their chances of winning a single game be as a to J ; the loser in each game is to give a counter to his adversary : required the chance that A will have won all B's counters on or before the a;*'' game. This is the most difficult problem which had as yet been solved in the subject. Montmort's formula is given on his pages 268, 269. 180. The history of this problem up to the current date will be found by comparing the following pages of Montmort's book, 275, 309, 315, 324, 344, 368, 375, 380. It appears that Montmort worked at the problem and also asked Nicolas Bernoulli to try it. Nicolas Bernoulli sent a solution to Montmort, which Montmort said he admired but could not understand, and he thought his own method of investi- gation and that of Nicolas Bernoulli must be very different : but after explanations received from Nicolas Bernoulli, Montmort came to the conclusion that the methods were the same. Before however the publication of Montmort's second edition, De Moivre had solved the problem in a different manner in the Be Mensura Sortis. 181. The general problem of the Duration of Play was studied by De Moivre with great acuteness and success ; indeed his inves- tigation forms one of his chief contributions to the subject. He refers in the following words to Nicolas Bernoulli and Montmort : Monsieur de Monmort, in the Second Edition of his Book of Chance.s, having given a very handsom Solution of the Problem relating to the duration of Play, (which Solution is coincident with that of Monsieur Nicolas Bernoidly, to be seen in that Book) and the demonstration of it being very naturally deduced from our first Solution of the foregoing Problem, I thought the Reader would be well pleased to see it trans- ferred to this place. Doctrine of Chances; first edition, page 122. 1 02 MONTMOET. ...the Solution of Mr Nicolas Bernoulli being very much crouded ■with Symbols, and the verbal Explication of them too scanty, I own I did not understand it thoroughly, which obliged me to consider Mr. de Monmorfs Solution with very great attention : I found indeed that he was very plain, but to my great surprizs I found him very erroneous; still in my Doctrine of Chances I printed that Solution, but rectified and ascribed it to Mr. de Monmort, without the least intimation of any alterations made by me ; but as I had no thanks for so doing, I resume my right, and now print it as my own Doctrine of Chances; second edition page 181, third edition, page 211. The language of De Moivre in his second and third editions would seem to imply that the solutions of Nicolas Bernoulli and Montmort are different ; but they are really coincident, as De Moivre had himself stated in his first edition. The statement that Montmort's solution is very erroneous, is unjustly severe ; Mont- mort has given his formula without proper precaution, but his example which immediately follows shews that he was right him- self and would serve to guide his readers. The second edition of the Doctrine of Chances appeared nearly twenty years after the death of Montmort ; and the change in De Moivre's language respecting him seems therefore especially ungenerous. 182. We shall not here give Montmort's general solution of the Problem of the Duration of Play; we shall have a better opportunity of noticing it in connexion with De Moivre's investiga- tions. We will make three remarks which may be of service to any student who examines Montmort's own work. Montmort's general statement on his pages 268, 269, might easily mislead ; the example at the end of page 269 is a safer guide. If the statement were literally followed, the second line in the example would consist of as many terms as the first line, the fourth of as many terms as the third, and the sixth of &s many terms as the fifth; but this would be wrong, shewing that the general statement is not literally accurate. Montmort's explanation at the end of his page 270, and the be- ginning of his page 271, is not satisfactory. It is not true as he intimates, that the four letters a and the eleven letters b must be MONTMOKT. 103 SO arranged that only a single h is to come among the four letters a : we might have such an arrangement as aaahbbbbbbhhbba. We shall return to this point in our account of De Moivre's in- vestigations. On his page 272 Montmort gives a rule deduced from his formula ; he ought to state that the rule assumes that the players are of equal skill : his rule also assumes that p — m is an even number. 183. On his pages 275, 276 Montmort gives without demon- stration results for two special cases. (1) Suppose that there are two players of equal skill, and that each starts with two counters ; then 1 — 35 is the chance that the match wiU be ended in 2a! games at most. The result may be de- duced from Montmort's general expression. A property of the Binomial CoeflScients is involved which we may briefly indicate. Let Mj, Mj, Mj, ... denote the successive tei-ms in the expansion of (1 + l)'"". Let B denote the sum of the following series u^ + 2m^_j+ m^2+ + M^_4+ 2m^5+ %.s+ + M^a+ ... ThenshaU5'=2='^'-2^-\ For let V, denote the r*"" term in the expansion of (1 + 1)'^"^ and w^ the r*" term in the expansion of (1 + 1)'*"^ Then Mr = v,. + %_i, Employ the former transformation in the odd terms of our pro- posed series, and the latter in the even terms; thus we find that the proposed series becomes «'« + V:^l + V«-2 + «:^3 + ««-4 + • • • + 2{w^i+ 2w^, + w^_3+ + w^3+ •••}. The first of these two series is equal to ^ (1 + 1)'^'' \ and the second is a series of the same kind as that which we wish to sum with X changed into x-\. Thus we can finish the demonstration by indiiction ; for obviously 2 (2^-i _ 2'"^') + 2'*"^ = 2 '''"' — 2*"'. lOi MONTMOHT. (2) Next suppose that each player starts with three counters ; then 1 - - is the chance that the match will be ended in 2a; + 1 games at most. This result had in fact been given by Montmort in his first edition, page 184. It may be deduced from Montmort's general expression, and involves a property of the Binomial Coeffi- cients which we will briefly indicate. Let Mj, Mj, Mj, ... denote the successive terms in the expansion of (1 + l)'*'^'. Let S denote the sum of the following series M,+ 2m^,-1- 2m^_,-1- M^3 + + + M^ + 2m^_,+2%_,+m^,+ 0-1-0+... Thenshallfif=2^-3". If w, denote the r"" term in the expansion of (1 + 1)**"' we can shew that m^ + 2m^.i+2m^+m^, + 3 (w^i + 2w^ + 2m^3 + w^. By performing a similar transformation on every successive four significant terms of the original series we transform it into ^ (1 + 1)'"""* + 32, where 2 is a series like S with x changed into x-1. Thus 8 = 2''^ + St. Hence by induction we find that 8=2^ — 3". 184. Suppose the players of equal skill, and that each starts with the same odd number of counters, say m ; let f= — ~- . Then Montmort says, on his page 276, that we may wager with advantage that the match will be concluded in 3f — 3f+ 1 trials. Montmort does not shew how he arrived at this approximation. 3 1 The expression may be put in the form t ''*''+ r • ^^ Moivre 4 4 spoke favourably of this approximation on page 148 of his first edi- tion; he says, "Now Mr de Montmort having with great Sagacity discovered that Analogy, in the case of an equal and Odd number of Stakes, on supposition of an equality of Skill between the MONTMOET. 105 Gamesters . . ." In his second and third editions De Moivre with- drew this commendation, and says respecting the rule " Which tho' near the Truth in small numbers, yet is very defective in large ones, for it may be proved that the number of Games found by his Expression, far from being above what is requisite is really below it." Doctrine of Chances, third edition, page 218. De Moivre takes for an example m = 45 ; and calculates by his own mode of approximation that about 1531 games are requisite in order that it may be an even chance that the match will be concluded ; Montmort's rule would assign 1519 games. We should differ here with De Moivre, and consider that the results are rather remarkable for their near agreement than for their dis- crepancy. The problem of the Duration of Play is fully discussed by Laplace, TMorie...des Prob. pages 225 — 238. 185. Montmort gives some numerical results for a simple problem on his page 277. Suppose in the problem of Art. 107 that the two players are of equal skill, each having originally n counters. Proceeding as in that Article, we have Hence we find u^= Ox+ 0^, where C and C, are arbitrary con- stants. To determine them we have hence finally, u^ = a-- Montmort's example is for w = 6 ; he gave it in his first edition, page 178. He did not however appear to have observed the gene- ral law, at which John Bernoulli expressed his surprise ; see Mont- mort's page 295. 186. Montmort now proposes on pages 278 — 282 four pro- blems for solution ; they were originally given at the end of the first edition. The first problem is svr le Jeu du Treize. It is not obvious why this problem is repeated, for Montmort stated the results on his pages 130 — 143, and demonstrations by Nicolas Bernoulli are given on pages 301, 302. IOC MONTMOET. The second problem is sur le Jew appelU le Her; a discussion respecting this problem runs through the correspondence between Montmort and Nicolas Bernoulli. See Montmort's pages 321, 334, 338, 348, 361, 376, 400, 402, 403, 409, 413. We wiU return to this problem in Art. 187. The third problem is sur le Jeu de la Ferme ; it is not referred to again in the book. The fourth Problem is sur le Jeu des Tas. We will return to this problem in Art. 191. Montmort's language in his Avertissement, page XXV, leads to the expectation that solutions of all the four problems will be found in the book, whereas only the first is solved, and indeed Montmort himself seems not to have solved the others ; see his page 321. 187. It may be advisable to give some account of the discus- sion respecting the game called Her. The game is described by Montmort as played by several persons ; but the discussion was confined to the case of two players, and we wUl adopt this limitation. Peter holds a common pack of cards ; he gives a card at random to Paul and takes one himself; the main object is for each to obtain a higher card than his adversary. The order of value is ace, two, three, ... ten, Knave, Queen, King. Now if Paul is not content with his card he may compel Peter to change with him ; but if Peter has a King he is allowed to retain it. If Peter is not content with the card which he at first obtained, or which he has been compelled to receive from Paul, he is allowed to change it for another taken out of the pack at random ; but if the card he then draws is a King he is not allowed to have it, but must retain the card with which he was dissatisfied. If Paul and Peter finally have cards of the same value Paul is considered to lose. 188. The problem involved amounts to a determination of the relative chances of Peter and Paul; and this depends on their using or declining their rights of changing their cards. Montmort communicated the problem to two of his friends, namely Walde- grave, of whom we hear again, and a person who is called some- MONTMORT. 107 times M. I'Abb^ de Monsoury and sometimes M. I'Abb^ d'Orbais. These two persons differed with Nicolas Bernoulli respecting a point in the problem ; Nicolas Bernoulli asserted that in a certain contingency of the game each player ought to take a certain course out of two which were open to him ; the other two persons con- tended that it was not certain that one of the courses ought to be preferred to the other. Montmort himself scarcely interfered until the end of the cor- respondence, when he intimated that his opinion was contrary to that of Nicolas Bernoulli ; it would seem that the latter intended to produce a fuller explanation of his views, but the correspondence closes without it. 189. "We wiU give some details in order to shew the nature of the dispute. It will naturally occur to the reader that one general principle must hold, namely, that if a player has obtained a high card it will be prudent for him to rest content with it and not to run the risk involved in changing that card for another. For example, it appears to be tacitly allowed by the disputants that if Paul has obtained an eight, or a higher card, he will remain content with it, and not compel Peter to change with him ; and, on the other hand, if Paul has obtained a six, or a lower card, he will compel Peter to change. The dispute turns on what Paul should do if he has obtained a seven. The numerical data for discussing this case will be found on Montmort's page 339 ; we wiU reproduce them with some explanation of the process by which they are obtained. I. Paul has a seven; required his chance if he compels Peter to change. Supposing Paul to change, Peter will know what Paul has and will know that he himself now has a seven ; so he remains content if Paul has a seven, or a lower card, and takes another card if Paul has an eight or a higher card. Thus Paul's chance arises from the hypotheses that Peter originally had Queen, Knave, ten, nine, or eight. Take one of these cases, for example, that of the ten. The 4 chance that Peter had a ten is rr ; then Paul takes it, and Peter 61 108 MONTMOET. gets the seven. There are 50 cards left and Peter takes one of these instead of his seven ; 39 cards out of the 50 are favour- able to Paul, namely 3 sevens, 4 Kings, 4 nines, 4 eights, 4 sixes, ... 4 aces. Proceeding in this way we find for Paul's chance 4 47 + 43 + 39+35 + 31 ^, ^ . 780 , that IS 51 ■ 50 ' 51 . 50 ■ In this case Paul's chance can be estimated without speculating upon the conduct of Peter, because there can be no doubt as to what that conduct wiU be. II. Paul has a seven; required his chance if he retains the seven. The chance in this case depends upon the conduct of Peter. Now it appears to be tacitly allowed by the disputants that if Peter has a nine or a higher card he will retain it, and if he has a seven or a lower card he will take another instead. The dispute turns on what he will do if he has an eight. (1) Suppose that Peter's rule is to retain an eight. Paul's chance arises from the hypotheses that Peter has a seven, six, five, four, three, two, or ace, for which he proceeds to take another card. We shall find now, by the same method as before, that Paul's chance is 3 24 . 4 27 , 4 27 , 4 27 , 4 27 , 4 27 . 4 27 ["50 720 51 ■ 50 ■•■ 51 ■ 50 "^ 51 ■ 50 "^ 51 ■ 50 "•■ 51 ■ 50 ■*" 51 ■ 50 "^ 51 ■ 50' that is 51.50" (2) Suppose that Peter's rule is to change an eight. 4 24 We have then to add kT ■ -iTr to the preceding result ; and thus we obtain for Paul's chance 51 50 816 51.50" 780 Thus we find that in Case I. Paul's chance is — — — : , and that 51 . 50 in Case II. it is either ■== — 37: or ^^ — =^ . If it be an even chance 51. oU 51 . 50 MONTMORT. 109 which rule Peter adopts we should take „ f^^; — =7^+ ~~, — =77) , that is, -Fi — ?7^ as Paul's chance in Case II. Thus in Case II. Paul's 51 . oU chance is less than in Case I. ; and therefore he should adopt the rule of changing when he has a seven. This is one of the argu- ments on which Nicolas Bernoulli relies. On the other hand his opponents, in effect, deny the correctness of estimating , it as an even chance that Peter will adopt either of the two rules which have been stated. We have now to estimate the following chance. Peter has an eight and Paul has not compelled him to change ; what is Peter's chance 1 Peter must argue thus : I. Suppose Paul's rule is to change a seven; then he now has an eight or a higher card. That is, he must have one out of a certain 23 cards. (1) If I retain my eight my chance of beating him arises only from the hypothesis that his card is one of the 3 eights; that is, my chance is ^ . (2) If I change my eight my chance arises from the five hypo- theses that Paul has Queen, Knave, ten, nine, or eight ; so that my chance is 4 3.47 4 11 4 15 3 22 that is 23 • 50 "*" 23 " 50 ■*■ 23 ■ 50 "*" 23 ■ 50 ■•" 23 ' 50 ' 210 23 . 50 ■ II. Suppose Paul's rule is to retain a seven. Then, as before, 7 (1) If I retain my eight my chance is -^ . (2) If I change my eight my chance is 27 ■ 50 "^ 27 ■ 50 "^ 27 ■ 50 "*" 27 ■ 50 "^ 27 ■ 50 ''' 27 ■ 50 ' ^1..^ • 314 *^^^^ 27750- 110 MONTMOET. 190. These numerical results were accepted by the disputants. We may sum them up thus. The question is whether Paul should retain a certain card, and whether Peter should retain a certain card. If Paul knows his adversary's rule, he should adopt the con- trary, namely retaining when his adversaiy changes, and changing when his adversary retains. If Peter knows his adversary's rule he should adopt the same, namely, retaining when his adversary re- tains and changing when his adversary changes. Now Nicolas Bernoulli asserted that Paul should change, and therefore of course that Peter should. The objection to this is briefly put thus by Montmort, page 405, En un mot, Monsieur, si je sgai que voiis ^tes le conseil de Pierre, il est Evident que je dois moi Paul me tenir au sept ; et de m^me si je suis Pierre, et qui je s^ache que vous fetes le conseil de Paul, je dois changer au huit, auquel cas vous aurls donn6 un mauvais con- seil k Paul. The reader will be reminded of the old puzzle respecting the veracity of the Cretans, since Epimenides the Cretan said they were liars. The opponents of Nicolas Bernoulli at first contended that it was indifferent for Paul to retain a seven or to change it, and also for Peter to retain an eight or to change it ; and in this Montmort considered they were wrong. But in conversation they explained themselves to assert that no absolute rule could be laid down for the players, and in this Montmort considered that they were right ; see his page 403. The problem is considered by Trembley in the Memoires de T Acad.... Berlin, for 1802. 191. The fourth problem which Montmort proposed for solu* tion is sur le Jeu des Tas. The game is thus described, page 281, Pour comprendre de quoi il s'agit, il faut SQavoir qu'aprls les reprises d'hombre un des Joueurs s'amuse souvent S, partager le jeu en dix tas composes chacun de quatre cartes couvertes, et qu'ensuite retournant la premiere de chaque tas, il 6te et met k part deux S. deux toutes celles qui se trouvent semblables, par exemple, deux Rois, deux valets, deux six, &c. alors il retoume les cartes qui suivent immSdiatement ceUes qui viennent de lui donner des doublets, et U continue d'6ter et de mettre k part celles qui viennent par doublet jusqu'S, ce quU en soit MONTMORT. Ill venu si la derniere de chaque tas, apr§s les avoir enlev6 toutes deux k deux, auquel cas seulement il a gagn6. The game is not entirely a game of pure chance, because the player may often have a choice of various methods of pairing and removing cards. In the description of the game forty cards are supposed to be used, but Montmort proposes the problem for solu- tion generally without limiting the cards to forty. He requires the chance the player has of winning and also the most ad- vantageous method of proceeding. He says the game was rarely played for money, but intimates that it was in use among ladies. 192. On his page 321 Montmort gives, without demonstration, the result in a particular case of this problem, namely when the cards consist of w pairs, the two cards in each pair being numbered alike ; the cards are supposed placed at random in n lots, each of two cards. He says that the chance the player has of winning is ^2 1 , , , , ■^ . On page 334 Nicolas Bernoulli says that this formula is correct, but he wishes to know how it was found, because he him- self can only find it by induction, by putting for n in succession 2, 3, 4,5, ...We may suppose this means that Nicolas Bernoulli veri- fied by trial that the formula was correct in certain cases, but could not give a general demonstration. Montmort seems to have overlooked Nicolas Bernoulli's inquiry, for the problem is never mentioned again in the course of the correspondence. As the result is remarkable for its simplicity, and as Nicolas Bernoulli found the problem difiScult, it may be interesting to give a solution. It will be observed that in this case the game is one of pure chance, as the player never has any choice of courses open to him. 193. The solution of the problem depends on our observing the state of the cards at the epoch at which the player loses, that is at the epoch at which he can make no more pairs among the cards exposed to view; the player may be thus arrested at the very beginning of the game, or after he has already taken some steps : at this epoch the player is left with some nwmber of lots, which are all unbroken, and the cards exposed to view present no pairs. This will be obvious on reflection. 112 MONTMOET. "We must now determine (1) the whole number of possible cases, and (2) the whole number of cases in which the player is arrested at the very beginning. (1) "We may suppose that 2w cards are to be put in 2a places, and thus [2w wiU be the whole number of possible cases. , (2) Here we may find the number of cases by supposing that the n upper places are first filled and then the n lower places. We may put in the first place any card out of the 2n, then in the second place any card of the 2n—2 which remain by rejectiug the companion card to that we put in the first place, then in the third place any card of the 2« — 4 which remain by rejecting the two companion cards, and so on. Thus the n upper places can be filled in 2''[» ways. Then the n lower places can be filled in [n ways. Hence we get 2* [w [w cases in which the player is arrested at the very beginning. We may divide each of these expressions by [w if we please to disregard the different order in which the n lots may be sup- l2w posed to be arranged. Thus the results become =j= and 2" \n respectively, we shall use these forms. Let M„ denote the whole number of unfavourable cases, and let fr denote the whole number of favourable cases when the cards consist of r pairs. Then ««=2"Lz?+^T-r^/h-'-2»-', the summation extending from r = 2 to r = n-l, both inclusive. For, as we have stated, the player loses by being left with some number of lots, all unbroken, in which the exposed cards contain no pairs. Suppose he is left with n — r lots, so that he has got rid of r lots of the original n lots. The factor — 1= edves the num- [»• !■" — " ° ber of ways in which r pairs can be selected from n pairs ; the factor fr gives the number of ways in which these pairs can be so arranged as to enable the player to get rid of them ; the factor \n — r 2""^ gives the number of ways in which the remaining n — r pairs can be distributed into n — r lots without a single pair occur- ring among the exposed cards. MONTMOET. Il3 It is to be observed that the case in which r = l does not occur, from the nature of the game ; for the player, if not arrested at the very beginning, will certainly be able to remove imjo pairs. We may however if we please consider the liummation to extend from r = l to r = m-l, since/ = when 9- = l. We have then The summation for m„_, extends to one term less; thus we shall find that But ''->+/-.=^; 2n\2n-2 therefore m, = — ' , • . 2\2n-2 , , . [^ n-1 Hence /„ = 1= - «, = -^^^^ ; and /. ^ t^ = __ . This is Montmort's result. 194. We now arrive at what Montmort calls the fifth part of his work, which occupies pages 283 — 414 It consists of the correspondence between Montmort and Nicolas Bernoulli, together with one letter from John Bernoulli to Montmort and a reply from Montmort. The whole of this part is new in the second edition. John Bernoulli, the friend of Leibnitz and the master of Euler, was the third brother in the family of brothers of whom James Bernoulli was the eldest. John was bom in 1667, and died in 1748. The second brother of the family was named Nicolas ; his son of the same name, the friend and correspondent of Montmort, was bom in 1687, and died in 1759. 195. Some of the letters relate to Montmort's first edition, and it is necessary to have access to this edition to study the letters with advantage; because although Montmort gives re- ferences to the qorresponding passages in, the second edition, yet 8 114- MONTMOET. as these passages have been modified or con-ected in accordance •with the criticisms contained in the letters, it is not always ob- vious what the original reading was. 196. The first letter is from John Bernoulli; it occupies pages 283 — 298; the letter is also reprinted in the collected edition of John Bernoulh's works, in four volumes, Lausanne and Geneva, 1742 ; see Vol. i. page 453. John Bernoulli gives a series of remarks on Montmort's first edition, correcting some errors and suggesting some improvements. He shews that Montmort did not present his discussion relating to Pharaon in the simplest form ; Montmort however did not modify this part of his work. John Bernoulli gave a general formula for the advantage of the Banker,, and this Montmort did adopt, as we have seen in Ai-t. 155. 197. John Bernoulli points out a curious mistake made by Montmort twice in his first edition ; see his pages 288, 296. Montmort had considered it practically impossible to find the numerical value of a certain number of terms of a geometrical progression ; it would seem that he had forgotten or never known the common Algebraical formula which gives the sum. The passages cited by John Bernoulli are from pages 35 and 181 of the first edition ; but in the only copy which I have seen of the first edition the text does not correspond with John Bernoulli's quotations : it appears however that in each place the original page has been cancelled and replaced by another in order to correct the mistake. After noticing the mistake, John Bernoulli proceeds thus in his letter : ...mais pour le reste, vous faites bien d'employer les logarithmes, je m'en suis servi utilement dans une pareille occasion il y a bien douze ans, oil il s'agissoit de .determiner combien il restoit de vin et d'eau mM ensemble dans im.stonneau, lequel gtant au commencement tout plein de vin, on en,tirer6it tous lis jours pendant une annge une certaine mesure, efi'-le remplissant incontinent apr§s cbaque ex- traction avec de I'eau pure. Yous trouvergs la solution de cette ques- tion qui est assSs curieuse dans ma dissertation De NutrUione, que Mr Varignon vous pourra communiquer. Je fis, cette question pour faire MONTMORT. 115 comprendre comment on pent determiner la quantity de vieille mar tiere qni reste dans nos corps mM6e aveo de la nouveUe qui nous vient tous les jours par la nourriturer, pour r^parer la perte que nos corps font insensiblement par la transpiration continuelle. The dissertation Be Nutritione will be found in the collected edition of John Bernoulli's works ; see Vol. i. page 275, 198. John Bernoulli passes on to a remark on Montmort's discussion of the game of Treize. The remark enunciates the following theorem. Let ^(„) = l-_+--_+...+L_^, and let t («)= , -n ''"■ , -D '"^^ 1 "i + -t 2 — + -tg -r+ • ■ • to L — \ terms > , So i s=- ■\x where Let P.= |p + A,-l M = l+- + -^ + S S ■ + ■ m ,1 cv n^ d^u Now u = -(?)■ : , m i-fi' MONTMOET. 119 '1.2 s^ (1-/*)""' X(X-l)(\-2) |X-3 l(l-l)(J-2),J:^ 1.2.3 s'^ (1-/*)''"'' =ll|{x.,[i,.,«i[=i),«fcM-),.,.]}, where the series between square brackets is to extend to X + 1 terms. We may observe that by the nature of the problem we have a + i + c + ...=0, and also s + i/ + x+ ... = 0. The problem simplifies very much if we may regard I as infinite or very great. For then let s denote the advantage of ^ ; if ^ ob- tains the next deal we may consider that his advantage is still s ; if A loses the next deal his advantage is the same as that of B originally. Thus mz + ny s Multiply by s and transpose ; therefore Similarly we have y = x-\-h^, x==u + cq, Hence we shall obtain , = ii^a{p-l) + bip-2) + c{p-3) + ...}, where p denotes the number of players ; and the values of y, x, , .. may be obtained by symmetrical changes in the letters. We may also express the result thus, s = -2|a+25 + 3c+...|. 120 MONTMORT, 203. The next letter is from Montmort to John Bernoulli ; it occupies pages 303 — 307. Montmort makes brief observations on the points to which John BemouUli had drawn his attention ; he suggests a problem on the Duration of Play for the consideration of Nicolas Bernoulli. 204. The next letter is from Nicolas Bernoulli to Montmort ; it occupies pages 308 — 314. Nicolas Bernoulli first speaks of the game of Treize, and gives a general formula for it ; but by accident he gave the formula in- correctly, and afterwards corrected it when Montmort drew his attention to it ; see Montmort's pages 315, 323. We win here investigate the formula after the manner given by Nicolas Bernoulli for the simple case already considered in Art. 161. Suppose there are n cards divided into p sets. Denote the cards of a set by a,b,c,,,. in order. The whole number of cases is \n. The number of ways in which a can stand first is p \n—l . The number of ways in which b can stand second without a standing first is p | w— 1 —jp' \n — 2 . The number of ways in which c can stand third without a standing first or h second is p | w — 1 — 2p^ |ra — 2 +p' | w — 3 . And so on. Hence the chance of winning by the first card is - ; the chance of winning by the second card is ^ t ^ ... ; the chance of win- ° •' n n{n—l) p 2»' »' nine bv the third card is — . ^ ... H — -. ^vt ?rr ; and so on. ° *' n w (n — 1) m (« — 1) (« — 2) Hence the chance of winning by one or other of the first m cards is mp m{m — V) p^ m{m — l){m— 2) p^ "n O n{n-iy 1.2.3 m(«- 1) (w-2) ""' And the entire chance of winning is found by putting m = - , so that it is P 1 n—p .2. MONTMORT, -p){n-2p) 3(«-l)(re-2) {n-p){n- -2p)(. >-3p) 1 1 .2(n- l)'l 1.2.3.4(m- l){n- ■2){n- •3) 121 + . 205. Nicolas Bernoulli then passes on to another game in which he objects to Montmort's conclusion. Montmort had found a certain advantage for the first player, on the assumption that the game was to conclude at a certain stage ; Nicolas Bernoulli thought that at this stage the game ought not to terminate, but that the players should change their positions. He says that the advantage for the first player should be only half what Montmort stated. The point is of little interest, as it does not belong to the theory of chances but to the conventions of the players ; Montmort, however, did not admit the justice of the remarks of Nicolas Bernoulli ; see Montmort's pages 309, 317, 327. 206. Nicolas Bernoulli then considers the problem on the Duration of Play which had been suggested for him by Mont- mort. Nicolas Bernoulli here gives the formulae to which we have already alluded in Art. 180 ; but the meaning of the formulae was very obscure, as Montmort stated in his reply. Nicolas Bernoulli gives the result which expresses the chances of each player when the number of games is unlimited ; he says this may be deduced from the general formulae, and that he had also obtained it pre- viously by another method. See Art. 107. 207. Nicolas Bernoulli then makes some remarks on the summation of series. He exemplifies the method which is now common in elementary works on Algebra. Suppose we require the sum of the squares of the first n triangular numbers, that is, the {r (r + D) " sum of M-terms of the series of which the r"" term is -l-Aj — a-^\ . Assume that the sum is equal to an^ + bn* + cd + dr^ + en +/; and then determine a, h, c, d, e, f by changing n into « + 1 in the assumed identity, subtracting, and equating coefficients. This method is ascribed by Nicolas Bernoulli to bis uncle John. 122 MONTMORT. Nicolas Bernoulli also indicates another method ; he resolves i 172-^1 ^^*° ^ r(r+l) (r+2) (r + 3) ^ r (r + 1) (r+ 2) ^ r (r + 1) ^ 1.2.3.4 1.2.3 "^ 1.2 ' and thus finds that the required sum is «(«+!) (w+2)(w+3)(w + 4) _ n(n + l) (w + 2) (w + 3) ^ 1.2.3.4.5 1.2.3.4 w (w + 1) (w + 2) "*■ 1.2.3 208. It seems probable that a letter from Montmort to Nicolas Bernoulli, which has not been preserved, preceded this letter from Nicolas Bernoiilli. For Nicolas Bernoulli refers to the problem about a lottery, as if Montmort had drawn his attention to it ; see Art. 180 : and he intimates that Montmort had offered to undertake the printing of James Bernoulli's unpublished Ars Conjectandi. Neither of these points had been mentioned in Montmort's preceding letters as we have them in the book. 209. The next letter is from Montmort to Nicolas Bernoulli ; it occupies pages 315 — 323. The most interesting matter in this letter is the introduction for the first time of a problem which has since been much discussed. The problem was proposed to Mont- mort, and also solved, by an English gentleman named Waldegrave ; see Montmort's pages 318 and 328. In the problem as originally proposed only three players are considered, but we will enunciate it more generally. Suppose there are n + 1 players ; two of them play a game ; the loser deposits a shilling, and the winner then plays with the third player; the loser deposits a shilling, and the winner then plays with the fourth player ; and so on. The player who lost the first game does not enter again until after the (n + 1)"' player has had his turn. The process continues until one player has beaten in continued succession all the other players, and then he receives all the money which has been deposited. It is required to determine the expectation of each of the players, and also the chance that the money will be won when, or before, a certain number of games has been played. The game is sup- MONTMOET. 123 posed a game of pure chance, or whicli is the same thing, the players are all supposed of equal skill. Montmort himself in the case of three players states all the required results, but does not give demonstrations. In the case of four players he states the numerical probability that the money will be won in any assigned number of games between 3 and 13 inclusive, but he says that the law of the numbers which he assigns is not easy to perceive. He attempted to proceed further with the problem, and to determine the advantage of each player when there are four players, and also to determine the pro- bability of the money being won in an assigned number of games when there are five or six players. He says however, page 320, mais cela m'a paru trop difficile, ou pl6t6t j'ai manqu^ de courage, car je serois stv d'en venir a bout. 210. There are references to this problem several times in the correspondence of Montmort and Nicolas Bernoulli; see Mont- mort's pages 328, 345, 350, 366, 376, 380, 400. Nicolas Bernoulli succeeded in solving the problem generally for any number of players ; his solution is given in Montmort's pages 381 — 387, and is perhaps the most striking investigation in the work. The following remarks may be of service to a student of this solution. (1) On page 386 Nicolas Bernoulli ought to have stated how many terms should be taken of the two series which he gives, namely, a number expressed by the greatest integer contained ft JL. in — J[ in +- . On page 330 where he does advert to this point he puts by mistake — instead of — . (2) The expressions given for a, b, c, ... on page 386 are 2 correct, except that given for a ; the value of a is ^ , and not n5 , as the language of Nicolas Bernoulli seems to imply. (3) The chief results obtained by Nicolas Bernoulli are stated at the top of page 329 ; these results agree with those afterwards given by Laplace. 124 MONTMORT. 211. Although the earliest notice of the problem occurs in the letter of Montmort's which we are now examining, yet the earliest publication of it is due to De Moivre ; it is Problem xv. of the Be Mensura Sortis. We shall however speak of it as Waldegrave's Problem, from the person whose name we have found first associated with it. The problem is discussed by Laplace', ThSorie . . . des Prob. page 238, and we shall therefore have to recur to it. 212. Montmort refers on page 520 to a book entitled TraitS du Jeu, which he says he had lately received from Paris. He says it is wn Livre de morale. He praises the author, but considers him to be wrong sometimes in his calculation of chances, and gives an example. Nicolas Bernoulli in reply says that the author of the book is Mr Barbeyrac. Nicolas Bernoulli agrees with Montmort in his general opinion respecting the book, but in the example in question he thinks Barbeyrac right and Mont- mort wrong. The difference in result arises from a difference in the way of understanding the rules of the game. Montmort briefly replied ; see pages 332, 346. Montmort complains of a dearth of mathematical memoirs ; he says, page 322, Je suis €tonn^ de voir les Joumeaux de Leipsio si d€gamis de morceaux de Mathematiques : Us doivent en partie leur reputation aux excellens Memoires que Meseieurs vos Oncles y envoyoient souvent : lea Geometres n'y trouvent plus depuis cinq ou six ans les m^es richesses qu' autrefois, faites-en des reproches ^ M. votre Oncle, et permettfe-inoi de vous en faire aussi, Zucecet lux vestra coram homiwSms. 213. The next letter is from Nicolas Bernoulli to Montmort ; it occupies pages 323—337. It chiefly relates to matters which we have already sufficiently noticed, namely, the games of Treize, Her, and Tas, and Waldegrave's Problem. Nicolas Bernoulli ad- verts to the letter by his uncle James on the game of Tennis, which was afterwards published at the end of the Ars Conjectandi, and he proposes for solution four of the problems which are con- sidered in the letter in order to see if Montmort's results will agree with those of James Bernoulli. ;ffiONTMOET. 125 Nicolas Bernoulli gives at the end of his letter an exaniiple of summation of series. He proposes to sum p terms of the series 1, 3, 6, 10, 15, 21, ... He considers the series 1 + 3a! + 6a!' + 10a;' + loa!* + 21a;'' + ... which he decomposes into a set of series, thus : 1 + 2a! + 33;" + 4.a!' + 5a;' + . . . + a! + 2a;' + 3a;' + 4a;*+... + ar' + 2a;' + 3a;*+... + ar' + 2a!'+... + 0!*+... + ... The series in each horizontal row is easily summed to "p terms ; the expression obtained takes the form - when a! = 1, and Nicolas Bernoulli evaluates the indeterminate form, as he says, ...en me servant de la regie de mon Oncle, que feu Monsieur le Marquis de I'Hfipital a insert dans son Analyse des infiniment petits, ... The investigation is very inaccurately printed. 214!. The next letter is from Montmort to Nicolas Bernoulli ; it occupies pages 337 — 347. Besides remarks on the game of Her and on Waldegrave's Problem, it contains some attempts at the problems which Nicolas Bernoulli had proposed out of his uncle's letter on the game of Tennis. But Montmort found the problems difficult to understand, and asked several questions as to their meaning. 215. Montmort gives on his page 342 the following equation, as the result of one of the problems, 4m» - %m^ + 14?w + 6 = 3""*', and he says that this is satisfied approximately by m = 5^ ; but there is some mistake, for the equation has no root between 5 and 6. The correct equation should apparently be 8m' - 12m'' + 16ot + 6 = 3'"", which has a root between o'l and 5'2. 126 MONTMORT. 216. One of the problems is the following. The skill of ^, that is his chance of success in a single trial, is p, the skill of H is g. A and B are to play for victory in two games out of three, each game being for two points. In the first game B is to have a point given to him, in the second the players are to be on an equaUty, and in the third also B is to have a point given to him. Eequired the skill of each player so that on the whole the chances may be equal, ^'s chance of success in the first game or in the third game is j^, and ^s chance is g^ + Iq-p. u4's chance of success in the second game is p' + Sp'g', and £'s chance is ^ + 2)fy. Hence ^'s chance of success in two games out of three is / (/ + 3/2) +/ {+60), C=54p-216, Hence the required sum of n terms is np (p -l)(p-2)- "-^^) (9/- 4>5p + 60) + -("-^)(;-^) (54, - 216) - "^^-^^^-^ -^^^ 162. This result is suflficiently near Montmort's to shew that he must have adopted nearly the same method; he has fallen into some mistake, for he gives a different expression for the terms inde- pendent of^. In the problem on chances to which this is subservient we should have to put for n the gi'eatest integer in -^ . o 9 130 MONTMOET. Montmort refers on his page 364 to a letter dated June 8* 1710, which does not appear to have been preserved. 223. The next letter is from Nicolas Bernoulli to Montmort ; it occupies pages 371 — 375. Nicolas Bernoulli demonstrates a property of De Beaune's curve ; he also gives a geometrical recti- fication of the logarithmic curve ; but his results are very in- correct. He then remarks on a subject which he says had been brought to his notice in HoUand, and on which a memoir had been inserted in the Philosophical Transactions. The subject is the argument for Divine Providence taken from the constant regu- larity observed in the births of both sexes. The memoir to which Bernoulli refers is by Dr John Arbuthnot ; it is in Vol. XXVII. of the Philosophical Transactions, and was published in 1710. Nicolas Bernoulli had discussed the subject in Holland with 'sGravesande. Nicolas Bernoulli says that he was obliged to refute the argu- ment. What he supposes to be a refutation amounts to this ; he examined the registers of births in London for the years from 1629 to 1710 inclusive; he found that on the average 18 males were born for 17 females. The greatest variations from this ratio were in 1661, when 4748 males and 4100 females were born, and in 1703, when 7765 males and 7683 females were born. He says then that we may bet 300 to 1 that out of 14,000 infants the ratio of the males to the females will fall within these limits ; we shall see in Art. 225 the method by which he obtained this result. 224. The next letter is also from Nicolas Bernoulli to Mont- mort ; it occupies pages 375 — 387. It contains some remarks on the game of Her, and some remarks in reply to those made by Montmort on De Moivre's memoir Be Mensura Sortis. The most important part of the letter is an elaborate discussion of Walde- grave's problem ; we have already said enough on this problem, and so need only add that Nicolas Bernoulli speaks of this discus- sion ag that which he preferred to every thing else which he had produced on the subject ; see page 381. The approbation which he thus bestows on his own work seems well deserved. 225. The next letter is also from Nicolas Bernoulli to Mont- mort ; it occupies pages 388 — 393. It is entirely occupied with MONTMORT. 131 the question of the ratio of male infants to female infants. We have already stated that Nicolas Bernoulli had refused to see any argument for Divine Providence in the fact of the nearly constant ratio. He assumes that the probability of the birth of a male is to the probability of the birth of a female as 18 to 17 ; he then shews that the chances are 43 to 1 that out of 14,000 infants the males wiU lie between 7037 and 7363. His investigation involves a general demonstration of the theorem of his uncle James called BernoulH's Theorem. The investigation requires the summation of terms of a binomial series ; this is effected approximately by a process which is commenced in these words : Or comme ces termes sont furieusement grands, il faut un artifice singulier pour trouver ce rapport : voici comment je m'y suis pris. The whole investigation bears some resemblance to that of James Bernoulli and may have been suggested by it, for Nicolas Bernoulli says at the end of it, Je me souviens que feu mon Oncle a d^montr^ une semblable chose dans son Traits De Arte Con- jectandi, qui s'imprime h present a BS,le, . . . 226. The next letter is from Montmort to Nicolas Bernoulli ; it occupies pages 395 — 400. Montmort records the death of the Duchesse d'AngoulSme, which caused him both grief and trouble ; he says he cannot discuss geometrical matters, but will confine himself to literary intelligence. He mentions a work entitled Primotion Physique, ou Action de Bieu sur les Creatures dimontree par raisonnement. The anonymous author pretended to follow the method of mathe- maticians, and on every page were to be found such great words as Definition, Axiom, Theorem, Demonstration, Corollary, &c. Montmort asks for the opinion of Nicolas Bernoulli and his uncle respecting the famous Commercium Epistolicum which he says M" de la Society Eoyale ont fait imprimer pour assurer S, M. Newton la gloire d'avoir invente le premier et seul les nou- velles methodes. Montmort speaks with approbation of a little treatise which had just appeared under the title of Mechanique du Feu. Montmort expresses his strong admiration of two investigations which he had received from Nicolas Bernoulli ; one of these was 9—2 132 MONTMOBT. the solution of Waldegrave's problem, and the other apparently the demonstration of James Bernoulli's theorem : see Arts. 224, 225. Montmort sajs, page 400, Tout cela Itoit en veriti bien rlifficile et d'un grand travail. Vous ^tes un terrible hommej je croyois que pour avoir pris les de- vants je ne serois pas si-tot ratrappi, mais je vois bien que je me suis trompS: je suis k present bieu derriere vousj et forc6 de mettre toute mon anjbition k vous suivre de lojn. 227. This letter from Montmort is interesting, as it records the perplexity in which the writer found himself between the claims of the rival systems of natural philosophy, the Cartesian and the Newtonian. He says, page 397, DIrange comme je le suis par I'autoritS de M. Newton, et d'un si grand nombre de s^avans Geometres Anglois, je serois presque tent6 de renoncer pour jamais 5, r6tude de la Physique, et de remettre k s§avoir tout cela dans le Ciel; mais non, I'autorite des plus grands esprits ne doit point nous faire de loi dans les choses oil la raisoa doit decider, 228. Montmort gives in this letter his views respecting a History of Mathematics ; he says, page 399, II seroit S, souhaiter que quelqu'un voulut prendre la peine de nous apprendre comment et en quel ordre les decouvertes en Mathe- matiquos se sont succedees les unes aux autres, et k qui nous en avons r obligation. On a fait FHistoire de la Peinture, de la Musique, de la Medecine, &c. TJne bonne Histoire des Mathematiques, et en par- ticulier de la Geometric, seroit un Ouvrage beaucoup plus curieux et p'.us utile : Quel plaisir n'auroit-on pas de voir la liaison, la connexion des methodes, Fenchainement des differentes theories, k commencer depuis les premiers temps jusqu'^au n6tre ou cette science se trouve port^e k un si £aut degr6 de perfection. II me semble qu'un tel Ouvrage bien fait pourroit Stre en quelque sorte regarde comme I'his- toire de I'esprit humaiu; puisque o'est dans cette science plus qu'en toute autre chose, que I'homme fait connoitre 1' excellence de ce don d'intelligence que Dieu lui a accord! pour I'llever au dessus de toutes les autres Creatures. MONTMORT. 133 Montmort himself had made some progress in the work which he here recommends ; see Art. 137. It seems however that hia manuscripts were destroyed or totally dispersed; see Montucla, Histoire des Mathematiques first edition, preface, page IX. 229. The next letter is from Nicolas Bernoulli to Montmort ; it occupies pages 401, 402. Nicolas Bernoulli announces that the Ars Conjectandi has just been published, and says, II n'y aura gueres rien de nouveau pour vous. He proposes five problems to Montmort in return for those which Montmort had proposed to him. He says that he had already proposed the first problem in his last letter ; but as the problem does not occur before in the correspondence, a letter must have been suppressed, or a portion of it omitted. The third problem is as follows. A and B play with a com- mon die, A deposits a crown, and B begins to play ; if B throws an even number he takes the crown, if he throws an odd number he deposits a crown. Then A throws, and takes a crown if he throws an even number, but does not deposit a crown if he throws an odd number. Then B throws again, and so on. Thus each takes a crown if he throws an even number, but B alone deposits a crown if he throws an odd number. The play is to continue as long as there is any sum deposited. Determine the advantage of A or B. The fourth problem is as follows. A promises to give to B a crown if B with a common die throws six at the first throw, two crowns if B throws six at the second throw, three crowns if B throws six at the third throw ; and so on. The fifth problem generalises the fourth, A promises to give B crowns in the progression 1, 2, 4, 8, 16, ... or 1, 3, 9, 27, ... or 1, 4, 9, 16, 25, ... or 1, 8, 27, 64, ... instead of in the progression 1, 2, 3, 4, 5, as in the fourth problem. 230. The next letter is the last; it is firom Montmort to Nicolas Bernoulli, and it occupies pages 403 — 412. It enters largely on the game of Her. With respect to the five problems proposed to him, Montmort says that he has not tried the first and second, that the fourth and fifth present no difficulty, but that the third is muph more difficult. He says that it took him 13i MONTMORT. a long time to convince himself that there would be neither advantage nor disadvantage for B, but that he had come to this conclusion, and so had Waldegrave, who had worked with him at the problem. It would seem however, that this result is obvious, for B has at every trial an equal chance of winning or losing ia crown. Montmort proposes on his page 408 a problem to Nicolas Bernoulli, but the game to which it relates is not described. 231. In the fourth problem given in Art. 229, the advantage of jB is expressed by the series 12 3 4 ■ ■ J, -, g + gi + gs + gj + ••• '^n infinitum. This series may be summed by the ordinary methods. We shall see that a problem of the same kind as the fourth and fifth of those communicated by Nicolas Bernoulli to Mont- mort, was afterwards discussed by Daniel Bernoulli and others, and that it has become famous under the title of the Petersburg Problem. 232. Montmort's work on the whole must be considered highly creditable to his acuteness, perseverance, and energy. The courage is to be commended which led him to labour in a field hitherto so little cultivated, and his example served to stimulate his more distinguished successor. De Moivre was certainly far superior in mathematical power to Montmort, and enjoyed the great advantage of a long life, extending to more than twice the duration of that of his predecessor ; on the other hand, the fortunate circumstances of Montmort's position gave him that abundant leisure, which De Moivre in exile and poverty must have found it impossible to secure. CHAPTER IX. DE MOrVRE. 233. Abraham De Moivre was bom at Vitri, in Champagne, in 1667. On account of the revocation of the edict of Nantes, in 1685, he took shelter in England, where he supported himself by giving iastruction in mathematics and answers to questions relating to chances and annuities. He died at London in 1754. John Bernoulli speaks thus of De Moivre in a letter to Leibnitz, dated 26 Apr. 1710 ; see page 847 of the volume cited in Art. 59 : ...Dominus Mojrvraeus, insignis certe Geometra, qui hand duHe adhuc haeret Londiiii, luctans, ut audio, cum fame et miseria, quas ut depellat, victum quotidianum ex informationibus adolescentum patera cogitur. O duram sortem hominis! et parum aptam ad excitanda ingania nobilia; quis non tandem succumberet sub tam iniquae fortunae vexationibus 1 vel quodnam inganium etiam fervidissimum non algeat tandem 1 Mirer certa Moyvraeum tantis angustiis pressum ea tamen adhuc praestare, quae praestat. De Moivre was elected a Fellow of the Royal Society in 1697 ; his portrait, strikingly conspicuous among those of the great chiefs of science, may be seen in the collection which adorns the walls of the apartment used for the meetings of the Society. It is recorded that Newton himself, in the later years of his life, used to reply to inquirers respecting mathematics in these words : " Go to Mr De Moivre, he knows these things better than I do." In the long list of men ennobled by genius, virtue, and mis- fortune, who have found an asylum in England, it would be 136 DE MOIVEE. difficult to name one who has conferred more honour on his adopted country than De Moivre. 234. Number 329 of the Philosophical Transactions consists entirely of a memoir entitled De Mensura Sortis, seu, de Prohahili- tate Eventuum in Ludis a Casu Fortuito Pendentihus. Autore Abr. De Moivre, K.S.S. The number is stated to be for the months of January, February, and March 1711 ; it occupies pages 213 — 26-1! of Vo- lume XXVII. of the Philosophical Transactions. The memoir was afterwards expanded by De Moivre into his work entitled The Doctrine of Chances : or, a Method of Galcwlating the Probabilities of Events in Play. The first edition of this work appeared in 1718 ; it is in quarto and contains xiv + 17o pages, besides the title-leaf and a dedication. The second edition appeared in 1738 ; it is in large quarto, and contains xiv + 258 pages, besides the title-leaf and a dedication and a page of corrections. The third edition appeared in 1756, after the author's death ; it is in large quarto, and contains xii + 348 pages, besides the title-leaf and a dedication. 235. I propose to give an account of the memoir De Mensura Sortis, and of the third edition of the Doctrine of Chances. In my account of the memoir I shall indicate the corresponding parts of the Doctrine of Changs ; and in my account of the Doctrine of Chances I shall give such remarks as may be suggested by compar- ing the third edition of the work with those which preceded it ; any reference to the Doctrine of Chances must be taken to apply to the third edition, unless the contrary is stated. 236. It may be observed that the memoir De Mensura Sortis is not reprinted in the abridgement of the Philosophical Transac- tions up to the year 1800, which was edited by Hutton, Shaw, and Pearson. The memoir is dedicated to Francis Eobartes, at whose recom- mendation it had been drawn up. The only works of any import- ance at this epoch, which had appeared on the subject, were the treatise by Huygens, and the first edition of Montmort's book. De Moivre refers to these in words which we have already quoted in Art. 142. DE MOIVRE. 137 De Moivre Bays that Problems 16, 17, 18 in his memoir were proposed to him by Robartes. In the Preface to the Doctrine of Chances, which is said to have been written in 1717, the origin of the memoir is explained in the following words : ' Tis now about Seven Years, since I gave a Specimen in the Philo- sopJiical Transactions, of what I now more largely treat of in this Book. The occasion of my then undertaking this Subject was chiefly owing to the Desire and Encouragement of the Honourable Francis Robartes Esq. (now Earl of Radnor); who, upon occasion of a French Tract, called B Analyse des Jeux de Hazard, which had lately been published, was pleased to propose to me some Problems of much greater diflRculty than any he had found in that Book ; which having solved to his Satisfaction, he engaged me to methodize those Problems, and to lay down the Pules which had led me to their Solution. After I had proceeded thus far, it was enjoined me by the Poyal Society, to communicate to them what I had discovered on this Subject : and thereupon it was ordered to be pub- lished in the Transactions, not so much as a matter relating to Play, but as containing some general Speculations not unworthy to be considered by the Lovers of Truth. 237. The membir consists of twenty-six Problems, besides a few introductory remarks which explain how probability is measured. 238. The first problem is to find the chance of throwing an ace twice or oftener in eight throws with a single die ; see Doctrine of Chances, page 13. 239. The second problem is a case of the Problem of Points. A is supposed to want 4 points, and B to want 6 points ; and ^'s chance of winning a single point is to £'s as 3 is to 2 ; see Doctrine of Chances, page 18. It is to be remembered that up to this date, in all that had been published on the subject, the chances of the players for winning a single point had always been assumed equal ; see Art. 173. 240. The third problem is to determine the chances of A and B for winning a single game, supposing that A can give B two games out of three ; the fourth problem is of a similar kind, supposing 138 DE MOIVRE. that A can give B one game out of three : see Problems i. and ii. of the Doctrine, of Chances. 241. The fifth problem is to find how many trials must be made to have an even chance that an event shall happen once at least. Montmort had already solved the problem ; see Axt. 170. De Moivre adds a useful approximate formula which is now one of the permanent results in the subject; we shall recur to it in noticing Problem III. of the Doctrine of Chances, where it is repro- duced. 242. De Moivre then gives a Lemma: To find how many Chances there are upon any number of Dice, each of them of the same number of Faces, to throw any given number of points ; see Doctrine of Chances, page 39. We have already given the history of this Lemma ia Art. 149. 243. The sixth problem is to find how many trials must be made to have an even chance that an event shall happen twice at least. The seventh problem is to find how many trials must be made to have an even chance that an event shall happen three times at least, or four times at least, and so on. See Problems III. and IV. of the Doctrine of Chances. 244. The eighth problem is an example of the Problem of Points with three players ; it is Problem VI. of the Doctrine of Chances. 245. The ninth problem is the fifth of those proposed for solution by Huygens, which Montmort had enunciated wrongly in his first edition ; see Axt. 199. Here we have the first publication of the general formula for the chance which each of two players has of ruining the other in an unlimited number of games ; see Art. 107. The problem is Problem vii. of the Doctrine of Chances. 246. The tenth problem is Problem Vlll. of the Doctrine of Ctiances, where it is thus enunciated : Two Gamesters A and B lay by 24 Counters, and play with three Dice, on this condition; that if 11 Points come up, A shall take one DE MOIVEE. 139 Counter out of the heap; if U, £ shall take out one; and he shall be reputed the winner who shall soonest get 12 Counters. This is a very simple problem. De Moivre seems quite un- necessarily to have imagined that it could be confounded with that which immediately preceded it ; for at the end of the ninth pro- blem he says, Maxime cavendum est ne Problemata propter speciem aliquam affinitatis inter se confundantur. Problema sequens videtur affine superior!. After enunciating his ninth problem he says, Problema istud a superiore in hoc differt, quod 23 ad plurimum tesserarum jactibus, Indus necessario finietvir ; cum Indus ex lege supe- rioris problematis, posset in aetemiun continuari, propter reciproca- tionem lucri et jactursB se invicem perpetuo destruentium. 247. The eleventh and twelfth problems consist of the second of those proposed for solution by Huygens, taken in two mean- ings ; they form Problems x. and xi. of the Doctrine of Chances. The meanings given by De Moivre to the enunciation coincide with the first and second of the three considered by James Ber- noulli ; see Arts. 35 and 199. 248. The thirteenth problem is the first of those proposed for solution by Huygens ; the fourteenth problem is the fourth of the same set : see Art. 35. These problems are very simple and are not repeated in the Doctrine of Chances. In solving the fourth of the set De Moivre took the meaning to be that A is to draw three white balls at least. Montmort had taken the meaning to be that A is to draw exactly three white balls. John Bernoulli in his letter to Montmort took the meaning to be that A is to draw three white balls at least. James Bernoulli had considered both mean- ings. See Art. 199. 249. The fifteenth problem is that which we have called Waldegrave's problem ; see Art. 211. De Moivre here discusses the problem for the case of three players : this discussion is re- peated, and extended to the case of four players, in the Doctrine of Chances, pages 132 — 159. De Moivre was the first in publishing a solution of the problem. 140 DE MOIVRE. 250. The sixteenth and seventeenth problems relate to the game of bowls ; see Art. 1,77. These problems are reproduced in a more general form in the Doctrine of Chances, pages 117 — 123. Eespecting these two problems Montmort says, on his page 366, Les Probl^mes 16 et 17 ne sont que deux cas tr§s simples d'un mSme ProblSme, c'eat presque le seul qui m'ait 6cliap6 de tous ceux que je troiive dans ce Livre. 251. The eighteenth and nineteenth problems are Problems XXXIX. and XL. of the Doctrine of Chances, where we shall find it more convenient to notice them. 252. The remaining seven problems of the memoir form a distinct section on the Duration of Play. They occur as Problems LViii, LX, LXi, LXii, LXiii, Lxv, Lxvi, of the Doctrine of Chances; and we shall recur to them. 253. It will be obvious from what we have here given that the memoir De Mensura Sortis deserves especial notice in the history of our subject. Many important results were here first published by De Moivre, although it is true that these results already existed in manuscript in the Ars Conjectandi and the correspondence between Montmort and the BemouUis. We proceed to the Doctrine of Chances. 254. The second edition of the Doctrine of Chances contains an Advertisement relating to the additions and improvements effected in the work; this is not reprinted in the third edition. The second edition has at the end a Table of Contents which neither of the others has. The third edition has the following Advertisement : The Author of this Work, by the failure of his Eye-sight in extreme old age, was obliged to entrust the Care of a new Edition of it to one of his Friends ; to whom he gave a Copy of the former, with some marginal Corrections and Additions, in his own hand writing. To these the Editor has added a few more, where they were thought necessary : and has disposed the whole in better Order; by restoring to their proper places some things that had been accidentally misplaced, and by putting all the Problems concerning Annuities together; as they stand in the late improved edition of the Treatise on that Subject. An Appendix DE MOIVEE. 141 of several useful Articles is likewise subjoined : the whole according to a Plan concerted with the Author, above a year before his death. . 255. The following list will indicate the parts which are new in the third edition. The Remark, pages 80 — 33 ; the Remark, pages 48, 49 ; the greater part of the second Corollary, pages 64 — 66 ; the Examples, page 88 ; the Scholium, page 95 ; the Remark, page 116 ; the third Corollary, page 138 ; the second Corollary, page 149 ; the Remark, pages 151—159 ; the fourth Corollary, page 162; the second Corollary, pages 176—179; the Note at the foot of page 187 ; the Remark, pages 251 — 254. The part on life annuities is very much changed, according to the plan laid down in the Advertisement. In the second and third editions the numbers of the Problems agree up to Problem XI ; Problem xii. of the third edition had been Problem Lxxxix. of the second; from Problem xii. to Problem LXix. of the third edition inclusive, the number of each Problem exceeds by unity its number in the second edition ; Pro- blem LXix. of the second edition is incorporated in the third edition with Problem vi; Problems Lxx. and LXXI. are the same in the two editions, allowing for a misprint of LXXI. for LXX. in the second edition. After thi» the numbering differs consider- ably because in the second edition Problems respecting life annui- ties are not separated from the other Problems as they are in the third edition. The first edition of the work was dedicated to Newton : the second was dedicated to Lord Carpenter, and the dedication of the second edition is reprinted at the beginning of the third ; the dedication to Newton is reprinted on page 329 of the third edition. 256. The first edition of the Doctrine of Chances has a good preface explaining the design and utility of the book and giving an account of its contents; the preface is reproduced in the other editions with a few omissions. It is to be regretted that the fol- lowing paragraphs were not retained, which relate respectively to the first and second editions of Montmort's work : However, had I allowed my self a little more time to consider if, I had certainly done the Justice to its Author, to have owned that lie had not only illustrated Huygens'a Method by a great variety of well 142 DE MOIVEE. chosen Examples, but that he had added to it several curious things of his own Invention. Since the printing of my Specimen, Mr. de Monmort, Author of the Analyse desjeux de Hazan-d, Published a Second Edition of that Book, in which he has particularly given many proofs of his singular Genius, and extraordinary Capacity; which Testimony I give both to Truth, and to the Friendship with which he is pleased to Honour me. The concluding paragraph of the preface to the first edition refers to the Ars Conjectandi, and invites Nicolas and John Ber- noulli to prosecute the subject begun in its fourth part; this paragraph is omitted in the other editions. We repeat that we are about to analyse the third edition of the Doctrine of Chances, only noticing the previous editions in cases of changes or additions in matters of importance. 257. The Doctrine of Chances begins with an Introduction of 83 pages, which explains the chief rules of the subject and illus- trates them by examples ; this part of the work is very much fuller than the corresponding part of the first edition, so that our remarks on the Introduction do not apply to the first edition. De Moivre considers carefully the following fundamental theorem : suppose that the odds for the happening of an event at a single trial are as a to h, then the chance that the event will happen r times at least in n trials is found by taking the first n — r + l terms of the expan- sion of (a 4- 6)" and dividing by (a+ 5)". We know that the result can also be expressed in another manner corresponding to the second formula in Art. 172 ; it is curious that De Moivre gives this without demonstration, though it seems less obvious than that which he has demonstrated. To find the chance that an event may happen just r times, De Moivre directs us to subtract the chance that it will happen at least r—1 times from the chance that it will happen at least r times. He notices, but less distinctly than we might expect, the modem method which seems more simple and more direct, by which we begin with finding the chance that an event shall happen just r times and deduce the chance that it shall happen at least r times. DE MOIVKE. 143 258. De Moivre notices the advantage arising from employing a single letter instead of two or three to denote the probability of the happening of one event. Thus if x denote the probability of the happening of an event, 1 —x will denote the probability of its failing. So also y and z may denote the .probabilities of the hap- pening of two other events respectively. Then, for example, x{\-y){l-z) wiU represent the probability of the first to the exclusion of the other two. De Moivre says in conclusion, " and innumerable cases of the same nature, belonging to any number of Events, may be solved without any manner of trouble to the imagination, by the mere force of a proper notation." 259. In his third edition De Moivre draws attention to the convenience of approximating to a fraction with a large numerator and denominator by continued fractions, which he calls "the Method proposed by Dr Wallis, ITuygens, and others." He gives the rule for the formation of the successive convergents which is now to be found in elementary treatises on Algebra ; this rule he ascribes to Cotes. 260. The Doctrine of ChaTices contains 74 problems exclusive of those' relating to life annuities ; in the first edition there were 53 problems. 261. We have enunciated Problems i. and ii. in Art. 240. Suppose p and q to represent the chances of A and 5 in a single game. Problem i. means that it is an even chance that A will win three games before B wins one ; thus p' = ^ . Hence ^ = -^ , and _Jl_ will win three games before B wins two. Thus p* + 4p'q = ^ ; which must be solved by trial. These problems are simple examples of the general formula in Art. 172. 262. Problems in, iv, and v. are included in the following 2' = 1 — T7^ . Problem li. means that it is an even chance that A lii DE MOIVRE. general enunciation. Suppose a the number of chances for the happening of an event in a single trial, and 6 the number of chances for its failing : find how many trials must be made to have an even chance that the event will happen r times at least. For example, let r = 1. Suppose X the number of trials. Then the chance that V the event fails x times in succession is , tn,, • -^i^i l^y suppo- sition this is equal to the chance of its happening once at least in X trials. Therefore each of these chances must be equal toi. Thus V 1 (a \}}f 2 ' from this equation x may be found by logaiithms. De Moivre proceeds to an approximation. Put - = j- Thus a; log (1 + -)= log 2. If 2 = 1, we have x=\. If £ be greater than 1, we have by pandinglog (l + -j' 11 1 = log2. (2 22''^"'' " *^ \ 1_ where log 2 will mean the logarithm to the Napierian base. Then if 2 be large we have approximately 7 a; = 2 log 2 = Y?; 2 iiearly. De Moivi-e says, page 37, Thus we have assigned the very narrow limits within which the ratio of a; to 5' is comprehended ; for it begins with unity, and terminates at last in the ratio of 7 to 10 vei-y near. But X soon converges to the limit 0.7g', so that this value of x may be assumed in all cases, let the value of q be what it will. The fact that this result is true when 2 is moderately large is the DE MOIVRE. 145 element of truth in the mistake made by M. de M^re ; he assumed that such a result should hold for all values of q : see Art. 14. 263. As another example of the general enunciation of Art. 262, let r=3. The chance that the event will happen at least 3 times in x trials is equal to the first x — 2 tenns of the expansion of \.a + b^a + bj' and this chance by hypothesis is ^ . Hence the last three terms of the expansion wiU also be equal to ^ , that is, If 2 = 1 we find a; = 5. If 2 be supposed indefinitely great, and we put - =z, we get ^=2(1+.+ !), where e is the base of the Napierian logarithms. By trial it is found that a = 2675 nearly. Hence De Moivre concludes that x always lies between hg and 2'675g'. 264. De Moivre exhibits the following table of results ob- tained in the manner shewn in the two preceding Articles. A Table of the Limits. The Value of x wiU always be Tor a single Event, betweeli Iq and 0'693g'. For a double Event, between 3q and 1-6782'. For a triple Event, between Bq and 2'67 5q. For a quadruple Event, between 7q and 3-672g. For a quintuple Event, between 9q and 4-670g'. For a sextuple Event, between llg^ and 5-668g'. &c. 10 146 DE MOIVRE. And if tlie number of Events contended for, as well as tte number q be pretty large in respect to Unity; the number of Trials requisite for those Events to happen n times will be — y~ 2'' °^ barely nq. De Moivre seems to have inferred the general result enun- ciated in the last sentence, from observing the numerical values obtained in the six cases which he had calculated, for he gives no further investigation. 265. In Art. 263 we have seen that De Moivre concludes that - always lies between 5 and 2'675. This may appear very probable, but it is certainly not demonstrated. It is quite con- ceivable, in the absence of any demonstration to the contrary, that - should at first increase with q, and so be greater than 5, and then decrease and become less than 2 675, and then increase again to its limit 2'675. The remark applies to the general pro- position, whatever be the value of r, as well as to the particular example iu which r- = 3. It would not be very easy perhaps to shew from such an equation as that in Art. 263, that x increases continually with q ; and yet from the nature of the question we may conclude that this must be the case. For if the chance of success in a single trial is diminished, it appears obvious that the number of trials must be increased, ia order to secure an even chance for the event to happen once at least. 266. On pages 39 — 43 of the Doctrine of Chances, we have the Lemma of which we have already given an account; see Art. 242. 267. Problem vi. of the Doctrine of Chances is an example of the Problem of Points with three players. De Moivre gives the same kind of solution as Fermat : see Arts. 16 and 18. In the third edition there is also a discussion of some simple cases according to the method which Pascal used for two players ; see Art. 12. De Moivre also gives here a good rule for solving the problem for any number of players; the rule is founded on DE MOIVKE. 147 Fermat's method, and is intended to lighten as much as possible the labour which must be incurred in applying the method to complex cases. The rule was first published in the Miscellanea Analytica, in 1730; it is given in the second edition of the Doctrine of Chances on pages 191, 192. ■ 268. Problem vil. is the fifth of those proposed by Huygens for solution ; see Art. 35. We have already stated that De Moivre generalises the problem in the same way as James Bernoulli, and the result, with a demonstration, was first published in the De Mensura Sortis; see Arts. 107, 245. De Moivre's demon- stration is very ingenious, but not quite complete. For he finds the ratio of the chance that A will ruin B to the chance that B will ruin A ; then he assumes in effect that in the long nm one or other of the players must be ruined : thus he deduces the absolute values of the two chances. See the first Appendix to Professor De Morgan's Essay on Probabilities in the Cabinet Cyclopcedia. We have spoken of Problem vill. in Art. 246, 269. Problem ix. is as follows. Supposing A and B, whose proportion of skill is as a to 6, to play together, till A either wins the number q of Stakes, or loses the number p of them; and that B sets at every Game the sum G to the sum L; it is required to find the Advantage or Disadvantage of A. This was Problem XLiii. of the first edition of the Doctrine of Chances, in the preface to which it is thus noticed : The 43d Problem having been proposed to me by Mr. Thomas Wood- cock, a Gentleman whom I infinitely respect, I attempted its Solution with a very great desire of obtaining it; and having had the good Fortune to succeed in it, I returned him the Solution a few Days after he was pleased to propose it. This Problem is in my Opinion one of the most curious that can be projios'd on this Subject ; its Solution containing the Method of determining, not only that Advantage which results from a Superiority of Chance, in a Play confined to a certain number of Stakes to be won or lost by either Party, but also that which may result from an unequality of Stakes ; and even compares those two Advantages together, when the Odds of Chance being on one side, the Odds of Money are on the other. 10—2 148 DE MOIVRE. In the Miscellanea Analytica, page 204, the problem is again said to have been proposed by Thomas Woodcock, spectatissimo vim, but he is not mentioned in the second or third edition of the Doctrine of Chances ; so that De Moivre's infinite respect for him seems to have decayed and disappeared in a finite time. The solution of the problem is as follows : Let R and S respectively represent the Probabilities which A and B have of winning all the Stakes of their Adversary ; which Probabilities have been determined in the vii* Problem. Let us first suppose that the Sums deposited by A and B are equal, viz. G, and G : now since A is either to win the sum qG, or lose the sum pG, it is plain that the Gain of A ought to be estimated by EqG — SpG; moreover since the Sums deposited are G and G, and that the proportion of the Chances to win one Game is as a to h, it follows that the Gain of A for each individual Game is j— ; and for the same reason the Gaia of each individual a + b Game would be — , if the Sums deposited by A and B were re- spectively L and G. Let us therefore now suppose that they are L and G; then in order to find the whole Gain of .4 in this second cir- cumstance, we may consider that whether A and B lay down equal Stakes or unequal Stakes, the Probabilities which either of them has of winning all the Stakes of the other, sufier not thereby any alter- ation, and that the Play will continue of the same length in both cir- cumstances before it is determined in favour of either; wherefore the Gain of each individual Game in the first case, is to the Gain of each individual Game in the second, as the whole Gain of the first case, to the whole Gain of the second; and consequently the whole Gain of the second case will be Bq-Hpy. — — — , or restoruig the values of R and S, qa''xa''-b''-pb'xa''-b'' ,^. ,. ^ , aG-bL ^^u_f^u multiplied by ^_^ . 270. In the first edition of the Doctrine of Chances, pages 136 — 142, De Moivre gave a very laborious solution of the preceding Problem. To this was added a much shorter solution, communicated by Nicolas Bernoulli from his uncle. This solution was founded on an artifice which De Moivre had himself used in DE MOIVBB. 149 the ninth problem of the De Menswa Sortis. De Moivre how- ever renounces for himself the claim to the merit of the solu- tion. This renunciation he" repeats in the Miscellanea Anah/tica, page 206, where he names the author of the simple solution which we have already giyen. He says, Ego vere illud ante libenter fassus sum, idque ipsum etiammim libenter fateor, quamvis solutio Problematis mei noni causam fortasse dederit hujus solutionis, me taiiien nihil juris in eam habere, eamque 01. illius Autori ascribi sequum esse. Septem aut octo abbiric annis D. Stevens Int. Tempi. Socius, Vir ingenuus, singular! sagacitate prseditus, id sibi propositum habens ut Problema superius allatum solveret, hac ratione solutionem facile asse- cutus est, quam mihi his verbis exhibuit. Then foUows the solution, after which De Moivre adds, Doctissimus adolescens D. Cranmer, apud Genevenses Mathematics Professor dignissimus, cujus recordatio seque ac Collegse ejus peritissLmi D. Galandrin mihi est perjucunda, cum superiore anno Londini com- moraretur, narravit mihi se ex literis D. Nic. Bernoulli ad se datis acce- pisse 01. Yirum novam solutionem hujus Problematis adeptum esse, quam prioribus autor anteponebat ; cum vero nihil de via solutionis dixerit, si mihi conjicere liceat qualis ea sit, banc opinor eandem esse atque illam quam modo attuli. 271. We have already spoken of Problems x. and xi. in Art. 247. In his solution of Problem X. De Moivre uses the theorem for the summation of series to which we have referred in Art. 152. A corollary was added in the second edition and was expanded in the third edition, on which we wiU make a remark. Suppose that A, B, and C throw in order a die of n faces, and that a faces are favourable to A, and h to B, and c to C, where a+b+c = n. Eequired the chances which A, B, and O have respectively of being the first to throw a corresponding face. li) may be easily shewn that the chances are proportional to an^ (h + c) In, and (b + c) (a + c) c, respectively. De Moivre, in his third edition, page 65, seems to imply that before the order was fixed, the chances would be proportional to a, b, c. This must of course mean that such would be the case if there were 150 DE MOIVRE. no order at all; that is if the die were to be thrown and the stake awarded to A, B, or G, according as the face which appeared was one of the a, h, c respectively. If there is to be an order, but the order is as likely to be one as another, the result will be different. The chance of A for example will be one sixth of the sum arising from six possible and equally likely cases. It will be found that ^'s chance is a {6a' + 9a (5 4 c) + 3 {1' + c") + 86c} 6{n=-(& + c)(c + a) (« + &)} 272. Problem xil. appeared for the first time in the second edition, page 248, with this preliminary notice. "A particular Friend having desired of me that to the preceding Problems I would add one more, I have thought fit to comply with his desire ; the Problem was this." The problem is of no great importance ; it is solved by the method often used in the Ars Conjectandi, which we have explained in Art. 106. 278. Problem xiii. relates to the game of Bassette, and Problem xrv. to the game of Pharaon ; these problems occupy pages 69 — 82 of the work. We have already sufficiently noticed these games ; see Arts. 154, 163. De Moivre's discussion is the same in all his three editions, except that a paragraph on page 37 of the first edition, extending from the words " Those who are . . ." to the end of the page, is omitted in the following editions. The paragraph is in fact an easy example of the formulae for the game of Bassette. 274. Problems XV. to xx. form a connected series. De Moivre solves simple examples in chances and applies his results to esta- blish a Theory of Permutations and Combinations ; in modem times we usually adopt the reverse order, establish the Theory of Permutations and Combinations first, and afterwards apply the theory in the discussion of chances. We will take an example of De Moivre's method from his Problem xv. Suppose there are six things a, b, c, d, e, f, and let two of them be taken at random ; required the chance that a shall stand first, and h second. The DE MOrVEE. ,151 1 chance of taking a first is ^ ; and there are then five things left, and the chance of now taking S is ^ . Therefore the required chance is ^k ■ Then De Moivre says, Since the taking a in the first place, and h in the second, is but one single Case of those by which six Things may change their order, being taken two and two ; it follows that the number of Changes or Permu- tations of six Things, taken two and two, must be 30. 275. In his Preface De Moivre says, Having explaiued the common Rules of Combinations, and given a Theorem which may be of use for the Solution of some Problems re- lating to that Subject, I lay down a new Theorem, which is properly a contraction of the former, whereby several Questions of Chance are resolved with wonderful ease, tho' the Solution might seem at first sight to be of insuperable difficulty. The iMw Theorem amounts to nothing more than the simplifi- cation of an expression by cancelling factors, which occur in its numerator and denominator ; see Doctrine of Chances, pages ix. 89. 276. Problems xxi. to xxv. consist of easy applications to questions concerning Lotteries of the principles established in the Problems xv. to xx. ; only the first two of these questions con- cerning Lotteries appeared in the first edition. A Scholium is given on page 95 of the third edition which deserves notice. De Moivre quotes the following formula : Sup- pose a and n to be positive integers ; then 1 1 1 1 , 1 n n+1 n+z n+3 a—1 ~ ^^n'^2^ 2a ■*" 2 U' «V * U' aV W~aV + Q{-e--s] + ' where ^ = |, ^ = -30, 0=^^^, 152 -DE MOIVBE. As De Moivre says A, B, 0, ... are "the numbers of Mr. James Bernoulli in his excellent Theorem for the Summing of Powers." See Art. 112. De Moivre refers for the demonstration of the formula to the Supplement to the Miscellanea Analytica, where the formula first appeared. We shall recur to this in speaking of the Miscellanea Analytica. 277. Problems xxvii. to xxxii. relate to the game of Quad- rille ; although the game is not described there is no difficulty in understanding the problems which are simple examples of the Theory of Combinations : these problems are not in the first edition. 278. Problem xxxiii. is To find at Pharaon how much it is that the Banker gets per Cent, of all the Money that is adventured. De Moivre iu his Preface seems to attach great importance to this solution ; but it scarcely satisfies the expectations which are thus raised. The player who stakes against the bank is in fact sup- posed to play merely by chance without regard to what would be his best course at any stage of the game, although the previous investigations of Montmort and De Moivre shewed distinctly that some courses were far less pernicious than others. The Banker's adversary in De Moivre's solution is therefore rather a machine than a gambler with liberty of choice. 279. Problem xxxiv. is as follows : Supposing A and JB to play together, that the Chances they have respectively to win are em a to b, and that B obliges himself to set to A so long as A wins without interruption : what is the advantage that A gets by his hand? The result is, supposing each to stake one, that is, "~^. 280. Problems xxxv. and xxxvi. relate to the game dis- cussed by Nicolas Bernoulli and Montmort, which is called Treize or Rencontre; see Art. 162. , DE MOIVKE. 153 J&e Moivre treats the subject with great ingenuity and with more generality than his predecessors, as we shall now shew. 281. Problem xxxv. is thus enunciated : Any number of Letters a, h, c, d, e,f, &c., all of them different, being taken promiscuously as it happens : to find the Probability that some of them shall be found in their places according to the rank they obtain in the Alphabet; and that others of them shall at the same timg be displaced. Let n be the number of the letters; suppose that^ specified letters are to be in their places, q specified letters out of their places, and the remaining n —p — q letters firee from any restric- tion. The chance that this result will happen is 1 U s ^ , g(g-i) 1 \ n{n—l)...{n—p+l)\ 1 n~p 1.2 (n—p)(n—p — l) "•]• This supposes that p is greater than ; if ^ = 0, the result is g 1 g(g-l) 1 In"^ 1.2 n{n-l) '" If we suppose ia this formula g = m — 1, we have a result already implicitly given in Art. 161. In demonstrating these formulae De Moivre is content to ex- amine a few simple cases and assume that the law which presents itself will hold universally. We will indicate his method. The chance that a is in the first place is - ; the chance that a is in the first place, and 5 in the second place is — 7 ^r- : hence the ^ ^ n{n — T) chance that a is in the first place and b not in the second place is 1 1 n « (re — 1) ' Similarly the chance that a, b, c are all in their proper places is subtract this from the chance that a and b are in n(n-l)(n-2)' their proper places, and we have the chance that a and b are in their proper places, and c not in its proper place : thus this chance is 1 1 n(n — l) w (n — 1) (n — 2) ■ 154 DE MOIVEE. De Moivre uses a peculiar notation for facilitating this process. Let +a denote the chance that a is in its proper place and —a the chance that it is out of it ; let + & denote the chance that 5 is in its proper place and — h the chance that it is out of it ; and so on. And in general let such a symbol as +a + h + c — d — e denote that a, b, c are in their proper places, and d, e out of theirs. T+1- 1 1 , •^® w"*"' n{n-l) *' n(n-l){n-2)~ ' 1 n{n-l)(n-2){n-3)~^''" Then we have the following results : + h =r +h+a=s + b — a = r—s (1) +c+h =s +c+b+a=t + c + b-a = s-t (2) + c-a =r-s by (1) ■^c-a + b= s-t by (2) + c-a-b= r-2s + t (3) + cZ+c + 6 =< +d+c+h+a=v + d+c + b-a = i-v (4) + d+c-a =s-t by (2) + d + c — a + b= t—v by (4) + c?+c — a — 5= s—^t + v (5) + d-b-a =r-2s + « by (3) + d-b-a + c= s-2t + v by (5) d—b-a-G= r-3s + St — v.,...(6) It is easy to translate into words any of these S3rmbolicaI pro- cesses. Take for example that which leads to the result (2) : + c + b = s; DE MOIVEE. 155 this means that the chance that c and h are in their proper places is s ; and this we know to be true ; this means that the chance that c, i, a are all in their proper places is t ; and this we know to be true. From these two results we deduce that the chance that c and h are in their proper places, and a out of its place is s — < ; and this is expressed symbolically thus, + c + 'b — a = s—t. Similarly, to obtain the result (3) ; we know from the result (1) that r — s is the chance that c is in its proper place, and a out of its proper place ; and we know from the result (2) that s — t is the chance that c and h are in their proper places, and a out of its pro- per place ; hence we infer that the chance that c is in its proper place, and a and i out of their proper places is r — 2s + < ; and this result is expressed symbolically thus, -\-c — a — h = r — 2s-\-t. 282. De Moivre refers in his Preface to this process in the fol- lowing terms : In tie 35th and 36th Problems, I explain a new sort of Algebra whereby some Qiiostions relating to Combinations are solved by so easy a Process, that their Solution is made in some measure an immediate consequence of the Method of Notation. I will not pretend to say that this new Algebra is absolutely necessary to the Solving of those Ques- tions which I make to depend on it, since it appears that Mr. Montmorf Author of the Analyse des Jeux de Hazard, and Mr. Nicholas Bernoulli have solved, by another Method, many of the cases therein proposed : But I hope I shall not be thought guilty of too much Confidence, if I assure the Reader, that the Method I have followed has a degree of Simplicity, not to say of Generality, which will hardly be attained by any other Steps than by those I have taken. 283. De Moivre himself enunciates his result verbally ; it is of course equivalent to the formula which we have given in Art. 281, but it will be convenient to reproduce it. The notation being that already explained, he says, 156 DE MOIVEE. ...then let all the quantities 1, r, s, t,v, &c. be written down with Signs alternately positive and negative, beginning at 1, i£p be = j at r, if ^ be = 1 J at «, if p be = 2 ; &c. Prefix to these Quantities the Co- efficients of a Binomial Power, whose index is = g'j this being done, those Quantities taken all together will express the Probability re- quired. 284. The eminciation and solution of Problem xxxvi. are as follows : Any given number of Letters a, b, c, d, e, f, &c., being each repeated a certain number of times, and taken promiscuously as it happens : To find the Probability that of some of those sorts, some one Letter of each may be found in its place, and at the same time, that of some other sorts, no one Letter be found in its place. Suppose n be the number of all the Letters, I the number of times that each Letter is repeated, and consequently j the whole number of Sorts : suppose also that f be the number of Sorts of which some one Letter is to be found in its place, and q the number of Sorts of which no one Letter is to be found in its place. Let now the prescriptions given in the preceding Problem be followed in all respects, saving that I P V r must here be made = — , « = — ; =-; , t = -— zrr-. ttt- , (fee, and n n{n — \) n(n—i){ri — 2) the Solution of any particular case of the Problem will be obtained. Thus if it were required to find the Probability that no Letter of any sort shall be in its place, the Probability thereof would be expressed by the Series \-qr+^^-j-s 1.2.3 ^ + 1.2.3.4 " ^''- of which the number of Terms is equal to g'+ 1, But in this particular case q would be equal to -j , and therefore, the foregoing Series might be changed into this, viz. 1 n-l )_ {n-l){n-2T) l^ {n-l){n-'21){n-^l) . 2 n-l 6 (w-l)(w-2) "^24:(«-l)(7i-2)(w-3) of which the number of Terms is equal to — -j— . DE MOIVBE. 157 285. De Moivre then adds some Corollaries. The following is the first of them : From hence it follows, that the Probability of one or more Letters, indeterminately taken, being in their places, -will be expressed as fol- lows : . 1 n-l 1 (n-l)(n-2l) 1 (n - T) (n - 21) (n - SI) „ 2«-l"^6 (n-l)(«_2) 24. (m- 1) («-2) (n-3) This agrees with what we have already given from Nicolas Bernoulli ; see Art. 204. In the next three Corollaries De Moivre exhibits the pro- bability that two or more letters should be in their places, that three or more should be, and that four or more should be. 286. The four Corollaries, which we have just noticed, are examples of the most important part of the Problem; this is treated by Laplace, who gives a general formula for the proba- bility that any assigned number of letters or some greater number shall be in their proper places. ThAorie. . .des Prob. pages 217 — 222. The part of Problems xxxv. and xxxvi. which' De Moivre puts most prominently forward in his enunciations and solutions is the condition that p letters are to be in their places, q out of their places, and n—p — q free from any restriction; this part seems peculiar to De Moivre, for we do not find it before his time, nor does it seem to have attracted attention since. 287. A Remark is given on page 116 which was not in the preceding editions of the Doctrine of Chances. De Moivre shews that the sum of the series 1 — ^+^ — H2;+ '-'in infinitum, is equal to unity diminished by the reciprocal of the base of the Napierian logarithms. 288. The fifth Corollary to Problem xxxvi. is as foUows : If A and S each holding a Pack of Cards, pull them out at the same time one after another, on condition that every time two like Cards are 158 t)E MOIVEE. pulled out, A shall give £ a Guinea; and it were required to find what consideration £ ought to give A to play on those Terms : the Answer wiU be one Guinea, let the number of Cards be what it wUl. Altho' this be a CoroUary from the preceding Solutions, yet it may more easily be made out thus ; one of the Packs being the Rule where- by to estimate the order of the Cards in the second, the Probability that the two first Cards are alike is -^ , the Probability that the two second are alike is also -^ , and therefore there being 52 such alike com- 52 binations, it follows that the value of the whole is ^o ~ ^' It may be interesting to deduce this result from the formulsB already given. The chance that out of n cards, p specified cards will be in their places, and all the rest out of their places will be obtained by making q= n —p in the first formula of Art. 281. The chance that any p cards will be in their places, and all the rest out of their places wiU be obtained by multiplying the pre- \n ... . ceding result by — . And since in this case B receives ^ \n-p \p p guineas, we must multiply by p to obtain JB's advantage. Thus we obtain \P-^ 1 2 [3 [i \n-v j Denote this by (p) ; then we are to shew that the sum of the values of ^ (p) obtained by giving to p aU values between 1 and n inclusive is unity. Let 1^ (w) denote the sum ; then it may be easily shewn that i/r (w + 1) - i/r (n) = 0. Thus ^fr (n) is constant for all values of n ; and it = 1 when m = 1, so that i|r (n) is always = 1. 289. The sixth Corollary to Problem xxxvi. is as follows : If the number of Packs be given, the Probability that any given number of Circumstances may happen in any number of Packs, wUl DE MOIVEE. 159 easily be found by our Method : thus if the number of Packs be le, the Probability that one Card or more of the same Suit and Name in every one of the Packs may be in the same position, will be expressed as fol- lows, _1 1 1 w*-^ 2 {« (« - 1)}'-'' ^ ^{n{n-\){n- 2)]'-=' 1 [4{n(n-l) (w-2) (ti-S)]*" &c. Laplace demonstrates this result ; see TMorie . . . des Prdb. page 224. 290. Problems xxxvii. and xxxvili. relate to the game of Bowls ; see Arts. 177, 250. De Moivre says, page 120, Having given formerly the Solution of this Problem, proposed to me by the Honourable Francis Bohartes, Esq;, in the Philosophical Trans- itions Number 329; I there said, by way of Corollary, that if the proportion of Skill in the Gamesters were given, the Problem might also be solved : since which time M. de Monmort, in the second Edition of a Book by him published upon the Subject of Chance, has solved this Problem as it is extended to the consideration of the Skill, and to carry his Solution to a great number of Cases, giving also a Me- thod whereby it might be carried farther: But altho' his Solution is good, as he has made a right use of the Doctrine of Combinations; yet I think mine has a greater degree of Simplicity, it being deduced from the original Principle whereby I have demonstrated the Doctrine of Permutations and Combinations:... 291. Problems xxxix. to XLli. form a connected set. Pro- blem XXXIX. is as foUows : To find the Expectation of A, when with a Die of any given num- ber of Paces, he undertakes to fling any number of them in any given number of Casts. Let ^ + 1 be the number of faces on the die, n the number of casts, /the number of faces which A undertakes to fling. Then A's expectation is 160 DE MOIVBE. _ /(/-W-^) (^_,).^....|. De Moivre infers this general result from the examination of the simple cases in which / is equal to 1, 2, 3, 4 respec- tively. He says in his Preface respecting this problem. When I began for the first time to attempt its Solution, I had nothing else to guide me but the common Rules of Combinations, such as they had been delivered by Dr. Wallis and others; which when I endeavoured to apply, I was surprized to find that my Calculation swelled by degrees to an intolerable Bulk : For this reason I was forced to turn my Views another way, and to try whether the Solution I was seeking for might not be deduced from some easier considerations; whereupon I happUy fell upon the Method I have been mentioning, which as it led me to a very great Simplicity in the Solution, so I look upon it to be an Improvement made to the Method of Com- binations. The problem has attracted much attention; we shall find it discussed by the following writers : Mallet, Acta Helvetica, 1772 ; Euler, Opuscula Analytica, Yol. ii. 1785 ; Laplace, Memoires... par divers Savans, 1774, TMorie... des Prob. page 191 ; Trembley, Memoires de I'Acad... Berlin, 1794, 1795. We shall recur to the problem when we are giving an account of Euler's writings on our subject. 292. Problem XL. is as foUows : To find in how many Trials it wUl be probable that A with a Die of any given number of Faces shall throw any proposed number of them. Take the formula given in Art. 291, siippose it equal to ^ , and seek for the value of n. There is no method for solving this equation exactly, so De Moivre adopts an approximation. He supposes that p + 1, p, p — l, p — 2, are in Geometrical DE MOIVRE. 161 Progression, which supposition he says "will very little err from the truth, especially if the proportion oip to 1, be not very small." » + 1 Put r for ; thus the equation becomes Ir""^ 1.2 »•» \2 r"'^'" 2' that is 1 1 — =)=■;;. 1 Hence ,1 = 1- (1)'^ and then n may be found by logarithms. De Moivre says in his Preface respecting this problem, The 40tli Problem is the reverse of the preceding; It contains a very remarkable Method of Solution, the Artifice of which consists in changing an Arithmetic Progression of Numbers into a Geometric one; this being always to be done when the Numbers are large, and their Intervals small. I freely acknowledge that I have been indebted long ago for this useful Idea, to my much respected Friend, That Ex- cellent Mathematician Dr. Mallei/, Secretary to the Royal Sodely, whom I have seen practise the thing on another occasion: For this and other Instructive Notions readily imparted to me, during an un- interrupted Friendship of five and Twenty years, I return him my very hearty Thanks. Laplace also notices this method of approximation in solving the problem, and he compares its lesult with that furnished by his own method ; see TMorie ... des Prob. pages 198 — 2.00. 293. Problem XLI. is as follows : Supposing a regular Prism having a Faces marked I, b Faces marked ii, c Faces marked iii, d Faces marked iv, &o. what is the Probability that in a certain number of throws n, some of the Faces marked I will be thrown, as also some of the Faces marked ii ? This is an extension of Problem xxxix ; it was not in the first editifon of the Doctrine of Chances. Let a + h + c+d+ ...=s; then the Probability required will be ^ i;^« _ i(s - ay + is- by] + {s-a- byi 11 162 DE MOIVRE. If it be required that some of the Faces marked i, some of the Faces marked li, and some of the Faces marked iii be thrown, the Probability required will be 1 1^5" _|(, _«)»+(, _J)»+(s- c)"} 4 (s-a-J)"+(s-6-c)"+(s-c-a)" -(s-a-J-c)"] . And so on if other Faces are required to be thrown. De Moivre intimates that these results follow easily from the method adopted in Problem xxxix. 294. Problem XLii. first appeared in the second edition; it is not important. Problem XLiii. is as follows : Any number of Chances being given, to find tte Probability of their being produced in a given order, without any limitation of the number of times in which they are to be produced. It may be remarked that, for an approximation, De Moivre proposes to replace several numbers representing chances by a common mean value ; it is however not easy to believe that the result would be very trustworthy. This problem was not in the first edition. 295. Problems XLiv. and XLV. relate to what we have called Waldegrave's Problem ; see Art. 211. In De Moivre's first edition, the problem occupies pages 77 — 102. De Moivre says in his preface that he had received the solution by Nicolas Bernoulli before his own was published ; and that both solutions were printed in the Philosophical Traiisactions, No. 341. De Moivre's solution consists of a very full and clear discussion of the problem when there are three players, and also when there are four players ; and he gives a little aid to the solution of the general problem. The last page is devoted to an explanation of a method of solving the problem which Brook Taylor communicated to De Moivre. In De Moivre's third edition the problem occupies pages 132 — 159. The matter given in the first edition is here reproduced, omitting. DE MOIVRE. 163 however, some details which the reader might be expected to fill up for himself, and also the method of Brook Taylor. On the other hand, the last nine pages of the discussion in the third edition were not in the first edition ; these consist of explanations and investigations with the view of enabling a reader to determine numerical results for any number of players, supposing that at any stage it is required to stop the play and divide the money deposited equitably. This part of the problem is pecuUar to De Moivre. The discussions which De Moivre gives of the particular cases of three players and four players are very easy and satis- factoiy; but as a general solution his method seems inferior to that of Nicolas Bernoulli. We may remark that the investigation for three players given by De Moivre will enable the student to discover how Montmort obtained the results which he gives with- out demonstration for three players; see Art. 209. De Moivre determines a player's expectation by finding first the advantage resulting from his chance of winning the whole sum deposited, and then his disadvantage arising from the contributions which he may have had to make himself to the whole sum deposited ; the expectation is obtained by subtracting the second result from the first. Montmort determined the expectation by finding, first the advantage of the player arising from his chance of winning the deposits of the other two players, and then the disadvantage arising from the chance which the other two players have of winning his deposits ; the expectation is obtained by subtracting the second result from the first. The problem will come before us again as solved by Laplace. 296. Problem XLVi. is on the game oi Hazard; there is no description of the game here, but there is one given by Montmort on his page 177; and from this description, De Moivre's solution can be understood : his results agree with Montmort's. Pro- blem XLVii. is also on Hazard ; it relates to a point in the game which is not noticed by Montmort, and it is only from De Moivre's investigation itself that we can discover what the problem is, which he is considering. With respect to this problem, De Moivre says, page 165, 164 DE MOIVBE. After I had solved the foregoing Problem, which is about 12 years ago, I spoke of my Solution to Mr. Hewry Stuart Stevens, but with- out communicating to him the manner of it: As he is a Gentleman who, besides other uncommon Qualifications, has a particular Sagacity in reducing intricate Questions to simple ones, he brought me, a few days after, his Investigation of the Conclusion set down in my third Corollary; and as I have had occasion to cite him before, in another Work, so I here renew with pleasure the Expression of the Esteem which I have for his extraordinary Talents : Then follows the investigation due to Stevens. The above passage occurs for the first time in the second edition, page 140 ; the name however is there spelt Stephens : see also Art. 270. Problem XLVII. is not in the first edition ; on the other hand, a table of numerical values of chances at Hazard, without ac- companying explanations, is given on pages 174, 175 of the first edition, which is not reproduced in the other editions. 297. Problems XLVIII. and XLIX. relate to the gamfe of Raffling. If three dice are thrown, some throws will present triplets, some doublets, and some neither triplets nor doublets; in the game of Raffles only those throws count which present triplets or doublets. The game was discussed by Montmort in his pages 207 — 212 ; but he is not so elaborate as De Moivre. Both writers give a numerical taJble of chances, which De Moivre says was drawn up by Francis Robartes, twenty years before the publica- tion of Montmort's work ; see Miscellanea Analytica, page 224. Problem XLIX. was not in De Moivre's first edition, and Problem XLViii. was not so fully treated as in the other edi- tions. 298. Problem L. is entitled Of Whisk; it occupies pages 172 — 179. This is the game now called Whist. De Moivre determines the chances of various distributions of the Honours in the game. Thus, for example, he says that the probability that there are no Honours .,, -J • 650 , . on either side is ., .„_ ; this of course means that the Honours looo are equally divided. The result would be obtained by considering two cases, namely, 1st, that in which the card turned up is an DE MOIVRE. 1(55 Honour, and 2nd, that in which the card turned up is not an Honour. Thus we should have for the required probability ± 3 25 ■ 26 ■ 25 9 4.3 25. 24. 26.25 13 • 1 ■ 51 . 50. 49 13 ' 1 . 2 "51.50.49.48' and this will be found equal to . ^ 1666 De Moivi-e has two Corollaries, which form the chief part of his investigation respecting Whist. The first begins thus : From what we have said, it wiU not be difficult to solve this Case at Whisk; viz. which side has the best, of those who have viii of the Game, or of those who at the same time have ix? In order to which it will be necessary to premise the following Principle. 1° That there is but 1 Chance in 8192 to get vn. by Triks. 2° That there are 13 Chances in 8192 to get vi. 3° That there are 78 Chances in 8192 to get v. 4° That there are 286 Chances ia 8192 to get iv. 5° That there are 715 Chances in 8192 to get in. 6° That there are 1287 Chances in 8192 to getii. 7° That there are 1716 Chances in 8192 to get I. All this will appear evident to those who can raise the Binomial o + 6 to its thirteenth power. But it must carefully be observed that the foregoing Chances ex- press the Probability of getting so many Points by Triks, and neither more nor less. De Moivre states his conclusion thus : From whence it follows that without considering whether the viii are Dealers or Eldest, there is one time with another the Odds of somewhat less than 7 to 5j and very nearly that of 25 to 18. The second Corollary contains tables of the number of chances for any assigned number of Trumps in any hand. De Moivre says. By the help of these Tables several useful Questions may be re- solved; as 1°. If it is asked, what is the Probability that the Deale.' has precisely iii Trumps, besides the Trump Card? The Answer , „ , . 4662 by Tab. I. IS jgg^; ... 166 DE MOIVRE. In the first edition there was only a brief notice of Whist, occupying scarcely more than a page. 299. Problems LI. to LV. are on Piquet. The game is not described, but there is no difficulty in understanding the problems, which are easy examples of combinations. The following Rema/rk occurs on page 186 ; it was not in the first edition : It may easily be perceived from tlie Solution of the preceding Problem, that the number of variations which there are in twelve Cards make it next to impossible to calculate some of the Probabili- ties relating to Piquet, such as that which results from the priority of Hand, or the Probabilities of a Pic, Repic or Lurch ; however not- withstanding that difficulty, one may from observations often repeated, nearly estimate what those Probabilities are in themselves, as will be proved in its place when we come to treat of the reasonable conjec- tures which may be deduced from Experiments; for which reason I shall set down some Observations of a Gentleman who has a very great degree of Skill and Experience in that Game, after which I shall make an application of them. The discussion of Piquet was briefer in the first than in the following editions. 300. We will give the enunciation of Problem LVi. and the beginning of the solution. Problem LVi. Of Saving Clauses. A has 2 Chances to beat £, and £ has 1 chance to beat A ; but there is one Chance which intitles them both to withdraw their own Stake, which we suppose equal to s ; to find the Gain of A. Solution. This Question tho' easy in itself, yet is brought in to caution Be- ginners against a Mistake which they might commit by imagining that the Case, which intitles each Man to recover Ms own Stake, needs not be regarded, and that it is the same thing as if it did not exist. This I mention so much more readily, that some people who have pretended great skill in these Speculations of Chance have themselves fallen into that error. DE MOIVKE. Igy This problem was not in the first edition. The gain of A is rs. 301. Problem LVn, which was not in the first edition, is as follows : ^ and 5 playing together deposit £s apiece; A has 2 Chances to win s, and -S 1 Chance to win s, whereupon A tells £ that he will play with him upon an equality of Chance, if he 5 will set him 2s to Is, to which B assents : to find whether A has any advantage or disad- vantage by that Bargain. In the first case A's expectation is ^ s, and in the second, o it is 2 « ; so that he gains ^ s by the bargain. S02. We now arrive at one of the most important parts of De Moivre's work, namely, that which relates to the Duration of Play ; we will first give a full account of -what is contained in the third edition of the Doctrine of Chances, and afterwards state how much of this was added to the investigations originally published in the De Mensura Sortis. De Moivre himself regarded his labours on this subject with the satisfaction which they justly merited; he says in his Preface, When I first began to attempt the general Solution of the Problem concerning the Duration of Play, there was nothing extant that could give me any light into that Subject; for altho' Mr de Monmort, in the first Edition of his Book, gives the Solution of this Problem, as limited to three Stakes to be won or lost, and farther limited by the Suppo- sition of an Equality of Skill between the Adventurers; yet he having given no Demonstration of his Solution, and the Demonstration when discovered being of very Httle use towards obtaining the general Solu- tion of the Problem, I was forced to try what my own Enquiry would lead me to, which having been attended with Success, the result of what I found was afterwards published in my Specimen before men- tioned. The Specimen is the Essay De Mensura Sortis. 168 DE MOIVBE. 303. The general problem relating to the Duration of Play may be thus enunciated : 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 probability that wh&n. or before a certain nwmher of games hxis been played, one of the players will have won all the counters of his adversary. It will be seen that the woi-ds in italics constitute the advance which this problem makes beyond the more simple one discussed in Art. 107. De Moivre's Problems LVin. and Lix. amount to solving the problem of the Duration of Play for the case in which m and n are equal. After discussing some cases in which w = 2 or 3, De Moivre lays down a General Eule, thus: A General Rule for determining what Probability there is that tfie Play shall not be determined in a given number of Gfames. Let n be the number of Pieces of each Gamester. Let also n+d be the number of Games given; raise a + b to the Power n, then cut off the two extream Terms, and miultiply the remainder by aa + 2ab + bb : then cut off again the two Extreams, and multiply again- the remainder by aa + 2ab + bb, stiU rejecting the two Extreams j and so on^ making as many Multiplications as there are Units in ^d ; make the last Pro- duct the Numerator of a Fraction whose Denominator let be (a + S)"**, and that Fraction wiU express the Probability required, ; still ob- serving that if d be an odd number^ you write d—1 in its room. For an example, De Moivre supposes w = 4, d=6. Raise a -1- 6 to the fourth power, and reject the extremes ; thus we have Wb + QaV + ^aW. Multiply by a' 4- 2aJ + b", and reject the extremes ; thus we have iWb' + 20a'b' + Ua'b\ Multiply by a" + 2ab + W, and reject the extremes ; thus we have 48a'6'-|-68a*6*-)-48a'&^ Multiply by a" -I- 2a6 + 6", and reject the extremes; thus we h-ave lUa%^+ 232aW-|- 164a*&». Thus the probability that the Play will not be ended in 10 games is DE MOIVBE. 169 164a°y + 232 aW + 164a'6° {a + by De Moivre leaves his readers to convince themselves of the accuracy of his rule ; and this is not difficult. De Moivre suggests that the work of multiplication may he abbreviated by omitting the a and b, and restoring them at the end ; this is what we now call the method of detached coefficients. 304. The terms which are rejected in the process of the preceding Article will furnish an expression for the probability that the play wiU be ended in an assigned number of games. Thus if m = 4 and d = 6, this probability will be found to be a* + b* 4o°6+4.g5° 14a°y + Ua'b^ 48a'y+48a'y («+&)*■*" {a + bf "*" (a + by ^ (a + by ' Now here we arrive at one of De Moivre's important results ; he gives, without demonstration, general formulae for determining those numerical coefficients which in the above example have the values 4, 14, 48. De Moivre's formulae amount to two laws, one connecting each coefficient with its predecessors, and one giving the value of each coefficient separately. We can make the laws most intelligible by demonstrating them. We start from a result given by Laplace. He shews, ThSorie . . . des Proh., page 229, that the chance of A for winning precisely at the (n + 2a;)"' game is the coefficient of f^^ in the expansion of l + ^(l-4,g60 i" ( 1 - V(l - 4a6<') I"' \ 2 Ml 2 I where it is supposed that « + & = !. Now the denominator of the above expression is known to be equal to where c = abf ; see Differential Calculm, Chapter ix. 170 . DE MOIVRE. We can thus obtain by the ordinary doctrine of Series, a linear relation between the coefficient of t"*^ and the coefficients of the preceding powers of t, namely, i"*^^, e^-*, ... This is De Moivre's first law; see his page 198. Again ; we may write the above fraction in the form where N N^il + c'N^")' _ 1 + V(l - ^abf) 2 and then by expanding, we obtain a'r {N-^ - {abf)" iV"*' + (dffN-^ -...]. The coefficient of f" in N~* is known to be ^^j^ n (w +a; + 1) (w + a; + 2) ■■■ (w + 2a; - 1) [X see Differential Calculus, Chapter ix. Similarly we get the coefficient of f^" in JV""'", N "^'', and so on. Thus we obtain the coefficient of f*^ in the expansion of the original expression. This is De Moivre's second law ; see his page 199. 305. De Moivre's Problems LX. LXi. LXii. are simple ex- amples formed on Problems LViil. and Lix. They ai-e thus enunciated : LX. Supposing A and B to play together till such time as four Stakes are won or lost on either side ; what must be their proportion of Skill, otherwise what must be their proportion of Chances for wia- ning any one Game assigned, to make it as probable that the Play will be ended in four Games as not ? LXI. Supposing that A and B play till such time as four Stakes are won or lost : What must be their proportion of Skill to make it a Wager of three to one, that the Play will be ended in four Games? LXII. Supposing that A and B play till such time as four Stakes are won or lost; What must be their proportion of Skill to make it an equal Wager that the Play will be ended in six Games 2 DE MOIVEE. 171 306. Problems Lxm. and Lxrv. amount to the general enun- ciation we have given in Art. 303 ; so that the restriction that m and n are equal which was imposed in Problems LViii. and LIX. is now removed. As before De Moivre states, without de- monstration, two general laws, which we will now give. Laplace shews, ThSorie...des Prob. page 228, that the chance of A for winning precisely at the (n + 2a;)*'' game is the coefficient of *"■"* in the expansion of a'f j l + V(l-4c) r f l-V(l-4c) [" ( l + V(l-4c) 1"'^' f l-V(l-4c) 2 JCl — 4c") Let — - be denoted by h ; then the fractional expression which multiplies a"*" becomes by expansion, and striking out 2/j from numerator and denominator, tV^-"- wi(m-l)(m-2) (V^\^ m(w-l)(m-2)(w-3 )(OT-4) l\\'-\^ A2J + 13 \^) ^+ 5 I2J ^ + m™-^--! (OT+»i)(m+M-l)(m+«-2) /n""^-%. (m+«)(^2j + ^ M ^+- We have to arrange the denominator according to powers of t, and to shew that it is equal to where Z= m + n — 2. Now, as in Art. 304, we have f i+V(i-4c) r I i-v(i-4c) r ~ ^ 1.2 [3 c-f..., and the left-hand member is equivalent to 172 DE MOIVRE. Differentiate both sides with respect to t observing that —^ = — alt. Thus, dt Now put »• = Z + 3 ; and we obtain the required result. Thus a linear relation can be obtained between the coefficient^ of successive powers of t. This is De Moivre's first law ; see his page 205. Again ; let .?/"= ^ ^ ; then the original expression, becomes aTfN'' (1 - c^iV^-"") = a^fN-^ (1 - c'"iV^"') (1 - c-^-^JV-*"-^')-'. We may now proceed as in the latter part of Art. 304, to de-i termine the coefficient of f*^. The result will coincide with De Moivre's second law ; see his page 207. 307. Problem LXV. is a particular case of the problem of Duration of Play ; m is now supposed infinite : in other words A has wnUmited capital and we require his chance of ruining B in an assigned number of games. De Moivre solves this problem in two ways. We will here give his first solution with the first of the two examples which ac- company it. Solution. Supposing n to be the number of Stakes which A is to win of B, and n + d the number of Games; let a + 6 be raised to the Power whose Index is M + i^; then if c? be an odd number, take so many Terms of that Power as there are Units in. — ^ — ; take also so many of the Terms next following as have been taken already, but prefix to them in an inverted order, the Coefficients of the preceding Terms. But if d be an even number, take so many Terms of the said Power as there DE MOIVKE. 173 are TJnits in^d+l; then take as many of the Terms next following as there are Units in ^ d, and prefix to them in an inverted order the Coefficients of the preceding Terms, omitting the last of them; and those Terms taken all together will compose the Numerator of a Frac- tion expressing the Probability required, the Denominator of which Fraction ought to be (a + 6)""^''. Example I, Supposing the number of Stakes, which -4 is to win, to be Three, and the given number of Games to be Ten; let a + b be raised to the tenth power, viz. a'°+ lOa'6 + 45a=66 + 120a'6'+ 210a°6*+ 252a=6° + 210a*6« + 120a'6' + iSaab" + lOab' + b'". Then, by reason that n=3, a,ndn + d=lQ, it follows that d is =7, and — ;r— = 4. Wherefore let the Four first Terms of the said Power be taken, viz. a" + lOa'b + 4:5a'bb + l20a''¥, and let the four Terms next following be taken likewise without regard to their Coefficients, "then prefix to them in an invei-ted order, the Coefficients of the preceding Terms : thus the four Terms following with their new Coefficients will be 120a°b* + 4:5a^¥ + 10a*b"+la"b''. Then the Probability which ji has of winning three Stakes of 5 in ten Games or sooner, will be expressed by the following Fraction a'°+ lOa'b + iSa^bb + I20a'b' + UOa'b' + i5a'b'+ lOa'b' + a'^ (a + bf which in the Case of an Equality of Skill between A and B will be A A. 352 11 reduced to ^^ or 32- 308. In De Moivre's solution there is no difficulty in seeing the origin of his first set of terms, but that of the second set of terms is not so immediately obvious. We will take his example, and account for the last four terms. The last term is aW. There is only one way in which B's capital may be exhausted while A wins only three games ; namely, A must win the first three games. The next term is 10a'b\ ■ There are ten ways in which B's capital may be exhausted while A wins only four games. For let there be ten places ; put b in any one of the first three places. 174 DE MOIVRE, and fill up the remaining places with the letters aaaabhhhb in this order ; or put a in any one of the last seven places, and fill up the remaining places with the letters aaahhlhhb in this order ; we thus obtain the ten admissible cases. The next term is 45a^6°. There are forty-five ways in which B's capital may be exhausted while A wias only five games. For let there be ten places. Take any two of the first three places and put h in each, and fill up the remaining places with the letters aaaaahhh in this order. Or take any two of the last seven places and put a in each, and fill up the remaining places with the letters aaahlhhh in this order. Or put h in any one of the first three places and a in any one of the last seven ; and fill up the remaining places with the letters aaaabhbb in this order. On the whole we shall obtain a number equal to the num- ber of combinations of 10 things taken 2 at a time. The following is the general result : suppose we have to arrange r letters a and s letters b, so that in each arrangement there shall be n more of the letters a than of the letters b before we have gone through the arrangement ; then if r is less than s+n the number of different arrangements is the same as the number of combina- tions of r + s things taken r — w at a time. For example, let r=6, s = 4, n = 3; then the number of different arrangements is 10x9 X 8 ,, , ■ .TOf. 1x2x3 ' '^^' '' ^^^- The result which we have here noticed was obtained by Mont- mort, but in a very unsatisfactory manner : see Art. 182. De Moivre's first solution of his Problem LXV. is based on the same principles as Montmort's solution of the general problem of the Duration of Play. 309. De Moivre's second solution of his Problem LXV. con- sists of a formula which he gives without demonstration. Let us return to the expression in Ai-t. 306, and suppose m infinite. Then the chance of A for winning precisely at the (n + 2a:)*'' game is the coefficient of f^^ in the expansion of T+ V(l - 4c) BE MOIVRE. J 175 that is „n>»(>^ + ^+l)( w+a; + 2) (n + 2x-l') ,,.. see Art. 304. ^ ^«^S^ The chance of A for winning at or before the (m + 2a;)'" game is therefore . n(n + x + l) (n+ x + 2)... (n + 2x-l) „ ] + ^ -^aJ^f. Laplace, ThSorie...des Proh.,Tpa,ge 235. 310. De Moivre says with respect to his Problem lxv. In the first attempt that I had ever made towards solving the general Problem of the Duration of Play, which was in the year 1708, I began with the Solution of this lxv"" Problem, well knowing that it might be a Foundation for what I farther wanted, since which time, by a due repetition of it, I solved the main Problem : but as I found afterwards a nearer way to it, I barely published in my first Essay on those matters, what seemed to me most simple and elegant, still pre- serving this Problem by me in order to be published when I should think it proper. De Moivre goes on to speak of the investigations of Montmort and Nicolas Bernoulli, in words which we have aheady quoted ; see Art. 181. 311. Dr L. Oettinger on pages 187, 188 of his work entitled Die Wahrscheinlichkeits-Rechnung, Berlin, 1852, objects to some of the results which are obtained by De Moivi-e and Laplace. Dr Oettinger seems to intimate that in the formula, which we have given at the end of Art. 309, Laplace has omitted to lay down the condition that A has an unlimited capital ; but Laplace has distinctly introduced this condition on his page 234. Again, speaking of De Moivre's solution of his Problem Lxrv. Dr Oettinger says, Er erhalt das namliche unhaltbare Resultat, welches Laplace nach ihm aufstellte. But there is no foundation for this i-emark ; De Moivre and 176 DE MOIVRE. Laplace are correct. The misapprehension may have arisen from reading only a part of De Moivre's page 205, and so assuming a law of a series to hold universally, which he distinctly says breaks off after a certain number of terms. The just reputation of Dr Oettinger renders it necessary for me to notice his criticisms, and to record my dissent from them. 312. De Moivre's Problems Lxvi. and Lxvii. are easy deduc- tions from his preceding results ; they are thus enunciated : LXVI. To find what Probability there is that in a given number of Gaines A may be a winner of a certain number q of Stakes, and at some other time B may likewise be winner of the number p of Stakes, so that both circumstances may happen. LXVII. To find what Probability there is, that in a given number of Games A may win the niunber q of Stakes ; with this farther con- dition, that B during that whole number of Games may never have been winner of the number p of Stakes. 313. De Moivre now proceeds to express his results relating to the Duration of Play in another form. He says, page 215, The Rules hitherto given for the Solution of Problems relating to the Duration of Play are easily practicable, if the number of Games given is but small; but if that number is large, the work will be veiy tedious, and sometimes swell to that degree as to be in some manner impracticable : to remedy which inconvenienoy, I shall here give an Extract of a paper by me produced before the Royal Society, wherein was contained a Method of solving very expeditiously the chief Pro- blems relating to that matter, by the help of a Table of Sines, of which I had before given a hint in the first Edition of my Doctrine ofChcmces, pag. 149, and 150. The paper produced before the Royal Society does not appear to have been published in the Philosophical Transactions; pro- bably we have the substance of it in the Doctrine of Chances. De Moivre proceeds according to the announcement in the above extract, to express his results relating to the Duration of Play by the help of Trigonometrical Tables ; in Problem Lxviii. he supposes the players to have equal skill, and in Problem LXix. he supposes them to have unequal skill. DE MOIVEE. 177 The demonstrations of the formulEe are to be found in the Mis- cellanea Analytica, pages 76 — 83, and in the Doctrine of Chances, pages 230 — 234. De Moivre supposes the players to start with the same number of counters ; but he says on page 83 of the Miscel- lanea Analytica, that solutions similar but somewhat more complex could be given for the case in which the original numbers of counters were different. This has been effected by Laplace in his discussion of the whole problem. 314. De Moivre's own demonstrations depend on his doctrine of Recurring Series ; by this doctrine De Moivre could effect what we should now call the integration of a linear equation in Finite Differences : the equation in this case is that furnished by the first of the two laws which we have explained in Arts. 304, 306. Cer- tain trigonometrical formulae are also required ; see Miscellanea Analytica, page 78. One of these, De Moivre says, constat ex jEquationibus ad circulum vulgo notis ; the following is the pro- perty : in elementary works on Trigonometry we have an expan- sion of cos nO in descending powers of cos 6 ; now cos nO vanishes when n6 is any odd. multiple of ^ , and therefore the equivalent ex- pansion must also vanish. The other formulae which De Moivre uses are in fact deductions from the general theorem which is called De Moivre's property of the Circle; they are as follows ; TT let a = s- J then we have 1 = 2""' sin a sin 3a sin 5a . . . sin (2wa - a) ; also if n be even we have cos n<^ = 2"-' {sin' a - sin'' ^} {sin' 3a - sin' ^} . . . ... {sin' (w - 3) a - sin' ^} {sin' (w - 1) a - sin' ^} : see Pla-ne Trigonometry, Chap, xxiil. De Moivre uses the first of these formulas ; and also a formula which may be deduced from the second by differentiating with respect to ^, and after differentiation putting j> equal to a, or 3a, or 6a, ... 315. De -Moivre applies his results respecting the Duration 12 178 BE MOIVEE. of Play to test the value of an approximation proposed by Mont- mort ; we have already referred to this point ia Art. 184. 316. It remains to trace the history of De Moivre's investi- gations on this subject. The memoir De Mensura Sortis contains the following Pro- blems out of those which appear in the Doctrine of Chances, LViii, LX, LXii, LXiii, the first solution of Lxv, Lxvi. The first edition of the Doctrine of Chances contains all that the third does, except the Problems LXVIII. and LXIX ; these were added in the second edition. As we proceed with our history we shall find that the subject engaged the attention of Lagrange and Laplace, the latter of whom has embodied the researches of his prede- cessors in the ThSorie. . .des Prob. pages 22-5 — 238. 317. With one slight exception noticed in Art. 322, the re- mainder of the Doctrine of Chances was not in the first edition but was added in the second edition. 318. The pages 220—229 of the Doctrine of Chanties, form a digression on a subject, which is one of De Moivre's most valuable contributions to mathema,tics, namely that of Recurring Series. He says, page 220, The Reader may have perceived that the Solution of several Pro- blems relating to Chance depends upon the Summation of Series; I have, as occasion has offered, given the Method of summing them up; but as there are others that may occur, I think it necessary to give a summary View of what is most requisite to be known in this matter; desiring the Reader to excuse me, if I do not give the Demonstrations, which would swell this Tract too much ; especially considering that I have already given them in my Miscellanea Analytica. 319. These pages of the Doctrine of Chances will not present any difficulty to a student who is acquainted with the subject of Recui-ring Series, as it is now explained in works on Algebra ; De Moivre however gives some propositions which are not usually reproduced in the present day. 320. One theorem may be noticed which is enunciated by De Moivre, on his page 224, and also on page 167 of the Miscellanea Analytica. DE MOIVRE. Ji^g The general term of the expansion of (1 - »•)-* in powers of • 2* (P + 1) ■••(?' + w — 1) ^ ^^ r^ »•" ; the sum of the first n terms of the expansion is equivalent to the following expression 1 » n/i \ n(n+l) ,,, ,„ \n+p — 2 This may be easily shewn to be true when w = 1, and then, by induction, it may be shewn to be generally true. For r»+' = r"{l-(l-r)}, so that r-' + («+ l)r-« (1-r) +^?i+ll%±^^- (1 -r-V+ ... = ^"{1 - (1 -r)} + (w+ 1) r" (1 -r){l- (1 -r)} \n + l+p-2 \n[p- (l-ry Thus the additional term obtained by changing n into n + 1 . \n+p—l IS -, — j :f- r" as it should be ; so that if De Moivre's theorem is |_n \p — l true for any value of n, it is true when n is changed into n+1. 321. Another theorem may be noticed ; it is enunciated by De Moivre on his page 229. Having given the scales of relation of two Recurring Series, it is required to iindthe scale of relation of the Series arising from the product of corresponding terms. For example, let M„r" be the general term in the expansion according to powers of r of a proper Algebraical fraction of which the denominator is 1 —fr + gr° ; and let v„a" be the general term in the expansion according to powers of a of a proper Algebraical 12—2 180 DE MOIVRE. fraction of which the denominator is 1-ma+pa'. We have to find the scale of relation of the Series of which the general term is «„«„ (»•«)". We know by the ordinary theory of decomposing Recurring Series into Geometrical Progressions that uy X v^a" = rV (R^p," + B^^) (^ a" + J a")> where p^ and p^ are the reciprocals of the roots of the equation l-fr + gr'^a, and a^ and a^ are the reciprocals of the roots of the equation 1 — ma +pc^ = ; and i?j, R^, A^, A^ are certain constants. Thus v„ = R^A^ {p^ay + R,A, (p,«^" this shews that the required scale of relation will involve four terms besides unity. The four quantities p^a.^, p^a^, p^a^, p^a^ will be the reciprocals of the roots of the equation in s which is found by eliminating r and a from l—Jr+gr^=0, l—ma+pa^ = 0, ra = z; this equation therefore is 1 -fms + {pf + gm^ - 2gp) z^ -fgmpz^ 4-//^* = 0. Thus we have determined the required scale of relation ; for the denominator of the fraction which by expansion produces u„v„ {raf as its general term will be 1 -fmra + {pP+gm^ - '2.gp) rV -fgrnpr'a^+g^p^^a*. De Moiva-e adds, page 229, But it is very observable, that if one of the differential Scales be the Binomial \ — a raised to any Power, it will be sufficient to raise the other differeDtial Scale to that Power, only substituting ar for r, or leaving the Powers of r as they are, if a be restrained to Unity; and that Power of the other differential Scale wiU constitute the differential Scale required. DE MOIVRE. 181 This is very easily demonstrated. For suppose that one scale of relation is (l-a)'; then by forming the product of the cor- responding terms of the two Recurring Series, we obtain for the general term This shews that the general term will be the coefficient of r" in the expansion of (1 - rap;)' ^ (1 - rap^' ^ (1 - rap,y "^ ' ' ' ' and by bringing these fractions to a common denominator, we obtain De Moivre's result. 322. De Moivre applies his theory of Recurring Series to demonstrate his results relating to the Duration of Play, as we have ah-eady intimated in Art. 313 ; and to illustrate still further the use of the theory he takes two other problems respecting play. These problems are thus enunciated : Lxx M and Jf, whose proportioa of Chances to win one Game are respectively as a to h, resolve to play together till one or the other has lost 4 Stakes : two Standers by, Ji and S, concern themselves in the Play, a takes the side of M, and S of JST, and agree betwixt them, that B shall set to S, the sum Z to the sum G on the first Game, 2Z to 2G on the second, 3£ to 3G on the third, AL to iG on the fourth, and in case the Play be not then concluded, 5L to BG on the fifth, and so increasing perpetually in Arithmetic Progression the Sums which they are to set to one another, as long as M and If play ; yet with this farther con- dition, that the Sums, set down by them S and S, shall at the end of each Game be taken up by the Winner, and not left upon the Table to be taken up at once upon the Conclusion of the Play: it is demanded how the Gain of H is to be estimated before the Play begins. Lxxi. If M and Jf, whose number of Chances to win one Game are respectively as a to b, play together till four Stakes are won or lost on either side ; and that at the same time, J? and S whose number of Chances to win one Game are respectively as c to d, play also together tm five Stakes are won or lost on either side; what is the Probability that the Play between M and N will be ended in fewer Games, than the Play between B and S. 182 DE MOIVRE. The particular case of Problem LXXI. in which a = J, and c = d, was given in the first edition of the Doctrine of Chances, page 152. 323. Problems Lxxil. and Lxxiil. are important ; it will be sufficient to enunciate the latter. A and B playing together, and having a different number of Chancea to win one Game, which number of Chances I suppose to be respectively as a to J, engage themselves to a Spectator ^S', that after a certain number of Games is over, A shall give him as many Pieces as he wins Games, over and above 7 n, and £ as many as he wins Games, over and above a+b ' •' the number j n ; to find the Expectation of 3. Problem Lxxii. is a particular case of Problem LXXlil. obtained by supposing a and b to be equal. These two problems first appeared in the Miscellanea Ana- lytica, pages 99 — 101. We there find the following notice respect- ing Problem LXXII : Cum aliquando labente Anno 1721, Vir Clarissimus Alex. Cuming Eq. Au. Regise Soeietatis Socius, qusestionem infra subjeotam mihi proposuisset, solutionem problematis ei postero die tradideram. After giving the solution De Moivre proceeds to Problem Lxxiii. which he thus introduces : Eodem procedendi modo, solutum fuerat Problema sequens ab eodem CI. viro etiam propositum, ejusdem generis ac superius sed multo latius patens. We will give a solution of Problem Lxxiii ; De Moivre in the Doctrine of Chances merely states the result. Let n = c{a + l); consider the expectation of 8 so far as it depends on A. The chance that A will win all the games is a" -. — -T^ , and in this case he gives ch to 8. The chance that A will {a + 0) 1 • noT'^i .... , . , _ „ wm n — 1 games is . .^ , and m this case he gives co — 1 to o. And so on. DE MOIVRE. 183 Thus we have to sum the series d'hc + ««"-& {be - 1) + ^-^:ill ar-^h' (5c - 2) + ... , the series extending so long as the terms in brackets are positive. We have dric - na"~'b = a"~'S {ac - w) = - a""'6 U ; thus the first iwo terms amount to (n-l)a"-'&Jc. Now combine this with !^j — ~i- a"~V2 ; we get (n - 1) oT-^F {ac - n), that is - (« - 1) a'^'^Vhc ; thus the first three terms amount to (.-!)(.- 2) ^„_.^.^^_ This process may be carried on for any number of terms ; and we shall thus obtain for the sum of he terms {n-l){n-2)...(n-hc + l) ^„-^:j..ij^_ oc — 1 This may be expressed thus [3 \bc ao a'"Wachc, which is equivalent to De Moivre's result. The expectation of 8 from B wiU be found to be the same as it is from A. -324. When the chances of A and B for winning a single game are in the proportion of a to 5 we know, from Bernoulli's theorem, that there is a high probability that in a large mimber of trials the number of games won by A and B respectively will be nearly in the ratio of a to h. Accordingly De Moivre passes naturally from his Problem LXXlil. to investigations which in fact amount to what we have called the inverse use of BernouUi's theorem; see Art. 125. De Moivre says, ISiJ! DE MOIVRE. ...I'll take the liberty to say, that this is the hardest Problem that can be proposed on the Subject of Chance, for which reason I have re- served it for the last, but I hope to be forgiven if my Solution is not fitted to the capacity of all Readers; however I shall derive from it some Conclusions that may be of use_ to every body : in order thereto, I shall here translate a Paper of mine which was printed Wovemher 12, 1733, and communicated to some Friends, but never yet made public, reserving to myself the right of enlarging my own Thoughts, as occasion shall require. Then follows a section entitled A Method of approwimating the Svrni of the Terms of the Binomial (a + b)° expanded into a Series, from whence are deduced some practical Rules to estimate the Degree of Assent which is to be given to Experiments. This section occupies pages 243 — 254 of the Doctrine of Chances; we shall find it convenient to postpone our notice of it until we examine the Miscellanea Analytica. 325. De Moivi-e's Problem Lxxrv. is thus enunciated : To find the Probability of throwing a Chance assigned a given number of times without intermission, in any given number of Trials. It was introduced in the second edition, page 243, in the fol- lowing terms : When I was just concluding this Work, the following Problem was mentioned to me as very diflS.cult, for which reason I have considered it with a particular attention. De Moivre does not demonstrate his results for this problem ; we will solve the problem in the modem way. Let a denote the chance for the event in a single trial, h the chance against, it ; let n be the number of trials, p the number of times without intermission for which the event is required to hap- pen. We shall speak of this as a rwn of p. Let M„ denote the probability of having the required run of p in n trials ; then M«+i = w„+ (1 -«„_;,) ha^: for in w + 1 trials we have all the favourable cases which we bave in n trials, and some more, namely those in which after having failed in n —p trials, we fail in the (w —p -(- 1)"» trial, and then have a run of p. DE MOIVEK. 185 Let w„ = 1 — «„, and substitute in the equation ; thus The Generating Function of v„ will therefore be where ^ (t)- is an arbitrary function of ^ which involves no powers of t higher than f. The Generating Function of m„ is therefore 1 it) . 1-t 1-t + ba^f^" we may denote this by ^(0 0--t)(l-t + ba''f^')' where yfr (t) is an arbitrary function of t which involves no powers of t higher than f^\ Now it is obvious that m„ = if n be less than J), also u^ = (f, and m^^j = a* + baF. _ Hence we find that so that the Generating Function of m„ is aFf{l-at) (l-t)(l-t + baFr')- The coefficient of f in the expansion of this function will therefore be obtained by expanding a^ (1 - at) l-t + ba^r" and taking the coefficients of all the powers of t up to that of f "* inclusive. It may be shewn that De Moivre's result agrees with this after allowing for a slight mistake. He says we must divide unity by 1 - £c - ax" - aV - . . . - a*''^x^, take n-p+l terms of the series, multiply by ^^ , and finally put a; = ^-^ . The mistake here 186 DE MOIVRE. is that In the series 1 -a- aa5'-aV- ... -a^'V instead of a we ought to read ^ . De Moivre is correct in an example which he gives on his page 255. Let t = c, then according to De Moivre's rule corrected we have to expand 1 aF +Vi t ■ 1 — ca; a** , l-c^a^ (^+67' ^ ^^■l-x(l + c)+c^£c^^ (« + &)*■ \ — x ^j 1 — ca; This will be seen to agree with our result remembering that we took a + & = 1. De Moivre himself on his page 256 practically gives this form to his result by putting \ — c^ of 1 — 03 —^ for \—x—c7? — c^x^ — ... — cT^ai'. \ — cx De Moivre gives without demonstration on his page 259 an approximate rule for determining the number of trials which must be made in order to render the chance of a run of ja equal to one half. De Moivre's Problem Lxxiv. has been extended by Condorcet, Essai...de ^Analyse... pages 73 — 86, and by Laplace, Theorie...des Prob. pages 247—253. 326. De Moivre's pages 261 — 328 are devoted to Annuities on Lives; an Appendix finishes the book, occupying pages 329 — 348 : this also relates principally to annuities, but it contains a few notes on the subject of Probability. As we have already stated in Art. 53, we do not profess to give an account of the investigations relating to mortality and life insurance. We may remark that there is an Italian translation of De Moivre's treatise on Annuities, with notes and additions ; the title is La Dottrina degli Azzardi...de Abramo Moivre: Trasportata daW Idioma Inglese,...dal Padre Don Roberto Gaeta...sotto Vassis- tenza del Padre Don Gregorio Fontana...In Milano 1776. This translation does not discuss the general Theory of Probability, but only annuities on lives and similar subjects. DE MOIVEE. 187 In the Advertisement to the second edition of the Doctrine of Ghanoes, page xiii, De Moivre says, There is in the World a Gentlemaa of an older Date, who in the year 1726 did assure the Public that he could calculate the Values of Lives if he would, but that he would not, . . . De Moivre proceeds tp make some sarcastic remarks ; a manu- script note in my copy says that the person here meant was "John Smart of Guildhall, who in that year published Tables of Interest, Discount, Annuities, &c. 4to." 327. We have now to notice De Moivre's work entitled Mis- cellanea Analytica de Serielms et Quadraturis... J^OTiAon., 1730. This is a quarto volume containing 250 pages, a page of Errata, a Supplement of 22 pages, and two additional pages of Errata; besides the title page, dedication, preface, index, and list of sub- scribers to the work. We have already had occasion to refer to the Miscellanea Analytica as supplying matter bearing on our subject; we now however proceed to examine a section of the work which is entirely devoted to controversy between Montmort and De Moivre. This section is entitled Responsio ad quasdam Criminationes ; it occu- pies pages 146 — 229, and is divided into seven Chapters. 328. In the first Chapter the design of the section is ex- plained. De Moivre relates the history of the publication of Montmort's first edition, of the memoir DeMensura Sortis, and of Montmort's second edition. De Moivre sent a copy of the De Men^ura Sortis to Montmort, who gave his opinion of the memoir in a letter to Nicolas Bernoulli, which was published in the second edition of Montmort's book; see Art. 221. De Moivre states briefly the animadversions of Montmort, distributing them under nine heads. The publication of Montmort's second edition however does not seem to have produced any quarrel between him and De Moivre; the latter returned his thanks for the present of a copy of the work, and after this a frequent interchange of letters took place between the two mathematicians. In 1715 Montmort visited England, and was introduced to Newton and other dis- 18S DE MOIVKE. linguished men ; he was also admitted as a member of the Koyal Society. De Moivre sent to Montmort a copy of the Boctrine of Chances when it was published, and about two years afterwards Montmort died. De Moivre quotes the words of Fontenelle which we have already given in Art. 136, and intimates that these words induced him to undertake a comparison between his own labours and those of Montmort, in order to vindicate his own claims. As the Boctrine of Chances was written in English it was not readily accessible to all who would take an interest in the dispute; and this led De Moivi-e to devote a section to the subject in his Mis- cellanea Analytica. 329. The second Chapter of the Responsio... is entitled Be Methodo Bifferentiarum, in qua exhibetur Solutio Stirlingiana de media Coefficiente Binomii. The general object is to shew that in the summation of series De Moivre had no need for any of Montmort's investigations. De Moivre begins by referring to a certain theorem which we have noticed in Ajrt. 152; he gives some examples of the use of this theorem. He also adverts to other methods of summation. Montmort had arrived at a very general result in the summa- tion of series. Suppose mV* to denote the n^ term of a series, where m„ is such that A^m^ is zero, m being any positive integer ; then Montmort had succeeded in summing any assigned number of terms of the series. De Moivre shews that the result can be easily obtained by the method of Differences, that is by the method which we have explained in Art. 151. The investigations by Montmort on the summation' of series to which De Moivre refers were published in Vol. xxx. of the Philo- sophical Transactions, 1717. This Chapter of the Responsio... gives some interesting details respecting Stirling's Theorem including a letter from Stirling himself. 330. The third Chapter of the Responsio... is entitled Be Me- thodo Combinationum; the fourth Be Permutationibtos ; the fifth Combinationes et Permutationes ulterius consideratce: these Chap- DE MOIVRE. 189 ters consist substantially of translations of portions of the Doctrine of Chances, and so do not call for any remark. The sixth Chapter is entitled Be Nwmero Punctorwm in Tesseris; it relates entirely to the formula of which we have given the history in Art. 149. 331. The seventh Chapter of the Responsio. . . is entitled Solu- tiones variorum Problematum ad Sortem spectantiuvi. This Chapter gives the solutions of nine problems in Chances. The first eight of these are in the Doctrine of Chances ; nothing of importance is added in the Miscellanea Analytica, except in two cases. The first of these additions is of some historical interest. Suppose we take an example of the Binomial Theorem, as {p + qf ; one term will be 'iSp^ct- ^^^^ De Moivre says, page 218, ...at fortasse nesciveram hujus termini coefficientem, nimiruni 28, designaturam numerum permutationum quas literse p, p, p, p, p, p, q, q, productum ^° g" constituentes Subire possintj imm5 vero, hoc jam diu mihi erat exploratum, etenim ego fortasse primus omnium detexi co- efficientes annexas productis Binomii, vel Multinomii cujuscunque, id denotare quotenis vaiiationibus literse producti positiones suas inter se permutent: sed utrum illud facile fuerit ad inveniendum, postquam lex coefficientium ex productis continuis -zr x — =— x — = — x — - — he. 1. Z O 4: jam perspecta esset, aut quisquam ante me hoc ipsum detexerit, ad rem praesentem non magni interest, cum id monere suffecerit banc proprie- tatem CoeflB.cientium a me assertam fuisse et demoustratam in Actis Phi- losopJiicis Anno 1697 impressis. The second addition relates to Problem XLix. of the Doctrine of Chances; some easy details relating to a maximum value are not given there which may be found in the Miscellanea Analytica, pages 223, 224. 332. The ninth problem in the seventh Chapter of the Re- sponsio ... is to find the ratio of the sum of the largest p terms in the expansion of (1 + 1)" to the sum of all the terms ; p being an odd number and n an even number. De Moivre expresses this ratio in terms of the chances of certain events, for which chances he had already obtained formulae. This mode of ex- pressing the ratio is not given in the Doctrine of Chances, being rendered unnecessary by the application of Stirling's Theorem ; 190 DE MOIVRB. but it involves an interesting fact in approximation, and we will therefore explain it. Suppose two players A and B of equal skill ; let A have an infinite number of counters, and B have the number p. Let {n,p) denote the chance that B will be ruined in n games. Then the required ratio is 1 — («, p) ; this follows from the first form of solution of Problem LXV; see Art. 307. Again, suppose that each of the players starts with p counters ; and let ■yjr [n, p) then denote the chance that B will be ruined in n games ; similarly if each starts with 3p counters let ■yfr (n, 3p) denote the chance that £ wiU be ruined in n games ; and so on. Then De Moivre says ' that approximately (n, p) = 'f (n, p)+y}r (n, Sp), and still more approximately ^ {n,p) = yfr (n,p) + i/r (n, Sp) - i|r (n, 5p) + i/r (n, 7p). The closeness of the approximation will depend on n being large, and p being only a moderate fraction of n. These results follow from the formula given on pages 199 and 210 of the Doctrine of Chances... The second term of •^ {n, p) is negative, and is numerically equal to the first term of t/t (m, 3p), and so is cancelled ; similarly the third term of i|c (m, p) is cancelled by the first of — "<|r (n, 5p), and the fourth term of ■<^ {n, p) by the first of ^fr (n, 7p). The terms which do not mutually cancel, and which we therefore neglect, involve fewer factors than that which we retain, and are thus com- paratively small. 333. We now proceed to notice the Supplement to the Mis- cellanea Analytica. The investigations of problems in Chances had led mathematicians to consider the approximate calculation of the coefficients in the Binomial Theorem ; and as we shall now see, the consequence was the discovery of one of the most striking results in mathematics. The Supplement commences thus : Aliquot post diebiia quam Liber qui inscribitur, Miscdlomea Avjaly- tica, in lucem prodiisset, Doctissimus StirlingiiLS me Uteris admonuit Tabulam ibi a me exhibitam de summis Logarithmorum, non satis au- toritatis habere ad ea fiimauda qnse in speculatione niterentur, utpote DE MOIVRE. 191 cui Tabulae subesset error perpetuus in quinta quaque figura decimal! auminarum : quae cum pro humanitate sua monuisset, his subjunxit seriem celerrime convergentem, cujus ope summse logarithmorum tot numerorum naturalium quot quis sumere voluerit obtineri possent ; res autem sic exposita fuerat. Then follows a Theorem which is not quite coincident in form with what we now usually call Stirling's Theorem, but is practically equivalent to it. De Moivre gives his own investiga- tion of the subject, and arrives at the following result : log 2 + log 3 + log 4 + ... + log (m - 1) = (m - 2) logm -m + ^2^ - 3g^3 + 1260^5- i68(W + ... , 1111 12^360 1260^ 1680 •■■ With respect to the series in the last line, De Moivre says on page 9, of the Supplement to the Miscellanea Analytica... quae satis commode convergit in principio, post terminos quinque pri- mes convergentiam amittit, quam tamen postea recuperat... The last four words involve an error, for the series is divergent, as we know from the nature of Bernoulli's Numbers. But De Moivre by using a result which Stirling had already obtained, 111 arrived at the conclusion that the series 1 ~ Tg + ggo ~" 1260 ^ ' " is equal to 5 log 2^ ; and thus the theorem is deduced which we now call Stirling's Theorem. See Miscellanea Analytica, page 170, Supplement, page 10. 834. De Moivre proceeds in the Supplement to the Miscellanea Analytica to obtain an approximate value of the middle coefficient of a Binomial expansion, that is of the expression (wz + l) (ot+2)... 2ot m (m — 1) ... 1 He expends nearly two pages in arriving at the result, which 192 DE MOIVRE. he might have obtained immediately by putting the proposed ex- |2m pression in the equivalent form ■, — ■. — . De Moivre then gives the general theorem for the approximate summation of the series 1 1 1 1 m" "*" (jiH- 1/ ■*■ (« + 2)" "^ (w + 3)° "*■ • " ' We have already noticed his use of a particular case of this summation in Art. 276. De Moivre does not demonstrate the theorem ; it is of course included in the wellknown result to which Euler's name is usually attached, See Novi Comm....Petrop. Vol. xiv. part 1, page 137; 1770. The theorem however is also to be found in Maclaurin's Treatise of Fluodons, 1742, page 673. 335. We return to the Doctrine of Chances, to notice what is given in its pages 243 — 254 ; see Art. 324. In these pages De Moivre begins by adverting to the theorem obtained by Stirling and himself. He deduces from this the following result : suppose w to be a very large number, then the /I IN" logarithm of the ratio which a term of (H+n) > distant from ^.2 ' 2J the middle term by the interval I, bears to the middle term, 2r is approximately . This enables him to obtain an approximate value of the sum of the I terms which immediately precede or follow the middle term. Hence he can estimate the numerical values of certain chances. For example, let n = 3600 : then, supposing that it is an even chance for the happening or failing of an event in a single trial, De Moivre finds that the chance is -682688 that in 3600 trials, the number of times in which the event happens, will lie between 1800 + 30 and 1800-30. DE MOIVRK. 193 Thus by the aid of Stirling's Theorem the value of Bernoulli's Theorem is largely increased. De Moiyre adverts to the controversy between Nicolas Ber- noulli and Dr Arbuthnot, respecting the inferences to be drawn from the observed fact of the nearly constant ratio of the number of births of boys to the number of births of girls ; see Art. 223. De Moivre shews that Nicolas Bernoulli's remarks were not re- levant to the argument really advanced by Dr Arbuthnot. 336. Thus we have seen that the principal contributions to our subject from De Moivre are his investigations respecting the Duration of Play, his Theory of Kecurring Series, and his extension of the value of Bernoulli's Theorem by the aid of Stirling's Theorem. Our obligations to De Moivre would have been still gi-eater if he had not concealed the demonstrations of the important results which we have noticed in Art. 306 ; but it will not be doubted that the Theory of Probability owes more to him than to any other mathematician, with the sole exception of Laplace. 13 CHAPTER X. MISCELLANEOUS INVESTIGATIONS Between the years 1700 anb 1750. 337. The present Chapter will contain notices of various con- tributions to our subject whicb were made between the years 1700 and 1750. 338. The first work which claims our attention is the essay by Nicolas Bernoulli, to which we have already alluded in Art. 72 ; it is entitled Specimina Artis conjectandi, ad quoestiones Juris ap- plicatce. This is stated to have been published at Basle in 1709; see Gouraud, page 36. It is reprinted in the fourth volume of the Act. Eruditorwm... Supplementa, 1711, where it occupies pages 159 — 170. Allusion is made to the essay in the volume which we have cited in Art. 59, pages 842, 844, 846. 339. In this essay Nicolas Bernoulli professes to apply mathe- matical calculations to various questions, principally relating to the probability of human life. He takes for a foundation some facts which his uncle James had deduced from the comparison of bills of mortality, namely that out of 100 infants born at the same time 64 are alive at the end of the sixth year, 40 at the end of the sixteenth year, and so on. Nicolas Bernoulli considers the following questions : the time at the end of which an absent man of whom no tidings had been received might be considered as dead ; the NICOLAS BEENOTJLLI. 195 value of an annuity on a life ; the sum to be paid to assure to a child just bom an assigned sum on his attaining a certain age ; marine assurances ; and a lottery problem. He also touches on the probability of testimony ; and on the probability of the innocence of an accused person. The essay does not give occasion for the display of that mathe- matical power which its author possessed, and which we have seen was called forth in his correspondence with Montmort ; but it indi- cates boldness, originality, and strong faith in the value and extent of the applications which might be made of the Theory of Pro- bability. We will take two examples from the Essay. 340. Suppose there are h men who wiU all die within a years, and are equally Ukely to die at any instant within this time : re- quired the probable duration of the life of the last survivor. Nicolas Bernoulli really views the problem as equivalent to the following : A line of length a is measured from a fixed origin ; on this hue h points are taken at random : determine the mean dis- tance from the origin of the most distant point. Let the line a be supposed divided into an indefinitely large number n of equal parts ; let each part be equal to c, so that nc = a. Suppose that each of the i points may be at the distance c, or 2c, or Sc, ...up to nc; but no two or more at exactly the same distance. Then the whole number of cases will be the number of combi- nations of n things taken 5 at a time, say ^ (w, h). Suppose that the most distant point is at the distance xc ; then the number of ways in which this can happen is the number of ways in which the remaining 5—1 points can be put nearer to the origin ; that is, the number of combinations of a; - 1 things, taken 6 — 1 at a time, say (x— 1, b — 1). Hence the required mean distance is 'txc (x— 1 , 5 — 1) ^ (n, h) where the summation extends from x = htox = n. 13—2 196 BAEBEYRAC. AEBUTHNOT. TlCu It is easily seen that the limit, when n is infinite, is , ^ , that ab IS -= — - . b + 1 The above is substantially the method of Nicolas Bernoulli. 341. Nicolas BernoulU has a veiy curious mode of estimating the probability of innocence of an accused person. He assumes that any single evidence against the accused person is twice as likely to be false as true. Suppose we denote by m„ the probability of innocence when there are n different evidences against him ; there are two chances out of three that the n*^ evidence is false, and then the accused prisoner is reduced to the state in which there are n — 1 evidences against him ; and there is one chance out of three that the evidence is true and his innocence therefore impos- sible. Thus _ 2u^, + 2 Hence ^„=(|).. This is not the notation of Nicolas ; but it is his method and result. 342. In the correspondence between Montmort and Nicolas Bernoulli allusion was made to a work by Barbeyrac, entitled Traits du Jeu; see Art. 212. I have not seen the book myself. It appears to be a dissertation to shew that religion and morality do not prohibit the use of games in general, or of games of chance in particular. It is stated that there are two editions of the work, published respectively ia 1709 and 1744. Barbeyrac is also said to have published a discourse Bur la nature du Sort. See the English Cyclopcedia, and the Biographie Universelle, under the head Barbeyrac. 343. We have next to notice a memoir by Arbuthnot to whom we have already assigned an elementary work on our subject; see Art. 79. The memoir is entitled An Argument for Divine Providence, ARBUTHNOT. 1917 taken from the constant Regularity observed in the Births of both Seoces. By Br John Arbuthnott, Physitian in Ordinary to Her Majesty, and Fellow of the College of Physitians- and the Royal Society. This memoir is published in Vol. xxvii. of the Philosophical Transactions; it is the volume for 1710, 1711 and 1712: the memoir occupies pages 186 — 190. 344. The memoir begins thus : Among innumerable Footsteps of Divine Providence to be found in the Works of Nature, there is a very remarkable one to be observed in the exact BaJlance that is maintained, between the Numbers of Men and Women; for by this means it is provided, that the Species may never fail, nor perish, since every Male may have its Female, and of a proportion- able Age. This Equality of Males and Females is not the Effect of Chance but Divine Providence, working for a good End, which I thus demonstrate : 345. The registers of births in London for 82 years are given ; these shew that in every year more males were born than females- There is very little relating to the theory of probability in the memoir. The principal point is the following. Assume that it is an even chance whether a male or female be bom ; then the chance that in a given year there will be more males than females is ^ ; and the chance that this wiU happen for 82 years in succession is ■^. This chance is so small that we may conclude that it is not ati even chance whether a male or female be bom. 346. The memoir attracted the attention of Nicolas Bernoulli, who in his coiTespondence with Montmort expressed his dissent from Arbuthnot's argument ; see Art. 223. There is also a letter from Nicolas Bernoulli to Leibnitz on the subject ; see page 989 of the work cited in Art. 59. De Moivre repHed to Nicolas Bernoulli, as we have already intimated in Art, 335. 347. The subject is also discussed in the Oemres Phih- sophiques et MathSmatiques of 'sGravesande, published at Amster- dam, 1774, 2 vols. 4to. The discussion occupies pages 221—248 of the second volume. 198 'SGKAVESANDE. It appears from page 237, that when Nicolas Bernoulli travelled in Holland he met 'sGravesande. In this discussion we have first a memoir by 'sGravesande. This memoir contains a brief statement of some of the elements of the theory of probability. The following result is then obtained. Assume that the chance is even for a male or female birth, and find the chance that out of 11429 births the males shall lie between 5745 and 6128. By a laborious arithmetical calculation this is found to be about -r . Then the chance that this should 4 happen for 82 years in succession will be j^ . But in fact the event for which the chance is so small had happened in London. Hence it is inferred that it is not an even chance that a male or female should be bom. It appears that 'sGravesande wrote to Nicolas Bernoulli on the subject ; the reply of Nicolas Bernoulli is given. This reply contains a proof of the famous theorem of James Bernoulli ; the proof is substantially the same as that given by Nicolas Ber- noulli to Montmort, and published by the latter in pages 389 — 393 of his book. Then 'sGravesande wrote a letter giving a very clear account of his views, and, as his editor remarks, the letter seems to have impressed Nicolas Bernoulli, judging from the reply which the latter made. Nicolas Bernoulli thus sums up the controversy : Mr. Arhutknot fait consister son argument en deux choses; 1°. en ce que, supposge une ggaliti de naissance entre les filles et les gargons, il y a peu de probability que le nombre des gargons et des filles se trouve dans des limites fort proches de I'egalite : 2°. qu'il y a peu de proba- bility que le nombre des gargons surpassera un grand nombre de fois de suite le nombre des filles. C'est la premiere partie que je refute, et non pas la seconde. But this does not fairly represent Arbuthnot's argument. Nicolas Bernoulli seems to have imagined, without any adequate reason, that the theorem known by his uncle's name was in some way contradicted by Arbuthnot. 348. Two memoirs on our subject are published in Vol. BROWNE. jgg XXIX. of the Philosophical Transactions, whicli is the volume for 1714, 1715, 1716 the memoirs occupy pages 133 — 158. They are entitled Solwtio Oeneralis Problematis xv. propositi ct D. de Moivre in tractatu de Mensura Sortis...Solulio generalis altera prcece- dentis Problematis, ope Combinationum et Seriervm injmitarum.... These memoirs relate to the problem which we have called Waldegrave's ; see Art. 211. The first memoir is by Nicolas Bernoulli; it gives substantially the same solution as he sent to Montmort, and which was printed in pages 381 — 387 of Montmort's work. The second memoir is by De Moivre; it gives the solution which was reproduced in the Doctrine of Chances. 349. We have next to notice a work which appeared under the following title : Christiani Hugenii Libellus de Eatiociniis in Ludo Alese. Or, the value of all chances in games of fortune; cards, dice, wagers, lotteries, &c. mathematically demonstrated. London : Printed by S. Keimer, for T. Woodward, near the Inner Temple-Gate in Fleet-street. 1714. This is a translation of Huygens's treatise, by W. Browne. It is in small octavo size; it contains a Dedication to Dr Richard Mead, an Advertisement to the Reader, and then 24 pages, which comprise the translation. The dedication commences thus : Honoured Sir, When I consider the Subject of the following Papers, I can no more forbear dedicating them to Your Name, than I can refuse giving my assent to any one Proposition in these Sciences, which I have already seen clearly demonstrated. The Reason is plain, for as You have contributed the greatest Lustre and Glory to a very consider- able part of the Mathematicks, by introducing them into their noblest Province, the Theory of Physick ; the Publisher of any Truths of that Nature, who is desirous of seeing them come to their utmost Perfection, must of course beg Your Patronage and Application of them. By so prudent a Course as this, he may perhaps see those Propositions which it was his utmost Ambition to make capable only of directing Men in the Management of their Purses, and instructing them to what Chances and Hazards they might safely commit their Money ; tum'd some time or other to a much more glorious End, and made instrumental likewise towards the securing their Bodies from the Tricks of that too successful 200 MAIEAN. Sharper, Death, and counterminLng the tuiderhand Dealings of secret and overreaching Distempers. In his Advertisement to tlie Reader, Browne refers to a trans- lation of Huygens's treatise which had been made by Axbuthnot ; he also notices the labours of Montmort and De Moivi'e. He says further, My Design in publishing this Edition, was to have made it as useful as possible, by an addition of a very large Appendix to it, containing a Solution of some of the most serviceable and intricate Problems I cou'd think of, and such as have not as yet, that I know of, met with a par- ticular Consideration: But an Information I have within these few Days receiv'd, that M. Montmort's French Piece is just newly reprinted at Paris, with very considerable Additions, has made me put a Stop to the Appendix, tUl I can procure a Sight of what has been added anew, for fear some part of it may possibly have been honour'd with the Notice and Consideration of that ingenious Author. I do not know whether this proposed Appendix ever ap- peared. 350. In the Hist, de I' Acad.... Paris for 1728, which was published in 1730, there is a notice respecting some results ob- tained by Mairan, Sur le Jeu de Pair ou Won. The notice occupies pages 53 — 57 of the volume; it is not by Mairan himself Suppose a heap of counters ; a person takes a number of them at random, and asks another person to guess whether the number is odd or even. Mairan says that the number is more likely to be odd than even ; and he argues in the following way. Sup- pose the number in the heap to be an odd number, for example 7; then a person who takes from the heap may take 1, or 2, or 3, ... or 7 counters ; thus there are 7 cases, namely 4 in which he takes an odd number, and 3 in which he takes an even number. The advantage then is in favour of his having taken an odd number. If the number in the heap be an even number, then the person who takes from it is as likely to take an even number as an odd number. Thus on the whole Mairan concludes that the guess should be given for an odd number. The modern view of this problem is different from Mairan's. NICOLE. 201' If tlie original heap contains n counters we should say that there are n ways of drawing one counter, ^^^ ^^y^ ^f drawing two counters, and so on. Mairan notices this view but con- demns it. Laplace treated this problem in the MSmoires . . . pwr divers Savans... Tome Vl., Paris, 1774, and he arrives at the ordinary result, though not by the method of combinations ; he refers to Mairan's result, and briefly records his dissent. The problem is solved by the method of combinations in the Thiorie...des Prob. page 201. In the article Pair ou Non of the original French Encyclo- pedie, which was published in 1765, Mairan's view is given ; this article was repeated in the Encyclopedie M^thodique, in 1785, without any notice of Laplace's dissent. 351. On page 68 of the volume of the Hist, de I'Acad.... Paris, which contains Mairan's results, is the following paragraph : M. L'Abbg Sauveur, fils de feu M. Sauveur Acad6mioien, a fait voir une Methode qu'il a trouvie pour dgterminer au Jeu de Quadrille quelle est la probability de gagner sans prendi-e plusieurs Jeux diffgreats, dont il a calcul6 une Table. On a trouve que la matigre 6pineuse et dglicate des Combinaisons €toit trgs-bien entendiie dans cet ouyrage. 352. "We have next to notice a memoir by Nicole, entitled Examen et Resolution de quelques questimis sur les Jewc. This memoir is published in the volume for 1730 of the Hist, de V Acad.... Paris; the date of publication is 1732 : the memoir occupies pages 45 — 56 of the part devoted to memoirs. The problem discussed is really the Problem of Points ; the method is very laborious, and the memoir seems quite superfluous since the results had already been given in a simpler manner by Montmort and De Moivre. One point may be noticed. Let a and h be proportional to the respective chances of A and B to win a single game ; let them play for an even number of games, say for example 8, and let 8 be the sum which each stakes. Then As advantage is ^ «° + %a'h + 28a°5' + 56a°y - 56g°&° - ^Mb" - 8aV - h" 202 NICOLE. This supposes that if each wins four games, neither receives nor loses any thing. Now it is obvious that the numerator of the expression is divisible by a + b; thus we may simplify the ex- pression to p a' + 7a^b + 21a'F + 35aV - 35a%* - IWF - 7a5° - V ^ {a + hy This is precisely the expression we should have if the players had agreed to play seven games instead of eight. Nicole notices this circumstance, and is content with indicating that it is not unreasonable ; we may shew without difficulty that the result is universally tme. Suppose that when A and B agree to play 2« — 1 games, p^ is the chance that A beats B by just one game, p^ the chance that A beats B by two or more games ; and let ffij 9.2 ^® similar quantities with respect to B, then ^'s advantage is S {p^+ p^ — q.^ — q^. Now consider 2w games : A's chance of beating B by two or more games, _is p^ + "' , ; B's chance of beating A by two or more games is q^ + " . Hence ^'s ad- vantage is V^»^a+6 ^^ a+b) Now we know that ^ = ^ = /i say; therefore a + b a + b ^^ i fi ii Hence the advantage of A for 2n games is the same as for 2n — 1 games. 353. In the same volume of the Hist, de V Acad.... Paris, on pages 331 — 344, there is another memoir by Nicole, entitled Mithode pour dSterminer le sort de tant de Joileurs que Von voudra, et Vavantage que les uns ont sur les autres, lorsqu'ils joiient h qui gagnera le plus de parties dans un nombre de parties d^termin^. This is the Problem of Points in the case of any number of players, supposing that each player wants the same number of BUFFON. 203 pointe. Nicole begins in a laborious way ; but he sees that the chances of the players are represented by the terms in the ex- pansion of a certain multinomial, and thus he is enabled to give a general rule. Suppose for example that there are three players, whose chances for a single game are a, b, c. Let them play a set of three games. Then the chance that A has of winning the whole stake is a' + Sa" {b + cj ; and similar expressions give the chances of B and G; there is also the chance 6abc that the three players should each win one game, and thus no one prevail over the others. Similarly, if they play four games, ^'s chance of winning the whole stake is a* + 4a' (& + c) + 12a^Jc; there is also the chance Qa'F that A and B should share the stake between them to the exclusion of C ; and so on. But all that Nicole gives was already weU known; see Montmort's page 353, and De Moivre's Miscellanea Analytica, page 210. 354. In the year 1733 Buffon communicated to the Academy of Sciences at Paris the solution of some problems in chances. See Hist, de V Acad.... Paris for 1733, pages 43 — 45, for a brief account of them. The solutions are given in Buffon's Essai d'Ariihm^tique Morale, and we shall notice them in speaking of that work. 355. We now return to the work entitled Of the Laws of Chance, the second part of which we left for examination until after an account had been given of De Moivre's works; see Arts. 78, 88. According to the title page this second part is to be attributed to John Ham. Although De Moivre is never named, I think the greater part of Ham's additions are taken from De Moivre. Ham considers the game of Pharaon in his pages 53 — 73. This I think is all taken from De Moivre. Ham gives the same in- troductory problem as De Moivre; namely the problem which is XI. in De Moivre's first edition, and x. in his third edition. In pages 74—94 we have some examples relating to the game of Ace of Hearts, or Fair Chance, and to Lotteries. Here we 204 HAM. have frequent use made of De Moivre's results as to the nuniher of trials in which it is an even chance that an event mil happen once, or happen twice ; see Art. 264. 356. There is however an addition given without demon- stration, to De Moivre's results, which deserves notice. De Moivre made the problem of finding the number of trials in which it is an even chance that an event will occur twice depend on the following equation : (l + ^J = 2(l+.). 1^ If we suppose g- infinite this reduces to « = log 2 + log (1 + s) ; from which De Moivre obtained z = 1-678 approximately. But let us not suppose g infinite ; put f 1 + -j =6°; so that our equation becomes e«=2(l+«). Assume z = 2 —y, thus e'^'^=6-2y. Assume 2c = 7 + s where & = 6. Thus, e'-" =1- -y. Take the logarithms of both sides, then 18 ^ ~ 81 1 1 2 1 3 s-cy = -^y-~f-^y -, that is '"3' - 13 .y" - 8][ / - ••• = » ; where r = c— ^. Hence by reversion of series we obtain = * + J_ /'*Y 4. 1 + 2'' /iV HAM. 205 This is Ham's formula, given as we have said without de- monstration. Since we assumed we have 7 = Napierian log of 6 = 1-791759 ; thus s = 2c -7= 2c- 1-791759. Ham says that this series will determine the value of z in all cases when q is greater than 4-1473. This limit is doubtless obtained by making 2c - 7 = 0, which leads to f 1 + -) = VG ; and this can be solved by trial. But Ham seems to be un- necessarily scrupulous here ; for if 2c be less than 7 we shall still have - mrnierically less than wnity, so long as 7 — 2c is less than c — K . that is so long as c is greater than ? + 77 . o 3 9 357. The work finishes -with some statements of the nu- merical value of certain chances at Hazard and Backgammon. 358. We have next to notice a work entitled Galcul du Jeu appeUi par les Frangois le trente-et-quarante, et que Von nonvme d Florence le trente-et-wn.... Par Mr D. M. Florence, 1739. This is a volume in quarto. The title, notice to the reader, and preface occupy eight pages, and then the text follows on pages 1 — 90. The game considered is the following : Take a common pack of cards, and reject the eights, the nines, and the terhs, so that forty cards remain. Each of the picture cards counts for ten, and each of the other cards counts for its usual number. The cards are turned up singly until the number* formed by the sum of the values of the cards falls between 31 and 40, both inclusive. The problem is to determine the chances in favour of each of the numbers between 31 and 40 inclusive. The problem is solved by examining all the cases which can occur, and counting up the number of ways. The operation is most laborious, and the work is perhaps the most conspicuous 206 SIMPSON. example of misdirected industry which the literature of Games of Chance can furnish. The author seems to refer on page 80 to another work which I have not seen. He says, ...j'en ai d^ja fait la demonstration dans mon Calcul de la Loterie de Eome,... It will be observed from our description of the game that it does not coincide with that which has been called in more recent times by the same name. See Poisson's memoir in Ger- gonne's Annales de Mathematiques, Vol. 16. 359. A treatise on the subject of Chances was published by the eminent Thomas Simpson, Professor of Mathematics at the Royal Military Academy, Woolwich. Simpson was bom in 1710, and died in 1761 ; an account of his life and writings is prefixed to an edition of his Select Exercises for Young Proficients in the Mathematicks, by Charles Hutton. Simpson's work is entitled The Nature and Laws of Chance. . . The whole after a new, general, and conspicuous Manner, and illustrated with a great variety of Exawjples ... 1740. Simpson implies in his preface that his design was to produce an introduction to the subject less expensive and less abstruse than De Moivre's work ; and in fact Simpson's work may be con- sidered as an abridgement of De Moivre's. Simpson's problems are nearly all taken from De Moivre, and the mode of treatment is substantially the same. The very small amount of new matter which is contributed by a writer of such high power as Simpson shews how closely De Moivre had examined the subject so far as it was accessible to the mathematical resources of the period. We will point out what we find new in Simpson. He divides his work into thirty Problems. 360. Simpson's Problem VI. is as follows : There is a given Number of each, of several sorts of Things, (of the same Shape and Size); as (a) of the first Sort, (6) of the second, etc. put promiscuously together; out of which a given Number (m) is to be taken, as it happens : To find the Probability that there shall come out precisely a given Number of each sort, as (p) of the first, (g) of the second, (r) of the third, &o. SIMPSON. 207 The result in modem notation is a fraction of which the nume- rator is — 1^— X— ii— X— k_x \p\a-p \q \b~q \r \G-r "•' and the denominator is ■ 1^ \m \n — m ' where n = a + i + c+ ... This is apparently the problem which Simpson describes in his title page as "A new and comprehensive Problem of great Use in discovering the Advantage or Loss in Lotteries, Raffles, &c." 361. Simpson's Problem x. relates to the game of Bowls ; see Art. 177. Simpson gives a Table containing results for the case of an indefinitely large number of players on each side, but he does not fully explain his Table ; a better account of it will be found in Samuel Clark's Laws of Chance, pages 63 — 65. 362. Simpson's Problem xv. is to find in how many trials one may undertake to have an equal chance for an event to occur r times, its chance at a single trial being known. Simpson claims to have solved this problem "in a more general manner than hitherto ;" but it does not seem to me that what he has added to De Moivre's result is of any importance. We will however give Simpson's addition. Suppose we require the event to happen r times, the chance for it in a single trial being , . Let q = -; and suppose that q is large. Then De Moivre shews that in order to have an even chance that the event shall occur r times we must make about q (r- -^j trials ; see Art. 262. But ii q=l the required number of trials is exactly 2r — 1. Simpson then / 3\ 7 . proposes to take as a universal formula q(r— r^ 1 + »* ~ Tq ! *^i^ is accurate when q = l, and extremely near the truth when q is large. 208 SIMPSON. 363. Simpson's Problem xx. is the same as De Moivre's Pro- blem VII ; it is an example of the Duration of Play : see Art. 107 ; Simpson's method is less artificial than that which De Moivre used, and in fact much resembles the modern method. 364. Simpson's Problem xxii. is that which we have explained in Art. 148 ; Simpson's method is very laborious compared with De Moivre's. Simpson however adds a useful Corollary. By introducing or cancelling common factors we may put the result of Art. 148 in the following form : (^-l)(y-2)...(p-w+l) _„ (y-l)(g-2)...(g-w+l) I w — 1 1 \n—l w (w - 1) {r-l)(r-2) ...(r-n + 1) "^ 1.2 \n-l "••■' where q =p —f, r =p — 2/ ... ; and the series is to continue so long as no negative factors appear. Simpson's Corollary then assigns the chance that the sum of the numbers exhibited by the dice shall not exceed p. We must put successively 1, 2, 3, ... up to p for p in the preceding expression, and sum the results. This gives, by an elementary proposition respecting the summation of series, the following expression for the required chance : p{p-l)...{p-n + \) n g(g-l)... (g-w + 1) n{n—\) r{r—l)...{r — n + l) ■*" 1.2 \n "•■•' where, as before, the series is to continue so long as no negative factor appears. 365. Simpson's Problem xxiv. is the same as De Moivre's Lxxrv., namely respecting the chance of a run of p successes in n trials ; see Art. 325. De Moivre gave the solution without a demonstration; Simpson gives an imperfect demonstration, for having proceeded some way he says that the " Law of Continuation is manifest." SIMPSON. 209 "We have shewn in effect that the solution is obtained by taking the coefficient of f^ in the expansion of a" (1 - at) that is in the expansion of We can thus express the result as the sum of two series, which will be found to agree with the form given by Simpson, 366. Simpson's Problem xxv. is on the Duration of Play. Simpson says in his Preface respecting his Problems xxil. and xxv, that they " are two of the most intricate and remarkable in the Subject, and both solv'd by Methods entirely new." This seems quite incorrect so far as relates to Problem xxv. Simpson gives results without any demonstration ; his Case I. and Case ii. are taken from De Moivre, his Case iii. is a particular example of his general statement which follows, and this general statement coin- cides with Montmort's solution ; see Montmort, page 268, Doctrine of Chances, pages 193 and 211. 367. We will give the enunciation of Simpson's Problem xxvii, together with a remark which he makes relating to it in his Preface. In a Parallelopipedon, whose Sides are to one another in the Ratio of a, b, c; To find at how many Throws any one may undertake that any given Plane, viz. ah, may arise. The 27th is a Problem that was proposed to the Public some time ago in Latin, as a very difficult one, and has not (that I know of) been answered before. We have seen the origin of this problem in Art. 87. Simpson supposes that a sphere is described round the parallelepiped, and that a radius of the sphere passes round the boundary of the given plane; he considers that the chance of the given plane being 14 210 SIMPSON. uppermost in a single throw is equal to the ratio which the spheri- cal surface bounded by the moving radius bears to the whole surface of the sphere. Thus the problem is reduced to finding the area of a certain portion of the surface of a sphere. 368. Simpson gives two examples of the Summation of Series on his pages 70 — 73, which he claims as new in method. (1) Let (a + xy be denoted hy A + Bx+ Ca? + Dn? + ...; required the sum of A Bx Cx' 1.2...r^2.3... (r + l)^3.4...(r- + 2) ■ Integrate both sides of the identity, and determine the con- stant so that both sides may vanish when x=0; thus (a+^)"« «^ _ . Ba^ C^ Bx' ^H^l~~^rnL~ "^"2"+ 3 +^+"" Repeat the operation ; thus {a + x) {n + l){n+2) n + 1 {n+l){n + 2) ~1.2'^2.3"'"3.4"^4.5"''"" Proceed thus for r operations, then divide both sides by x"", and the required sum is obtained. ■ (2) Required the sum of 1" + 2" -f 3" -F . . . -l- a;". Simpson's method is the same as had been already used by Nicolas Bernoulli, who ascribed it to his uncle John ; see Art. 207. 369. Simpson's Problem xxix. is as follows : A and £, whose Chances for winning any assigned Game are in the proportion of a to 6, agree to play until n stakes are won and lost, on Condition that A, at the Beginning of every Game shall set the Sum p to the Sum p x -, so that they may play without Disad- vantage on either Side ; it is required to find the present Yalue of all the Winnings that may be betwixt them when the Play is ended. The investigation presents no difficulty. SIMPSON. 211 870. Simpson's Problem xxx. is as follows : Two Gamesters, A and B, equally skilful, enter into Play together, and agree to continue the same tiU (w) Games are won and lost. 'Tis required to find the Probability that neither comes off a Winner of rjn Stakes, and also the Probability that B is never a Winner of that Number of Stakes during the whole Time of the Playj r being a given, and n any very great. Number. Simpson says in his Preface relating to his Problems XXIV. and xxx. that they "are the same with the two new ones, added in the End of Mr De Moivre's last Edition, whose Demonstrations that learned Author was pleased to reserve to himself, and are here fully and clearly in- vestigated...." The same two problems are thus referred to in Simpson's title page : EuU and clear Investigations of two Problems, added at the end of Mr. De Moivre's last Edition ; one of them allowed by that great Man to be the most useful on the Subject, but their Demonstrations there omitted. Simpson is quite wrong in claiming the solution of Pro- blem xxx, and saying that De Moivre had reserved his demon- stration to himself The investigation is that for determining the approximate value of terms near the largest in the expansion of (a + &)" ; it is given in the Doctrine of Chcmces, second edition, pages 233 — 243, third edition pages 241 — 251 : the method of Simpson is in fact identical with De Moivre's. 371. We may remark that Simpson published a work in 1757 under the title of Miscellaneous Tracts on some curious, ai^ very interesting Subjects in Mechanics, Physical-Astronomy, and Speculative Mathematics ;... In this work on pages 64 — 75 we have a section entitled An Attempt to shew the Advantage arising by Taking the Mean of a Nvmher of Observations, in Practical Astronomy. This is a very interesting section; the problems solved by Simpson were reproduced by Lagrange in a memoir in the fifth volume of the Miscellanea Taurinensia, without any allusion how- ever to Simpson. 14—2 212 JOHN BERNOULLI. It will be more convenient to defer any account of the section in Simpson until we examine Lagrange's memoir, and then we will state what Simpson gave in 1757. 372. The fourth volume of the collected edition of John Ber- noulli's works, which was published in 1742 has a section entitled De Alea, sive Arte Gonjectandi, Problemata qucedam; this section occupies pages 28 — 33 : it contains seven problems. 373. The first and second problems are simple and well- known ; they are solved completely. The third problem relates to the game of Bowls ; John Bernoulli gives, without demonstration, the result which had already been published ; see Montmori, page 248, and the Doctrine of Chances, page 117. 374. The fourth problem contains an error. John Bernoulli says that if 2w common dice are thrown, the number of ways in which the sum of the marks is ^n is (7w- 1) (7w- 2) (7w - 3) ... (5w+ 1) 1.2.3.4...(2w-l) • this amounts to asserting that the expression here given is the co- efficient of a;'" in the expansion of {x+!i^ + x' + x* + x\+ xj" : in fact however the coefficient is a series of which the above ex- pression is only the first term. 375. The fifth and sixth problems involve nothing new in principle ; John Bernoulli gives merely the numerical results which would require long calculation to verify. The seventh problem does not seem intelligible. CHAPTER XL DANIEL BERNOULLI. 376. Daniel Beenoxilli was the son of the John BernoulK to whom we have often referred ; Daniel was bom in 1700, and died in 1782 : he is the author of some important memoirs on our subject, remarkable for their boldness and originality, which we shall now proceed to examine. 377. The first memoir which we have to notice is entitled Specimen Theorice Novce de Mensura Sortis. This memoir is contained in the Commentarii Acad. ...Petrop. Vol. v., which is the volume for the years 1730 and 1731 ; the date of pubHcation of the volume is 1738 : the memoir occupies pages 175 — 192. 378. This memoir contains the theory of Moral expectation proposed by Daniel Bernoulli, which he considered would give results more in accordance with our ordinary notions than the theory of Mathematical expectation. Laplace has devoted to this subject pages 432 — 445 of his TMorie...des Prob., in which he reproduces and developes the hypothesis of Daniel Bernoulli. 379. Mathematical expectation is estimated by the product of the chance of obtaining a sum of money into that sum. But we cannot in practice suppose that a given sum of money is of equal importance to every man ; a shilling is a matter of small moment to a person who possesses a thousand pounds, but it is of great moment to a person who only possesses a few shillings. Various hypotheses may be proposed for taking into account the 214 DANIEL BERNOULLI. relative value of money ; of these Daniel Bernoulli's has attracted most notice. Suppose a person to possess a sum of money x, then if it re- ceive an increment dos, Daniel Bernoulli estimates the relative value of the increment as proportional to dx directly and x in- Isdos versely; that is, he takes it equal to where k is some con- stant. Put this equal to dy; so that , Jcdx therefore y = k log x + constant = k log - say. Laplace calls x the fortune physique and y the fortune morale. We must suppose a some positive quantity, for as Daniel Bernoulli remarks, no man is absolutely destitute unless he is dying of hunger. Daniel Bernoulli calls y the emolumentum, a he calls summa bonoru/m, and x — a he calls lucrum. 380. Suppose then that a person, starting with a for his fortune physique, has the chance p^ of gaining x^, the chance p^ of gaining x^, the chance p^ of gaining x^, and so on ; and suppose the sum of these chances to be unity. Let Y=kp^log{a + x^ + kp^\og{a + x^ +kp^log {a+x^ +...—kloga. Then Bernoulli calls Y the emolumentum medium, and Laplace stiU. calls Y the fortune morale. Let X denote the fortune physique which corresponds to this fortwne morale; then F=AlogX— ^loga. Thus X= (a + xf' (a + xf' {a + xf" ... And X—a will be according to Laplace Vaccroissement de la fortune physique qui procurerait d, Vindividu le m^me avantage moral qui r^sulte pour lui, de son expectative. Daniel Bernoulli calls X—a the lucrum legitime expectandum seu sors qucesita. DANIEL BERNOULLI. 215 381. Daniel Bernoulli in his memoir illustrates his hy- pothesis by drawing a curve. He does not confine himself to the case in which y = ^ log - , but supposes generally y = (x). Thus the ordinary theory of mathematical expectation amounts to supposing that the curve becomes a straight line, or

^^^^ is to 2 '9 approximately; and Cramer considers this to be nearer tlie common notion on the subject than his former value 13. 392. It is obvious that Cramer's suppositions are entirely arbitrary, and that such suppositions might be multiplied to any extent. Montucla alludes on his page 403 to an attempt made by M. Fontaine to explain the paradox. This attempt seems to con- sist in limiting the game to 20 throws at most, instead of allowing it theoretically to extend to infinity. But the opponents of the mathematical theory would assert that for the game as thus under- stood the value of the expectation assigned by the theory is still far larger than common sense can admit. 393. The Petersburg Problem will come under our notice again as we advance with the subject. We may remark that Laplace adopts Daniel Bernoulli's view; Thiorie...des Proh. page 439. Poisson prefers to reconcile mathematical theory with common sense by the consideration that the fortune of the person whom we represent by B is necessarily finite so that he cannot pay more than a certain sum ; this in result practically coincides with the first of Cramer's two suppositions ; see Poisson, Mecherches sur la Prob... page 73; Coumot, Exposition de la ThAorie des Chances... page 108. 394. We pass to another memoir by Daniel Bernoulli. The Academy of Sciences of Paris proposed the following question as a prize subject for 1732, Quelle est la cause physique de I'inclinaison des Plans des Orbit-es des Planetes par rapport au plan de I'Equateur de la revolution, du Soleil autour de son axe; Et d'oii vient que les inclinaisona de ces Orbites sont differentes entre elles. None of the memoirs sent in appeared to the judges to be worthy of the prize. The Academy then proposed the subject again for 1734, with a double prize. The prize was divided be- tween Daniel Bernoulli and his father John Bernoulli. The memoirs of both are contained in the Recueil des pieces qui ont remporU le prix de l' Academic Boy ale des Sciences, Tom. 3, 1734. DANIEL BEBNOULLI. 223 A French translation of Daniel Bernoulli's memoir occupies pages 95 — 122 of the volume ; the original memoir in Latin occu- pies pages 125 — 144. 395. The portion of the memoir with which we are concerned occurs at the beginning. Daniel Bernoulli wishes to shew that we cannot attribute to hazard the small mutual inclinations of the planetary orbits. He puts the calculation in three forms. (1) He finds that the greatest mutual inclination of any two planetary orbits is that of Mercury to the Ecliptic, which is 6° 54'. He imagines a zone of the breadth of 6° 54' on the surface of a sphere, which would therefore contain about :r^ of the whole sur- face of the sphere. There being six planets altogether he takes 1^ for the chance that the inclinations of five of the planes to one plane shall all be less than 6° 54'. (2) Suppose however that all the planes intersected in a common line. The ratio of 6° 54' to 90° is equal to ^g nearly ; 1 and he takes — , for the chance that each of the five inclinations 13^ would be less than 6° 54'. (3) Again ; take the Sun's equator as the plane of reference. The greatest incliaation of the plane of any orbit to this is 7° 30', which is about ^5 of 90° ; and he takes ^^ as the chance that each of the six inclinations would be less than T 30'. 396. It is difficult to see why in the first of the three pre- 1 2 ceding calculations Daniel BemouUi took j,^ instead of — ; that is why he compared his zone with the surface of a sphere instead of with the surface of a hemisphere. It would seem too that he should rather have considered the poles of the orbits than the planes of the orbits, and have found the chance that all the other poles should lie within a given distance from one of them. 224 DANIEL BERNOULLI. 397. We shall find hereafter that D' Alembert did not admit that there was any value in Daniel Bernoulli's calculations. Laplace proposes to find the probability that the sum, of all the inclinations should not exceed an assigned quantity ; see Thiorie... des Prob. page 257. The principle of Daniel Bernoulli's attempt seems more natural, because it takes more explicit account of the fact that each inclination is small. 398. The next memoir by Daniel Bernoulli is entitled Ussai d'une nouvelle analyse de la mortality causae par la petite V4role, et des avantages de VInoculation pour la pr^venir. This memoir is contained in the Hist, de I'Acad. ...Paris, for 1760 ; the date of publication of the volume is 1766 : the memoir occupies pages 1 — 45 of the part devoted to memoirs. 399. The reading of the memoir commenced on April 30th, 1760, as we learn from its seventh page. Before the memoir was printed, a criticism on it appeared, which Daniel Bernoulli ascribes to a grand matMmaticien ; see his pages 4 and 18. In consequence of this, an introduction apolog4tique was written on April 16th, 1765, and now forms the first six pages of the whole. The critic was D' Alembert; see Montucla, page 426, and our Chapter xiiL 400. Daniel Bernoulli's main object is to determine the mor- tality caused by the smaU-pox at various stages of age. This of course could have been determined if a long series of observations had been made ; but at that time such observations had not been made. Tables of mortality had been formed, but they gave the total number of deaths at various ages without distinguishing the causes of death. Thus it required calculation to determine the result which Daniel Bernoulli was seeking. 401. Daniel Bernoulli made two assumptions : that in a year on an average 1 person out of 8 of all those who had not pre- viously taken the disease, would be attacked by small-pox, and that 1 out of eveiy 8 attacked would die. These assumptions he supported by appeal to observation ; but they might not be uni- DANIEL BERNOULLI, 225 versally admitted. Since the introduction of vaccination, the memoir of Bernoulli will have no practical value ; but the mathe- matical theory which he based on his hypotheses is of sufficient interest to be reproduced here. 402. Let X denote the age expressed in years ; let ^ denote the number who survive at that age out of a given number who were bom ; let s denote the number of these survivors who have not had the small-pox. Assume that in a year the small- pox attacks 1 out of every n who have not had the disease, and that 1 out of every m who are attacked dies. The number of survivors who have not had the small-pox continually diminishes ; partly because the small-pox continually attacks some whom it had previously left unattacked, and partly because some persons die of other diseases without ever being attacked by the small-pox. The number of those attacked by the smaU-pox during the sdcC element dx of time is by hypothesis — - : because we suppose - to be attacked in one year, and therefore in the element n •' n dx oi a year. The number of those who die of the small-pox is bv hypothesis —^ ; and therefore the number of those who die of other diseases is -d^-- — . Bui this last number must be diminished in the ratio of s to ^, because we only want the diminution of those who have not yet had the small-pox, of whom the number is s. _ , sdx s ( ,f. , sdx\ Thus - = (^) *** ■ Therefore m, = ^ fl -^ + -7+5 f 1 -- +-T+ cfl - - + ^y, ■where A, B, C are constants, and a, A 7 are the three cube roots of unity. Then from the above equations we obtain ' V n nJ \ n nj ' \ n nj Similarly The three constants A, B, G are not all arbitrary, for we require that M^ + v^ + w, = n, with this condition and the facts that we shall obtain A = B= 0=-^. 418. The above process will be seen to be applicable if the number of urns be any whatever, instead of being limited to three. We need not investigate the distribution of the balls of the other colours ; for it is evident from symmetry that at the end of x DANIEL BERNOULLI. 233 operations the black balls will be probably distributed thus, u^ in the second urn, v^ in the third, and w^ in the first ; similarly the red balls will be probably distributed thus, u^ in the third urn, v^ in the first, and w^ in the second. It should be observed that the equations in Finite Differences and the solution will be the same whatever be the original distri- bution of the balls, supposing that there were originally n in each urn ; the only difference will be in the values to be assigned to the arbitrary constants. Nor does the process require n white balls originally. Thus in fact we solve the following problem : Suppose a given number of urns, each containing n balls, m of the whole number of balls are white and the rest not white ; the original distribution of the white balls is given : required their probable distribution after x operations. 419. Daniel Bernoulli does not give the investigation which we have given in Art. 417. He simply indicates the following result, which he probably obtained by induction : , x{x-l)(x-2){x-S){x-^){x-5 ) ( 1\^/1\' 1 ^ [6 V n) W"^-}' together with similar expressions for v^ and w^. These can be obtained by expanding by the Binomial Theorem the expressions we have given, using the known values of the sums of the powers of a, ^, 7- 420. Now a problem involving the Differential Calculus can be framed, exactly similar to this problem of the urns. Suppose three equal vessels, the first fiUed with a white fluid, the second with a black fluid, and the third with a red fluid. Let there be very small tubes of equal bore, which allow fluid to pass from the first vessel into the second, from the second into the thirdj and from the third into the first. Suppose that the fluids have the property of mixing instantaneously and completely. Required at the" end of the time t the distribution of the fluids in the vessels. 234 DANIEL BERNOULLI. Suppose at the end of the time t the quantities of the white fluid in the three vessels to be u, v, w respectively. We obtain the following equations, du = kdt (w — u), dv = kdt (u — v), dw = Jidt (« — w), where A is a constant. Daniel Bernoulli integrates these equations, by an unsym- metrical and difficult process. They may be easUy integrated by the modem method of separating the symbols. Put i) for -^ ; thus (Z> + A) M = kw, (D + A) w = ku, {D + k)w = kv, therefore {D + A)' « = ¥u. Hence u = e"*' [A^"^ + 5e"" + Ce"^'], where A, B, C are arbitrary constants, and a, ^, y are the three cube roots of unity. The values of v and w can be deduced from that of M. Let us suppose that initially u=h, « = 0, w = 0; we shall find that A=B= C=K, so that o Laplace has given the result for any number of vessels in the Theorie...des Prob. page 303. 421. Now it is Daniel Bernoulli's object to shew, that when x and n are supposed indefinitely large in the former problem its results correspond with those of the present problem. Here indeed we do not gain any thing by this fact, because we can solve the former problem ; but if the former problem had been too difficult to solve we might have substituted the latter problem for it. And thus generally Daniel Bernoulli's notion is that we may often ad- vantageously change a problem of the former kind into one of the latter kind. If we suppose n and x very large we can obtain by the Bino- mial Theorem, or by the Logarithmic Theorem, DANIEL BERNOTILLI. • 235 ( 1-1)--.- nl Hence when n and x are very large, we find that the value of u^ given in Art. 419 reduces to ne ■M^-ff©"-5©"---}- Daniel Bernoulli sums the series in the brackets by the aid of the Integral Calculus. We know however by the aid of the theorem relating to the value of the sums of the powers of % A 7) that this series is equal to Hence the analogy of the value of u^, when x and n are in- definitely large, with the value of u in Art. 420 is sufficiently obvious. Daniel Bernoulli gives some numerical applications of his general results. Daniel Bernoulli's memoir has been criticised by Malfatti, in the Memorie . . . della Bocieta Italiana, Vol. i. 1782. 422. The next memoir by Daniel Bernoulli is entitled. Men- sura Sortis ad fortuitam successionem rerwn naturaliter contin- gentiwm applicata. This memoir is in the same volume of the Ifovi Comm Petrop. as the preceding; it occupies pages 26 — 45. 423. The memoir begins by noticing the near equality in the numbers of boys and girls who are born ; and proposes to consider whether this is due to chance. In the present memoir only thus much is discussed : assuming that the births of a boy and of a girl are equally likely, find the probability that out of a given number of births, the boys shall not deviate from the half by more or less than a given number. The memoir gives some calcu- lations and some numerical examples. Daniel Bernoulli seems very strangely to be unaware that all which he effects had been done better by Stirling and D© Moivre long before; see De Moivre's Doctrine of Chxmces^ pages 243—254. u = V = '236 DANIEL BERNOULLI. I The following is all that Daniel Bernoulli contributes to the theory. Let m and n be laVge numbers ; let \2m 1 He shews that approximately u _ /4m+ 1 v~V 4« + 1 ■ /I IN'" He also stofes the following: in the expansion of (s + oj the fj}^ term from the middle is approximately equal to -^ . These results are included in those of Stirling and De Moivre, so that Daniel Bernoulli's memoir was useless when it appeared ; see Art. 337. 424. The next memoir by Daniel Bernoulli is entitled Di- judicatio maayime prohabilis plurium observationum discrepantium atque verisimillima inductio inde formanda. This memoir is con- tained in the Acta Acad Petrop. for 1777, pars prior; the date of publication of the volume is 1778 : the memoir occupies pages 3 — 23 of the part devoted to memoirs. 425. The memoir is not the first which treated of the errors of observations as a branch of the Theory of Probability; for Thomas Simpson and Lagrange had already considered the sub- ject ; see Art. 371. Daniel Bernoulli however does not seem to have been ac- quainted with the researches of his predecessors. Daniel Bernoulli says that the common method of obtaining a result from discordant observations, is to take the arithmetical mean of the result. This amounts to supposing all the observa- tions of equal weight. Daniel Bernoulli objects to this supposition, and considers that small errors are more probable than large errors. Let e denote an error ; he proposes to measure the pro- bability of the error by >^{r^ — e^, where r is a constant. Then DANIEL BERNOULLI. 237 the best result from a number of observations will be that which makes the product of the probabilities of all the errors a maximum. Thus, suppose that observations have given the values a,h, c, ... for an element ; denote the true value by x ; then we have to find x so that the following product may be a maximum : V{r= - (a; - of] ^J[r' -{x- ly] >J[t^ -{x- c)'} . . . Daniel Bernoulli gives directions as to the value to be assigned to the constant r. 426. Thus Daniel Bernoulli agrees in some respects with modem theory. The chief difference is that modem theory takes for the curve of probability that defined by the equation ^l^' y- while Daniel Bernoulli takes a circle. Daniel Bernoulli gives some good remarks on the subject ; and he illustrates his memoir by various numerical examples, which however are of little interest, because they are not derived from real observations. It is a fatal objection to his method, even if no other existed, that as soon as the number of observations surpasses two, the equation from which the unknown quantity is to be found rises to an unmanageable degree. This objection he himself recognises. 427. Daniel Bernoulli's memoir is followed by some remarks by Euler, entitled Observationes in praecedentem dissertationem ; these occupy pages 24 — 33 of the volume. Euler considers that Daniel Bernoulli was quite arbitrary in proposing to make the product of the probabilities of the errors a maximum. Euler proposes another method, which amounts to making the sum of the fourth powers of the probabilities a maximum, that is, with the notation of Art. 425, {r» -{x- o)T +{r'-(x- byy + {/ -{x- cYY +... is to be a maximum. Euler ^ays it is to be a maximum, but 238 DANIEL BERNOULLI. he does not discriminate between a maximum and a minimum. The equation which is obtained for determining a; is a cubic, and thus it is conceivable that there may be two minima values and one maximum, or only one minimum and no maximum. Euler seems to have objected to the wrong part of Daniel Bernoulli's method ; the particular law o'f probability is really the arbitrary part, the principle of making the product of the pro- babilities a maximum is suggested by the Theory of Probability. Euler illustrates his method by an example derived from real observations. CHAPTER XII. EULER. 428. EuLER was born in 1707, and died in 1783. His industry and genius have left permanent impressions in every field of mathematics; and although his contributions to the Theory of Probability relate to subjects of comparatively small importance, yet they wiU be found not unworthy of his own great powers and fame. 429. Euler's first memoir is entitled Calcul de la ProbahiliU dans le Jeu de Rencontre. This memoir is published in the volume for 1751 of the Histoire de I' A cad... Berlin; the date of pub- lication is 1753 : the memoir occupies pages 255 — 270 of the volume. 430. The problem discussed is that which is called the game of Treize, by Montmort and Nicolas Bernoulli ; see Art. 162. Euler proceeds in a way which is very common with him; he supposes first one card, then two cards, then three, then four, and exhibits definitely the various cases which may occur. After- wards, by an undemonstrated inductive process, he arrives at the general law. The results obtained by Euler had been given more briefly and simply by Nicolas Bernoulli, and published by Montmort in his page 301 ; so we must conclude that Euler had not read Montmort's book. When n is infinite, the expression given in Art. 161 for the 240 EULEK. chance that at least one card is in its right place becomes equal to 1 — e'^, where e is the base of the Napierian logarithms ; this is noticed by Euler : see also Art. 287. 431. The next memoir by Euler is entitled Recherches g^ne- rales sur la morialiU et la multiplication du genre humavn. This memoir is published in the volume for 1760 of the Histoire de V Acad. ...Berlin; the date of publication is 1767: the memoir occupies pages 144 — 164. 432. The memoir contains some simple theorems concerning the mortality and the increase of mankind. Suppose iV" infants bom at the same time ; then Euler denotes by (1) iV" the number of them alive at the end of one year, by (2) N the number of them alive at the end of two yeai-s, and so on. Then he considers some ordinary questions. For example, a certain number of men are alive, all aged m years, how many of them will probably be alive at the end of n years ? According to Euler's notation, (pi) N' represents the number alive aged m years out of an original number N"; and (m + n) N represents the number of those who are ahve at the end of n more years ; so that — . . is the fraction of the number •^ (m) aged m years who will probably be alive at the end of n years. Thus, if we have a number M at present aged m years, there will ('yn, — L- ^ J probably be -^ — r-^ M of them alive at the end of n years. 433. Then Euler gives formulae for annuities on a life. Sup- pose M persons, at present each aged m years, and that each of them pays down the sum a, for which he is to receive a; annually as long as he lives. Let - be the present worth of the unit of money due at the end of one year. Then at the end of a year there will be M ^ , . ^ of the (ot) persons alive, each of whom is to receive x : therefore the present - , 11 .1 . , . a; ,, ()n+ 1) worth of the whole sum to be received is - M ^ . . A. \^) EULER. 241 Similarly, at the end of the second year there -will be M — - — ; — of the persons alive, each of whom is to receive x : H therefore the present worth of the whole sum to be received is * 11^ (»i+ 2) . , -5 M ^ . ■ ■ . And so on. X^ (m) The present worth of all the sums to be received ought to be equal to Ma ; hence dividing by M we get Euler gives a numerical table of the values of (1), (2), ... (95), which he says is deduced from the observations of Kerseboom. 434. Let N denote the number of infants born in one year, and rN the number bom in the next year ; then we may suppose that the same causes which have changed N into rN will change rN into i^N, so that rW will be the number bom in the year succeeding that in which rN were born. Similarly, r'N will be bom in the next succeeding year, and so on. Let us now express the number of the population at the end of 100 years. Out of the N infants born in the present year, there will be (100) N alive ; out of the rN bom in the next year, there will be (99) rN alive ; and so on. Thus the whole number of persons alive at the end of 100 years will be ^,..|i+ (!) + (? + (? + ... I. \ r r r ) Therefore the ratio of the population in the 100* year to the number of infants born in that year will be t + W4.(2) , (3) , ■I H 1 — T"^ — r + ••■ /p y /y If we assume that the ratio of the population in any year to the number of infants bom in that year is constant, and we know this ratio for any year, we may equate it to the expression just given : then since (1), (2), (3), ... are known by observation, we have an equation for finding r. 16 242 EULEE. 435. A memoir by Euler, entitled Sur les Rentes Viageres, immediately follows the preceding, occupying pages 165 — 175 of the volume. Its principal point is a formula for facilitating the calculation of a life annuity. Let A,„ denote the value of an annuity of one pound on the life of a person aged m years, A^^ the value of an annuity of one pound on the life of a person aged m + 1 years. Then by the preceding memoir. Art. 433, , X \yiio-r±i yin-r^J l/«-ro; ) 1_ ( (w + l) , (m + 2) (m + S) 1 r(»i + 2) (m + S) (to + 4> "■« " {m + 1) }~X~ "*" X^ + \» "^ ' therefore (m) X J„ = (w + 1) + (to + 1)^,;^.,. Thus when .4,„ has been calculated, we can calculate A^^^ easily. Euler givSs a table exhibiting the value of an annuity on any age from to 94. But with respect to the ages 90, 91, 92, 93, 94, he says, Mais je ne voudrois pas conseiller S, un entrepreneur de se meler avec de tels vieillards, ^ moins que leur nombre ne fut assez consider- able; ce qui est une regie genSrale pour tous les 6tablissemens fond6s Bur les probabilitls. Euler is of opiaion that the temptations do not appear suf- ficient to induce many persons to buy annuities on terms which would be advantageous to the sellers. He suggests that deferred annuities might perhaps be more successful ; for it follows from his calculations, that 350 crowns should purchase for a new born infant an annuity of 100 crowns to commence at the age of 20 years, and continue for life. He adds, ...et si Ton y vouloit employer la somme de 3500 ecus, ce seroit toujours un bel gtablissement, que de jouii- dSs I'age de 20 ans d'une pension fixe de 1000 Sous. Cependant il est encore douteux, s'il se trouveroit plusieurs parens qui voudroient bien faire un tel sacrifice pour le bien de leurs enfans. EULER. 243 486. The next memoir by Euler is entitled 8ur Vavantage du Banquier au jeu de Pharaon. This memoir was published in the volume for 1764 of the Histoire de V Acad.... Berlin; the date of publication is 1766 : the memoir occupies pages 144 — 164. 437. Euler merely solves the same problem as had been solved by Montmort and Nicolas Bernoulli, but he makes no refer- ence to them or any other writer. He gives a new form however to the result which we will notice. Consider the equation in Finite Differences, m (m— V) (n — m) (n — m — 1) ""^2«(w-l)^ n{n-l) '"''-'- By successive substitution we obtain m(m — l) 8 " 2n{n-r)(n-2)...{n-m + l)' where 8 denotes the sum (n) + (j> (n — 2) + cji (n — i) + ... , ithatis^:^f^7-'. (1 +«)■'- 1 2a3 + fl3 Thus we require the coefficient of a;""' in the expansion of (l+a^r-l 2 + a; ■ 16—2 244 EULEE. This coefficient is n{n-T) ... (n-m+2) n(n-l) ....(n-m+S) 2 \ m- 1 4 |to — 2 n(n — l)... (w — »n + 4) _ "•■ 8 \m-S ■■■ Then 8 = \m — 2 times this coefficient. Hence with this expression for 8 we find that 1 m 1 m (m — 1) 4< n — m + 1 8 (n—in+l) {n — tn+2') 1 m (m — T) {m — 2) "^16 (n-m+l) {n-m + 2) (n-m + S) ~ + (_!)'» ^ m (OT-l)...2 w„ = 2'" (w-m + 1) ...(ji-1)' This is the expression for the advantage of the Banker which was given by Nicolas Bernoulli, and to which we have referred in Art. 157. Now the form which Euler gives for u„ is m { m — 1 (m — 1) (m — 2) (m — 3) ( OT-1 tl(n-] 2"" |l(re-l) 1.2.3(w-3) (m-1) {m-2) (w-3)(w-4)(m-5) ■*" 1.2.3.4.5(w-5) "*" ...J, Euler obtained this formula by trial from the cases in which m = 2, 3, 4, . . . 8 ; but he gives no general demonstration. We will deduce it from Nicolas Bernoulli's formula. By the theory of partial fractions we can decompose the terms in Nicolas Bernoulli's formula, and thus obtain a series of fractions having for denominators n — 1, n — 2,n—3,...n — m+l; and the numerators wiU be independent of n. We will find the numerator of the fraction whose denominator is n — r. From the last term in Nicolas Bernoulli's formula we obtain (-ir' m(w-l)...2 2"" Im-l-rlr-l' EULEE. 245 from the last term but one we obtain (-ly m{m-l)...S r-2' and proceeding in this way we find for the sum (-ir\m I r-1 (r-l){r-2) ) This vanishes if r be an e«;e« number ; and is equal to 2'^\r \m-l-r ' if r be odd. Thus Euler's formula foUows from Nicolas Bernoulli's. 438. The next memoir by Euler is entitled Sur la probabiliM des sequences dans la Lotterie Q^noise. This memoir was published in the volume for 1765 of the Histoire de V Acad.... Berlin; the date of publication is 1767 ; the memoir occupies pages 191 — 230. 439. In the lottery here considered 90 tickets are numbered consecutively from 1 to 90, and 6 tickets are drawn at random. The question may be asked, what is the chance that two or more consecutive numbers should occur in the drawing? Such a result is called a sequence; thus, for example, if the numbers drawn are 4, 5, 6, 27, 28, there is a sequence of three and also a sequence of two. Euler considers the question generally. He supposes that there are n tickets numbered consecutively from 1 to n, and he determines the chance of a sequence, if two tickets are drawn, or if th/ree tickets are drawn, and so on, up to the case in which six tickets are drawn. And having successively investigated all these cases he is able to perceive the general laws which would hold in any case. He does not formally demonstrate these laws, but their truth can be inferred from what he has previously given, by the method of induction. 246 EULER. 440. As an example of Euler's method we will give his iaves- tigation of the case in which three tickets are drawn. There are three events which may happen which may be repre- sented as follows : I. a, a+1, a + 2, that is a sequence of three. II. a, a+1, i, that is a sequence of two, the number b being neither a + 2 nor a — 1. III. a, h, c, where the numbers a, h, c involve no sequence, I. The form a, a+1, a + 2. The number of such events' is n — 2. For the sequence may be (1, 2, 3), or (2, 3, 4), or (3, 4, 5), up to (n—2, n— 1, n). II. The form a, a + \, h. In the same way as we have just shewn that the number of sequences of three, like a, a+\, a+ 2, is w — 2, it foUows that the number of sequences of two, like a, a + 1, is w — 1. Now in general h may be any number between 1 and n inclusive, except a — 1, a, a + \, a + 2; that is, h may be any number out of w — 4 numbers. But in the case of the first sequence of two, namely 1, 2, and also of the last sequence w — 1, «, the number of admissible values of 5 is n — 3. Hence the whole number of events of the form a, a+1, h,\s(n—l) (n — 4e) + 2, that is r? — 5n + 6, that is (n, - 2) (n — 3). III. The form a, h, c. Suppose a to be any number, then b and c must be taken out of the numbers from 1 to a — 2 inclusive, or out of the numbers from a + 2 to n inclusive ; and b and c must not be consecutive. Euler investigates the number of events which can arise. It will however be sufiScient for us here to take another method which he has also given. The total number of events is the number of combinations of n things taken 3 at a time, n (n—1) (n— 2) that is -, ^ n ■ The number of events of the third kind can be obtained by subtracting from the whole number the num- ber of those of the firslfand second kind; it is therefore n{n—l) (n — 2) , „, , .. ^ 1.2\3 ^--{n-2){n-Z)-{n-2). EtILER. 247 It will be found that this is (w - 2) (w - 3) (w - 4) 1.2.3 The chances of the three events will be found by dividing the number of ways in which they can respectively occur by the whole number. Thus we obtain for i, ii, iii, respectively 2-3 2.3 (n-3) ^^^ (n-3)(n-4) n(n—l)' n(n — l) ' n (n—1) 441. Euler's next memoir also relates to. a lottery. This memoir is entitled Solution d'une question tres difficile dans le Calcul des ProhabiUtSs. It was published in the volume for 1769 of the Histoire de I' Acad. ...Berlin; the date of publication is 1771 : the memoir occupies pages 285 — 302 of the volume. 442. The first sentences give a notion of the nature of the problem. C'est le plan d'une lotterie qui m'a fourni cette question, que je me propose de divelopper. Cette lotterie §toit de cinq classe,s, chacune de 10000 billets, parmi lesquels il y avoit 1000 prix dans chaqne classe, et par consequent 9000 blancs. Chaque billet devoit passer par toutes les cinq classes; et cette lotterie avoit cela de particulier qn'outre les prix de cbaque classe on s'engageoit de payer un ducat ^ chacun de ceux dont lea billets auroient pass6 par toutes les cinq classes sans rien gagner. 443. We may put it perhaps more clearly thus. A man takes the same ticket in 5 different lotteries, each having 1000 prizes to 9000 blanks. Besides his chance of the prizes, he is to have £1 returned to him if he gains no prize. The question which Euler discusses is to determine the pro- bable sum which will thus have to be paid to those who fail in obtaining prizes. 444. Euler's solution is very ingenious. Suppose h the num- ber of classes in the lottery ; let n be the number of prizes in each class, and m the number of blanks. 2-i8 EULEK. Suppose the tickets of the first class to have been drawn, and that the prizes have fallen on certain n tickets A, B, ... Let the tickets of the second class be now drawn. Required the chance that the prizes will fall on the same n tickets as before. The chance is 1.2 n (m + 1) (m+2) {m + n) ' And in like manner the chance that the prizes in all the classes will fall on the same tickets as in the first class, is obtained by raising the fiaction just given to the power & — 1. Let {(OT + l)(m + 2) {m + n)Y^^=M, and {1.2 nf-^ = ai. Then -j^ is the chance that all the prizes will fall on the same n tickets. In this case there are m persons who obtain no prize, and so the managers of the lottery have to pay m ducats. 445. Now consider the case in which there are m — 1 persons who obtaia no prize at all. Here besides the n tickets A, B, C, ... which gained in the first class, one of the other tickets, of which the number is m, gains in some one or more of the remaining classes. Denote the number of ways in which this can happen by /Sot. Now M denotes the whole number of cases which can happen after the first class has been drawn. Moreover /3 is in- dependent of m. This statement involves the essence of Euler's solution. The reason of the statement is, that all the cases which can occur will be produced by distributing in various ways the fresh ticket among ^, B, C, ... excluding one of these to make way for it. In like manner, in the case in which there are m—2 persons who obtain no prize at all, there are two tickets out of the m which failed at first that gain prizes once or oftener in the remain- ing classes. The number of ways iu which this can occur may be denoted by ^m (m — 1), where 7 is independent of m. Proceeding in this way we have from the consideration that the sum of all possible cases is M M= OL + ^m + 7»t (to - 1) + Sto (to - 1) (to - 2) -1- .. . EULEK. 249 Now a, j8, 7, . . . are all independent of m. Hence we may put in succession for m the values 1, 2, 3, . . . ; and we shall thus be able to determine /3, 7 446. Euler enters into some detail as to the values of ;8, 7 . . . ; but he then shews that it is not necessary to find their values for his object. For he proposed to find the probable expense which will fall on the managers of the lottery. Now on the first hypothesis it is m ducats, on the second it is m — 1 ducats, on the third it is ?re — 2 ducats, and so on. Thus the probable expense is -^ \a.m + /3m (m — 1) + 7m (m — 1) (to — 2) + . . . K- = j|a+^(m-l)+7(m-l)(m-2) + ...|. The expression in brackets is what we shall get if we change m into TO — 1 in the right-hand member of the value of M in Art. 445 ; the expression therefore is what M becomes when we change to into to — 1. Thus a + /8 (m - 1) + 7 (to - 1) (to - 2) + ... = {to (to + 1) ...{m + n- 1)]*"'. Thus finally the probable expense is / TO N*-' TO \TO + nj Euler then confirms the truth of this simple result by general reasoning. 447. We have next to notice a memoir entitled ^claircisse- mens sur le m6moire de Mr. Be La Orange, ins^r^ dams le V volwme de Mdlanges de Turin, concemant la m^thode de prendre le milieu entre les rhultats de pkisieurs observations, <&c, Pr4sent^ cl I'Acadimie le 27 J!^ov. 1777. This memoir was pubhshed in the I^ova Acta Acad. ...Petrop. Tom. 3, which contains the history of the Academy for the year 1785 ; the date of publication of the volume is 1788 : the memoir occupies pages 289 — 297. 250 EULEE. The memoir consists of explanations of part of that memoir by Lagrange to which we have alluded in Art. 371 ; nothing new is given. The explanations seem to have been written for the benefit of some beginner in Algebra, and would be quite un- necessary for any student unless he were very indolent or very duU. 448. The next contribution of Euler to our subject relates to a lottery ; the problem is one that has successively attracted the attention of De Moivre, Mallet, Laplace, Euler and Trembley. We shall find it convenient before we give an account of Euler's solution to advert to what had been previously published by De Moivre and Laplace. In De Moivxe's Doctrine of Chances, Problem xxxix. of the third edition is thus enunciated: To find the Expectation oi A, when with a Die of a.ny given number of Faces, he undertakes to fling any number of them in any given number of Casts. The problem, as we have already stated, first appeared in the De Men- sura Sortis. See Arts. 251 and 291. Let n be the number of faces on the die ; x the number of throws, and suppose that m specified faces are to come up. Then the number of favourable cases is -.V, m (m — l) , „.^ n''-m.{n-iy+ ^^ ^ ' {n-^)"-... where the series consists of m + 1 terms. The whole number of possible cases is n', and the required chance is obtained by di- viding the number of favourable cases by the whole number of possible cases. 449. The following is De Moivre's method of investigation. First, suppose we ask in how many ways the ace can come up. The whole number of cases is n" ; the whole number of cases if the ace were expunged would be (w — 1)" ; thus the whole number of cases in which the ace can come up is n" — [n — Vf, Next, suppose we ask in how many ways the ace and deux can come up. If the deux were expunged, the number of ways in which the ace could come up would be [n — 1)' — (n — 2)", by EULEE. 251 what we have just seen ; this therefore is the number of ways in which with the given die the ace can come up without the deux. Subtract this number from the number of ways in which the ace can come up with or without the deux, and we have left the number of ways in which the ace can come up with the deux. Thus the result is rf-in- ly -{{n- 1)» -{n- 2/] ; that is, n" - 2 (m - 1)=" + (w - 2)". De Moivre in like manner briefly considers the case in which the ace, the deux, and the tray are to come up ; he then states what the result will be when the ace, the deux, the tray, and the quatre are to come up ; and finally, he enunciates verbally the general result. De Moivre then proceeds to shew how approximate numerical values may be obtained from the formula ; see Art. 292. 450. The result may be conveniently expressed in the nota- tion of Finite Differences. The number of ways in which m specified faces can come up is A" (w — m)" ; where m is of course not greater than n. It is also obvious that if m be greater than x, the event required is impossible ; and in fact we know that the expression A"* (w — mY vanishes when m is greater than x. Suppose n = m; then the number of ways may be denoted by A"0"' ; the expression written at full is n'-n{n-ir+'^^^^ {n-2Y-... 451. One particular case of the general result at the end of the preceding Article is deserving of notice. If we put x = n, we obtain the number of ways in which all the n faces come up in n throws. The sum of the series when x = n \s known to be equal to the product 1 . 2 . 3 ... w, as may be shewn in various ways. But we may remark that this result can also be obtained by the Theory of Probability itself; for if all the n faces are to appear in n throws, there must be no repetition ; and thus the 252 EULEE. number of ways is the number of permutations of n things taken all together. Thus we see that the sum of a certain series might be inferred indirectly by the aid of the Theory of Probability ; we shall hereafter have a similar example. 452. In the M^moires . . . par divers Savans, Vol. VI., 1775, page 363, Laplace solves the following problem : A lottery con- sists of n tickets, of which r are drawn at each time ; find the probability that after x drawings, all the numbers will have been drawn. The numbers are supposed to be replaced after each drawing. Laplace's method is substantially the same as is given in his Theorie . . . des Prob., page 192 ; but the approximate numerical calculations which occupy pages 193 — 201 of the latter work do not occur in the memoir. Laplace solves the problem more generally than he enunciates it ; for he finds the probability that after x drawings m specified tickets will all have been drawn, and then by putting n for m, the result for the particular case which is enunciated is obtained. 453. The most interesting point to observe is that the pro- blem treated by Laplace is really coincident with that treated by De Moivre, and the methods of the two mathematicians are sub- stantially the same. In De Moivre's problem n'' is the whole number of cases ; the corresponding number in Laplace's problem is (^ (n, r)}", where by (j) (n, r) we denote the number of combinations of n things taken r at a time. In De Moivre's problem (n — 1)" is the whole number of cases that would exist if one face of the die were expunged; the corresponding number in Laplace's problem is \^ (n-l.r)}". Similarly to (n - 2y in De Moivre's problem corresponds [^ (n — 2, r)}"' in Laplace's. And so on. Hence, in Laplace's problem, the number of cases in which m specified tickets will be drawn is { (n, r)r-m { (n- 1, r)r + ^L^?^) (^ (n-2, r)}"- ... ; and the probability will be found by dividing this number by the whole number of cases, that is by {0 (n, r)}". EULER. 253 454. With the notation of Finite Differences we may denote the number of cases favourable to the drawing of m specified tickets by A"'{(^(w — m, r)}"'; and the number of cases favourable to the drawing of all the tickets by A" {^ (0, r)]". 455. In the Histoire de I' Acad. ... Paris, 1783, Laplace gives an approximate numerical calculation, which also occurs in page 195 of the ThSorie . . . des Prob. He finds that in a lottery of 10000 tickets, in which a single ticket is drawn each time, it is an even chance that all will have been drawn in about 95767 drawings. 456. After this notice of what had been published by De Moivre and Laplace, we proceed to examine Euler's solution. The problem appears in Euler's Opuscula Analytica, VoL ii., 1785. In this volume pages 331 — 346 are occupied with a memoir entitled Solutio quarundam quaestionum difficiliorum in calculo probabilium. Euler begins thus : His quaestionibus occasionem dedit ludus passim publioe institutus, quo ex nonaginta schedulis, numeris 1, 2, 3, 4,... 90 signatis, statis tem- poribus quinae schedulae sorte extrahi sclent. Hinc ergo hujusmodi quaestiones oriuntur : quanta scilicet sit probabilitas ut, postquam datus extractionum mimerus fuerit peractus, vel omnes nonaginta rnimeri exierint, vel saltern 89, vel 88, vel pauciorea. Has igitur quaestiones, utpote difficillimas, bio ex principiis calculi Probabilium jam pridem usu receptis, resolvere constitui. Neque me deterrent objectiones Illustris IfAlemhert, qui bunc calculum suspectum reddere est conatus. Post- quam enim summits Geometra studiis mathematicis valedixit, iis etiam bellum indixisse videtur, dum pleraque fundamenta solidissime stabilita evertere est aggressus. Quamvis enim hae objectiones apud ignaros maximi ponderis esse debeant, baud tamen metuendum est, inde ipsi scientiae ullum detrimentum allatum iri. 457. Euler says that he finds a certain symbol very useful in these calculations ; namely, he uses 1.2 q P .9.. 458. Euler makes no reference to his predecessors De Moivre and Laplace. He gives the formula for the chance that all the 254 EULER. tickets shall be drawn. This formula corresponds with Laplace's. We have only to put »i = m in Art. 453. Euler then considers the question in which n—1, or w — 2, ... tickets at least are to be drawn. He discusses successively the first case and the second case briefly, and he enunciates his general result. This is the following; suppose we require that n — v tickets at least shall be drawn, then the number of favour- able cases is { {n, r)Y-,}> {n, v + 1) [,p (n- v-1, r)Y + {v + !)<}, {n,v + 2) {(j,(n-v-2, r)Y _ttA(^^(rv.v + B){cf>{n-v-3,r)r-... This result constitutes the addition which Euler contributes to what had been known before. 459. Euler's method requires close attention in order to gain confidence in its accuracy; it resembles that which is employed in treatises on Algebra, to shew how many integers there are which are less than a given number and prime to it. We will give another demonstration of the result which wUl be found easier to follow. The number of ways in which exactly m tickets are drawn is ^ («, m) A*" { (0, r)}". For the factor A"" { (n, m) is the number of combinations of n things taken m at a time. The number of ways in which n — v tickets at least will appear, will therefore be given by the formula S ^ {n, m) A"' {^ (0, r)]", where 2 refers to m, and m is to have all values between n and n—v, both inclusive. Thus we get A" {0 (0, r)r + n A"-' {(n,v+ 2) E'^'-' - ^^^±^p^ ^in,v + 3) £— + ...} {i-p r ^^"nr^ 17273 +;••]' where the series ia brackets is continued to y + 1 terms, unless p be less than v+1 and then it is continued to p + 1 terms only. In the former case the sum of the series can be obtained by taking the coefficient of x" in the expansion of (1 — xY (1 — x)~^, that is in the expansion of (1 — x)^\ In the latter case the sum would be the coefficient of x^ in the same expansion, and is there- fore zero, except when^ is zero and then it is unity, 460. Since r tickets are drawn each time, the greatest number of tickets which can be drawn in x drawings is xr. Thus, as Euler remarks, the expression [^{n,r)Y -n[j>{ri-l,r)Y ^"^^^^ {{n-2,r)Y- ... must be zero if n be greater than xr ; for the expression gives the number of ways in which n tickets can be drawn in r drawings. Euler also says that the .case in which n is equal to xr is re- markable, for then the expression just given can be reduced to a product of factors, namely to 256 EULEE. Euler does not demonstrate this result; perhaps he deduced it from the Theory of Probability itself. For if xr = w, it is obvious that no ticket can be repeated, when aU the tickets are drawn in r drawings. Thus the whole number of favourable cases which can occur at the first drawing must be the number of combinations of n things taken r at a time ; the whole number of favourable cases which can occur at the second drawing is the number of combinations cii n — r things taken r- at a time ; and so on. Then the product of all these numbers gives the whole number of favourable cases. This example of the summation of a series indirectly by the aid of the Theory of Probability is very curious ; see also Art. 451. 461. Euler gives the following paragraph after stating his formulas, In his probabilifcatibus aestimandis utique assumitur omnes litteras ad extrahendum aeque esse proclivea, quod autem 111. D'Alemhert negat assTimi posse. Arbitratur enim, simul ad omnes tractus jam ante per- actos respici oporterej si enim quaepiam litterae nimis crebro fuerint extractae, turn eas in sequentibus tractibns rarius exituras ; contrarium vero evenire si quaepiam litterae nimis raro exierint. Haec ratio, si valeret, etiam valitura esset si sequentes tractus demum post annum, vel adeo integrum saeculum, quia etiam si in alio quocunque loco instituerentur ; atque ob eandem rationem etiam ratio haberi deberet omnium tractuum, qui jam dim in quibuscunque terrae locis fuerint peracti, quo certe vix quicquam absurdius excogitari potest. 462. In Euler's Opuscula Analytica, Vol. il., 1785, there is a memoir connected with Life Assurance. The title is Solutio quaestionis ad calcuhim prohabilitatis pertinentis. Quantum duo conjuges persolvere debeant, ut suis haeredibus post utriusque mortem certa argenti summa persohatur. The memoir occupies pages 315 — 330 of the volume. Euler repeats a table which he had inserted in the Berlin Memoirs for 1760 ; see Art. 433. The table shews out of 1000 infants, how many will be alive at the end of any given year. Euler supposes that in order to ensure a certain sum when both a husband and wife are dead, x is paid down aad z paid EULEE. 257 annually besides, until both are dead. He investigates the re- lation which must then hold between x, z and the sum to be ensured. Thus a calculator may assign an arbitrary value to two of the three quantities and determine the third. He may sup- pose, for example, that the sum to be ensured is 1000 Rubles, and that a; = 0, and find z. Euler does not himself calculate numerical results, but he leaves the fomiulse quite ready for application, so that tables might be easily constructed. 17 CHAPTER XIII. D'ALEMBERT. 463. D'Alembebt was born in 1717 and died in 1783. This great mathematician is known in the history of the Theory of Pro- bability for his opposition to the opinions generally received ; his high reputation in science, philosophy, and literature have secured an amount of attention for his paradoxes and errors which they would not have gained if they had proceeded from a less distin- guished writer. The earliest publication of his peculiar opinions seems to be in the article Gi-oix ou Pile of the Encyclopddie ou Dictionnaire Raisonnd We will speak of this work simply as the Encyclopddie, and thus distinguish it from its successor the EificyclopMie M^thodique. The latter work is based on the former ; the article Croix ou Pile is reproduced unchanged in the latter. 464. The date of the volume of the Encyclopedic containing the article Croix ou File, is 17-54. The question proposed in the article is to find the chance of throwing head in the course of two throws with a coin. Let H stand for head, and T for tail. Then the common theory asserts that there are four cases equally likely, namely, HH, TH, HT, TT; the only unfavourable case is the 3 last; therefore the required chance is -r- D'Alembert however 4 doubts whether this can be correct. He says that if head appears at the first throw the game is finished and therefore there is no d'alembeut. 259 need of the second throw. Thus he makes only three cases, 2 namely, H, TH, TT: therefore the chance is 5. o Sunilarly in the case of three throws he makes only four cases, namely, H, TH, TTE, TTT: therefore the chance is ? The common theory would make eight equally likely cases, and obtain - for the chance. o 463. In the same article D'Alembert notices the Petersburg Problem. He refers to the attempts at a solution in the Com- mentarii Acad Petrop. Vol. V, which we have noticed in Arts. 389 — 393 ; he adds : mais nous ne savons si on en sera satis- fait ; et il y a ici quelque scandale qui m^rite bien d'occuper les Algdbristes. D'Alembert says we have only to see if the expecta- tion of one player and the corresponding risk of the other really is infinite, that is to say greater than any assignable finite number. He says that a little reflexion will shew that it is, for the risk augments with the number of throws, and this number may by the conditions of the game proceed to any extent. He concludes that the fact that the game may continue for ever is one of the reasons which produce an infinite expectation. D'Alembert proceeds to make some further remarks which are repeated in the second volume of his Opuscules, and which will come under our notice hereafter. We shall also see that in the fourth volume of his Opuscules D'Alembert in fact contradicts the conclusion which we have just noticed. 466. We have next to notice the article Gageure, of the EncyclopMie; the volume is dated 1757. D'Alembert says he will take this occasion to insert some very good objections to what he had given in the article Croix ou Pile. He says, EUes sont de M. Necker le fils, citoyen de Genfeve, professeur de Math^matiques en cette ville, . . . nous les avons extraits d'une de ses lettres. The objections are three in number. First Necker denies that D'Alem- bert's three cases are equally likely, and justifies this denial. Secondly Necker gives a good statement of the solution on the 17—2 260 d'alembert. oi'dinary theory. Thirdly, he shews that D'Alembert's view is inadmissible as leading to a result which is obviously untrue : this objection is given by D'Alembert in the second volume of his Opuscules, and will come before us hereafter. D'Alembert after giving the objections says, Ces objections, sur-tout la demiere, mfiritent sans doute beaucoup d'attention. But still he does not admit that he is convinced of the soundness of the common theory. The article Gageure is not reproduced in the Encyclopidie M^thodique. 467. D'Alembert wrote various other articles on our subject in the Encyclopidie ; but they are unimportant. We will briefly notice them. Ahs3nt. In this article D'Alembert alludes to the essay by Nicolas Bernoulli ; see Art. 338. Avantage. This article contains nothing remarkable. Bassette. This article contains a calculation of the advantage of the Banker in one case, namely that given by Montmort on his page 145. Carreau. This article gives an account of the sorte de jeu dont M. de Buffon a donn4 le calcul in 1733, avant que d'etre de V Academic des Sciences; see Art. 354. D4. This article shews all the throws which can be made with two dice, and also with three dice. Loterie. This is a simple article containing ordinary remarks and examples. Pari. This article consists of a few lines giving the ordinary rules. At the end we read : Au reste, ces rfegles doivent ^tre modi- fi^es dans certains cas, ou la probability de gagner est fort petite, et celle de perdre fort grande. Voyez Jeu. There is however nothing in the article Jeu to which this remark can apply, which is the more curious because of course Jeu precedes Pari in alpha- betical order; the absurdity is reproduced in the Encyclopidie M^thodique. The article Prohabiliti in the Encyclopidie is apparently by Diderot. It gives the ordinary view of the subject with the excep- tion of the point which we have noticed in Art. 91. d'alembert. 261 468. In various places in his Opuscules Maihematiques D'Alem- bert gives remarks on. the Theory of Probabilities. These remarks are mainly directed against the first principles of the subject which D'Alembert professes to regard as unsound. We will now examine all the places in which these remarks occur. 469. In the second volume of the Opuscules the first memoir is entitled Reflexions sur le calcul des Prohahilites; it occupies pages 1 — 25. The date of the volume is 1761. D'Alembert begins by quoting the common rule for expectation in the Theory of Probability, namely that it is found by taking the product of the loss or gain which an event will produce, by the probability that this event will happen. D'Alembert says that this rule had been adopted by aU analysts, but that cases exist in which the rule seems to fail. 470. The first case which D'Alembei-t brings forward is that of the Petersburg Problem ; see Art. 389. By the ordinary theory A ought to give B an infinite sum for the privilege of playing with him. D'Alembert says, Or, independamnieut de ce qu'une somme infinie est une cliimere, il n'y a personae qui voulut dormer pour jouer \ ue jeu, je ne dis pas une somme infinie, mais m^me une somme assez modique. 471. D'Alembert notices a solution of the Petersburg Problem which had been communicated to him by un G^ometre c^Iebre de I'Acad^mie des Sciences, plein de savoir et de sagacitd He means Fontaine I presume, as the solution is that which Fontaine is known to have given ; see Montucla, page 403 : in this solution the fact is considered that B cannot pay more than a certain sum, and this limits what. J. ought to give to induce B to play. D'Alem- bert says that this is unsatisfactory ; for suppose it is agreed that ^ the game shall only extend to a finite number of trials, say 100 ; then the theory indicates that A should give 50 crowns. D'Alem- bert asserts that this is too much. The answer to D'Alembert is simple ; and it is very well put in fact by Condorcet, as we shall see hereafter. The ordinary rule is entitled to be adopted, because in the long run it is equally fair to 262 d'alembeet. both parties A and B, and any other rule would be unfair to one or the other. 472. D'Alembert concludes from his remarks that when the probability of an event is very small it ought to be regarded and treated as zero. For example he says, suppose Peter plays with James on this condition ; a coin is to be tossed one hundred times, and if head appear at the last trial and not before, James shall give 2"° crowns to Peter. By the ordinary theory Peter ought to give to James one crown at the beginning of the game. D'Alembert says that Peter ought not to give this crown because he wiU certainly lose, for head will appear before the hundredth trial, certainly though not necessarily. D'Alembert's doctrine about a small probability being equi- valent to zero was also maintained by Buffon. 473. D'Alembert says that we must distinguish between what is metaphysically possible, and what is physically possible. In the first class are included all those things of which the existence is not absurd ; in the second class are included only those things of which the existence is not too extraordinary to occur in the ordinary course of events. It is metaphysically possible to throw two sixes with two dice a hundred times running ; but it is physically impos- sible, because it never has happened and never will happen. This is of course only saying in another way that a very small chance is to be regarded and treated as zero. D'Alembert shews however,, that when we come to ask at what stage in the diminu- tion of chance we shall consider the chance as zero, we are in- volved in difficulty ; and he uses this as an additional argument against the common theory. See also Mill's Logic, 1862, Vol. ii. page 170. 474. D'Alembert says he will propose an idea which has occurred to him, by which the ratio of probabilities may be estimated. The idea is simply to make experiments. He ex- emplifies it by supposing a coin to be tossed a large number of times, and the results to be observed. We shall find that this has been done at the instance of Buffon and others. It is need- less to say that the advocates of the common Theory of Proba- DALEMBEUT. 263 bility would be quite -williiig to accept D'Alembert's reference to experiment ; for relying on the theorem of James Bernoulli, they would have no doubt that experiment would confirm their calcula- tions. It is however curious that D'Alembert proceeds in his very next paragraph to make a remark which is quite inconsistent with his appeal to experiment. For he says that if head has arrived three times in succession, it is more likely that the next arrival will be tail than head. He says that the oftener head has arrived in succession the more likely it is that tail will arrive at the next throw. He considers that this is obvious, and that it furnishes another example of the defects of the ordinary theory. In the Opuscules, Vol. iv. pages 90 — 92, D'Alembert notices the charge of inconsistency which may be urged against him, and attempts to reply to it. 475. D'Alembert then proceeds to another example, which, as he intimates, he had already given iu the Encyclop^die, under the titles Croix ou Pile and Gageure; see Art. 463. The question is this : required the probability of throwing a head with a coin in two trials. D'Alembert came to the- conclusion in the Encyclopidie that 2 3 the chance ought to be ^ instead of -r . In the Opuscules how- ever he does not insist very strongly on the correctness of the 2 result g , but seems to be content with saying that the reasoning t> 3 which produces -j is unsound. D'Alembert urges his objections against the ordinary theory with great pertinacity ; and any person, who wishes to see all that a great mathematician could produce on the wrong side of a question should consult the original memoir. But we agree with every other writer on the subject in thinking that there is no real force in D'Alembert's objections. 476. The following extract will shew that D'Alembert no 2 longer insisted on the absolute accuracy of the result k : 26i DALEMBERT. Je ne voudrois pas cependant regarder en toute rigueur les trois coups dont il s'agit, comme ^galement possibles. Car 1°. il pourroit se faire en effet (et je suis m^me port! ^ le croire), que le cas pUe croix ne fut pas faaotement aussi possible que le cas croix seul ; mais le rapport des possibUites me par6it inapprltiable. 2°. II pourroit se faire encore que le coup pile croix fut un peu plus possible que pile pile, par cette seule raison que dans le dernier le mSme effet arrive deux fois de suite; mais le rapport des possibilitis (suppose qu'elles soient inegales), n'est pas plus facile S, 6tablir dans ce second cas, que dans le premier. A in si il pourroit tr§s-bien se faire que dans le cas propose, le rapport des probability ne fut ni de 3 ^ 1, ni de 2 I, 1 (comme nous I'avons sup- pose dans Y Encyclopedie) mais un incommensurable ou inappr6tiable, moyen entre ces deux nombres. Je crois cependant que cet incommen- surable approchera plus de 2 que de 3, parce qu'encore tine fois il n'y a que trois cas possibles, et non pas quatre. Je crois de m^me et par les mSmes raisons, que dans le cas oii Ton joueroit en trois coups, le rapport de 3 & 1, que donne ma mgthode, est plus prSs du vrai, que le rapport de 7 ^ 1, donn6 par la m€thode ordinaire, et qui me paroit exorbitant. 477. D'Alembert returns to the objection which had been urged against his method, and which he noticed under the title Oagewe in the Encyclop4die ; see Art. 466. Let there be a die with three faces, A, B, C; then according to D'Alembert's original method in the Uncyclopidie, the chances would always be rather against the appearance of a specified face A, however great the number of trials. Suppose n trials, then by D'Alembert's method the chance for the appearance of A is to the chance against it as 2" — 1 is to 2". For example, suppose « = 3 : then the favourable cases are A, BA, CA, BBA, BOA, CCA, OBA ; the unfavourable cases are BBB, BBC, BCB, BCG, CBB, CBC, GGC, COB: thus the ratio is that of 7 to 8. D'Alembert now admits that these cases are not equally likely to happen ; though he believes it difficult to assign their ratio to one another. Thus we may say that D'Alembert started with decided but erroneous opinions, and afterwards passed into a stage of general doubt and uncertainty ; and the dubious honour of effecting the transformation may be attributed to Necker. DALEMBERT. 265 478. D'Alembert thus sums up his results, on his page 24 : Concluons de toutes ces rgfl6xionsj 1°. que si la rSgleque j'ai donate dans Y Encyclopedie' (faute d'ea conn6itre une meilleure) pour deter- miner le rapport des probability au jeu de croix et pile, n'est point €xacte ^ la rigueur, la rSgle ordinaire pour determiner ce rapport. Test encore moins ; 2°. que pour parvenir a une theorie satisfaisante du cal- cul des probabilit6s, il faudroit rlsoudre plusieurs ProblSmes qui sent peut-etre insolubles; savoir, d'assigner le vrai rapport des probabilitls dans les cas qui ne sont pas 6galement possibles, ou qui peuvent n'etre pas regardfe comme tels ; de determiner quand la probability doit ^tre regard6e comme nullej de fixer enfin comment on doit estimer respirance ou I'enjeu, selon que la probabilite est plus ou moins grande. 479. The next memoir by D'Alembert which we have to notice is entitled Sur I'application du Galcul des ProbabilitSs d, I'inoculation de la petite Verole; it is published in the second volume of the Opuscules. The memoir and the accompanying notes occupy pages 26 — 95 of the volume. 480. We have seen that Daniel Bernoulli had written a memoir in which he had declared himself very strongly in favour of Inoculation ; see Art. 398. The present memoir is to a certain extent a criticism on that of Daniel Bernoulli. D'Alembert does not deny the advantages of Inoculation ; on the contrary, he is rather in favour of it : but he thinks that the advantages and disadvantages had not been properly compared by Daniel Ber- noulli, and that in consequence the former had been overestimated. The subject is happily no longer of the practical importance it was a century ago, so that we need not give a very full account of D'Alembert's memoir ; we shall be content with statiag some of its chief points. 481. Daniel Bernoulli had considered the subject as it related to the state, and had shewn that Inoculation was to be recom- mended, because it augmented the mean duration of life for the citizens. D'Alembert considers the subject as it relates to a private individual : suppose a person who has not yet been attacked by small-pox ; the question for him is, whether he will be inoculated, and thus run the risk, small though it may be, of dying in the course of a few days, or whether he wiU take his 266 d'alembert. cliaace of escaping entirely from an attack of small-pox during his life, or at least of recovering if attacked. D'Alembert thinks that the prospect held out to an individual of a gain of three or four years in the probable duration of his life, may perhaps not be considered by him to balance the im- mediate danger of submitting to Inoculation. The relative value of the alternatives at least may be too indefinite to be estimated ; so that a person may hesitate, even if he does not altogether reject Inoculation. 482. D'Alembert lays great stress on the consideration that the additional years of life to be gained form a remote and not a present benefit ; and moreover, on account of the infirmities of age, the later years of a life must be considered of far less value than the years of early manhood. D'Alembert distinguishes between the physical life and the real life of an individual. By the former, he means life in the ordinary sense, estimated by total duration in years ; by the latter, he means that portion of existence during which the individual is free from suffering, so that he may be said to enjoy life. Again, with respect to utility to his country, D'Alembert dis- tinguishes between the physical life and the civil hfe. During infancy and old age an individual is of no use to the state ; he is a burden to it, for he must be supported and attended by others. During this period D'Alembert considers that the indi- vidual is a charge to the state ; his value is negative, and becomes positive for the intermediate periods of his existence. The civil life then is measured by the excess of the productive period of existence over that which is burdensome. Relying on considerations such as these, D'Alembert does not admit the great advantage which the advocates for Inoculation found in the fact of the prolongation of the mean duration of human life effected by the operation. He looks on the problem as far more difiicult than those who had discussed it appeared to have supposed. 483. "We have seen that Daniel Bernoulli assumed that the small-pox attacked every year 1 in n of those not previously d'alembeet. 2G7 attacked, and that 1 died out of every in attacked ; on these hypotheses he solved definitely the problem which he undertook. D'Alembert also gives a mathematical theory of inoculation ; but he does not admit that Daniel Bernoulli's assumptions are established by observations, and as he does not replace them by others, he cannot bring out definite results like Daniel Bernoulli does. There is nothing of special interest in D'Alembert's mathematical investigation ; it is rendered tedious by several figures of curves which add nothing to the clearness of the process they are sup- posed to illustrate. The following is a specimen of the investigations, rejecting the encumbrance of a figure which D'Alembert gives. Suppose a large number of infants born nearly at the same epoch ; let y represent the number alive at the end of a certain time ; let u represent the number who have died during this period of small-pox : let s represent the number who would have been alive if small-pox did not exist : required s in terms of y and u. Let dz denote the decrement of « in a small time, dy the decrement of y in the same time. If we supposed the s individuals subject to small-pox, we should have dz = - dy. But we must subtract from this value of dz the decrement arising from small-pox, to which the a individuals are by hypo- thesis not liable : this is - du, y Thus, dz— - dy + - du\ y '^ y z z we put + - du and not du, because z and y diminish while ■ y y " u increases. Then dz dy du, — = — + — ; ^ y y therefore log s = log y -1- I — ; /du ^^ ^.^ ^ — IjZ. ^ . 268 d'alembebt. The result is not of practical use because the value of the integral I — is not known. D'Alembert gives several formulsB which involve this or similar unfinished integrations. 484. D'Alembert draws attention on his page 74 to the two distinct methods by which we may propose to estimate the espS- rance de vivre for a person of given age. The mean duration of life is the average duration in the ordinary sense of the word average ; the probable duration is such a duration that it is an even chance whether the individual exceeds it or falls short of it. Thus, according to Halley's tables, for an infant the mean Hfe is 26 years, that is to say if we take a large number N of infants the sum of the years of their lives will be 26N; the probable N life is 8 years, that is to say ^ of the infants die under 8 years N old and -^ die over 8 years old. The terms mean life and probable life which we here use have not always been appropriated in the sense we here explaia ; on the contrary, what we call the mean life has sometimes been called the probable Hfe. D'Alembert does not propose to distinguish the two notions by such names as we have used. His idea is rather that each of them might fairly be called the duration of life to be expected, and that it is an objection against the Theory of Proba- biUty that it should apparently give two different results for the same problem. 485. We will illustrate the point as D'Alembert does, by means of what he calls the curve of mortality. Let X denote the number of years measured from an epoch ; let yfr (x) denote the number of persons alive at' the end of x years from birth, out of a large number born at the same time. Let "^ (x) be the ordinate of a curve ; then yjr (x) diminishes from a; = to a; = c, say, where c, is the greatest age that persons can attain, namely about 100 years. This curve is called the curve of mortality by D'Alembert. d'alembert. 2G9 The mean duration of life for persons of the age a years is ■\fr (x) dx J a Theprolahle duration is a quantity h such that ^ (S) = 2 ■t W- This is D'Alembert's mode. We might however use another curve or function. Let

(x) dx. Thus the mean duration is represented by the abscissa of the centre of gravity of a certain area ; and the probable duration is represented by the abscissa corresponding to the ordinate which bisects that area. This is the modern method of illustrating the point; see Art. 101 of the Theory of Probability in the Encyclopcedia Metro- politana. 486. We may easily shew that the two methods of the pre- ceding Article agree. For we have (}> (x) = — k '\jr' (x), where k is some constant. Therefore I (x — a)^ (x) dx j (x — a) ■\{r' (x) dx J a J a I ^ (x) dx I ■x^' (x) J jx.. where S denotes the number alive at the beginning of the year; for s is a variable gradually diminishing during the year from the value 8 with which it began. But -prr is the result which Daniel Bernoulli professed to take from observation ; therefore Daniel Bernoulli is inconsistent with himself D'Alembert's objection is sound ; Daniel Bernoulli would no doubt have admitted it, and have given the just reply, namely that his calculations only professed to be approximately correct, and that they were approximately correct. Moreover the error arising in taking I sdx and S to be equal in value becomes very small if we suppose S to be, not the value of s when x = n or ti + 1 but, the intermediate value when x = n+ -^ ; and nothing in Daniel Bernoulli's investigation forbids this sup- position. 517. We have put the objection in the preceding Article as D'Alembert ought to have put it in fairness. He himself however really assumes n = 0, so that his attack does not strictly fall on the whole of Daniel Bernoulli's table but on its first line ; see Art. 403. This does not affect the principle on which D'Alembert's objection rests, but taken in conjunction with the remarks in the preceding Article, it will be found to diminish the practical value of the ob- jection considerably. See D'Alembert's pages 312 — 314. 518. Another objection which D'Alembert takes is also sound ; see his page 315. It amounts to saying that instead of using the Differential Calculus Daniel Bernoulli ought to have used the Calculus of Finite Differences. We have seen in Art. 417 that Daniel Bernoulli proposed to solve various problems in the Theory of Probability by the use of the Differential Calculus. The reply to be made to D'Alembert's objection is that Daniel Bernoulli's investigation accomplishes what was proposed, namely an approxi- mate solution of the problem ; we shall however see hereafter in examining a memoir by Trembley that, assuming the hypotheses of Daniel Bernoulli, a solution by common algebra might be effected. 284 d'alembeet. 519. D'Alembert thinks that Daniel Bernoulli might have solved the problem more simply and not less accurately. For Daniel Bernoulli made two assumptions ; see Art. 401. D'Alembert says that only one is required ; namely to assume some function of 3^ for M in Art. 483. Accordingly D'Alembert suggests arbi- trarily some functions, which have apparently far less to recom- mend them as corresponding to facts, than the assumptions of Daniel Bernoulli. 520. D'Alembert solves what he calls mm probleme assez cu- rieux ; see his page 325. He solves it on the assumptions of Daniel Bernoulli, and also on his own. We will give the former solution. Return to Art. 402 and suppose it required to determine out of the number s the number of those who will die by the small-pox. Let o) denote the number of those who do not die of small-pox Hence out of this number = l—d^ 1 - mnj ^ ' ,, . day dp sdx therefore — = -r + i: • Substitute the value of s in terms of x and ^ from Art. 402, and integrate. Thus we obtain ^ e" (m - 1) + 1 where G is an arbitrary constant. The constant may be deter- mined by taking a result which has been deduced from observa- tion, namely that ? = ot when x = 0. 521. D'Alembert proposes on his pages 326 — 328 the method which according to his view should be used to find the value of s at the time x, instead of the method of Daniel Bernoulli which DALEMBERT. 285 we gave in Art. 402. D'Alembert's method is too arbitrary in its hypotheses to be of any value. 522. D.'Alembert proposes to develop his refutation of the Savant Geometre whom we introduced in Art. 487. He shews decisively that this person was wrong ; but it does not seem to me that he shews distinctly how he was wrong. 523. D'Alembert devotes the last ten pages of the memoir to the development of his own theory of the mode of comparing the risk of an individual if he undergoes Inoculation with his risk if he declines it. We have already given in Art. 482, a hint of D'Alembert's views ; his remarks in the present memoir are ingenious and interesting, but as may be supposed, his hypotheses are too arbitrary to allow any practical value to his investiga- tions. 524. Two remarks which he makes on the curve of mortality may be reproduced ; see his page 340. It appears from Buffon's tables that the mean duration of life for persons aged n years is always less than ^ (100 — n). Hence, taking 100 years as the extreme duration of human Hfe, it wiU follow that the curve of mortality cannot be always concave to the axis of abscissae. Also from the tables of Buffon it follows that the probable duration of life is almost always greater than the mean duration. D'Alem- bert applies this to shew that the curve of mortality cannot be always convex to the axis of abscissae. 525. The fifth volume of the Opuscules was published in 1768. It contains two brief articles with which we are con- cerned. Pages 228 — 231 are 8ur les Tables de mortalitS. The numeri- cal results are given which served for the foundation of the two remarks noticed in Art. 524. Pages 508 — 510 are Sur les calculs relatifs A V inoculation... These remarks form an addition to the memoir in pages 283 — 341 of the fourth volume of the Opuscules. D'Alembert notices a reply which had been offered to one of his objections, and enforces the 233 d'alembert. justness of his objections. Nevertheless he gives his reasons for regarding Inoculation as a useful practice. 526. The seventh and eighth volume of the Opiiscules were published in 1780. D'Alembert says in an Advertisement pre- fixed to the seventh volume, " . . . Ce seront vraisemblablement, h peu de chose prfes, mes derniers Ouvrages Math^matiques, ma t^te, fatigu^e par quarante-cinq ann^es de travail en ce genre, n'dtant plus gufere capable des profondes recherches qu'il exige." D'Alem- bert died in 1783. It would seem according to his biographers that he suffered more from a broken heart than an exhausted brain during the last few years of his life. 527. The seventh volume of the Opuscules contains a memoir Sur le calcul des Prohabilites, which occupies pages 39 — 60. We shall see that D'Alembert still retained his objections to the ordinary theory. He begins thus : Je demande pardon aux Geometres de revenir encore sur ce sujet. Mais j'avoue que plus j'y ai pens6, phis je me suis confirm^ dans mes doutes surles principes de la thSorie ordinaire; je desire qu'on ^claircis.se ces doutes, et que cette th6orie, soit qu'on j change quelques principes, soit qu'on la conserve telle qu'elle est, soit du moins expos6e d&ornaais de maniere ^ ne plus laisser aucun nuage. 528. We will not delay on some repetition of the old remarks ; but merely notice what is new. We find on page 42 an error which D'Alembert has not exhibited elsewhere, except in the article Cartes in the EncychpSMe M4thodique, which we shall notice hereafter. He says that taking two throws there is a chance -^ of 1 ^ head at the first throw, and a chance „ of head at the second throw ; and thus he infers that the chance that head will arrive at least once is ^ -I- ^ or 1. He says then, Or je demande si cela est vrai, ou du moins si un pareil r^sultat, fond^ sur de pareils prin- cipes, est bien propre k satisfaire I'esprit. The answer is that the result is false, being erroneously deduced : the error is exposed in elementary works on the subject. 529. The memoir is chiefly devoted to the Petersburg Problem. D'Alembert refers to the memoir in Vol. VI. of the Memoires...par d'alembert. 287 divers Savans... in whicli Laplace had made the supposition that the coin has a greater tendency to fall on one side than the other, but it is not known on which side. Suppose that 2 crowns are to be received for head at the first trial, 4 for head at the second, 8 for head at the third, . . . Then Laplace shews that if the game is to last for X trials the player ought to give to his antagonist less than X crowns if x be less than 5, and more than x crowns if x be greater than 5, and just x crowns if x be equal to 5. On the com- mon hypothesis he would always have to give x crowns. These results of Laplace are only obtained by him as approximations ; D'Alembert seems to present them as if they were exact. 530. Suppose the probability that head should fall at first to be (o and not ^ ; and let the game have to extend over n trial g Then if 2 crowns are to be received for head at the first trial, 4 for head at the second, and so on ; the sum which the player ought to give is 2a) (1 + 2 (1 - «) + 2^^ (1 - to)' + . .. + 2"- (1 - a))"-'}, which we will call O. D'Alembert suggests, if I understand him rightly, that if we know nothing about the value of w we may take as a solution of the problem, for the sum which the player ought to give I ildco. Jo But this involves all the difficulty of the ordinary solution, for the result is infinite when n is. D'Alembert is however very obscure here ; see his pages 45, 46, He seems to say that | ndco will be greater than, equal to, or less than n, according as n is gi-eater than, equal to, or less than 5. But this result is false ; and the argument unintelligible or incon- clusive. We may easily see by calculation that I ildm = n when J n — 1; and that for any value of n from 2 to 6 inclusive ildo) is less than n ; and that when w is 7 or any greater number I fldo) is greater than n. ^ n 288 d'alembeet. 531. D'Alembert then proposes a method of solving the Peters- burg Problem which shall avoid the infinite result ; this method is perfectly arbitrary. He says, if tail has arrived at the first throw, let the chance that head arrives at the next be — ^ , and not ^ , where a is some small quantity ; if tail has arrived at the first throw, and at the second, let the chance that head arrives at the next throw be ^ , and not „ ; if tail has arrived at the first throw, at the second, and at the third, let the chance that head arrives at the next throw be ^ , and not 5 ; and so on. The quantities a, h, c, ... are supposed small positive quantities, and subjected to the limitation that their sum is less than unity, so that every chance may be less than unity. On this supposition if the game be as it is described in Art. 389, it may be shewn that A ought to give half of the following series : 1 + (!+«) + (l-a)(l + a + J) + (1 - a) (1 - a - J) (1 + a + i + c) + (1 - a) (1 - OS - Z> - c) (1 + a + & + c + -' + r'+..., r being = l — a — h— c — d. 532. Thus on his arbitrary hypotheses D'Alembert obtains a finite result instead of an infinite result. Moreover he performs what appears a work of supererogation ; for he shews that the suc- cessive terms of the infinite series which he obtains form a con- tinually diminishing series beginning from the second, if we suppose that a, h,c,d,... are connected by a certain law which he gives, namely, 1—a — h — c — d — e—,.. = -. . , qr- , where p is a small fraction, and m — 1 is the number of the quan- tities a, b, c, d, e, ... Again he shews that the same result holds if we merely assume that a,b, c, d, e... form a continually diminish- ing series. We say that this appears to be a work of supereroga- tion for D'Alembert, because we consider that the infinite result was the only supposed difficulty in the Petersburg Problem, and that it was sufiicient to remove this without shewing that the series substituted for the ordinary series consisted of terms con- tinually decreasing. But D'Alembert apparently thought differ- ently ; for after demonstrating this continual decrease he says. En veils, assez pour faire voir que les termes de Tenjeu vont en diminuant des le troisiSme coup, jusqu'au dernier. ITous avons prouv6 d'ailleurs que I'enjeu total, somme de ces termes, est fini, en supposanfc m^me le nombre de coups infini. Ainsi le r6sultat de la solution que nous donnons ici du problSme de Petersbourg, n'est pas sujet a la diffi- culte insoluble des solutions ordinaires. 533. We have one more contribution of D'Alembert's to our subiect to notice; it contains errors which seem extraordinary, 19 290 d'alembeet. even for him. It is the article Cartes in the Encyclopedie MStho- dique. The following problem is given, Pierre tient huit cartes dans ses mains qui sent : un as, un deux, un trois, un quatre, un cinq, un six, un sept et un huit, qu'U a mll6es : Paul parie que les tirant I'une apres I'autre, il les devinera a, mesure qu'U les tirera. L'on demande combien Pierre doit parier contra un que Paul ne rlussira pas dans son enterprise ? It is correctly determined that Paul's chance is 1111111 Then follow three problems formed on this ; the whole is ab- surdly false. We give the words : Si Paid parioit d'amener ou de deviner juste 3, un des sept coups seulement, son esp6rance seroit 5 +=+■■■+ 7; , et par consequent I'enjeu de Pierre 8, celui de Paul, comme 11 111 1 - + -+...+-^1-----...--. Si Paid parioit d'amener juste dans les deux premiers coups seule- ment, son esp^rance seroit o + 7 > et le rapport des enjeux celui de 1 1 . 1 1 1 8 + 7 ^ i-g-y. S'il parioit d'amener juste dans deux coups quelconques, son esp6- .,11 11 11 ranee seroit 5—=- +-5 — 5 + ... +3 — s + s — n+ ••• + s r + s — ?+ ••• 8x7 8x6 8x2 7x6 7x2 6x5 The first question means, I suppose, that Paul undertakes to be right once in the seven cases, and wrong six times. His chance then is 1 /I 1 1 1 1 1 _ 8 (,7 + 6 + 5+1 + 3 + 2 + ^ )■ For his chance of being right in the first case and wrong in the other six is 1654321^,^. 1 X=X7;X^X7X = X=;> that IS 8 7654 3 2' 8x7' d'alembeet. 291 his chance of being right in the second case and wrong in all the others is 7164321,,^. 1 x^^x^x-gXTX^Xs, that is 8765432' 8x6' and so on. If the meaning be that Paul undertakes to be right once at 7 least in the seven cases, then his chance is ^ . For his chance of being wrong every time is 76543211 gX^XgXgXjXgX^;tnatis-g; 1 . 7 therefore his chance of being right once at least is 1 — „ , that is ^ • The second question means, I suppose, that Paul undertakes to be right in the first two cases, and wrong in the other five. His chance then is 1154321,,,. 1 -x^XgX^x^XgX^.thatis^^^-^-^. Or it may mean that Paul undertakes to be right in the first two cases, but undertakes nothing for the other cases. Then his chance is 3 x = . The third question means, I suppose, that Paul undertakes to be right in two out of the seven cases and wrong in the other five cases. The chance then wiU be the sum of 21 terms, as 21 .combi- nations of pairs of things can be made from 7 things. The chance that he is right in the first two cases and wrong in all the others is 1154321,,,. 1 x = X7:X-=XjX^x^, that is 8''l'"'^''l i I r ""'^•'^"8x7x6' similarly we may find the chance that he is right in any two assigned cases and wrong in all the others. The total chance wiU be found to be 19—2 292 d'alembert. Or the third question may mean that Paul undertakes to be right twice at least in the course of the seven cases, or in other words he undertakes to be right twice and undertakes nothing more. His chance is to be found by subtracting from unity his chance of being never right, and also his chance of being right only once. Thus his chance is 1_1/1 11 8 8 17 "^ 6 ^ 5 l-s-3fs + ^+^ + - + l)- 534. Another problem is given unconnected with the one we have noticed, and is solved correctly. The article in the EncyclopMie Methodique is signed with the letter which denotes D'Alembert. The date of the volume is 1784, which is subsequent to D'Alembert's death ; but as the work was published in parts this article may have appeared during D'Alem- bert's life, or the article may have been taken from his manu- scripts even if published after his death. I have not found it in the original Uncyclop&die : it is certainly not under the title Cartes, nor under any other which a person would naturally consult. It seems strange that such errors should have been admitted into the JEncyclopSdie Methodique. Some time after I read the article Cartes and noticed the errors in it, I found that I had been anticipated by Binet in the Comptes JRendus ... Vol. xix. 1844. Binet does not exhibit any doubts as to the authorship of the article ; he says that the three problems are wrong and gives the correct solution of the first. 535. We will in conclusion briefly notice some remarks which have been made respecting D'Alembert by other writers. 536. Montucla after alluding to the article Croix ou Pile says on his page 406, D'Alembert ne s'est pas bom6 a cet exemple, il en a accunnil6 plu- sieurs autres, soib dans la quatrilme volume de ses Opuscules, 1768, page 73, et page 283 du ciaqui^me; il s'est aussi fetayg dn suffrage de divers ggomStres qu'il qualifie de distingu6s. Condorcet a appuy6 ces objec- tions dans plusieurs articles de I'EiicyclopSdie m6thodique ou par ordre de matifires. D'un autre c6t6, divers autres geomltres ont entrepris D ALEMBERT. 293 de rgpondre aux raisonnemens de d'Alembert, et je orois qu'en par- ticulier Daniel Bernoulli a pris la defense de la thgorie ordinaire. In this passage the word ciTiqui^me is wrong; it should be quatri^me. It seems to me that there is no foundation for the statement that Condorcet supports D'Alembert's objections. Nor can I find that Daniel Bernoulli gave any defence of the ordinary- theory ; he seems to have confined himself to repelling the attack made on his memoir respecting Inoculation. 537. Gouraud after referring to Daniel Bernoulli's controversy with D'Alembert says, on his page 59, ...et quant au reste des matliSmatioiens, ce ne fut que par le silence ou le d6dain qu'il r^pondit aux doutes que d'Alembert s'etait permis d'6metfcre. M§pris injuste et malhabile oil tout le monde avait h perdre et qu'une post6rit6 moins pr6venue ne devait point sanctionner. The statement that D'Alembert's objections were received with silence and disdain, is inconsistent with the last sentence of the passage quoted from Montucla in the preceding Article. According to D'Alembert's own words which we have given in Art. 490, he was attacked by some indifferent mathematicians. 538. Laplace briefly replies to D'Alembert ; see TMorie...des Proh. pages vii. and x. It has been suggested that D'Alembert saw his error respecting the game of Croix ou Pile before he died; but this suggestion does not seem to be confirmed by our examination of all his writings : see Cambridge Philosophical Transactions, Vol. ix. page 117. CHAPTER XIV. BAYES. 539. The name of Bayes is associated with one of the most important parts of our subject, namely, the method of estimating the probabilities of the causes by which an observed event may have been produced. As we shall see, Bayes commenced the in- vestigation, and Laplace developed it and enunciated the general principle in the form which it has since retained. 540. We have to notice two memoirs which bear the fol- lowing titles : An Essay towards solving a Problem, m the Doctrine of Chances. By the late Rev. Mr. Bayes, F.R.S. communicated by Mr Price in a Letter to John Canton, A .M. F.R.S. A Demonstration of the Second Rule in the Essay towards the Solution of a Problem, in the Doctrine of Cha/nces, published in the Philosophical Transactions, Vol. liii. Com- municated by the Rev. Mr. Richard Price, in a Letter to Mr. John Canton, M. A. F.R.S. The first of these memoirs occupies pages 370 — 418 of Vol. Llll. of the Philosophical Transactions; it is the volume for 1763, and the date of publication is 1764. The second memoir occupies pages 296 — 325 of Vol. LIV. of the Philosophical Tramsactions ; it is the volume for 1764, and the date of publication is 1765. 541. Bayes proposes to establish the following theorem: If BAYES. 295 an event has happened p times and failed q^ times, the probability that its chance at a single trial lies between a and J is a^(l-a;)'(fo J a •'o Bayes does not use this notation ; areas of curves, according to the fashion of his time, occur instead of integrals. Moreover we shall see that there is an important condition implied which we have omitted in the above enunciation, for the sake of brevity: we shall return to this point in Art. 552. Bayes also gives rules for obtaining approximate values of the areas which correspond to our integrals. 542. It will be seen from the title of the first memoir that it was published after the death of Bayes. The Rev. Mr Eichard Price is the well known writer, whose name is famous in connexion with politics, science and theology. He begins his letter to Canton thus : Dear Sir, I now send you an essay wMch I have found among tie papers of our deceased friend Mr Bayes, and wMch, in my opinion, has great merit, and well deserves to be preserved. 543. The first memoir contains an introductory letter from Price to Canton ; the essay by Bayes follows, in which he begins with a brief demonstration of the general laws of the Theory of Probability, and then establishes his theorem. The enuncia- tions are given of two rules which Bayes proposed for finding approximate values of the areas which to him represented our integrals ; the demonstrations are not given. Price himself added An Appendix containing an Application of the foregoing Rules to some particular Cases. The second memoir contains Bayes's demonstration of his prin- cipal rule for approximation ; and some investigations by Price which also relate to the subject of approximation. 544. Bayes begins, as we have said, with a brief demonstra- tion of the general laws of the Theory of Probability ; this part of his essay is excessively obscure, and contrasts most unfavourably with the treatment of the same subject by De Moivre. 296 BATES. Bayes gives the principle by which we must calculate the probability of a compound event. Suppose we denote the probability of the compound event by P j^, the probability of the' first event by z, and the probability of the second on the supposition of the happening of the first by -^ . Then our principle gives us '^==^'^'^> ^^. Hence we 298 BAYES. have for the probability, before the first ball is thrown, that the second event will happen^ times and fail ^ times 549. We now arrive at the most important point of the essay. Suppose we only know that the second event has happened p times and failed q times, and that we wish to infer from this fact the probable position of the line ^F which is to us unknown. The probability that the distance of EF from AB lies between h and c is I x" {a- xy dx \ x^ia-xydx Jo This depends on Bayes's Proposition 5, which we have given in our Art. 544. For let a denote the required probability; then 2 X probability of second event = probability of compound event. The probability of the compound event is given in Art. 547, and the probability of the second event in Art. 548 ; hence the value of a follows. 550. Bayes then proceeds to find the area of a certain curve, or as we should say to integrate a certain expression. We have f ^ + 1 !_??+ 2 ' 1.2 J9 + 3 ■■■ This series may be put in another form ; let u stand for 1 — x, then the series is equivalent to x'^^u^ q x^'^u"-' q(q-l) x"^' 1^^ ^ + l'^j?+l p + 2 '^ {p + l){p + 2) p + 3 q(q-l)(q-2) x/^'u^-' ip + l){p + ^){p + 3} p + 4> ■^••• This may be verified by putting for u its value and rearranging according to- powers of x. Or if we differentiate the series with BATES. 299 respect to x, we shall find that the terms cancel so as to leave only a^vF. 551. The general theory of the estimation of the probabilities of causes from observed events was first given by Laplace in the Memoires ...par divers Savans, Vol.. vi. 1774. One of Laplace's results is that if an event has happened p times and failed q times, the probability that it will happen at the next trial is /, 1 {'x^ll-xYdx J Lubbock and Drinkwater think that Bayes, or perhaps rather Price, confounded the probability given by Bayes's theorem with the probability given by the result just taken from Laplace ; see Luhbock and Drinkwater, page 48. But it appears to me that Price understood correctly what Bayes's theorem really expressed. Price's first example is that in which j? = 1, and q = Q. Price says that "there would be odds of three to one for somewhat more than an even chance that it would happen on a second trial." His demonstration is then given ; it amounts to this : ' x^il-xYdx „ ( l\''0.-xydx J i 4' 3 where p = l and q = 0. Thus there is a probability - that the chance of the event lies between -^ and 1, that is a probability 3 -j- that the event is more Ukely to happen than not. 552. It must be observed with respect to the result in Art. 549, that in Bayes's own problem we know that a priori any position of .EF between AB and CD is equally likely ; or at least we know what amount of assumption is involved in this supposition. In the applications which have been made of Bayes's theorem, and of such I'esults as that which we have taken from Laplace in 800 BAYES. Art. 651, there has however often been no adequate ground for such knowledge or assumption. 553. We have already stated that Bayes gave two rules for approximating to the value of the area which corresponds to the integral. In the first memoir. Price suppressed the demonstrations to save room ; in the second memoir, Bayes's demonstration of the principal rule is given : Price himself also continues the subject. These investigations are very laborious, especially Price's. The following are among the most definite results which Price gives. Let n =p + q, and suppose that neither p nor q is small ; let h = ,/ -,< • Then if an event has happened p times and failed q times, the odds are about 1 to 1 that its chance at a sinde trial lies between - + -t^ and *- — j^; the odds are about ° n V^ w V 2 2 to 1 that its chance at a single trial lies between - + h and n -£—/»: the odds are about 5 to 1 that its chance at a single n trial lies between -^ + ^ V2 and ^ — h \/2. These results may be verified by Laplace's method of approximating to the value of the definite integrals on which they depend. 554. We may observe that the curve y = a;" (1 —a;)' has two points of inflexion, the ordinates of which are equidistant from the maximum ordinate ; the distance is equal to the quantity Ji of the preceding Article. These points of inflexion are of importance in the methods of Bayes and Price. CHAPTER XV. LAGRANGE. 555. Lagrange was bom at Turin in 1736, and died at Paris in 1813. His contributions to our subject will be found to satisfy the expectations which would be formed from his great name in mathematics. 556. His first memoir, relating to the Theory of Probability, is entitled Memoire sur I'utilite de la methode de prendre le milieu entre les resultats de plusieurs observations ; dans lequel on examine les avantages de cette methode par le calcul des probahiliUs ; et oh Von resoud differens probUmes relatifs a cette matihre. This memoir is published in the fifth volume of the Miscellanea Taurinensia, which is for the years 1770 — 1773 : the date of publication is not given. The memoir occupies pages 167 — 232 of the mathematical portion of the volume. The memoir at the time of its appearance must have been extremely valuable and interesting, as being devoted to a most important subject ; and even now it may be read with ad- vantage. 557. The memoir is divided into the discussion of ten pro- blems ; by a mistake no problem is numbered 9, so that the last two are 10 and 11. The first problem is as follows : it is supposed that at every observation there are a cases in which no error is made, b cases in which an error equal to 1 is made, and b cases in which an 302 LAGRANGE. error equal to — 1 is made ; it is required to find the probability that m taking the mean of n observations, the result shall be exact. In the expansion of {a + h(x + a;"^)}" according to powers of x, find the coefficient of the term independent of x; divide this coefficient by (a + 2J)" which is the whole number of cases that can occur ; we thus obtain the required probability. Lagrange exhibits his usual skill in the management of the algebraical expansions. It is found that the probability diminishes as n increases. 558. We may notice two points of interest in the course of Lagrange's discussion of this problem. Lagrange arrives indirectly at the following relation _ 1.3.5...(2,»-1) 1.2.3...M and he says it is the more remarkable because it does not seem easy to demonstrate it a priori. The result is easily obtained by equating the coefficients of the term independent of x in the equivalent expressions {i+xr(. ■-i)"-"^ This simple method seems to have escaped Lagrange's notice. Suppose we expand . in powers of z; let the Vl — 2az — cz' result be denoted by Lagrange gives as a known result a simple relation which exists between every three consecutive coefficients ; namely . 2w-l . ,w-l . LAGEANGE. 303 This may be establislied by differentiation. For thus — -|±^ = ^,+ 2^,« + ... + «A«-+... (1 — zaz — cz Y that is {a + cz) {l + A^z-\-A/ + ... + As" + ••■} = (1 - 2az - cz') {A, + 2A^z + ... + « ^s""' +...}; then by equating coefficients the result follows. 559. In the second problem the same suppositions are made as in the first, and it is required to find the probability that the error of the mean of n observations shall not surpass + — . Like the first problem this leads to interesting algebraical ex- pansions. We may notice here a result which is obtained. Suppose we expand {a + h{x + x^)}" in powers of x; let the result be de- noted by A, + A,ix+ X-') + A, {a? + x-^) +A,{o^ + x-^) + ...; Lagrange wishes to shew the law of connexion between the co- efficients A^, A^, A^, ... This he effects by taking the logarithms of both sides of the identity and differentiating with respect to x. It may be found more easUy by putting 2 cos ^ for a; + a;"', and therefore 2 cos r6 for x'' -I- x^. Thus we have (ffl + 25 cos 0)" = J.„ + 2^j cos e + 2A^ cos 29 + 2 J, cos 35 + . . . Hence, by taking logarithms and differentiating, nb sin _ A,sme + 2^, sin 25 + 3^3 sin S0+... a + 2bcose~ A„ + 2A^cose + 2A^cos20+... Multiply up, and arrange each side according to sines of mul- tiples of ; then equate the coefficients of sin r0 : thus nh {A,., - A, J = raA, + h{{r-l) A,_, + (r+l) A, J ; therefore A,, = h(n-r + l)A -raA. 304 LAGRANGE. 560. In the third problem it is supposed that there are a cases at each observation in which no error is made, b cases in which an error equal to — 1 is made, and c cases in which an error equal to r is made ; the probability is required that the error of the mean of n observations shall be contained within given limits. In the fourth problem the suppositions are the same as in the third problem ; and it is required to find the most probable error in the mean of n observations ; this is a particular case of the fifth problem. 561. Inthe fifth problem it is supposed that every observation is subject to given errors which can each occur in a given number of cases ; thus let the errors be p, q, r, s, ... , and the numbers of cases in which they can occur be a, h, c,d, ... respectively. Then we require to find the most probable error in the mean of n ob- servations. In the expansion of (ax^ + ia? + cx'^ +...)" let M be the coeffi- cient of a^ ; then the probability that the sum of the errors is fi, and therefore that the error in the mean is - is n M {a+b + c+.-.y Hence we have to find the value of fi for which Mis greatest. Suppose that the error p occurs a times, the error q occurs /Q times, the error r occurs 7 times, and so on. Then a + yS + 7+ =n, pa + q^ + ry+ = fj,. It appears from common Algebra that the greatest value of /jl is when a_/3_7_ _ n a b c a + 6 + c+...' so that f,^ pa+qb + rc + ... ^ n a + 6 + c+ ... This therefore is the most probable error in the mean result. 562. With the notation of Art. 561, suppose that a, h, c, ... LAGRANGE. 305 are not known ^ priori; but that a, /3, 7, ... are known by ob- servation. Then in the sixth problem it is taken as evident that the most probable values of a,h, c, ... are to be determined from the results of observation by the relations a_ 5 _ c _ «~^~7~ ■■■' so that the value of - of the preceding Article may be written ft, _ pa. + q^ +r'y+ ... n~ a + /8 + 7+... Lagrange proposes further to estimate the probability that the values of a, h, c, ... thus determined from observation do not differ from the true values by more than assigned quantities. This is an investigation of a different character from the others in the memoir; it belongs to what is usually called the theory of in- verse probability, and is a difficult problem. Lagrange finds the analytical difficulties too great to be over- come; and he is obliged to be content with a rude approxi- mation. 563. The seventh problem is as follows. In an observation it is equally probable that the error should be any one of the following quantities -a, - (a - 1), ... - 1, 0, 1, 2 ... /3 ; required the probability that the error of the mean of n observations shall have an assigned value, and also the probability that it shall lie between assigned limits. We need not delay on this problem; it really is coincident with that in De Moivre as continued by Thomas Simpson : see Arts. 148 and 364. It leads to algebraical work of the same kind as the eighth problem which we will now notice. 564. Suppose that at each observation the error must be one of the following quantities — c, - (a — 1), • • • 0, 1, . . . a ; and that the chances of these errors are proportional respectively to 1, 2, ... a -1- 1, a, ... 2, 1 : required the probability that the error in the mean of n observations shall be equal to — . 20 306 LAGRANGE. We must find the coefficient of a?^ in the expansion of {x-" + 2a;-""'' + ... + ax-' + (a + l) x" + ax + ... + 2a;«-' + «"}", and divide it by the value of this expression when as = 1, which is the whole number of cases ; thus we obtain the required pro- bability. Now l+2x + 3a?+... + {a+l)x''+... + 2x'"'-' + a?" = (X + a= + ^+... + a^r=[^^) . Hence finally the required probability is the coefficient of a^ in the expansion of 1 a;-"° (1 - g!°^')'" . (a + I)''" (1 - a;)'" ' that is the coefficient of a;'^'^" in the expansion of (^i)-.(i-)-(i-0-. Lagrange gives a general theorem for effecting expansions, of which this becomes an example ; but it will be sufficient for our purpose to employ the Binomial Theorem. We thus obtain for the coefficient of of'^'^ the expression -— — 2^ {<^.(«« + M + 1) - 2« (wa+ /. + 1 - a - 1) 2n (2n - 1) , , , „ + — \ ^ ^ ,0(wa + /^+l-2a-2) 2« {2n - 1) (2« - 2) _, , , „ „, ^ 12 3 -^H + /* + l-3a-3)+... where ^ (r) stands for the product r{r+l) (r+2) ... (r-|-2M-2) ; the series within the brackets is to continue only so long as r is positive in (p (r). 565. We can see d, priori that the coefficient of of- is equal to the coefficient of x''^, and therefore when we want the former we may if we please find the latter instead. Thus in the result of LAGRANGE. 307 Art. 564, we may if we please put — /u, instead of fi, without changing the value obtained. It is obvious that this would be a gain in practical examples as it would diminish the number of terms to be calculated. This remark is not given by Lagrange. 566. We can now find the probability that the error in the mean result shall lie between assigned limits. Let us find the probability that the error in the mean result shall lie between and - , both inclusive. We have then to substitute in the n n expression of Article 564 for /* in succession the numbers - na, - (wa - 1), ... 7 - 1, 7, and add the results. Thus we shall find that, using S, as is customary, to denote a summation, we have 2n 2^(na + /* + l) = ;s- ■f (wa + 7+ 1), where "^(j) stands for r{r + l) (r + 2) ... (r+2w-l). When we proceed to sum ^ {no. +/* — «) we must remember that we have only to include the terms for which na + /j, — a is positive; thus we find 2^ {na + /J, — a) = -^ yjr {no. + y — a). Proceeding in this way we find that the probabihty that the error in the mean result will lie between and - , both in- n n elusive, is in- Jiln («« 4- 7 + 1) — 2re -vt- (ma + 7 + 1 — a — 1) (a+irii'^l ' ' r ^ ' 2n(2n-l) , , - _ „> + —^—2 — ' i/f (wa + 7 + l-2a-2) 2n(2«-l)(2«-2) ,. ^ ,. „ „v, \, ■ I 2 2 ■f{m+y+l-3ix-3) + ...y, 20—2 308 LAGRANGE. the series within the brackets is to continue only so long as r is positive in \jr (r). We will denote this by F{'y). The probability that the mean error will lie between /3 and y, where 7 is greater than /3, is ^^(7) - i^(/S) if we include 7 and exclude ^ ; it is F{ry — 1) — F{^ — 1) if we exclude 7 and include /3; it is F{y)-F{^-1) if we include both 7 and /3 ; it is F{y-1) -F{/3) if we exclude both 7 and ^. It is the last of these four results which Lagrange gives. We have deviated slightly from his method in this Article in order to obtain the result with more clearness. Our result is i^(7 — 1) -F{0); and the number of terms in ^(7-!) is de- termined by the law that ?• in -v^ (r) is always to be positive : the number of terms in F(ff) is to be determined in a. similar manner, so that the number of terms in. F (/3) is not necessarily so great as the number of terms in F (y — l). Lagrange gives an incorrect law on this point. He determines the number of terms in F{y — 1) correctly; and then he prolongs F{^) until it has as many terms as F [y — 1) by adding fictitious terms. 567. Let us now modify the suppositions at the beginning of Art. 564. Suppose that instead of the errors - a, — (a — 1), ... we are liable to the errors — ha, —h{o. — l), ... Then the investi- gation in Art. 564 gives the probability that the error in the mean result shall be equal to — ; and the investigation in Art. 566 gives the probability that the error in the mean result shall lie Bh yk between ^-— and — . Let a. increase indefinitely and k diminish n n indefinitely, and let c7c remain finite and equal to h. Let 7 and y8 also increase indefinitely ; and let 7 = ca and ^ = ha. where c and b are finite. We find in the limit that F{y)—F (/S) becomes . 1 j(5 + nr -2n{b + n- 1)" + ^" ^f'^~ ^\ b + n- 2)'" -. . .| ; each series is to continue only so long as the quantities which are raised to the power 2w are positive. LAGRANGE. 309 This result expresses the probability that the error in the mean result will lie between — and — on the following hv- n n 6 J pothesis ; at every trial the error may have any value between — Ji and + h ; positive and negative errors are equally likely ; the probability of a positive error z is proportional to h — z, and {h — z)hz. in fact ^ ~ — is the probability that the error will lie be- tween z and« + hz. We have followed Lagrange's guidance, and our result agrees with his, except that he takes 7i = \, and his formula involves many misprints or errors. 568. The conclusion in the preceding Article is striking. We have an exact expression for the probability that the error in the mean result will lie between assigned limits, on a very rea- sonable hypothesis as to the occurrence of single errors. Suppose that positive errors are denoted by abscissae measured to the right of a fixed point, and negative errors by abscissae measured to the left of that fixed point. Let ordinates be drawn representing the probabilities of the errors denoted by the re- spective abscissae. The curve which can thus be formed is called the curve of errors by Lagrange ; and as he observes, the curve becomes an isosceles triangle in the case which we have just discussed. 569. The matter which we have noticed in Arts. 563, 564, 566, 567, 568, had all been published by Thomas Simpson,''in his Miscellaneous Tracts, 1757; he gave also some numerical illus- trations : see Art. 371. 570. The remainder of Lagrange's memoir is very curious; it is devoted to the solution and exemplification of one general problem. In Art. 567 we have obtained a result for a case in which the error at a single trial may have any value between fixed limits ; but this result was not obtained directly : we started with the supposition that the error at a single trial must be one of a certain specified number of errors. , In other words we started with the hypothesis of errors changing per saltum, and passed on 310 LAGEANGE. to the supposition of contimimis errors. Lagrange wishes to solve questions relative to continuous errors without starting with the supposition of eiTors changing per scdtwm. Suppose that at every observation the error must lie between h and c; let (a;) dx denote the probability that the error will lie between x and x + dx: required the probability that in n obser- vations the sum of the errors wiU lie between assigned limits say /S and 7. Now what Lagrange effects is the following. He trans- forms \\ <^{x)a!°dx\- into \f{z)a'dz, where f{z) is a known function of s which does not involve a, and the limits of the integral are known. When we say that f{z) and the limits of s are known we mean that they are determined from the known function ^ and the known limits i and c. Lagrange then says that the probability that the sum of the errors will lie between yS and 7 is I f(z) dz. He apparently concludes that his readers will admit this at once ; he certainly does not demonstrate it. We will indicate presently the method in which it seems the de- monstration must be put. 571. After this general statement we will give Lagrange's first example. Suppose that ^ (a;) is constant = K say ; then (x) a" dxV = [ f{z) a' dz, where /(a) = - — — ^ \ {nc - a)""' -n{nc-z- f)""' n(n—V), _ .„_, ] + \ 2 ' {nc-z-2tY'-...Y, the series within brackets being continued only so long as the quantities raised to the power w — 1 are positive. Lagrange then says that the probability that the sum of the errors in n observations will lie between /S and 7 is /, V(«) dz. (3 572. The result is correct, for it can be obtained in another way. We have only to carry on the investigation of the problem enunciated in Art. 563 in the same way as the problem enunciated in Art. 564 was treated in Art. 567; the result will be very similar to those in Art. 567. Lagrange thus shews that his process is verified in this example. 312 LAGRANGE. 573. In the problem of Art. 670 it is obvious that the sum of the errors must lie between nh and nc. Hence f{z) ought to vanish if z does not lie between these limits; and we can easily shew that it does. For if s be greater than nc there is no term at all in f{z), for every quantity raised to the power w — 1 would be nega;tive. And if z be less than nb, then f{z) vanishes by virtue of the theorem in Finite Differences which shews that the n* difference of an algebraical function of the degree w — 1 is zero. This remark is not given by Lagrange. 574. We will now supply what we presume would be the demonstration that Lagrange must have had in view. Take the general problem as enunciated in Art. 570. It is not difficult to see that the following process would be suitable for our purpose. Let a be any quantity, which for convenience we may suppose greater than unity.. Find the value of the ex- pression j 1^ (£K J o^'i dx\ 1 1^ {x^) a'^^dxj^ Ucfy (a;„) a^» dxj^ , where the integrations are to be taken under the following limitations ; each variable is to lie between b and c, and the sum of the variables between z and z + Bz. Put the result in the form Pa'Sz ; then 1 Pdz is the required probability. Now to find P we proceed in an indirect way. It follows from our method that (re )n rnc \ 4>{x)(fdx[ =J Pa'dz. But Lagrange by a suitable transformation shews that I r

o It will be remembered that a may be ani/ quantity whicli LAGRANGE. 313 is greater than unity. We shall shew that we must then have Suppose that a„ is less than nh, and Sj greater than nc. Then we have rvb rm /•« f{z) a'dz + {f(z) - P] adz + /(a) a' dz = 0, for all values of a. Decompose each integral into elements ; put a^" = p. We have then ultimately a result of the following form a^o r„ + T^p + T^p' + T,p'+ ... in inf. ... = 0, where T^, T^,... are independent of p. And p may have any positive value we please. Hence by the ordinary method of in- determinate coefficients we conclude that T,=o, ?; = o, 2; = o, ... Thus -P=/(2)- The demonstration will remain the same whatever supposition be made as to the order of magnitude of the limits z^ and », compared with nh and nc. 575. Lagrange takes for another example that which we have already discussed in Art. 567, and he thus again verifies his new method by its agreement with the former. He then takes two new examples; in one he supposes that (x) = K "^ c' — af, the errors lying between — c and c; in the other he supposes that (j) (x) = Kcos x, the errors lying between - 2 and 2 . 576. We have now to notice another memoir by Lagrange which is entitled Recherches sur les suites recurrentes dont les termes varient de plusieurs manieres diff^rentes, ou sur Tintegra- tion des Equations lin/aires aux differences finies et partielles ; et sur Vusage de ces Equations dans la tMorie des hazards. This memoir is published in the Nouveaux Memoires de I'Acad. ... Berlin. The volume is for the year 1775; the date of pub- 314 LAGRANGE. lication is 1777. The memoir occupies pages 183 — 272 ; the ap- plication to the Theory of Chances occupies pages 240 — 272. 577. The memoir begins thus ; J'ai donn6 dans le premier Volume des MImoires de la Soci6t6 des Sciences de Turin une m6tliode nouvelle pour traiter la th6orie des suites recurrentes, en la faisant dgpendre de I'intggration des Equations liniaires aux diffierences finies. Je me proposois alors de pousser ces recherches plus loin et de les appliquer principalement k la solution de plusieurs problemes de lathSorie des hasards; mais d'autres objets m'ayant depuis fait perdre celui IS, de vue, M. de la Place m'a pr6venu en grand partie dans deux excellens MImoires sur les suites recurro-recurrentes, et sur I' integration des equations differentielles firms et leur usage dans la theorie des hasards, imprimis dans les Volumes Ti et vii des M6moires pr€sent6s S, I'Acadlmie des Sciences de Paris. Je crois cependant qu'on peut encore aj outer quelque chose au travail de cet illustre GSometre, et traiter le m^me sujet d'une maniere plus directe, plus simple et surtout plus g6nerale ; o'est I'objet des Recherches que je vais donner dans ce M6moire; on y trouvera des ni6thodes nouvelles pour I'intlgration des Equations linlaires aux differences finies et partielles, et 1' application de ces mSthodes 5, plusieurs problemes int6ressans du calcul des probability ; mais U n'est question ici que des Equations dont les coefficiens sont con- stants, et je reserve pour un autre MImoire I'examen de celles qui ont des coefficiens variables. 578. We shall not delay on the part which relates to the Integration of Equations ; the methods are simple but not so good as that of Generating Functions. We proceed to the part of the memoir which relates to Chances. 579. The first problem is to find the chance of the happening of an event h times at least in a trials. Let p denote the chance of its happening in one trial ; let ?/j,_j denote the probability of its happening t times ia x trials ; then Lagrange puts down the equation y^ct =py^.x,t.^ + (1 -p)yz.i,f He integrates and determines the arbitrary quantities and thus arrives at the usual result. In a Corollary he applies the same method to determine the LAGRANGE. 315 chance that the event shall happen just h times ; he starts from the same equation and by a different determination of the arbi- trary quantities arrives at the result which is well known, namely, \b \a-h Lagrange refers to De Moivre, page 15, for one solution, and adds : mais celle que nous venons d'en donner est non seulement plus simple, mais elle a de plus I'avantage d'etre d^duite de prin- cipes directs. But it should be observed that De Moivre solves the problem again on his page 27; and here he indicates the modern method, which is self-evident. See Art. 257. It seems curious for Lagrange to speak of his method as more simple than De Moivre's, seeing it involves an elaborate solution of an equation in Finite Differences. 580. Lagrange's second problem is the following : On suppose qu'& chaque coup il puisse arriver deux 6venemens dont les probabilitSs respectives sclent p et q; et on demande le sort d'un joueur qiu parieroit d'amener le premier de ces Ivenemens b fois au moins et le second c fois au moins, en un nombre a de coups. The enunciation does not state distinctly what the suppositions really are, namely that at every trial either the first event happens, or the second, or neither of them ; these three cases are mutually exclusive, so that the probability of the last at a single trial is 1 —p — q. It is a good problem, well solved ; the solution is presented in a more elementary shape by Trembley in a memoir which we shall hereafter notice. 581. The third problem is the following : Les m^mes choses Itant suppos6es que dans le Probleme ii, on de- mande le sort d'un joueur qui parieroit d'amener, dans un nombre de coups ind6termin6, le second des deux ^venemens b fois avant que le premier fdt arriv6 a fois. Let y3,,t be the chance of the player when he has to obtain the second event i times before the first event occurs x times. Then 316 LAGEANGE. This leads to \t + x-2 + ... ^ ;- l \x- This result agrees with the second formula in Art. 172. 582. The fourth problem is like the third, only three events may now occur of which the probabilities are p, q, r respectively. In a Corollary the method is extended to four events; and in a second Corollary to any number. To this problem Lagrange annexes the following remark : Le Probleme dont nous venons de donner une solution tr§s g6n6rale et trSs simple renferme d'une maniere g6nerale celui qu'on nomme com- mun^ment dans I'analyse des hasards le probleme des partis, et qui n'a encore 6t6 r&olu complettement que pour le cas de deux joueurs. He then refers to Montmort, to De Moivre's second edition, Problem VI, and to the memoir of Laplace. It is very curious that Lagrange here refers to De Moivre's second edition, while elsewhere in the memoir he always refers to the third edition ; for at the end of Problem vi. in the third edition De Moivre does give the general rule for any number of players. This he first published in his Miscellanea Analytica, page 210 ; and he reproduced it in his Doctrine of Chances. But in the second edition of the Doctrine of Chances the rule was not given in its natural place as part of Problem vi. but appeared as Problem LXIS. There is however some difference between the solutions given by De Moivre and by Lagrange ; the difference is the same as that which we have noticed in Art. 175 for the case of two players. De Moivre's solution resembles the first of those which are given in Art. 172, and Lagrange's resembles the second. It is stated by Montucla, page 397, that Lagi-ange intended to translate De Moivre's third edition into French. 683. Lagrange's fifth problem relates to the Duration of Play, in the case in which one player has unlimited capital ; this is De Moivre's Problem LXV: see Art. 807. Lagrange gives three solu- tions. Lagrange's first solution demonstrates the result given LAGRANGE. 317 without demonstration in De Moivre's second solution ; see Art. 309. We will give Lagrange's solution as a specimen of Ms methods. We may remark that Laplace had preceded Lagrange in the discussion of the problem of the Duration of Play. La- place's investigations had been published in the Memoires . . . par Divers Savans, Vols. vi. and vii. Laplace did not formally make the supposition that one player had unlimited capital, but we arrive at this case by supposing that his symbol i denotes an infinite number ; and we shall thus find that on page 158 of Laplace's memoir in Vol. VII. of the M^7noires...par Divers Savans, we have in effect a demonstration of De Moivre's result. We proceed to Lagrange's demonstration. 584. The probability of a certain event in a single trial is^ ; a player bets that in a trials this event will happen at least b times oftener than it fails : determine the player's chance. Let 2/xt represent his chance when he has x more trials to make, and when to ensure his success the event must happen at least t times oftener than it fails. Then it is obvious that we re- quire the value of y„_j. Suppose one more trial made ; it is easy to obtain the follow- ing equation 2/..t=P2/x-i,t-i + (1 -p) y^-i.t+1- The player gains when t = and x has any value, and he loses when X = and t has any value greater than zero ; so that y^, „ = 1 for any value of a-, and i/^,t= for any value of t greater than 0. Put 2 for 1 —p, then the equation becomes To integrate this assume y = Aa''^' ; we thus obtain p-a^ + ql3^= 0. From this we may by Lagrange's Theorem expand ^ in powers of a ; there will be two series because the quadratic equation gives two values of /8 for an assigned value of a. These two series are 318 LAGRANGE. '^"a'"*' a'*" "^ 1.2 a'"^ 1.2.3 a"^ "^ ' <-« g' <;pa'-^ <(i-3) p'a'-^ <(<-4) (< - 5) pV 2' 2'' 1-2 2*"' 1-2.3 2'-^ '^•" If then we put in succession these values of ^ in the ex- pression A(f /S' we obtain two series in powers of a, namely, A-p' L'-' + t'pqtr'-^ + '-^p^ /gV-'-* + . . .| , and ^2"' ja"* - ^p^a"^*^ + ^-y^^ p^a^^ - • • j ■ Either of these series then would be a solution of the equation in Finite Differences, whatever may be the values of A and a ; so that we should also obtain a solution by the sum of any number of such series with various values of A and a. Hence we infer that the general solution will be y.., = P' {/(aJ -t) + tpqfix -t-2)+ ^.^t^pYfix - < - 4) + 1.2.3 ■yg/(a' - < - 6) + . . . ■ + q-'U {x + t)-tpq{x + t-2) + *\~^^ pY ^{x+t-i) Here f{x) and ^{x) represent functions, at present arbitrary, which must be determined by aid of the known particular values of y-.o and y„.,. Lagrange says it is easy to convince ourselves, that the con- dition y„_,=0 when t has any value greater than leads to the following results : all the functions with the characteristic ^ must be zero, and those with the characteristic f must be zero for all negative values of the quantity involved. [Perhaps this will not appear very satisfactory ; it may be observed that q^ will become indefinitely great with t, and this suggests that the series which multiplies c[* should be zero.] Thus the value of y^^t becomes a series with a finite number of terms, namely, y^,i =p* LAGRANGE. 319 fix- t) + tpqf{x - « - 2) + ^Jt^j>Yf(^^ _ « _ 4) the series extends to ^ {x-t + 2) terms, or to ^ (a;- < + 1) terms, according as a; — i is even or odd. The other condition is that ?/^_„ = 1, for any value of x. But if we put < = we have y^.o =/('*')• Hence /(a?) = 1 for every positive value of x. Thus we obtain 1 1 the series is to extend to 5 (tc — < + 2) terms, or to ^ (x — t + 1) ^ 2 terms. This coincides with the result in De Moivre's second form of solution: see Art. 309. 585. Lagrange gives two other solutions of the problem just considered, one of which presents the result in the same form as De Moivi-e's first solution. These other two solutions by Lagrange differ in the mode of integrating the equation of Finite Differences ; but they need not be further examined. 586. Lagrange then proceeds to the general problem of the Duration of Play, supposing the players to start with different capitals. He gives two solutions, one similar to that in De Moivre's Problem LXIII, and the other similar to that in De Moivre's Problem LXVIII. The second solution is very remarkable ; it demonstrates the results which De Moivre enunciated without demonstration, and it puts them in a more general form, as De Moivre limited himself to the case of equal capitals. 587. Lagrange's last problem coincides with that given by Daniel Bernoulli which we have noticed in Art. 417. Lagrange supposes that there are n urns ; and in a Corollary he gives some modifications of the problem. 588. Lagrange's memoir would not now present any novelty tp a student, or any advantage to one who is in possession of the method of Generating Functions. But nevertheless it may be read 320 LAGRANGE. with ease and interest, and at the time of publication its value must have been great. The promise held out in the introduction that something would be added to the labours of Laplace is abundantly fulfilled. The solution of the general problem of the Duration of Play is conspicuously superior to that which Laplace had given, and in fact Laplace embodied some of it subsequently in his own work. The important pages 231 — 233 of the Theorie ...des Prob. are substantially due to this memoir of Lagrange's. 589. We may notice a memoir by Lagrange entitled Mi- moire sur une. question concerncmt les annuites. This memoir is published in the volume of the Memoires de V Acad. ...Berlin for 1792 and 1793; the date of publication is 1798 ; the memoir occupies pages 235 — 246. The memoir had been read to the Academy ten years before. 590. The question discussed is the following: A father wishes to pay a certain sum annually during the joint continuance of his own life and the minority of all his children, so as to ensure an annuity to his children after his death to last until all have attained their majority. Lagrange denotes by A, B, C, ... the value of an annuity of one crown for the minority of the children A, B, C ... respectively. Then by AB he denotes the value of an annuity of one crown for the joint minority of two children A and B ; and so on. Hence he obtains for the value of an annuity payable as long as either ^ or jB is a minor, l + B-AM. Lagrange demonstrates this ; but the notation renders it almost obviously self evident. Similarly the value of an annuity payable as long as one of three children A, B, remains a minor is A + B + G - AB - AC - BC + ABC. De Moivre however had given this result in his Treatise of Annuities on Lives, and had used the same notation for an annuity on joint lives. Lagrange adds two tables which he calculated from his formulae, using the table of mortality given in the work of Sussmilch. CHAPTER XVI. MISCELLANEOUS INVESTIGATIONS Between the Yeaks 1750 and 1780. 591. The present Chapter will contain notices of various con- tributions to our subject which were made between the years 1750 and 1780. 592. We first advert to a work bearing the following title : Piece qui a remporU le prix sur le sujet des Evenemens Fortuits, propose par VAcademie Royale des Sciences et Belles Lettres de Berlin pour I'ann^e 1751. Avec les pieces qui ont concouru. This work is a quarto volume of 238 pages ; we notice it because the title might suggest a connexion with our subject, which we shall find does not exist. The Academy of Berlin proposed the following subject for dis- cussion : Les Evenemens heureux et mallieureux, ou ce que nous appellons Bonheur et Malheur dependant de la volont6 ou de la permission de Dieu, de sorte que le terme de fortune est un nom sans r6alitl; on de- mande si ces Evenemens nous obligent k de certains devoirs, quels sont ces devoirs et quelle est leur Itendue. The prize was awarded to Kaestner professor of Mathematics at Leipsic ; the volume contains his dissertation and those of his competitors. There are nine dissertations on the whole ; the prize disserta- tion is given both in French and Latin, and the others in French 21 322 DODSON. HOYLE. or German or Latin. The subject was perhaps unpromising ; the dissertations are not remarkable for novelty or interest. One of the best of the writers finishes with a modest avowal which might have been used by all : Ich mache hier den Schluss, well ich ohnehin mit gar zu guten Griinden. furchte, zu weitlaufig gewesen zu seyn, da ich so wenig neues artiges und scharfsinniges gesagt habe. Ich. finde auch in dieser Probe, das3 mein "Wille noch einmalil so gut als meine ubrige Fahigkeit, ist. 593. A work entitled the Mathematical Repository, in three volumes, was published by James Dodson, Accomptant and Teacher of the Mathematics. The work consists of the solution of Mathe- matical problems. The second volume is dated 1753 ; pages 82 — 136 are occupied with problems on chances : they present nothing that is new or important. The remainder of this volume is devoted to annuities and kindred subjects ; and so also is the whole of the third volume, which is dated 1755. 594. Some works on Games of Chance are ascribed to Hoyle in Watt's Bihliotheca Britannica. I have seen only one of them which is entitled: An Essay towards making the Doctrine of Chances easy to those who understand Vulgar Arithmetick only: to which is added, some useful tables on annuities for lives (he. dsc. &c. By Mr Hoyle. .. It is not dated ; but the date 1754 is given in Watt's Bihliotheca Britannica. The work is in small octavo size, with large type. The title, preface, and dedication occupy viil pages, and the text itself occu- pies 73 pages. Pages 1 — 62 contain rules, without demonstration, for calculating chances in certain games ; and the remainder is de- voted to tables of annuities, and to Halley's Breslau table of life, with a brief explanation of the latter. I have not tested the rules. 695. We advert in the next place to a work which is en- ■ titled DeW Azione del Caso nelle Invenzioni, e delV influsso degli Astri ne Corpi Terrestri Bissertazioni due. This is a quarto volume of 220 pages, published anonymously at Padua, 1757. It is not connected with the Theory of Pro- bability; we notice it because the title might perhaps suggest BAMUEL CLARK. 323 Buch connexion, especially when abbreviated, as in the Catalogues of Booksellers. The first dissertation is on the influence of chance in inven- tions, and the second on the influence of the celestial bodies on men, animals, and plants. The first dissertation recognises the influence of chance in inventions, and gives various examples ; the second dissertation is intended to shew that there is no influence produced by the celestial bodies on men, animals, or plants, in the sense in which astrologers understood such influence. The author seems to have been of a sanguine temperament ; for he obviously had hopes that the squaring of the circle would be eventually obtained ; see his pages 31, 40, 85. On the other hand his confidence is not great in the Newtonian theory of gravitation ; he thinks it may one day follow its prede- cessor, the theory of vortices, into oblivion ; see his pages 45, 172. The following is one of his arguments against Lunar influence. If there be such influence we must conceive it to arise from exhala- tions from the Moon, and if the matter of these exhalations be supposed of appreciable density it will obstruct the motions of the planets, so that it will be necessary from time to time to clean up the celestial paths, just as the streets of London and Paris are cleaned from dust and dirt. See his page 164. The author is not very accurate in his statements. Take the following specimen from his page 74 : Jacopo III. Ke d'Inghilterra alia vista d'una spada ignuda, come riferisce il Cavaliere d'Igby, sempre era compreso d'un freddo, e ferale spavento. This of course refers to James I. Again ; we have on his page 81 : ...cib che disse in lode d'Aiistotile il Bemi : II gran Maestro de color che sanno. It is not often that an Italian ascribes to any inferior name the honour due to Dante. 596. We have next to notice a work by Samuel Clark en- titled The Laws of Chance : or, a Mathematical Investigation of the Probabilities arising from any proposed Circumstance of Play. London, 1758. This is in octavo; there is a Preface of 2 pages, and 204 pages of text. The book may be described as a treatise based on those of De Moivre and Simpson; the abstruse problems are 21—2 324 SAMUEL CLARK. omitted, and many examples and illustrations are given in order to render the subject accessible to persons not veiy far advanced in mathematics. The book presents nothing that is new and important. The game of bowls seems to have been a favourite with Clark ; he devotes his pages 44 — 68 to problems connected with this game. He discusses at great length the problem of finding the chance of throwing an assigned number of points with a given number of similar dice; see his pages 113 — 130. He follows Simpson, but he also indicates De Moivre's Method ; see Art. 364. Clark begins the discussion thus : In order to facilitate the solution of this and the folio-wing problem, I shall lay down a lemma which was communicated to me by my inge- nious friend Mr William Payne, teacher of mathematics. The Lemnia. The sum of 1, 3, 6, 10, 15, 21, 28, 36, &c. continued to (ji) number ., . ,,w + 2 n + \ n 01 terms is equal to — = — x — ^r- x ^ . It was quite unnecessary to appeal to William Payne for such a well-known result ; and in fact Clark himself had given on his page 84 Newton's general theorem for . the summation of series ; see Art. 152. Clark discusses in his pages 139 — 153 the problem respecting a run of events, which we have noticed in Art. 325. Clark detects the slight mistake which occurs in De Moivre's solution ; and from the elaborate manner in which he notices the mistake we may conclude that it gave him great trouble. , Clark is not so fortunate in another case in which he ventures to differ with De Moivre ; Clark discusses De Moivre's Problem IX. and an-ives at a different result ; see Art. 269. The error is Clark's. Taking De Moivre's notation Clark assumes that A must either receive q G from B, or pay pL to B. This is wrong. Sup- pose that on the whole A wins va. q + m trials and loses in m trials ; then there is the required difference of q games in his favour. In this case he receives from B the sum {q + m) G and pays to him the sum mL ; thus the balance is qG -\- m [G — L) and not qG as Clark says. MALLET. JOHN BERNOULLL 325 597. We have next to notice a memoir by Mallet, entitled Recherches sur les avantages de trois Joueurs qui font entr'eux une Poule au trictrac ou d un autre Jeu quelconque. This memoir is published in the Acta Helvetica... BasilecB, Vol. V. 1762 ; the memoir occupies pages 230 — 248. The problem is that of De Moivre and Waldegrave; see Art. 211. Mallet's solution resembles that given byDe Moivre in his pages 132 — 138. Mallet however makes some additions. In the problem as treated by De Moivre the fine exacted from each defeated player is con- stant; Mallet considers the cases in which the fines increase in arithmetical progression, or in geometrical progression. A student of De Moivre will see that the extensions given by Mallet can be treated without any difficulty by De Moivre's process, as the series which are obtained may be summed by well-known methods. 598. The same volume which contains Euler's memoir which we have noticed in Art. 438, contains also two memoirs by Beguelin on the same problem. Before we notice them it will be convenient to consider a memoir by John Bernoulli, which in fact precedes Beguelin's in date of composition but not in date of publication. This John Bernoulli was grandson of the John whom we named in Art. 194. John Bernoulli's memoir is entitled Sur les suites ou sequences dans la loterie de Genes. It was published in the volume for 1769 of the Histoire de I' Acad Berlin; the date of pub- lication is 1771 : the memoir occupies pages 234 — 253. The fol- lowing note is given at the beginning : Ce Memoire a 6te lu en 1765, aprSs le Mgmoire de Mr. Euler sur cette matiere ins6r6 dans les Mgmoires de I'Acadlmie pour cette annle. Comme les Memoires de Mr. Beguelin imprimis a la suite de celui de Mr. Euler se rapportent au mien en plusieurs endroits, et que la Loterie qui I'a occasion^ est plus en vogue que jamais, je ne le supprimerai pas plus longtems. Si ma mithode ne mene pas aussi loin que celle de Mrs. Euler et Beguelin, elle a du moins, je crois, I'avantage d'etre plus facile k saisir. 599. In the first paragraph of the memoir speaking of the question respecting sequences, John Bernoulli says : Je m'en occupai done de terns en terns jusqu'l, ce que j'appris de Mr. Euler qu'il traitoit le m^me sujet; e'en fut assez pour me faire 326 JOHN BERNOULLI. abandonner mon dessein, et je me reservai seulement de voir par le MImoire de cet illustre Geometre si j'avoia raisonng juste; il a eu la bonte de me le communiquer et j'ai vu que le peu que j'avois fait, 6toit fondS sur des raisonnemena qui, s'il* n'6toieiit, pas sublimes, n'etoieut du Oioins pas faux. 600. John Bernoulli does not give an Algebraical investiga- tion ; lie confines himself to the arithmetical calculation of the chances of the various kinds of sequences that can occur when there are 90 tickets and 2 or 3 or 4 or 5 are drawn. His method does not seem to possess the advantage of facility, as compared with those of Euler and Beguelin, which he himself ascribes to it. 601. There is one point of difference between John Bernoulli and Euler. John Bernoulli supposes the numbers from 1 to 90 ranged as it were in a circle ; and thus he counts 90, 1 as a binary sequence ; Euler does not count it as a sequence. So also John Bernoulli counts 89, 90, 1 as a ternary sequence; with Euler this would count as a binary sequence. And so on. It might perhaps have been anticipated that from the greater symmetry of John Bernoulli's conception of a sequence, the in- vestigations respecting sequences would be more simple than on Euler's conception ; but the reverse seems to be the case on ex- amination. In the example of Art. 440 corresponding to Euler's results ._2, (.-2)(„-3), (r.-2)in-Snn-^) ^ we shall, find on John Bernoulli's conception the results n, n(n- 4), -^^^ g ^ . 602. There is one Algebraical result given_ which we may notice. Euler had obtained the following as the chances that there would be no sequences at all in the case of n tickets; if twa tickets be drawn the chance is , if three ~ ' ^~ ' if n w(«-l)' ' . {n-^){n-5){n-Q) (n-5). (»^- 6) (^-7) (n- 8)' . n{n-l) in-2) ' " ^^^ n(»i-l) («,-2) (^-3) ' and so the law can be- easily seen. Now John. Bernoulli states JOHN BERNOULLI. 327 that on his conception of a sequence these formulsfe will hold if we change n into n — l. He does not demonstrate this statement, so that we cannot say how he obtained it. It may be established by induction in the following way. Let E in, r) denote the number of ways in which we can take r tickets out of n, free from any sequence, on Euler's conception of a se- quence. Let B (n, r) denote the corresponding number on John Bernoulli's conception. Then we have given E (n, r) = (n-r+l){n-r)...{n-2r + 2) ^ and we have to shew that „. . n{n-r-l) ... {n-2r + l) For these must be the valties of E (n, r) and B (n, r) in order that the appropriate chances may be obtained, by dividing by the total num.ber of cases. Now the following relation will hold : E {n,r) = B (n, r) + B (n -1, r -1) - E {n- 2, r - 1). The truth of this relation will be seen by taking an example. Suppose n is 10, and r is 3. Now every case which occurs in the total B {n, r) will occur among the total E [n, r) ; but some which do not occur in B («, r) will occur in E [n, r), and these must be added. These cases which are to be added are such as (10, 1, 3) (10, 1, 4) (10, 1, 8). We must then examine by what general law we can obtain these cases. We should form aU the binary combinations of the numbers 1 . 2, ... 9 which contain no Bemoullian sequence, and which do contain 1. And generally we should want all the combinations r - 1 at a time which can be made from the first w - 1 numbers, so as to con- tain no Bemoullian sequence, and to contain 1 as one of the num- bers. It might at first appear that ^(w-1, r-l)-B{n-2, r-l) would be the number of such combinations ; but a little con- sideration will shew that i\,isB{n-l,r-l)-E{n-2,r-l), as we have given it above. Thus having estabUsh^d the relation, and found the value Of B{n 1) independently we can infer iH succession the values of B{n,2), B{n,S), and so on. 328 BEGUELIN. 603. "We now consider Beguelin's two memoirs. These as we have stated are contained in the same volume as Euler's memoir noticed in Art. 438. The memoirs are entitled 8ur les suites ou sequences dans la lotterie de Genes ; they occupy pages 231 — 280 of the volume. 604. Beguelia's memoirs contain general Algebraical formulae coinciding with Euler's, and also similar formulse for the results on John Bernoulli's conception; thus the latter formulse constitute what is new in the memoirs. 605. We can easily give a notion of the method which Beguelin uses. Take for example 13 letters a, b, c, ...i,j, k, I, m. Arrange 5 files of such letters side by side, thus a a a a a h I b I b c c G c c m m m m m Consider first only two such files ; take any letter ia the first file and associate it with any letter in the second file ; we thus get 13" such associations, namely aa, ah, ac ... ba, bb, be, ... Here we have ah and ba both occurring, and so ac and ca, and the like. But suppose we wish to prevent such repetitions, we can attain our end in this way. Take any letter in the first file and associate it with those letters only in the second file, which are in the same rank or in a lower rank. Thus the a of the first file will be associated with any one of the 13 letters of the second file ; the b of the second file wUl be associated with any one of the 12 letters in the second file beginning with b. Thus the whole number of 13 X 14 such associations wUl be 13 + 12 + . . . + 1 ; that is — = — ^— . Similarly if we take three files we shall have 13' associations if we allow repetitions ; but if we do not allow repetitions we 13 X 14 X 15 shaU have -^j — ^ — 5— . Proceeding in this way we find that if J. X .Z X o there are five files and we do not allow repetitions the number of 13 X 14 X 15 X 16 X 17 associations is 1x2x3x4x5 BEGUELIN. 329 All this is well known, as Beguelin says, but it is introduced by him as leading the way for his further investigations. 606. Such cases as a, a, a, a, a cannot occur in the lottery because no number is there repeated. Let the second file be raised one letter, the third file two letters; and so on. Thus we have a h c d e h c d e f i 3 k I m ,7 h I m h I m I m m We have thus 13 — 4 complete files, that is 9 complete files ; and, proceeding as before, the number of associations is found to be 9 X 10 X 11 X 12 X 13 ^^^^ . ^j^g number is what we know to 1x2x3x4x6 ' be the number of the combinations of 13 things taken 5 at a time. 607. Suppose now that we wish to find the number of asso- ciations in which there is no sequence at all. Raise each file two letters instead of one, so that we now have a c e 9 % b d f h 3 c e 9 i k d f h 3 I e 9 { k m / h 3 I ff i k m h J I i k m J I k m I m 330 BEGUELIN. Here there are only 13 — 8, that is, 5 complete files ; and proceeding as in Art. 605, we find that the whole number of asso- 5x6x7x8x9 ciations IS 1x2x3x4x5 In this way we arrive in fact at the value which we quoted for E{n, r) in Art. 602. 608. The method which we have here briefly exemplified is used by Beguelin in discussing all the parts of the problem. He does not however employ letters as we have done ; he supposes a series of medals of the Roman emperors, and so instead of a, h, c,...he uses Augustus, Tiberius, Caligula, ... 609. It may be useful to state the results which are obtained when there are n tickets of which 5 are drawn. In the following table the first column indicates the form, the second the number of cases of that form according to Euler's conception, and the third the number according to John Ber- noulli's conception. Sequence of 5, w — 4, n. Sequence of 4, (^_5)(„_4), n{n-G). Sequence of 3 combined with a sequence of 2, („_5) («-4.), w(m-6). Sequence of 3, and the other numbers not (n -6)(«-5)(?i-4) 1.2 n(n-*7) (n-6) 1.2 in sequence, Two sequences of 2, (n -6)(?^-5)(«-4) 1.2 n{n-*7) in-6) 1.2 Single sequence of 2, (n- -7) (n-6) (w-5) (n-4y 1.2.3 n(w-8)(w-7)(«-6) 1.2.3 No sequence, see Art. 602. BEGUELIN. 331 The chance of any assigned event is found by dividing the corresponding number by the whole number of cases, that is by the number of combinations, of n things taken 5 at a time. 610. "We have now to notice another memoir by Beguelin. It is entitled, 8ur V usage du principe de la raison suffisawte dans h calcul des probabilitSs. This memoir is published in the volume of the Histoire de rAcad....BerM,n ior 1767; the date of publication is 1769: the memoir occupies pages 382 — 412. 611. Beguelin begins by saying, J'ai montrd dans un M^moire pr^c^dent que la doctrine des probabilit^s ^toit uniquement fondle sur le principe de la raison suffisante : this refers apparently to some remarks in the memoirs which we have just examined. Beguelin refers to D'Alembert in these words. Un illustre Auteur, G^ometre et Philosophe k la fois, a public depuis peu sur le Calcul des probabilit^s, des doutes et des questions bien dignes d'etre approfondies ... Beguelin proposes to try how far meta- physical principles can assist in the Theory of Probabilities. 612. Beguelin discusses two questions. The first he says is the question : ... si les 6v6iiemeiis simmltriques et r6guliers, attribu^s au hazard,, sent (toutes choses d'ailleurs 6gales) aussi probables que les 6vlnemen3 qui n'ont ni ordre ni r6gularit6, et au cas qu'ils aient le m^me degrS de probability, d'oii vient que leur r6gularit6 nous frappe, et qu'ils nous paroissent si singuliers ? His conclusions on this question do not seem to call for any remark. 613. His next question, he considers more difficult ; it is .... lorsqu'un m^me Iv&ement est deja arrivl une ou plusieurs fois de suite, on demande si. cet 6vlnement conserve autant de probability pour sa future existence, que I'dv&ement contraire qui aveo une Sgale probability primitive n'est point arriv6 encore. Beguelin comes to the conclusion that the oftener an event has happened the less likely it is to happen at the next trial ; 332 BEGUELIN. thus he adopts one of D'Alembert's errors. He considers that if the chances would have been equal according to the ordinary theory, then when an event has happened t times in succession it is i + 1 to 1 that it will fail at the next trial. 614. Beguelin applies his notions to the Petersburg Problem. Suppose there are to be n trials; then instead of -^ which the common theory gives for the expectation Beguelin arrives at 112 2^ 2' 2""^ 2^2^2 + 1^2.3 + 1 [4 + 1 ■■■ |w-l + l' The terms of this series rapidly diminish, and the sum to infinity is about 2^. 615. Besides the above result Beguelin gives five other solutions of the Petersburg Problem. His six results are not coincident, but they all give a small finite value for the expecta- tion instead of the large or infinite value of the common theory. 616. The memoir does not appear of any value whatever; Beguelin adds nothing to the objections urged by D'Alembert against the common theory, and he is less clear and interesting. It should be added that Montucla appears to have formed a different estimate of the value of the memoir. He says, on his page 403, speaking of the Petersburg Problem, Ce problSme a gte aussi le sujet de savantes considerations m6taphy- siques pour Beguelin... ce metaphysicien et analyste examine au flam- beau d'une m^taphysique profonde plusieurs questions sur la nature du calcul des probabilites... 617. We have next to notice a memoir which has attracted considerable attention. It is entitled An Inquiry into the pro- bable Parallax, and Magnitude of the fixed Stars, from the Quantity of Light which they afford us, and the particular Circwmstances of their Situation, by the Eev. John Michell, B.D., F.R.S. This memoir was published in the Philosophical Transactions, Vol. LVil. Part I., which is the volume for 1767 : the memoir occupies pages 234 — 264. MICHELL. 333 618. The part of the memoir with which we are concerned is that in which Michell, from the fact that some stars are very- close together, infers the existence of design. His method will be seen from the following extract. He says, page 243, Let us then examine what it is probable would have been the least apparent distance of any two or more stars, any where in the whole heavens, upon the supposition that they had been scattered by mere chance, as it might happen. Now it is manifest, upon this supposition, that every star being as likely to be in any one situation as another, the probabiUty, that any one particular star should happen to be within a certain distance (as for example one degree) of any other given star, would be represented (according to the common way of computing chances) by a fraction, whose numerator would be to it's denominator, as a circle of one degree radius, to a circle, whose radius is the diameter of a great circle (this last quantity being equal to the whole surface of (60')' the sphere) that is, by the fraction .^„ _ , ^ , or, reducing it to a deci- mal form, "000076154 (that is, about 1 in 13131) and the complement 13130 of this to unity, viz. '999923846, or the fraction , will represent the probability that it would not be so. But, because there is the same chance for any one star to be within the distance of one degree from any given star, as for every other, multiplying this fraction into itself as many times as shall be equivalent to the whole number of stars, of not less brightness than those in question, and putting n for this number, (13130\" j will represent the probability, that no one of the whole number of stars n would be within one de- gree from the proposed given star ; and the complement of this quan- tity to unity will represent the probability, that there would be some one star or more, out of the whole number n, within the distance of one degree from the given star. And farther, because the same event is equally likely to happen to any one star as to any other, and there- fore any one of the whole number of stars n might as well have been taken for the given star as any other, we must again repeat the last found chance n times, and consequently the number {(-999923846)"}", r/13130\"')° or the fraction < ( ■ j > will represent the probability, that no where, in the whole heavens, any two stars, amongst those in question, would be within the distance of one degi'ee from each other j and the 33i MICHELL. complement of this quantity to unity will represent the probability of the contrary. 619. Michell obtains the following results on his page 246, If now we compute, according to the principles above laid down, what the probability is, that no two stars, in the whole heavens, should have been within so small a distance from each other, as the two stars yS Capricorni, to which I shall suppose about 230 stars only to be equal in brightness, we shall find it to be about 80 to 1. For an example, where more than two stars are concerned, we may take the six brightest of the Pleiades, and, supposing the whole number of those stars, which are equal in splendor to the faintest of these, to be about 1500, we shall find the odds to be near 500000 to 1, that no six stars, out of that number, scattered at random, in the whole hea- vens, would be within so small a distance from each other, as the Plei- ades are. Michell gives the details of the calculation in a note. 620. Laplace alludes to Michell in the Thiorie . . . des Proh., page LXIII., and in the Gonnaissance des Terns for 1815, page 219. 621. The late Professor Forbes wrote a very interesting criti- cism on Michell's memoir; see the London, Edinburgh and Dublin Philosophical Magazine, for August 1849 and December 1850. He objects with great justice to Michell's mathematical calculations, and he also altogether distrusts the validity of the inferences drawn from these calculations. 622. Struve has given some researches on this subject in his Catalogus I^ovus Stellarum Duplicium et Multiplicium . . . Dorpati, 1827, see the pages xxxvii. — xlviii. Struve's method is very different from Michell's. Let n be the number of stars in a given area S of the celestial surface ; let

(w) in Art. 161. We get by this law ^, = 9, ^, = 44, A = 265, .4,= 1854., A = 14833, ... We can however easily demonstrate the law independently of Art. 16L Let A''[0 stand for fr - r |r - 1 + '' ^!'"^-^^ \r-2 - , so that the notation is analogous to that which is commonly used in Finite Differences. Then the fundamental relation (1) sug- gests that A=A-10; (3), and we can shew that this is the case by an inductive proof For we find by trial that A''[0 = L0 = 1 = .4„, A'[p= 1 -1 = = J„ A^10= 2 -2 + 1=^,; and then from the fundamental relation (1) it follows that if u4, = A*" [0 for all values of r up to w — 1 inclusive, then A^ = A" [0. Thus (3) is established, and from (3) we can immediately shew that (2) holds. 628. We now come to another memoir by the writer whom we have noticed in Art. 597. The memoir is entitled Swr le Calcul des ProhaMliUs, par Mr. Mallet, Prof. dJAstronomie d, Geneve. This memoir is published in the Acta Helvetica ... Basilece, Vol. vii. ; the date of publication is 1772 : the memoir occupies pages 133—163. 22 338 MALLET. 629. The memoir consists of the discussion of two problems : the first is a problem given in tbe Ars Gonjectandi of James Ber- noulli ; the other relates to a lottery. 630. The problem from the Ars Cotijectandi is that which is given on page 161 of the work ; we have given it in Art. 117. Mallet notices the fact that James Bernoulli in addition to the correct solution gave another which led to a different result and was therefore wrong, but which appeared plausible. MaUet then says, Mr. Bernoulli s'^tant contenti d'indiquer cette singularity apparente, sans en donner I'explication, j'ai cru qu'il ne seroit pas inutile d'entrer dans un plus grand difcail lS,dessus, pour 6claircir parfaitement cette petite difficult^, on verra qu'on peut imaginer une infinite de cas sem- Hables h, celui de Mr. Bernoulli, dans la solution desquels H seroit aussi ais6 d'etre induit en erreur. 631. Mallet's remarks do not appear to offer any thing new or important ; he is an obscure writer for want of sufficiently develop- ing his ideas. The following illustration was suggested on reading his memoir, and may be of service to a student. Suppose we refer to the theory of duration of life. Let abscissae measured from a fixed point denote years from a certain epoch, and the cor- responding ordinates be proportional to the number of survivors out of a large number bom at the certain epoch. Now suppose we wish to know whether it is more probable than not that a new born infant wiU live more than n years. James Bernoulli's plausi- ble but false solution amounts to saying that the event is more probable than not, provided the abscissa of the centre of gravity of the area is greater than n : the true solution takes instead of the abscissa of the centre of gravity the abscissa which corresponds to the ordinate bisecting the area of the curve. See Art. 485. 632. We pass to MaUet's second problem which relates to a certain lottery. The lottery is that which was called by Montmort la lotterie de Loraine, and which he discussed in his work ; see his pages 257—260, 313, 317, 326, 346. The following is practically the form of the lottery. The director of the lottery issues n tickets to MALLET. 339 n persons, charging a certain sum for each ticket. He retains for himself a portion of the money which he thus receives, say a; the remainder he distributes into n prizes which will be gained by those who bought the tickets. He also offers a further inducement to secure buyers of his tickets, for he engages to return a sum, say i, to every ticket-holder who does not gain a prize. The prizes are distributed in the following manner. In a box are placed n coun- ters numbered respectively from 1 to w. A counter is drawn, and a prize assigned to the ticket-holder whose number corresponds to the number of the counter. The cotmter is then replaced in the box. Another drawing is made and a prize assigned to the corresponding ticket-holder. The coimter is then replaced in the box. This pro- cess is carried on until n drawings have been made ; and the prizes are then exhausted. Hence, owing to the peculiar mode of drawing the lottery, one person might gain more than one prize, or even gain them all ; for the counter which bears his number might be drawn any number of times, or even every time. The problem proposed is to find the advantage or disadvantage of the director of the lottery. 633. Montmort solved the problem in the following manner. Consider one of the ticket-holders. The chance that this per- son's number is never drawn throughout the whole process is [!i^^ J . If it is not drawn he is to receive b from the director; /« — 1\" so that his corresponding expectation is b ( j . A similar ex- pectation exists for each of the ticket-holders, and the sum of these expectations is the amount by which the director's gain is di- minished. Thus the director's advantage is «-"K^r)- In the case which Montmort notices b was equal to a, and n was 20000 ; thus the director's advantage was negative, that is, it was really a disadvantage. Before Montmort made a complete investigation he saw that the director's position was bad, and he 22—2 340 MALLET. suspected that there was a design to cheat the public, which actually happened. 634. Mallet makes no reference to any preceding writer on the subject ; but solves the problem in a most laborious manner. He finds the chances that the number of persons without prizes should be 1, or 2, or 3, ... up to w ; then he knows the advantage of the banker corresponding to each case by multiplying the chance by the gain in that case ; and by summing the results he obtains the total advantage. 635. One part of MaUet's process amounts to investigating the following problem. Suppose a die with r faces; let it be thrown s times in succession : required the chance that all the faces have appeared. The number of ways in which the desired event can happen is ^-,(,_i).+!i^(,_2).- -(--_i)_(;-^) (.-3)'+... and the chance is obtained by dividing this number by r*. This is De Moivre's Problem xxxix ; it was afterwards dis- cussed by Laplace and Euler ; see Art. 448. Mallet would have saved himself and his readers great labour if he had borrowed De Moivre's formula and demonstration. But he proceeds in a different way, which amounts to what we should now state thus : the number of ways in which the desired event can happen is the product of \r by the sum of aU the homogeneous products of the degree s — r which can be formed of the numbers 1, 2, 3, ... r. He does not demonstrate the truth of this statement ; he merely examines one very easy case, and says without offering any evidence that the other cases will be obtained by following the same method. See his page 144. Mallet after giving the result in the manner we have just indi- cated proceeds to transform it ; and thus he arrives at the same formula as we have quoted from De Moivre. Mallet does not demonstrate the truth of his transformation generally; he contents himself with taking some simple cases. 636. The transformation to which we have just alluded, MALLET. 341 involves some algebraical work -which we will give, since as we have intimated Mallet himself omits it. Let there be r quantities a,}), c, ... h. Suppose a;' to be di- vided by {x — ci) (x — h) {x — c) ... (x — k). The quotient wiU be x^ + H^ 03*-^' + H^ x'-'-^ +...in infinitum, where H^ denotes the sum of all the homogeneous products of the degree r which can be formed from the quantities a,b,c, ... Jc, This can be easily shewn by first dividing a;* by x — a; then dividing the result by x — h, that is multiplying it by a;~M 1 ) , and so on. Again, if ^ be not less than r the expression (a; — a) (x—b) ... (x — k) will consist of an integral part and a fractional part ; if ^ be less than r there will be no integral part. In both cases the fractional part wiU be ABC K : + :r^: + :—: + ••• + X—a x—b x—c X—K where A = 7 =^7 s 7 tt , {a—b) {a — c) ... (a — k) and similar expressions hold for B, G, ...K. Now expand each of the fractions — — , r, ... according to negative powers of x; and equate the coefficient of x~'~^ to the coefficient in the first form which we gave for x^-¥{{x — d) (x — b) ... (x — k)}. Thus Aa'+ Bb'+ Oc' + ... + KU = H^^^^^. Put m forp — r + ^+1; then p + t = m + r — l; thus we may express our result in the following words : the sum of the homoge- neous products of the degree m, which can be formed of the r quan- tities a,b, 0, ... k, is equal to (a-b)(a-c)...(a-k)'^{b-a)(b~c)...(b-k) + 342 MALLET. This is the general theorem which Mallet enunciates, but only demonstrates in a few simple cases. If we put 1, 2, 3, ... r respectively for a, 5, c, ... ^ we obtain the theorem by which we pass from the formula of Mallet to that of De Moivre, namely, the sum of the homogeneous products of the degree s — r which can be formed of the numbers 1, 2, ... r is equal to ^|^-.(.-i)..H!±^(.-2).- -(--;)^;-^) (.-3y+... The particular case in which s = »• + 1 gives us the following result, l+2 + 3 + ...+r + 12 ^'^ ^> 17273 ^"^ ^> +•■•]' which is a known result. 637. When Mallet has finished his laborious investigation he says, very justly, il y a apparence que celui qui fit cette Lotterie ne s'dtoit pas donnd la peine defaire tous les calculs prdcedens. 638. Mallet's result coincides with that which Montmort gave, and this result being so simple suggested that there might be an easier method of arriving at it. Accordingly MaUet gives another solution, in which like Montmort he investigates directly not the advantage of the director of the lottery, but the expectation of each ticket-holder. But even this solution is more laborious than Mont- mort's, because Mallet takes separately the case in which a ticket- holder has 1, or 2, or 3, ... , or m prizes; while in Montmort's solution there is no necessity for this. 639. Mallet gives the result of the following problem : Ke- quired the chance that in p thi-ows with a die of »i faces a specified face shall appear just m times. The chance is [P (n - 1)^ \m \p —m m' WILLIAM EMERSON. 343 The formula explains itself; for the chance of throwing the specified face at each throw is -, and the chance of not throwing M — 1 it is . Hence by the fundamental principles of the subject the chance of having the specified face just tn times in p throws is \m \p — 'm \n/ \ n } ' Since the whole number of cases in the p throws is rf, it follows that the number of cases in which the required event can happen is \m\p^ {n-ir and the result had been previously given by Montmort in this form : see his page 307. 640. On the whole we may say that Mallet's memoir shews the laborious industry of the writer, and his small acquaintance with preceding works on the subject. 641. William Emerson published in 1776 a volume entitled Miscellanies, or a Miscellaneous Treatise ; containing several Mathe- matical Subjects. The pages 1 — 48 are devoted to the Laws of Chance. These pages form an outliue of the subject, illustrated by thirty-four problems. There is nothing remarkable about the work except the fact that in many cases instead of exact solutions of the problems Emerson gives only rude general reasoning which he considers may serve for approximate solution. This he himself admits ; he says on his page 47, It may be observed, that in many of these problems, to avoid more intricate methods of calculation, I have contented myself with a more lax method of calculating, by which I only approach near the truth. See also the Scholium on his page 21. Thus Emerson's work would be most dangerous for a beginner and quite useless for a more advanced student. We may remark that pages 49 — 138 of the volume are devoted to Annuities and Insurances. 314 BUFFON. 642. We have now to examine a contribution to our subject from the illustrious naturalist Buffon whose name has already occurred in Art. 354<. Buffon's Essai d' Anthm^tique Morale appeared in 1777 in the fourth volume of the Supplement d, VHistoire Natwrelle, where it occupies 103 quarto pages. Gouraud says on his page 54, that the Essay was composed about 1760. 643. The essay is divided into 35 sections. Buffon says that there are truths of different kinds ; thus there are geometrical truths which we know by reasoning, and physical truths which we know by experience ; and there are truths which we believe on testimony. He lays down without explanation a peculiar principle with respect to physical truths. Suppose that for n days in succession the Sun has risen, what is the probability that it will rise to- morrow 1 Buffon says it is proportional to 2""'. See his 6th section. This is quite arbitrary'; see Laplace ThSorie. . .des Prob. page xiil. 644. He considers that a probability measured by so small a fraction as cannot be distinguished from a zero proba- bility. He arrives at the result thus; he finds from the tables that this fraction represents the chance that a man 56 years old will die in the course of a day, and he considers that such a man does practically consider the chance as zero. The doctrine that a very small chance is practically zero is due to D'Alembert ; see Art. 472 : Buffon however is responsible for the value ; see his 8th section. 645. Buffon speaks strongly against gambling. He says at the end of his 11th section : Mais nous allons donner un puissant antidote centre le mal 6pi- dgmique de la passion du jeu, et en mSme-temps quelques pr6servatife contre I'illusion de cet art dangereux. He condemns all gambling, even such as is carried on under conditions usually considered fair; and of course still more BUFFON. 345 gambling in which an advantage is ensured to one of the parties. Thus for example at a game like Pharaon, he says : . . . le banqiiier n'est qu'un fripon avou6, et le ponte une dupe, dont on est convenu de ne se pas moquer. See his 12th section. He finishes the section thus : ... je dis qu'en general le jeu est tin pacte mal-entendu, un contrat d&avantageux aux deux parties, dont reflfet est de rendre la perte tou- jours plus grande que le gain; et d'6ter au bien pour ajouter au mal. La demonstration en est aussi ais6e qu'6Yidente. 646. The demonstration then follows in the 13th section. Buffon supposes two players of equal fortune, and that each stakes half of his fortune. He says that the player who wins will increase his fortune by a third, and the player who loses will diminish his by a half; and as a half is greater than a third there is more to fear from loss than to hope from gain. Buffon does not seem to do justice to his own argument such as it is. Let a denote the fortune of each player, and b the sum staked. Then the gaia is estimated by Buffon by the fraction 7 , and the loss by - ; but it would seem more natural to estimate the r loss by T, which of course increases the excess of the loss to be feared over the gain to be hoped for. The demonstration may be said to rest on the principle that the value of a sum of money to any person varies inversely as his whole fortune. 647. Buffon discusses at length the Petersburg Problem which he says was proposed to him for the first time by Cramer at Geneva in 1730. This discussion occupies sections 15 to 20 inclusive. See Art. 389. Buffon offers four considerations by which he reduces the ex- pectation of A from an infinite number of crowns to about five crowns only. These considerations are (1) The fact that no more than a finite sum of money exists to pay A. Buffon finds that if head did not fall until after the 346 BTJTFON. twenty-ninth throw, more money would be required to pay A than the whole kingdom of France could furnish. (2) The doctrine of the relative value of money which we have stated at the end of the preceding Article. (3) The fact that there would not be time during a life for playing more than a certain number of games; allowing only two minutes for each game including the time necessary for paying. (4) The doctrine that any chance less than is to be considered absolutely zero : see Art. 644. Buffon cites Fontaine as having urged the first reason : see Arts. 392, 393. 648. The 18th section contains the details of an experiment made by Buffon respecting the Petersburg Problem. He says he played the game 2084 times by getting a child to toss a coin in the air. These 2084 games he says produced 10057 crowns. There were 1061 games which produced one crown, 494 which produced two crowns, and so on. The results are given in De Morgan's Formal Logic, page 185, together with those obtained by a re- petition of the experiment. See also Cambridge Philosophical Transactions, Vol. ix. page 122. 649. The 23rd section contains some novelties. Buffon begins by saying that up to the present time Arith- metic had been the only instrument used in estimating probabilities, but he proposes to shew that examples might be given which would require the aid of Geometry. He accordingly gives some simple problems with their results. Suppose a large plane area divided into equal regular figures, namely squares, equilateral triangles, or regular hexagons. Let a round coin be thrown down at random; required the chance that it shall fall clear of the bounding lines of the figure, or fall on one of them, or on two of them ; and so on. These examples only need simple mensuration, and we need not delay on them ; we have not verified Buffon's results. Buffon had solved these problems at a much earlier date. "We find in the Hist, de I'Acad. . . . Paris for 1733 a short account of BUFFON. 347 them ; they were communicated to the Academy in that year ; see Art. 354. 650. Buffon then proceeds to a more difficult example which requires the aid of the Integral Calculus. A large plane area is ruled with equidistant parallel straight lines ; a slender rod is thrown down : required the probability that the rod will fall across a Une. Buffon solves this correctly. He then proceeds to con- sider what he says might have appeared more difficult, namely to determine the probability when the area is ruled with a second set of equidistant parallel straight lines, at right angles to the former and at the same distances. He merely gives the result, but it is wrong. Laplace, without any reference to Buffon, gives the problem in the TMorie...des Proh., pages 359 — 362. The problem involves a compound probability ; for the centre of the rod may be supposed to fall at any point within one of the figures, and the rod to take all possible positions by ttiming round its centre : it is sufficient to consider one figure. Buffon and Laplace take the two elements of the problem in the less simple order ; we will take the other order. Suppose a the distance of two consecutive straight lines of one system, 6 the distance of two consecutive straight lines of the other system ; let 2r be the length of the rod and assume that 2r is less than a and also less than b. Suppose the rod to have an inclination 6 to the line of length a ; or rather suppose that the inclination lies between 6 and 6 + d0. Then in order that the rod may cross a line its centre must fall somewhere on the area ah -{a- 2r cos 0) {b — 2r sin 6), that is on the area 2r (a sin ^ + & cos 6) — ^r' sin 6 cos 0. Hence the whole probability of crossing the lines_ is j\2r {a sin0 + b cos 0) - 4/ sin cos 0\ d0 ldbd0 I' 3-i8 BUFFON. The limits of 6 are and ^ . Hence the result is 4r (« + &)- 4/-' "Kob li a — h this becomes 8ar — ^r" _ Buffon's result expressed in our notation is 2 (a — »•) r Tra" K we have only one set of parallel lines we may suppose b iafinite in our general residt : thus we obtain — . 651. By the mode of solution which we have adopted we may easUy treat the case iu which 2r is not less than a and also less than h, which Buffon and Laplace do not notice. Let i be less than a. First suppose 2r to be greater than 6 but not greater than a. Then the limits of 6 iastead of being and ^ wiU be and sin"^ ^ . Next suppose 2r to be greater than a. Then the limits of 9 wiU be cos"' ^ and sin"' ^ ; this holds so long as cos"' ^ is less than sin"' ^ , that is so long as V'(4r''— a") is less than b, that is so long as 2r is less than V(«'+i'). which is geometrically obvious. 652. Buffon gives a result for another problem of the same kind. Suppose a cube thrown down on the area; required the probability that it will fall across a line. With the same meaning as before for a and b, let 2r denote the length of a diagonal of a face of the cube. The required probability is f ja5 - (a -2»- cos ^) (6-2»-cos^)[ d0 /« jabd9 TT the limits of 9 being and j- . Thus we obtain FUSS. 349 , TT irab ab -r 4 Buffon gives an incorrect result. 653. The remainder of Buffon's essay is devoted to subjects unconnected with the Theory of Probability. One of the sub- jects is the scaZes 0/ no tof ton; Buffon recommends the duodenary scale. Another of the subjects is the wait of length : Buffon re- commends the length of a pendulum which beats seconds at the equator. Another of the subjects is the quadrature of the circle : Buffon pretends to demonstrate that this is impossible. His de- monstration however is worthless, for it would equally apply to any curve, and shew that no curve could be rectified ; and this we know would be a false conclusion. 654. After the Essay we have a large collection of results connected with the duration of human life, which Buffon deduced from tables he had formerly pubhshed. Buffon's results amount to expressing in numbers the following formula : For a person aged n years the odds are as a to 5 that he will live x more years. Buffon tabulates this formula for all integral values of n up to 99, and for various values of x. After these results follow other tables and observations con- nected with them. The tables include the numbers of births, marriages, and deaths, at Paris, from 1709 to 1766. 655. Some remarks on Buffon's views will be found in Con- dorcet's Essai...de F Analyse... Tpage Lxxi., and in Dugald Stewart's Works edited by Hamilton, Vol. i. pages 369, 616. 656. "We have next to notice some investigations by Fuss under the following titles : Recherches sur un problhme du Galcul des Probabilit^s par Nicolas Fuss.. SuppUment au m^moire sur vm probUme du Galcul des Probability s... The Recherches... occupy pages 81 — 92 of the Pars Posterior of the volume for 1779 of the Acta Acad. ... Petrop.; the date of publication is 1783. 350 FUSS. The Supplement... occupies pages 91 — 96 of the Pars Posterior of the volume for 1780 of the Acta Acad. ...Petrop.; the date of publication is 1784. The problem is that considered by James Bernoulli on page 161 of the Ars Gonjectandi; see Art. 117. In the Mecherches ... Fuss solves the problem ; he says he had not seen James BernoulH's own solution but obtained his know- ledge of the problem from Mallet's memoir ; see Art. 628. Fuss published his solution because his results differed from that obtained by James Bernoulli as recorded by Mallet. In the Sup- plement... Fuss says that he has since procured James BemouUi's work, and he finds that there are two cases in the problem ; his former solution agreed with James Bernoulli's solution of one of the cases, and he now adds a solution of the other case, which agrees with James Bernoulli's solution for that case. Thus in fact Fuss would have spared his two papers if he had consulted James Bernoulli's own work at the outset. We may observe that Fuss uses the Lemma given by De Moivre on his page 39, but Fuss does not refer to any previous writer for it ; see Art. 149. CHAPTER XVII. CONDOECET. 657. CoNDOECET was born in 1743 and died in 1794. He wrote a work connected with our subject, and also a memoir. It will be convenient to examine the work first, although part of the memoir really preceded it in order of time. 658. The work is entitled Essai sur I' application de V analyse d la prohabilite des decisions vendues a la pluralite des voix. Par M. Le Marquis de Gondorcet . . . Paris 1785. This work is in quarto ; it consists of a Biscours Preliminaire which occupies cxci. pages, and of the Essai itself which occupies 304 pages. 659. The object of the Preliminary Discourse is to give the results of the mathematical investigations in a form which may be intelligible to those who are not mathematicians. It commences thus : Tin grand homme, dont je regretterai toujours les legons, les exem- ples, et sur-tout I'amitil, 6toit persuadS que les virites des Sciences morales et politiques, sent susceptibles de la m^me certitude que celles qui ferment le systlme des Sciences physiques, et mSme que les branches de ces Sciences qui, comme I'Astronomie, paroissent approcher de la certitude mathSmatique. Cette opinion lui 6toit chire, parce qu'elle conduit ^ l'esp6rance con- solante que I'esplce humaine fera ngcessairement des progrgs vers le bonheur et la perfection, comme elle en a fait dans la connoissance de la vSritl. C'6toit pour lui que j'avois entrepris cet ouvrage 352 CONDOECET. The great man to whom Condorcet here refers is named in a note : it is Turgot. Condorcet himself perished a victim of the French Eevolution, and it is to be presumed that he must have renounced the faith here expressed in the necessary progress of the human race to- wards happiness and perfection. 660. Condorcet's Ussai is divided into five parts. The Discours Preliminaire, after briefly expounding the funda- mental principles of the Theory of Probability, proceeds to give in order an account of the results obtained in the five parts of the Essai. We must state at once that Condorcet's work is excessively difficult ; the difficulty does not lie in the mathematical investi- gations, but in the expressions which are employed to introduce these investigations and to state their results : it is in many cases almost impossible to discover what Condorcet means to say. The obscurity and self contradiction are without any parallel, so far as our experience of mathematical works extends ; some examples will be given in the course of our analysis, but no amount of examples can convey an adequate impression of the extent of the evils. We believe that the work has been very little studied, for we have not observed any recognition of the repulsive peculi- arities by which it is so undesirably distinguished. 661. The Preliminary Discourse begins with a brief exposition of the fundamental principles of the Theory of Probability, in the course of which an interesting point is raised. After giving the mathematical definition of probability, Condorcet proposes to shew that it is consistent with ordinary notions ; or in other words, that the mathematical measure of probability is an accurate measure of our degree of belief See his page Vll. Unfortunately he is extremely obscure in his discussion of the point. We shall not delay on the Preliminary Discourse, because it is little more than a statement of the results obtained in the Essay. The Preliminary Discourse is in fact superfluous to any person who is sufficiently acquainted with Mathematics to study the Essay, and it would be scarcely intelligible to any other person. CONDOECET, 353 For in general when we hare no mathematical symbols to guide us in discovering Condorcet's meaning, the attempt is nearly hopeless. We proceed then to analyse the Essay. 662. Condorcet's first part is divided into eleven sections, devoted to the examination of as many Hypotheses; this part occupies pages 1 — 136. We -will consider Condorcet's first Hypothesis. Let there be 2^' + 1 voters who are supposed exactly alike as to judgment ; let v be the ptobability that a voter decides correctly, e the probability that he decides incorrectly, so that v + e =,1 : required the probability that there will be a majority in favour of the correct decision of a question submitted to the voters. We may observe, that the letters v and e are chosen from commencing the words xeritS and erreur. The required probability is found by expanding (v + e)"*^' by the Binomial Theorem, and taking the terms from v''^^ to that which involves v'^'e', both inclusive. Two peculiarities in Con- dorcet's notation may here be noticed. He denotes the required probability by V^; this is very inconvenient because this symbol has universally another meaning, namely it denotes V raised to the power g. He uses — to denote the coefficient of v'^'^e"' in m the expansion of (v + e)"; this also is very inconvenient because the symbol — has universally another meaning, namely it denotes a fraction in which the numerator is n and the denominator is m. It is not desirable to follow Condorcet in these two innovations. We wUl denote the probability required by i{i) = {v + ey,l>{a). Thus .^ fe + 1) - .^ (2) = <^ (2 + l)-(r, + er (q) very few terms will remain uncancelled. In fact it wiU be easily found that I 2ff + 1 I 2ff + 1 g + 1 ^ |g + l |g 2g + l |g + l|g („_e)t,'+'e^' ;(1). Hence we deduce f 3 5 4 7 7.6.5 ,, ve 2.3 I 2g - 1 1 •••+[zfeH '')■ 664. The result given in equation (2) is the transformation to which we alluded. We may observe that throughout the first part of his Essay, Condorcet repeatedly uses the method of trans- formation just exemplified, and it also appears elsewhere in the Essay ; it is in fact the chief mathematical instrument which he employs. It will be observed that we assumed v + e = l in order to obtain equation (2). We may however obtain a result analogous CONDOECET. S55 to (2) which shall be identically true, •vi^hatever v and e may be. "We have only to replace the left-hand member of (1) by and we can then deduce JL . Jii ...»mi.^... |g+i Li = u (t) + e)-« + (u - e) ive {v + e)^"* + | «V (v + e)'^'* This is identically true ; if we suppose v + e = 1, we have the equation (2). 665. We resume the consideration of the equation (2). Suppose V greater than e ; then we shall find that <^ (g') = 1 when q is infinite. For it may be shewn that the series in powers of ve which occurs in (2) arises from expanding - 2 + 2 (1 - *^^)"* in powers of ve as far as the term which involves t?*e'. Thus when 2 is infinite, we have ,^ (2) = „ + (v- e) |- 1 + J (1 - 4w)-*J . Now l — ^ve={v+ef — ^ve = (v—e)\ Therefore when q^ is infinite , . f v-e , v+e \ = v+e = l. The assumption that v is greater than e is introduced when we put v-e for (1 - 4!ue)*. 23—2 356 CONDORCET. Thus we have the following result in the Theory of Probability : if the probability of a correct decision is the same for every voter and is greater than the probability of an incorrect decision, then the probability that tbe decision of the majority wiU be correct becomes indefinitely nearly equal to unity by sufficiently in- creasing the number of voters. It need hardly be observed that practically the hypotheses on which the preceding conclusion rests cannot be realised, so that the result has very little value. Some important remarks on the subject will be found in Mill's Logic, 1862, Vol. ii. pages 65, 66, where he speaks of " misapplications of the calculus of probabilities which have made it the real opprobrium of mathematics." 666. We again return to the equation (2) of Art. 663. If we denote by i/r [q) the probability that there wiU be a majority in favour of an incorrect decision, we can obtain the value oi-^{q) from that of <^ {q) by interchanging e and v. We have also <^{q)+'^ (q) = 1. Of course if v=e we have obviously ^(s) =>]r {q), for all values of q ; the truth of this result when q is infinite is esta- blished by Condorcet in a curious way ; see his page 10. 667- We have hitherto spoken of the probabihty that the decision will be correct, that is we have supposed that the result of the voting is not yet known. But now suppose we know that a decision Jias been given and that m voters voted for that decision and n against it, so that m is greater than n. We ask, what is the probability that the de- cision is correct 1 Condorcet says briefly that the number of com- binations in favour of the truth is expressed by I 22- + 1 [m.[n and the number in favour of error by \m\n.. Thus the probabilities of the correctness and incorrectness of -the decision are respectively CONDORCET. 357 and See his page 10. 668. The student ' of Condorcet's work must carefully dis- tinguish between the probability of the correctness of a decision that has been given when we know the numbers for and against, and the probability when we do not know these numbers. Con- dorcet sometimes leaves it to be gathered from the context which he is considering. For example, in his Preliminary Discourse page XXIII. he begins his account of his first Hypothesis thus : Je considire d'abord le cas le plus simple, celui oil le nombre des Votans 6tant impair, on prononce simplement S, la plurality. Dans ce cas, la probabiIit6 de ne pas avoir uue decision fausse, celle d'avoir une decision vraie, celle que la dScision rendue est conforme a la v6iitg, sent les m^mes, puisqu'il ne peut y avoir de cas ou il n'y ait pas de decision. Here, although Condorcet does not say so, the words celle que la decision rendue est conforme ci la verit4 mean that we know the decision has been given, but we do not know the numbers for and against. For, as we have just seen, in the Essay Con- dorcet takes the case in which we do know the numbers for and against, and then the probability is not the same as that of the correctness of a decision not yet given. Thus, in short, in the Preliminary Discourse Condorcet does not say which case he takes, and he really takes the case which he does not consider in the Essay, excluding the case which he does consider in the Essay; that is, he takes the case which he might most naturally have been supposed not to have taken. 669. We will now proceed to Condorcfet's second Hypothesis out of his eleven ; see his page 14. Suppose, as before, that there are 2q + l voters, and that a certain plurality bf votes is required in order that the decision should be valid ; let 2q' + 1 denote this plurality. Let (q)+'^ (q) is not equal to unity. In fact 1 —^{q) — "^{q) consists of all the terms in the expansion of {v + ef^^ lying between those which involve v^^*^ e'""" and ^,5-SgS+s;+i ijotii exclusive. Thus 1 - ^ (^) - 1^ {q) is the probability that the decision wiU be invalid for want of the prescribed plurality. It is shewn by Condorcet that if ij-is greater than e the limit of j> {q) when q increases indefinitely is unity. See his pages 19 — 21. 670. Suppose we know that a valid decision has been given, but do not know the numbers for and against. Then the pro- bability that the decision is correct is , , , . , ^ , and the pro- bability that it is incorrect is ^^—^ (g) must be large. 3. That there shall be a valid decision, correct or incorrect ; that is ^ (,q)+'f' (2) must be large. 4. That a valid decision which has been given is correct, CONDORCET. 359 supposing the numbers for and against not to be known ; that is -t f \ .1 / N must be large. 5. That a valid decision which has been given is correct, supposing the numbers for and against to be known; that is -; must be large, even when m and n are such as to give it the least value of which it is susceptible. These appear to be what Condorcet means by the principal conditions, and which, in his usual fluctuating manner, he calls in various places five conditions, four conditions, and two con- ditions. See his pages sviii, xxxi, lxix. 672. Before leaving Condorcet's second Hypothesis we will make one remark. On his page 17 he requires the following result, {i+v(i-4^)rv(i-4^) ^ 1 ^ 1.2 "+- \n + 2r-l ■«- + ... Ir \n + r- On his page 18 he gives two ingenious methods by which the result may be obtained indirectly. It may however be obtained directly in various ways. For example, take a formula which may be established by the Differential Calculus for the expansion of {1 + \/(l — 4 (w) consists of the first third of all the terms of {v + e)", and thus if v is greater 2 than K the greatest term is included within times in succession, 2nd the probability that in r trials the event will happen^ times in succession before it fails p times in succession. It is the second of these problems which Condorcet wishes to apply, but he finds it convenient to begin with the solution of the first, which is much the simpler, and which, as we have seen, in Art. 325, had engaged the attention of De Moivre. 678. We have already solved the first problem, in Art. 325, but it will be convenient to give another solution. Let ^ (r) denote the probability that in r trials the event will happen^ times in succession. Then we shall have ^ (r) = v^+v^'^e^ (r-p) + v^e^(r -j? + 1) -I- ... ...+ve(r-l) (1). 362 CONDOHCET. To shew the truth of this equation we observe that in the first p trials the following 'p cases may arise ; the event may happen p times in succession, or it may happen p — \ times in succession and then fail, or it may happen p - 2 times in succes- sion and then fail, , or it may fail at the first trial. The aggregate of the probabilities arising from all these cases is if). The probability from the first case is «'. The probability from the second case is v*"' e<^{r —p) : for «*"' e is the probability that the event will happen p— 1 times in succession, and then fail ; and (j — p) is the probability that the event will happen p times in succession in the course of the remaining r—p trials. In a similar way the term d'"V (»•— p + 1) is accounted for ; and so on. Thus the truth of equation (1) is established. 679. The equation (1) is an equation in Finite Differences ; its solution is 4> W = G,y: + G,y: + C,y: + ...+ G,y; + C (2). Here G^, G^, ...G^ are arbitrary constants ; y^ y^f-Vs are the roots of the following equation in y, yp =.e{v^' +v'-^y + v'-' f + ... +y^') (3); and G is to be found from the equation that is C=v'' + e \~'"'' G; 1 — v and as e = 1 — w we obtain G= 1. We proceed to examine equation (3). Put 1—v for e, and assume y = - ; thus 1-V ^ g (1 - g") l~z (4). We shall shew that the real roots of equation (3) are nu- merically less than unity, and so also are the moduli of the im- aginary roots; that is, we shall shew that the real roots of CONDOECET. 363 equation (4) are numerically greater than v, and so also are the moduli of the imaginary roots. We know that v is less than unity. Hence from (4) if z be real and positive it must be greater than v. For if a be less than V, then is less than z -, and a fortiori - ^ ~^ ' . is less I — Z 1 —V '' \ — z than _ . If s be negative in (4) we must have 1 — »' nega- tive, so that p must be even, and z numerically greater than unity, and therefore numerically greater than v. Thus the real roots of (4) must be numerically greater than t>. Again, we may put (4) in the form D + 'y'+i;'+... = » + a' + ...+«* (5). Now suppose that z is an imaginary quantity, say z = 'k (cos d + V^ sin 6) j then if A; is not greater than v, we see by aid of the theorem »" = F (cos nQ + V^ sin nB), that the real terms on the right-hand side of (5) will form an aggregate less; than the left-hand side. Thus h must be greater than V. After what we have demonstrated respecting the values of the roots of (3), it follows from (2) that when r is infinite (f) = 1. 680. We proceed to the second problem. Let ^ if) now denote the probability that in *• trials the event will happen j> times in succession before it fails p times in suc- cession. Let i|^ {n) denote the probability that the event will happen p times in succession before it fails p times in succession, supposing; that one trial has just been made in which the. event failed, and that n trials remain to be made. Then instead of equation (1) we shall now obtain ^(r) =v^-\-v'^^e^lr{r-p) +t;*"''ei|r (r-j?-f- 1) + .... ... +wy^ {r -2) + ey^ {r -1) ... (6). This equation is demonstrated in the same manner as (1) was. 364 CONDORCET. We have now to shew the connexion between the functionsf lr {n -p)}. • The meaning of the factor e*"' is obvious, so that we need only explain the meaning of the other factor. And it will be seen that (j)(n — p + 1) — eyjr {n —p) expresses the probability of the desired result in the n—p + 1 trials which remain to be made; for here the rejected part ey}r (n —p) is that part which would coexist with failure in the first of these remaining trials, which part would of course not be available when p—1 failures had already taken place. Thus we may consider that (7) is established. In (6) change r into r—p; therefore ^ {r-p) = v^ + v^^f (r- 2p) + if'ef (r- 2p + 1) + ... ... + vei/r (r — ^ — 2) + ei/r (»• — p - 1) (8). Now multiply (8) by e" and subtract the result from (6), ob- serving that by (7) we have -vlr (m) -e"'^ in-p) = (w) - e^' (w -^ + 1) ; thus we obtain ^ (r) -e^^{r-p)=v''- eV + v'^e{<}){r-p)-e''-^(}>{r-2p + l)} + v"-' e {V (*• -p + 1)- e"-^ ^ (r - 2p + 2)} + ... + e{{r-l)-er' ^ (»• — 2p + 1), • • • will all be equal. Thus we can obtain the probability that the event will happen p times in succession before it fails 'p times in succession in an indefinite number of trials. Let V denote this probability ; then we have from (9), 7(1-6") =?;^(l-e*)+eF(«'-'+D'"'+. ..+« + !) -e"F(u*-'+D*"''+... + v+l). Hence after reduction we obtain 682. The problems which we have thus solved are solved by Laplace, TMorie...des Prob. pages 247 — 251. In the solution we have given we have followed Condorcet's guidance, with some deviations however which we will now indicate ; our remarks will serve as additional evidence of the obscurity which we attribute to Oondorcet. Our original equation (1) is given by Oondorcet ; his demon- stration consists merely in pointing out the following identity ; {v + eY = v'[v +e)'^ + iT^e {v + e)'-' + ir^e {v + e)*^' 4- ... ...+v'e{v + ey-' + ve{v + e)'"" + e{v+ e)'-\ He arrives at an equation which coincides with (4). He shews that the real roots must be numerically greater than v ; but with respect to the imaginary roots he infers that the moduli cannot be greater than unity, because if they were

{r - 2p)} -e*-' {^(r-3jj + l)-e^(r-3p)} 366 CONDOECET. On his page 75 Condorcet gives an equivalent result without explicitly using (7) ; but he affords very little help in establish- ing it. Let % (r) denote what in) - e'~' [^ [n-q + 1) -ei/r {n- q)] ; instead of (9) we shall have j> (r) -e^4>{r-q)=v'' (1 -e«) + w'"' e {^ {r -p) -e^'^{r-p-q+\)] + v'^e { {r -p + 1) -e'-' ^{r-p-q+ 2)} + ... + e{ir-l)-e'-'ir-q)], and instead of (10) we shall have v'^'- (1 - ep v=- ,*-» + e^-^ - v"'' e'"' 684i. We will introduce here two remarks relating to that part of Condorcet's Preliminary Discourse which bears on his ninth Hypothesis. On page xxxvi. he says, ...c'est qu'en supposant que Ton connoisse le nombre des decisions et la plurality de chacune, on peut avoir la somme des pluralites obte- nues centre ropinion qui Temporte, plus grande que celle des pluralitis conformes k cet avis. This is a specimen of a kind of illogical expression which is not uncommon in Condorcet. He seems to imply that the result depends on our knowing something, whereas the result might happen quite independently of our knowledge. If he will begin his sentence as he does, his conclusion ought to be that we may have a certain result and know that we have it. On page xxxvii. he alludes to a case which is not discussed in the Essay. Suppose that a question is submitted to a series CONDOBCET. 369 of tribunals until a certain number of opinions in succession on the same side has been obtained, the opinions of those tribunals being disregarded in which a specified plurality did not concur. Let V be the probability of an opinion for one alternative of the question, which we will call the affirmative; let e be the proba- bility of an opinion for the negative ; and let a be the probability that the opinion will have to be disregarded for want of the re- quisite plurality. Thus v + e + z = l. Let r be the number of opinions on the same side required, q the number of tribunals. Suppose (v + zy to be expanded, and let all the terms be taken between «' and v^ both inclusive ; denote the aggregate by ^ («). Let (e) be formed from ^ [v) by putting e for v. Then ^ {v) is the probability that there will be a decision in the affirmative, and ^ (e) is the probability that there will be a decision in the negative. But, as we have said, Condorcet does not discuss the case. 685. Hitherto Condorcet has always supposed that each voter had only two alternatives presented to him, that is the voter had a proposition and its contradictory to choose between ; Condorcet now proposes to consider cases in which more than two propo- sitions are submitted to the voters. He says on his page 86 that there will be three Hypotheses to examine ; but he really arranges the rest of this part of his Essay under two Hypotheses, namely the tenth on pages 86 — 94, and the eleventh on pages 95 — 136. 686. Condorcet's tenth Hypothesis is thus given on his page XLil : ...celle oii Ton suppose que les Yotans peuvent non-seulement voter pour ou contre une proposition, mais aussi declarer qu'ils ne se croient pas assez instruits pour prononcer. The pages 89 — 94 seem even more than commonly obscure. 687. On his page 94 Condorcet begins his eleventh Hypo- thesis. Suppose that there are 62-4-1 voters and that there are three propositions, one or other of which each voter affirms. Let V, e, i denote the probabilities that each voter will affirm these three propositions respectively, so that v + e-\-i=\. Condorcet indicates various problems for consideration. We may for example suppose that three persons A, B, are candidates for an office, 24 370 CONDOECET. and that v, e, iare the probabilities that a voter will vote for ^, 5, G respectively. Since there are 63' + 1 voters the three candidates cannot be bracketed, but any two of them may be bracketed. We may consider three problems. I. Find the probability that neither B nor G stands singly at the head. II. Find the probability that neither B nor G is before A. III. Find the probability that A stands singly at the head. These three probabilities are in descending order of magnitude. In III. we have all the cases in which A decisively beats his two opponents. In II. we have, in addition to the cases in III., those in which A is bracketed with one opponent and beats the other. In I. we have, in addition to the cases in II., those in which A is beaten by both his opponents, who are themselves bracketed, so that neither of the two beats the other. Suppose for example that q = l- We may expand (v + e + iy and pick out the terms which will constitute the solution of each of our problems. For III. we shall have v' + 7v' (e + i) + 2W (e + iy + 35w* (e + iy + SBv' QeH\ For II. we shall have in addition to these For I. we shall have in addition to the terms in II. 7v 20e'i'. These three problems Condorcet briefly considers. He denotes the probabilities respectively by W^, W/, and W'^. It will scarcely be believed that he immediately proceeds to a fourth problem in which he denotes the probability by W/\ which is nothing but the second problem over again. Such however is the fact. His enun- ciations appear to be so obscure as even to have misled himself. But it will be seen on examination that his second and fourth problems are identical, and the final expressions which he gives for the probabilities agree, after allowing for some misprints. CONDORCET. 371 688. .It may be interesting to give Cordorcet's own enun- ciations. I. ...soit W'' la probability que ni e ni i n'obtiendront sur les deux autres opinions la pluralite, . . . page 95, II. ...W/ exprimant la probability que e et t n'ont pas sur v la pluralit6 exigge, sans qu'il soit n6oessaire, pour rejeter un terme, que Fun des deux ait cette pluralit6 sur I'autre,... page 100. III. . . . W'^, c'est-^-dire, la probabilit6 que v obtiendra sur i et ela, pluraUtl exigge, . . . page 102. lY. ■■•W/", c'est-£l-dire, la probability que v surpassera un des deux i ou e, et pourra cependant ^tre 6gal S, I'autre,... page 102. Of these enunciations I., Ill,, and IV. present no difficulty; II. is obscure in itself and is rendered more so by the fact that we naturally suppose at first that it ought not to mean the same as IV. But, as we have said, the same meaning is to be given to II. as to IV. Before Condorcet takes these problems individually he thus states them together on his page 95 : . . .nous chercherons la probability pour un nombre donn6 de Votans, ou que ni e ni i ne I'emportent sur v d'une plurality exigge, ou que e et i I'emportent cbacun sur v de cette pluralite sans I'emporter I'un sur I'autre, ou enfin que v I'emporte k la fois sur e et sur i de cette plurality. Thus he seems to contemplate three problems. The last clause ou enfin ... plurality gives the enunciation of the third problem distinctly. The clause ou que ni ... exigee may perhaps be taken as the enunciation of the second problem. The clause ou que... I'autre will then be the enunciation of the first problem. In the Preliminary Discourse the problems are stated together in the following words on page XLiv : ...qu'on cherclie...ou la probability d'avoir la plurality d'un avis sur les deux,..., ou la probabilite que, soit les deux autres, soit un seul des deux, n'auront pas la plurality ;. . . In these words the problems are enunciated in the order III., II., I. ; and knowing what the problems are we can see that the words are not inapplicable. But if we had no other way of testing the meaning we might have felt uncertain as to what problems II. and I. were to be. 24—2 372 CONDOECET. 689. Condorcet does not discuss these problems with much detail. He gives some general considerations with the view of shewing how what he denotes by W^*^ may be derived from W; but he does not definitely work out his suggestions. We wiU here establish some results which hold when the number of voters is infinite. We will first shew that when q is infinite W/ is equal to unity, provided that v is greater than either e or i. Suppose (v+e+i)'^*^ expanded in the form (« + e)^^ + (62 + !)(« + e)"^ i + ^^S + ^^^S („ + e)^' T + J. . A \6q-\-l Now take the last term which we have here explicitly given, and pick out from it the part which it contributes to Wf. We have (v + e)"^ =(v + e)*^^ f-^ + -^P'. Expand ■! 1 } as far as the term which involves ^ [v + e v + e) I I , and denote the sum by fl , ). Then finally \v + ej \v + e v + ej •' the part which we have to pick out is / v s \ Now if V be greater than e, then /( , is equal to \v+e v + eJ ^ unity when q is infinite, as we have already shewn ; see Art. 665. Hence we see that when q is infinite the value of W/ is the limit of {v + ef^' + [Qq + !)(« + e)« i + ^^i'^^^^i (« + ^f-' i' + |6ff + l Now we are at liberty to suppose that i is not greater than e, and then r + e is greater than 2i; so that v + e must be greater CONDORCET. 373 2 than s- Hence by Art. 674 the value of TF"/ will be unity when o q is infinite. Let {v, ei) + ^ (e, vi) + ^ {i, ev) = 1. Now suppose q infinite. Let v be greater than e or t; then as we have just shewn if> {v, ei) = \, and therefore each of the other functions in the above equation is zero. Thus, in fact, ^ {x, yz) vanishes if a; be less than y or z, and is equal to unity if x be greater than hoik y and z. Next suppose v = e, and i less than v or e. By what we have just seen

{v, ei) = (e, vi) — (j) (i, ev). But such an equation will not be true except on the assumption 374 CONDOECET. that W" and F' are equal ioWf ultimately; and on this assump- tion we have the required results at once without the five Hues which Condorcet gives after the sentence we have just quoted. 690. In the course of his eleventh Hypothesis Condorcet examines the propriety of the ordinary mode of electing a person by votes out of three or more candidates. Take the following example ; see his page LViil. Suppose A, B, G are the candidates ; and that out of 60 votes 23 are given for A, 19 for B, and 18 for C. Then A is elected according to ordinary method. But Condorcet says that this is not necessarily satisfactory. For suppose that the 23 who voted for A would all consider G better than B ; and suppose that the 19 who voted for B would aU con- sider G better than A ; and suppose that of the 18 who voted for C, 16 would prefer B to A, and 2 would prefer A to B. Then on the whole Condorcet gets the following result. The two propositions in favour of G are G is better than A, G is better than B. The first of these has a majority of 37 to 23, and the second a majority of 41 to 19. The two propositions in favour of B are B is better than A, B is better than G. The first of these has a majority of 35 to 25, the second is in a minority of 19 to 41. The two propositions in favour of A are A is better than B, A is better than G. The first of these is in a minority of 25 to 35, and the second in a minority of 23 to 37. Hence Condorcet concludes that G who was lowest on the poll in the ordinary way, really has the greatest testimony in his favour ; and that A who was highest on the poU in the ordinary way, really has the least. Condorcet himself shews that his own method, which has just been illustrated, will lead to difificulties sometimes. Suppose, for example, that there are 23 voters for A, 19 for B, and 18 for G. Suppose moreover that all the 23 who voted for A would have preferred B to G; and that of the 19 who voted for B, there CONDOECET. 375 are 17 who prefer C to A, and 2 who prefer A to C; and lastly that of the 18 who voted for C there are 10 who prefer A to B, and 8 who prefer B to A. Then on the whole, the following three propositions are affirmed: B is better than C, by 42 votes to 18 ; C is better than A, by 35 votes to 25 ; A is better than B, by 33 votes to 27. Unfortunately these propositions are not consistent with each other. Condorcet treats this subject of electing out of more than two candidates at great length, both in the Essay and in the Preliminary Discourse ; and it is resumed in the fifth part of his Essay after the ample discussion which it had received in the first part. His results however appear of too little value to detain us any longer. See Laplace, ThSorie ... des Prob. page 274. 691. The general conclusions which Condorcet draws from the first part of his work do not seem to be of great importance ; they amount to little more than the very obvious principle that the voters must be enlightened men in order to ensure our con- fidence in their decision. We will quote his own words : On. voit done ici que la forme la plus propre k remplir toutes les conditions exigees, est en m^me temps la plus simple, celle oil une assemblee unique, composfee d'hommes gclair6s, prononce seule un juge- ment h une plurality telle, qu'on ait une assurance suffisante de la ■viAti du jugement, m^me lorsque la pluralit6 est la moindre, et il faut de plus que le nombre des Votans soit assez grand pour avoir une grande probability d'obtenir tine d6cision. Des Votans Sclaires et une forme simple, sont les moyens de reunir le plus d'avantages. Les formes compliqules ne remedient point au default de lumilres dans les Votans, ou n'y rem6dient qu'imparfaifcement, ou m^me entratnent des inoonveniens plus grands que ceux qu'on a voulu 6viter. Page xlii. . . . il faut, 1° dans le cas des decisions sur des questions compliquees, faire en sorte que le systSme des propositions simples qui les forment soit rigoureusement d6velopp6, que chaque avis possible soit bien expos6, que la voix de chaque Votant soit prise sur chacune des propositions qui forment cet avis, et non sur le r6sultat seul 376 CONDORCET. 2°. II faut de plus que les Votans soient Iclair^s, et d'autant plus €clair6s, que les questions qu'ils decident sont plus compliqu6es ; sans cela on trouvera bien une forme de decision qui preservera de la crainte d'une decision fausse, mais qui en m^me temps rendant toute decision presque impossible, ne sera qu'un moyen de perp6tuer les abus et les mauvaises loix. Page lxix. 692. We now come to Condorcet's second part, whicli occupies his pages 137 — 175. In the first part the following three elements were always supposed known, the number of voters, the hypothesis of plurality, and the probability of the correctness of each voter's vote. From these three elements various results were deduced, the principal results being the probabihty that the decision will be correct, and the probability that it will not be incorrect ; these probabilities were denoted by '"^^-^"'^- The chief point in the solution of this problem is the fact to which we have drawn attention in the latter part of Art. 697. We may remark that Condorcet begins his solution of the second part of his problem thus : Soit suppos^e maintenant la pro- bability changeante §, chaque ^vfenement. He ought to say, let the probability not be assumed constant. See Art. 698. 702. Condorcet's fifth problem is thus enunciated : Conservant les memes hypotheses, on demande quelle est, dans le cas du probleme premier, la probabilit6, 1°. que celle de I'evgnement A n'est pas au-dessous d'une quantit6 donnSe ; 2°. qu'elle ne differe de la valeur moyenne • que d'une quantity a; 3°. que la probability d'amener j4, n'est point au-dessous d'une limite a ; 4°. qu'elle ne diffSre de la pro- bability moyenne „ que d'une quantit6 moindre que a. On demande aussi, ces probabilit^s 6tant donn^es, quelle est la limite a pour laquelle elles ont lieu. The whole solution depends on the fact to which we have drawn attention in the latter part of Art. 697. CONDOBCET. 383 As is very common with Condorcet, it would be uncertain from his language what questions he proposed to consider. On examin- ing his solution it appears that his 1 and 3 are absolutely identical, and that his 2 and 4 differ only in notation. 703. In his sixth problem Condorcet says that he proposes the same questions as in his fifth problem, takiag now the hypothesis that the probability is not constant. Here his 1 and 3 are really different, and his 2 and 4 are really different. It seems to me that no value can be attributed to the discus- sions which constitute the problems from the second to the sixth inclusive of this part of Condorcet's work. See also Coumot's Exposition de la Theorie des Chances... -page 166. 704. The seventh problem is an extension of the first. Sup- pose there are two events A and N, which are mutually exclusive, and that in m -I- w trials A has happened m times, and N has hap- pened n times : required the probability that in the next p -h 2' trials A wiU happen^ times and JV happen q times. Suppose that x and 1 —x were the chances of A and ^ at a single trial; then the probability that in m + n trials A would happen m times and N happen n times would be proportional to as" (1 — a;)". Hence, by the rule for estimating the probabilities of causes from effects, the probability that the chance of A lies be- tween X and x+dx at a. single trial is f V (1 - xf dx Jo And if the chance of .4 at a single trial is x the probability that in p + q trials A will occur ^ times and N occur q times is IfM.x'^ii-xy. \p[i Hence finally the probability required in the problem is , ^^ i'x'^^a-xy'dx \P-il fx'^ii-xydx ■' 384 CONDOECET. This important result had been given in effect by Laplace in the memoir which we have cited in Art. 551 ; but in Laplace's me- moir we must suppose the ^ + g' events to be required to happen \P + 9. ■ in an assigned order, as the factor -; — -. — is omitted. ^ [e\i We shall see hereafter in examining a memoir by Prevost and Lhuilier that an equivalent result may also be obtained by an elementary algebraical process. 705. The remaining problems consist chiefly of deductions from the seventh, the deductions being themselves similar to the problems treated in Condorcet's first part. We will briefly illus- trate this by one example. Suppose that A has occurred on times and B has occurred n times ; required the probability that in the next 22' -hi trials there will be a majority in favour of A. Let F{q) denote this probability ; then [ a;" (1 - xy 4> (([) dx «" (1 - £c)" Jj; where <^ {q) stands for a;^« + (2^ + 1) x^ [\-x) + (^y + lj^y aM-^ (i _ -c)«+ J- • ^ I 2? + 1 ... + }=f==.x^'{\-xy. Hence if we use, as in Art. 663, a similar notation for the case in which q is changed into g -h 1, we have tx''{l-xY^{q-Vl)dx F(s + 1) =^-^-rr' . »•" (1 - x)" dx Therefore, as in Art. 663, Cx-{l-xrU{q+l)-iq)\dx F{q+1)-F{q)=^-^- I ) f x'-'il-xYdx Jo CONDOBCET. 385 where (1- 'f7r \p' - ■ - >- "' "•^ •'o as we have already remarked in Art. 704. Condorcet quotes this result ; he thinks however that better formulae may be given, and he proposes two. But these seem quite arbitrary, and we do not perceive any reason for preferrino- them to the usual formula. We will indicate these formulae pro- posed by Condorcet. I. Let t = m + n+p + q and put M= -^ — — ^ f : t then the proposed formula is ^j, + q Iff- . . M"^ (1 - u)"'' dx, dx^... dx, I^Li {U...u'"{l-uYdxJx^...dx The limits of each integration are to be and 1. II. Suppose an event to have happened n times in succession, required the probability that it wUl happen p times more in suc- cession. _ , x^ + x„ x,+x^ + x x, + x^ + ...+x„ Let u = x,-^- 3 ... ; let V be an expression similar to u but extended to n +p factors ; then Condorcet proposes for the required probability the formula /// ...V dxi dx. . . . dx„. \\\ ...u dx^ dx^ . . . dx„ The limits of each integration are to be and 1. Condorcet proposes some other formulae for certain cases ; they 400 CONDORCET. are as arbitrary as those which we have already given, and not fully intelligible ; see his pages 550 — 553. 735. The fifth part of Condorcet's memoir is entitled Sur la prohabilite desfaits extraordinaires. Suppose that p is the probability of an event in itself; let t denote the probability of the truth of a certain witness. This wit- ness asserts that the event has taken place ; required the proba- bility that the event did take place, and that it did not. The required probabilities are Pt and 0--P)0--*) Condorcet gives these formulae with very little explanation. The application of these formulae is not free from difficulty. Suppose for example a trustworthy witness asserts that one ticket of a lottei-y of 10000 tickets was drawn, and that the number of the ticket drawn was 297. Here if we put p = we obtain such a very small value of the truth of the witness's statement that we lose our confidence in the formula. See Laplace Theorie. . .des Proh. pages 446 — 451. De Morgan, Cambridge Philosophical Transactions, Vol. ix. page 119. 736. Condorcet makes remarks on two points, namely the mode of estimating p and the mode of estimating t. He recurs to the former point in the sixth part of his memoir, and we shall give an extract which will shew the view he advocated in his fifth part, and the view which he advocated in his sixth part. With respect to the second point Condorcet's chief remark is that the probabiUty of a witness is not the same for all facts. If we estimate it at u for a simple fact, then we should estimate it at m' for a compound fact consisting of two simple facts, and so on. One witness however may be as capable of observing a compound fact consisting of two or more simple facts as another is of observ- ing a simple fact. 737. The sixth part of Condorcet's memoir is entitled Appli- CONDORCET. 401 cation des principes de V article prScident d, quelques questions de critique. It is published in the Hist, de I' Acad. . . . Paris for 1784; it occupies pages 454 — 468. 738. In this part Condorcet begins by adverting to some remarks which he had made in his fifth part as to the mode of estimating the value of what we denoted hjp in Article 735. He says, J'ai observl en m^me-temps qu'il ne falloit pas dans ce cas entendre, par la probability propre d'un fait, le rapport du nonibre des combi- naisons ou il a lieu, avec le nombre total des combtnaisons. Par ex- emple, si d'un jeu de dix cartes on en a tir6 une, et qu'un timoin me dise que c'est telle carte en partioulier, la probability propre de ce fait, qu'il s'agit de comparer avec la probabilLtg qui nait du t6moignage, n'est pas la probabilitg de tirer cette carte, qui seroit -^k , mais la probability d'amener cette carte plut6t que telle autre carte determinfie en parti- culierj et comma toutes ces probabilites sont Igales, la probabilite . • ■ 1 propre eSt ici „ . Cette distinction 6toit nicessaire, et elle suffit pour expliquer la contrari6te d' opinions entre deux classes de pbilosophes. Les uns ne peuvent se persuader que les m^mes t6moignages puissent produire, pour un fait extraordinaire, une probabilite figale K celle qu'ils produi- sent pour un fait ordinaire; et que, par exemple, si je crois un bomme de bon sens qui me dit qu'une femme est accoucMe d'un garQon, je dusse le croire ggalement s'il me disoit qu'elle est accoucbfie de douze. Les autres au contraire sont convaincus que les tgmoignages conser- vent toute leur force, pour les faits extraordinaires et trSs-peu proba- bles, et ils sont frapp6s de cette observation, que si on tire une loterie de 100000 billets, et qu'un bomme, digne de foi, dise que lo num6ro 256, par exemple, a eu le premier lot, personne ne doutera de son tem- oignage, quoiqu'il j ait 99999 k parier centre 1 que cet 6vduement n'est pas arriv6. Or, au moyen de 1' observation pr6c6dente, on voit que dans le second cas la probabilite propre du fait etant -= , le timoignage conserve toute sa force, au lieu que dans le premier, cette probability etant tr§s-petite, riduit presque ^ rien celle du t6moignage. J'ai propos6 ensuite de prendre, pour la probability propre du fait, 2G 402 CONDOECET. le rapport du nombre de combinaisons qui donnent ce fait, ou un fait senablable au nombre total des combinaisons. Ainsi, par exemple, dans le cas oii on tire une carte d'un jeu de dix cartes, le nombre des combinaisons ou Ton tire une carte d6termin6e quelconque est un ; celui des combinaisons oii Ton tire une autre carte d6termin6e est aussi un ; done -^ exprimera la probability propre. Si on me dit qu'on a tirg deux fois de suite la mSme carte, alors on trouvera qu'il n'y a que dix combinaisons qui donnent deux fois une m^me carte, et quatre-vingt-dix qui donnent deux cartes diffSrentes : la proba- bilite propre du fait n'est done que zr^, et celle du timoignage com- mence ^ devenir plus foible. Mais je crois devoir abandonner cette maniSre de consid^rer la question, 1° parce qu'elle me par6it trop hypotbltique ; 2° parce que souvent cette comparaison d'6v§nemens semblables seroit difficile &, faire, ou, ce qui est encore jjis, ne se feroit que d'aprSs des suppositions arbi- traires ; 3° parce qu'en I'appliquant S, des exemples, elle conduit h des risultats trop 61oign6s de ceux que donneroit la raison commune. J'en ai done chei'chS une autre, et il m'a paru plus exact de prendre, pour probability propre d'un 6venement, le rapport de la probability de cet Ivenement prise dans le sens ordinaire, avec la pro- bability moyenne de tous les autres 6v§nemens. 739. Thus we see that Condorcet abandons the suggestion which he made in the fifth part of his memoir and offers another. It does not seem that the new suggestion escapes any of the objec- tions which Condorcet himself advances against the old suggestion, as will appear by the analysis we shall now give of Condorcet' s examples. 740. Suppose there are ten cards and it is asserted that a specified card has been drawn twice running ; we proceed to estimate the prdbahiliti propre of the event. There are 9 other ways in which the same card can be drawn twice, and the ordinary proba- bility of each drawing is r-^ ; there are 45 ways in which two dif- ferent cards are obtained in two drawings, and the ordinary proba- . . 2 bility of each drawing is ^^ . Hence the mean probability of all the other events is CONDORCET. 403 lOOj ' 45 X -I- + 9 X ^l , that is ^^ 54 I 100 '" '^ lOOj ' '"'"' ''' 5400 • Hence according to Condorcet's own words the probahiliU jyropre should be j^ -=- ^^ , that is — . But he himself says that the 54 1 ( QQ 1 1 prohabiUU propre is — , so that he takes joo ^ {slo + lOo} 1 1 99 and not j^ -;- g— — . That is, as is so frequently the case with Condorcet, his own words do not express his own meaning. Again suppose that there are ten cards and it is asserted that a specified card has been drawn thrice running ; we proceed to esti- mate the probability propre of the event. Here the mean proba- bility of all the other events is If 6 3 9 1 999 2l9r^^l000 + 9^^i000 + 100o|' *^^*^^ 219000- 219 Condorcet says that the probahiliU propre is yhtq > s° t^^* ^^ V 1 j 999 1 ] *^^®' 1000 ""1219000 + 1000] • 741. Condorcet now proceeds to apply these results in the following words : Ainsi supposons, par exemple, que la probability du timoignage soit 99 . . =-^, c'est-S,-dLre, que le temoin ne se trompe ou ne veuille tromper qu'une fois sur cent, on aura, d'aprls son tlmoignage, la probabilite TTm "'^ (\r\(\ ^ °^ * ® '°^^ carte aeterminee ; la probabiute 9540 qu'on a tirl deux fois la m^rne carte ; et la probability .. qu'on I'a tirle trois fois. We find some difficulties in these numbers. Let p denote the probabilite propre and t the probability of the testimony; then the formula to be applied is, we presume. pt In the first case it seems that Condorcet pt+{l-p){\-t)- 26—2 401 • CONDORCET. supposes p = 1, that is he takes apparently the probability propre to he j^ -^ -^ \9 X YF.\ ^ which agrees indeed with his own words but not with his practice which we have exhibited in Art. 740 ; if we follow that practice we shall have p = ^'■ 64 In the second case we have p = — ^ , and with this value the 54 formula gives ^ which is approximately '9818. , 219 In the third case we have p — ' , and with this value the formula gives 3^7: which however is very nearly '9560 instead of "9540 as Condorcet states. 742. Condorcet's next example seems very arbitrary and ob- scure. His words are, Supposons encore que I'observation ait constats que, sur vingt mil- lions d'hommes, un seul ait v6cu 120 ans, et que la plus longue vie ait 6te de 130 ; qu'un homme me disc que quelqu'un vient de mourir S, 120 ans, et que je cherche la probabilite propre de cet 6v§nement : je regarderai d'abord comme Tin fait unique, celui de vivre plus de 130 ans, fait que je suppose n'^tre pas arrivl ; j'aurai done 131 faits dif- ferens, dont celui de mourir a 120 ans est un seul. La probabilit6 de celui-ci sera j^ ; la probabilite moyenne des 130 autres sera 20000130 , , , ,.,.,. , , , 130 ; done la probabilite propre cnercnee sera 20000131 X 130' ^ " " 20000260 1 ou environ 15384' 743. Condorcet's next example seems also arbitrary. His words are, Cette mSthode s'appliquera Sgalement aux gvlnemens indeterminfe. Ainsi, en continuant le meme exemple, si le timoin a dit seulement que Ton a deux fois amen6 la m^me carte, sans la nommer, alors ces dix gvSnemens, ayant chacun la probability yftk > Ynn ^xprimera leur pro- CONDORCET. 405 2 babilite moyenne; ^Kn exprimera de mSine celle des 45 autres 6vSne- 2 mens ayant cliacun la probabilit6 y^ : ainsi la probabilite propre de revlnement sera ^ . o Condorcet himself observes that it may appear singular that the result in this case is less than that which was obtained in Art. 740 ; so that a man is less trustworthy when he merely says that he has seen the same card drawn twice, than when he tells us ia addition what card it was that he saw drawn twice. Condorcet tries to explain this apparent singularity; but not with any ob- vious success. The singularity however seems entirely to arise from Con- dorcet 's own arbitrary choice ; the rule which he himself lays down requires him to estimate la probabilite moyenne de, tous les autres ivhnemens, and he estimates this mean probability differently in the two cases, and apparently without sufficient reason for the dif- ference. 744. Condorcet's next example is as follows : We are told that a person with two dice has five times successively thrown higher than 10 ; find the probabilite propre. With two dice the number thrown may be 2, 3, ... up to 12 ; the respective probabilities are 1 2 ^±^^A.AA11. 36' 36' 36' 36' 36' 36' 36' 36' 36' 36' 36" X • 11x12x13x14x15 ^, ^ . The whole number oi events is r^ , that is 3003 ; and of these only 6 belong to the proposed combination. Since the probability of these 6 throws is j^i t^^eir mean proba- bility is Ts • The mean probability of the other throws will 11' ,,.,.. • 2997 ^® 2997 X 12° • ■^^''''^ probabilite propre is g^ii5^2997 ' It is obvious that all this is very arbitrary. When Condorcet says there are 6 throws belonging to the proposed combination he means that all the throws may be 12, or all 11, or four 12 and one 11, or three 12 and two 11, ... And he says the mean probability is 406 CONDORCET. ^ — =155 . But if we consider the different orders in which these throws can occur we may say that the whole number is 2^ and the 1/1 2 \° 1 mean probability -^^ [sa + or] > ^^^^ i^ 2» V36 ' 367' """'"'■" 2^2'' Again let us admit that there are 3003 cases in all, and that of these only 6 belong to the proposed combination. The other 2997 cases form two species, namely those in which every throw is below 11, and those in which some throws are below 11 and the ir others above 10 ; when Condorcet takes ^ — ^95 ^^ *^® mean probability, he forgets this division of species and only con- siders the first species. He should take ^ [l — thb) instead 11^ of 2997 X 12' • 745. Suppose two classes of events A and B; let the pro- bability of an ^ be a and the probability of a B he h; let there be m events A and n events B. The probability propre of an assigned event of the class B will be, according to Condorcet's practice, ^ that is (m + n-l)h ma + (n — 1) , ma + [m + zn — 2) OT + w — 1 2& If m and n be equal and very large this becomes ^ . If we suppose h extremely small and consequently a very nearly unity we obtain 25 as an approximate value. 746. Condorcet proceeds to apply his doctrine to the credi- bility of two statements in the History of Rome. He says, Je vais maintenant essayer de faire ^ une queation de critique I'application des principes que je viens d'6tablir. Newton par&it ^tre le premier qui ait eu I'idSe d'appliquer le calcul des probabilites k la critique des faits. II propose, dans son ouvrage sur la chronologie, d'employer la connoissance de la dur6e moyenne des generations et des regnes, telle que rexp6rience nous la donue, soit pour fixer d'une mani^re du moins approchle, des points de ckronologie fort inoertains, CONDORCET. 407 soit pour juger du plus ou du moins de coufiance que miritent les difigrens systSmes imagines pour conoilier entr'elles des 6poques qui paroissent se contredire. Condorcet names Fr^ret as having opposed this application of the Theory of Probability, and Voltaire as having supported it; but he gives no references. 747. According to some historians the whole duration of the reigns of the seven kings of Rome was 257 years. Condorcet pro- poses to examine the credibility of this statement. He assumes that in an elective monarchy we may suppose that a king at the date of his election will be between 30 years old and 60 years old. He adopts De Moivre's hypothesis respecting human mortaUty ; this hypothesis, as Condorcet uses it, amounts to assuming that the number of people at any epoch who are y years old is h (90 — y), where A; is some constant, and that of these h die every year. Let n denote the greatest number of years which the youngest elected king can live, m the greatest number of years which the oldest elected king can live ; then the probability that a single reign wiU last just r years is the coefiQcient of a' in the expan- sion of (w-w+l)a:(l-a;)-a;'"" + a;"^' m A few words will be necessary to shew how this formula can be verified. It follows from our hypothesis that the number of per- sons from whom the king must be elected is A; {w + (w - 1) -1- (w - 2) -1- . . . -)- ot}, that is h ^^-5 — (m — m + 1). And if r be less than m -|- 1 the num- ber of persons who die in the r"" year will be ^ (re — m + 1) ; if »• be between wi + 1 and n + 1, both inclusive, the number who die in the »-* year vdll be ^ (n — ?• + 1) ; if r be gi-eater than w -|- 1 the number who die in the r*"" year will be zero. Now the coefficient of a;' in the expansion of {n-m^\)x x'^'-^-x"^ T^ (1 - xf J;08 CONDOECET. will be found to be n-m+liiris less tban m + 1, and if r is greater tban n + 1, and in other cases to be w - r + 1. 748. Hence the probability that the duration of seven reigns will amount to just 257 years is the coefficient of x^' in the expan- sion of the seventh power of (1 - xY -^— (n - m + 1) Now Condorcet takes m = 60 and w = 30 ; and he says that the value of the required coefficient is -000792, which we will assume he has calculated correctly. Thus he has obtained the probability in the ordinary sense, which he denotes by P; he requires the j)rohabiUt4 propre. He considers there are 414 events possible, as the reigns may have any duration in years between 7 and 420. Thus the mean proba- 1 — P bility of all the other events is , ; and so the probabilite propre 413P , , 1 '' 1T412P'"^^°"*4- 749. Condorcet says that other historians assign 140 years in- stead of 257 years for the duration of the reigns of the kings. He says the ordinary probability of this is •008887, which we may denote by Q. He then makes the prohabilite propre to be 412«9 , . , . ,.1 which IS more than 7; . 1+4116' """""^"' " 2' He seems here to take 413, and not 414, as the whole number of events. 750. Condorcet then proceeds to compare three events, namely that of 257 years' duration, that of 140 years' duration, and what he calls un autre Mnement inddtermine quelconque qui auroit pu avoir lieu. He makes the prohahilites propres to be respectively 411P illQ . 1-P-Q and 410(P+§)+l' 410 (P+ + 1 410(P+e)+l' 3 37 10 which are approximately — , — ^ , -^ . CONDORCET. 409 Here again he seems to take 413 as the whole number of events. He proceeds to combine these probabilities with probabilities arising from testimony borne to the first or second event. 751. Condorcet considers another statement which he finds in Roman History, namely that the augur Accius Naevius cut a stone with a razor. Condorcet takes (^f^/^^/^/^ as the ordinary proba- bility, and then by Art. 745 makes the prohdbiliU propre to be 2 1000000 ■ 752. We have spent a long space on Condorcet's memoir, on account of the reputation of the author; but we fear that the reader will conclude that we have given to~ it far more attention than it deserves. It seems to us to be on the whole excessively arbitrary, altogether unpractical, and in parts very obscure. 753. We have in various places expressed so decidedly our opinion as to the obscurity and inutility of Condorcet's investigar- tions that it will be just to notice the opinions which other writers have formed. Gouraud devotes pages 89 — 104 of his work to Condorcet, and the following defects are noticed : Un style embarrasse, deuu^ de justesse et de coloris, une philosophic souvent obscure ou bizarre, une analyse que les meilleurs juges ont trouv^e confuse. With this drawback Condorcet is praised in terms of such extravagant eulogy, that we are tempted to apply to Gouraud the reflexion which Du- gald Stewart makes in reference to Voltaire, who he says " is so lavish and undistinguishing in his praise of Locke, as almost to justify a doubt whether he had ever read the book which he extols so highly." Stewart's Works, edited by Hamilton, Vol. I. page 220. Galloway speaks of Condorcet's Essay as " a work of great in- genuity, and abounding with interesting remarks on subjects of the highest importance to humanity." Article Probability in the Encyclopcedia Britannica. Laplace in his brief sketch of the history of the subject does not name Condorcet ; he refers however to the kind of questions 410 CONDORCET. which Condorcet considers and says, Tant de passions, d'int^r^fe divers et de circonstances compliquent les questions relatives a ces objets, qu'elles sont presque toujours insolubles. Theorie...des Prob. page cxxxviii. Poisson names Condorcet expressly; with respect to his Prelimi- nary Discourse, he says, . . . oti sont ddvelopp^es avec soin les con- siderations propres a montrer I'utilit^ de ce genre de recherches. And after referring to some of Laplace's investigations Poisson adds, . . . il est juste de dire que c'est a Condorcet qu'est due I'id^e ing^nieuse de faire d^pendre la solution, du principe de Bayes, en considerant successivement la culpabilite et I'innocence de I'accus^, comme une cause inconnue du jugement prononce, qui est alors le fait observd, duquel il s'agit de d^duire la probability de cette cause. Recherches sur la Prob. . . . page 2. We have already referred to John Stuart Mill, see Art. 665. One sentence of his may perhaps not have been specially aimed at Condorcet, but it may well be so applied. Mr Mill says, " It is obvious, too, that even when the probabilities are derived from ob- servation and experiment, a very slight improvement in the data, by better observations, or by taking into fuller consideration the special circumstances of the case, is of more use than the most elaborate application of the calculus to probabilities founded on the data in their previous state of inferiority." Logic, Vol. II. page 65. Condorcet seems really to have fancied that valuable results could be obtained from any data, however imperfect, by using formulae with an adequate supply of signs of integration. CHAPTER XVIII. TREMBLEY. ^oi. We have now to examine a series of memoirs by Trembley. He was bom at Geneva in 1749, and died in 1811. The first memoir is entitled Disquisitio Elementaris circa Cal- culum Probabilium. This memoir is published in the Com/mentationes Societatis RegicB Scientiarum, Gottingensis, Vol. xil. The volume is for the years 1793 and 1794 ; and the date of publication is 1796. The memoir occupies pages 99 — 136 of the mathematical portion of the volume. 755. The memoir begins thus : Plurimae extant hie et illic sparsae meditationes analyticae circa cal- culum Probabilium, quas hie recensere non est animus. Quae cum plerumque quaestiones particulares spectarent, summi Geometrae la Place et la Grange banc theoriam generalius tractare sunt aggressi, auxilia derivantes ex intimis calculi integralium visceribus, et eximios quidem fructus inde perceperunt. Cum autem tota Probabilium theoria principiis simplicibus et obviis sit innixa, quae nihil aliud fere requirunt quam doctriaam combinationum, et pleraeque difficultates in enume- randis et distinguendis casibus versentur, e re visum est easdem quaes- tiones generaliores methodo elementari tractare, sine uUo alieno auxilio. Cujus tentaminis primum specimen hae paginae complectuntur, continent quippe solutiones elementares Problematum generaliorum quae vir illustrissimus la Grange soluta dedit in Commentariis Academiae Regiae Berolinensis pro anno 1775. Si haec Geometris non displicuerint, alias deinde ejusdem generis dilucidationes, deo juvante ipsis proponam. 756. The intention expressed at the end of this paragraph was 412 TKEMBLEY. carried into effect in a memoir in the next volume of the Gottin- gen Commentationes. The present memoir discusses nine problems, most of which are to be found in De Moivre's Doctrine of Chances. To this work Trembley accordingly often refers, and his references obviously shew that he used the second edition of De Moivre's work ; we shall change these references into the coiTesponding references to the third edition. ..^ In this and other memoirs Trembley proposes to give elemen- tary investigations of theorems which had been previously treated by more difficult methods ; but as we shall see he frequently leaves his results really undemonstrated. 757- The first problem is, to find the chance that an event shall happen exactly h times in a trials, the chance of its happening in a single trial being p. Trembley obtains the well known result, la , , '— — J p (1 — «) ; he uses the modern method : see Art. 257. 758. The second problem is to find the chance that the event shall happen at least b times. Trembley gives and demonstrates independently both the formulae to which we have already drawn attention ; see Art. 172. He says, longum et taediosum foret has formulas inter se comparare a priori; but as we have seen in Art. 174 the comparison of the formulae is not really difficult. 759. The third problem consists of an application of the second problem to the Problem of Points, in the case of two players ; the fourth problem is that of Points in the case of three players ; and the fifth problem is that of Points in the case of four players. The results coincide with those of De Moivre ; see Aii. 267. 760. Trembley's next three problems are on the Duration of Play. He begins with De Moivre's Problem Lxv, which in effect supposes one of the players to have an unlimited capital; see Arts. 307, 309. Trembley gives De Moivre's second mode of solution, but his investigation is unsatisfactory; for after having found in succession the first six terms of the series in brackets, he says Perspicua nunc est lex progressionis, and accordingly writes down the general term of the series. Trembley thus leaves the main difficulty quite untouched. TREMBLEY. 413 761. Trembley's seventh problem is De Moivre's Problem LXiv, and he gives a result equivalent to that on De Moivre's page 207; see Art. 306. But here again after investigating a few terms the main difficulty is left untouched with the words Perspicua nunc est lex progressionis. Trembley says, Eodem redit solutio Cel. la Grange, licet eaedem formulae non prodeant. This seems to imply that Lagrange's formulae take a different shape. Trembley probably refers to Lagrange's second solution which is the most completely worked out ; see Art. 583. Trembley adds in a Scholium that by the aid of this problem we can solve that which is Lxvii. in De Moivre ; finishing with these words, in secunda enim formula fieri debet c =jP — 1, which appear to be quite erroneous. 762. Trembley's eighth problem is the second in Lagrange's memoir ; see Art. 580 : the chance of one event is p and of an- other q, find the chance that in a given number of trials the first shaU happen at least h times and the second at least c times. Trembley puts Lagrange's solution in a more elementary form, so as to avoid the Theory of Finite Differences. 763. Trembley's ninth problem is the last in Lagrange's me- moir ; see Art. 587. Trembley gives a good solution. 764. The next memoir is entitled Be Probabilitate Causarum ah effectibus oriunda. This memoir is published in the Comm. Soc. Reg....Oott. Vol. XIII. The volume is for the years 1795 — 1798 ; the date of publication is 1799. The memoir occupies pages 64 — 119 of the mathematical portion of the volume. 765. The memoir begins thus : Hanc materiam pertractarunt eximii Geometrae, ac potissimum Cel. la Place in Oommentariis Academiae Parisinensis. Cum autem in hujusce generis Problematibus solvendis sublimior et ardua analysis fuerit adMbita, easdem quaestiones methodo elementari ac idoneo usu doctrinae serierum aggredi operae pretium duxi. Qua ratione baeo altera pars calculi Probabilium ad tbeoriam combinationum reduceretur, sicut et primam reduxi in dissertatione ad Regiam Sooietatem transmissa. •ill TREMBLEY. Primarias quaestiones hie breviter attingere oonabor, methodo diluci- dandae imprimis intentus. 766. The first problem is the following. A bag contains an infinite number of white balls and black balls in an unknown ratio ; p white balls and g- black have been drawn out va. p-\-q^ drawings ; what is the chance that m + n new drawings will give m white and n black balls ? The known result is m 4- n J that is. Jo [tn + n \m+p | n + q |p + g+ 1 Im In IP |_£ Itn+p + n+q+l ' Trembley refers to the memoir which we have cited in Ai-t. 551, where this result had been given by Laplace ; see also Art. 704 Trembley obtains the result by ordinary Algebra ; the investi- gations are only approximate, the error being however inappreci- able when the number of balls is infinite. If each ball is replaced after being drawn we can obtain an exact solution of the problem by ordinary Algebra, as we shall see when we examine a memoir by Prevost and Lhuilier ; and of course if the number of the balls is supposed infinite it will be indifferent whether we replace each ball or not, so that we obtain indirectly an exact elementary demonstration of the important result which Trembley establishes approximately. 767. We proceed to another problem discussed by Trem- bley. A bag is known to contain a very large number of balls which are white or black, the ratio being unknown. In ^ -f- 5- drawings p white balls and q black have been drawn. Required the probability that the ratio of the white to the black lies between zero and an assigned fraction. This question Trembley proceeds to consider at great length ; he supposes j? and q very large and obtains approximate results. If the assigned fraction above referred to be denoted by TREMBLEY. 41 — £ 6, he obtains as the numerator of the required probability, approximately {p + S)0 1 (P + iY ^ The denominator would be , — ' -. . Trembley refers to two places in which Laplace had given this result; they are i\iQ Hist, de I' Acad.... Paris for 1778, page 270, and for 1783 page 445. In the Theorie...des Prob. Laplace does not reproduce the general formula ; he confines himself to suppos- » 1 ing --£— — ^ = - ■ see page 379 of the work. p+ q 2' ^ *= Trembley's methods are laborious, and like many other at- tempts to bring high mathematical investigations into more elementary, forms, would probably cost a student more trouble than if he were to set to work to enlarge his mathematical know- ledge and then study the original methods. 768, Trembley follows Laplace in a numerical application relating to the births of boys and girls at Vitteaux in Bourgogne. Laplace first gave this in the Hist, de l' Acad.... Paris for 1783, page 448 ; it is in the Thiorie . . . des Prob. page 380. It appears that at Vitteaux in five years 212 girls were born to 203 boys. It is curious that Laplace gives no information in the latter work of a more recent date than he gave in the Hist, de l' Acad.,., Paris for 1783 ; it would have been interesting to know if the anomaly still continued in the births at Vitteaux. 769. We may observe that Laplace treats the problem of births as analogous to that of drawing black and white balls from a bag. So he arrives at this result ; if we draw 212 black balls to 203 white balls out of a bag, the chance is about '67 that the black balls in the bag are more numerous than the white. It is not very easy to express this result in words relating to births ; Laplace says in the Hist, de I' Acad... .Paris, la difference '670198 sera la 416 TEEMBLEY. probability qu'a Viteaux, la possibility des naissances des fiUes est sup^rieure k ceUe des naissances des gar9ons ; in tbe Thiorie . . . des Proh. he says, la sup&iorit^ de la facility des naissances des fiUes, est done indiqu^e par ces observations, avec une probability ^gale h, "67. These phrases seem much better adapted to the idea to be expressed than Trembley's, Probabilitas numerum puellarum superaturum esse numerum puerorum erit = •67141. 770. Trembley now takes the following problem. From a bag containing white balls and black balls in a large number but in an unknown ratio j> white balls and q black have been drawn ; required the chance that if 2a more drawings are made the white balls shall not exceed the black. This problem leads to a series of which the sum cannot be found exactly. Trembley gives some investigations respecting the series which seem of no use, and of which he himself makes no application ; these are on his pages 103 — 105. On his page 106 he gives a rough approximate value of the sum. He says, Similem seriem refert Cel. la Place. This refers to the Hist, de I' Acad.... Paris for 1778, page 280. But the word similem must not be taken too strictly, for Laplace's approxi- mate result is not the same as Trembley's. Laplace applies his result to estimate the probability that more boys than girls will be born in a given year. This is not repeated in the Theorie...des Proh., but is in fact included in what is there given, pages 397 — 401, which first appeared in the Hist, de V Acad.... Paris for 1783, page 458. 771. Trembley now takes another of Laplace's problems, namely that discussed by Laplace in the MSm,oires . . . par divers Savans, Vol. vi. page 633. Two players, whose respective skills are unknown, play on the condition that he who first gains n games over his adversary shall take the whole stake ; at a certain stage when A wants / games and B wants h games they agree to leave off playing : required to know how the stake should be divided. Suppose it were given that the skill of -4 is a; and that of .S is 1—x. Then we know by Art. 172 that £ ought to have the fraction ^ (x) of the stake, where TEEMBLEY. 417 J / \ II sm f-. ^ mim — 1) a? m(m — 1)(ot — 2) as' + — '-— ; + •• ^2\ \L\lzl (1 where m =f+ h — 1. Now if X represents ^'s skill the probability that in 2« — /— h games A would win n —f and B would win w — A is as""-'' (1 — »)""*, disregarding a numerical coefficient which we do not want. Hence if A wins n —f games and B wins n—h, which is now the observed event, we infer that the chance that A's skill is x is x"-' {1 - xf-" dx Therefore the fraction of the stake to which B is entitled is 4>{x)x''-'{l-xY-''dx I •^ dx All this involves only Laplace's ordinary theory. Now the following is Trembley's method. Consider (as) ; the first term is (1 — a;)" ; this represents the chance that B will win m games running on the supposition that his skill is 1 — x. If we do not know his skiU a priori we must substitute instead of (1 — a?)" the chance that B will win m games running, computed from the observed fact that he has won n — h games to .4's n— /games. This chance is, by Art. 766, Again consider the term ma; (1 - a;)""' in <^{x). This represents the chance that B will win m — 1 games out of m, on the suppo- sition that his skill is 1—x. If we do not know his skill a priori we must substitute instead of this the chance that B will win 27 ■ilS ■ TBEMBLEY m — 1 games out of m, deduced from the observed fact that he has won n — h games to ^'s n —/games. This chance is, by Art. 766, m (n —f+ 1) n+f-1 M. It is needless to go farther, as the principle is clear. The final result is that the fraction of the stake to which B is entitled is [ ^^J^ J n+f-l 1.2 n+f-\n+f-2 {f+h-V)...{h + l) {n-f+l){n-f+2)...{n-l) I /-I {n+f-\){n+f-2)...{n + l) This process is the most interesting in Trembley's memoir. Laplace does not reproduce this problem in the TMorie . . . des Prob. 772. Trembley gives some remarks to shew the connexion between his own methods and Laplace's. These amount in fact to illustrations of the use of the Integral Calculus in the summa- tion of series. For example he gives the result which we may write thus : 1 q t q{q-V) f g(g-l)(g-2) f p + 1 lp + 1'^ 1.2 ^ + 3 1.2.3 p-\-^^'" ^ '" p + q + 1 = f a;* (1 - tx)' dx = ~f *x^ (1 - x)' dx. Jo ''Jo 773. Trembley remarks that problems in Probability consist of two parts ; first the formulae must be exhibited and then modes of approximate calculation found. He proposes to give one ex- ample from Laplace. Observation indicates that the ratio of the number of boys bom to the number of girls born is greater at London than at Paris. Laplace says : Cette difference semble indiquer h Londres une plus grande facility pour la naissance des gar9ons, il s'agit de deter- miner combien cela est probable. Bee Hist, de I' Acad.... Paris TREMBLEY. 419 for 1778, page 304, for 1783, page 449 ; and Theorie... des Prob. page 381. Trembley says, Supponit Cel. la Place natos esse Parisiis intra certum tempus, p pueros q puellas, Londini autem intra aliud temporis spatium p' pueros q' puellas, et quaerit Probabilitatem, causam quae Parisiis produoit pueros esse efficaciorem quam Londini. B supra dictis sequitur hanc Probabilitatem repraesentaii per formulam j L" (1 - xY x'^ (1 - ody dxdid /Lf (1 - a;)' a;'"' (1 - ai)* dxdaf Trembley then gives the limits of the integrations ; in the numerator for x' from a;' = to a;' = x, and then for x from ic = to x = l; in the denominator both integrations are between and 1. Trembley considers the numerator. He expands a;* (1 — x'Y in powers of x' and integrates from a;' = to a;' = x. Then he expands x' (1 — xY and integrates from a; = to a; = 1 ; he obtains a result which he transforms into another more convenient shape, which he might have obtained at once and saved a page if he had not expanded x" (1 — xY- Then he uses an algebraical theorem in order to effect another transformation ; this theorem he does not demonstrate generally, but infers it from examining the first three cases of it ; see his page 113. We will demonstrate his final result, by another method. We have jx {l X) ax-x |^,_^^ l^' + 2^ 1.2 y + 3 J' Multiply by x'^ (1 — xY and integrate from x = to a; = 1 ; thus we obtain by the aid of known formulae [g|p+p' + l Ji. £ 1 y?+y+2 p + 1 1 y + 2 p+p' + q-vB ,ISI^ 1 {p+p' + ^)(p+p+^) \ '•' 1.2 p+Sip+p' + q + S){p+p' + q + 4>) )' 27—2 420 TREMBLEY, This result as we have said Trembley obtains, though he goes through more steps to reach it. Suppose however that before effecting the integration with respect to x we use the following theorem : + i{i-V) X' g'(cf'-l)(q'-2) y + 1 l/ + 2"^ 1.2 y + 3 1.2.3 / + 4 = (l-^)''{y + ^'+l + (y + g' + l)(y + 2') r^^ g'(g'-l) ■ ^ , g'(g'-l)(g'-2) 1_ ) (/ + g'+i)(p'+2')(/+g'-i)(p'+2'-2) (1-xy^-r Then by integrating with respect to x, we obtain \q+q' \p+p'+l f 1 q^ p+p'+q+q'+2 \p+p'+q + q'+^ [p+q'+i {p + q +i) {p + q) q+s I g'(g'-l) (iP+p'+g+g'+2)(j,+/+g+g'+l) ) (/+g'+i) (/ +20 (y+g'-i) (g+g') (g+g'-i) J ■ It is in fact the identity of these two results of the final inte- gration which Trembley assumes from observing its truth when q' = 1, or 2, or 3. With regard to the theorem we have given above we may remark that it may be obtained by examining the coefficient of x'' on the two sides ; the identity of these coefficients may be estab- lished as an example of the theory of partial fractions. 774. Trembley then proceeds to an approximate summation of the series ; his method is most laborious, and it would not repay the trouble of verification. He says at the end, Series haec, quae similis est seriei quam refert Gel. la Place ... He gives no refer- ence, but he probably has in view the Hist, de l' Acad.... Paris for 1778, page 310. 775. We have next to consider a memoir entitled Recherches sur une question relative, au calcul des prohahilites. This memoir is published in the volume for 1794 and 1795 of the 'MSmoires de TREMBLEY. 421 V Acad.... Berlin; the date of publication is 1799: the memoir occupies pages 69—108 of the mathematical portion of the volume. The problem discussed is that which we have noticed in Art. 448. 776. Trembley refers in the course of his memoir to what had been done by De Moivre, Laplace and Euler. He says, L'analyse dont M. Euler fait usage dans ce MImoire est trSs-ing^- nieuse et digne de ce grand ggomltre, mais comme elle est un peu indirecte et qu'il ne seroit pas aisg de I'appliquer au problgme ggngral dont celui-ci n'est qu'un cas parfciculier, j'ai entrepris de traiter la chose directement d'aprSs la doctrine des combinaisons, et de donner S, la question toute I'gtendue dont elle est susceptible. 777. The problem in the degree of generality which Trembley gives to it had already engaged the attention of De Moivre ; see Art. 293. De Moivre begins with the simpler case in his Pro- blem XXXIX, and then briefly indicates how the more general question in his Problem XLI. is to be treated. Trembley takes the contrary order, beginning with the general question and then deducing the simpler case. When he has obtained the results of his problem Trembley modifies them so as to obtain the results of the problem discussed by Laplace and Euler. This he does very briefly in the manner we have indicated in Art. 453. 778. Trembley gives a numerical example. Suppose that a lottery consists of 90 tickets, and that 5 are drawn at each time ; then he obtains '74102 as the approximate value of the probability that all the numbers will have been drawn in 100 drawings. Euler had obtained the result 7419 in the work which we have cited in Art. 456. 779. Trembley 's memoir adds little to what had been given before. In fact the only novelty which it contains is the investi- gation of the probability that w — 1 kinds of faces at least should come up, or that w — 2 kinds effaces at least, or w — 3, and so on. The result is analogous to that which had been given by Euler and which we have quoted in Art. 458. Nor do Trembley's methods present any thing of importance ; they are in fact such as would naturally occur to a reader of De Moivre's book if he wished to 422 TREJIBLEY. reverse the order which De Moivre has taken. Trembley does not supply general demonstrations ; he begins with a simple case, then he proceeds to another which is a little more complex, and when the law which governs the general result seems obvious he enun- ciates it, leaving to his readers to convince themselves that the law is universally true. 780. Trembley notices the subject of the summation of a cer- tain series which we have considered in Art. 460. Trembley says, M. Euler remarque que dans ce cas la somme de la suite qui donne la probability, peut s'exprimer par des produits. Cela peut se d^- montrer par le calcul integral, par la m^thode suivante qui est fort simple. But in what follows in the memoir, there is no use of the Integral Calculus, and the demonstration seems quite unsatis- factory. The result is verified when a: = 1, 2, 3, or 4 and then is assumed to be universally true. And these verifications them- selves are unsatisfactory; for in each case r is put successively equal to 1, 2, 3, 4, and the law which appears to hold is assumed to hold universally. Trembley also proposes to demonstrate that the sum of the series is zero, if n be greater than rx. The demonstration how- ever is of the same unsatisfactory character, and there is this ad- ditional defect. Trembley supposes successively that w = r (a; -1- 1), n = r (x + 2), n = r (x+ S), and so on. But besides these cases n may have any value between rx and r (x+1), or between r{x+l) and r {x + 2), and so on. Thus, in fact, Trembley makes a most imperfect examination of the possible cases. 781. Trembley deduces from his result a formula suitable for approximate numerical calculation, for the case in which n and x are large, and r small ; his formula agrees with one given by La- place in the Hist, de l' Acad.... Paris 1783, as he himself observes. Trembley obtains his formula by repeated use of an approximation which he establishes by ordinary Algebraical expansion, namely Trembley follows Laplace in the numerical example which we have noticed in Art. 455. Trembley moreover finds that in TREMBLEY. 423 about 86927 drawings there is aa even chance that all the tickets except one will have been drawn ; and he proceeds nearly to the end of the calculation for the case in which all the tickets except two are required to be drawn. 782. The next memoir is entitled Recherches sur la mortalite de la petite v4role. This memoir is published in the Memoires de 11 Acad..., Berlin for 1796 ; the date of publication is 1799 : the memoir occupies pages 17 — 38 of the mathematical portion of the volume. 783. This memoir is closely connected with one by Daniel Bernoulli ; see Art. 398. Its object may be described as twofold; first, it solves the problem on the hypotheses of Daniel Bernoulli by common Algebra without the Integral Calculus ; secondly, it examines how far those hypotheses are verified by facts. The memoir is interesting and must have been valuable in a practical point of view at the date of publication. 784. Let m and n have the same signification as in Daniel Bernoulli's memoir ; see Art. 402 : that is, suppose that every year small-pox attacks 1 in w of those who have not had the disease, and that 1 in wi of those who are attacked dies. Let a„ denote a- given num.ber of births, and suppose that a^, dj, fflj, ... denote the number of those who are alive &,t the end of 1, 2, 3, . . . years : then Trembley shews that the number of per- sons alive at the beginning of the a;*'' year who have not had the small-pox is aJl i+i(i-ir m, m \ nj For let h^ denote the number alive at the beginning of the a* year who have not had the smaU-pox, and h^^^ the number at the beginning of the {x + l)*"" year. Then in the a?"' year small-pox attacks — persons ; thus l^\l ) would be alive at the begin- ning of the next year without having had the small-pox if none of them died by other diseases. We must therefore find how many of 424 TEEMBLEY. these 6^ (1 j die of other diseases, and subtract. Now the total number who die of other diseases during the a;"' year is ^' mn these die out of the number a^ —. Hence, by proportion, the mn number who die out of 5^ ( 1 j is 1> mn ^' mnj K.{\-- Therefore 5^, = ^1 -1) - ^1^ (.,_ «^^_^J mn mn We can thus establish our result by induction ; for we may shew in the manner just given that and then universally that mn 5. = . "^(^-3 m m\ nj 785. We may put our result in the form - ma. + (-l)(l-3' TREMBLEY. 425 Now there is nothing to hinder us from supposing the intervals of time to be much shorter than a year ; thus n may be a large number, and then / In-" ? (1 j =.e" nearly. The result thus agrees with that given by Daniel Bernoulli, see Art. 402 : for the intervals in his theory may be much shorter than a year. 786. Hitherto we have used Daniel Bernoulli's hypotheses ; Trembley however proceeds to a more general hypothesis. He supposes that m and n are not constant, but vary from year to year ; so that we may take m„ and n^ to denote their values for the x^ year. There is no difficulty in working this hypothesis by Trembley's method ; the results are of course more complicated than those obtained on Daniel Bernoulli's simpler hypotheses. 787. Trembley then compares the results he obtains on his general hypothesis with a table which had been furnished by ob- servations at Berlin during the years 1758 — 1774. The comparison is effected by a rude process of approximation. The conclusions he arrives at are that n is very nearly constant for all ages, its value being somewhat less than 6 ; but m varies considerably, for it be- gins by being equal to 6, and mounts up to .120 at the eleventh year of age, then diminishes to 60 at the nineteenth year of age, and mounts up again to 133 at the twenty-fifth year of age, and then diminishes. Trembley also compares the results he obtains on his general hypothesis with another table which had been furnished by obser- vations at the Hague. It must be confessed that the values of m and n deduced from this set of observations differ very much from those deduced from the former set, especially the values of m. The observations at Berlin were nearly five times as numerous as those at the Hague, so that they deserved more confidence. 788. In the volume for 1804 of the M6moires de VAcad.... Berlin, which was published in 1807, there is a note by Trem- bley himself on the memoir which we have just examined. This note is entitled Eclaircissement relatif au M4moire sur la 426 TEEMBLEY. mortaUte....&c.; it occurs on pages 80 — 82 of the mathematical portion of the volume. Trembley corrects some misprints in the memoir, and he says : Au reste, je dois avertir que la m6tliode d'approximation que j'ai donnle dans ce mimoire comme un essai, en attendant que des obser- vations plus detaillges nous missent en 6tat de proc6der avec plus de rggularitg, que cette mgthode, dis-je, ne vaut absolument rien, et je dois des excuses au public pour la lui avoir presentee. He then shews how a more accurate calculation may be made ; and he says that he has found that the values of n instead of remainiag nearly constant really varied enormously. 789. The next memoir is entitled Essai sur la manikre de trouver le terme gdn^ral des siries r^currentes. This memoir is published in the volume for 1797 of the M^- moires de V Acad.... Berlin; the date of publication is 1800. The pages 97 — 105 of the memoir are devoted to the solution of a pro- blem which had been solved by Laplace in Vol. vii. of the Memoir es. . .par divers Savans ; Trembley refers to Laplace. The problem is as follows : Suppose a solid having n equal faces numbered 1, 2, 3 ...p; required the probability that in the course of n throws the faces will appear in the order 1, 2, 3, ...p. This problem is very nearly the same as that of De Moivre on the run of luck ; see Art. 325. Instead of the equation we shall now have i^vi-i = ■«„ + (1 - u^p) a'' ; and a = - . P Trembley solves the problem in his usual incomplete manner ; he discusses in succession the cases in which ^ = 2, 3, 4 ; and then he asssumes that the law which holds in these cases will hold generally. 790. The next memoir is entitled Observations sur les calculs relatifs d, la dur^e des mariages et au nomhre des dpoux subsistans. This memoir is published in the volume for 1799 — 1800 of the MSmoires de U Acad... Berlin; the date of publication is 1803; the memoir occupies pages 110 — 130 of the mathematical portion of the volume. TREMBLEY. 427 791. The memoir refers to that of Daniel Bernoulli on the same subject which we have noticed in Art. 412. Trembley ob- tains results agreeing with those of Daniel Bernoulli so far as the latter was rigorous in his investigations ; but Trembley urges ob- jections against some of the results obtained by the use of the infinitesimal calculus, and which were only presented as approxi- mate by Daniel Bernoulli. 7.92. As is usual with Trembley, the formulae which occur are not demonstrated, but only obtained by induction from some simple cases. Thus he spends three pages in arriving at the re- sult which we have given in Art. 410 from Daniel Bernoulli ; he examines in succession the five most simple cases, for which m = 1, 2, 3, 4, 5, and then infers the general formula by analogy. 793. For another example of his formulae we take the follow- ing question. Suppose n men marry n women at the same time ; if m out of the 2k die, required the chance that m. marriages are dissolved. \n We may take m pairs out of n in , 1 ways. In each •> >■ \m \n — in •' of the m pairs only one person must die ; this can happen in 2'" ways. Thus the whole number of cases favourable to the result 2" I w ^ , is , -^= — . But the whole number of cases is the whole jm I w — wi number of ways in which m persons out of 2n may die ; that is I 2w ' Hence the required chance is 2>? —m 2"'[«_ \2n-m I 2/1 \n — m Trembley spends two pages on this problem, and then does not demoiistrate the result. 794. Trembley makes some applications of his formulee to the subject of annuities for widows. He refers to a work by Karstens, entitled Theorie von Wittwencassen, Halle, 1784; and also names Tetens. On the other hand, he names Michelsen as a writer who 428 TEEMBLEY. had represented the calculations of mathematicians on such sub- jects as destitute of foundation. Trembley intimates his intention of continuing his investi- gations in another memoir, which I presume never appeared. 795. The next memoir is entitled Observations sur la methode de prendre les milieux entre les observations. This memoir is published in the volume for 1801 of the Mimoires de T Acad. ... Berlin ; the date of publication of the volume is 1804 : the memoir occupies pages 29 — 58 of the mathe- matical portion of the volume. 796. The memoir commences thus : La maniSre la plus avantageuse de prendre les milieux entre les observations a 6t6 detailMe par de grands glomltres. M. Daniel Ber- noulli, M. Lambert, M. de la Place, M. de la Grange s'en sont occupls. Le dernier a donn§ la-dessus un tr§s-beau mimoire dans le Tome v. des Mgmoires de Turin. II a employ6 pour cela le calcul integral. Mon desaein dans ce mSmoire est de montrer comment on peut parvenir aux memes rgsultats par un simple usage de la doctrine des combinaisons. 797. The preceding extract shews the object of the memoir. We observe however that although Lagrange does employ the Integral Calculus, yet it is only in the latter part of his memoir, on which Trembley does not touch ; see Arts. 570 — 575. In the other portions of his memoir, Lagrange uses the Differential Cal- culus ; but it was quite unnecessary for him to do so ; see Art. 564. Trembley's memoir appears to be of no value whatever. The method is laborious, obscure, and imperfect, while Lagrange's is simple, clear, and decisive. Trembley begins with De Moivre's problem, quoting from him ; see Art. 149. He considers De Moivre's demonstration indirect and gives another. Trembley's demonstration occupies eight pages, and a reader would probably find it necessary to fill up many parts with more detail, if he were scrupulous about exactness. After discussing De Moivre's problem in this manner, Trem- bley proceeds to inflict similar treatment on Lagrange's problems. We may remark that Trembley copies a formula from La- TREMBLEY. 429 grange with all the misprints or errors which it involves; see Art. 567. 798. The last memoir by Trembley is entitled Observations sv/r le calcul d'un Jeu de hasard. This memoir is published in the volume for 1802 of the Mimoires de I' Acad. ...Berlin; the date of publication is 1804: the memoir occupies pages 86 — 102 of the mathematical portion of the volume. 799. The game considered is that of Her, which gave rise to a dispute between Nicolas Bernoulli and others; see Art. 187. Trembley refers to the dispute. Trembley investigates fully the chance of Paul for every case that can occur, and more briefly the chance of Peter. He states his conclusion thus : ...M. de Montmort et ses amis conoluoient de Ik centre Nicolas Bernoulli, que ce cas €toit insoluble, car disoient-ils, si Paul salt que Pierre se tient au huit, il changera au sept, mais Pierre venant h, savoir que Paul change au sept, changera au huit, ce qui fait un cercle vicieux. Mais il risulte seulement de IS, que chacun sera perpetuellement dans I'incertitude sur la manifere de jouer de son adversaire; des lors il con- viendra 5, Paul de changer au sept dans un coup donne, mais il ne pourroit suivre constamment ce systeme plusieurs coups de suite. II conviendra de m^me S, Pierre de changer au huit dans un coup donnS, sans pouToir le faire plusieurs coups de suite, ce qui s'accorde avec les conclusions de M. Nicolas Bernoulli contre celles de M. de Montmort. 800. It is hardly correct to say that the conclusion here obtained agrees with that of Nicolas Bernoulli against that of Montmort. The opponents of Nicolas Bernoulli seem only to have asserted that it was impossible to say on which rule Paul should wniformly act, and this Trembley allows. 801. In Trembley's investigation of the chance of Peter, he considers this chance at the epoch before Paul has made his choice whether he will exchange or not. But this is of little value for Peter himself ; Peter would want to know how to act under cer- tain circumstances, and before he acted he would know whether Paul retained the card he obtained at first or compelled an ex- 430 TEEMBLEY. change. Heace Trembley's investigation of Peter's chance differs from the method which we have exemplified in Art. 189. 802. Trembley makes an attempt to solve the problem of Her for three players ; but his solution is quite unsound. Sup- pose there are three players, Paul, James, and Peter. Trembley considers that the chances of Paul and James are in the propor- tion of the chance of the first and second players when there are only two players ; and he denotes these chances by x and y. He takes a; to ^ as 8496 to 8079 ; but these numbers are of no con- sequence for our purpose. He supposes that the chances of James and Peter are also in the same proportion. This would not be quite accurate, because when James is estimating his chance with respect to Peter he would have some knowledge of Paul's card ; whereas in the case of Paul and James, the former had no know- ledge of any other card than his own to guide him in retaining or exchanging. But this is only a minute point. Trembley's error is in the next step. He considers that is the chance that Paul will ■^ x + y beat James, and that — — is the chance that Peter will beat x^-y James ; he infers that -, — ^-^ is the chance that both Paul and [x + yf Peter will beat James, so that James will be thrown out at the first trial. This is false: the game is so constructed that the players are nearly on the same footing, so that - is very nearly o the chance that a given player will be excluded at the first trial. Trembley's solution would give j as the chance that James will be excluded H os=y; whereas ^ should then be the value. o X y The error arises from the fact that — ; — and — '■ — do not x+y x+y here represent independent chances ; of course if Paul has a higher card than James, this alone affords presumption that James will rather have a card inferior to that of Peter than superior. This error at the beginning vitiates Trembley's solution. TREMBLET. 431 803. As a subsidiary part of his solution Trembley gives a tedious numerical investigation which might be easily spared. He wishes to shew that supposing James to have a higher card than both Peter and Paul, it is an even chance whether Peter or Paul is excluded. He might have proceeded thus, which will be easily intelligible to a person who reads the description of the game in Monimort, pages 278, 279 : Let n denote the number of James's card. I. Suppose n — r and n — s the other two cards ; where r and s are positive integers and different. Then either Paul or Peter may have the lower of the two n — r and n — s; that is, there are as many cases favourable to one as the other. II. Peter's card may also be n; then Paul's must be 1, or 2, or 3, ... or w— 1. Here are n — 1 cases favourable to Peter. III. Peter and Paul may both have a card with the same mark n — r; this will give n — 1 cases favourable to Paul. Thus II. and III. balance. CHAPTER XIX. MISCELLANEOUS INVESTIGATIONS Between the Years 1780 and 1800. 804. The present Chapter will contain notices of various contributions to our subject which were made between the years 1750 and 1780. 805. We have first to mention two memoirs by Prevost, en- titled, Sur les principes de la TMorie des gains fortuits. The first memoir is in the volume for 1780 of the Nouveaux Memoires... Berlin; the date of publication is 1782: the memoir occupies pages 430 — 472. The second memoir is in the volume for 1781 ; the date of publication is 1783 : the memoir occupies pages 463 — 472. Prevost professes to criticise the account of the elementary principles of the subject given by James Bernoulli, Huygens, and De Moivre. It does not seem that the memoirs present anything of value or importance ; see Art. 103. 806. We have next to notice a memoir by Borda, entitled Memoire swr les Elections au Scrutin. This is in the Hist....de l' Acad.... Paris for 1781 ; the date of publication is 1784 : the memoir occupies pages 657 — 665. This memoir is not connected with Probability, but we notice it because the subject is considered at great length by Condorcet, who refers to Borda's view ; see Art. 719. BOEDA. 433 Borda observes that the ordinary mode of election is liable to error. Suppose, for example, that there are 21 voters, out of whom 8 vote for A, 7 for B, and 6 for (7; then A is elected. But it is possible that the 7 who voted 'for JB and the 6 who voted for G may agree in considering A as the worst of the three can- didates, although they differ about the merits of B and C. In such a case there ai'e 8 voters for A and 13 against him out of the 21 voters ; and so Borda considers that A ought not to be elected. In fact in this case if there were only A and B as candidates, or only A and C as candidates, A would lose ; he gains because he is opposed by two men who are both better than himself. Borda suggests that each voter should arrange the candidates in what he thinks the order of merit. Then in collecting the results we may assign to a candidate a marks for each lowest place, a + h marks for each next place, a + 2b marks for each next place, and so on if there are more than three candidates. Suppose for example that there are three candidates, and that one of them is first in the lists of 6 voters, second in the lists of 10 voters, and third in the lists of 5 voters ; then his aggregate merit is ex- pressed by 6 (a + 2,h) + 10 (a + b) + 5a, that is by 21a +225. It is indifferent what proportion we establish between a and h, be- cause in the aggregate merit of each candidate the coefficient of a wiU be the whole number of voters. Condorcet objects to Borda's method, and he gives the follow- ing example. Let there be three candidates. A, B, and G; and suppose 81 voters. Suppose that the order ABC is adopted by 30 voters, the order AGB by 1, the order GABhj 10, the order BAGhy 29, the order BGA by 10, and the order GBA by 1. In this case B is to be elected on Borda's method, for his aggregate merit is expressed by 81a + 1095, while that oi A is expressed by 81a + 101&, and that of G by 81a + 335. Condorcet decides that A ought to be elected ; for the proposition A is better than B is affirmed by 30 + 1 + 10 voters, while the proposition B is better than A is affirmed by 29 + 10 + 1 voters, so that A has the ad- vantage over B in the ratio of 41 to 40. Thus suppose a voter to adopt the order ABG; then Condorcet considers him to affirm with equal emphasis the three propositions A is better than B, B is better than G, A is better than G; but 28 434 MALFATTI. Borda considers him to affirm the first two with equal emphasis, and the last with double emphasis. See Condorcet's Discours Priliminaire, page CLXXVii, Laplace, TAeom ... (^es Pro&. page 274. 807. We have next to notice a memoir by Malfatti, entitled Esame Gritico di un ProUema di probability, del Sig. Daniele Bernoulli, e soluzione d'un altro Problema analogo al Bernulliano. Del Sig. Gio: Francesco Malfatti Professore di Matematica nell' University, di Ferrara. This memoir is published in the Memorie di Matematica e Fisica della Societd, Italiana, Tomo l. 1782 ; the memoir occupies pages 768 — 824. The problem is that which we have noticed in Art. 416. Malfatti considers the solution of the problem about the balls to be erroneous, and that this problem is essentially different fi-om that about the fluids which Daniel Bernoulli used to illustrate the former; see Art. 420. Malfatti restricts himself to the case of two urns. Malfatti in fact says that the problem ought to be solved by an exact comparison of the numbers of the various cases which can arise, and not by the use of such equations as we have given in Art. 417, which are only probably true ; this of course is quite correct, but it does not invalidate Daniel Bernoulli's process for its own object. Let us take a single case. Suppose that originally there- are two white balls in A and two black balls in B ; required the probable state of the urn A after x of Daniel Bernoulli's operations have been performed. Let u^. denote the probability that there are two black balls in ^ ; v^ the probability that there is one black ball and one white one, and therefore l—u^ — v^ the probability that there are two white balls. 808. We will first give a Lemma of Malfatti's. Suppose there are n — ^ white balls in A, and therefore p black balls ; then there are n —p black balls in B and p white balls. Let one of Daniel Bernoulli's operations be performed, and let us find the number of cases in which each possible event can happen. There are r? cases altogether, for any ball can be taken from A and any ball from B. Now there are three possible events ; for after the opera- tion A may contain n —p + 1 white balls, or n —p, or n—p — 1. MALFATTI. 435 For the first event a black ball must be taken from A and a -white ball from B ; the number of cases is p^. For the second event a black ball must be taken from A and a black one from B, or else a white one from A and a white one 'from B; the number of cases is 2p(»i— j>). For the third event a white ball must be taken from A and a black ball from B; the number of cases is {n-py. It is obvious that n^ =y + 2p {n -p) + (w - jj)" as should be the case. 809. Now returning to the problem in Art. 807 it will be easy to form the following equations ; _1 1 Integrating these equations and determining the constant by the condition that v^ = 1, we obtain 2 fi (-1)1 ifi_^(-in Daniel BernouUi's general result for the probable number of white balls in A after x trials if there were n originally would be Thus supposing x is infinite Daniel Bernoulli finds that the probable number is „ . This is not inconsistent with our result ; 2 1 for we have when x is infinite Vx = -=, m^ = ^ , and therefore o u l — v^ — u^ = -^, so that the case of one white ball and one black ball is the most probable. 810. Malfatti advances an objection against Daniel Bernoulli's result which seems of no weight. Daniel Bernoulli obtains as 28—2 436 MALFATTI. we see 5 for the probable number of white balls in A after an infinite number of operations. Now Malfatti makes Daniel Ber- noulli's statement imply conversely that it will require an infinite number of trials before the result | will probably be reached. But Daniel Bernoulli himself does not state or imply this con- verse, so that Malfatti is merely criticising a misapprehension of his own. 811. Malfatti himself gives a result equivalent to our value of Mj, in Art. 809 ; he does not obtain it in the way we use, but by induction founded on examination of successive cases, and not demonstrated generally. 812. The problem which Malfatti proposes to solve and which he considers analogous to Daniel Bernoulli's is the following. Let T be zero or any given integer not greater than n : required to determine the probability that in x operations the event will never occur of having just n — r white balls in A. This he treats in a most laborious way ; he supposes r = 2, 3, 4, 5 in succession, and obtains the results. He extracts by inspection certain laws from these results which he assumes will hold for all the other values of r between 6 and n inclusive. The cases »• = 0, and r = 1, require special treatment. Thus the results are not demonstrated, though perhaps little doubt of their exactness would remain in the mind of a student. The patience and acuteness which must have been required to extract the laws will secure high admiration for Malfatti. 813. We will give one specimen of the results which Malfatti obtains, though we shall adopt an exact method instead. of his in- duction from particular cases. Eequired the probability that in x trials the number n — 2 of white balls will never occur in A. Let ^ {x, n) represent the whole number of favourable cases in x trials which end with n white balls in ^ ; let (pa + 1, n) = (x, n-2) represent the number of favourable cases in x trials, where the final number of white balls in A is n, n — 1, n— 2, respectively. Then we have the following equations ^(x + l, n)=^ (x, n — 1), (»+ 1, » - 1) = w'(^ {x, w) + 2 (« - 1) ^ (cc, ra - 1) + 4(^6 {x, n-2), ^{x+l,n-2) = (n- iy(x, n-1) +4<(n-2) <}){x, n-2). If we denote (x, n — 2) by u^ we shall arrive by ehmination at the equation «^3- (Qn - 10) M,^,+ (3n' - 16w + 12) u^, + in' (n - 2) m, = 0. Then it will be seen that (p{x, n — 1) and -q-ty. See also Algebra, Chapter LVi. 820. Pages 111 — 133 contain the solution of some examples. Two of them are borrowed from Buffon, namely those which we have noticed in Art. 649, and in the beginning of Art. 650. One of Bicquilley's examples may be given. Suppose p and q to denote respectively the chances of the happening and failing of an event in a single trial. A player lays a wager of a to 5 that the event will happen ; if the event does not happen he repeats the wager, making the stakes ra to rb ; if the event fails again he repeats the wager, making the stakes r^a to r'b ; and so on. If the player is allowed to do this for a series of n games, required his advantage or disadvantage. The player's disadvantage is {qa -j>b) {l+qr + qV+...+ g""' r""'}. This is easily shewn. For qa —pb is obviously the player's dis- advantage at the first trial. Suppose the event fails at the first trial, of which the chance is q ; then the wager is renewed ; and the disadvantage for that trial is qai — pbr. Similarly q' is the chance that the event will fail twice in succession ; then the wager is renewed, and the disadvantage is qar^ — pbr\ And so on. If then qa is greater than pb the disadvantage is positive and in- creases with the number of games. Bicquilley takes the particular case in which a = l, and r = . ; his solution is less simple than that which we have ENCYCLOPfDIE M^THODIQUE. HI given. The object of tlie problem is to shew to a gambler, by an example, that if a wager is really unfavourable to him he suffers still more by increasing his stake while the same proportion is maintained between his stake and that of his adversary. 821. Pages 134 — 149 relate to the evaluation of probability from experience or observation. If an event has happened m times and failed n times the book directs us to take as its m + n chance in a single trial. 822. Pages 150 — 164 relate to the evaluation of probability from testimony. Bicquilley adopts the method which we have exhibited in Art. 91. Another of his peculiarities is the following. Suppose from our own experience, independent of testimony, we assign the probability P to an event, and suppose that a witness whose probability is p offers his evidence to the event, Bicquilley takes for the resulting probability P+ (1 — P) Fp, and not as we might have expected from him P + (1 — P) p. He says that the reliance which we place on a witness is proportional to our own previous estimate of the probability of the event to which he testifies. 823. "We will now notice the matter bearing on our subject which is contained in the Encyclopedie Mithodique; the mathema- tical portion of this work forms three quarto volumes which are dated respectively 1784, 1785, 1789. Ahsent. This article is partly due to Condorcet : he applies the Theory of Probability to determine when a man has been ab- sent long enough to justify the division of his property among his heirs, and also to determine the portions which ought to be assigned to the different claimants. Assurances. This article contains nothing remarkable. Prohabilite. The article from the original Encyclopedie is re- peated : see Art. 467. This is followed by another article under the same title, which professes to give the general principles of the subject. The article has not Condorcet's signature formally attached to it ; but its last sentence shews that he was the author. It may be described as an outline of Condorcet's own writings on 44-2 ENCYCL0P15DIE METHODIQUK. the subject, but from its brevity it would be far less intelligible than even those writings. Substitutions. Condorcet maintains that a State has the autho- rity to change the laws of succession to property ; but when such changes are made the rights which existed under the old laws should be valued and compensation made for them. In this article Condorcet professes to estimate the amount of compensation. The formulae however are printed in such an obscure, and repulsive manner that it would be very difficult to determine whether they are correct ; and certainly the attempt to examine them would be a waste of time and labour. 824. It should be observed that in the Encyclopedie M^tho- dique various threats are uttered which are never carried into execution. Thus in the article Assurances we are referred to Evinemens and to Societe ; and in the article Probabilite we are referred to Verite and to Votans. Any person who is acquainted with Condorcet's writings will consider it fortunate that no articles are to be found under the titles here named. 825. The only important article connected with our subject in the EncyclopMie MSthodique is that under the title Milieu,, which we will now proceed to notice. The article is by John Bernoulli, the same person, we presume, whom we have noticed in Arts. 598 and 624. The article gives an account of two memoirs which it asserts had not then been printed. The article says : Le premier memoire dont je me propose de donner I'extrait, est iin petit 6crit latin do M. Daniel Bernoulli, qu'il me communiqua, en 1769, et qu'il gardoit depuis long-tems parmi ses manuscrits dans le dessein sans doute de I'^tendre davantage. II a pour titre : Dijudicatio maxime probabilis pluriv/m ohservationum discrepa/ntium ; atque verisi- milli/ma inductio inde formwnda. The title is the same as that of the memoir which we have noticed in Art. 424 ; but this article Milieu gives an account of the memoir which does not correspond with what we find in the Acta Acad....Petrop., so we conclude that Daniel Bernoulli modi- fied his memoir before publishing it. ENCTCLOPi^biE miJthodique. 443, The following is the method given in the article Milieu. Let the numerical results of discordant observations be set off as abscissae from a fixed point ; draw ordinates to represent the pro- babilities of the various observations ; trace a curve through the extremities of these ordinates and take the abscissa of the centre of gi-avity of the area of the curve as the correct value of the element sought. The probabilities are to be represented by the ordinates of a certain semi-ellipse or semicircle. The article says that to determine analytically the centre of the semicircle would be very difficult, because we arrive at an equation which is almost unmanageable ; accordingly a method of approximation is pro- posed. First take for the centre the point corresponding to the mean of all the observations, and determine the centre of gravity of the area corresponding to the observations; take this point as a new centre of a semicircle, and repeat the operation ; and so on, until the centre of gravity obtained corresponds with the centre of the respective semicircle. The magnitude of the radius of the semicircle must be assigned arbitrarily by the cal- culator. This is ingenious, but of course there is no evidence that we thus obtain a result which is specially trustworthy. The other memoir which is noticed in this article Milieu is that by Lagrange, published in the Miscellanea Taurinensia ; see Art. .556. It is strange that the memoirs by Daniel Bernoulli and Lagrange should be asserted to be unprinted in 1785, when Daniel Bernoulli' had published a memoir with the same title in the Acta Acad....Petrop. for 1777, and Lagrange's memoir was published in the Miscellanea Taurinensia for 1770 — 1773. The date of publication of the last volume is not given, but that it was prior to 1777 we may infer from a memoir by Euler; see Art. 447. 826. We will now notice the portions of the Uncyclopedie MSthodique which relate to games of chance. The three volumes which we have mentioned in Art. 817 contain articles on various games ; they do not give mathematical investigations, with a slight exception in the case of Bassette : see Art. 467. The commence- ment of the article Breland is amusing : il se joue d, tant de 44-4! ENCYCLOP^DIE M^THODIQUE. personnes que Von veut : mais il n'est beau, c'est-ci-dire, trh-ruineuao^ qu'ct, trois ou cinq. There is however a distinct work on games, entitled Diction- naire des Jeux, faisant suite au Tome III. des Mathematiques. 1792. The Avertissement begins thus : Comme il y a, dit Mon- tesquieu, une infinite de choses sages qui sont menses d'une manibre trfes-foUe, il y a aussi des folies qui sont conduites d'une manifere tr^s-sage. The work contains 316 pages of text and 16 plates. There are no mathematical investigations, but in three cases the numerical values of the chances are given. One of these cases is the game of Trente et quarante ; but the results given are inaccurate, as Poisson shewed in the memoir which we have cited in Art. 358. The other two cases in which the results are given are the games Krabs and Passe-dix. The copy of the EncychpSdie Mithodique which belongs to the Cambridge University Library includes another work on games which is wanting in other copies that I have examined. This is entitled Dictionnaire des Jeux Mathematiques.... An. vil. The advertisement states that after the publication of the Dictionary of Games in 1792 many of the subscribers requested that this treatise should be enlarged and made more complete. The pre- sent Dictionary is divided into two parts ; first, the Dictionnaire des Jeux Mathetnatiques, which occupies 212 pages; secondly, a Dictionnaire de Jeux familiers, which is unfinished, for it extends only from A to Grammairien, occupying 80 pages. The Dictionnaire des Jeux Mathematiques does not contain any thing new or important in the calculation of chances. The investigations which are given are chiefly taken from Montmort, in some cases with a reference to him, but more often without. Under the title Joueur we have the names of some writers on the subject, and we find a very faint commendation of Montmort to whose work the Dictionary is largely ipdebted : Plusieurs anteurs se sont exerces sur I'analyse des jeux ; on en a un traits 61gmentaire de Huygens ; on en a un plus profond de Moivre ; on a des morceaux tris-savans de Bernoulli sur oette matiere. II y a un analyse des jeux de hasard par Montmaur, qui n'est pas sans mfirite. The game oi Draughts obtains 16 pages, and the game of CJiess d'anteres. 445 73 pages. Under the title Cartes (jeu de) we have the problem which we noticed in Art. 533, omitting however the part which is false. Under the title Whisk ou Wisth we have 8 pages, beginning thus : Jeu de cartes mi-parti de hasard et de science. II a ItS invent! par les Anglais, et continue depuis long terns d'etre en vogue dans la Grand-Bretagne. C'est de tous les jewx. de cartes le plus judicieux dans ses principea, le plus convenable S, la sociSte, le plus difficile, le plus int6ressant, le plus piquant, et celui qui est combin6 avec le plus d'art. The article quotes some of the results obtained by De Moivre in his calculations of the chances of this game : it also refers to Hoyle's work, which it says was translated into French in 1770. With respect to the Dictionnaire de Jeux familiers we need only say that it comprises descriptions of the most trifling games which serve for the amusement of children ; it begins with J'aime mon amantpar A, and it includes Golin-Maillard. 827. We next advert to a memoir by D'Anieres, entitled Mdfleodons sur les Jeux de hazard. This memoir is published in the volume of the N'ouveausc MSmoires de V Acad.... Berlin for 1784; the date of publication is 1786 ; the memoir occupies pages 391 — 398 of the volume. The memoir is not mathematical ; it alludes to the fact that games of hazard .are prohibited by governments, and shews that there are different kinds of such games, namely, those in which a man may ruin his fortune, and those which cannot produce more than a trifling loss in any case. There is a memoir by the same author, entitled, Sur les Paris, in the volume of the Nouveaux MSmoires de V Acad.... Berlin for 1786 ; the date of publication is 1788 : the memoir occupies pages 273^—278 of the volume. This memoir is intended as a supplement to the former by the same author, and is also quite unconnected with the mathematical Theory of Probability. 828. We have now to notice a curious work, entitled On the 446 WARING. Principles of translating Algebraic quantities into probable rela- tions and annuities, &c. By E. Waring, M.D. Lucasian Professor of Mathematics at Cambridge, and Fellow of the Royal Societies of London, Bononia and Gottingen. Cambridge, Printed by J. Arch- deacon, Printer to the University; For J. Nicholson, Bookseller, in Cambridge. 1792. This is an octavo pamphlet. Besides the leaf on which the title is printed there are 59 pages of text, and then a page with a few corrigenda. The work is excessively scarce ; for the use of a copy I am indebted to the authorities of Queens' College, Cambridge. 829. The author and the printer seem to have combined their efforts in order to render the work as obscure and repulsive as possible ; and they have attained a fair measure of success. The title is singularly inaccurate ; it is absurd to pretend to translate algebraical quantities into probable relations or into annuities. What Waring means is that algebraical identities may be trans- lated so as to afford propositions in the Theory of Probabilities or in the Theory of Annuities. 830. Waring begins with a Lemma. He proposes to sum the series 1 + 2'-^ r + S'-' r^ + 4^"V' + S""'/ +...in infinitvm. The sum will be A+BrJr Cr'+Br' + ... +y'"' (i-ry The coefficients A, B, C ... are independent of r ; they must be determined by multiplying up and equating coefficients. Thus A = l, B=2'^-z, B = ^-"^ - zW-"- + ^ ^^ ~ -^^ 2'-' _ g(^-l)(g-2) M 2.3 Proceeding in this way we shall find that in the numerator of the fraction which represents the sum the last term is r'"" ; that WAEING. 447 is there is no power of r higher than this power, and the coefficient of this power is unity. Waring refers to another work by himself for the demonstration ; the student will see that it may be deduced from the elementary theorem in Finite Differences respecting the value of A"a;" when n is not less than m. Waring does not apply his Lemma until he comes to the part of the work which relates to Annuities^ which forms his pages 27^ — 59. 831. Waring now proceeds to his propositions in the Theory of Probabilities ; one of his examples will suffice to indicate his method. It is identically true that ^ — ^^ — ~ W ~ Ivfa • Suppose -^ to represent the chance of the happening of an assigned event in N — a one trial, and therefore — ^^r— the chance of its failing : then the identity shews that the chance of the happening of the event in the first trial and its failing in the second trial is equal to-the dif- ference between the chance of the happening of the event once and the chance of its happening twice in succession. 832. There is nothing of any importance in the work respect- ing the Theory of Probability until we come to page 19. Here Waring says, Let the chances of the events A and B happening be respectively — 7 and =-; then the chance of the event A happening r times a+ 6 a + cif more than Binr trials will be (a + Vf' in »• + 2 trials will be dr ( ah 'y in »• + 4 trials will be dr ( ah r(r+S) a'b' ] and in general it will be 4'4!8 WAEING. or f ah r (r + 3) a°6' r {r + i) {r + 5) «'&' {a + hyi ia + h)''^ 2 {a + iy'^ [3 (a + 6)' r(r + l+l)(r + l + 2)...(r + 2l-l) a'b' ■ . n -. 1 + + -i ^-^ n— ^ — ^ '7 — TXS + m mfinitum >. \£ {a+by ) This may be deduced from the subsequent arithmetical theorem, viz. 2m{2m-l) {2m -2)... {2m -s) {2m-2){2m~S}...(2m- s -1) 1^ + 1 +r ^ r (r + 3) {2m -i){2m-5)... {2m-s -2) ■^—2 [^£1 r(r + 4)(r + 5) (2w.-6)...(2m-g-3) ■^ [3 |g-2 + ... r{r + s + 2) (r + g + 3)... (r+2a+ 1) "^ |8 + 1 ' _ {r + 2m) {r + 2m- 1) ... (r + 2m - s) ~ [7+T ■ Waring's words, "A happening r times more than B" are scarcely adequate to convey his meaniDg. We see from the for- mula he gives that he really means to take the problem of the Duration of Play in the case where B has a capital r and A has un- limited capital. See Art. 309. Waring gives no hint as to the demonstration of his arith- metical theorem. We may demonstrate it thus : take the formula in Art. 584, suppose « = !+«, p = l, 2' = 2; we shall find that ^ = 1+ .-(1- 2s -)-l Thus we get 1 = 1 + t (1 (< + 3) 2 z' -(1+.)' {1+z)'^ t{t + ^) (t + S) 2» "I" In /I . ^tta L? (1 + «) Multiply both sides by (1 + »)'"+' : thus t{t+S){t + Q){t + 1) s* . WAEING. 449 (1 + s)="*' = (1 + s)'" + te (1 + «)'^"-'+ ii^±^ 3= (1 + «)»■-* + ^^^-±1^ .»(! + .)--'+... If we expand the various powers oil+z and equate the coeffi- cients of z' we shall obtain the arithmetical theorem with t in place of r. But it is not obvious how Waring intended to deduce the theorem on the Duration of Play from this arithmetical theorem. If we put - for s we obtain ^ a (a + by^ =a'(a + IT + fe' (a + 5)="^ ab + i(^+^ «« (« + 5)^"-* a'J» + ^(^ + ^y + ^) «« (« + J)2-6 «3ja ^ _ _ _ and it was perhaps from this result that Waring considered that the theorem on the Duration of Play might be deduced ; but it seems difficult to render the process rigidly strict. 833. Waring gives another problem on the Duration of Play ; see his page 20. If it be required to find the chance of A's succeeding n times as oft as ^'s precisely : in w + 1 trials it will be fotmd in 2w + 2 trials it will be found 2nX2 in 3m + 3 it will be n{n+\){Zn+l) a'"V ^* ' 2 ia + hf'*'- Waring does not give the investigation ; as usual with him until we make the investigation we do not feel quite certain of the meaning of his problem. The first of his three examples is obvious. 29 450 WARING. In the second example we observe that the event may occur in the first w + 1 trials, and the chance of this is P; or the event may- have failed in the first w + 1 trials and yet may occur if we proceed to « + 1 more trials. This second case may occur in the following ways : B may happen twice in the first n+1 trials, or twice in the second n+1 trials ; while A happens in the remaining 2;i trials. Thus we obtain „ (w + 1) w g'V 2 {a + hf-'^' which must be added to P to give the chance in the second ex- ample. In the third example we observe that the event may occur in the first 2n + 2 trials, and the chance of this is ^ ; or the event may have failed in the first 2n + 2 trials, and yet may occur if we proceed to n + 1 more trials. This second case may occur in the following ways : B may happen three times in the first n + 1 trials, or three times in the second n+1 trials, or three times in the last w + 1 trials ; while A happens in the remaining Zn trials. Or B may happen twice in the first n + 1 trials and once in the second n + 1 trials, or once in the second n + 1 trials and twice in the third n + 1 trials ; while A happens in the remaining 3« trials. Thus we obtain f - (n + l)n{n~l) ^ „ (n + 1)^ n] a'"J^ l'^ [3 +^ 2" (a + &)»»«' which must be added to Q to give the chance in the third ex- ample. 834. The following specimen may be given of Waring's imper- fect enunciations ; see his page 21 : Let a, b, c, d, &c. be the respective chances of the happening of a, j8, y, 8, &c. : in one trial, and (aK" + 6a^ + cky + da^ + &c.)" = a"a3"» + . . . + Nx'' + &c. ; then will N be the chance of the happening, of tt in ?i trials. Nothing is said as to what tt means. The student will see that the only meaning which can be given to the enunciation is to WARING. 451 suppose that a, h, c, d, ... are the chances that the numbers "> A 7, S, . . . respectively will occur in one trial ; and then N is the chance that in n trials the sum of the numbers will be tt. 835. "Waring gives on his page 22 the theorem which we now sometimes call by the name of Vandermonde. The theorem is that {a + h) {a + b-l)...{a + h-n+l) = a (a — 1) ... (a— w+ 1) + na {a — 1) ... {a — n + 2) h + ^^-^^a (a - 1) ... (a-M + 3) 5 (5- 1) + ^'-^^f^^^^a(a-l)...(«-n + 4)J(5-l)(&-2) + + b{b-l)...{b-n + l). From this he deduces a corollary which we will give in our own notation. Let ^ (as, y) denote the sum of the products that can be made from the numbers 1. 2, 3, ... x, taken i/ together. Then will , d>(n — \,n-s) = — — != — — d) (n — r — 1, n — s) |_r \n-r ^ ^ ^ + , —i^^ -{n-r-2,n-s-l){n-r-3,n-s-2){r+l,2) ^—-^ ^0(«-r-4,«-s-3)<^(r+2,3) |r + 3 I w — r— 3 + It must be observed that s is to be less than n, and r less than s ; and the terms on the right-hand side are to continue until we arrive at a term of the form (a;, 0), and this must be replaced by unity. 29—2 432 WARING, This result is obtained by equating the coefficients of the term oT'^b'' in the two members of Vandermonde's identity. The result is enunciated and printed so badly in "Warings work that some difficulty arose in settling, what the result was and how it had been obtained. 836. I do not enter on that part of "Waring's work which relates to annuities. I am informed by Professor De Morgan that the late Francis Baily mentions in a letter the following as the interesting parts of the work : — the series 8 — m8'-\ ^^ 8" — ...., the Problem III, and the observations on assurances payable imme- diately at death. 837. Another work by Waring requires a short notice ; it is entitled An essay on the principles of human knowledge. Cam- hridge 1794. This is an octavo volume ; it contains the title-leaf, then 240 pages, then 3 pages of Addenda, and a page containing Corrigenda. 838. This work contains on pages 35 — 40 a few common theo- rems of probability; the first two pages of the Addenda briefly notice the problem discussed by De Moivre and others about a series of letters being in their proper places ; see Art. 281, and De Moivre Prob. xxxv. Waring remarks that if the number of letters is infinite the chance that they will occur all in their right places is infinitesimal. He gives page 49 of his work as that on which this remark bears, but it would seem that 49 is a misprint for 41. 839. Two extracts may be given from this book. I know that some mathematicians of the first class have endeavoured to demonstrate the degree of probability of an event's happening n times from its having happened m preceding times ; and consequently that such an event will probably take place; but, alas, the problem far ex- ceeds the extent of human understanding ; who can determine the time when the sun will probably cease to run its present course ? Page 35. . . .1 have myself wrote on most subjects in pure mathematics, and in iNCILLOX. 453 these books inserted nearly all the inventions of the moderns with which I was acquainted. In my prefaces I have given an history of the inventions of the dif- ferent writers, and ascribed them to their respective authors ; and like- wise some account of my own. To every one of these sciences I have been able to make some additions, and in the whole, if I am not mis- taken in enumerating them, somewhere between three and four hundred new propositions of one kind or other, considerably more than have been given by any English writer ; and in novelty and difficulty not inferior ; I wish I could subjoin in utility : many more might have been added, but I never could hear of any reader in England out of Cambridge, who took the pains to read and understand what I have written. Page 115. Waring proceeds to console himself under this neglect in Eng- land by the honour conferred on him by D'Alembert, Euler and Le Grange. Dugald Stewart makes a remark relating to Waring ; see his Works edited by Hamilton, Vol. iv. page 218. 840. A memoir by Ancillon, entitled Doutes sur les hoses du calcul des probabilitSs, was published in the volume for 1794 and 1795 of the Memoires de V Acad.... Berlin; the memoir occupies pages 3 — 32 of the part of the volume which is devoted to specu- lative philosophy. The memoir contains no mathematical investigations; its ob- ject is to throw doubts on the possibility of constructing a Theory of Probability, and it is of very little value. The author seems to have determined that no Theory of Probability could be con- structed without giving any attention to the Theory which had been constructed. He names Moses Mendelsohn and Garve as having already examined the question of the admissibility of such a Theory. 841. There are three memoirs written by Prevost and Lhuilier in conjunction and published in the volume for 1796 of the Memoires de V Acad.... Berlin. The date of publication is 1799. 842. The first memoir is entitled Sur les ProbaUlitts ; it was read Nov. 12, 1795. It occupies pages 117—142 of the mathe- matical portion of the volume. 454 PREVOST AND LHUILIER. 843. The memoir is devoted to the following problem. An urn contains m balls some of which are white and the rest black, but the number of each is unknown. Suppose that p white balls and q black balls have been drawn and not replaced ; required the probability that out of the next r + s drawings r shall give white balls and s black balls. The possible hypotheses as to the original state of the urn are, that there were q black balls, or g + 1 black balls, or q + % ... or m—p. Now form the probability of these various hypotheses according to the usual principles. Let p^= (rn — q — n + l) {m — q — n) to p factors, Q„=(g^-\-n — l){q-\-n — 2) to q factors; then the probability of the w*'^ hypothesis is where 2 denotes the sum of all such products as P„ Q^. Now if this hypothesis were certainly true the chance of drawing r white balls and s black balls in the next r + s drawings would be \r\lN' where R^= (rn — q — p — n + T) {m — q— p —n) to r factors, yS^ = (m — 1) (m — 2) to s factors, N= number of combinations oim—p-q things >• + s at a time. Thus the whole required probability is the sum of all the terms of which the type is 2[r [siV ■ We have first to find S. The method of induction is adopted in the original memoir ; we may however readily obtain % by the aid of the binomial theorem : see Algebra, Chapter L. Thus we shall find ^_ \Jt\l 1"^ + ^ [jP + g + 1 \m — p-q ' PREVOST AND LHUILIEE. 455 Now P„B„ differs from P„ only in having p + r instead of p ; and Q„8^ differs from Q„ only in having q + s instead of £. There- fore the sum of all the terms of the form P„ Q„Rn8„ is I p + r \q + s \m + l ]p + q + r + s+ 1 \m — p — q — r — s ' \m — p — q And N= i ' ^ ^ . l?- + g Im — p — q — r — s Thus finally the required probability is \'r + s [p + r \q + s |y+^ + l [r_[s [P !_£ \p + q + r + s + l 844. Let us suppose that r and s vary while their sum r+s remains constant; then we can apply the preceding general result to r + s + 1 different cases ; namely the case in which all the r + s drawings are to give white balls, or aU but one, or all but two, and so on, down to the case in which none are white. The sum of these probabilities ought to he unity, which is a test of the accuracy of the result. This verification is given in the original memoir, by the aid of a theorem which is proved by induction. No new theorem however is required, for we have only to apply again the formula by which we found S in the preceding Article. The variable part of the result of the preceding" Article is \p + r \q + s lull ' that is the product of the following two expressions, (r + 1) (r + 2) p factors, (s + 1) (s + 2) ...... q factors. The sum of such products then is to be found supposing r + s constant ; and this is IE ll \P + q + '>' + s + 1 \p + q + 1 \r + s Hence the required result, unity, is obtained by multiplying this expression by the constant part of the result in the preceding Article. ■ioG PREVOST AND LHUILIEB. This result had been noticed by Condorcet ; see page 189 of the Ussai .. . de l' Analyse... 845. Out of the r + s + 1 cases considered in the preceding Article, suppose we ask which has the greatest probability ? This question is answered in the memoir approximately thus. A quan- tity when approaching its maximum value varies slowly ; thus we have to find when the result at the end of Article 843 remains nearly unchanged if we put r — 1 for r and s + 1 for s. This leads to p+r q+s+1 , = ' -^j— , nearly ; r s + 1 •' ' therefore - = —~- nearly. r s +-1 ■' Thus if r and s are large we have - =-^ nearly. s c[ •' 846. It wiU be observed that the expression at the end of Art. 843 is independent of m the number of balls originally con- tained in the urn ; the memoir notices this and draws attention to the fact that this is not the case if each ball is replaced in the urn after it has been drawn. It is stated that another memoir will be given, which wiU consider this form of the problem when the number of balls is supposed infinite ; but it does not seem that this intention was carried into effect. 847. It will be instructive to make the comparison between the two problems which we may presume would have formed the substance of the projected memoir. Suppose that j) white balls have been drawn and q black balls, and not replaced; and suppose the whole number of balls to be infinite : then by Art. 704 the pro- bability that the next r + s drawings wUl give r white balls and s black balls is 1^+, foc'^'^ii-xy^'dx I ■ J_Q . tl^ fx^l-xydx ' and on effecting the integration we obtain the same result as in PEEVOST AND LHUILIER. 457 Art. 843. The coincidence of the results obtained on the two dif- ferent hypotheses is remarkable. 848. Suppose that »• = 1 and s = in the result of Art. 843 ; we thus obtain p + 1 p + q + 2' Again suppose r = 2 and s = ; we thus obtain (p + l)(p + 2) {p + q+2)ij, + q + S)' it? + 1 . The factor -^ — — - is, as we have just seen, the probability of drawing another white ball after drawing p white balls and q black balls ; the factor — — 5 expresses in like manner the probabUity of drawing another white ball after drawing^ + 1 white balls and q black balls : thus the formula makes the probability of drawing two white balls in succession equal to the product of the probability of drawing the first into the probability of drawing the second, as should be the case. This property of the formula holds generally. 849. The memoir which we have now examined contains the first discussion of the problem to which it relates, namely, the problem in which the balls are not replaced. A particular case of the problem is considered by Bishop Terrot in the Transactions of the Royal Society of Edinburgh, Vol. xx. 850. The other two memoirs to which we have referred in Art. 841 are less distinctly mathematical, and they are accordingly printed in the portion of the volume which is devoted to speculative philosophy. The second memoir occupies pages 3 — 24, and the third memoir pages 25 — 41. A note relating to a passage of the third memoir, by the authors of the memoir, is given in the volume for 1797 of the M^moires de I' Acad.... Berlin, page 152. 851. The second memoir is entitled Sur I'art d'estimer la probability des causes par les effets. It consists of two sections. The first section discusses the general principle by which, the 458 PREVOST AND LHUILIEE. probabilities of causes are estimated. The principle is quoted as given by Laplace in the M^moires...par divers Savans, Vol. Vi. : Si un ^vdnement pent Mre produit par un nombre n de causes differentes, les probabilit^s de I'existence de ces causes prises de I'^v^nement, sont entre elles comme les probabiUt^s de I'^v^ne- ment prises de ces causes. The memoir considers it useful and necessary to demonstrate this principle ; and accordingly deduces it from a simple hypothesis on which it is conceived that the whole subject rests. Some remarks made by Condorcet are criticised; and it is asserted that our persuasion of the constancy of the laws of nature is not of the same kind as that which is represented by a fraction in the Theory of Probability. See Dugald Stewart's Works edited by Hamilton, Vol. i. pages 421, 616. The second section of the memoir appUes Laplace's principle to some easy examples of the following kind. A die has a certain number of faces ; the markings on these faces are not known, but it is observed that out oi j) + q throws j> have given ace and q not-ace. Find the probability that there is a certain number of faces marked ace. Also find the probability that in p' + g' more throws there will be^ aces and q not-aces. It is shewn that the result in the last case is where % denotes a summation taken with respect to m from m = 1 to m = n; and n is the whole number of faces. This is the result if the aces and not-aces are to come in a prescribed order ; if they \p' + q' are not we must multiply by — , , , . The memoir states without demonstration what the approxi- mate result is when n is supposed very great; namely, for the case in which the order is prescribed, \q + q' \ p + p' \P + 1 + ^ ~\l iJ" lp + g+y' + g' + i ' 852. The third memoir is entitled Remarques sur I'utilite et I'etendue du jprincipe par lequel on estime la probabilite des caiises. This memoir also relates to the principle which we have quoted PREVOST AND LHUILIER. 459 in Art. 851 from Laplace. The memoir is divided into four sections. 853. The first section is on the utility of the principle. It is asserted that before the epoch when this principle was laid down many errors had occurred in the writers on Probability. The following paragraph is given : Dans I'apprgoiation de la valeur du temoignage de deux t^moins siinultangs, il paroit que, jusqu'^ Lambeet, on n'a point us6 d'un autr6 artifice, que de prendre le complement de la formule employee pour le temoignage successif. On suivoit ^ cet ggard la trace de rappr6ciation des argumens conspirans, telle que Tavoit faite Jac. Bernoulli. Si I'on avoit connu la vraie m^thode de 1' estimation des causes, on n'auroifc pas manqu6 d' examiner avant tout si ce cas s'y rapportoit ; et Ton auroit vu que I'accord entre les t^moins est un eyenement post^rieur k la cau.se quelconque qui a determine les depositions : en sorte qu'il s'agit ioi d'estimer la cause par I'effet. On seroit ainsi retombe tout naturelle- ment et sans effort dans la m6tliode que Lambert a trouvee par un effet de cette sagacite rare qui caract6risoit son g^nie. 854. The authors of the memoir illustrate this section by quoting from a French translation, published in Paris in 1786, of a work by Haygarth on the smaU-pox. Haygarth obtained from a mathematical friend the following remark. Assuming that out of twenty persons exposed to the contagion of the small-pox only one escapes, then, however violent the small-pox may be in a town if an infant has not taken the disease we may infer that it is 19 to 1 that he has not been exposed to the contagion ; if two in a family have escaped the probability that both have not been exposed to the contagion is more than 400 to 1 ; if three it is more than 8000 to 1. With respect to this statement the memoir says that M. de la Roche the French translator has shewn that it is wrong by a judi- cious discussion. The end of the translator's note is quoted ; the chief part of this quotation is the following sentence : Si I'on a observe que sur vingt personnes qui pontent ^ une table de pharaon il y en a dix-neuf qui se ruinent, on ne pourra pas en deduire qu'il y a un & parier contre dix-neuf que tout homme dont la fortune 460 HAYGAETH. n'est pas derarigee, n'apas pontg au pharaon, ni qu'il y ait dix-neuf 3, parier centre un, que cet homme est un joueur. This would be absurd, M. de la Roche says, and he asserts that the reasoning given by Haygarth's friend is equally absurd. "We may remark that there must be some mistake in this note ; he has put 19 to 1 for 1 to 19, and vice versl And it is difficult to see how Pre vest and Lhuilier can commend this note ; for M. de la Roche argues that the reasoning of Haygarth's friend is entirely absurd, while they only find it slightly inaccurate. For Prevost and Lhuilier proceed to calculate the chances according to Laplace's • • 1 J ^i, fi J ^v , , 20 400 8000 , . , prmciple ; and they nnd them to be ^ , j—r , Kn?ri > which, as they say, are nearly the same as the results obtained by Hay- garth's friend. 855. The second section is on the extent of the principle. The memoir asserts that we have a conviction of the constancy of the laws of nature, and that we rely on this constancy in our applica- tion of the Theory of Probability ; and thus we reason in a vicious circle if we pretend to apply the principle to questions respecting the constancy of such laws. 856. The third section is devoted to the comparison of some results of the Theory of Probability with common sense notions. In the formula at the end of Art. 843 suppose s = ; the for- mula reduces to {p+l){p + 2)...(p + r) {p+q+2){p + q + S) ...{p + q + r + 1)' it is this result of which particular cases are considered in the third section. The cases are such as according to the memoir lead to conclusions coincident with the notions of common sense ; in one case however this is not immediately obvious, and the memoir says, Ceci donne I'explication d'une espfece de paradoxe remarqu6 (sans I'expliquer) par M. De La Place ; and a reference is given to Ecoles normales, Gi^me cahier. We will give this case. Nothing is known ct pnori respecting a certain die ; it is observed on trial that in five throws ace occurs twice and not-ace three times ; find the probability that the next four throws will all give ace. Here PBEVOST AND LHUILIEE. 461 3.4.5.6 1 p=2, q^S, r=i ; the above result becomes iT-Q-"Q~yA > that is jj . If we knew ^ priori that the die had as many faces ace as not-ace 1 1 we should have ^4 , that is :r-g , for the required chance. The para- dox is that =-7 is greater than — ; while the fact that we have had only two aces out of five throws suggests that we ought to have a smaller chance for obtaining four consecutive aces, than we should have if we knew that the die had the same number of faces ace as not-ace. We need not give the explanation of the paradox, as it ■will be found in connexion with a similar example in Laplace, Theorie,..des Proh. page ovi. 857. The fourth section gives some mathematical develop- ments. The following is the substance. Suppose n dice, each having r faces ; and let the number of faces which are marked ace be »»', m", m", . . . respectively. If a die is taken at random, the probability of throwing ace is m' + w" + m"'+ ... nr If an ace has been thrown the probability of throwing ace again on a second trial with the same die is r{m' + m" + m:" + ...) ' The first probability is the greater ; for (m' + m" + ml" + ...f is greater than n («i" -h m'" + m"" + ...). The memoir demonstrates this simple inequality. 858. Prevost and Lhuilier are also the authors of a memoir entitled Mdmoire sur Vapplication du Galcul des probabiliUs d, la valeur du temoignage. This memoir is published in the volume for 1797 of the Mi- moires de T Acad.... Berlin; the date of publication is 1800: the memoir occupies pages 120 — 151 of the portion of the volume devoted to speculative philosophy. The memoir begins thus : Le but de ce mlmoire est plut6t de reconnoitre I'^tat actuel de cette theorie, que d'y rien aj outer de nouveau. 4i62 PKEVOST AND LHUILIER. The memoir first notices the criticism given in Lambert's Orga- non of James Bernoulli's formula which we have already given in Art. 122. It then passes on to the theory of concurrent testimony now commonly received. Suppose a witness to speak truth m times and falsehood n times out oim + n times ; let m! and n' have similar meanings for a second witness. Then if they agree in an assertion the probability of its truth is ; r • mm +nn The ordinary theory of traditional testimony is also given. Using the same notation as before if one witness reports a state- ment from the report of another the probability of its truth is mm' + nn {m + m') {n + n') ' for the statement is true if they both tell the truth or if they both tell a falsehood. If there be two witnesses in succession each of whom reverses the statement he ought to give, the result is true ; that is a double falsehood gives a truth. It is stated that this con- sequence was first indicated in 1794 by Prevost. The hypothesis of Craig is noticed ; see Art. 91. The only new point in the memoir is an hypothesis which is proposed relating to traditional testimony, and which is admitted to be arbitrary, but of which the consequences are examined. The hypothesis is that no testimony founded on falsehood can give the truth. The meaning of this hypothesis is best seen by an example : suppose the two witnesses precisely alike, then instead of taking m +n (m + n)' as the probability of the truth in the case above considered we should take -; r^ ; that is we reject the term w" in the (m + n) •' numerator which arises from the agreement of the witnesses in a falsehood. rm , 1 wi" ^ 2?WM + n" , . . , Ihus we take -, — ; — r~ and -; — ; — r^- to represent respectivelv {m + n) im + n) ^ ^ •' the probabilities of the truth and falsehood of the statement on which the witnesses agree. Suppose now that there is a second pair of witnesses inde- pendent of the former, of the same character, and that the same MATTHEW YOUNG. 463 statement is also afifirmed by this pair. Then the memoir combines the two pairs by the ordinary rule for concurrent testimony, and so takes for the probability arising from the two pairs Then the question is asked for what ratio of m to w this expres- sion is equal to , so that the force of the two pairs of wit- nesses may be equal to that of a single witness. The approximate value of — is said to be 4-864 so that is about -p; . n m+ n 6 859. In Vol. VII. of the Transactions of the Royal Irish Academy there is a memoir by the Rev. Matthew Young, d.d. S.F.T.c.D. and M.R.I.A., entitled On the force of Testimony in esta- blishing Facts contrary to Analogy. The date of publication of the volume is 1800 ; the memoir was read February 3rd, 1798 : it occupies pages 79 — 118 of the volume. The memoir is rather metaphysical than mathematical. Dr Young may be said to adopt the modem method of estimating the force of the testimony of concurrent witnesses ; in this method, supposing the witnesses of equal credibility, we obtain a formula coinciding with that in Art. 667. Dr Young condemns as erroneous the method which we noticed in Art. 91 ; he calls it "Dr Halley's mode," but gives no authority for this designation. Dr Young criticises two rules given by Waring on the subject ; in the first of the two cases however it would not be difficult to explain and defend Waring's rule. CHAPTER XX. LAPLACE. 860. Laplace was bom in 1749, and died in 1827. He wrote elaborate memoirs on our subject, which he afterwards embodied in his great work the ThSorie analytique des ProbabilitSs, and on the whole the Theory of Probability is more indebted to him than to any other mathematician. We shall give in the first place a brief account of Laplace's memoirs, and then consider more fully the work in which they are reproduced. 861. Two memoirs by Laplace on our subject are contained in the Mimoires...par divers Savans, Vol. vi. 1774. A brief notice of the memoirs is given in pages 17 — 19 of the preface to the volume which concludes thus : Ces deux M6moires de M. de la Place, ont et6 clioisis parmi un trSs-grand nombre qu'il a prIsentSs depuis trois ans, ^ rAcadlmie, oii il remplit actuellement une place de G6omltre. Cette Compagnie qui s'est empress6e de rlcompenser ses travaux et ses talens, n'avoit encore vu personne aussi jeune, lui presenter en si peu de temps, tant de Mtooires importans, et sur des matiSres si diverges et si difficiles. 862. The first memoir is entitled Mimoire sur les suites rS- curro-recwrentes et sur leurs usages dans la thiorie des hasards. It occupies pages 353 — 371 of the volume. A recurring series is connected with the solution of an equation in Finite Differences where there is one independent variable ; see Art. 318. A recurro-recurrent series is similarly connected with the solution of an equation in Finite Differences where there are two independent variables. Laplace here first introduces the term laplace: 4,q$ and the' subject itself; we stall not give any account of his investi- gations, but confine ourselves to the part of his memoir which relates to the Theory of Probability. 863. Laplace considers three problems in our subject. The first is the problem of the Duration of Play, supposing two players of unequal .skill and. unequaL capital ; Laplace, however, rather shews how the problem may be solved than actually solves it. He begins with the case of equal skill and equal capital, and then passes on to the case of unequal skill. He proceeds so far as to obtain an equation in Finite Differences with one independent variable which would present no diflSculty in solving. He does not actually discuss the case of unequal capital, but intimates that there Will bfe' no obstacle except the length of the process. The problem is solved completely in the ThSorie. . .des Prob. pages. 225—238 ; see Art. 588. , 864. The next problem is that connected with a lottery which appears in the Tkiorie...des Prob. pages 191 — 201. The mode of solution is nearly the same in the two places, but it is easier to follow in the I%Sorie...des Prob. The memoir does not contain any of the approximate calculation which forms a large part of the discussion in the ThSorie... des Prob. We have already given the history of the problem; see Arts. 448, 775. 865. The third problem is the following : Out of a heap of counters a number is taken at random ; find the chances that this number will be odd or even respectively. Laplace obtains what we should now call the (Ordinary results ; his metb.od however is more elaborate than is necessary, for he uses Fiuite Differences : in the ThSorie... des Prob. page 201, he gives a more simple solution. We have already spoken of the problem in Art. 350. 866. The next memoir is entitled M^moire sur la Probability des causes par les euhnemens ; it occupies pages 621 — 656 of the volume cited in Art. 861. The memoir commences thus : La Thiorie des.hasards est una des parties les plus curieuses et le^ 30 466 LAPLACE. plus d61!cates de I'analyse, par la finesse des combinaisons qu'elle exige et par la difficult^ de les soumettre au calcul ; celui qui paroit I'avoir traitee avec le plus de succSs est M. Moivre, dans un excellent Ouvrage qui a pour titre, Tlieory of Chances; nous devons \, cet habile G6omStre les premieres recherches que Ton git faites sur I'intlgration des Equa- tions differencielles aux difflreuces finies ; . . . 867. Laplace then refers to Lagrange's researches on the theory of equations in Finite Differences, and also to two of his own memoirs, namely that which we have just examined, and one which was about to appear in the volume of the Academy for 1773. But his present object, he says, is very different, and is thus stated : ...je me propose de dSterminer la probability des causes par les 6v§nemens, matiSre neuve k bien des Egards et qui mlrite d'autant plus d'etre cultivie que c'est principalement sous ce point d« vue que la science des hasards pent ^tre utile dans la vie civile. ; 868. This memoir is remarkable in the history of the silbject, as being the first which distinctly enunciated the principle for estimating the probabilities of the causes by which an observed event may have been produced. Bayes must have had a notion of the principle, and Laplace refers to him in the Th4orie...des Prob. page cxxxvii. though Bayes is not named in the memoir. See Arts. 539, 696. 869. Laplace states the general principle which he assumes in the following words : Si un evenement pent ^tre produit par un nombre n de causes dif- ferentes, les probabilites de 1' existence de ces causes prises de I'evSne- ment, sont entre elles comme les probabilitds de I'evlnement prises de ces causes, et la probability de 1' existence de chacune d'elles, est %ale S, la probabilite de r4Yenement prise de cette cause, divisee par la somme de toutes les probabilites de I'Svenement prises de chacune de ces 870. Laplace first takes the standard problem in this part of our subject : Suppose that an urn contains an infinite number of white tickets and black tickets in an unknown ratio ; p + q tickets XAPLACE. 467 are drawn of which p are white and q are black : required the pro- bability of drawing m white tickets and n black tickets in the next m + n drawings. Laplace gives for the required probability /, a^°'{l-x)'^dx i x^{l-xy J a dx so that of course the m white tickets and n black tickets are sup- posed to be drawn in an assigned order ; see Arts. 704, 766, 843. Laplace effects the integration, and approximates by the aid of a formula which he takes from Euler, and which we usually call Stirling's Theorem. The problem here considered is not explicitly reproduced in the Thiorie. ..des Proh., though it is involved in the Chapter which forms pages 363—401. 871. After discussing this problem Laplace says. La solution de ce ProblSme donne une mlthode directs pour deter- miner la probabilite des 6v6nemens futurs d'apr§s ceux qui sent dija arriv6s ; mais cette matiSre 6tant fort 6tendue, je me bornerai ici h, donner une d6monstration assez singnliere du th6or§me suivaht. On peut supposer les nombres p e< q tellement grands, qiCil devienne aussi approchant que Von voudra de la certitude, que le rapport du nomhre de billets blancs au nomhre total des billets renfermes dans I'ume, est compris entre les deux limites —^ m, et —^ — l-o), a pouvant etre supposS moindre qvlaucune grandeur donnee. The probability of the ratio lying between the specified limits is Ix'il-xydx I' j J a^ (1 - xY dx I where the integral in the numerator is to be taken between the limits — a and — — h eo, Laplace by a rude process of j> + q p + q f J i- , 30—2 468 XAPLACE. approximation arrives at the conclusion that this probability does not dififer much from unity. 872. Laplace proceeds to the Problem of Points. He quotes the second formula which we have given in Art. 172 ; he says that it is now demonstrated in several works. He also refers to his own memoir in the volume of the Academy for 1773 ; he adds the following statement : ...on y trouvera pareillement une solution gSnlrale du Probllme des partis dans le cas de trois ou d'un plus grand nombre de joueurs, problSme qui n'a encore 6te risolu par personne, que je sache, bien que les GreomStres qui out travaille sur ces matiSres en aient desiri la solution. Laplace is wrong in this statement, for De Moivre had solved the problem ; see Art. 582. 873. Let X denote the skill of the player A, and 1—x the skill of the player B; suppose that A wants / games in order to win the match, and that B wants h games : then, if they agree to leave off and divide the stakes, the share of B will be a certain quan- tity which we may denote by {x,f, h). Suppose the skill of each player unknown; let n be the whole number of games which A or B ought to win in order to entitle him to the stake. Then Laplace says that it follows from the general principle which we have given in Art. 869, that the share of B is /a;''-/(l-a;)--*^(«,/^) where a = - . 883. The next problem is thus enunciated: Je suppose un nombre n de joneurs (1), (2), (3), ... (n), jouanfc de cette maniSre ; (1) joue avec (2), et s'il gagne il gagne la partie j.s'il ne perd ni gagne, il continue de jouer avec (2), jusqu'jl ce que I'un des deux gagne. Que si (1) perd, (2) joue avec (3) ; s'il le gagne, il gagne la partie ; s'il ne perd ni gagne, il continue de jouer avec (3) ; mais s'il .perd, (3) joue avec (4), et ainsi de suite jusqu'H ce que I'un des joueurs ait vaincu celui qui le suit ; c'est-^-dire que (1) soit vainqtieiii: de (2), i^i LAPLACE. oil (2) de (3), ou (3) de (4), ... ou (» - 1) de (re), ou (n) de (1). De plus, la probability d'un quelconque des joueurs, pour gagner I'autre = 3 > et celle de ne gagner ni perdre = ^ . Cela posfi, il faut dlterminer la pro- o babilit€ que I'un de ces joueurs gagnera la partie au coup x. This problem is rather difficult; it is not reproduced in the Theorie...des Prob. The following is the general result: Let v^ denote the chance that any assigned player will win the match at the a:*'' trial ; then _n n (n-l) 1 w (w - 1) (w - 2) 1 -1 SSi. Laplace next takes the Problem of Points in the case of two players, and then the same problem in the case of three players ; see Art. 872. Laplace solves the problem by Finite Diifer- ences. At the beginning of the volume which contains the memoir some errata are corrected, and there is also another solution indi- cated of the Problem of Points for three players; this solution depends on the expansion of a multinomial expression, and is in fact identical with that which had been given by De Moivre. Laplace's next problem may be considered an extension of the Problem of Points; it is reproduced in the Theor{e...des Prob. page 214, beginning with the words Gonoevons encore. 885. The next two problems are on the Duration of Play; in the first case the capitals being equal, and in the second case unequal; see Art, 863. The solutions are carried further than in the former memoir, but they are still much inferior to those which were subsequently given in the Theorie...des Prob. 886. The next problem is an extension of the problem of Duration of Play with equal capitals. It is supposed that at every game there is the chance p for A, the chance q for B, and the chance r that neither wins; each player has m crowns originally, and the loser in any game gives a crown to the winner: required the probability that the play will be finished in x games. This problem is not reproduced in the ThSorie...des Prob. LAPLACE 475 887. The present memoir may be regarded as a collection of examples in the theory of Finite Differences ; the methods ex- emplified have however since been superseded by that of Gene- rating Functions, which again may be considered to have now given way to the Calculus of Operations. The problems involve only questions in direct probability; none of them involve what are called questions in inverse probability, that is, questions respecting the probability of causes as deduced from observed events. 888. In the same volume as the memoir we have just ana- lysed there is a memoir by Laplace entitled, Memoire sur Vincli- naison moyenne des orbites des comhtes ; sur la figure de la Terre, et sur les Fonctions. The part of the memoir devoted to the mean inclination of the orbits of comets occupies pages 503 — 524 of the volume. In these pages Laplace discusses the problem which was started by Daniel Bernoulli ; s^e Art. 395. Laplace's result agrees with that which he afterwards obtained in the Theorie...des Proh. pages 253— 260, but the method is quite different; both methods are extremely laborious. Laplace gives a numerical example; he finds that supposing 12 comets or planets the chance is '339 that the mean inclination of the planes of the orbits to a fixed plane wiU lie between 45° — 7^° and 45°, and of course the chance is the same that the mean inclination will lie between 45° and 45° + 7J^°, 889. The volume with which we have been engaged in Arti- cles 881 — 888 is remarkable in connexion with Physical Asti'onomy. Historians of this subject usually record its triumphs, but omit its temporaiy failures. In the present volume Lagrange affects to shew that the secular acceleration of the Moon's motion cannot be explained by the ordinary theory of gravitation ; and Laplace affects to shew that the inequalities in the motions of Jupiter and Saturn cannot be attributed to the mutual action of these planets : see pages 47, 213 of the volume. Laplace lived to correct both his rival's error and his own, by two of his greatest contributions to Physical Astronomy. 476 LAPLACE 890. Laplace's next memoir on our subject is entitled Me- wsire su'r les Prohahilites; it is contained in the volume for 1778 of the Histoire de F Acad.... Paris: the date of publication of the volume is 1781. The memoir occupies pages 227 — 332, In the notice of the memoir which is given in the introductory part of the volume the names of Bayes and Price are mentioned. Laplace does not allude to them in the memoir. See Art. 540. 891. Laplace begins with remarks, similar to those which we have already noticed, respecting the chances connected with the tossing of a coin which is not quite symmetrical ; see Arts. 877, 881. He solves the simple problem of Duration of Play in the way we have given in Art. 107. Thus let p denote ^'s skill, and 1—p de- note B's skill. Suppose A to start with m stakes, and B to start with n — m stakes : then ^'s chance of winning all ^'s stakes is F'"^{p'"-o--pr} p'^-ii-pT ' Laplace puts for p in succession ^ (1 + o) and k (1 — o.), and takes half the sum. Thus he obtains for ^'s chance I {(1 + a) "-"• + (1 - artjCl + a)" - (1 - a)"} (l + «)''-(l-a)" ' which he transforms into 1 1 (l+ar-^-a-a)-"" 2 2^ ' (l + a)"-(l-a)» " The expression for ^'s chance becomes — when a vanishes ; Laplace proposes to shew that the expression increases as a in- creases, if 2m be less than n. The factor (1 — a')"* obviously dimin- ishes as a. increases. Laplace says that if 2m is less than n it is clear that the fraction (l+a)''-""-(l-a)"-'°' •LAPLACE. 477 also diminishes as a increases. We will demonstrate this. Put r forn — 2m, and denote the fraction by u ; then Idu (l + a)-'+(l-ar (l + a)-'+(l-a)-' udoi~ {l + (xy-{l-ay (H-a)"-(l-a)" " Thus Idu rjz'-'+l) >i(g''-'+l) '■ '^Kda~ z'-l s"-l ' ■where z = :j . We have to shew that this expression is nega- r (z'~^ + 1) tive ; this we shall do by shewing that ,._ — -"^ increases as successive integral values are ascribed to r. We have {r + l){z'+l) r(g'-'+l) _ (r + 1) (z^ -l)-r (z'^' - 1) (z'-' + 1) . {z'-''-l)lz'--l) thus we must shew that s""— 1 is greater than r (a*^' — s'"'). Expand by the exponential theorem ; then we find we have to shew that (2}-)* is greater than r | (r + 1)" - (r - 1)* I , 'where j) is any positive integer ; that is, we must shew that 2^' r^' is greater than^r""' ^- P (j'-lHy- 2) ^^-3 ^ _ But this is obvious, for r is supposed greater than unity, and the two members would be equal if all the exponents of r on the right hand side of the inequality were p — 1. We observe that r must be supposed not less than 2 ; if r = 1 we have z''-l=r {/^ - s*^'). We have assumed that r and n are integers, and this limitation is necessary. For return to the expression (l + ar-(l-a)- (1 + «)"-(!-«)•" 478 LAPLACE. and put for a in succession and 1 ; then we Have to compare - with nH ; that is, we have to compare ^ with ^ . Now consider ^ ; the differential coefficient with respect to x is ^x > so that ^ increases as x changes from to , — ^, and then diminishes. Laplace treats the same question in the Th4orie...des Proh. page 406 ; there also the difficulty is dismissed with the words il est facile de voir. In the memoir prefixed to the fourth volume of Bowditch's Translation of the Micanique CSleste, page 62, we read : Dr Bowditch himself was accnstomed to temark, " Whenever I meet in La Place with the words ' Thus it plainly appears' I am sure that hours, and perhaps days of hard study will alone enahle me to discover how it plainly appears." 892. The pages 240 — 258 of the memoir contain the im- portant but difficult investigation which is reproduced in the Thiorie...des Proh. pages 262 — 272. Laplace gives in the memoir a reference to those investigations by Lagrange which we have noticed in Art. 570 ; the reference however is omitted in the Thiorie...des Proh. 893. Laplace now proceeds to the subject which he had con- sidered in a former memoir, namely, the probability of causes as deduced from events; see Art. 868. Laplace repeats the general principle which he had already enunciated in his foraier memoir; see Art. 869. He then takes the problem which we have noticed in Art. 870, enunciating it however with respect to the births of boys and girls, instead of the drawings of white and black balls. See Art. 770. 894. Laplace is now led to consider the approximate evalu- ation of definite integrals, and he gives the method which is repro- duced almost identically in pages 88 — 90 of the Thiorie...des Proh. He applies it to the example I oa' (i—x)"^ dx, and thus demon- strates the theorem he had already given ; see Art. 871 : the pre- sent demonstration is much superior to the former. tAPLACE; 479 895. There is one proposition given here which is not repro- duced in the Thiorie...des Prob., but which is worthy of notice. Suppose we require the value of lydx where y = x^[l — a;)', the integral being taken between assigned limits. Put p=- and q = -; and let a '-a' 1 dx a^ dy Then, by integrating by parts, lydx= \o^dy = ayz — a. \yds (I), ( , f ds , ds { d [. dz\ , ]y^'=^rii^y-°y'Tx-'']ydx\'TA^''-^ so that jydx = ayz-a^y..^^+a^Jyj^{.^£)dx (2). Now y vanishes with x. Laplace shews that the value of lydx when the lower limit is zero and the upper limit is any value of X less than , is less than ayz and is greater than ds ayz — o^yz -j—; so that we can test the closeness of the approxi- mation. This proposition depends on the following considera- dz . . . 1 tions : --r- is positive so long as x is less than , and there- fore lydx is less than ays by (1); and -— (« t-) is also positive, so r . dz that lydx is greater than ayz — i^yz -y- by (2). For we have a; (1 — 0") ^~l-(l+/^)x' and this can be put in the formi 480. EAPLACE /^ , a? I A* "- (l + /.)^ + l+/.+ (l+/.)^{l-(l+/.)a;}- dz Hence we see that z and ^ both increase with x so long as X is less than = : this establishes the requii-ed proposition. See also Art. 767. 896. Laplace then takes the following problem. In 26 years it was observed in Paris that 251527 boys were born and 241945 girls : required the probability that the possibility of the birth of a boy is greater than .^ . The probability is found to differ from unity by less than a fraction having for its numerator 1"1521 and for its denominator the seventh power of a million. This problem is reproduced in the Tlieorie...des Prdb. pages 377 — 380, the data being the numbers of births during 40 years instead of during 26 years. 897. Taking the same data as in the preceding Article, La- place investigates the probabihty that in a given year the number of boys born shall not exceed the number of girls born. He finds the probability to be a little less than ncq " The result of a similar calculation from data furnished by observations in London is a little less than ^^.^„ . In pages 397 — 401 of the Thiorie...des Prob. we have a more difficult problem, namely to find the probfl;bility that during a century the annual births of boys shall never be less than that of girls. The treatment of the simpler problem in the memoir differs from that of the more difficult problem in the ThSorie...des Prdb. In the memoir Laplace obtains an equation in Finite Differences hence he deduces ty^= constant + y„?„., |l - A2„_j + A (z^^ Aa„ J -A [a„.,A (a„_3A3„Jl +— }' S' LAPLACE. 481 which as he says is analogous to the corresponding theorem in the Integral Calculus given in Art. 895 ; and, as in that Article, he shews that in the problem he is discussing the exact result lies between two approximate results. See also Art. 770. 898. The memoir contains on page 287 a brief indication of a problem which is elaborately treated in pages 369^376 of the Theorie...des Prob. 899. Laplace now developes another form of his method of approximation to the value of definite integrals. Suppose we require \ydx; let T be the maximum value of 7/ within the range of the integration. Assume y = Ye~*', and thus change 1/dx into an integral with respect to t. The investigation is reproduced in the Theone...des Prob. pages 101 — 103. Laplace determines the value of / e'^'dt. He does this by Jo taking the double integral / I e-^^^+'''^dsdu, and equating the results which are obtained by considering the integrations in different orders. 900. Laplace also considers the case in which instead of as- suming 1/ = Ye^, we may assume y = Fe"**. Something similar is given in the Theorie...des Prob. pages 93 — 95. Some formulae occur in the memoir which are not reproduced in the TMorie...des Prob., and which are quite wrong: we will point out the error. Laplace says on pages 298, 299 of the memoir : /■/■ dxdz Consid^rons pr6sentement la double mtfegrale jj— — ^——j, prise depuis x = jusqu'^ a;=l, et depuis « = jusqu'a, «=1; en faisant — - — ,=x', elle se changera dans celle-ci I—- I- --, ces (1-»=)1 Jj(l-^J{l-x)i intllr(x)+x ^ ; but this is obviously true, for ^^ ^ is negative. The result stated on page 321 of the TMoric.des Prob., that Jc" . 1 under a certain condition -j- is less than - , is an example of this theorem. 923. In the Connaissance des Terns for 1813, which is dated July 1811, there is an article by Laplace on pages 213 — 223, entitled, Du milieu qu'ilfaiit choisir enfre les rhultats d'un grand nombre d' observations. The article contains the matter which is reproduced in pages 322 — 329 of the TMorie. . .des Prob. Laplace speaks of his work as soon about to appear. or LAPLACE. 491 924. In the Gormaissance des Terns for 1815, which is dated November 1812, there is an article on pages 215 — 221 relating to Laplace's TMo'ne.,.des Froh. The article begins with an extract from the work itself, containing Laplace's account of its object and contents. After this follow some remarks on what is known as Laplace's nebular hypothesis respecting the formation of the solar system. Reference is made to the inference drawn by Michell from the group of the Pleiades ; see Art. 619. 925. In the Oonnaissance des Terns for 1816, which is dated November 1813, there is an article by Laplace, on pages 213 — 220, entitled, 8ur les Comltes. Out of a hundred comets which had been observed not one had been ascertained to move in an hyperbola; Laplace proposes to shew by the Theory of Probability that this result might have been expected, for the probabiUty is very great that a comet would move either in an elhpse or parabola or in an hyperbola of so great a transverse axis that it would be undistinguishable from a parabola. The solution of the problem proposed is very difficult, from the deficiency of verbal explanation. We will indicate the steps. Laplace supposes that r denotes the radius of the sphere of the sun's activity, so that r represents a very great length, which may be a hundred thousand times as large as the radius of the earth's orbit. Let V denote the velocity of the comet at the instant when it enters the sphere of the sun's activity, so that r is the comet's radius vector at that instant. Let a be the semi- axis major of the orbit which the comet proceeds to describe, e its excentricity, D its perihelion distance, ot the angle which the direction of V makes with the radius r. Take the mass of the sun for the unit of mass, and the mean distance of the sun from the earth as the unit of distance ; then we have the well-known formulse; a r rF sin OT = Va (1 — e*^. 492 LAPLACE. From these equations by eliminating a and e we have 9 TV 21)-— +!)'¥'' sin'' •57 = jjrp . and from this we deduce 1 — cos ■ST = 1 — y(-?) rV J{r^V^{l + f)-2D Now if we suppose that when the comet enters the sphere of the sun's activity all directions of motion which tend inwards are equally probable, we find that the chance that the direction will make an angle with the radius vector lying between zero and OT is 1 — cos tn. The values of the perihelion distance which correspond to these limiting directions are and D. Laplace then proceeds thus : ...en supposant done toutes les valeurs de Z> 6galement possibles, on a pour la probabilit6 que la distance p6rili§Iie sera comprise entre ziro et Z), ,_yK) rV y{"K-f)--}- II faut multiplier cette valeur par dV ; en I'iiitlgrant ensuite dans des limites diterminles, et divisant I'intlgrale par la plus grande valeur de V, valeur que nous designerons par U ; on aura la probability que la valeur de F sera comprise dans ces limites. Cela pos6, la plus petite valeur de V est celle qui rend nuUe la quantity renfermle sous le radical pr§o6dent j ce qui donne J2I> rV = y(-7)- It would seem that the above extract is neither clear nor correct; not clear for the real question is left uncertain; not correct in what relates to U. "We will proceed in the ordinary way, and not as Laplace does. Let ■y^ ( V) stand for rV .y{.7-(x.5)_^, LAPLACE. 493 then we have found that supposing all directions of projection equally probable, if a comet starts with the velocity V the chance is ■\jr(V) that its perihelion distance will lie between and D. Now suppose we assume as a fact that the perihelion distance does lie between and D, but that we do not know the initial velocity: required the probability that such initial velocity lies between assigned limits. This is a question in inverse probability ; and the answer is that the chance is |^(F) dV fir{V)dv' where the integral in the numerator is to be taken between the assigned limits ; and the integral in the denominator between the extreme admissible values of V. Laplace" finds the value of j'^{V)dV; for this purpose he assumes \/r ' y{.P(x.f)-2.}=.ry(x.5)_.. For the assigned limits of V he takes . and * The value of ji^ (F) (?F between these limits he finds to be ap- proximately (7r-2)V2^ n 2r ir >Jr ' the other terms involve higher powers of r in the denominator, and so are neglected. The above expression is the numerator of the chance which we require. For the denominator we may suppose that the upper limit of the velocity is infinite, so that i will now be infinite. Hence we have for the required chance ( (tt - 2) ViP D \ ^ I 2r irsir) ' { (tt -2)^20 D \ . {ir-'i)'j2D 2r 494 LAPLACE, that is, If for example we supposed *' = 2, we should have the extreme velocity which would allow the orbit to be an ellipse. 1 2 In the equation - = — F" suppose a = — 100 ; then ^, /• + 200 ,, ^ r + 200 If we use this value of i we obtain the chance that the orbit shall be either an ellipse or a parabola or an hyperbola with transverse axis greater than a hundred times the radius of the earth's orbit. The chance that the orbit is an hyperbola with a smaller transverse axis will be V25 Laplace obtains this result by his process. Laplace supposes D = 2, r = 100000 ; and the value of i to be that just given: he finds the chance to be about ' ,i^.. , . Laplace then says that his analysis supposes that all values of D between and 2 are equally probable for such comets as can be perceived; but observation shews that the comets for which the perihelion distance is greater than 1 are far less numerous than those for which it lies between and 1. He proceeds to consider how this will modify his result. 926. In the Connaissance des Terns for 1818, which is dated 1815, there are two articles by Laplace on pages 361 — 381 ; the first is entitled, Sur I'a^plication du Calcul des Prohabilites cl la Philosophie naturelle ; the second is entitled, Su/r le Calcul des Prohabilites, applique d, la Philosophie naturelle. The matter is reproduced in the first Supplement to the Thiorie...des Prob. pages 1 — 25, except two pages, namely, 376, 377: these contain an application of the formulae of probability to determine from observations the length of a seconds' pendulum. LAPLACE. 495 927. In the Connaissance des Terns for 1820, which is dated 1818, there is an article by Laplace on pages 422 — '440, entitled. Application du Galcul des ProbaMlites, aux operations geodisiqms: it is reproduced in the second Supplement to the TMorie...des Prob. pages 1—25. 928. In the Connaissance des Terns for 1822, which is dated 1820, there is an article by Laplace on pages 346—348, entitled. Application du Calcul des ProbaMlites aux operations geodesiques de la meridienne de France: it is reproduced in th? third Supple- ment to the Theorie...des Proh. pages 1 — 7. 929. We have now to speak of the great work of Laplace which is entitled, Theorie analytique des Probabilites. This was published in 1812, in quarto. There is a dedication to Napol^on-le-Grand ; ■then foUow 445 pages of text, and afterwards a table of contents which occupies pages 446 — 464 : on another page a few errata are noticed. The. second edition is dated 1814, and the third edition is dated 1820. The second edition contains an introduction of cvi. pages ; then the text "paged from 3 to 484 inclusive ; then a table of contents which occupies pages 485^506 : then two pages of errata are given. The pages 9 — 444 of the first edition were not reprinted for the second or third edition ; a few pages were cancelled and re- placed, apparently on account of errata. The third edition has an introduction of CXLII. pages; and then the remainder as in the second edition. There are, however, four supplements to the work which appeared subsequently to the first edition. The exact dates of issue of these supplements do not seem to be given; but the first and second supplements were probably published between 1812 and 1820, the third in 1820, and the fourth after 1820. Copies of the third edition generally have the first three supplements, but not the fourth. 930. Since the bulk of the text of Laplace's work was wt reprinted for the editions which appeared during his life time. 496 LAPLACE. a reference to the page of the work will in general suffice for any of these editions : accordingly we shall adopt this mode of reference. An edition of the works of Laplace was published in France at the national expense. The seventh volume consists of the Theorie...des Prob.; it is dated 1847. This volume is a reprint of the third edition. The title, advertisement, introduction, and table of contents occupy cxcv. pages ; the text occupies 532 pages, and the four supplements occupy pages 533 — 691. It wUl be found that in the text a page n of the editions pub- lished by Laplace himself will correspond nearly to the page « + t": of the national edition : thus our references will be easily available for the national edition. We do not think that the national edition is so good as it ought to have been ; we found, for example, that in the second supplement the misprints of the original were generally reproduced. 931. We shall now proceed to analyse the work. We take the third edition, and we shall notice the places in which the introduc- tion differs from the introduction to the second edition. The dedication was not continued after the first edition, so that it may be interesting to reproduce it here. A Napol^on-le-Grand. Sire, La bienveillance avec laquelle Votre Majeste a daigu€ accueUlir rhommage de men Traits de Mecanique Celeste, m'a inspire le desir de Lui d6dier cet Ouvrage sur le Calcul des Probabilitls. Ce calcul d61icat s'ltend aux questions les plus impor- tantes de la vie, qui ne sont en effet, pour la plupart, que des problfemes de probabilitg. 11 doit, sous ce rapport, int^resser Yotre Majeste dont le g€nie sait si bien apprlcier et si dignement encourager tout ce qui pent contribuer au progrls des lumiSres, et de la prospSritl publique. J'ose La supplier d'agrler ce nouvel bommage dict6 par la plus vive reconnaissance, et par les sentimens profonds d'admiration et de respect, avec lesquels je suis, Sii'e, de Votre Majesty, Le trgs-humble et tres- oblissant serviteur et fiddle sujet, Laplace. Laplace has been censured for suppressing this dedication after the fall of Napoleon ; I do not concur in this censure. The dedi- cation appears to me to be mere adulation; and it would have LAPLACE. 497 been almost a satire to have repeated it when the tyrant of Europe had become the mock sovereign of Elba or the exile of St Helena : the fault was in the original publication, and not in the final supr pression. 932. We have said that some pages of the original impression were cancelled, and others substituted ; the following are the pages : 25, 26, 27, 28, 37, 38, 147, 148, 303, 304, 359, 360, 391, 392 ; we note them because a student of the first edition will find some embarrassing errata in them. 933. The introduction to the Theorie...des Prob. was pub- lished separately in octavo under the title of Sssai pkilosophique sur les Probabilites; we shall however refer to the introduction by the pages of the third edition of the Theorie...des Prob. 934. On pages i — xvi. of the introduction we have some gene- ral remarks on Probability, and a statement of the first principles of the mathematical theory; the language is simple and the illustrations are clear, but there is hardly enough space allotted to the subject to constitute a good elementary exposition for be-, ginners. 935. On pages xvi — xxxvii. we have a section entitled Des m^tkodes ancdytiques du Cahvl des Probabilites; it is principally devoted to an account of the Theory of Generating Functions, the account being given in words with a very sparing use of symbols. This section may be regarded as a complete waste of space ; it would not be intelligible to a reader unless he were able to master the mathematical theory delivered in its appropriate symboHcal language, and in that case the section would be entirely super- fluous. This section differs in the two editions ; Laplace probably thought he improved in his treatment of the difficult task he had undertaken, namely to explain abstruse mathematical processes in ordinary language. We will notice two of the changes. Laplace gives on pages xxiil. and xxiv. some account of De Moivre's treatment of Recurring Series; this account is transferred from page CI. of the second edition of the introduction : a student however' 32 498 LAPLACE. who wished to understand the treatment would have to consult the original work, namely De Moivre's Miscellanea Analytica, pages 28 — 33. Also some slight historical reference to Wallis and others is introduced on pages xxxv— xxxvii. ; this is merely an abridgement of the pages 3 — 8 of the Theorie...des Prob. 936. We have next some brief remarks on games, and then some reference to the unknown inequalities which may exist in chances supposed to be equal, such as would arise from a want of symmetry in a coin or die ; see Arts. 877, 881, 891. 937. We have next a section on the laws of probability, which result from an indefinite multiplication of eveuts ; that is the section is devoted to the consideration of James BernoulU's theorem and its consequences. Some reflexions here seem aimed at the fallen emperor to whom the first edition of the work was dedicated ; we give two sentences from page xliil Voyez au contraire, dans quel abime de malLeurs, les peuples ont 6t6 souvent pr6cipit6s par rambition et par la perfidie de leurs chefs. Toutes les fois qu'une grande puissance enivr^e de Tamour des conqu^tes, aspire k la domination universelle; le .sentiment de I'indlpendanoe pro- duit entre les nations menaces, une coalition dont elle devient presque toujours la victime. The section under consideration occurs in the second edition, but it occupies a different position there, Laplace having made some changes in the arrangement of the matter in the third edition. We may notice at the erfd of this section an example of the absurdity of attempting to force mathematical expressions into unmathematical language. Laplace gives a description of a certain probability in these words : La tliSorie des fonctions generatrices donne une expression trSs simple de cette probability, que I'on obtient en integrant le produit de la differentieUe de la quantite dont le residtat d6duit d'un grand nombre d'observations s'Icarte de la v6rit6, par uue constante moindre que I'unitg, dependante de la nature du problgme, et 61ev6e h une puissance dont I'exposant est le rapport du carr6 de cet gcart, au nombre des observations. L'intggrale prise entre des limites donnges, et divis6e LAPLACE. 4,99 par la mSme mt%rale ^tendue S, I'infini positif et nggatif, exprimera la probabilite que I'Scart de la v6rit6, est compris entre ces limites. A student familiar with the Tkeorie...des Proh. itself might not find it easy to say what formula Laplace has in view ; it must be that which is given on page 309 and elsewhere, namely T (, JJ^ 77— [are « . ; IT J Other examples of the same absurdity will be found on page LI. of the introduction, and on page 5 of the first supplement. 938. A section occupies pages XLIX — LXX. entitled Applica- tion du Calcul des Probabilit^s, d la Philosophie naturelle. The principle which is here brought forward is simple ; we will take one example which is discussed in the Tkeorie...des Prdb. If a large number of observations be taken of the height of a barometer at nine in the morning and at four in the afternoon, it is found that the average in the former case is higher than in the latter ; are we to ascribe this to chance or to a constant cause? The theory of probabilities shews that if the number of observations be large enough the existence of a constant cause is very strongly in- dicated. Laplace intimates that in this way he had been induced to undertake some of his researches in Physical Astronomy, be- cause the theory of probabilities shewed ii-resistibly that there were constant causes in operation. Thus the section contains in reality a short summary of La- place's contributions to Physical Astronomy ; and it is a memor- able record of the triumphs of mathematical science and human genius. The list comprises — the explanation of the irregularity in the motion of the moon arising from the spheroidal figure of the earth — the secular equation of the moon — the long inequalities of Jupiter and Saturn — the laws connecting the motions of the satellites of Jupiter — the theory of the tides. See Gouraud, page 115 ; he adds to the list — the temperature of the earth shewn to be constant for two thousand years : it does not appear that Laplace himself here notices this result. 939. In the second edition of the Theorie ...des Proh. 32—2 500 LAPLACE. [Laplace did not include the secular acceleration of the moon and the theo.ry of the tides in the list of his labours suggested by the Theory of Probability. Also pages LI — LVI. of the introduction seem to have been introduced into the third edition, and taken from the first supplement. Laplace does not give references in his TMorie...des Prob., so we cannot say whether he published all the calculations respecting probability which he intimates that he made; they would how- ever, we may presume, be of the same kind as that relating to the barometer which is given in page 350 of the Thiorie. . .desProb., and so would involve no novelty of principle. Laplace alludes on page Liv. to some calculations relating to the masses of Jupiter and Saturn; the calculations are given in the first supplement. Laplace arrived at the result that it was 1000000 to 1 that the error in the estimation of the mass of Jupiter could not exceed t^t^ of the whole mass. Nevertheless it has since been recognised that the error was as large as — ; see Poisson, Recherches sur la Proh. . ., page 316. diO. Laplace devotes a page to the Application du Calcul des Prohabilitds awx Sciences morales; he makes here some inter- esting remarks on the opposing tendencies to change and to con- servatism. 941. The next section is entitled. Be la Probability des Umoignages; this section occupies pages Lxxi — Lxxxii : it is an arithmetical reproduction of some of the algebraical investigations of Chapter xi. of the Theorie. ..des Prob. One of Laplace's discus- sions has been criticised by John Stuart Mill in his Logic; see Vol. II. page 172 of the fifth edition. The subject is that to which we have alluded in Art. 735. Laplace makes some observations on miracles, and notices with disapprobation the language of Kacine, Pascal and Locke. He examines with some detail a famous argument by Pascal which he introduces thus : Ici se presents naturellement la discussion d'un argument fameux de Pascal, que Craig, mathgmaticien anglais, a reproduit sovis une forme LAPLACE. 501 geomgtrique. Des tgmoins fittestent qu'ils tiennent de la Divinity mSme, qu'en se conformant I, telle chose, on jouira, non pas d'une ou de deux, mais d'une infinite de vies heureuses. Quelque faible que soit la proba- .bilit6 des tgmoignages, pourvu qu'elle ne soit pas infiniment petite; il est clair que I'avantage de ceux qui se conforment S. la chose prescrite, est infini, puisquil est le produit de cette probability par un bien infini ; on ne doit done point balancer a se procurer cet avantage. See also the Athencmm for Jan. 14th, 1865, page 55. 942. The next section is entitled, Des choix et des dicisions des assemblees; it occupies four pages: results are stated re- specting voting on subjects and for candidates which are obtained at the end of Chapter ii. of the Theorie...des Proh. - The next section is entitled, De la prohabilite des Jiigemens des trihunaux; it occupies five pages; results are stated which are obtained in the first supplement to the Theorie...des Proh. This section is nearly all new in the third edition of the TMorie. . .des Proh. The next section is entitled, Des Tables de morixdite, et des durees moyennes de la vie, des manages et des associations quel- conqms; it occupies six pages : results are stated which are ob- tained in Chapter viii. of the Theorie...des Proh. The next section is entitled, Des hemfices des etahlissemens qui dependent de la prohabilite des euhnemens; it occupies five pages. This section relates to insurances : results are given which are ob- tained in Chapter ix. of the Theorie...des Proh. 943. The next section is entitled, Des illusions dans I'esti- mation des Probabilites ; this important section occupies pages cii — cxxviii : in the second edition of the Theorie. . .des Proh. the corresponding section occupied little more than seven pages. The illusions which Laplace notices are of various kinds. One of the principal amounts to imagining that past events influence future events when they are really unconnected. This is illus- trated from the example of lotteries, and by some remarks on page CIV. relating to the birth of a son, which are new in the third edition. Another illusion is the notion of a kind of fatality which gamblers often adopt. Laplace considers that one of the great advantages of the 502 LAPLACE. theory of probabilities is that it teaches, us to mistrust our first impressions; this is iUustrated by the example which we have noticed in Art. 856, and by the case of the Chevalier de M6r^: see Art. 10. Laplace makes on his page CVIIL some remarks re- specting the excess of the births of boys over the births of girls; these remarks are new in the third edition. Laplace places in the list of illusions an application of the Theory of Probability to the summation of series, which was made by Leibnitz and Daniel Bernoulli. They estimated the infinite series 1-1+1-1 + .. . as equal to g-; because if we take an even number of terms we obtain 0, and if we take an odd number of terms we obtain 1, and they assumed it to be equally probable that an infinite number of terms is odd or even. See Dugald Stewart s Worhs edited hy Hamilton, Vol. iv. page 204. Laplace makes some remarks on the apparent verification which occasionally happens of predictions or of dreams ; and justly remarks that persons who attach importance to such coincidences generally lose sight Of the number of cases in which such antici- pations of the future are falsified by the event. . He says, Ainsi, le philosophe de I'autiquitS, anquel on montrait dans un temple, pour exalter la puissance du dieu qu'on y adorait, les ex voto de tous ceux qui apr§s ravoir invoqu6, s'ltaient sauv6s du naufirage, fit une remarque conforme au calcul des probabilites, en observant qu'il ne voyait point inscrits, les noms de ceux qui, malgrS cette invocatiou, avaient p6ri. 944. A long discussion on what Laplace calls Psychologie occupies pages cxiii — cxxviii of the present section. There is much about the sensorium, and from the close of the discussion it would appear that Laplace fancied all mental phenomena ought to be explained by applying the laws of Dynamics to the vibra- tions of the sensorium. Indeed we are told on page cxxiv. that faith is a modification of the sensorium, and an extract from Pascal is used in a manner that its author would scarcely have approved^ LAPLACE. 503 945. The next section is entitled, Bes divers moyens d'ap- procher de la certitude; it occupies six pages. Laplace says, L'induction, I'analogie, des hypotheses fondees sur les faits et recti- iiees sans cesse par de nouvelles observations, un tact heureux donni par la nature et fortifie par des comparaisons nombreuses de ses indi- cations avec I'experience; tels sont les principaux moyens de parvenir 3. la vfirite. A paragraph beginning on page cxxix. with the words IJ'ou^ jugeons is new in the third edition, and so are the last four lines of page cxxxii. Laplace cites Bacon as having made a strange abuse of induction to demonstrate the immobility of the earth. Laplace says of Bacon, II a d.onn6 pour la recherche de la vSritg, le pricepte et non I'ex- emple. Mais en insistant avec toute la force de la raison et de I'ilo- quence, siu* la n6cessil.6 d'abandonner les subtiUt&s insignifiantes de r6cole, pour se livrer aux observations et aux experiences, et en indi- quant la vi-aie mithode de s'6lever aux causes ginlrales des phSnomlnes; ce grand philosophe a contribu6 aux progrSs immenses que I'esprit liumain a faits dans le beau silcle ofi il a terming sa carriSre. Some of Laplace's remarks on Analogy are quoted with ap- probation by Dugald Stewart ; see his Works edited hy Hamilton, Vol. IV. page 290. 946. The last section of the introduction is entitled, N'otice historique sur le Galcul des Prohdbilites ; this is brief but very good. The passage extending from the middle of page cxxxix, to the end of page cxll is new in the third edition; it relates principally to Laplace's development in his first supplement of his theory of errors. Laplace closes this passage with a reference to the humble origin of the subject he had so much advanced; he says it is remarkable that a science which began with the consi- deration of games should have raised itself to the most important objects of human knowledge. A brief sketch of the plan of the Theone...des Proh., which appeared on the last page of the introduction in the second edi- tion, is not repeated in the third edition. 947. The words in which at the end of the introduction La- 504 LAPLACE. place sums up the claims of the Theory of Probability well deserve to be reproduced here: On Toit par cet Essai, que la tWorie des probabilites n'est au fond, que le bon sens rMuit au calcul: elle fait apprgcier avec exactitude, ce que les esprits justes sentent par une sorte d'instinct, sans qu'ils puissent souvent s'en rendre compte. Si Ton consid6re les mgthodes analytiques auxquelles cette thlorie a donne naissanoe, la v6rit6 des principes qui lui servent de base, la logique fine et delicate qu'exige leur emploi dans la solution des problSmes, les Itablissemens d'ntilit6 publique qui s'appiiient sur elle, et I'extension qu'elle a regue et qu'elle pent recevoir encore, par son application aux questions les plus impor- tantes de la Philosophie naturelle et des sciences morales; si Ton ob- serve ensuite, que dans les choses mSmes qui ne peuvent Stre soumises au calcul, elle donne les apergus les plus surs qui puissent nous guider dans nos jugemens, et qu'elle apprend k se garantir des illusions qui souvent nous 6garent; on verra qu'il n'est point de science plus digne de nos m§ditations, et qu'il soit plus utile de faire entrer dans le systSme de I'instruction publique, 948. We now leave the introduction and pass to the Theorie. . . des Prob. itself Laplace divides this into two books. Livre i. is •entitled Du Calcul des Fonctions Oeneratrices: this occupies pages 1 — 177 ; Livre u. is entitled Theorie genSrale des ProhahiUtes; this occupies pages 179 — 461. Then follow Additions on pages ,462^484. 949. The title which Laplace gives to his Livre i. does not •adequately indicate its contents. The subject of generating func- tions, strictly so called, forms only the first part of the book ; the second part is devoted to the consideration of the approximate calculation of various expressions which occur in the Theory of Probability. 950. The first part of Livre I. is almost a reprint of the me- moir of 1779 in which it originally appeared ; see Art. 906. This part begins with a few introductory remarks on pages 3 — 8 ; these pages 3 — 8 of the third edition do not quite agree with the pages 1 — 8 of the first edition, but there is nothing of consequence pecu- liar to the first edition. Laplace draws attention to the importance -of notation in mathematics ; and he illustrates the point by the LAPLACE. 505 advantage of the notation for denoting powers, which leads him to speak of Descartes and Wallis. Laplace points out that Leibnitz made a remarkable use of the notation of powers as applied to differentials ; this use we might describe in modern terms as an example of the separation of the symbols of operation and quantity. Lagrange followed up this analogy of powers and differentials ; his memoir inserted in the volume for 1772 of the memoirs of the Academy of Berlin is cha- racterised by Laplace as one of the finest applications ever made of the method of inductions. 951. The first Chapter of the first part of Livre I. is entitled Des Fonctions generatrices, d une variable; it occupies pages 9 — 49. The method of generating functions has lost much of its value since the cultivation of the Calculus of Operations by Professor Boole and others ; partly on this account, and partly because the method is sufficiently illustrated in works on the Theory of Finite Differences, we shall not explain it here. Pages 39—49 contain various formulae of what we now call the Calculus of Operations ; these formulse cannot be said to be de- monstrated by Laplace ; he is content to rely mainly on analogy. Lagrange had led the way here ; see the preceding Article. One of the formulse may be reproduced ; see Laplace's page 41. If we write Taylor's theorem symbolically we obtain where A indicates the difference in y^ arising from a difference h in X. Then ^^y,= \e^-l) y,. Laplace transforms this into the following result, / hd_ hay ^"y.^V'^-e'^^Jy.^l The following is his method : \e ^-l)y^ = e^f V "'' - e"' ^) y.. 2 506 LAPLACE. Now let h r j-j denote any term arising from the development of g2 (to _ g~ 2 dxj ^ Then ,(^),.^,^=,(^),,,^.; and the term on the right hand may be supposed to have arisen from the development of \e^ ''*— e ^'^J y ^'!^. Thus the formula 2 is considered to be established. We ought to observe that Laplace does not express the formula quite in the way which we adopt. His mode of writing Taylor's Theorem is and then he would write ( >fiy^ V He gives verbal directions as to the way in which the sjonbols are to be treated, which of course make his formulas really iden- tical with those which we express somewhat differently. We may notice that Laplace uses c for the base of the Napierian logarithms, which we denote by e. If in the formula we put A = 1 and change x into a; — „ we obtain f \ ±_ 1 A" 2 which Laplace obtains on his page 45 by another process. 952. The second Chapter of the first part of Liiire L is entitled Des fonctions generatrices a deux variables: it occupies pages 50—87. Laplace applies the theory of generating functions to solve equations in Finite Differences with two independent variables. He gives on his pages 63 — 65 a strange process for integrating the following equation in Finite Differences, tAPLACE. 507 "We might suppose that z^,^ is the coefiicient of f t* in the ex- pansion of a function of t and t ; then it would easily follow that this function must be of the form <^ (<) + ^ (t) «(s-M- y where (t) is an arbitrary function of t, and yjr (t) an arbitrary function of t. Laplace, however, proceeds thus. He puts lab „ --c = 0, Tt T t and he calls this the equation generatrice of the given equation in Finite Differences. He takes u to denote the function of t and r whifh when expanded in powers of t and t has s^, ^ for the co- ' efficient of provided we suppose that y is made zero after the operation denoted by S^ has been performed on -^ . Similarly in « ft - «Y the coefficient of «V will be a'A' T^") , provided we suppose that x is made zero after the operation de- noted by A' has been performed on -^ . In this way we obtain «... = yy^^ + rr'^-' (%") + ••• + ^^^».» F A c^ab "*'■ + ...+ "" F A==('?^»'l (c + a6)=° Thus we see that in order to obtain z^^^ we must know ^o,i> \i'--' "P *o ^0,,, and we must know «i,„, s^,„,... up to a^,,. Now we have to observe that this process as given by Laplace cannot be said to be demonstrative or even intelligible. His method of connecting the two independent variables by the equation gSnSratrice without explanation is most strange. But the student who is acquainted with the modern methods of the Calculus of Operations will be able to translate Laplace's process into a more familiar language. Let E denote the change of x into x + 1, and F the change of y into ^ + 1 : then the fundamental equation we have to integrate will be written {EF-- aF-.bE-c)z^,, = 0, or for abbreviation EF-aF-bF-c = 0. Then F'^F" will be expanded in the way Laplace expands — — and his result obtained from F^F^^,. Thus we rely on the foundations on which the Calculus of Operations is based. 510 LAPLACE. We may notice that we have changed Laplace's notation in order to avoid the dashes which are diflficult in printing. La- place uses X where we use y, U where we use t, and 'A where we use S. 953. Laplace takes another equation in Finite Differences. The equation we will denote thus Here A belongs to x of which the difference is unity; and S belongs to y of which the difference is a. Laplace says that the equation generatrice is He supposes that this equation is solved, and thus decomposed into the following n equations : 1-1 = 2 fi-i), t a V W 1 l=^fl_l), where g, g'l; S'a)"- ^'^^ ^^ ** roots of the equation Then, using the first root u u i^ (- 1)' I^^V^ - ^^ (l + I) TTiSrn + ■ • ■ } ■ Then passing from the generating functions to the coefficients, that is equating the coefficients of i°T°, we obtain LAPLACE. 511 The second member may be put in the form Denote the quantity f— 2— ja^j J, by the arbitrary function ^(^). Thus '•■.-(^+i)"'(-l)'^*«- This value of Zx,y will then satisfy the equation in Finite Dif- ferences. Each of the n roots q, g^, q^, ... gives rise to a similar ex- pression ; and the sum of the n particular values thus obtained for z^^y will furnish the general value, involving n arbitrary functions. The student will as before be able to translate this process into the language of the Calculus of Operations. Laplace continues thus : Suppose a indefinitely small, and equal to di/. Then 9.' as we may see by taking logarithms. Thus we shall obtain This is the complete integral of the equation Laplace next gives some formulae of what we now call the Cal- culus of Operations, in the case of two independent variables ; see his pages 68 — 70. 954. In his pages 70 — 80 Laplace offers some remarks on the transition from the finite to the indefinitely small ; his object is to shew that the process will furnish rigorous demonstrations. He illustrates by referring to the problem of vibrating strings, and this leads him to notice a famous question, namely that of the ad- missibility of discontinuous functions in the solution of partial dif- 012 LAPLACE". ferential equations; he concludes that such functions are ad- missible under certain conditions. Professor Boole regards the argv/ment as unsound ; see his Finite Differences, Chapter X. 955. Laplace closes the Chapter with some general considera- tions respecting generating functions. The only point to which we need draw attention is that there is an important error in page 82 ; Laplace gives an incomplete form as the solution of an equation in Finite Differences ; the complete form will be found on page 5 of the fourth supplement. We shall see the influence of the eiTor hereafter in Arts. 974, 980, 984. 956. We now arrive at the second part of Livre L, this is nearly a reprint of the memoir for 1782; the method of approxi- mation had however been already given in the memoir for 1778. See Arts. 894, 899, 907, 921. The first chapter of the second part of Livre i. is entitled De Vintigration par approximation, des diffirentieUes qui renferment des facteurs Sieves d de grandes puissances; this Chapter occupies pages 88—109. 957. The method of approximation which Laplace giyes is of great value : we will explain it. Suppose we require the value of ydx taken between two values of x which include a value for which y is a maximum. Assume y = Fe-'", where F denotes this maximum value of y. Then /^ jyclx=YJe -^^dt. at Let ^ = ^ (x) ; suppose a the value of x which makes y have the value Y : assume x=a-\-d. Thus <^(a + e)= Fe-*'; Y therefore <° = log -j-. — — ^ . ^\a-\o) From this equation we may expand Mn a series of ascending powers of Q, and then by revei-sion of series we may obtain in a series of ascending powers of t. Suppose that thus we have LAPLACE. 513 jt/dx =YJe-<^{B^ + 2BJ + SB/ + ...)dt. Such is the method of Laplace. . It will be practically advan- tageous in the cases where B^,B^,B^,... form a rapidly converging series ; and it is to such cases that we shall have to apply it, when we give some examples of it from Laplace's next Chapter. In these examples there will be no difficulty in calculating the terms B^, B^, B^, ..., so far as we shall require them. An investigation of the general values of these coefficients as far as Bs inclusive will be found in De Morgan's Differential and Integral Calculus, page 602. If we suppose that the limits of x are such, as to make the cor- responding values of y zero, the limits of t wiU be — oo and + oo , Now if J- be odd I e-^fdt vanishes, and if r be even it is equal to Thus we have Besides the transformation y = Ye~^ Laplace also takes cases in which the exponent of e instead of being — «" has other values. Thus on his page 88 the exponent is — t, and on his page 93 it is —t^; in the first of these cases Y is not supposed to be a maximum value of y. 958. Some definite integrals are given on pages 95—101, in connexion with which it may be usefiil to supply a few references. The formula marked {T) on page 95 occurs in Laplace's memoir of 1782, page 17. /, cos ra3 e"" * aa; = -5— e la" ; 2a this was given by Laplace in the Memoires...de VInstitut for 1810, page 290 ; see also Tahles dllntigralea Ddjmies,. 1858, by D. Bierens de Haan, page 376. S3 514 LAPLACE. /. " Sin rx , IT X I see D. Bierens de Haan, page 268. r "cos aa; , tt „ /""assinaa;, it _ where a is supposed positive ; these seem due to Laplace ; see D. Bierens de Haan, page 282, TMone...desProb., pages 99 — 134. We may remark that these two results, together with Jo sin ax dx ir ,, „, 1 + a;' a; 2 ^ " are referred by D. F. Gregory, in his Examples of the... Differential and Integral Calculus, to Laplace's memoir of 1782 ; but they are not exphcitly given there : with respect to the last result see D. Bierens de Haan, page 293. 959. Since the integral |e~*' dt occurs in the expressions of Art. 957, Laplace is led to make some observations on modes of approximating to the value of this integral He gives the follow- ing series which present no difficulty : j/ ''*~'^"3'^[2 5 [3^7'^"""' j. ^''^'=27(^-2?+2V--W- + -J' In the memoir of 1782 the second of these three expressions does not occur, Laplace also gives a development of I e**' dt into the form of a continued fraction, which he takes from his MScanique Celeste, Livre x. See also De Morgan's Differential and Integral Calculus, page 591, for this and some similar developments. LAPLACE. 515 960. Laplace extends the method of approximation given in Art. 957 to the case of double integrals. The following is substan- tially his process. Suppose we require I ly dx dx' taken between such limits of x and x as make y vanish. Let Y denote the maximum value of y, and suppose that a and a are the correspond- ing values of x and x'. Assume y=Ye-^'-(\ X =a+6, x' = a' + 6'. Y Substitute these values of x and x' in the function log — and expand it in powers of d and 6' ; then since Y is by hypothesis the maximum value of y the coefficients of 6 and 0' will vanish in this expansion : hence we may write the result thus that is M(^e + ^ffJ+(P-^)e"'=f + t'^ Since we have made only one assumption respecting the inde- pendent variables t and t' we are at liberty to make another ; we will assume and therefore & I [P— -^ ] = ^'■ Now by the ordinary theory for the transformation of double integrals we have Uy dx dx' = j^ — — jj , ^ , - „ dt di dt dt' where i) stands for addd'~de'dd- Thus far the process is exact. For an approximation we may suppose M, N, F to be functions of a and a' only ; then we have ,. 1 d'Y ,,_ 1 d'Y p 1 d^ ^-"¥Y~W ^ rYdadc^' ^~ %Y d^'' S3— 2 516 LAPLACE. Then we shaU- find that And the limits of t and t' will be — oo and + oo ; thus finally we have approximately I \y dx dx' = 2-irY^ /id^Y d''Y _ f d^Y V ) • y\da^ dci'' Kdaddl] See Art. 907. 961. The second Chapter of the second part of lAvre L is entitled De Vintegration par approodmation, des Equations UnSaires aux differences Jlnies et infiniment petites : this Chapter occupies pages 110—125. This Chapter exemplifies the process of solving linear differential equations by the aid of definite integrals. Laplace seems to be the first who drew attention to this subject : it is now fully dis- cussed in works on differential equations. See Boole's Differential Equations. 962. The third Chapter of the second part of Livre i. is entitled Application des mdthodes prMdentes, d I' approodmation de diverses fonctions de tres-grands nombres: this Chapter oc- cupies pages 126—^177. The first example is the following. Suppose we have to in- tegrate the equation in Finite Differences, ys^.y = (s + 1) y.- Assume y,= jafcpdx, where ^ is a function of x at present undetermined, and the limits of the integration are also unde- termined. Let 8^ stand for aj' ; then -^ = sx^\ Hence the proposed equation becomes 0= Udx{{l-x)By + dhy dx LAPLACE. ol'7 that is, by integrating by parts, Where by [as Zy (^] we mean that xSy^ is to be taken between limits. Assume such that and take the limits of integration such that lxSy — j- (as (—s) = 1. 964. Laplace, after investigating a formula sometimes de- duces another from it by passing from real to imaginary quantities. This method cannot be considered demonstrative ; and indeed Laplace himself admits that it may be employed to discover new formulae, but that the results thus obtained should be confirmed by direct demonstration. See his pages 87 and 471 ; also Art. 920. Thus as a specimen of his results we may quote one which he gives on his page 134. Let Q^coB^^ c^rir + V-lsin^ ^._^^y , then r..___2M^-^ J a f '' a x''e''dx A memoir by Cauchy on Definite Integrals is published in the Journal de I'Ecole Polytechnique, 28° Cahier; this memoir was' presented .|o the Academy of Sciences, Jan. 2nd, 1815,. but not printed until 1841. The memoir discusses very fully the results given by Laplace in the Chapter we are now considering. Cauchy says, page 148, ...je suis parvenu &, quelques r^sultats nouveaux, ainsi qu'S, la demonstration directe de plusieurs formules, que M. Laplace a diduites LAPLACE. 521 du passage du r^el S, I'imaginaii-e, dans le 3"* chapitre du Calcul dss ProhabilUes, et qu'il vient de confirmer paf des methodes rigoureuses dans quelques additions faites 5, cet ouvrage. The additions to which Oauchy refers occupy pages 464 — 484 of' the TMorie...des Prob., and first appeared in the second edi- tion, which is dated 1814. 965. An important application which Laplace makes of his method of approximation is to evaluate the coefl&cients of the terms in the expansion of a high power of a certain polynomial. Let the polynomial consist of 2k + 1 terms and be denoted and suppose the polynomial raised to the power s. First, let it be required to find the coefficient of the term independent of a. Substitute e*^^ for a; then we require the term which is independent of when |l4-2cos0 + 2cos20+ ... + 2cosn6X is expanded and arranged according to cosines of multiples of 0. This term will be found by integrating the above expression with respect to d from to tt, and dividing by tt. Sum the series of cosines by the usual formula ; then the required term -1 r sm — X — Y 9. I f-h^ sm^^ J ^2r4''/sin^y where — -0, and m = %n-\rl. Now the expression [ -^-^j vanishes when — or — or — , m m m /I 522 LAPLACE. and between each of these values it will be found that the ex- pression is numerically a maximum, and it is also a maximum when ^ = 0. Thus we may calculate by Art. 957 the value of the integral '—. — -] dd} when the limits are consecutive multiples of — . , sm ^ / ^ " m The equation which determines the maxima values of —. — y- ^ sm

— cos (f) sin mcf) _ sm (p It will be found that this is satisfied when <^ = ; the situation of the other values of (j> will be more easily discovered by putting the equation in the form tan m^ — m tan ^ = : now we see that the next solution will lie between md) = — :- and ^ 4 m = 0, and is therefore wi* ; assume therefore /sin in t/6 dt Vl«(w"-l)l' and f f^)'# = nl'i\.x h'<^^- The limits of t will be and oo . Hence approximately •7rj„ Vsm^/ ^ TT Vl«(»i -l)}io VfsTT (»n= - 1) j V{« (« + 1) 2s7r} ' Laplace next considers the value of the integral with respect 2 to d> between the limits — and — , and then the value between m m 9— o— the limits — and — , and so on ; he shews that when s is a very mm •'' large number these definite integrals diminish rapidly, and may be neglected in comparison with the value obtained for the limits and — . This result depends on the fact that the successive numerical maxima values of — : — r- diminish rapidly ; as we shall sm r J > now shew. At a numerical maximum we have sin nK^ _ m cos m^ _*"_"*. sin^ cos^ cos ^ \/(l + m" tan* ^) i^ (cos' sm

y „,, ■,, As before assume /sin mdA' . _,, V sm ^ / then = ,, / 2_jsi » approximately. Hence the integral becomes 2 toV6 [-a '^It'JQ , /. / 2 — rrr ^ cos ,, . ^ — rv-, at. LAPLACE. 525 As before we take and oo for the limits of t, and thus neglect all that part of the integral with respect to which is not included between the limits and — . Hence by Art. 958 we m have finally 2 mV6 Vtt -,^ (2n + l)V3 -^j^, 7rV{«K-l)} 2*^ ' Vl** (w + 1) 2s7r} Suppose now that we require the swm of the coefficients, from that of a-' to that of a} both inclusive ; we must find the sum of 2^j + 2^j_i + 2^z_a + . . . + 2^j + ^„ : this is best effected by the aid of Euler's Theorem ; see Art. 334. We have approximately 2o'"' M« = j uj,x - 2 % + 2 Mo ; ri 11 therefore X J % = I M^c?a; + 5 «« + « ''o 5 therefore 2So u^ — Ug=2J ujlx + u^. Hence the required result is (2w + l)V6 V l« (w + 1) st} We may observe that Laplace demonstrates Euler's Theorem in the manner which is now usual in elementary works, that is by the aid of the Calculus of Operations. 966. Laplace gives on his page 158 the formula / J a 1 ' He demonstrates this in his own way ; it is sufficient to observe that it may be obtained by putting x for sx in the integral in the numerator of the left-hand side. 526 LAPLACE. Hence he deduces /• CO I a;"e-™(e-'-l)'' dx I Laplace calculates the approximate value of this expression, supposing t very large. He assumes that the result which he obtains will hold when the sign of { is changed ; so that he obtains an approximate expression for AV; see page 159 of his work. He gives a demonstration in the additions; see page 474 of the ThSorie...des Prob. The demonstration involves much use of the symbol \/(- 1). Cauchy gives a demonstration on page 247 of the memoir cited in Art. 964. Laplace gives another formula for AV on his page 163 ; he arrives at it by the aid of integrals with imaginary limits, and then confirms his result by a demon- stration. 967. Laplace says, on his page 165, that in the theory of chances we often require to consider in the expression for AV only those terms in which the quantity raised to the power i is positive; and accordingly he proceeds to give suitable approximate formulae for such cases. Then he passes on to consider specially the ap- proximate value of the expression (n + r Vrt)" -n(n + r>Jn-2y+ \ ^ {n+r >Jn-4{x,0)f-t{0,y)r'+{0, 0) = uipt + qr)-ptt(x,0)f-qTtct> (0,^)7^ (2'), where 2 <^ {x, 0) f denotes a summation with respect to x from a; = inclusive to a; = oo ; and S ^ (0, y) r" denotes a summation with respect to y from y = inclusive to y= oo. In order to shew that (2) is true we have to observe two facts. First, the coefiicient of any such term as Ct", where neither m nor n is less than unity, is the same on both sides of (2) by virtue of (1). Secondly, on the left-hand side of (2) such terms as Tt", where m or n is less than unity, cancel each other ; and so also do such terms on the right-hand side of (2). Thus (2) is fully established. From (2) we obtain (1 -pt)t (0,y) T"- <}> (0, 0) . M = = , 1 ~pt — qr LAPLACE. 529 ■we may write this result thus, «-m±m (3), where F{{) and /(t) are functions of t and t respectively, which are at present undetermined. By supposing that the term in /(t) which is independent of t is included in F{^, we may write the result thus, „=%(0±itM ,4) y — 'pt — qr ^ ' Thus either (3) or (4) may be taken as the general solution of the equation (1) in Finite Differences; and this general solution involves two arbitrary functions which must be determined by special considerations. We proceed to determine these functions in the present case, taking the form (4) which will be the most convenient. Now A loses if B first makes up his set, so that ^ {x, 0) = for every value of x from unity upwards, and ^ (0, 0) does not occur, that is it may also be considered zero. But from . (4) it follows that ^ (aj, 0) is the coefficient of f in the development of Y~t ' therefore x (*) = 0- Again, A wins if he first makes up his set, so that ^ (0, y) = \ for every value of y from unity upwards. But from (4) it follows that ^ (0,2^) is the coefficient of t" in the development of _^ , so that T"^ (t) _ T therefore Thus finally "■~(l-T)(l-i7«-2T)' Now ^{x,y) ig the coefficient of fr* in the development of «:. First expand. according to powers of t; thus we obtain for the 34 1- qr 1 -t' t^{t) t(1- 1- -qr) - T r(l- Ar). 530 LAPLACE. coefficient of f the expression . ^J_ .^ . Then expand this expression according to powers of t, and we finally obtain for the coefficient of fi'" This is therefore the probability in" favour of A ; and that in favour of B may be obtained by interchanging p with j and x with y. The result is identical with the second of the two formulae which we have given in Art. 172. 974. The investigation just given is in substance Laplace's ; he takes the particular case in which ^ = g and g' = 5 > hut this makes no difference in principle. But there is one important difference. At the stage where we have 1 —pt — qr ' Laplace puts 1 —j)t — qr This is an error, it arises from a false formula given by Laplace on his page 82; see Art. 955. Laplace's error amounts to neg- lecting the considerations involved in the second of the facts on which equation (2) of the preceding Article depends ; this kind of neglect has been not uncommon with those who have used or expounded the method of Generating Functions. 975. We will continue the discussion of the Problem of Points, and suppose that there are more than two players. Let the first player want 33^ points, the second x^ points, the third x^ points, and so on. Let their respective chances of winning a single game be^j, 2?j,^3, ... Let ^ (x^, x^, x^, . . .) denote the probability in favour of the first player. Then, as in Art. 973, we obtain the equation ^ (a?,, x^, a?3, ...) =p^^ (a!i-l, x^, x,, ...) +p^4> {x^, x.,-1, x^, ...) +2\) (x-1) (1). Laplace takes s^ to denote the probability that the run will finish at the a;"' trial, and not before ; then he obtains «« = (1 P) |«»-i + P^^ +p\-s +...+p'-'zA (2). We may deduce (2) thus ; it is obvious that «» = <^(«)-<^(a'-i); hence in (1) change x into x—1 and subtract, and we ob- tain (2). Laplace proceeds nearly thus. If the run is first completed at the a;"" trial the (x—iy^ trial must have been unfavourable, and the following i trials favourable. Laplace then makes i distinct cases. L The (x — {— l)"" trial unfavourable. IL The {x-i-Vf favourable; and the (a!-i-2)"' un- favourable. Ill The {x - i- 1)'" and the {x - i- 2)* favourable, and the (a;— 1—3)"' unfavourable. IV. The {x-i- 1)*, the (a; - 1 - 2)* and the {x-i- 3)* favourable ; and the (a; — » — 4)"' unfavourable. And so on. Let us take one of these cases, say IV. Let P^ denote the probability of this case existing ; then will For in this case a run of 3 has been obtained, and if this be followed by a run of /— 3, of which the chance is p*"", we obtain a run of i ending at the (a; — 4)"" trial. Now the part of z^ which arises from this case IV. is P^(l —p) p*; for we require an unfavourable result at the (a; — t")'" trial, of LAPIACE. Ml which the chance is. 1 — p, and then a run of i. Thus the part of a, is ^aCl-i')/; 0'•y(l-i')^»-4• We have said that Laplace adopts nearly the method we have given ; but he is rather obscure. In the method we have given Pj denotes the probability of the following compound event : no run of i before the (as — «— 4)'" trial, the (a;— t— 4)"' trial un- favourable, and then the next three trials favourable. Similarly our Pj would denote the probability of the following compound event ; no run of i before the {x — i — 2)^ trial, the {x—i— 2)*^ trial unfavourable, and the next trial favourable. Laplace says, Nommons P' la probability qu'il n'arrivera pas au coup x—i— 2. Now Laplace does not formally say that there is to be no run of i before the {x~i— 2)"' trial ; but this must be understood. Then his P' agrees with our P^ if we omit the last of the three clauses which form our account of the probability represented by P^ ; so that in fact pP" with Laplace denotes the same as P^ with us. Laplace gives the integral of the equation (2), and finally ob- tains the same result as we have exhibited in Art. 325. 986. Laplace then proceeds to find the probability that one of two players . should have a nan of ^ successes before the other ; this investigation adds nothing to what Condorcet had given, but is more commodious in form. Laplace's result on his page 250 will be found on examination to agree with what we have given in Art. 680, after Condorcat. Laplace then supplies some new matter, in which he considers the expectation of each player supposing that after failing he deposits a franc, and that the sum of the deposits is taken by him who first has a run of i successes. 987. Laplace's next problem is the following. An urn con- tains n + 1 balls marked respectively 0, 1, ... w ; a ball ig drawn and replaced : required the probability that after { drawings the sum of the numbers drawn wUl be s. This problem and applica- tions of it occupy pages 253 — 261. See Arts. 888, 915. The problein is due to De Moivre ; see Arts. 149, 364. La- place's solution of the problem is very laborious. We will pass to 54.2 LAPLACE. the application which Laplace makes of the result to the subject of the planes of motion of the planets. By proceeding as in Art. 148, we find that the probability that after i drawings the sum of the numbers drawn will be s is the coefficient of of in the expansion of Thus we obtain for the required probability 1 ( \i + s-l i )t + s-w-2 (w+l)M |T-1 \s ~\ \i-\ \s~n^ i{i-l) \t + s-2n-3 + ■ 1.2 \i-l\s-2n-2 If the balls are marked respectively 0, 6, -26, Sd, ...nO, this expression gives the probability that after i drawings the sum of the numbers drawn will be sd. Now suppose 6 to become indefinitely small, and n and s to become indefinitely great. The above expression becomes ulti- mately ^!(r-i'e-r-^a-r- Let - be denoted by x, and - bv dx, so that we obtain n n •' ^p--!(a,-ir+i^)(^-2r-...|j:r; this expression may be regarded as the conclusion of the follow- ing problem. The numerical result at a single trial must lie between and 1, and all fractional values are equally probable : determine the probability that after i trials the sum of the results obtained wiU lie between x and x + dx, where dx is indefinitely smaU. Hence if we require the probability that after i trials the sum of the results obtained will lie between x^ and x^i we must inte- lAPLACE. 543 grate the above expression between the limits a;, and x^ ; thus we obtain im'-;(-.-i)'+^(^.-2)'-...} Each series, like the others in the present Article, is to be continued only so long as the quantities which are raised to the power i are positive. We might have obtained this result more rapidly by using Art. 364 as our starting point instead of Art. 148. At the beginning of the year 1801, the sum of the inclinations of the orbits of the ten planets to the ecliptic was 91"4187 French degrees, that is -914187 of a right angle ; suppose that for each planet any inclination between zero and a right angle had been equally likely : required the probability that the sum of the inclinations would have been between and '914187 of a right angle. By the preceding expression we obtain for the result ry^ '(•914187)" that is about -00000011235. Speaking of this probability, Laplace says : ... EUe est A&jk tris-petite; mais il faut encore la combin'er avec la probability d'une circonstance trSs-remarquable dans le systlme du monde, et qui consiste en ce que toutes les planStes se meuvent dans le mSme sens que la terre. Si les mouvemens directs et retrogrades sont /In" supposes ggalement possibles, cette derniSre probability est (-] ; il /1V° faut done multiplier -00000011235 par f^J , pour avojr la probability que tons les mouvemens des planStes et de la terre seront dirigis dans le m^me sens, et que la somme de leurs inclinaisons il I'orbite de la terre, 1-0972 (10)" sera comprise dans les limites z^ro et 2V'i\%7 ; on aura ainsi tyttvk,- 1-0972 pour cette probability ; ce qui donne 1 — .^ --i„ pour la probability que cela n'a pas dii avoir lieu, si toutes les incUnaisons, ainsi que les mouve- mens directs et retrogrades ont et6 Igal^ment faciles. Cette probability 54-i laplace; approche tellement de la certitude, que le resultat observe devient invraisemblable dans cette hypothese ; ce resultat indique done avec une trSs-grande probability, 1' existence d'une cause primitive qui a dlter- min6 les mouvemens des plangtes ^ se rapprooher du plan de I'gcliptique, ou plus naturellement, du plan de I'equateur solaire, et h, se mouvoir dans le sens de la rotation du soleil..,. Laplace then mentions other circumstances whicli strengthen his conclusion, such as the fact that the motion of the satellites is also in the same direction as that of the planets. A similar investigation applied to the observed comets does, not give any gi'ound for suspecting the existence of a primitive cause which has affected the inclination of their planes of motion to the plane of the ecliptic. See however Cournot's Ewposition de la Th^orie des Chances . . . page 270. Laplace's conclusion with respect to the motions of the planets has been accepted by very eminent writers on the subject ; for example by Poisson : see his Recherches sur la Prob. . . . page 302. But on the other hand two most distinguished philosophers have' recorded their dissatisfaction ; see Professor Boole's Laius of Thought, page 364, and a note by K. L. ElUs in The Works of Francis Bacon... Vol. I. 1857, page 343. 988. Laplace devotes his pages 262 — 274 to a very remark- able process and examples of it ; see Art. 892. The following is his enunciation of the problem which he solves : Soient i quantit6s variables et positives t, *,,.--^i-i dont la somme soit S, et dont la loi de possibilit6 soit connue; on propose de trouver la somme des produits de chaque valeur que pent reoevoir une fonction. donn6e, i^ {t, «,, t^, ifec.) de ces variables, multiplige par la probabilit6 correspondante % cette valeur. The problem is treated in a very general way; the laws of possibility are not assumed to be continuous, nor to be the same for the different variables. The whole investigation is a charac- teristic specimen of the great powers of Laplace, and of the brevity and consequent difficulty of-his expositions of his methods. . Laplace applies his result to determine the probability that the sum of the errors of a given number of observations shall li^ between assigned limits, supposing the law of the facility of error in LAPLACE. 545 a single observation to be known : Laplace's formula when applied by bim to a special case coincides with that which we have given in Art. 567 from Lagrange. 989. An example is given by Laplace, on his page 271, which we may conveniently treat independently of his general investi- gation, with which he himself connects it. Let there be a number n of points ranged in a straight line, and let ordinates be drawn at these points ; the sum of these ordinates is to be equal to s : moreover the first ordinate is not to be greater than the second, the second not greater than the third, and so on. Required the mean value of the r*^ ordinate. Let s, denote the first ordinate, let s^ + z^ denote the second, 2i + 22+ »3 the third, and so on : thus s^, z^, z^, ... a„ are all posi- tive variables, and since the sum of the ordinates is s we have nz,+ {n -l)s^+(n-2)z,+ ...+z„ = s (1). The mean value of the r^^ ordinate will be /// {z^ + s^+ ... +z^dz^dz^...dz^ jjj dz^dz^...dz^ where the integrations are to be extended over all positive values of the variables consistent with the limitation (1). Put w»j = (Cj, {n —V)z^ = tCj, and so on. Then our expression becomes \\\... f^' + ^ + A + ... + "'r \ ^^ax,...dx„ JJJ \n n — 1 n — 2 n — r + lj ' " jjj...dxj^dx^... dx„ with the limitation x^ + x^+ ... +x„=-s (2). The result then follows by the aid of the theorem of Lejeune Dirichlet : we shall shew that this result is sfl 1_ 1_ 1 [ n\n n-1 n-2'^"''^n-r-\-l]' 35 546 LAPLACE. For let us suppose that instead of (2) we have the condition that x^ + a!^+... + x„ shall lie between s and s + As. Then by the theorem to which we have just referred we have ///• .^,^....^..(i±M •'^"•"""i"""* •••"""" \n + l and jjj dx^dx^.,.d9!„= r^ Hence by division we obtain /// . . . xjlx^ dx^... dx„ ^^ ^ ^^^„+, _ ^„. dxdx dx («+^«r-«'' ■«+!■ The limit of this expression when As is indefinitely diminished ///■ s is -. Then by putting for m in. succession the values 1, 2, ... r, we obtain the result. Laplace makes the following application of the result. Sup- pose that an observed event must have proceeded from one of n causes A,B,G,...; and that a tribunal has to judge from which of the causes the event did proceed. Let each individual arrange the causes in what he considers the order of probability, beginning with the least probable. Then to the r**" cause on his list we must consider that he assigns the numerical value 111 n n—1 n—2 n -r- + l)' The sum of all the values belonging to the same cause, accord- ing to the arrangement of each member of the tribunal, must be taken ; and the greatest sum wiU indicate in the judgment of the tribunal the most probable cause. 990. Another example is also given by Laplace, which we will treat independently. Suppose there are n candidates for an office, and that an elector arranges them in order of merit ; let a denote the maximum merit : required the mean value of the merit of a candidate whom the elector places r* on his list. LAPLACE. 547 Let ?i, Jj, ... t„ denote the merits of the candidates, beginning with the most meritorious. The. problem differs from that just discussed, because there is now no condition corresponding to the sum of the ordinates being giv.en ; the elector may ascribe any merits to the candidates, consistent with -the conditions that the merits are in order, none being greater than that which imme- diately precedes it, and no merit being greater than a. The mean value of the merit of the f^ candidate will be jjj ...trdt^dt^...dt„ jjj...dtjt^...dt„ The integrations are to be taken subject to the following con- ditions : the variables are to be all positive, a variable t^ is never to be greater than the preceding variable t^^, and no variable is to be greater than a. Laplace's account of the conditions is not in- telligible ; and he states the result of the integration without explaining how it is obtaiined. We may obtain it thus. Put «„ = *„, «^i = «„+a;^,, t„_, = t^^ + x^^, ...; then the above expression for the mean value becordes jjj ... (x„ + x„_i+ ... + Xf) dxj_dx^ ... dx„ ///... ^. CCSCa • • < UftX/tt, with the condition that all the variables must be positive, and that x^ + x^+ ... + Xn must not be greater than a. Then we may shew in the manner of the preceding Article that the result is (n — r+l) a n + 1 Laplace suggests, in accordance with this result, that each elector should ascribe the number n to the candidate whom he thinks the best, the number n—1 to the candidate whom he thinks the next, and so on. Then the candidate should be elected who has the greatest sum of numbers. Laplace says, 35—2 548 LAPLACE. Ce mode d'61eotion serait sans doute le meilleur, si des considlratioTis 6traiig§res an in6rite n'influaient point soiivent sur le choix des 61ec- teurs, m^me les plus tonnStes, et ne les diterminaient point a placer aux derniers rangs, les candidats les plus redoutables h, celui qu'ils prg- ferent ; ce qui donne un grand avantage aux candidats d'un merite mediocre. Aussi I'explrience I'a-t-elle fait abandonner aux gtablissemens qui I'avaient adopte. It would be interesting to know where this mode of managing elections had been employed. The subject had been considered by Borda and Condorcet ; see Arts. 690, 719, 806. 991. Thus we close our account of the second Chapter of Laplace's work which we began in Art. 970 ; the student will see that comparatively a small portion of this Chapter is originally due to Laplace himself. 992. Laplace's Chapter III. is entitled Des his de la proha- bilitS, qui r&sultent de la multiplication indejinie des evenemens : it occupies pages 275 — 303. 993. The first problem is that which constitutes James Ber- noulU's theorem. We will reproduce Laplace's investigation. The probability of the happening of an event at each trial is p ; required the probability that in a given number of trials the number of times in which the event happens will lie between certain assigned limits. Let q = l-p and fj, = m + n; then the probabihty that the event wiU happen m times and fail ?i times in /m trials is equal to a certain term in the expansion of {p + qY, namely Now it is known from Algebra that if m and n vary subject to the condition that m + n is constant, the greatest value of the above term is when — is as nearly as possible equal to - , so that m and n are as nearly as possible equal to fip and /iq respectively. "We say as nearly as possible, because fip is not LAPLACE. 549 necessarily an integer, while m is. We may denote the value of m hy fip + z, whfere z is some proper fraction, positive or negative ;■ and then n= fiq —z. The r^ term, counting onwards, in the expansion of {p + q^ after -^»V is , ^ »'"-'o"^ jw \n^ ^ \m — r \n + r -^ ^ We shall now suppose that m and n are large numbers, and transform the last expression by the aid of Stirling's Theorem ; see Arts. 333, 962. We have [^ = /.-ie-V(27r)jl+3|^ + ...J, |7r+7 = ^" + '■^""'"^^"''7(2^ r ~ 12(n + r) ~ -J " We shall transform the term (to — r)'"""*. Its logarithm is (r-m-J)|logTO+.]og(l-£)|, and lGg(l-£)=-^-^,-^3- We shall suppose that r' does not surpass fi in order of mag- nitude, and we shall neglect fractions of the order - ; we shall r* thus neglect such a term as — g, because m is of the order /j-. Thus we have approximately ^^_^_l)|togTO + l0g(l-£)| and then, passing from the logarithms to the numbers, (to - J-)'"™-^ = to'-"-*.6'~5S ( 1 + __ _ -— J . 550 LAPLACE. Similajrly Thus we have approximately (»r' Now suppose that the values of m and n are those which we have already assigned as corresponding to the greatest term of the expansion of (p + qTi then m—z n+z thus we have approximately ^ ^ /A" V mnj Therefore finally we have approximately for the r"" term after the greatest We shall obtain the approximate value of the r* term before the greatest by changing the sign of r in the above expression ; by adding the values of the two terms we have A/(27r»rtn) If we take the sum of the values of this expression from r = to r = r, we obtain approximately the sum of twice the greatest term of a certain binomial expansion together with the r terms which precede and the r terms which follow the greatest term ; subtract the greatest term, and we have the approximate value of the sum of 2r + I terms of a binom,ial expansion which include the greatest term as their middle term. Now by Euler's theorem, given in Art. 334, ty=jydr-p + Y<^^ dy y^YLdr LAPLACE. 551 Here y= ,,„ , e"g^, and the differential coefficients of «/ with respect to r will introduce the factor ^— , and its powers; and ^-— is of the order -p at most, so that when multiplied by the constant factor in y we obtain a term of the oraer — . Thus as far as we need proceed, ^:!/ = jydr~^y + ^Y, where both the symbols % and I are supposed to Indicate opera- tions commencing with r = 0, and ^ Y denotes the greatest term I of the binomial expansion, that is the value of 5 y when r = 0. The expression 2y denotes as usual the sum of the values of y up to that corresponding to r — 1; adding the value of y correspond- ing to r we obtain /" 1 1 jydr+^y + :^Yi subtract the greatest term of the binomial, and thus we have /^ ydr + ^y. Put T = ,,^ s ; thus we obtain finally This expression therefore is the approximate value of the sum of 2r + 1 terms of the expansion of (p -(- q)v; these terms including the greatest term as their middle term. In the theory of proba- bility the expression gives the probability that the number of times in which the event will happen in /* trials will lie between m—v and m + r, both inclusive, that is between f^P^'- V/^ ^and/.p + ^-H— ^^ 552 LAPLACE. or, in other words, tlie expression gives the probability that the ratio of the number of times in which the event happens to the whole number of trials will lie between z T V(2«jm) , ,B T (2mw) ^x-i,r, and Zx,r+i, the equation becomes dJJ ^(^ 1\ 7-r « /-. 2\ dU 4 4\ d^U /, i^ 4 4\ d^U \ n n wj dfir :and thus the error seems to be of the order -, or even larger, since ft" may be as great as n. 1000. Laplace proceeds to integrate his approximate equation by the aid of definite integrals. He is thus led to investigate some auxiliary theorems in definite integrals, and then he passes on to other theorems which bear an analogy to those which occur in connexion with what are called Laplace's Functions. We will give two of the auxiliary theorems, demonstrating them in a wa,y which is perhaps simpler than Laplace's. To shew that, if i is a positive integer, / e-^-'"(s + fiJ^*dsdii, = 0. Transform the double integral by putting s = r cos 0, /i = r sin 0; we thus obtain r Te^ (cos ie + 7=a sin i 0) r*^' dr d0. •'o •' 0. It is obvious that the positive and negative elements in this integral balance each other, so that the result is zero. Again to shew that, if i and q are positive integers and q less than i, 560 LAPLACE. *' -00 *' -00 Transforming as before we obtain I 6-^ (cos ie+ nTTi sin z 0) sin« 6 r^^^ £?r cZ6'. Jo ""0 Now sin' 5 may be expressed in terms of sines or of cosines of multiples of 6, according as q is odd or even, and the highest multiple of 6 will be qd. And we know that if m and n are unequal integers we have sin m6 cos n9 d9 = 0, / J / I J a 2ir cos mO cos nd dd = 0, 2ir sin md sin n9 dO = 0\ thus the required result is obtained. Laplace finally takes the same problem as Daniel Bernoulli had formerly given ; see Ai-t. 420. Laplace forms the differential equations, supposing any number of vessels ; and he gives without demonstration the solutions of these differential equations : the demonstration may be readily obtained by the modern method of separating the symbols of operation and quantity. 1001. Laplace's Chapter IV. is entitled, De la prohabiliU des erreurs des risultats moyens d'un grand nomhre d! observations, et des rSsultats moyens les plus avantageux : this Chapter occupies pages 304.— 348. This Chapter is the most important in Laplace's work, and perhaps the most difficult ; it contains the remarkable theory which is called the method of least squares. Laplace had at an early period turned his attention to the subject of the mean to be taken of the results of observations ; but the contents of the pre- sent Chapter occur only in his later memoirs. See Arts. 874, 892, 904, 917, 921. Laplace's processes in this Chapter are very peculiar, and it is scarcely possible to understand them or feel any confidence in LAPLACE. 561 their results without translating them into more usual mathema- tical language. It has been remarked by R Leslie Ellis that, " It must be admitted that there are few mathematical investiga- tions less inviting than the fourth Chapter of the ThSorie des ProbdbiliUSj which is that in which the method of least squares is proved." Cambridge Phil Trans. Vol. viii. page 212. In the Connaissance des Terns for 1827 and for 1832 there are two most valuable memoirs by Poisson on the probability of the mean results of observations. These memoirs may be de- scribed as a commentary on Laplace's fourth Chapter. It would seem from some words which Poisson uses at the beginning — j'ai pensd que les remarques que j'ai eu I'occasion de faire en r^tudiant, — that his memoirs form a kind of translation, which he made for his own satisfaction, of Laplace's investigations. Poisson embodied a large part of his memoirs in the fourth Chapter of his Mecherohes sur la Prdb We shall begin our account of Laplace's fourth Chapter by giving Poisson's solution of a very general problem, as we shall then be able to render our analysis of Laplace's processes more intelligible. But at the same time it must be remembered that the merit is due almost entirely to Laplace; although his pro- cesses are obscure and repulsive, yet they contain all that is essential in the theory : Poisson follows closely in the steps of his illustrious guide, but renders the path easier and safer for future travellers. 1002. Suppose that a series of s observations is made, each of which is liable to an error of unknown amount ; let these errors be denoted by e^, e^, ... e^. Let E denote the sum of these errors, each multiplied by an assigned constant, say •^ = TA + %e^ + 73^3+ ••• +7A : required the probability that E will lie between assigned limits. Suppose that each error is susceptible of various values, posi- tive or negative, and that these values are all multiples of a given quantity co. These values will be assumed to lie between aw and /Sea, both inclusive ; here a and ^ will be positive or negative integers, or zero, and we shall suppose that a is algebraically 36 562 LAPLACE. greater than ^, so that a-/8 is positive. The chance of an as- signed error will not be assumed the same at each observation. If w be any integer comprised between a and ^ we shall denote the chance of an error nm at the first observation by N^, at the second observation by N^, at the third observation by N^, and so on. Let tn- be a factor such that all the products tsry^, OTy^, ■nr7g, . . . OT7, are integers ; such a factor can always be found either .exactly or to any required degree of approximation. Let where S denotes a summation with respect to n for aU values from /S to a, both inclusive ; and let then the probability that istE will be exactly equal to mai, where m is a given integer, is the coefficient of f^" in the development of T according to powers of t ; or, which is the same thing, the probability is equal to the term independent of t in the develop- ment of Tr"*". For t" put e*"^ and denote by X what T becomes ; then the required probability is equal to 1 r xe-^^-^-^de. 27r Let X and fi be two given integers, such that X — /i is positive ; then the probability that -aTE will lie between Xoj and jjuto, both inclusive, may be derived from the last expression by putting for m in succession the values fju, ji + l, /i + 2, ... X, and adding the re- sults. Since the sum of the values of e~'"*'^-i jg 7=^1 2sin-;^ 2 i_ fe-(x+4)flV=T _ g-(f.-4)eV=il the required probability is equal to m we shall denote this probability by P. 47r LAPLACE. 563 Let us now suppose that « is indefinitely small, and that A, and fi are infinite ; and let Xw = (c + '"i) '^, li-to = {c — rj) w, ■taO = cox. The limits of the integration with respect to x will be + oo . Also we have dQ = — ax, sm t; y = j;— . Thus, neglecting + ^ compared with \ and /t, we obtain At ■4/: „vrr • - zz)f, {z)fi {£) dz dz. Hence, by addition, ^h^=n\z-z'rfi{z)f,{z)dzdz'. J b J b Thus 4A/ is essentially a positive quantity which cannot be zero, for every element in the double integral is positive. It is usual to call_;^ {z) the function which gives the facility of error at the ^■"' observation ; this means that_^(z) dz expresses the chance that the error will lie between s and z + dz. If the function of the facility of error be the same at every LAPLACE. 567 observation we shall denote it by f{z) ; and then dropping those suflSxes which are no longer necessary, we have . h = I V(«) dz, h' = I V(s) dz, Such is the solution which we have borrowed from Poisson ; he presents his investigation in slightly varying forms in the places to which we have referred : we have not adopted any form ex- clusively but have made a combination which should be most ser- viceable for the object we have in view, namely, to indicate the contents of Laplace's fourth Chapter. " Our notation does not quite agree with that which Poisson has employed in any of the forms of his investigation; we have, for example, found it expedient to interchange Poisson's a and h. We may make two remarks before leaving Poisson's problem. I. We have supposed that the error at each observation lies between the same limits, a and h ; but the investigation will apply to the case in which the limits of error are different for different observations. Suppose, for example, it is known at the first observation that the error must lie between the limits a^ and \, which are within the limits a and h. Then f^ {z) will be a function of z which must be taken to vanish for all values of z between h and \ and between a^ and a. Thus in fact it is only necessary to suppose that a and 6 are so chosen, that no error at any observation can be algebraically greater than a or less than h. II. Poisson shews how to proceed one step further in the ap- proximation. We took y = lx; we have more closely y=lx — l^, where Hence, approximately, cos {y — ca^ = cos (te — cx)-V l^a? sin {Ix — ex). 568 LAPLACE. Therefore (2) becomes P = 2 r - — e TT J COS (ta; — ex) sin ija; — X TT J -K% sin {Ix — ccc) a;^ sin 7;a; <^a3. We formerly transformed the first term in this expression of P; it is sufficient to observe that the second term may be derived from the first by differentiating three times with respect to I and multiplying by ^ ; so tbat a transformation may be obtaiaed for the second term similar to that for the first term. 1003. Laplace gives separately various cases of the general result contained in the preceding Article. We will now take his first case. Let y^ = y^= ... =y,= l. Suppose that the function of the facility of error is the same at every observation, and is a constant; and let the limits of error be + a. Then /: f{z)dz = l. If G denote the constant value of /(z) we have then 2aC=l. Here Jc = 0, k'=—^ = ^, ^'=|-, Z = 0, K^=^h-'tr,t = sF=~. Letc = 0; then by equation (4) of the preceding Article the probability that the sum of the errors at the s observations will lie between — 17 and 17 V6 p -,^.^_ V6 3»2 Let — 2 = i^; then the probability that the sum of the errors will lie between — ra v'* and ra i/s VTJ„ '^ dt. This will be found to agree with Laplace's page 305. LAPLACE. 569 1004. We take Laplace's, next case. Let 7 = 7„ = . . . = 7s = 1- I^st the limits of error be + a ; sup- pose that the function of the facility of error is the same at every observation, and that positive and negative errors are equally likely : thus / (— x) =/(«). Here k = 0, h' = \k', 1=0, i<:' = ^k'. By equation (4) of Article 1002 the probability that the sum of the errors at the s observations will He between —r) and rj IS s> rv — ^ dv. 2 /•" V(2s4'7r)i/ This will be found to agree with Laplace's page 308. We have k' = f ^^(s) dz=2 f%'f{z) dz, J —a Jo and 1 = r f{z) de=2 ["/{z) dz ; J -a Jo hence if/(s) always decreases as z increases from to a we see, as 2 in Alt. 922, that k' is less than -^ . 1005. Laplace next considers the probability that the sum of the errors in a large number of observations will he between certain limits, the sign of the error being disregarded, that is all eiTors being treated as positive; the function of the facility of error is supposed to be the same at every observation. Since all errors are treated as positive, we in fact take nega- tive errors to be impossible ; we must therefore put 5 = in Poisson's problem. Take 7, = 7^ = . . . = 7, = 1. Then l = sk, ii'=^{k'-k'). Take c = l; then, by equation (4) of Art. 1002, the probability that the sum of the errors will lie between 1—7} and I + t) ia 570 LAPLACE. 2 mi' , e 2«;;/-»')( V{257r(A'-^-0}J„' This -will be foand to agree with Laplace's page 311. For an example suppose that the function of the facility of error is a constant, say C ; then since •J a we have aC=l. Thus ^ = 2' ^'^"3' ^'~^'^12' Therefore the probability that the sum of the errors will lie , , sa , sa between -^ — v and -^ +i) is 2V6 a i^lisir) L 1 _?f 1006. Laplace next investigates the probability that the sum of the squares of the errors will lie between assigned limits, sup- posing the function of the facility of error to be the same at every observation, and positive and negative errors equally likely. In order to give the result we must first generalise Poisson's problem. Let i{z)fi{s)dz, J b and 2«» = S J" !i + vCi+ (7). 37—2 580 LAPLACE, Let ji stand for ^^ ; tlien from (7) we can deduce the follow- ing system of equations : 1 = Xtaiji + fitaibji + v%afiji + ... = XZaihJi + iJ-thiji + v^hiCiji + ... = Xtafiiji + /jiZhiCiji + vtcfji + ... ,(8). To obtain the first of equations (8) we multiply (7) by aejij, and then sum for all values of i paying regard to (6) ; to ob- tain the second of equations (8) we multiply (7) by b^Jt and sum ; to obtain the third of equations (8) we multiply (7) by Ciji and sum ; and so on. The number of equations (8) will thus be the same as the number of conditions in (6), and therefore the same as the number of arbitrary multipliers \, /m, v, ... Thus equations (8) will determine X, /j,, v, . . . ; and then from (5) we have x = ty^2^+l (9). We shall now shew how this value of x may practically be best calculated. Take s equations of which the type is ttix' + biy' + dz' + =qi + ki. First multiply by aj^ and sum for all values of t ; then mul- tiply by bji and sum ; then multiply by cji and sum ; and so on : thus we obtain the following system x'ta^ji + ytahji + ^tafiiji -1- . . . = X (^-^ -1- A;,) a J^ a!tafiij\ + y'tbfiij\ + z'tc^j\ + ...=% (qi + Ja) Cij\ (10). Now we shall shew that if x' be deduced from (10) we shall have x' = Xyiqt + I, and therefore x = x'. For multiply equations (10) in order by \, fj,,v, ... and add; then by (8) LAPLACE. 681 a! = XS {qi + ^i) aji + /^X (g-* + ^i) SJi + vt (g'j + A;j)cji + . . . = 2 fe + ii)y* {XMi + /i6j + vcj + . . .} =.S7ife + ^i) by (7). The advantage of using equations (10) is twofold; in the first place we determine x, and thence x, by a systematic process, and in the next place we see that the equations (10) are sym- metrical with respect to x, y', z', ...: thus if we had proposed to find y, or z, or any of the other unknown quantities instead of X, we should, by proceeding in the same manner as we have already, arrive at the same system (10). Hence the same ad- vantage which we have shewn by the Theory of Probability to belong to the value of x by taking it equal to x, will belong to the value of y by taking it equal to y', and to the value of z by taking it equal to s', and so on. In fact it is obvious that if we had begun by investigating the value of y instead of the value of x the conditions (6) would have been changed in such a manner as to leave the proportion of the factors 7,, 7^, 7g, ... unchanged; and thus we might have anticipated that a sym- metrical system of equations like (10) could be formed. We have thus shewn how to obtain the most advantageous values for the required quantities x,y,z, ... Suppose now that we wished to find the values of x', y' , s, ... which render the foUow.ing expression a minimum, Sy* [aiX + %' + Ci»' + ... - g-i - hf; it will be found that we arrive at the equations (10) for deter- mining X, y, z ... Hence the values which have been found for x,y,z,... give a minimum value to the following expression Si(6,-A;,)Mhatis t{^^-^ If hi be zero, and Aj constant, for all values of i, the values which have been found for x,y, z, ... render the sum of the squares of the errors a minimum : as in Art. 1007 these conditions will hold if the function of the facility of error is the same at every ob- servation, and positive and negative errors are equally likely. 582 LAPLACE. Thus we have completed one mode of arriving at the result, and we shall now pass on to the other. If we proceed as in the latter part of Art. 1007 we shall find that the probability that the error in the value of x, when it is determined by (5), lies between t and t+dtis i^dt (11). For put c = 5? in equation (4) of Art. 1002. Then the proba- bility that S7i6i will lie between and 2?) Thus the probability that tyiSi will lie between t and r + dr is -e ^' dr, and therefore the probability that SYiCj will lie between Z + t and Z + t' + {a^- x) . (J3(a^- x) . ^ {a^- x). ... Then, by the ordinary principles of inverse probabiHty, the pro- bability that the true value lies between x and x + dx is Pdx I- Fdx the integral in the denominator being supposed to extend over all the values of which x is susceptible. Let She such that, with the proper limits of integration, HJPdx=l, and let y = S^ («i - x) . ^ {a^- x) . <]) {a^- x) . ... Laplace conceives that we draw the curve of which the ordi- nate is y corresponding to the abscissa x. He says that the value which we ought to take as the mean result of the observations is that which renders the mean error a minimum, every error being considered positive. He shews that this corresponds to the point the ordinate of which bisects the area of the curve just drawn ; that is the mean result which he considers the best is such that the true result is equally likely to exceed it or to fall short of it. See Arts. 876, 918. Laplace says, on his page 335, Des g^omStres calibres out pris pour le milieu qu'il faut ohoisir, celui qui rend le r6sultat observe, le plus probable, et par consequent LAPLACE. 585 I'abscisse qui r^pond k la plus grande ordonn^e de la oourbe ; mais le milieu que nous adoptons, est Ividemment iiidiqu6 par la th^orie des probabilitls. This extract illustrates a remark which we have already made in Art. 1008, namely that strictly speaking Laplace's method does not profess to give the most probable result but one which he con- siders the most advantageous. 1014. Laplace gives an investigation in his pages 335 — 340 which amounts to solving the following problem : if we take the average of the results furnished by observations as the most pro- bable result, and assume that positive and negative errors are equally likely and that the function of the facility of error is the same at every observation, what function of the facility of error is implicitly assumed 1 Let the function of the facility of an error a be denoted by e-"f'W, which involves only the assumption that positive and nega- tive errors are equally likely. Hence the value of y in the pre- ceding Article becomes ffe-', where Jr' (x - a^' = 0. The average value in this case is ta^ +(s — i) a^ 8 Substitute this value of x in the equation, and we obtain 586 LAPLACK This cannot hold for all values of - and a^ - a^ unless ■<^' (s) be s independent of z ; say T|r' (z) = c. Hence i^ (a) =cz + c', where c and c' are constants. Thus the function of the facility of error is of the form Ce~'"'' ; and since an error must lie between — oo and oo, we have J —00 therefore C = -7— . The result given by the method of least squares, in the case of a single unknown quantity, is the same as that obtained by taking the average. For if we make the following expression a minimum ix-a,y+ (a;-a,y+ ... + (x-a.y we obtain s Hence the assumption in the preceding investigation, that the average of the results furnished by observations will be the most probable result, is equivalent to the assumption that the method of least squares will give the most probable result. 1015. Laplace devotes his pages 340 — 342 to shewing, as he says, that in a certain case the method of least squares becomes necessary. The investigation is very simple when divested of the cumbrous unsymmetrical form in which Laplace presents it. Suppose we require to determine an element from an assem- blage of a large number of observations of various kinds. Let there be s^ observations of the first kind, and from these let the value ttj be deduced for the unknown quantity; let there be s^ observations of the second kind, and from these let the value a^ be deduced for the unknown quantity ; and so on. Take x to represent a hypothetical value of the unknown quan- tity. Assume positive and negative errors to be equally likely; then by Ai-t. 1007 the probability that the error of the result deduced from the first set of observations will lie between x — a^ and x + dx — a,is -^ e-^''^*""!'' dx. LAPLACE. 587 Here /3,' stands for ^J^Tp) ^^ tlie value of )8j will therefore depend on the values of the factors 7^, 7^, ... which we employ; for example we may take each of these factors equal to unity, which amounts to adopting the average of the results of observation ; or we may take for these factors the system of values which we have caUed the most advantageous system : if we adopt the latter we findA'=^2g. Similarly the probahiHty that the error of the result deduced from the second set of observations will lie between x — a^ and a;+efe-a„ is -^ e-^^'-'^^dx. And so on for the other sets of observations. Hence we shall find, in the manner of Aft. 1013, that the pro- babUity that x is the true value of the unknown quantity is pro- portional to e-", where o- = p,' {x - a^ + ^/ {x - a,)= + /3/ (x - a,)' + . . . Now detei-mine x so that this probabiHty shall have its greatest value ; o- must be a minimum, and we find that x = We may say then that Laplace obtains this result by deducing a value of the unknown quantity from each set of observations, and then seeking for the most probable inference. If Aj, a^, a,, ... are determined by the most advantageous method, this result is similar in form to that which is given in Art. 1007, if we suppose that positive and negative errors are equally likely, and that one function of facility of error applies to the first set of observations, another function to the second set, and so on. For the numerator of the value of x just given corresponds with the 't^, and the denominator mth the tj-t of Art. 1007. 588 LAPLACE. 1016. Laplace gives some remarks on his pages 343 — 348 relative to another method of treating errors, namely, that which consists in making the sum of the 2n*^ powers of the eiTors a minimum, n being supposed indefinitely great. He explains this method for the case of one unknown quantity, and he refers to the Mecanique Celeste, Livre ill. for the case in which there is more than one unknown quantity. The section intended of Livre ill. must be the 39th, in which Laplace gives some rules as in the present place, but without connecting his rules with the con- sideration of infinite powers of the errors. Another method is given in the next section of the Mecanique Celeste which Dr Bowditch in a note on the passage ascribes to Boscovich : Laplace takes up this method in the second Supplement to the Theorie...des Prdb., where he calls it the method of situation. 1017. Laplace gives on his pages 346 — 348 some account of the history of the methods of treating the results of observations. Cotes first proposed a rule for the case in which a single element was to be determined. His rule amounts to taking 'yi = 7i! = ---=7»=l in Art. 1007, so that Mi Laplace says that the rule was however not employed by mathe- maticians until Euler employed it in his first memoir on Jupiter and Saturn, and Mayer in his investigations on the libration of the moon. Legendre suggested the method of least squares as convenient when any number of unknown quantities had to be found ; Gauss had however previously used this method himself and communicated it to astronomers. Gauss was also the first who endeavoured to justify the method by the Theory of Proba- bihty. We have seen that Daniel Bernoulli, Euler, and Lagrange had studied the subject : see Arts. 424, 427, 556. Lambert and Bos- covich also suggested rules on the subject ; see the article Milieu, of the EncydopSdie Methodique and Dr Bowditch's translation of the Mecanique Celeste, Vol. ii. pages 434, 435. LAPLACE. 589 The titles of some other memoirs on the subject of least squares will be found at the end of the Treatise on Probability in the Encyclopcedia Britcmnica ; we would also refer the student to the work by the Astronomer Koyal On the Algebraical and Nmnerical Theory of Hrrors of Observations and the combination of Observa- tions. 1018. Laplace's fifth Chapter is entitled Application du Calcul des Probabilites, ci la recherche des phenomhnes et de leurs causes : it occupies pages 349 — 362. The example with which Laplace commences will give a good idea of the object of this Chapter. Suppose that observations were made on 400 days throughout which the height of the barometer did not vary 4 millimetres ; and that the sum of the heights at nine in the morning exceeded the sum of the heights at four in the afternoon by 400 millimetres, giving an average excess of one millimetre for each day : required, to estimate the probability that this excess is due to a constant cause. We must examine what is the probability of the result on the supposition that it is not due to any constant cause, but arises from accidental perturbations and from errors of ob- servation. By the method of Art. 1004, supposing that it is equally pro- bable that the daily algebraical excess of the morning result over the afternoon result wiU be positive or negative, the probability that the sum of s excesses wiU exceed the positive quantity c e ^**' dv = \ r V(2A;W) Jo 1 r = —r- e"^ dt, where t = V(2s&') Hence the probability that the sum will be algebraically less than c is 1 r 1 — f- e-t'df. 590 LAPLACE. Now, as in Art. 1004, we may take -^ as the greatest value of Te, so that the least value of r is - — tt^tt- ; also a = 4, c = 400, s = 400 : thus the least value of t is -~ , that is /^(STS). 1 r°° Hence 1 — j- I er*'dt is found to be very nearly equal to unity. We may therefore regard it as nearly certain that the sum of the excesses would fall below 400 if there were no constant cause : that is we have a very high probability for the existence of a constant cause. 1019- Laplace states that in like manner he had been led by the theory of probabiHties to recognise the existence of con- stant causes of various results in physical astronomy obtained by observation; and then he had proceeded to verify the existence of these constant causes by mathematical investigations. The remarks on this subject are given more fully in the Introduction, pages LVII — Lxx ; see Art. 938. 1020. Laplace on his pages 359 — 362 solves Buffon's problem, which we have explained in Art. 650. Suppose that there is one set of parallel lines ; let a be the distance of two consecutive straight lines of the system, and 2r the length of the rod : then the chance that the rod wiU fall across a line is — , Hence, bv Art. 993, if the rod be thrown down a very large number of times we may be certain that the ratio of the number of times in which the rod crosses a line to the whole number of trials wUl be very nearly — : we might therefore determine by experiment an approximate value of tr. Sr Laplace adds... et il est facile de voir que le rapport — 'qui, pour un nombre donn^ de projections, rend l*erreur a craindre la plus petite, est I'unit^. . . Laplace seems to have proceeded thus. Suppose^ the chance of the event in one trial j then, by Art. 993, LAPLACE. 591 the probability that in /* trials the number of times in which the event happens will lie between ^H — T V2/i^ (1 —p) and pfi + t V2/ip (1 — p) 2 f'' IS approximately -j- e~^'dt. Hence to make the limits as close as possible we must have p {1-p) as small as possible, and thus ^ = 5 ■ This, We say, ap- pears to have been Laplace's process. It is however wrong ; for P 0- ~p) is a mcuximum and not a minimum, w'hen p = -5 . More- over we have not to make t 'J'i.yi-p (1 — p) as small as possible, but the ratio of this expression to pfi. Hence we have to make VjB (1 — ip) . 1 -^-^ ^ as small as possible ; that is we must make 1 as small as possible : therefore p must be as great as possible. In the present case p= — ; we must therefore make this as great as possible : now in the solution of the problem 2r is assumed to be not greater than a, and -therefore we take 2r =-a as the most favourable length of the rod. Laplace's error is pointed out by Professor De Morgan in Art. 172 of the Theory of Prohahilities in the Encychpcedia Metropolitana. The most curious point however has I believe hitherto been unnoticed, namely, that Laplace had the correct result in his first edition, where he says ...et il est fecUe de voir 2r que le rapport — qui, pour un nombre donn^ de .projections, rend I'erreur h, craindre la plus petite, est I'unitd . . . The original leaf was cancelled, and a new leaf inserted in the second and third editions, thus causing a change from truth to error. See Art. 932. Laplace solves the second part of Buffon's problem correctly, in which Buffbn himself had failed; Laplace's solution is much less simple than that which we have given in Art. 650. 592 LAPLACE. 1021. Laplace's sixth Chapter is entitled De la prdbaUliU des causes et des 4-o4nemens futurs, tiree des Svenemena observSs: it occupies pages 363 — 401. The subject of this Chapter had engaged Laplace's attention from an early period, and to him we must principally ascribe the merit of the important extension thus given to the Theory of Probability, due honour being at the same time reserved for his predecessor Bayes. See Arts. 851, 868, 870, 903, 909. Let X denote the chance, supposed unknown, of a certain simple event ; let y denote the chance of a certain compound event depending in an assigned manner on this simple event: then y will be a known function of x. Suppose that this com- pound event has been obsei-ved; then the probability that the chance of the simple event lies between a. and /S is r^ ydx /' J a L y dx This is the main formula of the present Chapter: Laplace applies it to examples, and in so doing he evaluates the integrals by his method of approximation. In like manner if the compound event depends on two inde- pendent simple events, the probability that the chance of one lies between a, and /3 and the chance of the other between a' and /3' is ydx'dx n ny dx dx 1022. The examples in the present Chapter of Laplace's work exhibit in a striking way the advantage of his method of approxi- mation ; but as they present no novelty nor difficulty of principle we do not consider it necessary to reproduce any of them in detail. 1023. Laplace makes a remark on his page 366 which may deserve a brief examination. He says that if we have to take the integral je-^dt between the limits — t and t we may for an ap- LAPLACfi. 593 proximation take the' integral between the limits and • / ( — ^ — j , and double the result : he says this amounts to neglecting the square of t" — t". We may put the matter in the following form : suppose that a and h are positive, and we require x such that^ e-«V<+ e-^dt = 2\ e'^ dt. •'0 *' "^ Suppose a less than 6 ; then in fact we require that re-<^dt^p J a J x ■''dt. Laplace, in effect, tells us that we should take 35= a /( — 5 — ) as an approximation. He gives no reason however, and the more natural approximation would be to take a; = s (a + b), and this is certainly a better approximation than his. For since the function e~^ decreases as t increases, the true value of x is less than jj (a + S), while Laplace's approximation is greater than „ (a + h). 1024. Laplace discusses on his pages 369 — 376 a problem re- lating to play ; see Art. 868. A and £ play a certain number of matches ; to gain a match a player must win two games out of three ; having given that A has gained i matches out of a large number n, determine the probability that A's skill lies within as- signed limits. If a player wins the first and second games of a match the third is not played, being unnecessary; hence if n matches have been played the number of games must lie between 2« and 3n : Laplace investigates the most probable number of games. 1025. Laplace discusses in his pages 377 — 380 the problem which we have enunciated in Art. 896. The required proba- bility is ' x'il-xydx 1} I x^{l-oifdx where p and q have the values derived from observations during 88 594 LAPLACE. 40 years ; these values are given in Art. 902. Laplace finds that the probability is approximately 1 - 0030761 where /* is a very large number, its logarithm being greater than 72. Thus Laplace concludes that the probability is at least equal to that of the best attested facts in history. With respect to a formula which occurs in Laplace's solution see Art. 767. With respect to an anomaly observed at Vitteaux see Arts. 768, 769. 1026. Laplace discusses in his pages 381 — 384 the problem which we have noticed in Art. 902. He offers a suggestion to account for the observed fact that the ratio of the number of births of boys to girls is larger at London than at Paris. 1027. Laplace then considers the probability of the results founded on tables of mortality : he supposes that if we had observa- tions of the extent of Ufe of an infinite number of infants the tables would be perfect, and he estimates the probability that the tables formed from a finite number of infants will deviate to an assigned extent from the theoretically perfect tables. We shall hereafter in Art. 1036 discuss a problem like that which Laplace here considers. 1028. A result which Laplace indicates on his page 390 sug- gests a general theorem in Definite Integrals, which we will here demonstrate. Let «" = let e""' be integrated with respect to each of the n—1 variables 0j, a^, ...2^,, between the limits - oo and oo : then the result will be it-i 7 «» a»-i an-. a. "-1 LAPLACE. 595 Let us consider first the integration with respect to s^; we have aX + < {\ - h^rY = W + 2 _ 2 2 .2 , '„2i2 . that is, by fi\\ By proceeding in this way it is obvious that we shall arrive at the assigned result. 1029. Laplace devotes his pages 391 — 394 to a problem which we have indicated in Art. 911. The problem resembles that which we have noticed in Art. 1027, and the mode of solution will be illustrated hereafter in Art. 1036. The problems which Laplace considers in his pages 385 — 394 relate to the probabilities of future events ; and thus these pages are strangely out of their proper place : they should ha.ve followed the discussion which we are about to analyse in our next Article, and which begins thus, Considirons maintenant la probdbiliU des ivSnemens futurs, tir^e des 4v6nemens observes, 1030. Laplace considers in his pages 394 — 396 the impor- tant subject of the probability of future events deduced from observed events : see Arts. 870, 903, 909. Retaining the, notation of Art. 1021, suppose that z, which is a known function of x, represents the chance of some compound future event depending on the simple event of which x represents the chance : then the whole probabiUty, P, of this future event will be given by I yzdx •J a P = ' f •I ydx Laplace then suggests approximations for the integrals in the LAPLACE. 697 above expression. We will reproduce the substance of his remarks. In Art. 957 we have «' = logr-log^(a+^) = log Y- log 1^ (a) + e (a), and ^' (a) = 0, by hypothesis. Thus approximately Hence if y vanishes when x = and when x = l, we have approximately y M-i*/ V (~ rfaV Similarly if we suppose that yz is a maximum when x = a', and that then yz = Y'Z', we have Suppose that « is a function of y, say z = ^ (y), then yz is a maximum when y is a maximum, so that a' = a; and since — — = 0, we find that Hence we have approximately VFW 1031. Laplace discusses on his pages 397 — 401 the following problem. It has been observed during a certain number of years at Paris that more boys than girls are annually baptised : deter- mine the probability that this superiority will hold during a cen- tury. See Art. 897. 598 LAPLACE. Let p be the observed number of baptisms of boys during a certain number of years, q the observed number of baptisms of girls, 2n the annual number of baptisms. Let x represent the chance that an infant about to be born and baptised will be a Let (a3 + 1 — xf" be expanded in a series x'" + 2»»a='^»-' (1 - x) + ^"^^""^^ '^'""' {l-xy+...; then the sum of the first n terms of this series will represent the probability that in a year the number of baptisms of boys will predominate. Denote this sum by f ; then f* will be the probability that the superiority wiU be maintained during i years. Hence we put x" (1 — xY for y and ^* for z in the formula of the preceding Article, and obtain \ x^il-xY^'dx fx^{l- ■ xy dx Laplace applies his method of approximation with great success to evaluate the integrals. He uses the larger values of p and c[ given in Art. 902 ; and he finds that P = '782 approximately. 1032. Laplace's seventh Chapter is entitled Be V influence des inigalitSs inconnues qui peuvent exister entre des chances que Von suppose parfaitement igales : it occupies pages 402 — 407. The subject of this Chapter engaged the attention of Laplace at an early period ; see Arts. 877, 881, 891. Suppose the chance of throwing a head with a coin is either or — ^ — , but it is as likely to be one as the other. Then the chance of throwing n heads in succession will be lf/l + a\ 2- " ' + that IS, gs j 1 + J 2 "^ T^ " + ■ ■ • r ■ XAPLACE. 599 Thus there is an advantage in undertaking to throw n heads in succession beyond what there would be if the coin were per- fectly symmetrical. Laplace shews how we may diminish the influence of the want of symmetry in a coin. Let there be two coins A and £; let the chances of head and taU in A he p and q respectively, and in B let them be p' and q respectively : and let us determine the probability that in n throws the two coins shall always exhibit the same faces. The chance required is {pp + qqf. Suppose that 1+a 1-a , 1+a' , 1-a' p = -^> i = —2r'' then (i'/ + 2'?r = ^(l + ««')•• But as we do not know to which faces the want* of symmetry is favourable, the preceding expression might also be ^s (1 — aa')" by interchanging the forms of p and q or of p' and q. Thus the true value wiU. be i|l(l + aar + |.(l -««')"}, that is If n{n-l) , ™ n{n-r){n-2){n-S) 1 rr+ 1.2 ""^ + II "" +-r It is obvious that this expression is nearer to -^ than that which was found for the probability of securing n heads in n throws with a single coin. 1033. Laplace gives again the result which we have noticed in Art. 891. Suppose p to denote ^'s skill, and q to denote B's skill; let A have originally a counters and B have originally h counters. Then ^'s chance of ruining B is 600 LAPLACE. Laplace puts for p in succession ^ (1 + a) and g (•'■ ~" ")' ^'^'^ takes half the sum. Thus he obtains for ^'s chance l{(l + a)''_(l_ar}{(l+a)»+(l-an 2 (l+a)'"^*-(l-«)°^ ■ Laplace says that it is easy to see that, supposing a less than h, this expression is always greater than , , which is its limit when a = 0. This is the same statement as is made in Art. 8.91, but the proof will be more easy, because the trans- formation there adopted is not reproduced. T> 1 + a Put ^i = X, 1 — a and u- gA^_i • We have" to shew that u continually increases as x increases from 1 to 00 , supposing that a is less than h. It will be found that Idu aa!"(x'^-l)-5a^(a!^-l) udx~ x{x''- 1) (x* + 1) (a;"^- 1) " We shall shew that this expression cannot be negative. We have to shew that i a cannot be negative. ■ This expression vanishes when x=\, and its differential coeffi- cient is (a^^ — af^) (1 — x'^'*), which is positive if x lie between 1 and 00 ; therefore the expression is positive if x lie between 1 and 00 . Laplace says that if the players agree to double, triple, ... their respective original numbers of counters the advantage of A will continually increase. This may be easily shewn. For change a into Tea and h into Tcb i we have then to shew that LAPLACE. 601 continually increases with h. Let a^=y; then we have to shew that (/-i).(/ + i) continually increases as y increases from unity : and this is what we have already shewn. 1034. Laplace's eighth Chapter is entitled Bes durees moyennes de la vie, des mariages et des associations quelconques: it occupies pages 408—418. Suppose we have found from the tables of mortality the mean duration of the life of n infants, where n is a very large number. Laplace proposes to investigate the probability that the deviation of this result from what may be considered to be the true result will lie within assigned limits : by the true result is meant the result which would be obtained if n were infinite. Laplace's analysis is of the same kind as that in his fourth Chapter. 1035. Laplace then examines the effect which would be produced on the laws of mortality if a particular disease were ex- tinguished, as for example the small-pox. Laplace's investigation resembles that of Daniel Bernoulli, as modified by D'Alembert : see Arts. 402, 405, 483. We will give Laplace's result. In Art. 402, we have arrived at the equation dq _q 1 dx n mn ' f . 1 1 where q = -. Put i for - , and r for - ; and let i and r not be ■^ s n m assumed constant. Thus we have dq dx = '^-'^- Let V denote e"/**"; thus d therefore qv = constant — jirv dx. 602 lAPLACE. The constant is unity, if we suppose the lower limit of the integral to be 0, for ^ and v are each unity when a; = ; thus qv = 1—1 {r (1 - a;)"' (xy (1 - xy-" dx therefore P= -. — ^^—r- 1 • itZllH x-^{l-xr^dx 604 LAPLACE. Laplace however adopts neither of the above methods ; but forrns a mixture of them. His process may be described thus : Take the first form of solution, but use Bayes's theorem to deter- mine the value of ^, instead of putting j? equal to I ' — j . We will complete the second solution. The next step ought to consist in evaluating strictly the integrals which occur in the expression for P; we shall however be content with some rough approximations which are about equivalent to those which Laplace himself adopts. Assume, in accordance with Art. 993, that \u, e 2m»^(i-«') \v \UL-v ^ -^ V27r/xa;' (1 - x') ' where r is supposed to be not large, and to be such that nearly V = x''/i — r, /j.— v={l — af}fi + r. Jo ^2Tru,x' (1 - a;') Thus p^roV2^/^«^'(l-£l I a;*"' (1 - x)"i dx J a Then, as in Arts. 957, 997, we pui; a;"' (1 - a;)"' = Ye-*'. x = a+~ — *-V, nearly, where a = — . And finally we have approximately e~s,«t»(i-o«) \f2iriid' (1 - d') Then we have to effect a s.ummation for different values of r, XA^LACE. 603 like that given in Art. 993. The result is that there is approxi- mately the probability that the number of unbroken couples will lie between fia' - T v'2/ia'(l-a'') and /lO' + t VB/wi' (1 - a'). This substantially agrees with Laplace, observing that in the third line of his page 418 the equation ought to be simplified by the consideration that p has been assumed very great ; so that the equation becomes F = — 1— 2nf (1 - i, 1i^ = X'pigi; and 2ej becomes equal to the number of times in which an event happens out of s trials, the chance of the happening of the event 2 f '' being -p^ at *"' trial. Thus we have the probability -j- I e-^ dt that the number of times will lie between "Zpi— T V2%piqi and %pi + t V22pi j-j. This is an extension of James Bernoulli's theorem to the case in which the chance of the event is not constant at every trial ; if we suppose that pi is independent of i we have a result practically coincident with that in Art. 993. This extension is given by Poisson, who attaches great importance to it ; see his Reaherches sur la Prdb. ..., page 246. 1039. If instead of two values at the z"' trial as in Art. 1037, we suppose a larger number, the investigation wiU be similar to 608 LAPLACE. that already given. Denote these values by g, f,, ;;^, ... ; we shall have ■where 2>i + qi + Wi+ ... = 1'; 2«» = 2 |S>, + l.^q, + x>i + . . . - (Spi + lqi + Xi^i + • • •)'}• Laplace himself takes the particular case in -which the function fi {z) is supposed the same at every trial ; see his pages 423 — 425. 1040. Laplace proceeds to a modification of the problem just considered, which may be of more practical importance. Nothing is supposed known a priori respecting the chances, but data are taken from observations. Suppose we have observed that in /i, trials a certain result has been obtained v, times : if fi, more trials are made determine the expectation of a person who is to receive f each time the result is obtained, and to forfeit f each time the result fails. The analysis now is like that which we have given at the end of 2 f '■ Art. 1036. There is the probability -j- I e-*^ dt that the number of times the result is obtained will lie between But if the result is obtained 601,609, 614. Bernoulli, John, 325 to 328, 442, 469. Bicqujlley, 55, 438 to 441. Bienaym^, 608. Binet, 292. Boole, 7, 505, 512, 544, 617. Borda, 391, 432 to 434, 548. Boacovich, 588. Bowditch, 478, 588. Breslau Registers, 41, 226, 322. Browne, 23, 49, 199. Buckley, 26. Bu£fon, 203, 262, 275, 277, 285, 344 to 349> 376, 386, 44°- Bullialdus, 65. Buteo, 33. Calandrin, 149. Calculus of Operations, £05 to 511, 525, 534- Canton, 294, Caramuel, 44 to 46, Carcavi, 8. Cardan, i to 4, 33. Carpenter, Lord, 141. Castelli, 6, Cauchy, 20, 520, 526, Clark, 207, 323. Clavius, 33, 44. Combinations, 26 to 36, 64, 82, 150. Commerdvm Epistolicum, 131. Condorcet, 41, 186, 261, 292, 351 to 410, 432, 441, 4S6, 458, 539. 541. S48, 618, Cotes, 143, 588. Cournot, 222, 383, S44, 609. Craig, 54, 462, 500. Cramer, 221, 222, 345. Cranmer, 149. Cuming, 182. D'Alembert, 14, 23, 224, 227, 228, 253, 256, 258 to 293, 331, 344, 377, 601. Dangeau, Marquis of, 47, D'Anierea, 445. Dante, 1, 323. De Eeaune, 127, 130. De Ganiferes, 29. DeHaan, 513, 514. De la Hontan, 95. De la Boche, 459. De M&^, 7, 8, II, 63, 14s, 502. De Moivre, 43, 52, £4, 63, 78 to 94, 100 to 105, 128, 134 to 193, 199 to 211, 250 to 253, 30s, 31S to 325, 340, 361, 378, 412, 421, 466, 468, 497, 527, S35, S39» 541, 553- De Morgan, 26, 49, 147, 346, 379, 400, 452, 513, 514, 639. 552, 557. S91. 605, 612, 615. Descartes, 21, 59, 137,, 505. 622 INDEX, Da Witt, 37 to 41, 614. Diderot, 55, 260. Dodson, 322, Election, modes of, 374, 547, 618. Ellis, 544, 561, 578. !Emeraon, 343. Encyclopedic, 39, 55, 201, 258 to 265, 286, 2go, 441 to 445. Errors, Theory of, 236 to 238, 301 to 309, 428, 442, 468 to 470, 484, 488, 490, 561 to 589. Euler, go, 237 to 257, 325 to 328, 422, 443, 489. 553, 588. Expectation, 213, 261, 392, 609, 614. Eaulhaberus, 65. Fenu, 618. Fennat, 7 to 2i, 35, 97, 146. Fontaine, 222, 261, 346. Fontana, 186. Fontenelle, 46, 57, 78, 188. Forbes, 334. Fr^ret, 407. Frommicheu, 618. Fuss, 6g, 349. Gaeta, 186. Galileo, 4 to 6. Galloway, 48, 409, 552, 537, 589, 612. Games : Ace of Hearts, 49, 203. Backgammon, 49, 205. Bassette, 46, 69, 93, 116, 150, 260, 443. Bernoulli's, Nicolas, 116. Bowls, 100, 140, 159, 207, 212, 324. Breland, 443. Cartes, 290. Cinq et Neuf, 69. Croix ou Pile, 258 to 265, 279, 281, 292. Dice, 260. Esp^rance, 94. Ferme, 106. Gageure, 259, 263, 264. Hazard, 48, S3, 94, 163, 164, 205. Her, 106, 133, 429. Krabs, 444. Lansquenet, 91. Lotteries, 48, 53, 100, 151, 203, 206, 24S to 256, 260, 325, 338, 421, 465, 527- Noyaux, 95. Odd and Even, 200, 465, 473, 527. Oublieux, 100. Pari, 260. ' Fasse-Diz, 45, 94, 444. Paume, 75, 125. Pharaon, 48, 80, 87, 116, 150, 152, 203, 243, 345- Piquet, 94, 166. QuadrUle, 152, 201. Quinquenove, 94. Bafle, Baffling, 94, 164. Eoyal Oak, 52. Tas, 106, 110, 124. Treize or Rencontre, 91, 105, 115, 120, 152, 239, 335, 452, 535. Trente et Quarante, 205, 444. Trijaques, 69. Trois Dez, 94. Whist, 52, 164, 445. Garve, 453. Gauss, 489, 588. Generating Functions, 484, 497, 504, 530. 534- Gouraud, 1, 11, 16, 36, 37, 38, 39, 46, 77, 293, 344, 409, 499, 613- Graunt, 37. 's Gravesande, 79, 130, 197, 616. Gregory, 514. Halley, 41 to 43, 81, 161, 226, 268, 463. Ham, 49, 203 to 205. Hamilton, 127. Haygarth, 459. Hendriks, 614. Hermann, 57. Herschel, 335. Hoyle, 322, 445. Huygens, 14, 21 to 25, 40, 44 to 52, 58 to 62, 81, 138 to 141, 143, 199, 432, 444- Izquierdus, 44, INDEX. 623 Jones, Si. Justell, 41. Kaestner, 321, Kahle, 615. Karstens, 427. Kepler, 4. Eerseboom, 241, Lacroix, 377. Lagrange, 178, 211, 249, 301 to 320, 428, 466, 469, 478, 484, 505, S35, 613. Lambert, 71, 93, 335 to 337, 428, 459, 462, 588. Laplace, 7, 123, 157 to 163, i6g to 178, 186, 201, 213 to 224, 228, 230, 234, 250 to 253, 279, 287, 293, 299, 314, 317, 347. 379. 400, 409*0422, 428, 459, 464*0613. Laplace, Comte de, 532 to 534. Laplace's Functions, 559. Least Squares, 560, 57s. Leibnitz, 12, 14, 21, 22, 31 to 33, 39, 47, S6, 73, 79, 502, SOS- Leslie, 26. Lhtiilier, 618. See Prevost and Lhuilier. Libri, i, 2, 5, 6. Locke, 500. Longomontanus, 45. Lubbock and Drinkwater, 11, 23, 33, 37, 41, 48, SO, S4, 5S, 299- LuUy, 44. M., 205. Maolaurin, 192. Mairan, 200. Malebranche, 78, 12C. Malfatti, 235, 434 to 438. MaUet, 325, 337 to 343, 350. Marpurg, 618. Maseres, 34, 59, 65. Mayer, 588. Mead, 199. MScanique Celeste, 478, 487, 514, 588. Mechanigue du Feu, 131. Mendclsuhn, 4 5 Mercator, 65. Merian, -,6. Michaelis, 93. MicheU, 332 to 336, 393, 491. Michelaen, 427. MiU, 262, 356, 409, 500. Monsoury, L'Abb^ de, 107. Montmort, 2, 36, 44 to 47, 55, 58, 78 to 142, 159, 167, 174, 187, 195 to 203, 209, 212, 338 to 343, 429, 444. Montucla, 11, 12, 16, 22, 26, 38, 39, 42, 46, 48, 79, 133, 222, 224, 261, 292, 293, 316, 332, 618. Mortality, 37 to 43, 240, 268, 285. Motte, 23, 48. Napoleon, 495 to 497. Necker, 259, 264. Neumann, 41. Newton, 21, 54, 86, 126, 131, 132, 135, 141, 187, 324, 485. Nicole, 201 to 203. Nozzolini, 5. Numbers of Bernoulli, 65, 152, 191. Orbais, L'Abb^ d', 107. Oettinger, 175. Pacioli, I. Pascal, 7 to 21, 28 to 30,40,66, 96, 128, 277, Soo, 502. Payne, 324. Peacock, 26. Permutations, 34, 64, 67, 150. Peterson, 54. Petty, 39, 81. Peverone, i. PoissOD, 7, 206, 222, 410, 489, 544, 556 to 561, 567, 571, 576, 607, 609, 613. Promotion Physique, 131. Prestet, 28, 36, 64, 65. Prevost and Lhuilier, 54, 60, 71, 384, 414, 432, 453 to 461. Price, 294 to 300, 378, 476. Problems : Arbuthnot's, 53, 209. Bernoulli's, James, 67, 338, 350. Bernoulli's, Daniel, 231 to 235, 319, 434 to 436, S58, 560. Births of boys and girls, 130, 193, 196 to 198, 235, 415 to 420, 480 to 484, 593, 597- G24 INDEX. Problems : BufFon'a, 260, 347, 590. Cuming's, 182. Duration of Marriages, 229, 335, 426, 602. Duration of Hay, 61, 101 to 105, 147, 167 to 183, 2og, 317 to 320, 448, 46s. 474, 476, 489, S3S. Inclination of Planes of Orbits, 222 to 224, 273, 475, 487, 542. Laplace's on Comets, 491 to 494- Petersburg, 134, 220 to 222, 259 to 262, 27s, 280, 286 to 289, 332, 345, 393, 470. Points, 8 to 19, 59, 96 to 99, 137, 146, 201 to 203, 316, 412, 46S, 474, 528, 532. Poisson's, 561 to 568. Eun of Events, 184 to 186, 208, 324, 361 to 368, 473. Small-pox, 224 to 228, 265 to 286, 423, 601. Waldegrave's, 122 to 125, 139, 162, 199. 3«5> 535- Woodcock's, 147 to 149. Puteanus, 27, Eacine, 500. Eiccati, 614. Eizzetti, 614. Eoberta, 53, 136, 137, 159, 164. Eoberval, 8, 12 to 15. Eudiger, 615. Saurin, 58. Sauveur, 46, 201. Schooten, 22, 26, 30, 64, 67. Sohwenter, 33. Series, 65, 73 to 75, 85, 89, 121, 125, 178 to 181, 210, 313, 426, 464. Simpson, 53, 206 to 212, 236, 305, 309. Smart, 187. Stevens, 149, 164. Sbewart, Dugald, 4, 349, 409, 453, 458, 502. 603. 613. Stifel, 33. Struve, 334. Sussmilch, 320. Svanberg, 612. Sylvester, 618. Tacquet, 36. Tartaglia, i. Taylor, 162. Terrot, 457. Tetens, 427. Theorems : Bayes's, 73, 294 to 300, 398, 410, SS7, 603. Bernoulli's, James, 71 to 73, 131, 183, 198, 360, 393, S48. 554. 556, 607. Binomial, 65, 82. De Moivre's on Dice, 84, 138, 146, 189, 208, 305, 350, 428, S42. De Moivre's approximation, 138, 144, 207. Buler's, 192. Stirling's 72, 188, 235, 467, 485, 520, 649. 653- Vandermonde's, 451. Wallis's, 610. Thomson, 49. Thubeuf, 618. Titius, 54. Trembley, no, 160, 230, 250, 411 to 431. Trial by Jury, 388. Turgot, 352. Van Hudden, 38. Yarignon, 114. Vastel, 59. Voltaire, 407, 409. Vossius, 28. Waldegrave, 122, 134. Wallis, 21 to 28, 34 to 36, 59, 65, 143, 160, 498, 505. Waring, 446 to 452, 463, 618. Watt, 49, 322. Woodcock, 147, 148. Young, 463. THE END. CAMBRIDGE : PRINTED AT THE UNlVERaiTT PRESS. wONSEIWATION.19e9