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MRS. GRIFFITH C. EVANS

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HISTORY OF

THE THEORY OF PROBABILITY.

A HISTOEY

OF THE

MATHEMATICAL THEORY OF PROBABILITY

FROM THE TIME OF PASCAL TO TEAT

OF LAPLACE.

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

\XYr O' C P . UlMOAy^^ ^

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

I86^

CambritJcjc:

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

PREFACE.

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

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

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

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

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

Vi PREFACE.

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

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

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

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

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

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

PREFACE. Vii

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

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

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

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

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

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

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

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

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

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

vlii PREFACE.

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

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

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

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

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

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

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

PREFACE. IX

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

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

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

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

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

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

X PREFACE.

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

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

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

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

PREFACE. XI

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

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

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

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

Xll PREFACE.

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

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

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

I. TODHUNTER.

Cambridge,
May, 1865.

CONTENTS.

PAGE

Chapter I. Cardan. Kepler. Galileo . . l

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

Chapter II. Pascal and Fermat .... 7

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

Chapter III. Huygens 22

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

Chapter IV. On Combinations . . . .20

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

Chapter V. Mortality and Life Insurance . 37

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

Chapter YI. Miscellaneous Investigations between

THE YEARS 1670 AND 1700 -ii

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

Chapter YII. Jaihes Bernoulli . . . .56

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

XIV CONTENTS.

PAGE

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

Chapter VIII. Montmoet 78

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

Chapter IX. De Moivre 135

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

Chapter X. Miscellaneous Investigations BET^yEEN

THE YEARS 1700 AND 1750 191?

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

CONTENTS. XV

PAGB

Chapter XL Daniel Bernoulli . . . .213

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

Chapter XIL Euler 239

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

Chapter XIII. D'Alembert 258

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

Chapter XI Y. Bates 2.94

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

Chapter XV. Lagrange 301

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

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

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

Chapter XVIL Condorcet 351

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

XVI CONTENTS.

PAGR

Chapter XVIII, Teembley . . . . • . 411

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

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

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

Chapter XX. Laplace 464

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

Appendix . 614

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

CHAPTER I.

CARDAN. KEPLER. GALILEO.

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

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

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

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

du Calcul des Frohahilites, page 3.

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

2 CARDAN.

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

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

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

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

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

CARDAN, 3

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

12 34 5 6789 10 11 12

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

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

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

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

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

1—2

^ KEPLER.

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

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

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

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

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

GALILEO. b

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

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

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

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

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

6 GALILEO.

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

CHAPTER II.

PASCAL AND FERMAT.

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

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

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

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

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

8 PASCAL AND FERMAT.

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

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

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

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

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

PASCAL AND FERMAT. 9

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

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

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

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

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

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

10 PASCAL AND FERMAT.

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

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

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

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

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

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

■^2.4.6... 2/1 •

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

PASCAL AND FERMAT. 11

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

14. Pascal then proceeds to another topic. He says

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

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

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

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

12 PASCAL AND FERMAT.

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

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

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

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

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

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

PASCAL AND FERMAT. 13

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

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

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

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

14j pascal and fermat.

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

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

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

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

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

PASCAL AND FERMAT. 15

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

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

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

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

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

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

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

16 PASCAL AND FERMAT.

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

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

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

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

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

PASCAL AND FERMAT. 17

The Arithmetical Triangle in its simplest form consists of the

following

table :

1 1

2 3

3 G

30 ...

4 10

8-i..

5 15

120..

•

6 21

120.

■ •

7 28

81^.

• •

8 3G..

•

%J a • •

J. •

• •

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

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

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

r-1

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

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

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

18 PASCAL AND FERMAT.

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

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

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

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

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

- r (r — 1) r (r — l) ... (r — n-\-2)

I . z n — 1

and the chance of B proportional to

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

1.2 ^^1-1

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

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

PASCAL AXD FERMAT. 19

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

In the second example we have

M — N.= . -^ = X say.

\n— 1 I ;2 — 1 "^

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

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

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

n — \ ?i— 2 \n — 1 In — 1

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

stake.

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

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

20 PASCAL AND FERMAT.

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

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

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

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

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

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

PASCAL AND FEEMAT. 21

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

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

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

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

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

CHAPTER III.

HUYGENS.

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

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

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

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

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

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

huvgp:ns. 2.3

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

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

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

i^ + 2'

It has been stated with reference to the last proposition :

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

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

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

appear to have been any opposition to the elementary principle,

except at a much later period by D'Alembert.

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

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

24 ^ HUYGENS.

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

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

^2l, Thus
So 81?/

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

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

(ja-^SOx

81«

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

80cj

a — x=^

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

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

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

HUYGENS. 25

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

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

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

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

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

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

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

CHAPTER IV.

ON COMBIjSTATIONS.

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

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

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

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

lUUHUsius. 27

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

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

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

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

Tot tibi sunt dotes, Virgo, quot sidera copIo.

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

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

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

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

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

28 PASCAL.

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

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

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

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

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

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

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

PASCAL. 29

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

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

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

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

■P

After this problem Pascal adds.

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

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

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

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

■ {p ■

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

.so SniOOTEN.

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

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

h. ah.

c. ac. he. ahc.

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

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

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

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

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

a. aa.

h. ah. aab.

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

LEIBNITZ. 3 1

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

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

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

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

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

32 LEIBNITZ.

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

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

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

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

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

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

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

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

LEIBNITZ. S3

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

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

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

With respect to Schwenter it has been observ^ed,

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

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

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

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

34 ^yALLIS.

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

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

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

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

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

WALLIS. 35

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

Tot tibi sunt dotes, virgo, quot sidera caelo.

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

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

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

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

8G PRESTET.

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

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

CHAPTER V,

MORTALITY AND LIFE INSURANCE.

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

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

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

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

38 GRAUNT.

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

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

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

Gouraud says in a note on his page 16,

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

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

56. The names of two Dutchmen next present themselves,

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

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

JOHN DE WITT. 30

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

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

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

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

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

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

40 JOHN DE WITT.

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

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

In liis reply Leibnitz says, page 84,

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

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

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

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

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

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

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

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

HALLEY. 41

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

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

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

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

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

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

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

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

42 HALLEY.

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

2
3
4

1000
855

798
760

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

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

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

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

HALLEY

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

Till

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

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

D
1

E _C

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

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

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

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

CHAPTER VI.

MISCELLANEOUS INVESTIGATIONS
Between the yeaes 1670 and 1700.

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

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

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

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

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

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

CARAMUEL. 45

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

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

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

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

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

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

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

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

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

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

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

46 SAUVEUR.

5 5

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

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

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

of his first throw.

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

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

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

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

5 6

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

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

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

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

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

LEIBXITZ. 47

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

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

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

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

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

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

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

48 ARBUTHXOT.

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

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

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

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

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

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

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

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

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

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

ARBUTHNOT. 49

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

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

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

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

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

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

Of the laws of chance, or, a method of calculation of the hazards
of game, plainly demonstrated, and applied to games at present most
in use; which may be easily extended to the most intricate cases of
chance imaginable. The fourth edition, ro^is'd by John Ham. By
whom is added, a demonstration of the gain of the banker in any
circumstance of the game call'd Pharaon; and how to determine the
odds at the Ace of Hearts or Fair Chance; with the arithmetical
solution of some questions relating to lotteries; and a few remarks
upon Hazard and Backgammon. London. Printed for B. Motte and
C. Bathurst, at the Middle-Temple Gate in Fleet-street, jt.dcc.xxxviii.

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

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

50 ARBUTHNOT.

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

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

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

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

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

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

ARBUTHNOT. 51

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

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

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

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

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

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

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

4—2

52 ARBUTHNOT.

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

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

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

55 9

will be Y9^ » which is less than Yq •

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

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

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

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

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

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

86. Our author says on page 43,

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

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

ROBERTS. 53

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

]_)^- q + r

author demonstrates this, and intimates that it may be extended

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

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

87. The first part of the book concludes thus :

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

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

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

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

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

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

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

Oi CRAIG.

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

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

a 1

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

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

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

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

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

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

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

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

CEAIG. 55

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

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

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

The views do not agree with those now received.

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

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

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

CHAPTER VII.

JAMES BERNOULLI.

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

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

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

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

James Bernoulli in reply says, page 77,

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

JAMES BERNOULLI. 57

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

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

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

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

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

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

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

58 JAMES BERNOULLI.

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

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

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

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

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

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

JAMES BEENOULLT. 59

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

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

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

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

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

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

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

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

60 JAMES BERNOULLI.

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

104. James Bernoulli gives the general expression for the

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

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

b c

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

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

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

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

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

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

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

JAMES BERNOULLI. 61

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

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

II.

III.

IV.

YI.

YIL

YIIL...

2 >

hcf
a''

e'er

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

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

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

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

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

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

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

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

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

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

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

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

infinite geometrical progressions. By summing these progressions

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

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

the result in Art. 31.

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

62 JAMES BERNOULLI.

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

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

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

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

Hence

_ a h

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

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

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

0=0.+ a„ 1 = 0.+ c,g)

«x

«!+n

therefore ^i — ~ ^2~

m+n 1 m+n '

Hence u^ =

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

X rn-^n J vi+n

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

7/ =

JAMES BERXOULLI. 63

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

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

will be

h"" (g^ - If)

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

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

108. We may observe that Bernoulli seems to have found,
as most who have studied the subject of chances have also found,
that it was extremely easy to fall into mistakes, especially by
attempting to reason without strict calculation. Thus, on his
page 15, he points out a mistake into which it would have been
easy to fall, nisi nos calculus aliud clocuisset He adds,

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

Again, on his page 27,

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

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

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

64 JA^IES BERNOULLI.

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

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

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

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

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

JAMES BERNOULLI. 65

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

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

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

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

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

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

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

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

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

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

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

66 JAMES BERNOULLI.

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

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

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

where //< = m -f w — 1 .

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

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

JAMES BERNOULLI. 67

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

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

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

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

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

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

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

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

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

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

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

5—2

68 JAiyiES BERNOULLI.

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

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

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

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

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

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

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

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

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

JAMES BERNOULLI. G9

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

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

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

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

119. The nineteenth problem is thus enunciated,

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

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

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

70 JAMES BERNOULLI.

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

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

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

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

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

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

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

V. Solutio Prohlematis prcecedentis.

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

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

JAMES BERNOULLI. 71

c f %

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

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

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

the conclusion fails only when all the arguments fail.

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

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

Then James Bernoulli gives for the resulting probability

, cfiru

1 — -^

adg (ru + qt) '

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

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

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

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

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

72 JAMES BERNOULLI.

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

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

and logc + log(r-l) A^

loor(s+ l)-log5 V

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

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

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

7* + 1 T ~— 1.

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

t f

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

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

7* + 1

happens to the whole number of trials lies between and

r-1

t '
As an example James Bernoulli takes

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

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

31 29 . .

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

t)0 50

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

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

to make 36966 trials; and so on.

JA3IES BERNOULLI. 73

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

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

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

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

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

31 29

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

50 oO

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

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

important way. Suppose that in the preceding illustration we

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

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

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

then according to James Bernoulli we are to infer that the

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

mately — . To determine the precise numerical estimate of the

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

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

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

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

74 JAJVIES BERNOULLI.

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

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

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

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

2 ' 3 n + l '

a ..11 11

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

z 3 n 71 + 1

1 r ^ -, 111 11

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

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

, . Ill 1,1

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

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

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

r (r + 1) ' n + 1 '

ISO. James Bernoulli says that his brother first observed

1111

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

i. jLi O ^

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

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

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

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

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

JAMES BERNOULLI. 75

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

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

1111

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

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

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

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

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

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

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

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

76 JAMES BERNOULLI.

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

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

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

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

Thus X = pq S -{■ {p"^ -{- q^) X \

r pq S pq S S

therefore a?=., ^ ., 2= ^ =7T •

1 —p — q zpq A

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

are on an equal footing.

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

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

JAMES BERNOULLI. 77

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

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

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

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

CHAPTER VIII.

MONTMORT.

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

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

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

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

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

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

MONTMORT. 79

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

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

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

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

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

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

80 MONTMORT.

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

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

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

Montmort makes some judicious observations on the foolish
and superstitious notions which were prevalent among persons
devoted to games of chance, and proposes to check these by shew-
ing, not only to such persons but to men in general, that there
are rules in chance, and that for want of knowing these rules
mistakes are made which entail adverse results; and these results
men impute to destiny instead of to their own ignorance. Per-
haps however he speaks rather as a philosopher than as a gambler
when he says positively on his page vili,

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

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

MONTMORT. 81

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

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

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

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

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

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

82 MOXTMORT.

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

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

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

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

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

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

MONTMORT. 83

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

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

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

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

The result will be in modern notation

\a \b[G...

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

G— 2

8^ MONTMOET.

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

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

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

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

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

/I — x^y^

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

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

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

«-/

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

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

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

MONTMORT. 85

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

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

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

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

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

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

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

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

86 MONTMOET.

Then it is shewn in works on Finite Differences, that

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

, yi(?i--l) ...(??-m+l) .,„

i -» j 11 Uq .

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

By the aid of this formula the summation of an assigned

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

,. ^ . ^ ,., n (n—1) ... (n — r+1) .

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

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

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

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

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

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

153. Montmort's second part occupies pages 73 — 172 ; it 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 DoctHne of Chances, pages ix, 77, 105,

Montmort's second edition gives the general results more
compactly than the first.

15i. 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 ii^ denote the Banker's advantage, A the sum of money
which his adversary stakes. Montmort shews that

,. _ g (y - 1) \ A. (p-q){p-q-^) ,,

;'^-^.0;-l) 2^+ p[p-l) ^'-^

supposing that j9 — 2 is greater than q. That is Montmort should

have this; but he puts {pq — q^) 2 A + {(f — q)-^A, on his page 89,

by mistake for q^q — l) - A) he 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 ?/^
depend on u^ ; and then it follows from the laws of the game that

Wj is equal io A \i q is equal to 2, and to ^ ^ if ^ is greater

than 2. Thus we shall have, if q is an even number greater
than 2,

'88 MONTMORT.

, (;>-g)(p-<7-i)-'-i I

If ^ = 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 w^^^, and if q is greater than unity it

can be shewn that u.^, = ^-^^ -77 . Thus we shall have, if a is an

^^^ q+1 Z

odd number greater than unity,

,, _ g(^-^) 1 J f 1 4. (p-^)(p-g-l)
^^-^(^-1)2^|'+ (^-2)(p-3)

(p-g)(j9-g-l)(j9-g-2)(;.-g-3)

(^_2)(^-3)(;.-4)(2.-5)

(i>-g)(;>-g-i)...2

"^ (i5-2)(i>-3) ^

If ^ = 1 we have by a special investigation Up = — .

If we suppose q even and p — q not less than q — 1, or q 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

{p-2)(p-S)...{p-q + l);

the numerator consists of a series, the first term of which is the
same as the denominator, and the last term is

fe-2)(^-3)...2.1, or (^-l)(^-2)...3.2,

according as q is even or odd.

The matter contained in the present article was not given
by Montmort in his first edition ; it is due to John Bernoulli :
see Montmort's, page 287.

MONTMORT. 89

156. We are thus naturally led to consider the summation of
certain series.

Let 0 (71, r) = -^ ^

so that (j) {n, r) is the n^^ number of the (r + 1)"' order of figurate
numbers.

Let 8<f) (n, r) stand for <j) {n, r) + <^ (w — 2, ?•) + </> (^ — 4, r) + . . . ,
so that S(j> (n, r) is the sum of the alternate terms of the series of
figurate numbers of the (r + 1)**" order, beginning with the w"' and
going backwards. It is required to find an expression for /S'</) {n, r).
It is known that

0 (n, r) + (l)(n-l,r) +(f) {n - 2, r) + </> (71 - 3, r) +... = (/> (ji, r + 1) ;

and by taking the terms in pairs it is easy to see that

<j) (n, 7') — (j) (n — l,r) -{-(f) [n — 2, 7^) —(f>{n — 3, ?•) + ... = S(j) {n, r — 1) ;

therefore, by addition,

S(l> {n, r) = - (/) {71, r-\-l)+^S(i> (w, r- 1).

Hence, continuing the process, we shall have

1 11

^^ (w, ?•) = 2 ^ ^^' ** + ^) + 3 ^ (^^' ^') + ^ <^ {^h r - 1) + ...

and we must consider 8<\> (n, ^—-n, if 71 be even, and = - (n+1),
if n be odd.

We may also obtain another expression for 8<^ {n, r). For
change w into n + 1 in the two fundamental relations, and subtract,
instead of adding as before ; thus

>^(/) (71, r) = i <^ (n + 1 , r + 1) - ^ ^0 ( ; . + 1 , r - 1 ) .
Hence, continuing the process, we shall have
^(/)(7i,r)=-(/>(7i + l, r + 1)- ^ </,(« + 2,7')+^ <3«>(n + 3, r-1)

{- ly {- \Y

- -^ ^(u + r, 2) + 4^ Sct>(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 Montmort's result when q is odd, ^^ — 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 ?i + 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 ioY 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

w + 2

should be calculated from the formula -r — -, -. , by puttino^ for n

2n (n — 1) -^ ^ ^

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

3117 26

incorrectly ; it begins with ' ^ instead of ^ , 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 ^w tv-/ i^ ? ^Y putting for n in succession

z{n—l){n — 3) ,

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,
which 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. Ai'ticle 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 promiscuousl}^ 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 yi — 1. The number of cases in which h is second, but a not first,

n — 1 — 1 7i — 2. The number of cases in which c is third, but a

not first nor b second, is | w — 1 — | ^^ — 2 — ||?i — 2 — | n — 31

92 MONTMORT.

that is \n-l -2\n-2+\n-S. The number of cases in

which d is fourth, but neither a, b, nor c in its proper place is
\n-l-2\n-2 + \n-S-hn-2-2\n-S + | n - 41, that is

1/1 — 1 — 3 \n — 2 + 3 \n — S — \n — 4*. And generally the number

of cases in which the m^^ card is in its proper place, while none
of its predecessors is in its proper place, is

\n-l - (m - 1) 1 71-2 + ^ -^ ^ \n-S

{m -1) (m- 2) (m-3)

wt-l

w — m.

^ ,71-4 + + (-1)

"We may supply a step here in the process of Nicolas Bernoulli,
by shewing the truth of this result by induction. Let -v/r (771, n)
denote the number of cases in which the m"' card is the first that
occurs in its right place ; we have to trace the connexion between
^jr (m, n) and yjr {m + 1, n). The number of cases in which the
{m + l)**^ card is in its right place while none of the cards between
h and the W2*'^ 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 yjr {in + 1, n). The cases to
be rejected are in number '^ {m, n — 1). Thus

y^ (in + 1, w) = i/r {in, n) — yfr {in, n — 1).

Hence we can shew that the form assigned by Nicolas Bernoulli
to -^/r (m, n) is universally true.

Thus if a person undertakes that the m*^ card shall be the first
that is in its right place, the number of cases favourable to him is

'^ (m, n), and therefore his chance is . ' — - ,

If he undertakes that at least one card shall be in its right

place, we obtain the number of favourable cases by summing

^jr (m, n) for all values of m from 1 to n 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,

i_l+i_l, , (- 1)-

2 [3 li^'"^ \n '

MONTilORT. 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 (f> (n) denote the number of cases in which no card is in its right
place, we obtain

162. The game which Montmort calls Treize has sometimes
been called Rencontre. The problem which is here introduced for
the first time has been generalised and discussed by the following
writers : De Moivre, Doctrine of Chances, pages 109 — 117. Euler,
Hist, de T Acad.... Berlin, for 1751. Lambert, Kouveaux Memoires
de T Acad. ... Berlin, for 1771. Laplace, TJieorie . . . des Proh.
pages 217 — 225. Michaelis, Memoire sur la prohahilite du jeu de
rencontre, Berlin, 1846.

163. Pages 148 — 156 of Montmort relate to the game of Bas-
sette. This is one of the most celebrated of the old games : it
bears a great resemblance to Pharaon.

As we have already stated, this game was discussed by James
Bernoulli, who summed up his results in the form of six tables ;
see Art. 119. The most imi^ortant of these tables is in the fourth,
which is in effect also reproduced in De Moivre's investigations.
The reader who wishes to obtain a notion of the game may con-
sult De Moivre's Doctrine of Chances, pages 69 — 77.

164. James Bernoulli and De Moivre confine themselves to
the case of a common pack of cards, so that a particular card, an
ace for example, cannot occur more than four times. Montmort
however, considers the subject more generally, and gives formulae
for a pack of cards consisting of any number of suits. Montmort
gives a general formula on his page 153 which is new in his second
edition. The quantity which De Moivre denotes by y and puts
equal to ^ is taken to be | by Montmort.

Montmort gives a numerical table of the advantage of the
Banker at Bassette. In the second edition some fractions are
left unreduced which were reduced to their lowest terms in the
first edition, the object of the change being jDrobably to allow

94 MONTMORT.

the law of formation to be more readily perceived. The last
fraction, given in the table was wrong in the first edition ; see
Montmort's page 803. It would be advisable to multiply both
numerator and denominator of this fraction by 12 to maintain
uniformity in the table.

165. Montmort devotes his pages 157 — 172 to some pro-
blems respecting games which are not entirely games of chance.
He gives some preliminary remarks to shew that the complete
discussion of such games is too laborious and complex for our
powers of analysis ; he therefore restricts himself to some special
problems relating to the games.

The games are not described, so that it would be difficult to
undertake an examination of Montmort's investigations. Two of
the problems, namely, those relating to the game of Piquet, are
given by De Moivre with more detail than by Montmort ; see
Doctrine of Chances, page 179. These problems are simple exer-
cises in combinations ; and it would appear that all Montmort's
other problems in this part of his book are of a similar kind, pre-
senting no difficulty except that arising from a want of familiarity
with the undescribed games to which they belong.

166. Montmort's third part occupies pages 173 — 215 ; it
relates to games of chance involving dice. This part is almost
identically repeated from the first edition.

The first game is called Qicinqiienove ; it is described, and a
calculation given of the disadvantage of a player. The second
game is called Hazard; this is also described, and a calculation
given of the disadvantage of the player who holds the dice. This
game is discussed by De Moivre; see his pages 160 — 166. The
third game is cslled Fs2:>erance ; it is described and a particular
case of it with three players is calculated. The calculation is
extremely laborious, and the chances of the three players are
represented by three fractions, the common denominator being a
number of twenty figures. Then follow games called Trois Dez,
Passe-dix, Rafle ; these are described somewhat obscurely, and
problems respecting them are solved ; Raffling is discussed by De
Moivre; see pages 166 — 172 of the Doctrine of Chances.

MONTMORT. 95

167. The last game is called Le Jeu des Koyaux, which
Montmort says the Baron de la Hontan had found to be in use
among the savages of Canada ; see Montmort's pages xii and 213.
The game is thus described,

On y joue avec huit noyaux noirs d'un cote et blancs de I'autre : on
jette les noyaux en Fair : alors si les noirs se trouvent impairs, celui qui
a jette les noyaux gagne ce que I'autre Joueur a uiis au jeu : S'ils se
trouvent ou tous noirs ou tous blancs, il en gagne le double ; et hors de
ces deux cas il perd sa mise.

Suppose eight dice each having only two faces, one face black
and one white ; let them be thrown up at random. There are
then T, that is 256, equally possible cases. It will be found that
there are 8 cases for one black and seven white, 5Q cases for three
black and five white, 28 cases for two black and six white, and
70 cases for four black and four white ; and there is only one case
for all black. Thus if the whole stake be denoted by A, the chance
of the player who throws the dice is

_L j (8 + 8 + 56 + 50) .4 + 2 (.1 + I A) | ,

and the chance of the other player is

2^^1(28 + 28 + 70)^ + 2(0-1.1)1.

131 125

The former is equal to tt^. A, and the latter to 77^ A,

2ob loij

Montmort says that the problem was proposed to him by a
lady who gave him almost instantly a correct solution of it ; but
he proceeds very rudely to depreciate the lady's solution by in-
sinuating that it was only correct by accident, for her method was
restricted to the case in which there were only two faces on each
of the dice : Montmort then proposes a similar problem in which
each of the dice has ybi^r faces.

Montmort should have recorded the name of the only lady who
has contributed to the Theory of Probability.

96 MONTMORT.

168. The fourth part of Montmort's book occupies pages
216 — 282 ; it contains the solution of various problems respecting
chances, and in particular of the five proposed by Huygens in
1657 ; see Art. 35. This part of the work extends to about double
the length of the corresponding part in the first edition.

169. Montmort's solution of Hujgens's first problem is similar
to that given by James Bernoulli. The first few lines of Mont-
mort's Remarque on his page 217 are not in his first edition ; they
strongly resemble some lines in the Ars Coiijectandi, page 51.
But Montmort does not refer to the latter work, either in his
preface or elsewhere, although it appeared before his own second
edition; the interval however between the two publications may
have been very small, and so perhaps Montmort had not seen the
Ars Conjectandi until after his own work had been completely
printed.

The solution of Huygens's fifth problem is very laborious, and
inferior to that given by James Bernoulli ; and Montmort him-
self admits that he had not adopted the best method ; see his
page 223.

The solutions of Huygens's problems which Montmort gave
in his first edition received the benefit of some observations by
John Bernoulli ; these are printed in Montmort's fifth part,
pages 292 — 294, and by the aid of them the solutions in the second
edition were improved : but Montmort's discussions of the pro-
blems remain still far less elaborate than those of James Bernoulli.

170. Montmort next takes two problems which amount to
finding the value of an annuity, allowing compound interest.
Then he proceeds to the problem of which a particular example
is to find in how many throws with a single die it will be an
even chance to throw a six.

171. Montmort now devotes his pages 232 — 248 to the Pro-
blem of Points. He reprints Pascal's letter of August 14th, 1654,
to which we have alluded in Art. 16, and then he adds, page 241,

Le respect que nous avons pour la reputation et pour la memoire de
M. Pascal, ne nous permet pas de faire remarquer ici en detiiil toutes

MONTMORT. 97

les fautes de raisonnement qui sont dans cette Lettre ; il nous suffira
d'avertir que la cause de son erreur est de n'avoir point d'egard aux
divers arrangemens des lettres.

Montmort's words seem to imply that Pascal's letter contains
a large amount of error ; we have, however, only the single fun-
damental inaccuracy which Fermat corrected, as we have shewTi in
Art. 19, and the inference that it was not allowaVjle to suppose
that a certain number of trials will necessarily be made; see Art. 18.

172. Montmort gives for the first time two formulae either of
which is a complete solution of the Problem of Points when there
are two players, taking into account difference of skill. We will
exhibit these formulae in modern notation. Suppose that A wants
711 points and B wants n points ; so that the game will be neces-
sarily decided in m-\-n—l trials ; \etm + n—l = r. Let p denote
A's skill, that is his chance of winning in a single trial, and let
q denote J5's skill ; so that p + q = l.

Then ^'s chance of winning the game is

pr^ r-i _^r(r-l) ,_^ + ,— ,^^— fi^V";

^"^^ 1.2^ [m I ?? — 1

and Bs chance of winning the game is

q'+rr'p+^^-^Y^ 2-p=+ + ^^zi^ ?>"'- ■

This is the first formula. According to the second formula J's
chance of winning the game is

m f 1 . m (m + 1) « , , 1/ "" i_ ^K-i 1 .

and B's chance of winning the game is

„ f- , 7^ (n + 1) „ , l^~^ ,,--' I

^ r-^^'^"^ 172 ^+ ^\m-i.n-ij r

Montmort demonstrates the truth of these formulae, but we
need not crive the demonstrations here as they will be found in
elementary works; see Algebra, Chapter Llli.

173. In Montmort's first edition he had confined himself
to the case of equal skill and had given only the first formula,

93 MONTMORT.

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
BernouUi; 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

p' + rp''q-h \ ,^ ' J) <i +... + -—:::: rV 9.

r^V^ if-^ + m (2> + ^)--^ + !!^!i±i) (^ + g)'-'-/ +

r — 1

m — 1 n

This may be shewn by picking out the coefficients of the
various powers of ^ in the expression on the right-hand side,
making use of the relations presented by the identity

(1 - j)-"-»'(l- 2)-= (1 -?)"'.

Thus we see that \i 'p-\- c[ be equal to unity the two expres-
sions given in Art. 172 for ^'s chance are numerically equal.

175. If however ^ + ^^ be not equal to unity the two expres-
sions given in Art. 172 for ^'s chance are not numerically equal.
If we suppose jy-^- 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 jp, and i?'s chance is q, and that there is the
chance ^—jp — q that it is a drawn contest.

Then the formula

mi, w (??z + 1) „ \r —\

^ 1.2 -^ m— 1^1 — 1^

MONTMORT. 09

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 -{- q is equal to unity.

But the formula

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 (2^ + q + 1 —p — qY
in powers of j^, q, 1 — ^ — 5', a term such as Cj^^q^il —2^ — qy ex-
presses the chance that A wins p points, B wins a points, and r
contests are drawn.

Or we may treat this second case by using the transformation
in Art. 174. Then we see that {p + qy"^ expresses the chance
that there shall be no dra^\Ti 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 71 points ;
but this he says is not the case ; see his page 24? 7. 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 nearly in
the ratio of his skill ; so that if the ratio ofm to n be less than that
of the skill of A to the skill of B, we can, by increasing k, obtain as
great a probability as we please that A will win km points before
B wins hi points.

Montmort probably implies, though he does not state, the con-
dition which we have put in Italics.

7—2

1 00 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
Be Mensura Sortis ; see Montmort, page 866, 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 likely as the

.2 .1

others to be first, the chance of ^ is ^r and that of ^ is - . But by

3 3 -^

the incorrect solution Montmort arrives at a different result. For

suppose A to have delivered his bowl. Then B has the chance

^ 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
chance appears to be - • Montmort considers the error 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 will beat A with
his second bowl : for the fact that B failed with his first bowl
suggests that ^'s bowl has a position better than the average, so
that jB's 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 Oublieux. The fourth is an extension of
Huygens's eleventh problem, and is also given in the Ars Conjee-
tandi, page 34. These four problems are new in the second edition.

MOXTMORT. 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 i? to have n counters ; let
their chances of winning a single game be as a to ^ ; the loser in
each game is to give a counter to his adversary : required the chance
that A will have won all 5's counters on or before the x^^ game.

This is the most difficult problem which had as yet been solved
in the sulyect. 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 o^^TL 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 Moi\Te
had solved the problem in a different manner in the De 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 Monniort^ in the Second Edition of his Book of Chances,
having given a very handsom Solution of the Problem relating to the
duration of Play, (which Solution is coincident with that of Monsieur
Nicolas Bemoully, 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.

J 02 MONTMORT.

...the Solution of Mr Nicolas Bernoulli beiog 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 Monimort^s Solution with very great attention : I found indeed that
he was very plain, but to my great surpriza 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 2G8, 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 as 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 h must be

MONTMORT. IQ:

SO arranged that only a single h is to come among the four letters
a : we might have such an arrangement as aaahhhhhhhhhhha. We
shall return to this point in our account of De Moivi'e'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 — ^- is the chance that the

match will be ended in 2x games at most. The result may be de-
duced from Montmort's general expression. A property of the
Binomial Coefficients is involved which we may briefly indicate.

Let Wj, u^, u^, ... denote the successive terms in the expansion
of (I + l)'"^. Let >S' denote the sum of the following series

w. + ^ii.-i+ Ux-i+ 0 + u,_,+ 2it,_,+ u,_,+ 0 + w^_3+ ...
Then shall S=r'-'-2'-\

For let V, denote the r^^ term in the expansion of (1 + 1)"''"S and
lOy the ?'"' term in the expansion of (1 + 1)"''"^ Then

t'y t/j. "t" I- r—Xf

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

'^x + ^.r-1 + ^'x-2 + ^.r-3 + ^x--4 + ' ' '
+ 2 [W^_^ -h 2W^_^ + IC,_^ + 0 + 10, _, + ...}.

The first of these two series is equal to ^ (1 + I)'''"' ; and the

second is a series of the same kind as that which we wish to sum
with X chanced into x-1. Thus we can finish the demonstration
hy induction ; for obviously

101 MONTMORT.

(2) Next suppose that each player starts with three counters ;
ox

then 1 — — is the chance that the match will be ended in 2ic + 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 w^, u^, u^, ... denote the successive term-; in the expansion
of (1 + iy'^\ Let S denote the sum of the following series

Then shall 8=2'"'- 3^.

If w^ denote the r**^ term in the expansion of (1 + 1)'^''"^ we can
shew that

w^ + 2m^_i + 2w^_2 + u^_s

+ S (2^,_i + 2w;^_2 + 2w;^_3 + 2(7^ J.

By performing a similar transformation on every successive
four significant terms of the original series we transform it into

2 (1 + 1)'''"^ + 3S, where 2 is a series like S with x changed into

x-1. Thus

8 = 2^^-2 + 32.

Hence by induction we find that /S^= 2^"" - S''.

184. Suppose the players of equal skill, and that each starts

with the same odd number of counters, say m ; let /= '^'^^ ,

Then Montmort says, on his page 276, that we may wager with
adva.ntage that the match will be concluded in 3/' - 3/+ 1 trials.
Montmort does not shew how he arrived at this approximation.

The expression may be put in the form \m'^\, De 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

MONTMORT. 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, Theorie...des Proh. 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^= Cx+ C^, where C and (7^ are arbitraiy con-
stants. To determine them we have

^0=0, %„ = !;

hence finally, w« = ^ •

Montmort's example is for ?i = 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 sm-prise ; 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 sur le Jeu dii 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.

lOG MONTMORT.

The second problem is sur le Jen appelle 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 will 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 Avertisseynent, 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 will 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, tiuo, three, ... ten. Knave, Queen, Kmg.

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'Abbe de Monsoury and sometimes M. TAbbe 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 corresj)ondence
closes without it.

189. We will 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 Paid has
obtained an ei(jht, 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 discussino- this
case v/ill be found on Montmort's page 339 ; we will reproduce
them with some explanation of the process by which thev 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

chance that Peter had a ten is — - ; then Paul takes it, and Peter

lOS MONTMORT.

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, 4i 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 will 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
seve7i 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 ^ 27 j^ 27 _£ 27 4 27 ^ 27 ^ 27
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 -pr • ^t: to the preceding result ; and thus

51 oU

we obtain for Paul's chance -

51.50'

780
Thus we find that in Case I. Paul's chance is , and that

51 . 50

in Case II. it is either -,- ^ or .^ —.7: . If it be an even chance

51 .50 51 . oU

MOXTMOET. 109

1 / *720 SIP) ^

■which rule Peter adopts we should take ^ f --^ — p^ + --.-^j > that

is, 1^ — ^t; as Paul's chance in Case 11. Thus in Case II. Paul's
51 . oO

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 ? 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 h}^o-
theses that Paul has Queen, Knave, ten, nine, or eight; so that my
chance is

23 ■ 50 "^ 23 ■ 50 "^ 23 ■ 50 "^ 23 ' 50 "^ 23 ' 50 '

210

that is

23 . 50

II. Suppose Paul's rule is to retain a seven. Then, as before,

(1) If I retain my eight my chance is ^ .

(2) If I change my eight my chance is

4 3 4 7 4 11 4 15 3 22 4 26

314

27 ' 50 "^ 27 * 50 "^ 27 ■ 50 "^ 27 ' 50 "^ 27 ' 50 "^ 27 ■ 50 '

that is

, 27 . 50

1 I 0 MONTMORT.

190. These numerical results were accepted by the disjDutants.
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 adversary 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 SQai que vous etes le conseil de Pierre,
il est evident que je dois moi Paul me tenir au sept ; et de meme
si je suis Pierre, et qui je SQache que vous etes le conseil de Paul,
je dois changer au liuit, auquel cas vous aures donne un mauvais con-
seil a 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
V Acad.... Berlin, for 1802.

191. The fourth problem which Montmort proposed for solu^
tion is sur le Jen des Tas. The game is thus described, page 281,

Pour comprendre de quoi il s'agit, il faut s9avoir qu'apres les reprises
d'hombre un des Joueurs s'amuse sou vent a partager le jeu en dix tas
composes chacun de quatre cartes couvertes, et qu'ensuite retournant la
premiere de chaque tas, il ote et met a part deux ^ deux toutes celles
qui se trouvent semblables, par exemple, deux Pois, deux valets, deux
six, &c. alors il retourne les cartes qui suivent immediatement celles
qui viennent de lui donner des doublets, et il continue d'oter et de
mettre a part celles qui viennent par doublet jusqu'a ce qu'il en soit

MONTMORT. Ill

venu a la derniere de chaque tas, apres les avoir enleve toutes deux a
deux, auquel cas seulement il a gagne.

The game is not entirely a game of pure chance, because the
l^layer may often have a choice of various methods of pjairing 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 i^roceeding. He says the game was rarely
played for money, but intimates that it was in use aniong ladies.

192. On his page 821 Montmort gives, without demonstration,
the result in a particular case of this problem, namely when the
cards consist of ?2 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

92 — 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 jDutting for n in succession
2, 3, ^,o, ...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 difficult, 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 exj^osed to view ; the player may be thus arrested at the
very beginning of the game, or after he has already taken som^j
steps : at this epoch the player is left icitk some number of lots,
which are all unbroken, and the cards exposed to vieiu present no
pairs. This will be obvious on reflection.

1 1 2 MONTMORT.

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 2n cards are to be put in 2n
places, and thus [ 27i will 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 m the first place any card oat of the 2/2, then in the
second place any card of the 2n — 2 which remain by rejecting the
companion card to that we put in the first place, then in the third
place any card of the 2n — 4< which remain by rejecting the two
companion cards, and so on. Thus the n upper places can be
filled in 2" [n ways. Then the n lower places can be filled in [n
ways. Hence we get 2*" 1^2 [ji cases in which the player is arrested
at the very beginning.

We may divide each of these expressions by \n if we please

to disregard the different order in which the n lots may be sup-

\2n
posed to be arranged. Thus the results become M^ and 2" [n

respectively ; we shall use these forms.

Let u^ denote the whole number of unfavourable cases, and let
/,. denote the whole number of favourable cases when the cards
consist of r pairs. Then

u^=^r[n + t -—^ £ \n-r 2""'',

the summation extending from r = 2 to r = w — 1, 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

\ 71

of r lots of the original n lots. The factor != g-ives the num-

\r n — r

ber of ways in which r pairs can be selected from n pairs ; the
factor fi gives the number of ways in which these pairs can be so
arranged as to enable the player to get rid of them ; the fiictor
\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.

MONTMORT. 113

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 tivo pairs.
We may how^ever if we please consider the summation to extend
from r = lio r = n-l, since/. = 0 when r = 1.

We have then

u„=.T\n[l + 2J^.

The summation for w„_, extends to one term less ; thus we
shall find that

But «„., +/„_, =

71-1

therefore

2/2 I 2;i - 2

n — l

\1n 2\2?i-2 I 2/1 ,j_i

Hence /„ = i= - z/„ =-==-- ; and /„ -^

[w " 1 71 -2 ' ^" • |_^ 2/4 -l'
, This is Montmort's result.

19^. We noAv arrive at what Montmort calls the fifth part
of his work, which occupies pages 288 — 41-i. It consists of the
coiTcspondence 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 born in 1667, and died in
IT-iS. The second brother of the family was named Nicolas ; his
son of the same name, the friend and corres^oondent of Montmort,
was born in 16S7, 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 corresponding passages in the second edition, yet

114? MONTMORT.

as these passages have been modified or corrected 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 Bernoulli'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 Art. 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 logarithm es,
je m'en suis servi utilement dans une parcille occasion il y a bien
douze ans, ou il s'agissoit de determiner combien il restoit de vin et
d'eau mele ensemble dans un tonneau, lequel etant an commencement
tout plein de vin, on en tireroit tons les jours pendant une amice
une certaine mesure, en le remplissant incontinent apres cliaque ex-
traction avec de I'eau pure. Vous trouveres la solution de cette ques-
tion qui est asses curieuse dans ma dissertation De Nutritlone, que Mr
Varignon vous pourra communiquer. Jc fis cette question pour faire

MONTMORT. 115

comprendre comment on pent determiner la quantite de vieille ma-
tiere qui reste dans nos corps melee avec de la nouvelle qui nous
vient tous les jours par la nourriture, pour reparer la perte que nos
corps font insensiblement par la transpiration continuelle.

The dissertation De 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--^+^-^+...+-^

and let

t(«)=^(«) + J^("-l) + U(n-2)+...+ ^^<^(1);

111 1

then shall i/r {n) = --[-+_++. ..+_.

^ [^ [I [± \JL

We may prove this by induction. For we may write yjr (n) in
the following form,

T fl 1 1 1 1 1

^11

2 ] ^ ' 1 ' 12 ' [3_

1 r, 1 1 1 1 1

-o^ l + T + n5 + rT. + + ^^3^ J

Hence we can shew that

y}r (n + 1) = i/r {n) +

;i + 1 *

199. John Bernoulli next adverts to the solutions which
Montmort had given of the five problems proposed by Huygens ;
see Art. 35.

According to John Bernoulli's opinion, Montmort had not
understood the second and third problems in the sense which
Huygens had intended ; in the fifth problem Montmort had

8-^2

116 MONTMORT.

changed the enunciation into another quite different, and yet had
really solved the problem according to Huygens's enunciation. By
the corrections which he made in his second edition, Montmort
shewed that he admitted the justice of the objections urged against
his solutions of the second and fifth problems; in the case of
the third problem he retained his original opinion; see his
pages 292, 805.

John Bernoulli next notices the solution of the Problem of
Points, and gives a general formula, to which we have referred in
Art. 173. Then he adverts to a problem which Montmort had
not fully considered; see Art. 185.

200. John Bernoulli gives high praise to Montmort's work,
but urges him to extend and enrich it. He refers to the four
problems which Montmort had proposed for investigation ; the
first he considers too long to be finished in human life, and the
fourth he cannot understand : the other two he thinks might be
solved by great labour. This opinion seems singularly incorrect.
The first problem is the easiest of all, and has been solved without
difficulty; see Article 161 : perhaps however John Bernoulli took
it in some more general sense; see Montmort's page 308. The
fourth problem is quite intelligible, and a particular case of it is
simple ; see Art. 193. The third and fourth problems seem to be
far more intractable.

201. A letter to Montmort from Nicolas Bernoulli occupies
pages 299 — 303. This letter contains corrections of two mistakes
which occurred in Montmort's first edition. It gives without de-
monstration a formula for the advantage of the Banker at Pharaon,
and also a formula for the advantage of the Banker at Bassette ;
Montmort quoted the former in the text of his second edition ;
see Art. 157. Nicolas Bernoulli gives a good investigation of the
formulae which occur in analysing the game of Treize ; see Art. 161.
He also discusses briefly a game of chance which we will now
explain.

202. Suppose that a set of players A, B, C, D, ... undertake
to play a set of I games with cards. A is at first the dealer, there
are m chances out of on + n that he retains the deal at the next
game, and n chances out of m + 7i that he loses it ; if he loses the

MONTMORT. 117

deal the player on Lis right hand takes it ; and so on in order.
B is on the left of A, C is on the left of B, and so on. Let the
advantages of the players when A deals .be a, h, c, d, ... respec-
tively; these advantages are supposed to depend entirely on
the situation of the players, the game being a game of pure
chance.

Let the chances of A, B, C, D, ... bo denoted by z, y, x, u, ... ;
and let s = 7n + 7l

Then Nicolas Bernoulli gives the following values :

z = a + —^ + ~, + -, +...,

, 7nh + nc m^h 4- 2mnc-\-n^d m%-^Snfnc + 2mn^d+ n^e
2/ = i + -^- + p + p +...,

. mc + 7id 7n'^G-\-27n7id-{-7i^e 7ifc-h2m^7id+Sm7i^e + 72^f

^ = 0 + ^— + 7 + — ? - + ■■■'

_ , md + 7ie m^d-\-2miie+ri^f 7n^d-\-Sm^ne + Sm7}^f+7i'g
and so on.

Each of these series is to continue for I terms. If there are
not so many as I players, the letters in the set a, h, c, d, e,f,(/,...
will recur. For example, if there are only four players, then
e = a, f=h, g = c,....

It is easy to see the meaning of the separate terms. Take, for
example, the value of z. A deals ; the advantage directly arising
from this is a. Then there are m chances out of 5 that A will hav
the second deal, and 7i chances out of s that the deal will pass o.
to the next player, and thus put A in the position originally hek

by B. Hence we have the term . Again, for the third

deal ; there are (rti + 7iy, that is, s^ possible cases ; out of these

there are 711^ cases in which A will have the third deal, 2mn cases

in which the player on the right of A will have it, and n"^ cases in

which the player next on the right w^ill have it. Hence we

, ,1 , iii^a -\- Imnh -\- 7i^c . ,

nave the term z . And so on.

118

MONTMOET.

Nicolas Bernoulli then gives another form for these expressions ;
we will exhibit that for z from which the others can be deduced.

Let

^^i, ,= K ,= !i. Then
^ n \sj m

z = aq(l-r)-\-hq\l-r{l-\-tl]\-\-C(i\l-r

r _ fia-r)~\]

-{■dq \l — r

-^ ^ ^^ ^ tH{i-i) ^ fi{i-i)(i-2y

1.2

1.2.3

I • • • J

this series is to be continued for I terms.

The way in which this transformation is effected is the follow-
ing : suppose for example we pick out the coefficient of c in the
value of z, we shall find it to be

1 .Zs [ s s s'^ )

where the series in brackets is to consist of Z — 2 terms.
We have then to shew that this expression is equal to

«{'-'h"*'^']}-

We will take the general theorem of which this is a particular
case. Let

where

Let

P.=

p + X-1

1 + - + -T +

then S=T— -y-^ .
[Xdmr'

Now

- fmr

u —

say;

MOXTMORT. 119

1.2 s^ (l-At/"'

1.2.3 6'^ (l-/x)

\-2

-^^!i rli I til ^^^-^ tn{i-i)(i-2) ■

- ,,,U » i + ^^+ ^^2 + 1.2.3 "^•••

71^-^^

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 + Z> + c + ...=0, and also z+y + x+... = 0.

The problem simplifies very much if we may regard I as infinite
or very great. For then let z denote the advantage of -4 ; if ^ ob-
tains the next deal we may consider that his advantage is still z ; if
A loses the next deal his advantage is the same as that of B

originally. Thus

mz + n2/

MultijDly by s and transpose ; therefore

z = 7/-{- aq.
Similarly we have

y = x + hq, x = ii + cq,

Hence we shall obtain

^ = 2L(^-l) + &(p-2) + c(i?-3) + ...j,

where p denotes the number of players ; and the values of y, ^, . . .
may be obtained by symmetrical changes in the letters.
We may also express the result thus,

= _£|a+2^>+3c+...|.

120 MONTMORT.

203. The next letter is from Montmort to John Bernoulli ; it
occupies pages 803 — 307. Montmort makes brief observations on
the points to which John Bernouilli 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 808 — 814.

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 w^hen Montmort drew his
attention to it ; see Montmort's pages 815, 328.

We will here investigate the formula after the manner given by
Nicolas BernoulH 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,h,c,... in order.
The whole number of cases is \n.
The number of ways in which a can stand first is p \n — \.

The number of ways in which h can stand second without a
standing first is p \n — l — p\ n

The number of ways in which c can stand third without a
standing first or h second is p \n — \ — 2p^ |^ — 2 + p^ | n — 3.
And so on.

Hence the chance of winning by the first card is - ; the chance

of winning: by the second card is -, ^ ,. ; the chance of Avin-

° -^ n 7i{n— 1)

ning by the third card is — , ^ .,. H 7 =^-7 -^ ; and so on.

° '' n n{n—l) n{n— 1) [ii — z)

Hence the chance of winning by one or other of the first m
cards is

mjj m (m — 1) p^ m (m — 1) {m — 2) p^

"^"' O n {n - 1) "^ 1.2.3 7i (n - 1) (w -2) "" *'*

And the entire chance of winning is found by putting

m = - , so that it is
P

MONTMORT. 121

1 n —J) {n —p) (n — 2p)

i " 1 . 2 (7i - 1) "^ 1 . 2 . 3 (?i - 1) (n - 2)

(n —p) (n — 2p) (n — 8/;)
""1.2. 3. 4(7-^-1) (w-2) (/i^Ts) +•••

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 formulaa to which we have
already alluded in Art. ISO; but the meaning of the formuloB 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. Sujipose we require
the sum of the squares of the first n triangular numbers, that is, the

sum of n terms of the series of which the r^^ term is \---^ — —^

Assume that the sum is equal to

an^ + hn^ + cn^ + dn^ + en + /*;

and then determine a, h, c, d, e, f by changing n into n-\-\ in
the assumed identity, subtracting, and equating coefficients. This
method is ascribed by Nicolas Bernoulli to his uncle John,

122 MONTMOET.

Nicolas Bernoulli also indicates another method ; he resolves
^(f + l)finto

r (r+l)(r + 2) (r + 3) _ r (r + 1) (r+2) 7- (r + 1)
1.2.3.4 1.2.3 "^ 1.2 '

and thus finds that the required sum is

??0^4-l) 02+2) (M+3)(n + 4) __ ^ (72 + 1) (w + 2) (?z + 3)
1.2.3.4.5 1.2.3.4

w (^ 4- 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 Bernoulli. 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-{-l 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 -\-iy^ 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-

MONTMORT. 123

posed a game of pure chance, or wliicli is the same thing, the
jjlayers 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 j^roblem, 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 pltitot j'ai manque de courage,
car je serois stir 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, 34^5, 350, 3G6, 875, 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

in — . On page 330 where he does advert to this point

he puts by mistake — instead of .

(2) The expressions given for a, h, c, ... on j^age 386 are

correct, excej)t that given for a ; the value of « is ^ , and not

■^ , 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 tliose afterwards
given by Laplace.

124 MONTMORT.

211. Althougli the earliest iiotice of the problem occurs in
the letter of Montmort's which we are now examining, yet the
earliest piiblicatioyi of it is due to De Moivre ; it is Problem XV.
of the De Mensura Sortis. We shall however speak of it as
Waldegraves Problem, from the person whose name we have found
first associated with it.

The problem is discussed by Laplace, Theorie . . . des Froh.
page 238, and we shall therefore have to recur to it.

212. Montmort refers on page 320 to a book entitled Traite
dii Jeu, which he says he had lately received from Paris. He says
it is un 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, 3-i6.

Montmort complains of a dearth of mathematical memoirs ; he
says, page 322,

Je suis etonne de voir les Journeaux de Lei23sic si degarnis de
morceaux de Matliematiques : ils doivent en partie leur reputation aux
excellens Memoires que Messieurs vos Oiicles y envoyoient souveiit : les
Geometres n'y trouvent plus depuis cinq ou six ans les memes ricliesses
qu' autrefois, faites-en des reproches a M. votre Oncle, et permettcs-nioi
de vous en faire aussi, Luceat lux vestra coram hominihus.

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 Waldcgrave'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.

MONTMORT. 125

Nicolas Bernoulli gives at the end of his letter an example
)f summation of series. He proposes to sum p terms of the
jeries 1, 3, 6, 10, 15, 21, ... He considers the series

1 + 3:c + 6aj' + lOa;^ + lox" + 21^' + ...

which he decomposes into a set of series, thus :

1 + 2a; + 3x' + ^x^ + 5;c* + ...

+ a; + 2a;' + 3^' + 4ic*+ ...

+ ir' + 2a;' + 3^'+...

+ 33^+2^'+...

+ 0;*+...
+ ...
The series in each horizontal row is easily summed to p terms ;

he expression obtained takes the form - when x = l, 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'Hojoital a insere dans son Analyse des infiniment petits, ...

The investigation is very inaccurately printed.

21 -i. The next letter is from Montmort to Nicolas Bernoulli ;
it occupies pages 337 — 317. 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 312 the following equation
as the result of one of the problems,

4m'-8m'+llM + 6 = 3'"-'\

and he says that this is satisfied approximately by m = of-J^ ; but
there is some mistake, for the equation has no root between
5 and 6. The correct equation should apparently be

which has a root between 51 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 ^, the skill of B
is q. 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
equality, and in the third also B is to have a point given to
him. Required 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 p^, and i?'s chance is ^ + '^qp.
u4's chance of success in the second game is p^ + ^p^q, and ^'s
chance is <f + 3^^. Hence ^'s chance of success in two games
out of three is

/ (/ + 3/2) +/ (2' + ^P) (p' + 3/2) +/ (q" + 32» ;
and this by supposition must equal ^ .

This agrees with Montmort's result by putting , for ^

and 7 for g, allowinc^ for a mistake which was afterwards

corrected ; see Montmort's pages 34^3, 350, 352.

217. The letter closes with the following interesting piece of
literary history.

Je ne sgai si vous slaves qu'on reimprime la Recherche de la verite.
Le R. P. Malbranche m'a dit que cet Ouvi-age paroitroit au commence-
ment d'Avril. II y aura un grand nonibre d'additions sur des sujets
tres importans. Yous y verres entr'autres nouveautes une Disserta-
tion sur la cause de la pesanteur, qui apparemment fixera les doutes
de tant de Sgavans hommes qui ne sgavent a quoi s'en tenir sur
cette matiere. II prouve d'une maniere invincible la necessite de ses
petits tourbillons pour rendre raison de la cause de la pesanteur, de la
durete et fluidite des corps et des principaux plienomenes toucliant la
lumiere et les couleursj sa theorie s'accorde le niieux du monde avec
les belles experiences que M. Newton a rapporte dans son beau Traite
Be Natura Lucis et Colorum. Je peux me glorifier auprcs du Pub-
lic que mes prieres ardentes et reiterees depuis j)lusieurs annces, ont
contribue a determiner cet incomparable Philosoplie a ecrire sur cette

MONTMORT. 127

matiere qiii renfcrme toute la Physique generale. Vous verres avec
admiration que ce grand hiomme a porte dans ces matieres obscures
cette nettete d'idces, cette sublimite de genie et d'invention qui bril-
lent avec tant d'eclat dans ses Traites de Metaphysique.

Posterity has not adopted the high opinion which Montmort
here expresses respecting the physical speculations of his friend
and master ; Malebranche is now remembered and honoured for
his metaphysical works alone, Avhich have gained the following
testimony from one of the gi'eatest critics :

As a thinker, he is perhaps the most profound that France has
ever produced, and as a writer on i)liilosopliical subjects, there is not
another European author who can be placed before him.

Sir William Hamilton's Lectures on Metaphysics, Vol. i. page 262 ;
see also his edition of Reid's Works, page 266.

218. The next letter is from Montmort to Nicolas Bernoulli ;
it occupies pages 3-52 — 360. We may notice that Montmort here
claims to be the first person who called attention to the theorem
which is now given in elementary treatises on Algebra under the
following enunciation : To find the number of terms in the expan-
sion of any multinomial, the exponent being a positive integer.
See Montmort's page 355.

219. Montmort gives in this letter some examples of the recti-
fication of curves ; see his pages 35G, 357, 359, 360. In particular
he notices" one which he had himself discussed in the earty days
of the Integral Calculus, when, as he says, the subject was well
known only by five or six mathematicians. This example is the
rectification of the curve called after the name of its inventor De
Beaune ; see John Bernoulli's works. Vol. I. pages 62, 63. AMiat
Montmort gives in this letter is not intelligible by itself, but it can
be understood by the aid of the original memoii*, which is in the
Journal des Scavans, Vol. xxxi.

These remarks by Montmort on the rectification of curves are
of no great interest except to a student of the history of the Inte-
gral Calculus, and they are not free from errors or misprints.

128 MONTMORT.

220. Montmort quotes the following sentence from a letter
written by Pascal to Format.

Pour vous parler francliement de la Geometrie, je la trouve le plus
haut exercice de I'espritj mais en meme temps je la connois pour si
invitile, que je fais peu de difference entre un homme qui n'est que
Geometre et un habile Artisan; aussi je I'appelle le plus beau metier
du monde; mais enfin ce n'est qu'un metier: et j'ai souvent dit qu'elle
est bonne pour faire I'essai, mais non pas I'emploi de notre force.

Montmort naturally objects to this decision as severe and humi-
liating, and probably not that which Pascal himself would have
pronounced in his earlier days.

221. The next letter is also from Montmort to Nicolas Ber-
noulli; it occupies pages 361 — 370. Montmort says he has just
received Do Moivre's book, by which he means the memoir De
Mensura Sortis, published by De Moivre in the Philosophical
Transactions ; and he proceeds to analyse this memoir. Montmort
certainly does not do justice to De Moivre. Montmort in fact
considers that the first edition of his own work contained im-
plicitly all that had been given in the De Mensura Sortis; and he
seems almost to fancy that the circumstance that a problem had
been discussed in the correspondence between himself and the
Bernoullis was sufficient ground to deprive De Moivre of the credit
of originality. The opinion of Nicolas Bernoulli was far more favour-
able to De Moivre ; see Montmort's pages 862, 375, 378, 386.

De Moivre in his Miscellanea Analytica replied to Montmort,
as we shall see hereafter.

222. On his page 365 Montmort gives some remarks on the
second of the five problems which Huygens proposed for solution ;
see Art. 35.

Suppose there are three players ; lot a be the number of
white balls, and h of black balls ; let c = a-\- h. The balls are
supposed not to be replaced after being drawn ; then the chance of
the first player is

a h{h-V) (h-2)a b(h-l) .,.(h-5)a
,c+c(c-l)(c-2)(c-3) c(c-l) ... (c-6) '^ '"

MONTMORT. 129

Montmort takes credit to himself for summing this series, so as
to find its value when a and h are large numbers ; but, without
saying so, he assumes that a = 4. Thus the series becomes

4l&f|c-l \g- ^ |c — *7

\h_ ' |5-3 ' \h-Q

Let p = h + ^, then c=p+l] thus the series within brackets
becomes

+ (p-(i)(p->7)(p-8) + ...

Suppose we require the sum of n terms of the series. The
r^^ term is

(p-Sr + S) (^-'3r + 2) (^^-3r + l) ;

assume that it is equal to

where A, B, C, D are to be independent of r.
We shall find that

A=j>{p- 1) {p - 2),
B=-(9/-4.5;>+()0),
(7=54p-216,
i>=-162.

Hence the required sum of n terms is

np (p - 1) {p - 2) - '^^^ (V- ^op 4- 60)

n{n-l){n-^) __ n {n - 1) {n - 2) (7^ - S)

^ 1.2.3 K'^W--^^) 1.2.3.4 ^'"

This result is sufficiently 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 oip.

In the problem on chances to which this is subser\dent we

should have to put for ?i the greatest integer in -^ .

130 MONTMORT.

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 871 — 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 Holland, 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 's Gravesande.

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 800 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 875 — 887. It contains some remarks on
the game of Her, and some remarks in reply to those made by
Montmort on De Moivre's memoir De Mensura Soi'tis. The most
impoj'tant 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 as that which he preferred to every thing else which he had
produced on the subject; see page 881. The approbation which
he thus bestows on his own work seems well deserved.

225. Thie next letter is also from Nicolas Bernoulli to Mont-
mort ; it occupies pages 388 — 893. 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 ])rohahility of the hiHh of a male is to
the probability of the birth of a female as IS to 17 ; he then shews
that the chances are 43 to 1 that out of 14,000 infants the males
will lie between 7037 and 7363. His investigation involves a
general demonstration of the theorem of his uncle James called
Bernoulli'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 demontre une sembla])le chose dans son Traits De Arte Con-
jectandi, qui s'imprime a present a Bale, . . .

226. TJie next letter is from Montmort to Nicolas Bernoulli ;
it occupies pages 395 — 400. Montmort records the death of the
Duchesse d'Angouleme, 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 Pr emotion Physique ^ ou Action
de Dieu sur les Creatures demontree 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 Commerciiim Epistolicum which he
says M™ de la Societe Royale ont fait imprimer pour assurer a
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 MONTMORT.

the solution of Waldegrave's problem, and the other apparently
the demonstration of James Bernoulli's theorem : see Arts. 224, 225.
Montmort says, page 400,

Tout cela etoit en verite bien difficile et d'un grand travail.
Yous etes ini terrible homme; je croyois que pour avoir pris les de-
vants je ne serois pas si-tot ratrappe, mais je vois bien que je me suis
trompe: je suis a present bien derriere vous; et force de mettre toute
mon ambition a vous suivre de loin.

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,

Derange comme je le suis par I'autorite de M. Newton, et d'un
si grand nombre de sgavans Geometres Anglois, je serois presque tente
de renoncer pour jamais a I'etude de la Physique, et de remettre a
sgavoir tout cela dans le Ciel; mais non, I'autorite des plus grands
esprits ne doit Y>oijit nous faire de loi dans les clioses oil la raison
doit decider.

228. Montmort gives in this letter his views respecting a
History of Mathematics ; he says, page 399,

II seroit a soidiaiter que quelqu'un voulut prendre la peine de

nous apprendre comment et en quel ordre les decouvertes en Mathe-

matiqucs se sont succedees les unes aux autres, et h qui nous en avons

r obligation. On a fait I'Histoire de la Peinture, de la Musique, de

la Medecine, &c. line bonne Histoire des Matliematiques, et en par-

ticulier de la Geometrie, seroit un Ouvrage beaucoup plus curieux et

plus utile : Quel plaisir n'auroit-on pas de voir la liaison, la connexion

des raetliodes, 1' enchain em ent des difFerentes theories, a commencer

depuis les premiers temps jusqu'au notre ou cette science se trouve

portee a. un si haut degre de perfection. II me semble qu'un tel

Ouvrage bien fait pourroit etre en quelque sorte regarde comme I'his-

toire de I'esprit humaiiij puisque c'est dans cette science plus qu'eu

toute autre chose, que I'homme fait connoitre I'excellence de ce don

d'intelligence que Dieu lui a accorde pour I'clever au dessus de toutcs

les autres Creatures.

MONTMORT. 1.33

Montmort himself had made some progi'ess in the work which
he here recommends; see Art. 137. It seems however that his
manuscripts were destroyed or totally dispersed ; see Montucla,

Ilistoire 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 from 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 foTU'th and fifth present no difticulty, but
that the third is much more difficult. He says that it took him

134? 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 a 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 B is expressed by the series

77 + ^2 + 7^ + pi + • • • '^^^ 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 imj^jossible to secure.

CHAPTER IX.

DE MOIVRE.

233. Abraham De Moivre was bom at Vitri, in Cliampagne,
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 instruction in mathematics and answers to questions
relatiuGf to chances and annuities. He died at London in 1754.

John Bernoulli speaks thus of De Moi^Te in a letter to
Leibnitz, dated 26 Apr. 1710; see page 847 of the volume cited
in Art. 59 :

...Dominus Moy^Taeus, insignis certe Geometra, qui liaud dubie
adluic haeret Loudini, luctans, ut audio, cum fome et miseria, quas ut
depellat, victum quotidianum ex informationibus adolescentum petere
cogitur. O duram sortein hominis! et parum aptam ad excitanda
ingenia nobilia; quis non tandem succumberet sub tam iniquae foi-tunae
vexationibus ? vel quodnam ingenium etiam fervidissimum non algeat
tandem ? Miror certe MoyvTaeum 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 asvlum in England, it would be

18G DE MOIYRE.

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 Soi^tis, sen, de Probabili-
tate Eventuum in Ludis a Casu Fortuito Pendentihus. Autore
Abr. De Moivre, RS.S.

The number is stated to be for the months of January,
February, and March 1711 ; it occupies pages 213 — 261? 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 Calculating
the Pi'ohabilities of Events in Play. The first edition of this work
appeared in 1718 ; it is in quarto and contains xiv + 175 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 Chances ; 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 Robartes, 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 says 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-
sojyJiical 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 Kaclnor); who, upon occasion of a French Tract, called
L Analyse des Jeux de Hazard, which had lately been published, was
i)leased to propose to me some Problems of much greater difficulty than
any he had found in that Book ; which having solved to his Satisfaction,
he engaged me to methodize those Problems, and to lay down the 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 i)ub-
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 memoir consists of twenty-six Problems, besides
a few introductory remarks which exj^laiu 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 G points ; and ^-I'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 tlie 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 Avinning a single game, supposing that A can give B two games
out of three ; the fourth problem is of a similar kind, supjDosing

138 BE MOIVRE.

that A can give B one game out of three : see Problems I. and ii.
of the Doctrine of Chances.

24^1. 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 Art. 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 89. We have already given the history
of this Lemma in 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 Art. 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 vil. of the Doctrine of
Chances.

246. The tenth problem is Problem viii. of the Doctrine of
Chances, where it is thus enunciated :

Two Gamesters ^ and -5 lay by 24 Counters, and play with three
Dice, on this condition ; that if 1 1 Points come np, A shall take one

DE MOIVRE. 139

Counter out of tlie heap; if 14, ^ shall take out one; and he shall be
reputed the winner who shall soonest get 1 2 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 Prohlemata propter speciem aliquam
affinitatis inter se confundantur. Problema sequens videtur affine
superiori.

After enunciating his ninth problem he says,

Problema istud a superiore in hoc diifert, quod 23 ad pluriraum
tesserarum jactibus, ludus necessano finietur ; cum Indus ex lege supe-
rioris problematis, posset in aeternum continuari, propter reciproca-
tionem lucri et jacturse 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 fur
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. 177. These problems are reproduced in
a more general form in the Doctrine of Chances, pages 117 — 123.
Respecting these two problems Montmort says, on his page 366,

Les Problemes 16 et 17 ne sont que deux cas tres simples d'un
meme Probleme, c'est presque le seul qui m'ait echape de tous ceux que
je trouve 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 Bernoullis.

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 obUged to entrust the Care of a new Edition of it to one of
his Eriends ; 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 Aymuities together; as they stand in the
late imj-yroved edition of the Treatise on that Subject. An A'ppendix

DE MOIVRE. 14:1

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 30 — 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 this 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 it,
I had certainly done the Justice to its Author, to have owned that he
had not only illustrated Buy gens' s Method by a gi'cat variety of well

142 DE MOIVRE.

chosen Examples, but tliat be bad added to it several curious things of
bis own Invention.

Since the printing of my Specimen, Mr. de Monmort, Author of the
Analyse des jeux de Hazard^ 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
S3 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-i-1 terms of the expan-
sion of (a + hy and dividing by (a + by. 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 modern
method which seems more simple and more direct, by which we
begin with finding the chance that an event shall happen jvst r
times and deduce the chance that it shall happen at least r
times.

DE MOIVRE. 113

258. De Moivre notices the advantage arising from employing
a single letter instead of two or three to denote the probaljility of
the happening of one event. Thus if x denote the probability of
the happening of an event, \ —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){\-z)

will 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, Hiiygens, 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.

2G0. The Doctrine of Clicuices contains 7-i 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 ^ in a single
game. Problem I. means that it is an even chance that A "wall win

1 1

three o^ames before B wins one : thus p^ = cv- Hence x> = -^^^ , and

7 = 1 — 7777 . Problem li. means that it is an even chance that A

will win three games before B wins two. Thus p^ + Aip^q = ^ ; which

must be solved by trial.

These problems are simple examples of the general formula in
Art. 172.

262. Problems ill, IV, and V. are included in the followin

144 DE MOIVRE.

general enunciation. Suppose a the number of chances for the
happening of an event in a single trial, and h 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

If
the event fails x times in succession is -. tt^ . And by 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

to -X . Thus

}f 1

{a + hy 2 '
from this equation x may be found by logarithms.

De Moivre proceeds to an approximation. Put - = q. Thus

X log [ 1 4- - ) = log 2.

If ^ = 1, we have x=l. If 5' be gi'eater than 1, we have by

expanding log ( 1 + - J ,

where log 2 will mean the logarithm to the Napierian base. Then
if q be large we have approximately

x= q log 2 = zTTzq nearly.

De Moivre says, page 87,

Thus we have assigned the very narrow limits within which the ratio
of £c to q is comprehended ; for it begins with unity, and terminates at
last in the ratio of 7 to 10 very near.

But X soon converges to the limit 0.7^', 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 q is moderately large is the

DE MOIVRE. 145

element of truth in the mistake made by M. de Mdre ; 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 = S.

The chance that the event will happen at least 3 times in x
trials is equal to the first x — 2 terms of the expansion of

a h

\a + b a + bj *
and this chance by hypothesis is - . Hence the last three terms

of the expansion will also be equal to ^ , that is,

W + xV-' a + ^^^ I'-'' «' = I (« + W'

If ^ = 1 we find x = o.

If q be supposed indefinitely great, and we put - = z, we get

where e is the base of the Napierian logarithms.

By trial it is found that 2 =2675 nearly. Hence De Moivre
concludes that x always lies between oq and 2675(;^.

264. De Moivre exhibits the following table of results ob-
tained in the manner shewn in the two preceding Ai'ticles.

A Table of the Limits.

The Value of x will always be

For a single Event, between \q and O'GOSg'.
For a double Event, between 2)q and l-GTS^-.
For a triple Event, between 5q and 2 675^.
For a quadruple Event, between ^q aud 2>(dl2q.
For a quintuple Event, between 9^' and i-GTO^-.
For a sextuple Event, between \\q and b-ij^Sq.
ka.

146 DE MOIVRE.

And if the number of Events contended for, as well as the number

q be pretty large in respect to Unity; the number of Trials requisite for

.... ^n—\ . ,

those Events to happen n times will be — ^ q^ or 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 in 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, in 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 MOIVRE. 147

Fermat's metliod, and is intended to lighten as mucli 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.

2G8. Problem vii. 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 lonof 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
Prohahilities in the Cabinet Cyclopcedia.

We have spoken of Problem viii. 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
;; 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 ^.

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 jn-opose it. This Problem is in my Opinion one of
the most curious that can be propos'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, sjyectatissimo
viro, 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 ihe 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

aQ _ hQ

Game is ^ — j and for the same reason the Gain of each individual

a + 0

aG — hL
Game would be j- , if the Suras deposited bv A and B were re-

spectively L and G. Let us therefore now suppose that they are L
and Gj then in order to find the whole Gain of A 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

/-7 7 7-

second case will be Rq -^px or restoring the values of H and >S',

a — b ° '

qa^xa^-b^-pb^xa'^-b^ i.. -,. -, , aG-bL
^^^,_f^^, 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 MOIVRE. 14y

the ninth problem of the De Mensura Sortis. De Moivre how-
ever renounces for himself the claim to the merit of the solu-
tion. This renunciation he repeats in the Miscellanea Analytica,
page 206, where he names the author of the simple solution
which we have already given. He says,

Ego vere illucl ante libenter fassus sum, idque ipsum etiamnum
libenter fateor, quamvis solutio Problematis mei noni causam fortasse
dederit hiijus solutionis, me tamen nihil juris in eam habere, eamque
CI. ilhus Autori ascribi lequum esse.

Septem aut octo abliinc annis D. Stevens Int. Tempi. Socius, Yir
ingenuus, singulari sagacitate prseditus, id sibi propositum habens ut
Problema superius allatum solveret, hac ratione solutionem facile asse-
cutus est, quam mihi his verbis exhibuit.

Then follows the solution, after which De Moivre adds,

Doctissimus adolescens D. Cranmei', apud Genevenses MatliematicaB
Professor dignissiraus, cujus recordatio seque ac Collegae ejus i)eritissimi
D. Calandrin mihi est perjucunda, cum superiore anno Londini com-
moraretur, narravit milii se ex Uteris D. Nic. Bernoulli ad se datis acce-
pisse CI. Yirum novam solutionem hujus Problematis adeptum esse,
quani prioribus autor anteponebat ; cum vcro nihil de via solutionis
dixerit, si mihi conjicere liceat qualis ea sit, banc opinor eandem esse
atque illam quam raodo 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 refen^ed
in Art. 152. A corollary was added in the second edition and
was expanded in the third edition, on which we Avill 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 (7,
where a ■\- h + c = n. Required the chances wdiich A, B, and G
have respectively of being the first to throw a corresponding face.
It may be easily shewn that the chances are proportional to
air, (h + c) hn, and (h + c) {a + c) c, respectively. De Moivre, in
his third edition, page Qo, seems to imply tliat before the order
was fixed, the chances would be proportional to ^, h, 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 C, 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 Hkely 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 A's chance is

{6a' + 9a(h-i-c) + S (h' + c') + She}
Q{n'-{b + c) (c + a) (a + h)}

272. Problem xii. appeared for the first time in the second
edition, page 24^8, 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 Ai^s Co7vjectandi,
which we have explained in Art. 106.

273. Problem xiii. relates to the game of Bassette, and
Problem XIV. to the game of Pharaon; these problems occupy
pages 69 — 82 of the work. We have already sufficiently noticed
these games ; see Arts. 154, 168. 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 modern
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, h, 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 MOIVRE. 151

chance of taking a first is ^ ; and tliere are then five things left,

and the chance of now taking 5 is ^ . Therefore the required

chance is -^ . Then De Moivre says,

Since the taking a in the first i)lace, and h in the second, is hut 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 explained 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 difiiculty.

The mil} 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.

27C. 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

1111 1

- + — r-T + — -^ + — 7-i. + ...+

n n-\-l 71 + 2 ?i + 3 a — 1

_ a 1 1 A (l^ 1\ B (1 r

~ ^""^7^ + 2;^ "" 2^ "^ 2 \n' " ^; "^ I W a\

where A=\. ^^'W ^=A' -

152 DE MOIVRE.

As De Moivre says A, B, (7, ... 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 xxxil. 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 xxxiil. is To find at Pharaon how much it iiT
that the Banker gets per Cent of all the Money that is adventured.
De Moivre in 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 £ to play together, that the Chances they have
respectively to win are as a to 6, 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,

Itj 1 ^ + ^ + FTi)^ + ^TTif + • • • '" *"-^'"'^"™ } '

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 MOIVRK. 153

De 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,/, &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 time
be displaced.

Let n be the number of the letters ; suppose that j) specified
letters are to be in their places, q specified letters out of their
places, and the remaining n —p — q letters free from any restric-
tion. The chance that this result will happen is

n{n—l)...(n—j)+l)\ 1 n—p 1.2 (n— p)(?i— ^ — 1)
This supposes that p is greater than 0 ; if ^ = 0, the result is

Iw"^ 1.2 n{n-l) '"

If we suppose in this formula q=^m — 1, we have a result akeady
implicitly given in Art. IGl.

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 h in the second place is — , — -:r^ : hence the

^ ^ n{n—i)

chance that a is in the first place and h not in the second place is

1 1

n n (ii — 1) *

Similarly the chance that a, h, c are all in their proper places is
1

/ IN / — K^ ; subtract this from the chance that a and h are in
n [71 — 1) (n — 2) '

their proper places, and we have the chance that a and h are in
their proper places, and c not in its proper place : thus this chance is

1 1

(n - 1) n («. - 1) {n - 2)

15^ DE MOIVRE.

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 + J denote the chance that h 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, h, c are in their proper places, and d, e out of theirs.

n ' n(n -1) ' n{n — l)(n — 2)

1 ^

n(n-l){n-2){n-B)~'^'"'

Then we have the following results :

+ h =r

+ h + a = s

+ h — a = r—s (1)

+c+h =s
+ c-\-h + a = t

+ c + h-a = s-t •. (2)

+ c - a =r-s by (1)

+ c - a + J = s-t by (2)

+ c-a-b= r-2s + t (3)

+d+c+h =t

■Yd+G^-h-{-a = v

+ d + c-\-h-a = t-v (4)

+ J+c-a ^s-t by (2)

+ c?+c — a + &= t — v by (4)

■\-d-\-c-a-h= s-2t^v (5)

■\-d-h-a =r-2s-\-t by (3)

-\-d—h — a-\-c= s-2t-\-v by (5)

d — h — a — c— r— 35 + 8^—1? (6)

It is easy to translate into words any of these S3rmbolical pro-
cesses. Take for example that which leads to the result (2) :

DE MOIYRE. 155

this means that the chance that c and h are in their proper places
is s ; and this we know to be true ;

-\- c-\-'b-\- a = t,

this means that the chance that c, h, 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 5 — ^ ; and this
is expressed symbolically thus,

-{-c-\-h — a = s — t

Similarly, to obtain the result (8) ; we know from the result (1)
that r — 5 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 5 — /5 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 h out of their proper places is r — 2s + ^ ; and this
result is expressed symbolically thus,

282. De Moivre refers in his Preface to this process in the fol-
lowing terms :

111 the 3oth and 36th Problems, I explain a new sort of Algebra,
whereby some Questions 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. Montmort,
Author of the Analyse cles 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 Moi^Te 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,

loG DE MOIVRE.

...tlien let all the quantities I, r, s, f,v, &c. be written down with
Signs alternately positive and negative, beginning at 1, if]) be = 0; at r,
if^9 be = 1; at 5, if p be = 2; &c. Prefix to these Quantities the Co-
efficients of a Binomial Power, whose index is = q; this being done,
those Quantities taken all together will express the Probability re-
quired.

284. The enunciation and solution of Problem xxxvi. are as
follows :

Any given number of Letters a, h, c, d, e, /, &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 : supj)ose also that p 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

r must here be made = — , s = —, =^- , t — — ; 7T-7 777 , &c., and

n n {ii — 1) n {n —1) [71 — 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

^ ^"^^ 1.2 ^ 1.2.3 ^ 1.2.3.4 '^'^''•

of which the number of Terms is equal to q+ 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 I {n-l){n-2l) 1 {71-I) {n-2l) (n-Sl)

2 n-l 6 {n - 1) {n - 2) '^ 24: (n-l) {n - 2) (n - 3) '^*

of which the number of Terms is equal to — j— .

DE MOIVRE. loT

285. De Moivre then adds some Corollaries. The follo^\ing
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 iij-l 1 ill - I) {n - 21) _ j_ {n -l)(n- 21) {n - 3Q
2 n-1^ ^\r-1){ji-2) 24< (n- 1) {n-2) (n-S)

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. Theorie. . .des Proh. pages 217 — 222.
The part of Problems xxxv. and xxxvi. which' De Moivi-e 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 — ij — q free from any restriction ; this part
seems peculiar to De MoIato, 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

111

1 — o + ^ " oT + • • • wi 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 follows :

If A and B each holding a Pack of Cards, pull them out at the same
time one after another, on condition tliat every time two like Cards are

158 DE MOIVRE.

pulled out, A shall give B a Guinea; and it were required to find what
consideration £ ought to give A to play on those Terms : the Answer
will be one Guinea, let the number of Cards be what it will.

Altho' this be a Corollary 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 r-:: == 1.

It may be interesting to deduce this result from the formulae
already given. The chance that out of ti cards, 2^ 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 cmy p cards will be in their places, and all the
rest out of their places will be obtained by multiplying the pre-

ceding result by - — ^= . And since in this case B receives

\n — p I p

p guineas, we must multiply by p to obtain 5's advantage. Thus
we obtain

p-l \ '2 [3 ' [^

n — p

Denote this by <^ {p) ; then we are to shew that the sum of
the values of <^ {p) obtained by giving to p all values between
1 and n inclusive is unity.

Let y^r (ti) denote the sum ; then it may be easily shewn that

'f(n + l)-'f {71) = 0.

Thus -yjr (n) is constant for all values of n ; and it = 1 when
71 = 1, so that -^ {71) 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 iu any number of Packs, will

DE MOIVRE. 159

easily be found by our Metliod : thus if tbe number of Packs be ^, 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

n'-" 2[n(n- l)]''' ' [3 {w (ji - 1) (w - 2)]^-^

[4 [n (71-1) {n-2) (n-S)\

—, &c.

k-2.

Laplace demonstrates this result; see Theorie . . . des Prob.
page 224.

290. Problems xxxvii. and xxxviii. 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 Rohartes, Esq;, in the FhilosopTiical Trans-
actions 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 w^hich 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 altlio' 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 XLil. form a connected set. Pro-
blem XXXIX. is as follows :

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 j9 -f 1 be the number of faces on the die, n the number
of casts, /the number of faces which A undertakes to fling. Then
-4's expectation is

ICO DE MOIVKE.

{p + 1)"

(_p+l)"_/^« + /01^(^_l)»

_/(/-lK/-2)(^_,)„^..

Be Moivre infers this general result from the examination
of the simple cases in which f is equal to 1, % 8, 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 Kules 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 happily 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, Vol. ii. 1785; Laplace, Memoir es..,
2Mr clivers Savans', 1774, Theorie... cles Proh. page 191 ; Trembley,
Memoires de V 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 follows :

To find in how many Trials it will 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, suppose 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 ^ + 1, ^, ^ — 1, j9 — 2, are in Geometrical

DE MOIYRE. 161

Progression, which supposition he says "will very little err from
the truth, especially if the proportion of ^ to 1, be not very small."

Put r for ; thus the equation becomes

1 / 1 /(/- 1) 1 /(/- 1) (/- 2) 2_ . ^1.
1?^""^ 1.2 r''' \S r''''^'" 2'

that is ^1__)=2.

Hence -■ = 1 _ f _ ] ,

and then n may be found by logarithms.

De Moivre says in his Preface respecting this problem,

The 40th 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. Halley, Secretary to the Royal Society,
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 result with that furnished by his
own method ; see Theorie ... des Proh. pages 198 — 200.

293. Problem XLI. is as follows :

Supposing a regular Prism having a Faces marked i, h Faces
marked ii, c Faces marked iii,* d Faces marked iv, kc. 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
edition of the Doctrine of Chances.

Let a + h ■{- c -{■ d + ...=s\ then the Probability required
will be

1 [," _ {(, _ „). + (, _ j)«j + {s-a- in

162: BE MOIVRE.

If it be required that some of the Faces marked I, some of
the Faces marked ii, and some of the Faces marked ill be
thrown, the ProbabiUty required will be

-f (s-a-hy+ {s-h-cy+{s-c-ay

— {s — a — h — cy

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 XLiil. is as follows :

Any number of Chances being given, to find the 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 PhilosojyJiical Transactions, 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 peculiar to
De Moivre.

The discussions which De Moivre gives of the particular
cases of three players and four players are very easy and satis-
factory ; 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 pla^^er'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 of Hajzard; 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 wliat the problem is,
which he is considering. With respect to this problem, De Moivre

says, page 165,

11—2

164< DE MOIVRE.

After I had solved the foregoing Problem, which is about 12 years
ago, I spoke of my Solution to Mr. Henry 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 V7% 175 of the first
edition, which is not reproduced in the other editions.

297. Problems XLVIII. and XLix. relate to the game 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 table of chances, which De Moivre says was
drawn up by Francis Eobartes, 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 Honouy^s in the game. Thus,
for example, he says that the ^probability that there are no Honours

on either ride is ^c ' ^^^ of course means that the Honours

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. 165

Honour, and 2nd, that in which the card turned up is not an
Honour. Thus we should have for the required probability

_4 8 25 . 26 . 25 9^ 4^ 25 . 24 . 26 . 25 ^
13 ■ T * 51750 . 49 "*" 13 * 1 . 2 ' 51 . 50 . 49 . 48 ^

and this will be found equal to -^^ .

loob

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 will 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 vii. 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 in 8192 to get iv.

5° That there are 715 Chances in 8192 to get iii.

6° That there are 1287 Chances in 8192 to get n.

7" That there are 1716 Chances in 8192 to get i.

All this will appear evident to those who can raise the Binomial
a + b 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 5; 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 Dealer
has precisely iii Trumps, besides the Trump Card ] The Answer,

. rp . . 4662
by Tab. i. is .^^^^ ; ...
158y5

166 DE MOIVRE.

In tlie 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 Remark
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, Kepic or Lurch j 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
followinof 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 B^ and B has 1 chance to beat A ; but
there is one Chance which intitlcs them both to withdraw their own
Stake, which we suppose equal to s j 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 his 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 gi-eat skill in these Speculations of Chance have themselves
fallen into that error.

DE MOIVRE. 1G7

This problem was not in the first edition. The gain of A

. 1

IS 75.

301. Problem LVir, which was not in the first edition, is as
follows :

A and B playing together deposit £s apiece ; A lias 2 Chances to
win s, and B 1 Chance to win 5, whereupon A tells B that he will
play with him. upon an equality of Chance, if he B will set him 2s to I5,
to which B assents : to find whether A has any advantage or disad-
vantage by that Bargain.

In the first case ^'s expectation is - s, and in the second,

it is ^ 5 ; so that he gains ^ s by the bargain.

802. 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 little 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 MOIVRE.

803. 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 Z> ; the loser in each game is to give a counter to his
adversary : required the probability tliat when or before a certain
number of games has been played, one of the players will have won
all the counters of his adversary. It will be seen that the words
in italics constitute the advance whi'ch this problem makes beyond
the more simple one discussed in Art. 107.

De Moivre's Problems LVIII. 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 n = 2 or 3, De Moivi*e
lays down a General Rule, thus :

A General Rule for determining what Probability there is that
the Play shall not be determined in a given number of Games.

Let 71 be the number of Pieces of each Gamester. Let also n-hd
be the number of Games given; raise a + h to the Power n, then cut off
the two extream Terms, and multiply the remainder by aa + 2ab + hb :
then cut off again the two Extreams, and multiply again the remaiiM^er
by aa + 2ab + hb, still rejecting the two Extreams; and so on, makiog

as many Multiplications as there are Units in ^d ; make the last Pro-

duct the Numerator of a Fraction whose Denominator let be (a + b)"'^'\
and that Fraction will express tlie Probalnlity required, ; still ob-
serving that if d be an odd number, you wj-ite d—1 in its room.

For an example, De Moivre supposes n = 4<, d= 6.

Raise a+h to the fourth power, and reject the extremes ; thus
we have 4<a^b + MV + ^a¥.

Multiply by a^ + 2ab + V^, and reject the extremes ; thus we
have l^a'h' + 2^a%' + Ua^h\

Multiply by <^ + 2ah + W, and reject the extremes ; thus we
have 48a'Z>' + Q%a%' + ^Mh\

Multiply by a^+2«& + Z>^ and reject the extremes; thus we
have lUa%' + 232a'Z>'^ + l(j^a'h\

Thus the probability that the Play will not be ended in
10 games is

DE MOIVRE. 169

,10

(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 be
abbreviated by omitting the a and h, 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 ivill be ended in an assigned number of games.
Thus if ?i = 4 and d = ^, this probability will be found to be

a' + b' ^a'b + ^ab' l^a%'' -\-\^a%' 48a^6^ + 48a^^>^
{a-^by^ {a + bf "^ {a + hf "^ . {a-\-bf' *

Now here we arrive at one of De Moi\T:e's important results ;
he gives, luithout demonstj^ation, 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, Theorie . . . des Prob., page 229,
that the chance of A for winning precisely at the (n + 2x)*'* game
is the coefficient of T"^^ in the expansion of

( i + ^{i--^abf) r M - V(l - -^abt') r '

I 2 l^-j 2 J

where it is supposed that a + b = 1.

Now the denominator of the above expression is known to be
equal to

^ 1.2 [3 0 -t ...

where c = abt^ ; see Differential Calculus, Chapter ix.

170 . DE MOIVRE.

We can tlius obtain by the ordinary doctrine of Series, a linear
relation between the coefficient of f^^"" and the coefficients of the
preceding powers of t, namely, r'^^^ ^""^^"^ ... This is De
Moivre's first law; see his page 198.

Again ; we may wiite the above fraction in the form

JV" (1 + c"iY"^") '

, 1 + ^(1 _ 4^ahe)
where N = --^ ;

and then by expanding, we obtain

The coefficient of f'' in N'" is known to be

^j^n {n-{-.x-\-l) {n -{-x + 2) ... {n ■\- 2x — 1) ^

' X '

see Differential Calculus, Chapter ix.

Similarly we get the coefficient of f^'' in N-'\ of i'^'*" in
iV"^**, 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.

805. De Moivre's Problems LX. LXi. LXii. are simple ex-
amples formed on Problems LVIII. and Lix. They are 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 win-
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. Su2:>posing 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 ?

DE MOIVRE. 171

806. Problems LXiii. and LXiv. 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, tuithout de-
monstration, two general laws, which we will now give.

Laplace shews, T]ieorie...des Prob. page 228, that the chance
of A for winning precisely at the {n + 2u?)*'^ game is the coefficient
of f^'^'^ in the expansion of

f 1 + V(l - 4c) ) " f 1 - V(l - 4c) 1 "^

r-{

1 9 9

j 1 + ^(1 _ 4c) r"^'^ ( 1 - v(i - 4c) I '"^'* •

Let s) — ^^ denoted by h \ then the fractional expression

which multiplies a"^*" becomes by expansion, and striking out 2h
from numerator and denominator,

ay^-' m{m-V){m-2) /1\'""%2 w(772-l)(w-2)(m-3)(m-4) [ly^,

. ^lY"--^"-^ , (m+^2)(m+^^-l)(m+?i-2) /l^""^"-",, ,
(^+^)(2J + g %) ^^+-

We have to arrange the denominator according to powers of
t, and to shew that it is equal to

where 7 = m + n — 2.

Now, as in Art. 30-A, we have

{

1 + V(l - 4c) I" ^ 1 1 - V(l - 4c) r

= l_,e + '-(p^8)^,_r(,-4)(r-o)^3_^ __.
and the left-hand member is equivalent to

172 DE MOIVRE.

Differentiate both sides with respect to t observing that

——-=:^aot. inns,
dt

^ 2 { r - '-^^ aU + ' ^' -f ^^ - ^^ (abtr -...

Now put r = Z + 3 ; and we obtain the required result.
Thus a linear relation can be obtained between the coefficients
of successive powers of t.

This is De Moivre's first law ; see his page 205.

1 _l_ a/(1 — 4c)
Again ; let iV= ^ ~ ; then the original expression

becomes

'\Jni+n /-| m+n 7vr-2wt-2n\

= d'fN-'' (1 - c'"iV^"'"0 (1 - c"*^"^-'"^-'")"'-
We may now proceed as in the latter part of Art. 304, to de-
termine the coefficient of r"*"^"".

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 unlimited cajntal 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 on- 6 be raised to the Power whose
Index is 7i + d; then if d be an odd number, take so many Terms of

that Power as there are Units in — ^ — j take also so many of the

Terms next following as have been taken already, but prefix to them
in an inverted oi^der, the Coefficients of the preceding Terms. But if
d be an even number, take so many Terms of the said Power as there

DE MOIVRE. 173

are Units in-d+l; then take as many of tlie Terms next following

as there are Units in ^ d, and prefix to them in an inverted order the

Coefiicients 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 + S)"^**.

Example I.

Supposing the number of Stakes, which A is to win, to be Three,
and the given number of Games to be Ten; let a+h be raised to the
tenth power, viz. a'' + lOa'h + ioa'bh + UOa' b' + 210a'b' + 252a'b'
+ 210a'' 6' + 120a' 6' + 45aa&' + lOab' + b'". Then, by reason that n^3,

and w + 6^=10, 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 + 120a''b^, and let the four Terms next following be taken
likewise without regard to their Coefficients, then prefix to them in an
inverted order, the Coefficients of the preceding Terms : thus the four
Terms following with their new Coefficients will be 120a^b^ + 4:5a^b^
+ 10a^b^+la^b^. Then the Probability which ^ has of winning three
Stakes of £ in ten Games or sooner, will be expressed by the following
Fraction

a'o ^ iQ^,9 J ^ 45^8^5 ^ UOa'b^ + UOa'b' + iSa'b' + lOa'b' + a'b'

' {a + by '

which in the Case of an Equality of Skill between A and B will be

A w 352 11

reducea to r— --r- or ^r^ .
1024 32

808. In De Moivre's solution there is no difficulty in seeing
the origin of his fii'st 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 a^b\ There is only one way in which ^'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 h in any one of the first three places,

17-i BE MOIVKE.

and fill up the remaining places with the letters aaaahlhhh in this
order ;- or put a in any one of the last seven places, and fill up the
remaining places with the letters aaahbhhhh in this order ; we thus
obtain the ten admissible cases.

The next term is 4t5a%^. There are forty -five ways in which
i?'s capital may be exhausted while A wins 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 aaaaabhh 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 aaahhhhh 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 aaaabhhh 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 h, so that in each arrangement there shall be n more
of the letters a than of the letters h 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 T -\-s things taken r — w at a time. For example, let
r = 6, s = 4, w = 3 ; then the number of different arrangements is

10 X 9 X 8 ,, , . ^^_

— — rr . that IS 120.

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.

809. 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 + 2ic)*'' game is
the coefficient of f^^"" in the expansion of

(l + V(i-4c)

n >

DE MOIVRE, 175

that IS a -^ — \ — ^^ a^ ;

[x '

see Art.. 804.

The chance of A for winning at or he/ore the (n -f 2^)*^' game
is therefore

a- I l+nah + '-^^-^^^ a'lf -\- ...

niii^-x + 1) (?i + a? + 2) ... {7i^2x-l) ,-,,
+ — ^ ^^ ^ ^^ -' ah

Laplace, T]ieorie...des Proh., page 235.

310. De Moivre says with respect to his Problem LXV,

In the first attempt that I had ever made towards solving tlie
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 mattei'S, 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 akeady 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 Moivre 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 LXiv.
Dr Oettinger says, Er erhiilt das namliche unhaltbare Resultat,
welches Laplace nach ihm aufstellte.

But there is no foundation for this remark ; 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.

812. 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 Games 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 number q of Stakes ; with this farther con-
dition, that £ during that whole number of Games may never have
been winner of the number j^ of Stakes.

813. De Moivre now proceeds to express his results relating
to the Duration of Play in another form. He says, page 215,

The Pules 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 very
tedious, and sometimes swell to that degree as to be in some manner
impracticable : to remedy which inconveniency, I shall here give an
Extract of a paper by me produced before the Poyal 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 of Chances,
pag. 149, and 150.

The paper produced before the Poyal Society does not appear
to have been published in the Philosophical Transactions; pro-
bably we have the substance of it in the Docty^ine 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 MOIYRE. 177

The demonstrations of the formulae 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
-^quationibus ad circulum vulgo notis ; the following is the pro-
perty : in elementary works on Trigonometry we have an expan-
sion of cos 7x6 in descending powers of cos 6 ; now cos nO vanishes

when nO 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 ;

let a = ^7- , then we have
2?i

1 = 2""^ sin a sin 3a sin 5a ... sin (2/ia — a) ;

also if n be even we have

cos n(^ = 2""^ [sin^ a — sin^ </>} {sin^ 3a — sin^ 0} . . .

. . . { sin^ {n — 3) a — sin^ <^} { sin^ (ti — 1) a — sin^ 0} :

see Plane Trigonometry, Chap, xxiii.

De Moivre uses the first of these formulae ; and also a formula
which may be deduced from the second by differentiating with
respect to (f), and after differentiation putting </> equal to a, or
3a, or 5a, ...

315. De Moivre applies his results respecting the Duration

178 DE MOIVRE.

of Play to test the value of an approximation proposed by Mont-
mort ; we have already referred to this point in 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 Theorie...des Proh. pages 225 — 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 Chances, form
a digression on a subject, which is one of De Moivre's most
valuable contributions to mathematics, namely that of Recurring
Series. He says, page 220,

The E-eader may have perceived that the Sohition 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 Yiew 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
Recurring 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 MOIYUE. 179

The general term of the expansion of (1 — r)~^ in powers of

r is ^-^ '" r^ ; the sum of the first n terms of

the expansion is equivalent to the following expression

^ ^ 1.2 ^ ^ n—\ i^J— 1

(l-r)"

This may be easily shewn to be true when n= 1, and then,
by induction, it may be shewn to be generally true. For

r«+i = r"|l-(l-r)},
so that

r-^^ + (^ + 1) r"^^ (1 - r) + ^'' "^ ^ ^l "^ ^^ r""^^ (1 _ r)^ + . . .
= r« |l _ (1 _,.)} +(«+!) r» (1 - r) |l - (1 - r)}

n{n-\-V\ \n + p— 2

= ^" + ^^" (1 -r) + ""^V ^'" (1 -0'+ • • • + ^ Ti 1 ''" (1 - ^')'

^ ^ l.z ^ ^ ?i — Iw — 1 ^

^ ,.« (1 _ ^)p.

\n\j) — 1

Thus the additional term obtained by changing oi into n + 1

\n -\-p— 1

is I — , r- r" as it should be ; so that if De Moivre's theorem is

\np—\. '

true for any value of ?2, it is true when n is changed into ii-\-l.

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 find the scale of relation
of the Series arising from the product of corresponding terms.

For example, let w^^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 4- gr'^ ; and let t'„a" be the general term
in the expansion according to powers of a of a proper Algebraical

12—2

180 I>E 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 u^Vn {ro)'\

We know by the ordinary theory of decomposing Recurring
Series into Geometrical Progressions that

where p^ and p^ are the reciprocals of the roots of the equation

and a^ and a^ are the reciprocals of the roots of the equation

1 —ina -\-pa^ = 0 ;

and R^, R^, A^, A^ are certain constants.

Thus u^v^ = R^A^ (p/^X + ^1^2 {pi^:T

this shews that the required scale of relation will involve four
terms besides unity. The four quantities p^a^, p^a^, p^a^, p^7^ will
be the reciprocals of the roots of the equation in z which is found
by eliminating r and a from

1 —fr + gr"^ = 0, 1 — ma + pa^ = 0, ra — z\

this equation therefore is

1 -fmz + {pP + gm^ — 2^p) z^ —fgmjpz^ -^-g^fz^ — 0.

Thus we have determined the required scale of relation ; for
the denominator of the fraction which by expansion produces
w„t;„ (ra)" as its general term will be

1 —fmra + {pf^+gm^ — ^gjp) ^V — fgmj^r^d ■\- g^j^r^a^.

De Moivre 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
differential Scale to that Power, only substituting ar for ?•, or leaving
the Powers of r as they are, if a be restrained to Unity; and that
Power of the other differential Scale will constitute the differential
Scale required.

DE MOIVEE. 181

This is very easily demonstrated. For suppose that one scale
of relation is (1 — of ; then by forming the ]3roduct of the cor-
responding terms of the two Recurring Series, we obtain for the
general term

=== a" {Rj>: + E,p: + B,p: +...]

Tliis shews that the general term will be the coefficient of
r" in the expansion of

(l-rapy {1-rap;)' (1 - rap^y '" '

and by bringing these fractions to a common denominator, we
obtain De Moivre's result.

822. De Moivre applies his theory of Recurring Series to
demonstrate his results relating to the Duration of Play, as we
have already intimated in Art. 313 ; and to illustrate still further
the use of the theory he takes two other problems respecting j)lay.
These problems are thus enunciated :

Lxx. M and N, whose proportion 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, j5 and S, concern themselves in the
Play, R takes the side of M, and S of N, and agree betwixt them, that R
shall set to S, the sum L to the sum G on the first Game, 2L to '2G on
the second, 3Z to ?>G on the third, 4Z to AG on the fourth, and in case
the Play be not then concluded, 5L to 5G 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 iV play; yet with this farther con-
dition, that the Sums, set down by them R and aS', 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 R is to be estimated before the Play begins.

Lxxi. If M and iV, whose number of Chances to win one Game
are respectively as a to h, play together till four Stakes are won or lost
on either side ; and that at the same time, R and S whose number of
Chances to win one Game are respectively as c to d, play also together
till five Stakes are won or lost on either side ; what is the Probability
that the Play between M and iV will be ended in fewer Games, than the
Play between R and S.

182 DE MOIVKE.

The particular case of Problem LXXI, in which a = h, and
c = d, was given in the first edition of the Doctrine of Chances,
13age 152.

823. Problems LXXII. and LXXIII. are important ; it will be
sufficient to enunciate the latter.

A and B playing together, and having a different number of Chances
to win one Game, which number of Chances I suppose to be respectively
as a to h, engage themselves to a Sj)ectator S, that after a certain, number
of Games is over, A shall give him as many Pieces as he wins Games,

over and above ^ n, and B as many as he wins Games, over and above

the number n ; to find the Expectation of S.

Problem LXXII. is a particular case of Problem LXXIII. obtained
by supposing a and h 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, Yir Clarissimus Alex. Cuming
Eq. Au. Pegi?e Societatis Socius, quaestionem infra subjectam 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 latins
patens.

We will give a solution of Problem LXXIII ; De Moivre in the
Doctrine of Chances merely states the result.

Let n = c {a-\-h) ', consider the expectation of 8 so far as it

depends on A. The chance that A will win all the games is

ia + hy

- , and in this case he gives ch to S. The chance that A will

.n-lj

win n—1 games is 7 r-r , and in this case he gives cl — l to S.

And so on.

DE MOIVRE. 18.3

Thus we have to sum the series

a%c + ndr-'l {he - 1) + !?L^LdO ^-252 (^c - 2) + . . . ,

the series extending so long as the terms in brackets are positive.
"We have

a%c - noT-'h = a''~'h (ac -n)=- aJ'-'b he ;
thus the first two terms amount to

{n-l)d;'''hhc.

f2, (fi 1^

Now combine this with ^- — -~ a''~%^2 ; we ^et

1.2 > &

(n - 1) a'^-'h' (ac - 7i), that is - (?i - 1) a'^-'h'hc ;

thus the first tJu^ee terms amount to

1.2

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-ic + l) ^,.-^,,...j^_
\oc — l

This may be expressed thus

n I he I ae

a'^'h^'aehc,

which is equivalent to De Moivre's result. The expectation of S
from B will be found to be the same as it is from A.

82-i. When the chances of A and B for winning a single game
are in the proportion of a to & we know, from Bernoulli's theorem,
that there is a high probability that in a large number 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 Lxxiii. to investigations which in fact amount to what
we have called the inverse use of Bernoulli's theorem ; see
Art. 125. De Moivre says,

184 I>E MOIVRE.

...I'll take the liberty to say, tliat tliis is tlie hardest Problem that
can be projDosed 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 Headers; 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 November 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 a2yproximating the
Sum of the Terms of the Bmomial (a -1- b)" expanded into a Series,
from lohence are deduced some practical Rules to estimate the
Degree of Assent which is to he 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 Moivre's Problem LXXIV. 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 difficult, 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 modern 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 nvimber of
times without intermission for which the event is required to hap-
pen. We shall speak of this as a run of p.

Let Un denote the probability of having the required run of ^?
in n trials ; then

«^,i+i = y-n + (1 - '^fn-j^ 'bdF :

for in n + 1 trials we have all the favourable cases which we have
in n trials, and some more, namely those in which after having
failed in n—p trials, we fail in the (n— ^? + l)"' trial, and then
have a run of p.

DE MOIVRE. 185

Let Un=l —Vn, and substitute in the equation ; thus
The Generating Function of v^ will therefore be

where 0 (f) is an arbitrary function of t which involves no powers
of t higher than f.

The Generating Function of w„ is therefore

1 </>© .

1-^ i-t + hd'r''

we may denote this by

t(0

{1-t) {\-t + ha''r')'

where -y^r (t) is an arbitrary function of t which involves no powers
of t higher than f^^. Now it is obvious that w„ = 0 if n be less
than J), also u^ = a^, and Up^_^ = a^ + ha^.

Hence we find that
so that the Generating Function of u„ is

(1 - t) {1 - t -{- haH'"-') '

The coefficient of f in the expansion of this function will
therefore be obtained by expanding

a^ (1 - at)

i-t + loFr^ '

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 agi'ees with this after
allowing for a slight mistake. He says we must divide unity by
\—x — ax^ — a^x^ — ... — a^~'^x^, take n—jp + 1 terms of the series,

multiply by ^^ , and finally put x = j . The mistake here

we ou^lit to read 7- . De Moivre is correct in an example which

18G I)E MOIVRE.

is that in the series 1 - a?- aa?^-aV - ... -a^~V instead of a

a
I

he gives on his page 255. Let j =Cy then according to De Moivre's

rule corrected we have to expand

1 a^ , . 1 — ex c?

\ — X

\ — cx

This will be seen to agree with our result remembering that we
took a^h = \.

De Moivre himself on his page 256 practically gives this form
to his result by putting

\ — c^ d^

1 — X — i for 1 —x^cx"^ — c^ x^ — . . . — (f ^x^.

I — ex

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 _p equal to
one half.

De Moivre's Problem LXXiv. has been extended by Condorcet,
Essai...de V Analyse... pages 73 — 86, and by Laplace, Theorie...des
Frob. 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 Ahramo Moivre: Trasportata
dalV Idionia 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 MOIVKE. 187

In the Advertisement to the second edition of the Doctrine of
Chances, page xiii, De Moiwe says,

There is in the World a Gentleman 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 to 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 Seriehus et Quadratmns...Ijondoii, 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 Chaj)ters.

328. In the first Chapter the design of the section is ex-
plained. De Moivre relates the history of the ]3^iblication of
Montmort's first edition, of the memoir De Mensura Sor^tis, and
of Montmort's second edition. De Moi\Te sent a copy of the De
Mensura 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-

188 DE MOIVRE.

tingiilshed men ; he was also admitted as a member of the Koyal
Society. De Moivre sent to Montmort a copy of the Doctrine of
Chances when it was pubKshed, 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 Doctrine 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 Moivre to devote a section to the subject in his Mis-
cellanea Analytica.

329. The second Chapter of the Responsio... is entitled De
Methodo Diferentiarum, in qua exhihetur 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 Art. 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 u\'^ to denote the n^^ term of a series,
where u^ is such that A'^w^ is zero, ni 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, 171 7.

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 De Me-
thodo Comhinatio7ium ; the fourth De Permutationibus ; the fifth
Combinationes et Permutationes idterius consideratce : these Chap-

DE MOIVRE. 189

ters consist substantially of translations of portions of the DoctHne
of Chances, and so do not call for any remark. The sixth Chapter
is entitled De Kumero PiDictorum 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 Prohlematum ad Sortem spectantium. 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 + qY', one term w^ill
be 28p^q^: then De Moivre says, page 218,

...at fortasse nesciveram hujus termini coefficientem, nimiruni 28,
designaturam numerum permutationuni quas literse ^9, p, p, p, p, p, q, q,
productum /)^ q' constituentes subire possint ; immb vero, hoc jam din
mihi erat exploratum, etenim ego fortasse primus omnium detexi co-
efficientes annexas productis Binomii, vel Multinomii cujuscunque, id
denotare quotenis variationibus literse producti positiones suas inter se
permuteut: sed utrum illud facile fuerit ad inveniendum, j)ostquani

lex coefficientium ex productis continuis :r- x — ^r— x —= — x (tc.

1 z o 4

jam perspecta esset, aut quisquam ante me hoc ipsum detexerit, ad rem

prassentem non magni interest, cum id monere suffecerit banc proprie-

tatem Coefficientium a me assertam fuisse et demonstratam in Actis FJd-

losophicis 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 7i 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 moivrp:.

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 j^- Let
(j) [n,p) denote the chance that B will be ruined in n games. Then
the required ratio is 1 — (/) (ii, 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 -v/r [n, p) then
denote the chance that B will be ruined in 7i games ; similarly if
each starts with Zp counters let -v/r {71, Sp) denote the chance that
B will be ruined in n games ; and so on. Then De Moivre says
that approximately

^ (??, p) = 'f (n, p)+'^ (n, Sp),

and still more approximately

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 formulse given on pages 199
and 210 of the Doctrine of Chances... The second term of
T^r {n, p) is negative, and is numerically equal to the first term
of A|r {n, Sj)), and so is cancelled ; similarly the third term of
'ylr[n,p) is cancelled by the first of — 'sfr [n, 5p), and the fourth
term of ^fr (n, p) by the first of -v/r (/z, 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 diebus qiiam Liber qui inscribitur, Miscellanea Analy-
tica, in lucem prodiisset, Doctissimus Stlrlingius me liteiis admonuit
Tabulam ibi a me exhibitam de summis Logarithmorum, non satis aii-
toritatis habere ad ea firmanda quse in speculatione nitercntur, utpote

DE MO IV RE. 191

cui Tabulae subesset error perpetuus in quinta quaque figura decimali
sum mar um : quae cum pro bumauitate sua monuisset, bis subjunxit
seriem celerrime convergentem, cujus ope summse logaritbmorum 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 + ^^-3g^3+j260;^.-l^
, 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 A nalytica . . . quae
satis commode convergit in principle, 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 — :r^ + -rr^ — =-^— r + ...

12 360 1260

is equal to - log 27r ; and thus the theorem is deduced which

we now call Stirling's Theorem. See Miscellanea Analytica,
page 170, Supplement, page 10.

334. De Moivre proceeds in the Siqyplement to the Miscellanea
Analytica to obtain an approximate value of the middle coefficient
of a Binomial expansion, that is of the expression

(W2+1) (m+2)... 2m
m [m — 1) ... 1

He expends nearly two pages in arriving at the result, which

192 BE MOIVRE.

he might have obtained immediately by putting the proposed ex-

I 2)n
pression in the equivalent form — - .

De Moivre then gives the general theorem for the approximate
summation of the series

1 1,1,1,

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,

^ _ r , 1 11 du^ 1 1 d^u^

See Novi Coram.... Petrop. Vol. xiv. part 1, page 137 ; 1770.
The theorem however is also to be found in Maclaurin's
Treatise of Fluxions, 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 7i to be a very large number, then the

/I 1\"
logarithm of the ratio which a term of I ^ + ^ ) , distant from

the middle term by the interval I, bears to the middle term,

2P
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 MOIVEE. 193

Thus by the aid of Stirling's Theorem the value of Bernoulli's
Theorem is largely increased.

De Moivre 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.

886. 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 Recurring 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.

CHAPTER X.

MISCELLANEOUS INVESTIGATIONS
Between the years 1700 and 1750.

837. The present Chapter will contain notices of various con-
tributions to our subject which 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 quwstiones Juris ap-
plicatce. This is stated to have been published at Basle in 1709;
see Gouraud, page STG.

It is reprinted in the fourth volume of the Act. Eruditorum..,
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.

839. 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 BERNOULLI. 195

value of an annuity on a life ; the sum to be paid to assure to a
child just born 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.

34:0. Suppose there are h men who will all die within a years,
and are equally likely to die at any instant ^\dthin 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 line 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 h 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 </> {n, 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 ^ — 1 points can be put nearer to the
origin ; that is, the number of combinations of a; — 1 things, taken
J — 1 at a time, say </> (a? — 1, h — 1).

Hence the required mean distance is

^ xc (j) {x — \, h — V)

where the summation extends from x = h to x = n.

13—2

196 BAPvBEYRAC. ARBUTHNOT.

TlCu

It is easily seen that the limit, when n is infinite, is j— ^ , that

The above is substantially the method of Nicolas Bernoulli.

341. Nicolas BernouUi has a very 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 ii^ 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

_ 2w„_,jf0 _ 2

. n

Hence ^"~ (sj *

This is not the notation of Nicolas ; but it is his method and
result.

842. In the correspondence between Montmort and Nicolas
Bernoulli allusion was made to a work by Barbeyrac, entitled
Traits dii 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 in 1709 and 1744.

Barbeyrac is also said to have published a discourse Sur la
nature du Sort

See the English Cyclopoidia, 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 A7i Argument for Divine Providence,

ARBUTHNOT, 1Q7

taken from the constant Regularity ohservd in the Births of both
Sexes. By Br John Arbuthnott, Physitian in Ordinary to Her
Majesty, and Felloiu 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.

844. 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 Ballance 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 J thus
demonstrate :

845. 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 born ; then
the chance that in a given year there will be more males than

females is ^ ; and the chance that this will happen for 82 years in

succession is ^ . This chance is so small that we may conclude
that it is not an even chance whether a male or female be born.

846. The memoir attracted the attention of Nicolas Bernoulli,
who in his correspondence with Montmort expressed his dissent
from Ai'buthnot'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 replied to Nicolas Bernoulli,
as we have already intimated in Art. 835.

847. The subject is also discussed in the Oemres Philo-
sophiques et Mathematiques of 's Gravesande, 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

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 born.

It appears that 'sGravesande wrote to Nicolas Bernoulli on
the subject; the reply of Nicolas Bernoulli is given. This reply
contains a jDroof 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. Arhutlmot fait consister son argument en deux cliosesj 1°. en
ce que, supposee une egalite de naissance entre les filles et les gargons,
il y a peu de probahilite que le n ombre des garyons et des filles se trouve
dans des limites fort proches de I'egalite: S''. qu'il y a peu de proba-
hilite que le nombre des gargous 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. 199

XXIX. of the Philosopliical Transactions, which is the volume for
Vjl^, 1715, 1716 the memoirs occupy pages 133 — 158. They are
entitled Solutio Generalis Prohlematis XV. 2)ro2:)ositi a D. de Moivre,
in tractatii de Mensura Sortie... Solutio generalis altera prcece-
dentis Prohlematis, ope Comhinationum et Serierum infinitarum....

These memoirs relate to the problem which we have called
Waldegraves ; 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 Hugeuii Libellus de Ratiociniis in Liido 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 Eichard
Mead, an Advertisement to the Header, and then 24 pages, which
comprise the translation. The dedication commences thus :

Honour'd 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
ib 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 ; turn'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 MAIRAN.

Sharper, Death, and counterminmg the underhand Dealings of secret and
overreaching Distempers.

In his Advertisement to the Reader, Browne refers to a trans-
lation of Huygens's treatise which had been made by Arbuthnot ;
he also notices the labours of Montmort and De Moivre. 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, till 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.

850. In the Hist de V Acad.... Paris for 1728, which was
published in 1730, there is a notice respecting some results ob-
tained by Mairan, Siir le Jeu de Fair ou Non. 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 the original heap contains n counters we should say that there

are n ways of drawing one counter, -— ^ — ^-^ ways of drawing

two counters, and so on. Mairan notices this view but con-
demns it.

Laplace treated this problem in the Memoires . . . par divers
Savans...TomeYl., 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 Theori'e...des Proh. 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 Methodique, in 1785,
without any notice of Laplace's dissent.

351. On page 68 of the volume of the Hist de VAcad....
Paris, which contains Mairan's results, is the following paragraph :

M. L'Abb6 Sauvenr, fils de feu M. Sauveur Academicien, a fait voir
une Metliode qu'il a trouvee pour determiner au Jeu de Quadrille quelle
est la probabilite de gagner sans prendre plusieurs Jeux differents, dont
il a calcule une Table. On a trouve que la matiere ej^ineuse et delicate
des Combinaisons etoit tres-bien entendlie dans cet ouvrage.

352. We have next to notice a memoir by Nicole, entitled
Examen et Resolution de quelques questions sur les Jeux.

This memoir is published in the volume for 1730 of the Hist,
de T Acad.... Paris; the date of publication is 1732 : the memoir
occupies pages 45 — oQ 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
S be the sum which each stakes. Then ^'s advantage is

^ ft8 + 8a^6 + 'iMh'' + 56a^^^ - b^a'lf - 28a'h' - 8ah' - b'

^ {a-^hy : •

202 NICOLE.

This supposes tliat 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 + h ; thus we may simplify the ex-
pression to

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 true. Suppose that when A and B agree to play
2n — l games, p^ is the chance that A beats B by just one game,
^2 the chance that A beats B by two or more games ; and let
^j, q^ be similar quantities with respect to B, then ^'s advantage
is S {p^-\- p^ — q^ — q^. Now consider 2n games : ^'s chance of

beating B by two or more games, is j?2 + ? ; -S's chance of

beating A by two or more games is q^ + ^^ . Hence A's ad-
vantage is

S(p.+

^2-

a + b ^2 a+ bj '
Now we know that ^ = ^ = fj, say; therefore

p^a-qj) ^{a ~b) .

a + b a + b ^^ i Lx i\

Hence the advantage of A for 2/^ games is the same as for
2/1 — 1 games.

853. In the same volume of the Hist, de TAcad....Pa7^is, on
pages 331 — 344, there is another memoir by Nicole, entitled
Methode jwur determiner le sort de taut de Joueurs que Von
voudra, et Vavantage que les iins out sur les autres, lorsqitils
joilent h qui gagnera le plus de parties dans un nomhre de parties
determine.

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

points. Nicole begins in a laborious way; but he sees tliat 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, h, c. Let them play a
set of three games. Then the chance that A has of winning
the whole stake is a + 3a^ (^ + c) ; and similar expressions give
the chances of B and (7; there is also the chance ^ahc 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^ -\-^a {h + c) + 12a^hc\ there is also the chance
Wlf that A and B should share the stake between them to the
exclusion of G\ and so on.

But all that Nicole gives was already well known ; see
Montmort's page 353, and De Moivi'e's Miscellanea Analytica,
page 210.

854^. In the year 1733 Bufifon communicated to the Academy
of Sciences at Paris the solution of some problems in chances.
See Hist, de V Acad.... Pains for 1733, pages 43 — 45, for a brief
account of them. The solutions are given in Buffon's Essai
dArithm^tique Morale, and we shall notice them in speaking
of that work.

So 5. 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 Moivi^e ; 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

201 HAM.

have frequent use made of De Moivre's results as to the number
of trials in which it is an even chance that an event will happen
once, or happen twice ; see Art. 264.

856. 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 + ^)' = 2 (1 + z).

If we suppose q infinite this reduces to

^ = log 2 + log (1 + ^) ;

from which De Moivre obtained z = 1-678 approximately. But let
us not suppose q infinite ; put ( 1 + -j =6"; so that our equation

becomes

6*'^= 2(1+^).

Assume z=2 —y, thus

Assume 2c = 7 + s where e*^ = 6.

Thus, e*-^ = 1 - g y.

Take the logarithms of both sides, then

1 1 , 1 3

that is ''i/ - Yg .V' - gj 3/' - ••• = ^ ;

where r = c — ^.

Hence by reversion of series we obtain

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

5 = 2c -7= 2c- 1-791759.

Ham says that this series will determine the value of z in
all cases when ^ is greater than 4-1473. This limit is doubtless

obtained by making 2c - 7 = 0, which leads to (l + - j = V6 ;

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 - numerically less than unity, so long as 7 — 2c is less than
r

c - -^ , that is so long as c is greater than k + q .

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 Calcul du Jeu
appellS par les Frangois le trente-et-quarante, et que Von nomme
d Florence le trente-et-un.,,. 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 tens, 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 deja fait la demonstration
dans mon Calcul de la Loterie de E-ome,...

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 Mathe7natiques, Vol. 16.

859. 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 born 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 Kature and Laws of Chance. . .
The whole after a new, general, and conspicuous Manner, and
illustrated luith a great variety of Exam^jles ... 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.

SCO. Simpson's Problem VI. is as follows :

There is a given Number of each of several sorts of Things, (of the
s£i,me Shape and Size); as {a) of the first Sort, (h) of the second, &c.
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, {<j) of
the second, (r) of the third, &c.

SIMPSON. 207

The result in modern notation is a fraction of which the nume-
rator is

\a \h [c

X "; ; :; X X . • . j

\p \<^-p \q\'b-q

c — r

and the denominator is : — —•-

\m\n — m

where ?z = a + & + 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."

861. 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.

S62. 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 j. Let

2' = - ; 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 ( ^ ~ tt; ) trials ; see Ai't. 262. But if ^ = 1

the required number of trials is exactly 2r — 1. Simpson then

proposes to take as a universal formula 2'(^'~t7v)+^~t^j this

is accurate when g[ = 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 xxil. 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 :

(p'-l){p-2) ... (p-n-\-l) _n fe-1) fa- 2). ..(^-72+1)
\n — l 1 1^ — 1

n{n — l) (r— 1) (r — 2) ... (r — n + 1)

where g^—'p-f, r — 'p—^f, ...; 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-\-l) n q{q-V) ... (q-n + l)
[n 1 [n

n(n—l) r (r—1) ... (r — n + 1)

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
LXXiv., 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. 203

We have shewn in effect that the solution is obtained by taking
the coefficient of f~^^ in the expansion of

a^ (1 — at)
(1-0 [I - t -^^ baH^'-'Y

that is in the expansion of

a^ (1 - at)

{1-ty

,T 1-at 1 (l-a)t 1 ht

Now y, ^=- -+Vt— Tv.=^^ : +

(i-ty i-t' (1-ty i-t ' (1-0''

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 xxii. 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 ill, 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 XX VI I,
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, 6, cj 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 Ai't. 87. Simpson
supposes that a sphere is described round the paralleleiDiped, 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

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.

868. 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 hj A-\-Bx-\- Cx^ + Dx"" + . . . ;
required the sum of

A Bx Cx'

1.2...r"^2.3... (r + l)"^S.4...(r + 2)"^**'*

' Integrate both sides of the identity, and determine the con-
stant so that both sides may vanish when a? = 0 ; thus

{a + xY^' g"-^^ _ . Bx^ Cx^ Bx"^
71 + 1 n-\-\ 2 3 4

Repeat the operation ; thus

(?i + l)(n+2) n + 1 (n+l)(7^ + 2)

_A^ B^ C'^ Dx^
~'1.2"^2.3"^3T4"^4.5"^""

Proceed thus for r operations, then divide both sides by a?*", and
the required sum is obtained.

(2) Required the sum of 1" + 2« + 8" + . . . + ic".

Simpson's method is the same as had been already used by
Nicolas Bernoulli, who ascribed it to his uncle John ; see Art. 207.

869. Simpson's Problem xxix. is as follows :

A and B, whose Chances for winning any assigned Game are in
the proportion of a to 6, agree to play until 7i stakes are won and
lost, on Condition that A, at the Beginning of every Game shall set

the Sum p to the Sum ^x-, so that tliey may play without Disad-

vantage on either Side; it is required to find the present Value 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 £, equally skilful, enter into Play together,
and agree to continue the same till (n) Games are won and lost. 'Tis
required to find the Probability that neither comes off a Winner of
Q'Jn Stakes, and also the Probability that B is never a Winner of
that Number of Stakes during the whole Time of the Play; r being
a given, and n any very great, Number,

Simpson says in bis Preface relating to bis Problems XXIV. and
XXX. tbat tbey

" are the same with the two new ones, added in the End of Mr
Be 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 :

Full 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 tbe 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 + hy ; it is given in the Doctrine of Chances, second edition,
pages 233 — 243, third edition pages 241 — 2ol : the method of
Simpson is in fact identical with De Moivre's.

871. We may remark that Simpson published a work in l7o7
under the title of Miscellaneous Tracts on some curious, and
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
Number 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 17 o7.

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 Conjectandi, Prohlemata 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 Montmort,
page 248, and the Doctrine of Chances, page 117.

374. The fourth problem contains an error. John Bernoulli
sa3"s that if 2n common dice are thrown, the number of ways in
which the sum of the marks is 7n is

(7n-l) (7^-2)(7n-3)...(5yz + l) .
1.2.3.4 ... (2?z-l) *

this amounts to asserting that the expression here given is the co-
efficient of x"" in the expansion of

. (ic + a?' -I- a;' + a;' + x\+ x^ :

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. Daxiel Beexoulli was the son of the John Bernoulli
to whom we have often referred ; Daniel was born 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. Tlie first memoir which we have to notice is entitled
Specimen TJieornce Xovcb 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 publication
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 — 415 of his Theorie...des Proh., 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 hj^otheses 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 dx, Daniel Bernoulli estimates the relative
value of the increment as proportional to dx directly and x in-

ndx

versely ; that is, he takes it equal to where Jc is some con-

stant. Put this equal to Jt/ ; so that

■J ruCtX

dy = ;

therefore y — T^ log ^ + constant

= Ti 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
honorum, and x — a he calls lucrum.

880. Suppose then that a person, starting with a for his fortune
physique, has the chance p^^ of gaining a?^, 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= hp^ log {a + x^ + hp^ log {a-\-x,^ -\- hp^ log (a + i^Cg) + . . . — ^ log a.

Then Bernoulli calls Y the emolumentum medium, and Laplace
still calls Y the fortune morale. Let X denote the fortune
physique which corresponds to this fortune morale ; then

Y=h log X—h log a.

Thus X = (a + cc/^ (« + xf^ {a + x^""' . . .

And X—a will be according to Laplace V accroissement de la
fortune physique qui procurerait a Tindividu le menie avantage
moral qui r4sidte pour lui, de son expectative. Daniel Bernoulli
calls X—a the liicru7n legitime expectandum seu so7^s quwsita.

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 ^ = 7c log - , but supposes generally 2/ = </> (x).

Thus the ordinary theory of mathematical expectation amounts to
supposing that the curve becomes a straight line, or (p (x) a
linear function of x.

382. After obtaining the value of X which we have given
in Art. 380, the remainder of Daniel Bernoulli's memoir consists
of inferences drawn from this value.

383. The first inference is that even a fair game of chance
is disadvantageous. Suppose a man to start with a as his fortune
physique, and have the chance p^ of gaining x^, and the chance
p^ of losing x^. Then by Art. 380, the fortune physique which he
may expect is

{a + a?/' (a - x^^^ ;

we have to shew that this is less than a, supposing the game to be
mathematically fair, so that

Daniel Bernoulli is content with giving an arithmetical ex-
ample, supposing i>i =/>2 = 2 • I^aplace establishes the proposition

generally by the aid of the Integral Calculus. It may be proved
more simply. We have

x^ x^

■^^ ~ iCj + a?/ ^^"aj^ + iCg'

and we have to shew that

[[a-^x^'^'ia-x^"''^^' is less than a.

Now we may regard x^ and x^ as integers. Thus the result
we require is true by virtue of the general theorem in inequalities
that the geometrical mean is less than tlie arithmetical mean. For

216 DANIEL BERNOULLI.

here we may suppose that there are x^ quantities, each equal to
a + iCj, and x^ quantities each equal to a — iCg. The arithmetical
mean is

'^2 (^ + ^l) + ^1 (<^ - ^2)

— J

^1 + ^2

that is a. The geometrical mean is the quantity which we had
to shew to be less than a.

884. Daniel Bernoulli proposes to determine what a man
should stake at a wager, in order that the wager may not be

disadvantageous to him. He takes the case in which 'p^—]p^ — -^ .
Then we require that

(a + icj^ {a — x^^ — a.

This leads to x^ — — .

Thus x^ is less than x^ and less than a.

885. Daniel Bernoulli now makes an application to in-
surances. But this application will be more readily understood if
we give first a proposition from Laplace which is not in Daniel
Bernoulli's memoir. Suppose that a merchant has a fortune
physique equal to a, and that he expects the sum x to arrive
by a ship. Also let p be the chance that the ship will arrive
safely, and lei q = l —p.

Suppose that he insures his ship on the ordinary terms of
mathematical equity ; then he pays qx to the insurance company,
so that he has on the whole a + x — qx, that is a -\-px.

Suppose however that he does not insure ; then his fortune
physique is (a + xfa'^. We shall shew that a-\-px is greater
than {a + xYa^.

Laplace establishes this by the aid of the Integral Calculus,
with which however we may dispense. We have to shew that

(a + xYa^ is less than a +px,

that is that (1 + - ) is less than 1 + -^ .

\ a/ a

DANIEL BERNOULLI. 2l7

Let » = where m and n are integers.

Then we know that [{l + ^)" l'^ 1^^ is less than

m +n

by the theorem respecting the geometrical mean and the arith-

metrical mean which we quoted in Art. 383 ; and this is what we

had to establish.

It follows that the merchant can afford without disadvantage

to increase his payment to the insurance company beyond the

sum qx. If we suppose f to represent the extreme additional

sum, we have

f = a +jyx — (a + ic) V.

886. We now return to Daniel Bernoulli. We have seen
that a merchant can afford to pay more than the sum qx for
insuring ; but it may happen that the insurance company demand
more than the merchant can afford to pay. Daniel Bernoulli
proposes this question : for a given charge by the insurance com-
pany required to find the merchant's fortune, so that it may
be indifferent to him whether he insures or not.

Retaining the notation of the last Article, let e be the charge
of the insurance company ; then we have to find a from the

equation

a-\-x — e = {a + xYa^

Daniel Bernoulli takes for an example a?= 10000, e=800,^= ^ ;

whence by approximation a— 5043. Hence he infers that if the
merchant's fortune is less than 5043 he ought to insure, if greater
than 5043 he ought not to insure. This amounts to assuming
that the equation from which a is to be found has only one
positive root. It may be interesting to demonstrate this. We
have to compare

a-\-x — e with {a + ic)^a^

where a is the variable, and x is greater than e.

218 DANIEL BERNOULLI.

Let p = — ; — SLiid q= — ; — , where m and n are inteofers :

then we have to compare

{a + x- eY^*" with {a + xY a\

Wlien a = 0 the right-hand member is the less ; when a is
infinite the right-hand member is the greater, provided mx is
greater than (m -^ n) [x — e) : we will assume that this is the case.
Thus the equation

{a-^x- e)'""-" = (a + xY oT

has one positive root. We must examine if it has another.

Let log {a + x- ef'^'' = y, log (a + xY dr = z\

. dy m-\-n dz m n

then - -f = — , , -7- = — , h - .

da a + X — e da x + a a

d z d II

Thus when a is zero -j- is greater than j- , so that z begins

by increasing more rapidly tlian y does. If we suppose

dy dz
da da

, , . nx (x — e)

we obtam a = - — — ^r '■— .

(??i -\-n) e — nx

Now begin with a = 0, and let a gradually increase until we
have y = z\ then it is obvious that we have not yet reached the
value of a just given. And if by increasing a we could arrive
at a second value at which y = z, we should have passed beyond
the value of a just given. Then after that value z would increase
more slowly than y, and the final value of z would be less than
the final value of y, which is impossible. Thus there is only one
value of a which makes y = z, and this value is less than

nx {x — e)
{m ■\- n) e — nx'

If mx is less than (m -\-n) {x — e) the original equation has
no positive root; for then we have z always increasing more
rapidly than y, and yet the final value of z less than that of y ;
so that it is impossible that any value of a can make y = z.

DANIEL BERNOULLI. 219

387. Daniel Bernoulli also inquires what capital the in-
surance company must have so that they may safely undertake
the insurance. Let y denote the least value of the capital ; then
y must be found from

This is merely the former equation with y in place of a + ic — e.
Thus, taking the same example as before, we have^ = ltt2^3.

888. Daniel Bernoulli now lays down the important principle

that it is more advantageous for a person to expose his fortune

to different independent risks than to expose it all to one risk.

He gives this example : suppose a merchant to start with a

9
capital of 4000, and that he expects 8000 by a ship ; let —

be the chance of the safe arrival of the ship. The merchant's
fortune 'physiqiie is thus

(4000 + 8000)T^ (4000)^=10751 approximately.

But suppose him to put half of his merchandize in one ship

and half in another. The chance that both ships will arrive safely

81
is r7^\ the chance that one of the two will amve safely is
100 ''

9 1 18

2 X Y^ X — r , that is —— ; the chance that both will be lost is

r— X . Hence the merchant's fortune 'physique is

(4000 + 8000)tV(7 (4000 + 4000)^^ (4000)^^= 11033
approximately.

Subtract the original capital 4000, and we find the expectation
in the former case to be 6751, and in the latter to be 7033.

Daniel Bernoulli says that the merchant's expectation con-
tinually increases by diminishing the part of the merchandize
which is intrusted to a single ship, but can never exceed 7200.

This number is — of 8000 ; so that it expresses the Mathematical

expectation. The result which Daniel Bernoulli thus enunciates.

220 DANIEL BERNOULLI.

without demonstration is demonstrated by Laplace, Theorie . . . des
Froh., pages 435 — 437 ; the proposition is certainly by no means
easy, and it is to be wished that Daniel Bernoulli had explained
how he obtained it.

389. Daniel Bernoulli now applies his theory to the problem
which is known as the Petershurg Problem, probably from its first
appearing here in the Coiiimentarii of the Petersburg Academy.
The problem is similar to two which Nicolas Bernoulli proposed to
Montmort; see Art. 231.

A throws a coin in the air ; if head appears at the first throw
he is to receive a shilling from B, if head does not appear until the
second throw he is to receive 2 shillings, if head does not appear
until the third throw he is to receive 4 shillings, and so on : re-
quired the expectation of A.

The expectation is

1 2 4 8, ... V

2 + 22 + 2^ + 2^ + • • • ^^^ 'infinitum,

that is ^ + ^ + 2 + 9 + • • . ^'^^ infinitum.

Thus ^'s expectation is infinite, so that he ought to give an
infinite sum to B to induce B to play with him in the manner
proposed. Still no prudent man in the position of A would be
willing to pay even a small number of shillings for the advantage
to be gained.

The paradox then is that the mathematical theory is apparently
directly opposed to the dictates of common sense.

390. We will now give Daniel Bernoulli's application of his
theory of Moral expectation to the Petersburg Problem.

Suppose that A starts with the sum a, and is to receive 1 if
head appears at the first throw, 2 if head does not appear until the
second throw, and so on. ^'s fortune physique is

{a + 1)^ {a + 2)^ {a + 4)^ (a + 8)^^ ... - a.

This expression is finite if a be finite. The value of it when
a = 0 is easily seen to be 2. Daniel Bernoulli says that it is about
8 when a = 10, about 4 J when a = 100, and about 6 when a = 1000.

DANIEL BEKNOULLI. 221

Let X represent the sum which a person with the capital a
might give without disadvantage for the expectation of A \ then x is
to be found from

(a + 1 — a?)^ (a + 2 — a?)^ (a + 4 — x)^ (a + 8 — x)^ . .. = a.

Put a — X — a \ thus

{a + 1)^ [a + 2)^ (a' + 4)^ {a + 8)tV ... - a' = a;.

Then if a is to have any large value, from what we have
already seen, x is small compared with a, so that we may put a for
a \ and we have approximately

a; = (a + 1)^ (a + 2)^ (a + 4)^ (a + 8)^... -a.

Laplace reproduces this part of Daniel Bernoulli's memoir with
developments in pages 439 — 442 of the Theorie...des Proh.

391. Daniel Bernoulli's memoir contains a letter addressed to
Nicolas Bernoulli by Cramer, in which two methods are suggested
of explaining the paradox of the Petersburg Problem.

(1) Cramer considers that the value of a sum of money is not
to be taken uniformly proportional to the sum ; he proposes to
consider all sums greater than 2^"^ as practically equal. Thus he
obtains for the expectation of B

1 2 4 2^^

2*^2^

"■" 02 "■" 03 "•"••• • '^"•5

924 924 924

' 926 "^ 927 "' 928 "1" ••••

The first twenty-five terms give 12 J; the remainder constitute

a geometrical progression of which the sum is ^ . Thus the total
is 13.

(2) Cramer suggests that the pleasure derivable from a sum
of money may be taken to vary as the square root of the sum.
Thus he makes the moral expectation to be

2 a/I + J V2 + g v/4 + ^ V8 +■ . . . ,

that is j^ . This moral expectation corresponds to the sum

222 DANIEL BERNOULLI.

rg-, that is to 2 "9 approximately; and Cramer considers

(2 — V2)

this to be nearer the comm.on notion on the subject than his former
value 13.

892. 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 ; Theorie . . . 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, RechercJies
sur la Proh... page 73; Cournot, Exposition de la Theorie des
Chances... page 108.

894. 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 rinclinaison des Plans des Orbites
des Planetes par rapport au plan de I'Equateur de la revolution du
Soleil autour de son axe; Et d'oii vient que les inclinaisons 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 BernoulH. The
memoirs of both are contained in the Recueil des pieces qui ont
remporte le prix de VAcademie Roy ale des Sciences, Tom. 3, 1734.

DANIEL BERNOULLI. 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 —z of the whole sur-
face of the sphere. There being six planets altogether he takes
|i^ 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 ^q iiearly ;

and he takes -r—n 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 inclination of the plane of any orbit to this is 7° 30',

which is about r=-^ of 90" ; and he takes — r^ as the chance that each
12 12*^

of the six inclinations would be less than 7" 30'.

896. It is difficult to see why in the first of the three pre-

1 . 2

ceding calculations Daniel Bernoulli took ^^ 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 Theorie...
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 Essai
dune nouvelle analyse de la mortalite causee par la petite Verole,
et des avantages de V Inocidation pour la prevenir.

This memoir is contained in the Hist de FA cad. ... 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 mathematicien ; see his pages 4 and 18.
In consequence of this, an introduction apologetique 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 xiii.

400. Daniel Bernoulli's main object is to determine the mor-
tality caused by the small-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 every 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 f denote
the number who survive at that age out of a given number
who were born ; 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 small-pox during the

sdx
element dx of time is by hypothesis — - : because we suppose

o sdx

- to be attacked in one year, and therefore in the element

n n

dx of a year. The number of those who die of the small-pox is

sdx
by hypothesis ; and therefore the number of those who die

^dx
of other diseases is — d^— - — . But this last number must be

diminished in the ratio of s to f, because we only want the

diminution of those who have not yet had the small-pox, of whom

the number is s.

Thus „ds = —-i(d^-^--).

n g V 7)inJ

This equation is to be integrated. We have

s^dx

•

, _ O..C ..^^ ^..^ dx

therefore

- ds ■•

sdx
n

sd^-

^ds

^dx

226

DANIEL BERNOULLI.

Put q for - ; thus, da = — ^ dx ;

^ s nin

therefore
therefore

and

n log {mq — l)—x-\- constant ;

^m^

-1 =e^^^.

e " + 1

To determine the constant C, we observe that when x
we have s = f ; thus, finally,

= 0,

s =

(m - 1) ^' + 1

403. By this formula Daniel Bernoulli calculates a table on
the basis of Halley's table, derived from the Breslau Observations,
assuming that m and n each equal 8 ; Halley's table gives the
values of f corresponding to successive integer values of x, and
Daniel Bernoulli's formula then gives the values of s. The fol-
lowing is an extract from the table :

1300

1000

896

855

685

798

571

760

485

732

416

710

359

692

311

680

272

670

237

661

208

653

182

646

160

640

140

634

123

628

108

622

DANIEL BERNOULLI. 227

Halley's table begins with 1000 at the end of the first year,
and does not say to what number of births this corresponds.
Daniel Bernoulli gives reasons for assuming this to be 1300,
which accordingly he takes ; see Art. 64?.

404. On page 21 of the memoir, Daniel Bernoulli says that
the following question had been asked: Of all persons alive
at a given epoch what fractional part had not been attacked
by the small-pox ? The inquirer himself, who was D'Alembert,
estimated the number at one-fourth at most. Daniel Bernoulli
himself makes it about two-thirteenths. He intimates that it
would be desirable to test this by observation. He adds,

Voici un autre theoreme qui pourroit servir h la verification de
nos principes. Si de tous les vivans on ne prend que Tenfance et la
jeunesse, jusqu'a I'age de seize ans et demi, on trouvera le nombre
de ceux qui auront eu la petite verole a pea-pres egal au nombre de
ceux qui ne I'auront pas eue.

405. Daniel Bernoulli gives another interesting investigation.
Bequired to find the number of survivors at a given age from
a given number of births, supposing the small-pox altogether
extinguished. Retain the notation of Article 402 ; and let z be
the number who would have been alive at the age x if there had
been no small-pox, the original number of births being supposed
the same.

The whole mortality during the element dx of time being

9 fix
— d^, and the mortality caused by the small-pox being , we

II tit

sd'Oc
have for the mortality in the absence of small-pox — d^ .

But this mortality arises from a population f ; and we must mul-
tiply it by g to obtain the mortality which would arise from a
population z. Hence, finally,

mn,>

dz d^ s dx

therefore — = -p -\r -z. — •

z ^ g tnn

15—2

228 DANIEL BERNOULLI.

Substitute for s from the result in Art. 402 ; then integrate,
and determine the arbitrary constant by the condition that 2=^
when x = 0. Hence we shall obtain

z me^

^ (m - 1) e" + 1
Thus as X increases, the right-hand member approaches the

limit

m — 1

406. After discussing the subject of the mortality caused by
the small-pox, Daniel Bernoulli proceeds to the subject of In-
oculation. He admits that there is some danger in Inoculation,
but finds on the whole that it is attended with large advantages.
He concluded that it would lengthen the average dur^ation of life
by about three years. This was the part of the memoir which
at the time of publication would be of the greatest practical
importance ; but that importance happily no longer exists.

407. We shall find hereafter that DAlembert strongly ob-
jected to the justness of Daniel Bernoulli's investigations. La-
place speaks very highly of Daniel Bernoulli ; Laplace also briefly
indicates the method of treating the problem respecting Inocula-
tion, but as he does not assume ?/^ and w to be constant, he rather
follows DAlembert than Daniel Bernoulli; see Theoiie...des Proh.,
pages cxxxvii. and 413.

408. The next memoir by Daniel Bernoulli is entitled De usu
algoritlimi infinitesimaUs in arte conjectandl specimen.

This memoir is contained in the Novi Comm...Petrop. Vol. xil,
which is the volume for the years 17C6 and 1767 ; the date
of publication of the volume is 1768 ; the memoir occupies
pages 87 — 98.

409. The object of the memoir is twofold. A certain problem
in chances is to be solved, which is wanted in the next memoir to
which we shall come ; and the introduction of the Differential
Calculus into the Theory of Probability is to be illustrated. The
reader will see in Art. 402 that Daniel Bernoulli had already really

DANIEL BERXOULLI. 229

employed the Differential Calculus, and the present memoir con-
tains remarks which would serve to explain the process of Art. 402 ;
but the remarks are such as any student could easily supply
for himself We shall see the point illustrated in another memoir.
See Art. 417.

410. The problem which Daniel Bernoulli solves is in its
simplest form as follows : In a bag are 2n cards ; two of them are
marked 1, two of them are marked 2, two of them are marked 3, ...
and so on. We draw out m cards ; required the probable number
oi pairs which remain in the bag.

We give the solution of Daniel Bernoulli with some changes of
notation. Suppose that a?,,, pairs remain after m cards have been
drawn out ; let a new drawing be made. The card thus drawn out
is either one of the cards of a pair, or it is not ; the probabilities
for these two cases are proportional to 20?,,^, and 2n — 2x,,^ — m re-
spectively : in the former case there remain x^^ — 1 pairs in the bag,
and in the latter case there remain x,^ pairs. Thus by ordinary
principles

^ ^^m G^., - 1) + (2>i - 2^,^ - m) x^
"^1 2/1 - m

2n — m— 2

Zn — m

iC„j

We can thus form in succession x^, x^, ^3> ••• As x^=n we
find that

(2)1 — on) (2n - m — 1)

•^ni "~

2 {2n - 1)

411. The problem is extended by Daniel Bernoulli afterwards
to a greater generality ; but we have given sufficient to enable the
reader to understand the nature of the present memoir, and of that
to which we now proceed.

412. The next memoir is entitled De duratione media matri-
moniorum, pro qitacunque conjugum aetate, aliisque quaestionihus
affinibus.

This memoir is closely connected with the preceding ; it fol-
lows in the same volume of the JS'ovi Comm...Petrop., and occupies
pages 99—126.

230 DANIEL BERNOULLI.

413. Suppose 500 men of a given age, as for example 20 years,
to marry 500 women of the same age. The tables of mortality
will shew at what rate these 1000 individuals gradually diminish
annually until all are dead. But these tables do not distinguish
the married from the unmarried, so that we cannot learn from them
the number of unbroken couples after the lapse of a given number
of years. Daniel Bernoulli applies the result of Art. 410 ; the pairs
of cards correspond to the married couples. From that article
knowing- the number of cards which remain undrawn we infer the
probable number of pairs. The number of cards remaining un-
drawn corresponds to the number of persons remaining alive at a
given age ; this is taken from the tables of mortality, and by the
formula the probable number of unbroken couples is calculated.
Daniel Bernoulli calculates such a table for the numbers we have
supposed above.

414. Daniel Bernoulli then proceeds to the case in which the
husband and wife are supposed of different ages ; this requires the
extended problem to which we have referred in Art. 411. Daniel
Bernoulli calculates a table for the case in which 500 men aged
40 years marry 500 women aged 20 years.

Daniel Bernoulli allows that his results must not claim im-
plicit confidence. He has taken the same laws of mortality for
both men and women, though of course he was aware that on an
average women live longer than men. With respect to this fact he
says, page 100, ...neque id diversse vivendi rationi tribui potest,
quia ista sequioris sexus praerogativa a primis incunabilis constan-
tissime manifestatur atque per totam vitam in illo manet.

Daniel Bernoulli's process is criticised by Trembley in the
M^moires de V Acad.... Berlin, 1799, 1800.

The problem respecting the mean duration of marriages is con-
sidered by Laplace, Theorie...des Proh. page 415.

415. The memoir which we have noticed in Arts. 412 — 414
bears a close analogy to the memoir which we have noticed in
Arts. 398 — 406. In both cases theory is employed to supply the
lack of observations, in both cases the questions discussed are of the
same kind, and in both cases the use of the Differential Calculus is
illustrated.

DANIEL BERNOULLI. 231

416. The next memoir by Daniel Bernoulli is entitled Dis-
quisitiones Analyticce de novo prohlemate conjectwrali.

This memoir is contained in the Novi Comm...Petrop...Yo\. 14,
1769, pars prior. The date 1759 occurs by mistake in the title-
page. The date of publication of the volume is 1770. The
memoir occupies pages 1 — 25 of the part devoted to memoirs.

417. The object of the memoir is to illustrate the use of the
Differential Calculus, and it is thus analogous to memoirs which we
have already noticed by Daniel Bernoulli.

Suppose three urns ; in the first are n white balls, in the second
n black balls, in the third n red balls. A ball is taken at random
from each urn ; the ball taken from the first urn is put into the
second, the ball taken from the second is put into the third, and
the ball taken from the third is put into the first ; this operation
is repeated for any assigned number of times : required the proba-
ble distribution of the balls at the end of these operations.

Suppose that after x operations the probable numbers of white
balls in the three urns are denoted by u^., v^y w^ respectively. Then

1t/~.,t — U,f ~~ ~1 •

"'■^^ "^ n n
For — is the probability of drawing one white ball out of the

first urn, and -^ is the probability that a white ball will be drawn

from the third urn and so put into the first. Similarly

By eliminating, using the condition u^-\-v^-\-w^= n, we may
obtain an equation in Finite Differences of the second order for

Ujc, namely,

/^ 3\ /^ 3 3\ 1

'^x^'i = "^^ar+l (2. ] —f^'x 1 H— )+-•

x+2 x+i \^ ^J X \^ ^^ ^^y ^

But the following process is more symmetrical. Put w^^^ = Eu^,
and separate the symbols in the usual way ;

232 DANIEL BERNOULLI.

thus •i^-(^-D^'^"^''^'

E-(l--] [ v^ = - u^,

n I n

{

^_(1__) [^^= ^;^,

n n

therefore \ E- ( 1 - - ) \ u^^iAux-

nj I ^ \nj

Therefore w, = ^ f 1 - - + -)\ ^ f 1 - - + -)V (7 fl - - + '^V,

\ n nJ \ n nJ \ n nj

where A, B, C are constants, and a, A 7 ^^^ the three cube roots
of unity.

Then from the above equations we obtain

therefore

\ n nJ \ 71 7iJ \ n nJ

Similarly

\ n nJ \ n nJ \ n nJ

The three constants A, B, C are not all arbitrary, for we
require that

with this condition and the facts that

Uo = n, ^0 = 0, Wq=0,

we shall obtain A = B= G=-^.

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, ii^ in
the second urn, v^ in the third, and ic^ in the first ; similarly the
red balls will be probably distributed thus, ii^ 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 Avhole
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 :

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, P, y.

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 filled 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 third, 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.

23 i 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 — y),

dv = kdt (u — v),
dw — kdt {v — w)y
where Zj is a constant.

Daniel Bernoulli integrates these equations, by an unsym-
metrical and difficult process. They may be easily integrated by

the modern method of separating the symbols. Put i) for -7- ; thus

{D + ^) w = kw, (J) ■\-k) v= kuj (I)-\-k)w = kv,

therefore (D -\-Tcf u = Hu.

Hence u = e"^* [Ad"^ + Be""^' + Ce^*^'},

where A, B, G are arbitrary constants, and a, /3, 7 are the three cube
roots of unity. The values of v and w can be deduced from that of
u. Let us suppose that initially u — h, v = 0, i^ = 0 ; we shall find

that A =B= C=^, so that

Laplace has given the result for any number of vessels in the
Theorie...des Proh. 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 BERNOULLI. 235

(

1 - - = e '*
n

Hence when n and x are very large, we find that the value of u^
ofiven in Art. 419 reduces to

«e-"U + ,4-f-T+7^l'-U...^

13 \nj ' 16 V**>

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, 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 Meniorie ... della Societa Italiana, Vol. I. 1782.

422. The next memoir by Daniel Bernoulli is entitled, 2Ien-
sura Sortis ad fortuitam successionem rerum naturaliter contin-
gentium applicata. This memoir is in the same volume of the
I^ovi 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 Do
Moivre long before ; see De Moivre's Doctrine of Chances^
pages 243—254.

236 DANIEL BERNOULLI.

The following is all that Daniel BeiTioulli contributes to the
theory. Let m and n be lai-ge numbers ; let

|2n 1

u =

V =

2m 1

in f

He shews that approximately

II /^m-\- 1

V V 4w + 1 *

/I IN""
He also states the following : in the expansion of f ^ + „ I

the jj}'^ term from the middle is approximately equal to —2 .

These results are included in those of Stirling and De Moivre,
so that Daniel Bernoulli's memoir was useless when it appeared;
see Art. 837.

424. The next memoir by Daniel Bernoulli is entitled Di-
judicatio maxime prohabilis plurium ohservationum discrepantitwi
Clique verisimillima inductio inde formanda. This memoir is con-
tained in the Acta Acad. ...Petrop. for 1777, pay^s piHor ; the
date of publication of the volume is 1778 : the memoir occupies
pages 8 — 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 ^(j-'^ — e^), where 7- 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 bv x ;
then we have to find x so that the following product may be a
maximum :

^y- _ (^ _ ay] sjy -{x- hy] s/y -{x- cy] . . .

Daniel Bernoulli gives directions as to the value to be assigned
to the constant ?\

426. Thus Daniel Bernoulli agrees in some respects with
modern theory. The chief difference is that modern theory takes
for the curve of probability that defined by the equation

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 Ohservationes in pj'aecedeiitem 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,

y _ (a: - ayY + [r' - {x - ly]' 4- 17-^ -{x- c)^ + . . .
is to be a maximum. Euler sa3^s it is to be a maximum, but

288 DANIEL BERNOULLI.

he does not discriminate between a maximum and a minimum.
The equation which is obtained for determining iK 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 of 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 bora 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 will be found not unworthy of his own great
powers and fame.

429. Euler's first memoir is entitled Calcul de la Prohahilite
dans le Jeu de Rencontre. This memoir is published in the volume
for 1751 of the Histoire de V Acad ... 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 EULER.

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 mortalite et la multiplication du genre humain. 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 N infants
born at the same time ; then Euler denotes by (1) N the number
of them alive at the end of one year, by (2) N the number of
them alive at the end of two years, 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, (m) 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 alive at the end of n

more years ; so that — , . is the fraction of the number

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

probably be — -. — —- 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 x

annually as long as he lives. Let - be the present worth of the

unit of money due at the end of one year.

(m -f 1)
Then at the end of a year there will be M ' . ^ of the

(in)

persons alive, each of whom is to receive x : therefore the present

worth of the whole sum to be received is - M —- -r — .

X {m)

EULEE. 241

Similarly, at the end of the second year there will be

(?/i + 2)
M —7 — -^ of the persons alive, each of whom is to receive x :

(w)

therefore the present worth of the whole sum to be received is
-5 M ^ , . . And so on.

The present worth of all the sums to be received ought to be
equal to Ma ; hence dividing by M we get

_ X ((m + 1) (m + 2) (m + 3)

Euler gives a numerical table of the values of (1), (2), ... (95),
which he says is deduced from the observations of Kerseboom.

434. Let iV denote the number of infants born in one year,
and r'i\^ the number born in the next year ; then we may suppose
that the same causes which have changed N" into riV will change
rN into rW, so that r^N will be the number born in the year
succeeding that in which rN were born. Similarly, r^N will be
born 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 born 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

[ ^. ^. ^

Therefore the ratio of the population in the 100*'' year to the
number of infants born in that year will be

If we assume that the ratio of the population in any year to the
number of infants born 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.

212 EULEn.

435. A memoir by Euler, entitled Stir les Rentes Viageres,
immediately follows tlie preceding, occupying pages 1G5 — 175 of
the volume.

Its principal point is a formula for facilitating tlie calculation
of a life annuity.

Let A.,^^ denote the value of an annuity of one pound on the
life of a person aged 7n 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,

1 {{m-^l) 0)^ + 2) U+3) I

. _ _1 {{m^ (m + 3) (»z + 4) ]

"'^^ (m+1) 1 \ ^ X' "^ \' "^ ""J '

therefore (m) X A,,, = [m + 1) + {m + 1) J,

Wi+l"

Thus when A.^^ has been calculated, Ave can calculate A„^^^
easily.

Euler gives a table exhibiting the value of an annuity on
any age from 0 to 94. But with respect to the ages 90, 91, 92,
93, 94, he says,

Mais je ne voudrois pas consei'der a un entrepreneur de se mekr
avec de tels vieillards, a nioins que leur nombre ne fut assez consider-
able; ce qui est une regie generale pour tons les etablissemens fondes
sur les probabilites.

Euler is of opinion 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 defended
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 I'on y vouloit employer la somme de 3500 ecus, ce seroit
toujours un bel etablissement, que de jouir cles I'age de 20 ans d'une
pension fixe de 1000 ecus. Ce^^endant il est encore douteux, s'il se
trouveroit plusieurs parens qui voiuh"oicnt bien faiie uu tel sacrifice
pour le lien de leurs enfans.

EULER. 2i3

436. The next memoir by Euler is entitled Sur Vavantage du
Banqiiier cm jeu de Fharaon. 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 hoAvever
to the result which we will notice.

Consider the equation in Finite Differences,

m {in — 1) (;? — m) (n — ??2 — 1 )
^'" " 2n (ii - 1) ^ 71 (n - 1) ""-' •

By successive substitution we obtain

m (m — 1)S

u„ =

" 2n{?i-l){n-2) ... (n-m + 1)'

where S denotes the sum (f) (u) + </> (?i — 2) + (^ (/i — 4) + . . . ,

(f> (>i) being (ii — 2) (n — S) ... (n — m + 1).

This coincides with what we have given in Art. 155, supposing
that for A we put unity.

AYe shall first find a convenient expression for S. We see that

^= coefficient of x"'~^ in the expansion of (1 +£c)"~^

Hence S is equal to | m — 2 times the coefficient of cc'" " in the
expansion of

(1 + xy-' + (1 + xy-' +(14- xy-" + .,.

Now in the game of Pharaon we have n always even ; thus we
may suppose the series to be continued down to 1, and then its
sum is

(i+^)"-i ^, . . (1+^r-i

(1 -\-'xy^l ^^'""^ '' 2x + x' '
Thus we require the coefficient of x"'"^ in the expansion of

(1 + xy - 1

2-^x '

16— 2

2H EULER.

This coefficient is

n{n-l) ... {n-m-\- 2) n {n - 1) ... (n-m + 3)

2 1 on - 1 ~ 4 I ?/z - 2

n (n — 1) ... {n — m-\- 4)
■^ 8 ! m - 3

Then 8 — \m — 2 times this coefficient.
Hence with this expression for S we find that

1 772 1 771 {m — 1)

^i'n = T

'" 4 7i - «i + 1 8 (n - ??i + 1) {n - ??z + 2)

1 m {in — 1) (w — 2)

IG (?i - wi + 1) {n - m + 2) (« - m + 3)

. (_ 1 V^ -1 m(m-l)...2
■^^ ^ 2"^ (yi-m + 1) ...(vi-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 w„ is

m { m — 1 {m — l){m — 2){m — 2)

2'" \ l{n-l) 1.2.3(/i-3)

• • • ( •

{m — 1) {m — 2) {m — 3) (m — 4) {m — 5)
+ ^^ — ^ o A — ^7 ^^ i-

1.2. 3. 4. 5(;z-5)

Euler obtained this formula by trial from the cases in which
w = 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 w — 1, w — 2, n — 3, . . . ?z — ?7i + 1 ;
and the numerators will be independent of n.

We will find the numerator of the fraction whose denominator
is w — r.

From the last term in Nicolas Bernoulli's formula we obtain

{-ly^^ w(m-l)...2

m — ^ — r ?' — 1 '

EULER. 245

from the last term but one we obtain

2"'"' \m-l-r\r-2'

and proceeding in this way we find for the sum

_1 m-l-r |^~TT2 "^^ 17273 ''+-"j

a^ir

2»'+i 1^

(- 1)''"" I m

-^ 1 - (1 - 2)' .

n — 1 — r I )

This vanishes if r be an et;e?i number ; and is equal to

'2r\r \m-l-r'
if r be odd.

Thus Euler's formula follows from Nicolas Bernoulli's.

438. The next memoir by Euler is entitled Sur la prohabilite
des sequences dans la Lotterie Genoise. 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 5 tickets are drawn at random.
The question may be asked, what is the chance that two or
more consecutive numbers should occur in the drawings? 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 three 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 w^hich 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.

24 G EULEE.

440. As an example of Euler's method we will give his inves-
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-\-l, a-\-2, that is a sequence of three.

II. a, a + 1; h, that is a sequence of two, the number h
being neither a + 2 nor a — 1.

III. a, h, c, where the numbers a, Jj, c involve no sequence.

I. The form a, a-\-l, a + 2. The number of such events is
n — 2. For the sequence may be (1, 2, 3), or (2, S, 4), or (3, 4, 5),
up to (71 —2,n—l, n).

II. The form a, a + 1, h. In the same way as we have just
shewn that the number of sequences of three, like a, a + 1, « + 2,
is 71 — 2, it follows that the number of sequences of two, like
<2, a + 1, is 7z — 1. Now in general h may be any number between
1 and n inclusive, except a—1, a, a + 1, a + 2; that is, h may be
any number out of ?? — 4 numbers. But in the case of the first
sequence of two, namely 1, 2, and also of the last sequence n — 1, 71,
the number of admissible values of J is n — 3. Hence the whole
number of events of the form a, « + 1, I, is (w — 1) (71 — 4) + 2, that
is 71^ — 5n + Q), that is {n — 2) {ii — 3).

III. The form a, h, c. Suppose a to be any number, then h
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 investiofates the number of events
which can arise. It will however be sufficient for us here to take
another method which he has also given. The total number of
events is the number of combinations of 7i things taken 3 at a time,

that is — ^^ — — . The number of events of the third kind

can be obtained by subtracting from the whole number the num-
ber of those of the first and second kind ; it is therefore

71 (n — 1) Cii — 2) , J,. , ,,. .

17273 — ~ ^'' ~ ^ ^" ~ ) ~ ^'' " ^-

EULER. 94.7

It will be found tliat tliis is

(n - 2) (n - 3) {n - 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(^-3) {n - 3) {n - 4)

71 (a -1)' n{a-l) ' n (n ~ 1) '

441. Euler's next memoir also relates to a lottery. This
memoir is entitled Solution d'lme question tres difficile dans le
Calcid des Prohahilites. It w^as published in the volume for
1769 of the Histoire de VAcad. ... 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 qni ni'a fourni cette question, que je
me propose de developper. Cette lotterie etoit de cinq classes, chacuue
de 10000 billets, parmi lesquels il y avoit 1000 prix dans chaque
classe, et par consequent 9000 bJancs. Chaque billet devoit passer
par toutes les cinq classes; et cette lotterie avoit cela de particulier
qu'outre les prix de chaque classe on s'engagooit de payer un ducat
a cliacun de ceux dont les billets auroient passe 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 jmzes.

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.

24iS EULER.

Suppose tlie tickets of the first class to have been drawn, and
that the prizes have fallen on certain n tickets A, B, G ...

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 71

(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 fraction just given to the power k — 1.

Let {(m + 1) (m + 2) {m + n)Y~'=M,

and {1.2 nY'' = a.

Then -^ 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 obtain no prize at all. Here besides the n tickets A, B, G, ...
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
^m. Now If 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 A, B, G, ... 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 in which this can occur may
be denoted by <ym {in — 1), where 7 is indejjendent of m.

Proceeding in this way we have from the consideration that
the sum of all possible cases is M

M= a + jSin + ym (in - 1) + 3m [m — 1) {in - 2) -f . . ..

EULER. 249

Now % 13, 'y, ... are all independent of m. Hence we may put
in succession for on the values 1, 2, 3, ... ; and we shall thus be
able to determine /S, y —

446. Euler enters into some detail as to the values of /3, 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 m — 2 ducats, and so on. Thus the probable expense is

-r> \am + /3m (m - 1) + jm {m - 1) (m — 2) + . . . L

= -^ ja + /3(w-l)+7 (771-1) (m-2) + ...L

The expression in brackets is what we shall get if we change
m into m — 1 in the right-hand member of the value of M in
Art. 445 ; the expression therefore is what M becomes when Ave
change m into m — 1. Thus

a + y5(m-l) +7(7??- 1) {m-2) + ...

= [m (771 + 1) . . . (m + n - 1) }'"\

Thus finally the probable expense is

m Y ^

/)n + Uj

Euler then confirms the truth of this simple result by general

reasoning.

447. We have next to notice a memoir entitled Eclaircisse-
mens sur le memoire de Mr. De La Gra^ige) inserS dans le V'^

volume de Melanges de Turin, concernant la methode de prendre le

milieu entre les residtats de plusieurs observations, <^c. Presente

a VAcademie le 27 Nov. 1777. This memoir was published in the

Nova 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 EULEPu

The memoir consists of explanations of psn't of that memoir
by Lagrange to which we have aUuded 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
dull.

4i8. 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 Moivre's Doctrine of Chances, Problem xxxix. of the
third edition is thus enunciated: To find the Expectation of J.,
when with a Die of any given number of Faces, he undertakes
to fling any numxber 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 71 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

,f _ ,„ u -Vf-\- ^^'^lH (,, _ 2)-^ - . . .
^ ^ 1.2^ ^

where the series consists of m + 1 terms. The whole number of
possible cases is if, and the required chance is obtained by di-
viding the number of favourable cases by the whole number of
possible cases.

44^9. 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 7f ; the whole number of cases
if the ace were expunged would be {n — iy ; thus the whole number
of cases in which the ace can come up is ;?/*"— (n — ly.

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 tlie ace could come up would l)e [n — ly — {n — 2y, by

EULER. 251

what we have just seen ; this therefore is the niunber 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 luith the deux.
Thus the result is

n^-{n-ir-\{n-\r-{n-^.rr,
that is, if -2{n- Vf -f {n - 2f.

De Moivre in like manner briefly considers the case in wdiich
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'" (71 — mY ; 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 knoAv that the expression
A"' (?z — my vanishes when ?n is greater than x.

Suppose 71 = m ; then the number of ways may be denoted by
A^O"^ ; the expression written at full is

,f _ ,, (,, _. 1)- a. ^^j-^-^ {^ri-Tf-...

451. One particular case of the general result at the end
of the preceding Article is deserving of notice. If we jDut x = n,
we obtain the number of ways in which all the 71 faces come up
in n throws. The sum of the series wdien x = 7i is known to be
equal to the product 1.2.3...??, 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 7i faces are
to appear in ii throws, there must be no repetition ; and thus the

'252 EULER.

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 Memoires ... 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 7i^ is the whole number of cases ; the
corresponding number in Laplace's problem is [^ (n, r)}'', where
by (/) (71, r) we denote the number of combinations of n things
taken r at a time. In De Moivre's problem (n — ly 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
j^(7i-l, r)]^ Similarly to (n — 2y in De Moivre's problem
corresponds [(f) (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

{<P (n, r)Y-m {4, («- 1, r)}' + "^"~^^ (<^ (»- 2, r)}' - ... ;

and the probability will be found by dividing this number by the
whole number of cases, that is by {</> (?i, 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'" {(/> (n — 7?^, r)}^; and the number of cases favourable
to the drawing of all the tickets by A" {(/> (0, r)Y.

455. In the Histoire de VAcad. ... Paris, 1783, Laplace gives
an approximate numerical calculation, which also occurs in
page 195 of the Theorie ... des Proh. 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 957()7
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 Opuscida Analytica, Vol. Ii.,
1785. In this volume pages 331 — 346 are occupied with a memoir
entitled Solutio quarundam quaestionum dijjiciliorum in calculo
prohabilium. Euler begins thus :

His quaestionibus occasionem dedit ludus passim publice institutus,
quo ex nonaginta scliedulis, 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 numerus fuerit peractus, vel omnes nonaginta numeri
exierint, vel saltern 89, vel 88, vel pauciores. Has igitur quaestiones,
utpote difficillimas, hie ex principiis calculi Probabilium jam pridem usu
receptis, resolvere constitui. Neque me deterrent objectiones Illustris
lyAlembert, qui huuc calculum suspectum reddere est conatiis. Post-
quam enim summus Geometra studiis mathematicis valedixit, lis etiam
helium indixisse videtur, dum pleraque fundamenta solidissinie stabilita
evertere est aggressus. Quamvis enim hae objectiones apud ignaros
maximi ponderis esse debeant, hand 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

_q] 1.2 q

458. Euler makes no reference to his predecessors De Moivre
and Laplace. He gives the formula for the chance that all the

254 EULEE.

tickets shall be drawn. This formula corresponds with Laplace's.
We have only to put 771 = w in Art. 453.

Euler then considers the question in which n — 1, or ?i — 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 tlie following ; suppose we require that
71 — V tickets at least shall be drawn, then the number of favour-
able cases is

+ {i> + l)<j> {», v+2){,j>{n-v- 2, r)}'

- i^^H^-±^ ^ („, ^ + 3){ct,in-v- 3, 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 will be found easier
to follow.

The number of ways in which exactly m tickets are drawn
is (^ (n, m) A'" {</) (0, r)Y. For the factor A"^ [(\> (0, r)Y is, by
Art. 454, the number of ways in which in a lottery of m tickets,
all the tickets will appear in the course of x drawings ; and
0 (n, 77i) is the number of combinations of ii 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 </> {ii, m) A"* {^ (0, r)Y,
where S refers to m, and m is to have all values between n and
n — v, both inclusive.

Thus we get

A" [^ (0, r)Y + « A'- [</> (0, r)r + ^^^-^ A»-[<^ (0, ryf
the series extending^ to i^ + 1 terms.

EULER. 255

We may write this for shortness thus,
{a-+ n A-- + 4;^ A- + " ^"^^;'- '' A-. ...} j^ (0, ,.;}:

Now put E—1 for A, expand, and reariange in powers of E;
we shall thus obtain

1^" - (/> {n, V + 1) E''-'-' -\-{v + l)^ {n, V + 2) E"-'"'

- (^^±1)^^ ^ (., . + .3) E-^ + ...} {^ (0, .^ ;

and this coincides with Euler's result.

We shall find in fact that when we put E—1 for A, the
coefficient of E''~^ is

(- '^y \J1 U n ^P^P-^') p{r-V){p-^-.^ I

\p\n-p\ ^ "^ 1.2 1.2.3 ■^•••j'

where the series in brackets is continued to z^ + 1 terms, unless
p be less than z^ + 1 and then it is continued to jj + 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 - xy\
that is in the expansion of (1 — xY~^. In the latter case the sum
would be the coefficient of x^ in the same expansion, and is there-
fore zero, except when ^9 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

[<^ («. r)Y - n [4> [n - 1, r)Y + ^^^ {<i> (u - 2. r)]' - ...

must be zero if n be greater than xr ; for the expression gives the
number of ways in which 71 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 EULER

Euler does not demonstrate this result; perhaps he deduced
it from the Theory of ProbabiHty itself. For if xr = n, it is
obvious that no ticket can be repeated, when all 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 of ?2 — 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
formulae,

In his probabilitatibiis aestimandis utique assiimitur omnes litteras
ad extrahendum aeque esse proclives, quod autem 111. D^Alemhert negat
assumi posse. Arbitratur enim, simul ad omnes tractus jam ante per-
actos respici oportere; si enim quaepiam litterae nimis crebro fuerint
extractae, turn eas in sequentibus tractibus rarius exituras; contrarium
vero eveniie si quaepiam litterae nimis raro exierint. Haec ratio, si
valeret, etiam valitura esset si sequentes tractus demum post annum,
vel adeo integrum speculum, quin etiam si in alio quocunque loco
instituerentur ; atque ob eandem rationem etiam ratio haberi deberet
omnium tractuum, qui jam olim in quibuscunque terrae locis fuerint
peracti, quo certe vix quicquam absurdius excogitari potest.

462. In Euler's Opuscula Analytica, Yol. ii., 1785, there is
a memoir connected with Life Assurance. The title is Solutio
quaestionis ad calculum j)7vhahilitatis pertinentis. Quantum duo
conjuges per^solvere deheant, ut suis haeredihus post utriusque
mortem certa argenti summa persolvatiir. 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 and 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 x = 0, and find z.

Euler does not himself calculate numerical results, but he
leaves the formulae quite ready for application, so that tables
might be easily constructed.

CHAPTER XIII.

D'ALEMBERT.

463. D'Alembert was born in 1717 and died in 1788. 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 Croix ou Pile of the Fncyclopedie ou

Dictionnaire Raisonne We will speak of this work simply as

the Encyclopedie, and thus distinguish it from its successor the
Encyclopedie M^thodique. The latter work is based on the former ;
the article Croiw ou Pile is reproduced unchanged in the latter.

464. The date of the volume of the Encyclopedie containing
the article Croix ou Pile, is 1754. 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, II2\ TT\ the only unfavourable case is the

last ; therefore the required chance is - . D'Alembert however

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'alembert. 259

need of the second throw. Thus he makes only three cases,

2
namely, H, TH, TT\ therefore the chance is ^.

Similarly 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.

465. 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 w^e have noticed in

Arts. 889 — 393 ; he adds : mais nous ne savons si on en sera satis-
fait ; et il y a ici quelque scandale qui merite bien d'occuper les
Algebristes. 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 wdth 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
wdiich produce an infinite expectation.

D'Alembert proceeds to make some further remarks w^hich are
repeated in the second volume of his Ojniscides, 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 w^e have just noticed.

466. We have next to notice the article Gageure, of the
Encyclopedie; the volume is dated 1757. D'Alembert says he wall
take this occasion to insert some ver}^ good objections to what he
had given in the article Croix ou Pile. He says, Elles sont de
M. Necker le fils, citoyen de Geneve, professeur de Mathematiques
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

2(j0 d'alembert.

ordinary 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 derniere,
meritent 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 Encyclopklie
M<itliodique.

467. D'Alembert wrote various other articles on our subject
in the E ncy dope die ; but they are unimportant. We will briefly
notice them.

Als3nt. 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 sorts de jeu dont
M. de Buff on a donne le calcid in 1733, avant que d'etre de
V Academie des Sciences ; see Art. 354.

De. 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 regies doivent ^tre modi-
fiees dans certains cas, ou la probabilite 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 Encyclopedic
M4thodi€[ue.

The article Prohahilite in the Encyclopedie 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 Ojyuscules Mathematiques D'Alem-
bert gives remarks on the Theory of ProbabiUties. 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 silt le calcul des Prohabilites ; 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 all analysts, but that cases exist in which the rule
seems to fail.

470. The first case wdiich D'Alembert 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, independamment de ce qu'une somme infinie est line cliimere,
11 n'y a personne qui vouliit douner pour jouer a ce jeu, je ne dis pas
une somme infinie, mais meme une somme assez modique.

471. D'Alembert notices a solution of the Petersburg Problem
which had been communicated to him by un Geometre celebre
de I'Academie des Sciences, plein de savoir et de sagacite. 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 A 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 unfah^ 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 will 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. DAlembert 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-

d'alembeht. 263

bility would be quite willing 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 in the Encyclopedie, 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 Encyclopedie that

2 3

the chance ought to be ^ instead of -r . In the Opuscides how-

ever he does not insist very strongly on the correctness of the

2
result ^ , but seems to be content with saying that the reasoning
o

which jDroduces 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 gi'eat mathematician could produce on the wrong side of a
question should consult the original memoir. But we agree with
every other ^^Titer on the subject in thinking that there is no
real force in D'Alembert's objections.

476. The folloAving extract will shew that D'Alembert no

2
longer insisted on the absolute accuracy of the result ^ :

264^ d'alembekt.

Je ne voudrois pas cependant regarder en toute rigueur les trois coups
dont il s'agit, comme egalement possibles. Car 1°, il povirroit se faire
en effet (et je suis meme porte a le croire), que le cas pile croix ne fut
pas exactement aussi possible que le cas croix seiil; mais le rapport des
possibilites me paroit inappretiable. 2". II poiirroit se faire encore que
le coup 2)il& croix fiit un peu plus possible que pile pile, par cette seule
raison que dans le dernier le meme effet arrive deux fois de suite; mais
le rapport des possibilites (suppose qu'elles soient inegales), n'est pas
plus facile a etablir dans ce second cas, que dans le premier. Ainsi
il pourroit tres-bien se faire que dans le cas propose, le rapport des
probabilites ne fut ni de 3 a 1, ni de 2 a 1 (comme nous I'avons sup-
pose dans VUncyclopeclie) mais un incommensurable ou inappretiable,
moyen entre ces deux nombres. Je crois cependant que cet incommen-
surable approcliera plus de 2 que de 3, parce qu' encore une fois il n'y
a que trois cas possibles, et non pas quatre. Je crois de meme et par
les memes raisons, que dans le cas ou Ton joueroit en trois coups, le
rapport de 3 a 1, que donne ma methode, est plus pres du vrai, que
le rapport de 7 h 1, donne par la methode 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
Gageure in the EncyclopMie ; see Art. 466. Let there be a
die with three faces, A, B, C ; then according to DAlembert's
original method in the Encydopedie, 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 DAlembert's
method the chance for the appearance of A is to the chance
against it as 2" — 1 is to 2".

For example, suppose ti = 8 : then the favourable cases are
A, BA, CA, BBA, BOA, CCA, CBA ; the unfavourable cases are
BBB, BBC, BOB, BCG, CBB, GBG, GGG, GGB: 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.

d'alembert. 265

478. D'Alembert thus sums up his results, on his page 24 :

Concluons de toutes ces reflexions j 1". que si la regie que j'ai donnee
dans Y Encyclopedie (faute d'en connoitre une meilleure) pour deter-
miner le rapport des probabilites au jeu de croix et pile, n'est point
exacte a la rigueur, la regie ordinaire pour determiner ce rapport, Test
encore moins; 2". que pour parvenh a une tlieorie satisfaisante du cal-
cul des probabilites, il faudroit resoudre plusieurs Problemes qui sont
peut-etre insolubles; savoir, d'assigner le vrai rapport des probabilites
dans les cas qui ne sont pas egalement possibles, ou qui peuvent
n'etre pas regardes conime tels; de determiner quand la j)robabilite
doit etre regardee comme nulle ; de fixer enfin comment on doit estimer
I'esperance ou I'enjeu, selon que la probabilite est 2)lus ou moins grande.

479. The next memoir by D'Alembert which we have to
notice is entitled Sur^ Vwpplication du Calcid des Prohahilites a
Vinocidation 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 stating 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 will take his

2GG d'aleivibert.

cliauc3 of escaping entirely from an attack of small-pox during
his life, or at least of recovering if attacked.

D'Alembert thinks that the prosj^ect held out to an individual
of a gain of three or four years in the probable duration of his
hfe, 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 p)liysical life and the civil life. 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 difficult 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 ii of those not previously

d'alembeht. 2G7

attacked, and that 1 died out of every m 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 follomng 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 ti represent the number who have died during this
period of small-pox : let z represent the number who would have
been alive if small-pox did not exist : required z 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 z individuals
subject to small-pox, we should have

dz = - dii.

y ^

But we must subtract from this value of dz the decrement

arising from small-pox, to which the z individuals are by hypo-

z
thesis not liable : this is - du.

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 dif du

^ y y

therefore log z = \ogy + \ —

/du

268 d'alembert.

The result is not of practical use because the value of the

C Ciu
integral 1— is not known. D'Alembert gives several formulie

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 espe-

7'ance 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 j^^^^ohahle 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 7nea)i life 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 2QN; 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 mea/i life and probable life which we here use have
not always been appropriated in the sense we here explain ; on the
contrary, what we call the mean life has sometimes been called
the probable life. 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-
bility 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 mortaliti/.

Let X denote the number of years measured from an epoch ; let
yjr (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
'yjr (x) be the ordinate of a curve ; then yjr (x) diminishes from
X = 0 to X = 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

I -v/r (x) dx

The probable duration is a quantity h such that

This is D'Alembert's mode. We might however use another
curve or function. Let cf) (x) be such that <^ (x) dx represents the
number who die in an element of time dx. Then the mean dura-
tion of life for j)ersons aged a years is

I (a? — a) (f) (x) dx

I </) (x) dx

J a

The probable duration is a quantity h such that

I (f) (x) dx = I cj) (x) dx,

that is I (f> (x) dx = - j (f) (x) dx.

Thus the mean duration is represented by the abscissa of the
centre of gi'avity 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 <f)(x) —— h -^fr' (x), where Jc is some constant.
Therefore

I {x — a)<f> (x) dx j (x — ^) ^' (^) <^'^

. ' a J_a .

I <^ (x) dx I -v/r' (x) dx

J a -'a

270 d'alemfsErt.

iind I (x — «) -v/r' (x) dx = (x — a) ^jr {x) — I yfr (x) dx,
therefore I (x — a) yjr' (x) dx = — j yjr (x) dx;

and \//' (x) dx=: — yjr (a).

Thus

I (a? — a) (p (x) dx I yjr (x) dx

a .' a

V (.X) dx ^ ^"^

This shews that ihe two methods give the same mean duration.
In the same way it may be shewn that they give the same pyvhahle
duration.

487. D'Alembert draws attention to an erroneous solution of
the problem respecting the advantages of Inoculation, which he
says was communicated to him by un savant Geotnetre. D'Alem-
bert shews that the solution must be erroneous because it leads to
untenable results in two cases to which he applies it. But he does
not shew the nature of the error, or explain the principle on which
the pretended solution rests ; and as it is rather curious we will
now consider it.

Suppose that N infants are born at the same
epoch, and let a table of mortality be formed by
recording how many die in each year of all dis-
eases excluding small-pox, and also how many die

of small-pox. Let the table be denoted as here ;

so that u^ denotes the number who die in the r*'^ year excluding
those who die of small-pox, and v,. denotes the number who die of
small-pox. Then we can use the table in the following way : sup-
pose M any other number, then if u^ die in the r*'' year out of N

M
from all diseases except small-pox, -^^ w,. would die out of M; and

so for any other proportion.

Now suppose small-pox eradicated from the list of human dis-
eases ; required to construct a new table of mortality from the
above data. The savant Geometre proceeds thus. He takes the

^'l

^'■2

^'2

«'3

^^4

^'4

d'alembert. 271

preceding table and destroys the colmnn v^, v^, ^3> ••• Then he
assumes that the remaining column will shew the correct mortality
for the number N—n at starting, where n is the total number who
died of small-pox, that is n — v^-\- v^-\-v^+ ...

Thus if we start w4th the number M of infants ^r^ ?<,. would

N-n '

die on this assumption in the r^^ year.

There is a certain superficial plausibility in the method, but it

is easy to see that it is unsound, for it takes too unfavourable a view

of human life after the eradication of small-pox. For let

u^ + ^^2 + • • • ^^r = ^r >

then we know from the observations that at the end of r years
there are N —U^— V^ survivors of the original N \ of these w.^^ die
in the next year from all diseases excluding small-pox. Thus
excluding small-pox

^r+i

N-U,- V/

is the ratio of those who die in the year to those who are aged
r years at the beginning of the year. And this ratio will be the
ratio which ought to hold in the new tables of mortality. The
method of the savant Geometre gives instead of this ratio the
greater ratio

'•+1

N- U.-

488. Thus we see where the savant Geometre was wrong, and
the nature of the error. The pages in D'Alembert are 88 — 92 ;
but it will require some attention to extricate the false principle
really used from the account which D'Alembert gives, which is also
obscured by a figure of a curve. In D'Alembert's account regard
is paid to the circumstance that Inoculation is fatal to some on
whom it is performed ; but this is only a matter of detail : the
essential principle involved is that which we have here exhibited.

489. The next publication of D'Alembert on the subject of
Probabilities appears to consist of some remarks in his Melanges

272 d'alembert.

de Philosophie, Vol. v. I have never seen the original edition of
this work ; but I have no doubt that the remarks in the Melanges
de Philosopliie were those which are reprinted in the first volume
of the collected edition of the literary and philosophical works of
D'Alembert, in 5 Vols. 8vo, Paris, 1821. According to the cita-
tions of some writers on the subject I conclude that these remarks
also occur in the fourth volume of the edition of the literary and
philosophical works in 18 Vols. 8vo, Paris, 1805.

490. In the first volume of the edition of 1821 there are two
essays, one on the general subject of Probabilities, and the other on
Inoculation.

The first essay is entitled Doutes et questions sur le Calcul des
Prohahilites. These occupy pages 451 — 466 ; the pages being
closely printed.

D'Alembert commences thus :

On se plaint assez communement que les formules des matli^ma-
ticiens, appliquees aux objets de la nature, ne se trouvent que trop
en defaut. Personne neanmoins n'avait encore apergu ou cru aper-
cevoir cet inconvenient dans le calcul des prohahilites. J'ai os6 le
premier proposer des doutes sur quelques principes qui servent de base
ii ce calcul. De grands geometres ont juge ces doutes dignes d' attention;
d'autres grands geometres les ont trouves ahsurdes; car pourquoi adou-
cirais-je les termes dont ils se sont servis ? La question est de savoir
s'ils ont eu tort de les employer, et en ce cas ils auraient doublement
tort. Leur decision, qu'ils n'ont pas juge a propos de motiver, a en-
courage des mathematiciens mediocres, qui se sont hates d'ecrire sur ce
sujet, et de m'attaquer sans m'entendre. Je vais tacher de m'expliquer
si clairement, que presque tous mes lecteurs seront a portee de me
jager.

491. The essay which we are now considering may be described

in general as consisting of the matter in the second volume

of the Opuscides divested of mathematical formulae and so adapted

to readers less versed in mathematics. The objections against

the ordinary theory are urged perhaps with somewhat less con-

2
fidence ; and the particular case in which - was proposed in-

3 .

stead of 7 ^s the result in an elementary question does not appear.

But the other errors are all retained.

d'alembert. 273

492. There is some additional matter in the essay. D'Alem-
bert notices the calculation of Daniel Bernoulli relative to the
small inclination to the ecliptic of the orbits of the planets ;
see Art. 394. D'Alembert considers Daniel Bernoulli's result
as worthless.

DAlembert says with respect to Daniel Bernoulli,

Ce qu'il y a de singulier, c'est que ce grand geometre dont je parle,
a trouve ridicules, du moins ^ ce qu'on m'assure, mes raisonnemens
sur le calcul des 2^Tobahilites.

493. D'Alembert introduces an illustration which Laplace
afterwards adopted. D'Alembert supposes that we see on a table
the letters which form the word Constantinojyolitanensibus, ar-
ranged in this order, or arranged in alphabetical order ; and he
says that although mathematically these distributions and a third
case in which the letters follow at hazard are equally possible,
yet a man of sense would scarcely doubt that the first or second
distribution had not been produced by chance. See Laplace,
Theorie . . . des Proh. page xi.

494. D'Alembert quotes the article Fatalite in the Ency do-
pe die, as supporting him at least partially in one of the opinions
which he maintained ; namely that which we have noticed in the
latter part of our Art. 474. The name of the writer of the article
Fatalite is not given in the Encyclopedia.

495. The other essay Avhich we find in the first volume
of the edition of D'Alembert's literary and philosophical works
of 1821, is entitled Reflexions sur T Inoculation ; it occupies
pages 463 — 514.

In the course of the preface D'Alembert refers to the fourth
volume of his Opuscules. The fourth volume of the Opuscules is
dated 1768 ; in the preface to it D'Alembert refers to his Me-
langes de Philosophie, Vol. V.

We may perhaps infer that the fifth volume of the Melanges.,,
and the fourth volume of the Opuscules appeared at about the
same date.

496. The essay may be said to consist of the same matter

27^ d'alembert.

as appeared on the subject in the second volume of the Opuscules,
omitting the mathematical investigations, but expanding and
illustrating all the rest.

D'Alembert's general position is that the arguments which
have hitherto been brought forward for Inoculation or against it
are almost all unsound. His own reflexions however lead to the
conclusion that Inoculation is advantageous, and that conclusion
seems more confidently maintained in the essay than in the
Opuscules. Some additional facts concerning the subject are re-
ferred to in the essay ; they had probably been published since
the second volume of the Opuscules.

497. D'Alembert retains the opinion he had formerly held as
to the difficulty of an exact mathematical solution of the problem
respecting the advantages of Inoculation. He says in summing
up his remarks on this point : S'il est quelqu'un a qui la solution
de ce probl^me soit reservee, ce ne sera st\rement pas a ceux qui
la croiront facile.

498. D'Alembert insists strongly on the want of ample col-
lections of observations on the subject. He wishes that medical
men would keep lists of all the cases of small-pox which come
under their notice. He sa3"s,

...ces registres, donnes an public par les Facultes de medecine ou
par les particuliers, seraient certainement d'une utilite plus palpable
et plus prochaine, que les recueils d'observations meteorologiques pub-
lies avec tant de soin par nos Academies depuis 70 ans, et qui pour-
tant, ^ certains egards, ne sont pas eux-memes sans utilite.

Combien ne serait-il pas a souhaiter que les medecins, au lieu de
se quereller, de s'injurier, de se dechirer mutuellement au sujet de
I'inoculation avec un aeharnement tlieologique, au lieu de supposer
ou de degniser les faits, voulussent bien se reunir, pour faire de bonne
foi toutes les experiences necessaires sur une matiere si interessante
pour la vie des homrnes ]

499. We next proceed to the fourth volume of D'Alemherfs
Opuscules, in which the pages 73 — 105 and 283—341 are de-
voted to our subject. The remarks contained in these pages are
presented as extracts from letters.

U^ALEMBERT. 275

500, We will now take the first of the two portions, which
occupies pages 73 — 105,

D'Alembert begins with a section Siir le calcul des Prohahilites.
This section is chiefly devoted to the Petersburg Problem. The

chance that head wdll not appear before the n^^' throw is ^

on the ordinary theory. D'Alembert proposes quite arbitrarily to
change this expression into some other which will bring out a

finite result for ^'s expectation. He suggests , ^-^ where

/8 is a constant. In this case the summation wdiich the problem re-

quires can only be effected approximately. He also suggests ^^^^^

^^^d „^^.a(»-i) where a is a constant.

He gives of course no reason for these suggestions, except
that they lead to a finite result instead of the infinite result of
the ordinary theory. But his most curious suggestion is that of

1 ^ 1 , where B and K are constants

and 17 an odd integer. He says,

Nous mettons le nombre pair 2 au denominateur de Texposant, afiu
que quand on est arrive au nombre n qui donne la probabilite ^gale
a zero, on ne trouve pas la probabilite negative, en faisant n plus
grand que ee nombre, ce qui seroit clioquant ; car la probabilite ne
sauroit jamais ^tre au-dessous de zero, II est vrai qu'en faisant n
plus grand que le nombre dont il s'agit, elle devient imaginaire; mais
cet inconvenient me paroit moindre que celui de devenir negative;...

501. D'Alembert's next section is entitled Bur Tanalyse des
Jeux.

D'Alembert first proposes une consideration tr^s-simple et
tr^s-naturelle a faire dans le calcul des jeux, et dont M. de Buffon
m'a donne la premiere idee, . . . This consideration we will explain
when noticing a w^ork by Buffon. D'Alembert gives it in the
form which Buffon ought to have given it in order to do justice
to his own argument. But soon after in a numerical example

18—2

276 d'alembert.

D'Alembert falls back on Biiffon's own statement ; for he supposes
that a man has 100000 crowns, and that he stakes 50000 at an
equal game, and he says that this man's damage if he loses is
greater than his advantage if he gains ; jDuisque dans le premier
cas, il s'appauvrira de la moitie ; et que dans le second, il ne
s'enrichira que du tiers.

502. If a person has the chance ■ of gaining x and the

chance — - — of losing y, his expectation on the ordinary theory

is ~ ^ . D'Alembert obtains this result himself on the ordi-

nary principles ; but then he thinks another result, namely

— , miofht also be obtained and defended. Let jz denote the

sum which a man should give for the privilege of being placed
in the position stated. If he gains he receives x, so that as he

paid z his balance is x — z. Thus — is the correspondincj

expectation. If he loses, as he has already paid z he will have
to pay y — ^ additional, so that his total loss is y, and his con-
sequent expectation - — ^'^- . Then ^— — — is his total ex-

p -^ q p + q

pectation, which ought to be zero if z is the fair sum for him

to pay. Thus z = ^ ^^ . It is almost superfluous to observe

that the words which we have printed in Italics amount to as-
signing a new meaning to the problem. Thus D'Alembert gives
us not two discordant solutions of the sa77ie problem, but solu-
tions of two different problems. See his further remarks on his
page 283.

503. D'Alembert objects to the common rule of multiplying
the value to be obtained by the probability of obtaining it in
order to determine the expectation. He thinks that the pro-
bability is the principal element, and the value to be obtained
is subordinate. He brings the following example as an objection
against the ordinary theory; but his meaning is scarcely intel-
ligible :

d'alembert. 277

Qu'on propose de choisir entre 100 combinaisons, dont 99 feroiit
gagner mille ecus, et la 100® 99 mille ecus; quel sera I'liomme assez
iusense pour preferer celle qui donnera 99 mille ecus. Uesperance dans
les deux cas n'est done pas reellement la meme; quoiqu'elle soit la
meme suivant les regies des probabilites.

504^. D'Alembert appeals to the authority of Pascal, in the
following words :

Un homme, dit Pascal, passeroit pour fou, s'il hesitoit a so laisser
donner la mort en cas qu'avec trois dez on fit vingt fois de suite trois
six, ou d'etre Empereur si on y manquoit ? Je pense absolument comme
lui j mais pourquoi cet homme passeroit-il pour fou, si le cas dont il
s'agit, est 'pliysiquement j)ossible %

See too the edition of D'Alembert's literary and philosophical
works, Paris, 1821, Vol. I. page 553, note.

505. The next section is entitled 8ur la duree de la vie.
D'Alembert draws attentioD to the distinction between the mean
duration of life and the probable duration of life ; see Art. 484.
D'Alembert seems to think it is a great objection to the Theory
of Probabihty that there is this distinction.

D'Alembert's objection to the Theory of Probability is as
reasonable as an objection to the Theory of Mechanics would be
on the ground that the centre of gi'avity of an area does not
necessarily fall on an assigned line which bisects the area.

D'Alembert asserts that a numerical statement of Buffon's,
which Daniel Bernoulli had suspected of inaccuracy, was not really
inaccurate, but that the difference between Buffon and Daniel
Bernoulli arose from the distinction between what we call meaii
duration and probable duration of life.

506. The last section is entitled Sur un Memoire de M. Ber-
noulli concernant F Inoculation.

Daniel Bernoulli in the commencement of his memoir had
said, il seroit a souhaiter que les critiques fussent plus reserves
et plus circonspects, et sur-tout qu'ils se donnassent la peine de se
mettre au fait des choses qu'ils se proposent d'avance de critiquer.
The words se mettre au fait seem to have given great offence to

»'=7C' -tn'

27S D ALEMBEllT.

D'Alembert as he supposed they were meant for him. He refers
to them in the Opuscules, Vol. IV. pages IX, 99, 100 ; and he
seems with ostentatious deference to speak of Daniel Bernoulli
as ce grand Geometre; see pages 99, 101, 315, 821, 323 of the
volume,

507. D'Alembert objects to the hypotheses on which Daniel
Bernoulli had based his calculation ; see Art. 401. D'Alembert
brings forward another objection which is quite fallacious, and
which seems to shew that his vexation had disturbed his judg-
ment. Daniel Bernoulli had found that the average life of all
who die of small-pox is 6^^ years ; and that if small-pox were
extinguished the average human life would be 29^^ years. More-
over the average human life subject to small-pox is 26^^ years.
Also Daniel Bernoulli admitted that the deaths by small-pox

were — ^^ ^^^ ^^® deaths,

J-O

Hence D'Alembert affirms that the folloAving relation ought

to hold;

1 12

but the relation does not hold; for the terms on the left hand side
will give 27{^ nearly instead of 263^. D'Alembert here makes the
mistake which 1 have pointed out in Art. 487 ; when that Article
was written, I had not read the remarks by D'Alembert which
are now under discussion, but it appeared to me that D'Alembert
was not clear on the point, and the mistake which he now makes
confirms my suspicion.

To make the above equation correct we must remove 29^,
and j)ut in its place the average duration of those who die of
other diseases while small-pox still prevails ; this number will be
smaller than 29j^2'

508. We pass on to the pages 283—341 of the fourth volume
of the Opuscules. Here we have two sections, one Sur le Calcul
des probahilites, the other Sur les Calculs relatifs a I' Inoculation,

609. The first section consists of little more than a repetition

D ALEMBEllT. 279

of the remarks which have akeady been noticed. D'Alembei-t
records the origin of his doubts in these words :

II y a pres de trente ans que j'avois forme ces doutes en Hsaiit
Texcellent livre de M. BernoulU de Arte conjectandi ; . . .

He seems to have returned to his old error respecting Croix
ou Pile with fresh ardour : he says,

...si les trois cas, croix, pile et croix, pile et 2^ile, les seuls qui
puissent arriver dans le jeu propose, ne sont pas egalement possibles,
ce n'est point, ce me semble, par la raison qu'on en apporte commu-

nement, que la probabilite du premier est - , et celle des deux autres

Q X - ou - . Plus j'y pense, et plus il me paroit que Tiiathematique-
nient parlaut, ces trois coups sont egalement possibles...

510. D'Alembert introduces another point in which he ob-
jects to a principle commonly received. He will not admit that
it is the same thing to toss one coin m times in succession, or
to toss m coins simultaneously. He says it is perhaps physically
speaking more possible to have the same face occurring simul-
taneously an assigned number of times with m coins tossed at
once, than to have the same face repeated the same assigned
number of times when one coin is tossed ')n times. But no person
will allow what D'Alembert states. We can indeed suppose circum-
stances in which the two cases are not quite the same ; for example
if the coins used are not perfectly symmetrical, so that they
have a tendency to fall on one face rather than on the other.
But we should in such a case expect a run of resemblances rather
in using one coin for m throws, than in using m coins at once.
Take for a simple example m — 2. We should have rather more

than -r as the chance for the former result, and only - lor tlie

latter; see Laplace, Theorie...des Proh. page 402.

511. D'Alembert says on his page 290, II y a quelque temps
qu'un Joueur me demanda en combien de coups consecutifs on
pouvoit parier avec avantage d'amener une face donnee d'un de —
This is the old question proposed to Pascal by the Chevalier de

280 d'alembert.

Mere. D'Alembert answered that according to the common theory
in n trials, the odds would be as &" — 5" to 5". Thus there would
be advantage in undertaking to do it in four throws. Then
D'Alembert adds, Ce Joueur me repondit que Texperience lui avoit
para contraire a ce resultat, et qu'en jouant quatre coups de
suite pour amener une face donnee, il lui etoit arrive beaucoup
plus souvent de gagner que de perdre. D'Alembert says that
if this be true, the disagreement between theory and observation
may arise from the fact that the former rests on a supposition
which he has before stated to be false. Accordingly D'Alembert
points out that on his principles the number of favourable cases
in n throws instead of being 6" — 5^ as by the ordinary theory,
would be 1 + 5 + 5^4-... + 5'^~\ This is precisely analogous to what
we have given for a die with three faces in Art. 477. D'Alembert
however admits that we must not regard all these cases as equally
likely.

512. D'Alembert quotes testimonies in his own favour from the
letters of three mathematicians to himself; see his pages 296, 297.
One of these correspondents he calls, un tr^s-profond et trt^s-habile
Analyste ; another he calls, un autre Mathematicien de la plus
grande reputation et la mieux meritee ; and the third, un autre
Ecrivain tres-eclaire, qui a cultive les Mathematiques avec succ^s,
et qui est connu par un excellent Ouvrage de Philosophie. But
this Ecrivain tres-dclaire is a proselyte whose zeal is more con-
spicuous than his judgment. He says "ce que vous dites sur la
probabilite est excellent et trer-evident ; I'ancien calcul des pro-
babilitcs est ruine . . . D'Alembert is obliged to add in a note,
Je n'en demande pas tant, a beaucoup pres ; je ne pretends point
ruiner le calcul des probabilites, je desire seulement qu'il soit
eclairci et modifie.

513. D'Alembert returns to the Petersburg Problem. He
says,

Yous dites, Monsieur, que la raison pour laqiielle on trouve I'enjeu
infini, c'est la supposition tacite qu'on fait que le jeu peut avoir
une duree infinie, ce que n'est pas admissible, attendu que la vie des
hommes ne dure qu'un temps.

d'alembert. 281

D'Alembert brings forward four remarks which shew that thi^
mode of explaining the difficulty is unsatisfactory. One of theoi
is the following : instead of supposing that one crown is to be
received for head at the first throw, two for head at the second
throw, four for head at the third throw, and so on, suppose that in
each case only one crown is to be received. Then, although theo-
retically the game may endure to infinity, yet the value of the
expectation is finite. This remark may be said to contradict a
conclusion at which D'Alembert arrived in his article Croix oil
Pile, which we noticed in Ai"t. ^^d.

51-i. The case just brought forward is interesting because
D'Alembert admits that it might supply an objection to his prin-
ciples. He tries to repel the objection by saying that it only leads
him to suspect another principle of the ordinary theory, namely
that in virtue of which the total expectation is taken to be equal
to the sum of the partial expectations ; see his pages 299 — 301.

515. D'Alembert thus sums up his objections against the
ordinary theory :

Pour resumer en un mot tons ines doutes sur le calcul des pro-
babilites, et les mettre sous les yeux des vrais Juges; voici ce que
j'accorde et ce que je nie dans les raisonnemens explicites ou implicites
sur lesquels ce calcul me paroit fonde.

Premier raisonnement. Le u ombre des combinaisons qui amenent
tel cas, est au nombre des combinaisons qui amenent tel autre cas,
comme p est ^ q. Je conviens de cette verite qui est purement ma-
thematique; done, conclut-on, la probabilite du premier cas est a celle
du second comme j^ est a q. Yoila ce que je nie, ou du moins de
quoi je doute fort; et je crois que si, par exemple, ^^ = 5', et que dans
le second cas Je meme evenement se trouve un tres-grand nombre de
fois de suite, il sera moins probable physiquement que le premier,
quoique les probabilites mathematiques soient egales.

Second raisonnement. LaprobabiUte - est a la probabilite — comme

^ m n

np ecus est a mp ecus. J'en conviens; done— x mp ecus = - x np ecus;

j'en conviens encore; done Vesperance, ou ce qui est la meme chose,

282 d'alembert.

le sort d'un Joueur qui aura la probabilite — de gagner mp ecus,

sera egale a I'esperance, au sort d'un Joueur qui aura la probabilite

- de gagner np ecus. Yoila ce que je nie; je dis que Vesperaiice est

plus grande pour celui qui a la plus grande probabilite, quoique la
somme esperee soit moindre, et qu'on ne doit pas balancer de preferer

le sort d'un Joueur qui a la probabilite - de gagner 1000 ecus, au

sort d'un Joueur qui a la probabilite ^ttftt. d'en gagner 1000000.

Trolsieme raisonnement qui nest qiiimplicite. Soit p + q le nombre
total des cas, 2^ 1^ j^robabilite d'un certain nombre de cas, q la proba-
bilite des autres; la probabilite de cliacun sera a la certitude totale,
comme p et q sont k p + q. Viola ce que je nie encore; je conviens,
ou plutot j'accorde, que les probabilites de cliaque cas sont comme p
et q ; je conviens qu'il arrivera certainement et infailliblement un
des cas dont le nombre est jy + q', mais je nie que du rapport des pro-
babilites entr'elies, on puisse en conclure leur rapport a la certitude
absolue, parce que la certitude absolue est infinie par ra^iport a la plus
grande probabilite.

Vous me demanderez peut-etre quels sont les principes qu'il faut,
selon moi, substituer a ceux dont je revoque en doute I'exactitude ? Ma
reponse sera celle que j'ai deja faite; je n'en sais rien, et je suis meme
tres-porte a croii'e que la matiere dont il s'agit, ne peut etre soumise,
au moins a plusieurs egards, a un calcul exact et precis, egalement net
dans ses principes et dans ses resultats.

516. D'Alembert now returns to the calculations relating to
Inoculation. He criticises very minutely the mathematical in-
vestigations of Daniel Bernoulli.

The objection which D'Alembert first urges is as follows. • Let
s be the number of persons alive at the commencement of the

SOi'JC

time X ; then Daniel Bernoulli assumes that — -r- die from small-

pox during the time dx. Therefore the whole number who die
from small-pox during the (n -f- 1)*^ year is

""^^ sdx

d'alembeht. 2bo

S
But this is not the same thing as 7,-7 , where S denotes the

number aUve at the beginning of the year ; for s is a variable
gradually diminishing during the year from the value S with

which it began. But ^7 ^^ ^^^ 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 sdx and S to be equal in

value becomes very small if we suppose S to be, not the value of

s when x = 7i ov n + 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 DAlembert'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.

618. 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 In^otheses of
Daniel Bernoulli, a solution by common algebra might be effected.

28 i d'alembert. -

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 y for u 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 un prohleme assez cu-

rieux ; see his page 325. He solves it on the assumptions of Daniel

Bernoulli, and also on his own. We wdll 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 ft) denote the number of those who do not die of small-pox.

Hence out of this number w during the time dx none will die

of small-pox, and the number of those who die of other diseases

/ sdx\ ft)

will be, on the assumptions of Daniel Bernoulli, [ — d^ —

Hence, — dco = (— d^ —

mnj ^ *

sdx\ ft)
mnJ f '

,1 r dco dP sdx

thereiore — = —34. - — . .

Substitute the value of s in terms of x and | from Art. 402,
and integrate. Thus we obtain

ft) Ce"

^ e" (/>^ - 1) + 1

where C is an arbitrary constant. The constant may be deter-
mined by taking a result which has been deduced from observa-
tion, namely that ^ = 97 when a? = 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

d'alembeet. 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 Jiow 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 DAlembert's views ; his remarks in the present memoir are
ingenious and interesting, but as may be supposed, his h}^otheses
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 840. 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 life, it will follow that the curve of
mortality cannot be always concave to the axis of abscissse. Also
from the tables of Buffon it follows that the pivhahle 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 Bur les Tables de mortalite. 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 d 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

28(> d'alembeut,

justness of his objections. Nevertheless he gives his reasons for
regarding Inoculation as a useful practice.

526. The seventh and eighth volume of the Opuscules were
published in 1780. D'Alembert says in an Advertisement pre-
fixed to the seventh volume, ''... Ce seront vraisemblablement, ^
peu de chose pr^s, mes derniers Ouvrages Mathematiques, ma tete,
fatiguee par quarante-cinq annees de travail en ce genre, n'etant
plus gu^re capable des profondes recherches qu'il exige." D'Alem-
bert died in 1788. 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 Prohahilites, 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 j)lus j'y ai pense, plus je me suis confirme dans mes
<k>utes sur les principes de la theorie ordinaire; je desire qu' on eclaircisse
ces doutes, et que cette tlieorie, soit qu'on y change quelques principes,
soit qu'on la conserve telle qu'elle est, soit du moins exposee desormais
de maniere a ne plus laisser aucuii nnage.

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 Encyclopedie Methodique, 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 - +-^ or 1. He says then, Or je demande si cela est

vrai, ou du moins si un pareil r^sultat, fonde sur de pareils prin-
cipes, est bien propre a 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 Yol. vi. of the Memoires...par

d'alembeht. 287

divers Savans... in which Laplace had made the supposition that
the coin has a gTeater 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 crow^ns if x be equal to 5. On the com-
mon hj^pothesis 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 ft) and not ^ ; and let the game have to extend over n trial s

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
ouQ:ht to orive is

2(o ;i + 2 (1 - ft)) + 2^ (1 - ft>)^ + ... + 2"-^ (1 - ft))"-'},

which we will call H.

D'Alembert suggests, if I understand him rightly, that if we
know nothing about the value of co we may take as a solution of

the problem, for the sum which the player ought to give I fldo).

But this involves all the difficulty of the ordinary solution, for the
result is infinite wdien n is. D'Alembert is however very obscure
here ; see his pages 45, 46.

He seems to say that I Cldco will be greater than, equal to, or

•^ 0

less than ??, according as n is greater than, equal to, or less than 5.
But this result is false ; and the argument unintelligible or incon-

elusive. We may easily see by calculation that I D-dw = n when

n = l\ and that for any value of n from 2 to 6 inclusive

I Hc^ft) is less than n ; and that when n is 7 or any greater number

I Q.du> is greater than n.

' 0

288 d'alembert.

531. D'Alembert then proposes a method of solving the Peters-
hiirg Prohlem 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 - ; and so on.

^ Ji

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 :

+ (!+«)

-f (1 - a) (1 + a + ^)

+ (1 - «) (1 - a - Z>) (1 + a + Z> + c)

■^(l-a){l- a-h-c) {l + a-\-h + c-\-d)

It is easily shewn that this is finite. For

(1) Each of the factors 1+a, \ -\-a-\-h, l + a + Z>+c, ...is less
than 2.

(2) \ — a — h is less than 1 — a\

1 — a — 5 — c is less than 1 — a — h, and a fortiori less than
1 — a ;

and so on.

Thus the series excluding the first two terms is less than the
Geometrical Progression

2 {1 - a + (1 - a)^ -f (1 - ay + (1 _ a)\ . .),
and is therefore finite.

d'alembert. 289

This is D'Alembert's principle, only he uses it thus: he shews
that all the terms beginning with

are less than

2{l-a){l-a-'h){l-a'-h'-e){l-a-h-c-d)s,
where s denotes the geometrical progression

r being = l^a — b-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,Cjd, ... are connected by a certain law which he gives,
namely,

where p is a small fraction, and m — 1 is the number of the quan-
tities a, h, c, d, e, ,.. Again he shews that the same result holds if
we merely assume that a,h,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
Avas the only supposed difficulty in the Petersburg Problem, and
that it was sufficient 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 voila assez pour faire voir que les termes de Tenjeu vont en
diininuant des le troisieme coup, jusqu'au dernier. Nous avons prouve
d'ailleurs que I'enjeu total, somme de ces termes, est fini, en supposant
meme le nombre de coups infini. Ainsi le resultat de la solution que
nous donnons ici du probleme de Petersbourg, n'est pas sujet a la diffi-
culte insoluble des solutions ordinaires.

583. We have one more contrilnition of D'Alembert's to our
subject to notice; it contains errors which seem extraordinary,

290 d'alembert.

even for him. It is the article Cartes in the Encyclo^edie Metho-
dique. The following problem is given,

Pierre tient huit cartes dans ses mains qui sont : tin as, lin deux,
un trois, im quatre, un cinq, un six, un sept et un huit, qu'il a melees :
Paul parie que les tirant Tune apres I'autre, il les devinera k mesure
qu'il les tirera. L'on demande combien Pierre doit parier centre un
que Paul ne reussira 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 Paul parioit d'amener ou de deviner juste a un des sept coups

.11 1

seulement, son esperance seroit -+=+...+ -^ , et par consequent

I'enjeu de Pierre a celui de Paul, comme

11 1 . - 1 1 1

— 1 — 4- -I — a, I

8 7 2 8 7 "* 2*

Si Paul parioit d'amener juste dans les deux premiers coups seule-

1 1
ment, son esperance seroit o + ^ > et le rapport des enjeux celui de

1111
g + ^ a i-g-7.

S'il parioit d'amener juste dans deux coups quelconques, son espe-

.^11 11 11

ranee seroit ——+—— + ... ^ — ^ — + ... ^ — ^ — ^+...

8x7 8x6 S xz 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

l/llllllN

8(,7"^6 + 5 + 4 + 3'^2 + V-

For his chance of being right in the first case and wrong in the
other six is

16 5 4 8 2 1.. 1

8^r6^.5^4^3^2-''^'^^^^^^^'

d'alembert. 291

his chance of being right in the second case and wrong in all the

others is

7154321,,,. 1

^X;iX^x-^XjX-Xx, that IS

8 76 5 432' 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
8^7^6^5^4^3^2' ^^ 8 '

therefore his chance of being right once at least is 1 — - , that is ^ .

o 8

Tlie 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,,,. ■»

^XsX-X-XtX^Xt;, that IS

8 7 6 5 4 3 2' 8x7x6*

Or it may mean that Paul undertakes to be right in the first
two cases, but undertakes nothing for the other cases. Then his

. 1 1

chance is ^ 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 will 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

gX^x^x^x-x^x^, that IS 3 ^ ^ ^ g ;

similarly we may find the chance that he is right in any two
assigned cases and wrong in all the others. The total chance will
be found to be

8{7(6 + 5- + 4 + 3 + 2 + V + 6(5+4 + S + 2 + V

19—2

292 d'alembekt.

Or tlie 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 1 1

8 8 1 7 "^ 6 "^ 5

^ + 5+9 + ... + i).

53-i. Another problem is given unconnected with the one we
have noticed, and is solved correctly.

The article in the Encyclopedie Metliodique 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 EncycloiJedie : 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
Encyclopedie 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 Rendus ... 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 borne a cet exemple, il en a accumule plu-
sieurs autres, soit dans le qiiatrieme volume de ses Opuscules, 1768, page
73, et page 283 du cinquieme; il s'est aussi etaye dii suffrage de divers
geometres qu'il qiialifie de distingues. Condorcet a appuye ces objec-
tions dans plusieurs articles de rEiicyclopedie methodique ou par ordre
de matieres. D'un autre cote, divers autres geometres out entrepris

d'alembert. 293

de r^pondre aux raisonnemens de d'Alembert, et je crois qu en par-
ticulier Daniel Bernoulli a pris la defense de la theorie ordinaire.

In this passage tlie word cinquieme is wrong; it should be
quatrihne. 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 mathematiciens, ce ne fut que par le silence
ou le dedain qu'il r^jDondit aux doutes que d'Alembert s'etait permis
d'emettre. Mepris injuste et malhabile ou tout le monde avait a perdre
et qu'une posterite moins prevenue 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 Theorie... 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, Yol. ix.
page 117.

CHAPTER XIV.

BAYES.

589. The name of Bayes is associated with one of the most
important parts of our subject, namely, the method of estimating
the probabihties 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 Prohlem in the Doctrine of Chances.
By the late Rev. Mr. Bayes^ F.R.S. communicated hy 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 Prohlem in the Doctrine of
Chances, published in the Philosoi')hical Transactions, Vol. liii. Com-
municated hy the Rev. Mr. Richard Price, in a Letter to Mr. John
Canton, M. A. F.R.S.

The first of these memoirs occupies pages 870 — 418 of Vol. Liii.
of the Philosophical Transactions ; it is the volume for 1763, and
the date of publication is 1764.

The second memoir occupies pages 296 — 825 of Vol. Liv. of the
Philosophical Transactions; 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 ^ times, the probability
that its chance at a single trial lies between a and h is

/■

x^ (1 - xy cix

i)(f{l-xydx

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 Richard
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 which I have found among the
papers of our deceased friend Mr Bayes, and which, in my opinion, has
gi-eat 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 Rides
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

-^y the probability of the first event by z, and the probability

of the second on the supposition of the happening of the first

7 P h

by -^ . Then our principle gives us ^^T = ^ '^Jf> ^^^ therefore

z = — . This result Bayes seems to present as something new

and remarkable ; he arrives at it by a strange process, and enun-
ciates it as his Proposition 5 in these obscure terms :

If there be two subsequent events, the probability of the 2nd -^

P . .

and the probability of both together -^, and it being 1st discovered

that the 2nd event has happened, from hence I guess that the 1st event

. . P
has also happened, the probability I am in the right is -r-.

Price himself gives a note which shews a clearer appreciation
of the proposition than Bayes had.

b^o. We pass on now to the remarkable part of the essay.

Imagine a rectangular billiard table ABCD. Let a ball be rolled on

it at random, and when the ball comes to rest let its perpendicular

distance from A She measured ; denote this by x. Let a denote the

distance between AB and CD. Then the probability that the

. c — h

value of X lies between two assio^ned values J and c is . This

we should assume as obvious ; Bayes, however, demonstrates it

very elaborately.

54<6. Suppose that a ball is rolled in the manner just ex-
plained ; through the point at which it comes to rest let a line EF
be drawn parallel to AB, so that the billiard table is divided into
the two portions AEFB and EDCF. A second ball is to be rolled
on the table ; required the probability that it will rest within the

BATES. 297

space AEFB. If x denote the distance between AB and ^i^tlie

required probability is - : this follows from the preceding Article.

547. Bayes now considers the following compound event :
The first ball is to be rolled once, and so EF determined ; then
p +q^ trials are to be made in succession with the second ball :
required the probability, before the first ball is rolled, that the
distance of EF from AB will lie between h and c, and that the
second ball will rest p times within the space AEFB, and q times
without that space.

We should proceed thus in the solution : The chance that EF

falls at a distance x from AB is — ; the chance that the second

event then happens p times and fails q^ times is

hence the chance of the occurrence of the two contino^encies is

a \p_\q_ \«/ \ «/ *
Therefore the whole probability required is

a\p[q

(i-I)''^-

Bayes's method of solution is of course very different from the
above. With him an area takes the place of the integral, and
he establishes the result by a rigorous demonstration of the ex
ahsurdo kind.

548. As a corollary Bayes gives the following: The proba-
bility, before the first ball is rolled, that EF will lie between AB
and CD, and that the second event will happen p times and fail q
times, is found by putting the limits 0 and a instead of h and c.
But it is certain that EF will lie between AB and CD. Hence we

298 BATES.

have for the probability, before the first ball is thrown, that the
second event will happen^ times and fail ^ times

P <1

C©'('-3'

Cv«X/»

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 ^i^" 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
•h

x^(a-xydx

J 0

This depends on Bayes's Proposition 5, which we have given
in our Art. 544. For let z denote the required probability ;
then

z 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 j hence the
value of z 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

^i>+i ^ ^P^ q{q-l) x'^'

X I jL ^ X) CLX —~ — — ^ ~I~

p + 1 1 J9+ 2 ' 1.2 ^ + 3 "•

This series may be put in another form ; let u stand for 1 — x,
then the series is equivalent to

jj + l'^p+l p + 2 '^ (p + 1) {p-^2) p + S

q(q-l)(q-2) x^^r'

{p + l){p-\-'2){p + '6) iJ + 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

BAYES. 299

respect to x^ we shall find that the terms cancel so as to leave
only ^u^,

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 8avans, 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

J 0

x''^' (1 - xY dx

J 0

x'' (1 - xY dx

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
Lubbock 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 p = 1, and ^ = 0. 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 :

af{l-xYdx .^

I x^Q.- xy dx

J 0

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

7 that the event is more likely to happen than not.

552. It must be observed with respect to the result in Art. 549,
that in Bayes's own problem we hnonj that a priori any position
of ^jF 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 results as that which we have taken from Laplace in

300 BAYES.

Art. 551, 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 = — //IN • Then if an event has happened p times and

failed q times, the odds are about 1 to 1 that its chance at

a single trial lies between - + -7^ and 7^ ; the odds are about

2 to 1 that its chance at a single trial lies between - ■\- h and

-^ — h'. the odds are about 5 to 1 that its chance at a sinsfle
n °

trial lies between ^ + A V2 and ^ — h J% These results may be

n n "^

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 — x^ (1— xy has two
points of inflexion, the ordinates of which are equidistant from the
maximum ordinate ; the distance is equal to the quantity h of the
preceding Article. These points of inflexion are of importance in
the methods of Bayes and Price.

CHAPTER XV.

LAGRANGE.

555. Lagrange was born 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 VutiliU de la methode de prendr^e le milieic
entre les resultats de plusieurs observations ; dans lequel on examine
les avantages de cette methode par le calcul des prohahilites ; et ou
Von resoud differens prohlenies relatifs a cette matiere.

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-
vantasfe.

"O"

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, h cases
in which an error equal to 1 is made, and h cases in which an

302 LAGEANGE.

error equal to — 1 is made ; it is required to find the probability
that in taking the mean of n observations, the result shall be
exact.

In the expansion of {« + 5 (ic + cc~^)}" according to powers of x,
find the coefficient of the term independent of x\ divide this
coefficient by {a + 2^)" 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

„ [n (n — 1)1 ^ {n(n — 1) (n —

2.3 J"^*"

^ 1.3.5... (2n~l) ,
1.2.3...71

and he says it is the more remarkable because it does not seem
easy to demonstrate it a i^riori.

The result is easily obtained by equating the coefficients of the
term independent of x in the equivalent expressions

(1 + ^rfl + iy, and^li^

^2n
XI ' X''

This simple method seems to have escaped Lagrange's notice.

Suppose we expand ■ in powers of z ; let the

V 1 — 2as — cz^

result be denoted by

1 + A^z + A/ + A/ -f ... ;

Lagrange gives as a known result a simple relation which exists
between every three consecutive coefficients ; namely

. _2n — l . n — 1 J

LAGEAXGE. 303

This may be established by differentiation. For thus

that is

(a+cz) {l-{-A^z + A,/ + .., +Ay+ ...}

= (l-2a2-cs') [Aj^ + 2A^z + .,, +nA^z''~'-^ ...};
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 + 5 (ic + £c"^)}" in powers of x\ let the result be de-
noted by

A^ + A^ (x-^x-') +A^ {x'^x-') -{-A, {x'+x-^ + ... ;

Lagrange wishes to shew the law of connexion between the co-
efficients Aq, 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 easily b}" putting 2 cos ^ for a? + x~^, and
therefore 2 cos rO for a?*" + x~'\ Thus we have

(a + 25 cos ey = ^0 + ^A cos 6 + 2A^ cos 26 + 2A^ cos 8^ + . . .

Hence, by taking logarithms and differentiating,

Tib sin 6 _ A^ sin 6 + 2A^ sin 26 + ^A^ sin 3^ +. . .
a-^2bco^6~ A^ + 2A^ cos 6 + 2A^ cos 2^ + . . .

Multiply up, and arrange each side according to sines of mul-
tiples of 6 \ then equate the coefficients of sin r6 : thus

rib [A,_^ - A,^,] = raA, + b [{r - 1) A,_^ -f (r + 1) A,^,] ;
therefore A^^, = b {n-r^rl) A -raA^

30i LAGEANGE.

560. In the third problem it is supposed that there are a
cases at each observation in which no error is made, h 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. In the 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, Cyd, ... respectively. Then
we require to find the most probable error in the mean of n ob-
servations.

In the expansion of {ax^ + hx^ + cic** + ...)" let M be the coeffi-
cient of iC* ; then the probability that the sum of the errors is yit,

and therefore that the error in the mean is — is

Hence we have to find the value of //. for which M is greatest.

Suppose that the error p occurs a times, the error q occurs
/3 times, the error r occurs 7 times, and so on. Then

a + /3 + 7+ =n,

pOL + ql3 + ry -\- = fi.

It appears from common Algebra that the greatest value of fjL

is when

a_/3_7_ _ n

a h c a4-& + c+..''

^T , Lb pa-^- qh + rc-\- ...

so that - —^ S •

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 d priori; but that ct, /S, y, ... 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 = ;8 = 7"--"
so that the value of - of the jjreceding Article may be written

fM _pa. + ql3 +ry+ ...
n a + /3 + 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 ... 13 ; 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. 14^8 and S64<. It leads to algebraical work of the same kind
as the eighth problem which we will now notice.

oQ4^. Suppose that at each observation the error must be
one of the following quantities — &, - (a — 1), ... 0, 1, ... a ; and
that the chances of these errors are proportional respectively to
1, 2, ... a + 1, a, ... 2, 1 : required the probability that the error in

the mean of n observations shall be equal to - .

SOG LAGRANGE.

We must find the coefficient of x>^ in the expansion of

{^-« + 2a?-"+^ + ... + aa?-' + (a + 1) x° + ao; + .. . + 2a;"-' + a;"}",

and divide it by the vahie of this expression when aj = 1, which is
the whole number of cases ; thus we obtain the required pro-
babiHty.

Now 1+ 2x + 3a;'+ ... + (a+ 1) a;"+ ... + 2a;""-' + x^

Jia

a+i\ 2

= {iJ,x+x'-\-... + xy=(Kj^^ .

Hence finally the required probability is the coefficient of
a;'* in the expansion of

1 a;-"" (1 - a;"-^y" ^
that is the coefficient of a;'^"^"" in the expansion of

^,(i-.r(i-o-

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 a?'^"^"" the expression

+ \ ^ ^ </)(7ia+At + l-2«-2)

%i {271 - 1) {2n - 2) ^ , . o ON 1
12 3 " 0(wa+/^ + l-3a-S)4-...|;

where 0 (r) stands for the product

r (r+1) (r + 2) . . . (r + 2w - 2) ;

the series within the brackets is to continue only so long as r is
positive in (j) (r).

565. We can see a priori that the coefficient of xf^ 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 — //, instead of /jl, without
changing the vakie 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 56^ for //, in succes.sion the numbers

— ny, —(iia — l), ...7 — 1, 7,

and add the results. Thus we shall find that, usingf ^, as is
customary, to denote a summation, we have

S^ (7ia + /^ + 1) = — -^/r (7ia + 7 + 1),

where -^/r (?-) stands for

r (r + 1) (?' + 2) . . . (r + %i - 1).

When we proceed to sum <^ (?ia + //- — ct) we must remember
that we have only to include the terms for which noL + fi — a is
positive; thus we find

Xcj) (na + fJ' — OL) = ~ yjr {iiOL 4- 7 - a).
Proceeding in this way we find that the probability that the

710L "V

error- in the mean result will lie between and - , both in-

n n

elusive, is

r^r- J-v/r (nOL + 7 + 1) — 2/1 -v/r (nct. + 7 + 1— a— 1)

+ \ ^ — ' -^ (71a + 7 + 1 - 2a - 2)

2w i^n - 1) (2?i - 2) , , , , . „ ^s . \
^ ^ ./ ^^ ^ ^ {no, + 7 + 1 - 3a - 3) + . . .| ;

20—2

308 LAGRANGE.

the series tuithin the brackets is to continue only so long as r is
positive in yjr (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 F[y) — F {^) if we include 7 and
exclude ^ ; it is F{y — 1) — F{/3 — 1) if we exclude 7 and include
/9; it is F{y)—F[(3—1) if we include both 7 and yS ; it is
F{y — 1) —F{/3) if we exclude both 7 and /3.

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
F {y — 1) — F {^) ; and the number of terms in F (y-l) is de-
termined by the law that r in ^fr (r) is always to be positive :
the number of terms in F ((3) 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 j^'^ohngs F {,6) 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^. Suj^pose that instead of the errors — a, — (a — l), ...
we are liable to the errors — ka, — k {a — l), ... Then the investi-
gation in Art. 5Q4! gives the probability that the error in the mean

ak
result shall be equal to — ; and the investigation in Art. 5G6

gives the probability that the error in the mean result shall lie

between — and - . Let a increase indefinitely and k diminish
71 n

indefinitely, and let o.k remain finite and equal to h. Let 7 and /3

also increase indefinitely ; and let 7 = c« and I3 = h(x where c and h

are finite. We find in the limit that F [y] — F {/S) becomes

J- L + ny''-2n (c + n-ir^ + ^"^ ^^"^ ~ ^^ (c + n- 2^- ..

each series is to continue only so long as the quantities which
are raised to the power 2n are positive.

LAGRANGE. 309

This result expresses the probability that the error in the

mean result will lie between — and — on the followinof hy-

n n o J

pothesis ; at every trial the error may have any value between
— h and + h ; positive and negative errors are equally likely ;
the probability of a positive error z is proportional to h — z, and

in fact ~ — — - is the probability that the error will lie be-
tween z and z + Sz.

We have followed Lagrange's guidance, and our result agrees
with his, except that he takes 7i = l, 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 7^ea-
sonahle hypothesis as to the occiirrence of single errors.

Suppose that positive errors are denoted by abscissse 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, dQ>^,
566, 567, 568, had all been published by Thomas Simpson, in his
Miscellaneous Tracts, 1757 ; he gave als(3 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

10 LAGKANGE.

to the supposition of continuous errors. Lagrange wishes to solve
questions relative to continuous errors without starting with the
supposition of errors changing per saltum.

Suppose that at every observation the error must lie between h
and c; let ^ {x) dx denote the probabiUty that the error will lie
between x and x-\-dx\ required the probability that in n obser-
vations the sum of the errors will lie between assigned limits say
/3 and 7. Now what Lagrange effects is the following. He trans-
forms \\ ^{pc)a^dx\ into \f{z)(idz, where f{z) is a known

function of z which does not involve a, and the limits of the
integral are known. When we say that f {£) and the limits of
z are known we mean that they are determined from the known
function ^ and the known limits h and c. Lagrange then says
that the probability that the sum of the errors will lie between

/? and 7 is 1 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 ^ {x) is constant = K say ; then

6 (x) «* dx = — i^ ,

J h loof a

b\n

therefore \j (b(x)a'dx[ = — —^ — r-f—

Vb^' j (log a)"

Now we may suppose that a is greater than unity, and then it
may be easily shewn that

J 0

n-l

(log ay'

thus \ (f){x) a^ dx\ = -^ {a' - a^ j if ''a'" dy.

LAGRANGE. 311

Let c-'h = t, and expand {a' — a^y by the Binomial Theorem ;
thus ' W (fi {x) a" dx\

^j— yja -na + 2. 2 ~***l/ ^ «^ «^-

Now decompose I ^" ^oP'dy into its elements; and multiply

0
them by the series within brackets. AVe obtain for the coefficient

of a^'^ the expression

where the series within brackets is to continue only so long as the
quantities raised to the power n — 1 are positive.

Let nc—y = z ; then dy — — dz\ when y—^ we have z = nc,
and when y = 00 we have 0 = — 00 . Substitute nc — z for ?/, and
we obtain finally

where f{z) = \ [nc — zy ^ - n {nc - z -ty

+ \,2 ' {nc-z-2ty'-..

the series within brackets being continued only so long as the
quantities raised to the power n — 1 are positive.

Lagrange then says that the probability that the sum of the
errors in n observations will lie between /3 and 7 is

f V(.) dz.

J/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. 570 it is obvious that the sum
of the errors must lie between nb 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 z be greater than 7ic there is no term at all in f{z),
for every quantity raised to the power n — 1 would be negative.

And if z be less than nh, 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 n—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 v/ould 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

\ i(f) {xj a^i dxjr \ Icj) {x^) a^2 dxA J j</) {x^) a^» dxA ,

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 I Pdz is the required probability.

J/3

Now to find P we proceed in an indirect way. It follows from
our method that

'ne

(f> {x) d" dx\ =i Pddz.

J b ) J lib

But Lagrange by a suitable transformation shews that

' (f> {x) d'dxi = I ^f{z) a'dz,

h ) J Zo

where z^ and ^^ are known. Hence

rnc fz

Pa'dz= f[z)a'dz.

J nb J Xo '

It will be remembered that a may be ani/ quantity which

LAGRANGE. 813

is greater than unity. We shall shew that we must then have

Suppose that z^ is less than nh^ and z^ greater than nc. Then
we have

1 f{z)a'dz+ {f{z)-F} a'dz'-\- f{z)a'dz = ^,

J SSq J nb '1 nc

for all values of a. Decompose each integTal into elements ; put
a^^ = p. We have then idtimately a result of the following
form

^0 jr, + T^p + V + ^3P'+ - ^^^ ^V- •••} = 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

Thus P=f{^)-

The demonstration will remain the same whatever supposition
be made as to the order of magnitude of the limits z^ and z^
compared with nh and 7ic.

57o. Lagrange takes for another example that which we have
akeady 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

<f) {x) = K \/ c^ — x\ the errors lying between — c and c; in the
other he supposes that cj) (x) = Kcosoo, the errors lying between

- :=: and ^ .
2 2

576. We have now to notice another memoir by Lagrange
which is entitled Becherches sw les suites reciirrentes dont les
termes varient de plusieurs manieres di0rentes, qu sur Vintegra-
tion des equations lineaires aux differences jinies et partielles ; et
sur Tusage de ces equations dans la theorie des hazards.

This memoir is published in the Nouveaux Menioires de VAcad.
... 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-
l^lication to the Theory of Chances occupies pages 240 — 272.

577. The memoir begins thus ;

J'ai clonne dans le premier Volume des Memoires de la Societe des
Sciences de Turin une metliode nouvelle pour traiter la theorie des suites
recurrentes, en la faisant dependre de Tintegration des equations lineaires
aux differences finies. Je me proposois alors de pousser ces recherches
l>lus loin et de les appliquer principalement a la solution de plusieurs
problemes de la theorie des liasards; mais d'autres objets m'ayant depuis
fait perdre celui la de vue, M. de la Place m'a prevenu en grand partie
dans deux excellens Memoires sur les suites recurro-recurrentes, et sur
V integration des equations differentielles finies et leur iisage dans la
theorie des liasards^ imprimes dans les Volumes vi et vii des Memoires
pr^sentes a 1' Academic des Sciences de Paris. Je crois cependant qu'on
peut encore aj outer quelque chose au travail de cet illustre Geometre, et
traiter le meme sujet d'une maniere plus directe, plus simple et surtout
plus generale ; c'est I'objet des Recherches que je vais donner dans ce
Memoire; on y trouvera des methodes nou veil es pour I'integration des
equations lineaires aux differences finies et partielles, et I'application de
ces methodes a plusieurs problenies interessans du calcul des probabilites ;
mais il n'est question ici que des equations dont les coefficiens sont con-
stants, et je reserve pour un autre Memoire 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 tlie chance of the happening
of an event h times at least in a trials.

Let j9 denote the chance of its happening in one trial ; let
?/^^^ denote the probability of its happening t times in x trials ;
then Lagrange puts down the equation

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. 31o

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,

y (1 -pr' \a_

\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 sujDpose qu'a chaqiie coup il puisse arriver deux 6venemens dont
les probabilites respectives soient p et q; et on demande le sort d'un
joueur qui parieroit d'amener le premier de ces evenemens h 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 memes choses etant supposees que daus le Probleme li, on de-
mande le sort d'un joueur qui parieroit d'amener, dans un nombre de
coups indetermine, le second des deux Evenemens h fois avant que le
premier fut arrive a fois.

Let 7/^ J be the chance of the player when he has to obtain the
second event t times before the first event occurs x times. Then

316 LAGl^ANGE.

This leads to

yx,t - q |i + f/? + — 2 — P + 273 ' ^

+ ...

(5 + ic- 2

^ - 1 1 .-r - 1

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 tres generale
et tres simple renferme d'luie maniere generale celui qu'on nomme com-
munement dans I'analyse des liasards le probleme des partis, et qui
n'a encore ete resolu 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 LXix.

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 Lagrange intended
to translate De Moivre's third edition into French.

583. 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. 307. Lagrange gives three solu-
tions. Lagrange's first solution demonstrates the result given

LAGRANGE. 31

without demonstration in De Moivres second solution ; see
Art. 309. We will give Lagrange's solution as a specimen of his
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 Yii.

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
Memoires... 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
h times oftener than it fails : determine the player's chance.

Let y^i 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 7/„^j^.

Suppose one more trial made ; it is easy to obtain the follow-
ing equation

The player gains when ^ = 0 and x has any value, and he loses
when x=0 and t has any value greater than zero ; so that y^^^=\
for any value of x, and y^^t^ 0 for any value of t gTeater
than 0.

Put ^ for 1 —p, then the equation becomes

To integrate this assume y = ^a'^/3* ; we thus obtain

p - a/B + q/3' = 0.

From this we may by Lagrange's Theorem expand /S' in powers
of a ; there will be two series because the quadratic equation
gives two values of /S for an assigned value of a. These two
series are

318 LAGRANGE.

^~a''^ 0.'^' ^ 1.2 a'^' '^ 1.2.3 ^^^^ +•••

0.^ fpo!-' t{t-Z) p'o!-' t{t-^) (t-5) for"
i i~^ 1.2 2'"' 1-2.3 2'"'

If then we put in succession these values of /3* in the ex-
pression Ao^ ^^ we obtain two series in powers of a, namely,

Af |a^- + ipq^^'-' + ^-^^ //a--^ + . . . J ,

and Aq-' \o^' - tjpqa''^'-' + *-^j^ fi^'^'" - . •

Either of these series then would be a solution of the equation
in Finite Differences, whatever may he 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' {/ {^-t) + tpqfix -t-2)+ ^^t^pYfix - < - 4)

+ ^^T4^i'W(— 6) + ...}
+ q-' U, ix + t)- tfq ,^ (a; + « - 2) + '-^^ pY <l>{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 Vx,^ and ?/„,,.

Lagrange says it is easy to convince ourselves, that the con-
dition 2/^^=0 when t has any value greater than 0 leads to the
following results : all the functions with the characteristic 0 must
be zero, and those with the characteristic / 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 whicli
multiplies q~^ should be zero.]

Thus the value of y^^t becomes a series with a finite number
of terms, namely,

LAGRANGE. 819

y... =P' {/(^ -t) + tpqfix - < - 2) + '-^J^^Y/(^ - < - 4)

the series extends to ^ (a? — ^ + 2) terms, or to ^ (a? - ^ + 1) terms,

according as ic — ^ is even or odd.

The other condition is that j/a;,o— ^y ^''^^ ^^7 vakie of x. But if
we put ^ = 0 we have yx,Q=f{p^)' Hence f{x) = l for every
positive value of x. Thus we obtain

the series is to extend to 3 (x — t + 2) terms, or to ^ (x — t-\-l)

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 Moivre's first solution. These other two solutions by Lagi^ange
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
Moivi'e 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
to a student, or any advantage to one who is in possession of the
method of Generating Functions. But nevertheless it may be read

820 LAGRANGE.

with ease and interest, and at the time of pubhcation 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 Proh. are substantially due to this memoir of Lagrange's.

589. We may notice a memoir by Lagrange entitled Me-
moire sur une question concernant les annuiies.

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, G, ... the value of an annuity of
one crown for the minority of the children A, B, G ... 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 ^ is a minor,

~A + B- AB.

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, G remains a minor is

A + B + C - AB - AG - BG + ABG.

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 Years 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 remport^ le prix sur le sujet des Evenemens Fortuits,
propose par VAcademie Roy ale des Sciences et Belles Lettres de
Berlhi pour Vann^e 1751. Avec les p)ieces 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 malheurenx, ou ce que nous apj)elIons
Bonheur et Malheur dependant de la volonte ou de la permission de
Dieu, de sorte que le terme de fortune est un nom sans realite; on de-
mande si ces Evenemens nous obligent a de certains devoirs, quels sont
ces devoirs et quelle est leur etendue.

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

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 maclie hier den Schluss, weil ich ohnehin mit gar zii guten
Griinden fiirchte, zu weitliiufig gewesen zu seyn, da ich so wenig neues
artiges und scharfsinniges gesagt habe. Ich finde auch in dieser Probe,
dass mein Wille noch einmahl so gut als meine iibrige Tahigkeit, 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
Avhole 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 Vidgar Arithmetick only:
to which is added, some useful tables on annuities for lives (&c. &c. &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 tjrpe. The title,
preface, and dedication occupy viii 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.

595. We advert in the next place to a work which is en-
titled DelV Azione del Caso nelle Invenzioni, e deW influsso degli
Astri ne Corpi Terrestri Dissertazioni 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

SAMUEL CLARK. 323

such connexion, especiallysvLen 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. Re 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 : ...ci5
che disse in lode d'Aristotile il Berni : II gran Maestro de color
die 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 TheLaius of Chance : or, a Mathematical Investigation of the
Prohabilities arising from amj 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 very 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 — ISO. 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 following problem,
I shall lay down a lemma which was communicated to me by my inge-
nious friend Mr William Fayne, teacher of mathematics.

The Le7tima.

The sum of 1, 3, 6, 10, 15, 21, 28, 36, &c. continued to (n) number

J, ^ . , ^ n + 2 n + 1 n

01 terms is equal to — ^ — x — ^— x - .

It was quite unnecessary to appeal to William Pa3riie 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 arrives 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 jiL to B. This is wrong. Sup-
pose that on the whole A wins in 5' -F 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 {(i + 111) G and pays to him
the sum m.L ; thus the balance is qG + m {G — L) and not qG SiS
Clark says.

IklALLET. JOHN BERNOULLI. 325

597. We have next to notice a memoir by Mallet, entitled
Recherches sur les avantages de trots Joueurs qui font entreux une
Poule au trictrac ou a un autre Jeu quelconque.

This memoir is published in the Acta Helvetica... Basilece,
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 Histoii^e de VAcad 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 ete In en 1765, apres le Memoire de Mr. Euler sur
cette matiere insere dans les Memoires de I'Academie pour cette annee.
Comme les Memoires de Mr. Beguelin imprimes a la suite de celui de
Mr. Euler se rapportent au mien en plusieurs endroits, et que la Loterie
qui I'a occasione est phis en vogue que jamais, je ne le supprimerai pas
plus longtems. Si ma methode ne mene pas aussi loin que celle de
Mrs. Euler et Beguelin, elle a du moins, je crois, I'avantage d'etre plus
facile a 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 tems jusqu'a ce que j'appris de
Mr. Euler qu'il traitoit le meme sujet; e'en fiit assez pour me faire

326 JOHN BERNOULLI.

abandonner mon dessein, et je me reservai seulement de voir par le
Memoire de cet illustre Geometre si j'avois raisonue juste; il a eu la
bonte de me le commuDiquer et j'ai vu que le peu que j'avois fait, etoit
fonde sur des raisonnemens qui, s'ils n'etoient pas sublimes, n'etoient du
moins pas faux.

600. Jobn 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

o / ON/ ON (n-2) (n-S) {n-4<)
n-2, {n - 2) (n - 3), -^^ ^ ^ ^ g ,

we shall find on John Bernoulli's conception the results

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 two

tickets be drawn the chance is , if three -^^ ^'-W^ — , if

n n{n — l)

four (^-4)(^-5)(n-6) (^,5)(.^-6)(^-7)(/^-8) .

n{n-\){n-2) ' " "^^ n{n-l) {n-2) {n-^) '

and so the law can be easily seen. Now John Bernoulli states

JOHN BERNOULLI. 327

that on his conception of a sequence these formulae will hold if we
change n into n — 1. 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
^ (n, 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
BernoulH's conception. Then we have given

^, . (n—r-{-l)(n — r)...(n — 2r-\-2)
E{n, r) = ^ ^ ^ ^ ^ ,

and we have to shew that

^ , . n(n — T — 1) ... (71— 2r + 1)
B in, r) = — ^ '- ^ .

For these must be the values of E (w, r) and B [n, r) in order
that the appropriate chances may be obtained, by dividing by the
total number of cases. Now the following relation will hold :

E{n,r)=B{n,r) + B{7i-l, r -1) -E{n-2, r-l).

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{n,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 all the
binary combinations of the numbers 1 . 2, ... 9 which contain no
Bernoullian 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 n—1 numbers, so as to con-
tain no Bernoullian sequence, and to contain 1 as one of the num-
bers. It might at first appear that B (n — 1, r — V)—B(7i—2, r — 1)
would be the number of such combinations ; but a little con-
sideration will shew that it is B {n — 1, r — 1) — E {n — 2, 7* — 1), as
we have given it above.

Thus having established the relation, and found the value of
B {ti, 1) independently we can infer in succession the values of
B (n, 2), B {n, 3), 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 Bur les suites ou
sequences dans la lotterie de Genes ; they occupy pages 231 — 280
of the volume.

604. Beguelin's memoirs contain general Algebraical formulae
coinciding with Euler's, and also similar formulae for the results on
John Bernoulli's conception; thus the latter formulae 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, h, c, ... i,j, h, l, m.
Arrange 5 files of such letters side by side, thus

• • ■

,m m m 771 m

Consider first only two such files ; take any letter in the first
file and' associate it with any letter in the second file ; we thus
get 13^^ such associations, namely aa, ah, ac ... ha, hh, he, ...

Here we have ah and ha 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 h of
the second file will be associated with any one of the 12 letters
in the second file beginning with h. Thus the whole number of

13 X 14

such associations will be 13 + 12 + . .. + 1; that is .. ^ .

' 1.2

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

shall have — ^ — ^— , Proceeding in this way we find that if

JL X ^ X o

there are five files and we do not allow repetitions the number of
13x14x15x16x17

associations is

1x2x3x4x5

BEGUELIN. 829

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 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 ,. , , .

— — ^ — -. z — ; that IS, the number is what we know to

1x2x3x4x0

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

•

k
«

•

830 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{7i, r) in Art. 602.

608. The method which we have here briefly exemplified is
used by Begnelin 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, n — 4, n.

Sequence of 4, {n — 5) {n — 4), n{n — Q).

Sequence of 3

combined with (n-5) {n — 4<), n{n-6).

a sequence of 2,

Sequence of 3,

and the other (n — 6) {n— o) [n — 4) n {n — 7) {n — 6)

numbers not 1.2 ' 1.2

in sequence,

Two sequences {n — 6) {n— 5) {n — 4) n{n — 7) (n — 6)

of 2, 1.2 ' 172 •

Single sequence {n-7) (w-6) (n-5) (m-4) n {n-8) {n-7) (w-6)
of 2, 1.2.3 ' 17273 •

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 Beguehn.
It is entitled, Sur V usage du principe de la raison suffisante dans
le calcul des jprohabilites.

This memoir is published in the volume of the Histoire de
V Acad.... Berlin iox 1767; the date of publication is 1769: the
memoir occupies pages 382 — 412.

611. Beguelin begins by saying, J'ai montre dans un Memoire
precedent que la doctrine des probabilites etoit uniquement fondee
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,
Geometre et Philosophe a la fois, a public depuis peu sur le
Calcul des probabilites, 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 evenemens simmetriques et regnliers, attribues an hazard,
sont (toutes choses d'ailleurs egales) aiissi probables que les evenemens
qui n'ont ni ordre ni regularity, et au cas qu'ils aient le nieme degre de
probabilite, d'ou vient que leur regularite 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 meme evenement est deja arrive nne ou plusieurs fois
de suite, on demande si cet evenement conserve autant de probabilite
pour sa future existence, que Tevenement contraire qui avec une egale
probabilite primitive n'est point arrive encore.

BegTielin 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 ^ + 1 to 1 that it will fail at the next trial.

(314. Beguelin applies his notions to the Petersburg Problem.

n
Suppose there are to be n trials ; then instead of 3 which the

common theory gives for the expectation Beguelin arrives at
112 2' 2' 2"-'

2"^2"^2 + l'^2.3 + l^[4 + l ••• l^-l + l'

The terms of this series rapidly diminish, and the sum to
infinity is about 2 J.

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 probleme a 6te aussi le siijet de savantes considerations metapliy-
siques pour Beguelin... ce metaphysicien et analyste examine au flam-
beau d'une metapliysique 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 particidar Circumstances of
their Situation, by the Rev. John Michell, B.D., F.RS.

This memoir was published in the Philosophical Transactions,
Vol. LVII. Part I., which is the volume for 1767 : the memoir
occupies pages 234 — 264.

MICHELL. 333

(il8. 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 ^vill be
seen from the following extract. He says, page 243,

Let us then examine what it is jDrobable would have been the least
apj)arent distance of any two or more stars, any where in the whole
heavens, ui^on 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 probability, 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 racUus is the diameter
of a great circle (this last quantity being equal to the whole surface of

the sphere) that is, by the fraction . ..[^, ^ j or, reducing it to a deci-

( Oo / D'O )

mal form, -000076154 (that is, about 1 in 13131) and the complement

13130
of this to unity, viz. -999923846, or the fraction y^Yqt will represent

1 0 1 Ox

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, multiiDlying 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 jDutting oi for this number,

(-999923846)", or the fraction (^|^)" 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 oi, within the distance of
one degree from the given star. And farther, because the same event
is equally likely to ha^Dpen to any one star as to any other, and there-
fore any one of the whole number of stars 7i 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 I ( ^kj^. ) [ 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; and the

S34< MICHELL.

complement of this quantity to unity will represent tlie 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
P 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, supj)osing 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 ofi ves the details of the calculation in a note.

620. Laplace alludes to Michell in the Theorie . . . des Proh.,
page LXiii., and in the Connaissa^ice des Terns for 1815, page 219.

621. The late Professor Forbes wrote a very interesting criti-
cism on Michell's memoir; see the London, Edinhui^gh and Buhlin
Philosophical Magazine, for August 181^9 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 Kovus Stellarum Duplicium et Midtipliciiim . . . 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 (/> represent the area of a small

circle of x" radius. Then Struve takes ^ — - ^ as the chance

of having a pair of the n stars within the distance x", supposing
that the stars are distributed by chance. Let 8 represent the
surface beginning from —15'' of declination and extending to the
north pole; let n = 10229, and ic = 4 : then Struve finds the above
expression to become '007814.

JOHN BERNOULLI. 335

See also Struve's SteUarum Duplicium et Multiplicium Men-
surce Micrometricce ...Petrop. 1837, page xci., and his SteUarum
Fixarum ... Positiones Medice ... Petroj:^. 1852, page CLXXXViii.

Sir John Herschel in his Outlines of Astronomy , 1849, page 565,
ofives some numerical results which are attributed to Struve ; but
I conclude that there is some mistake, for the results do not
appear to agree with Struve's calculations in the works above cited.

628. For a notice of some of the other subjects discussed in
Michell's memoir, see Struve's Etudes d'Astronomie Stellaire,
St Peter shourg, 1847.

624. We have next to notice another memoir by John Ber-
noulli ; it is entitled Memoire sur un prohleme de la Doctrine du
Hazard.

This memoir is published in the volume of the Histoire de
V Acad. ... Berlin for 1768; the date of publication is 1770: the
memoir occupies pages 384 — 408.

The problem discussed may be thus generally enunciated.
Suppose n men to marry 7i women at the same time ; find the
chance that when half the 2n people are dead all the marriages
will be dissolved ; that is, find the chance that all the survivors
will be widows or widowers. John Bernoulli makes two cases ;
first, when there is no limitation as to those who die ; second, when
half of those who die are men and half women.

The memoir presents nothing of interest or importance ; the
formulas are obtained by induction from particular cases, but are
not really demonstrated.

625. We have next to notice a memoir by Lambert, en-
titled Exameri d'une espece de Superstition ramenee au calcul
des prohahilites.

This memoir is published in the volume for 1771 of the
Nouveaux Memoir es ... Berlin ; the date of publication is 1773:
the memoir occupies pages 411 — 420.

626. Lambert begins by adverting to the faith which many
people in Germany had in the predictions of the almanack makers
respecting the weather and other events. This suggests to him to

336 LAMBERT.

consider what is the chance that the predictions will be verified
supposing the predictions to be thrown out at random.

The problem which he is thus led to discuss is really the old
problem of the game of Treize, though Lambert does not give this
name to it, or cite any preceding writers except Euler's memoir of
1751 : see Arts. 162, 280, 430.

627. We may put the problem thus : suppose n letters to be
written and n corresponding envelopes to be directed ; the letters
are put at random into the envelopes : required the chance that
all, or any assigned number, of the letters are placed in the wrong
envelopes.

The total number of ways in which the letters can be put into
the envelopes is n. There is only one way in which all can be

placed in the right envelopes. There is no way in which just one
letter is in the wrong envelope. Let us consider the number of
ways in which just two letters are in the wrong envelopes : take

a pair of letters ; this can be done in — ^- — ^-^ ways j then find

in how many ways this pair can be put in the wrong envelopes

without disturbing the others : this can only be done in one way.

Next consider in how many ways just three letters can be put in

the wrong envelopes ; take a triad of letters ; this can be done

7X in 1 ) in 2 )

in -^ L \ ways, and the selected triad can be put in

wrong envelopes in 2 ways, as will be seen on trial.
Proceeding thus we obtain the following result,

A A ^ ^ (*^ — 1)

A ^ (^ - 1) (^ - ^) A \jl

1.2.3

where A^ expresses the number of ways in which r letters, for
which there are r appropriate envelopes, can all be placed in wrong
envelopes. And

Ao = ^, A^ = \), A^=l, A^ = 2, ..,

Now Aq, A^, A^, ... are independent of 7i ; thus we can deter-
mine them by putting for w in succession the values 1, 2, 3, ... in

LAMBERT. S37

the above identity. This last remark is in fact the novelty of
Lambert's memoir.

Lambert gives the general law which holds among the quan-
tities -4j, A^, ... , namely

A,= rA,_^+(-iy (2).

He does not however demonstrate that this law holds. "We
have demonstrated it implicitly in the value which we have found
for (f) {n) in Art. 161.

We get by this law

A, = 9, X=44, ^, = 265, ^7=1854, A= 14833, ...

We can however easily demonstrate the law independently of
Art. 161.

T \V — 1 )

Let A** I 0 stand for \r — r r — \ -\ ^^^ — ^

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-[0; (3),

and we can shew that this is the case by an inductive proof For
we find by trial that

A"Lo = Lo = i = A>

A^ [0 = 1 -1=0 = ^,,
A^[0= 2 -2 + 1=^,;

and then from the fundamental relation (1) it follows that if
-4^ = 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 Sur le Calcul
des Prohahilites, par Mr. Mallet, Prof. d'Astronomie a Geneve.

This memoir is published m the Acta Helvetica ... Basilece,
Vol. VII. ; the date of publication is 1772 : the memoir occupies
pages 133—163.

338 MALLET.

629. The memoir consists of the discussion of two problems r
the first is a problem given in the Ars Conjectandi of James Ber-
noulli ; the other relates to a lottery.

630. The problem from the Ars Conjectandi 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. Mallet
then says,

Mr. Bernoulli s'etant contente d'indiquer cette singularity apparente,
sans en donner Texplication, j'ai cru qu'il ne seroit pas inutile d'entrer
dans un plus grand detail ladessus, pour eclaircir parfaitement cette
petite difficulte, on verra qu'on peut imaginer une infinite de cas sem-
hlables a celui de Mr. Bernoulli, dans la solution desquels il seroit aussi
aise 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 born at the certain epoch. Now suppose we
wish to know whether it is more probable than not that a new
born infant will 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 Mallet'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
5, to every ticket-holder who does not gain a prize. The jDrizes are
distributed in the following manner. In a box are placed n coun-
ters numbered respectively from 1 to n. A counter is drawn, and
a prize assigned to the ticket-holder whose number corresponds to
the number of the counter. The counter is then replaced in the box.
Another drawing is made and a prize assigned to the corresponding
ticket-holder. The counter 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

) . If it is not drawn he is to receive h from the director ;

so that his corresponding expectation is h ( J • ^ 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

a — nh I j .

In the case which Montmort notices h 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 n ; 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 Mallet'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

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 all the homogeneous
products of the degree s — r which can be formed of the numbers
1, 2, 3, ... 7\ 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,

IVIALLET. 341

involves some algebraical work which we will give, since as we
have intimated Mallet himself omits it.

Let there be r quantities a,h,c, ... h. Suppose x^ to be di-
vided by (x — a) (x — b) (x — c) ... {x — h). The quotient will be

x'-" + H^ x^-' + H^ x^-'^ + . . . in infinitum,

where E^ denotes the sum of all the homogeneous products of the
degree r which can be formed from the quantities a,h,c, ... Ic. 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 J , and

so on.

Again, if ^ be not less than r the expression

(x — 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 will be

ABC K

X — a x — b x — c x — k'

where A =

(a — b)(a — c),.. (a — k)^

and similar expressions hold for B, C, ... K. Now expand each of

A B

the fractions , 7 , . . . according to negative powers of x ;

X "~" a X ~~ 0

and equate the coefficient of £c~*~^ to the coefficient in the first

form which we gave for x^ -^[(x — a) (x — b) ... (x — k)]. Thus

Aa'+BU+ Cc' + ... + Kk' = E,

?-*•+«+!•

Put m for ^ — r + ^ + 1 ; then p + ^ = m + r — 1; 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, c, ... k, is equal to

m+r-l 7,m+r- 1

+ 77 TTI ^ 77 fT+ ...

(a — b) (a — c) ... (a — k) (b —a) (b — c) ... (b — k)

12 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, h, c, .,,k 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

ij._,,_.,,rfc3),..,_r(t^«fcS,_a,.,...),

The particular case in which s — r+l gives us the following

result,

l+2 + S-\- ...+r

,^(r-l) r(r^l)(r- 2) ^^, ]

+ 1.2 ^"^ ^^ 1.2.3 ^"^ ^^ ^"V

which is a known result.

687. When Mallet has finished his laborious investigation he
says, very justly, il y a apparence que celui qui fit cette Lotterie ne
setoit pas donne la peine defaire tons les calculs pr^cedens.

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 Mallet 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 ?^ prizes ; while in Montmort's
solution there is no necessity for this.

639. Mallet gives the result of the following problem : Re-
quired the chance that in p throws with a die of n faces a specified
face shall appear just m times. The chance is

\m p —m n^

WILLIAM EMERSON. 34;3

The formula explains itself; for the chance of throwing the
specified face at each throw is -, and the chance of not throwing

71 — 1

it is . Hence by the fundamental principles of the subject

the chance of having the specified face just m times in p throws is

I m \p — m \nj

Since the whole number of cases in the p throws is if, it follows
that the number of cases in which the required event can happen is

I m I p — m

(n - 1)

p-m .

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 outline 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 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 j^ossibility 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 Theorie...des Pi^oh. pages
877 — 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 probability 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 ^—^ . The

result of a similar calculation from data furnished by observations

in London is a little less than t^^tzti . In pa<?es 397 — 401 of the

12410 ^ °

Theorie...des Prob. we have a more difficult problem, namely to
find the probability 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 Theorie...des Froh. In the memoir
Laplace obtains an equation in Finite Differences

hence he deduces

%m= constant -f ?/,„.^,„_, |l - A^^,„_2 + A {z,,,_^i^z^,,_^

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
TJi eo rie . . . cles P) 'ob.

899. Laplace now developes another form of his method of
approximation to the value of definite integrals. Suppose we

require lydx; let Y be the maximum value of ?/ within the

range of the integration. Assume ?/ = Ye~^\ and thus change

li/dx into an integi'al with respect to t. The investigation is

reproduced in the Theor{e...des Frob. pages 101 — 103.

n 00

Laplace determines the value of / e~^'dt. He does this by

taking the double integral / 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 y = Fe"*^^, we may assume y = Ye~^. Something similar is
given in the TJieorie...des Proh. pages 93 — 95.

Some formulae occur in the memoir which are not reproduced
in the Theorie...des Proh., and which are quite wrong: we will
point out the error. Laplace says on pages 298, 299 of the
memoir :

, , , , . -iff d.c dz

Considerons presentement la double integrale 1 1 ■- —3, prise

;y (1 -z -x^'Y

depuis a: = 0 jusqu'a £c = l, et depiiis z = ^ jusqu'a ^=1; en faisant

X , „ , , . [ dz [ dx

,=x, elle se cnano;era dans ce.le-ci /-— / — , ces

(l-z')^ Jj{l-z')J{l-x")i

integrales etant prises depuis x' -0 et ;^ - 0, ju^-qu'a x' —1 et z=l,

482 LAPLACE.

Then, as ,,., "" .,, = ^ , Laplace infers that

r r dxch _'TT f' dx

Bat this is wrono^ ; for the limits of x are 0 and 7 , and

° (1 - £y

not 0 and 1, as Laplace says ; and so the process fails.

Laplace makes the same mistake again immediately after-
wards ; he puts -jj^ -^ — z\ and thus deduces

dxdz [^ dx [^ dz

J„ (1 _ ^^ _ ^^)i J, (1 _ x^)k!, (1 _ ^-)t •

But the upper limit for z should be -j-Fx — ~T\ > ^^^ ^^^ ^ ^^

'Y ( X X j

Laplace assumes ; and so the process fails.

901. -Laplace applies his method to evaluate approximately

I a?^ (1 — xY dx ; and he finds an opportunity for demonstrating

•J 0

Stirling's Theorem. See Art. 333.

902. Laplace discusses in pages 304 — 313 of the memoir the
following problem. Observation shews that the ratio of the num-
ber of births of boys to that of girls is sensibly greater at London
than at Paris ; this seems to indicate a greater facility for the birth
of a boy at London than at Paris : required to determine the
amount of probability. See Art. 773.

Let w be the probability of the birth of a boy at Paris, p the
number of births of boys observed there, and q the number of births
of girls ; let z^ — ic be the possibility of the birth of a boy at Lon-
don, p the number of births of boys observed there, and q the
number of births of girls. If P denote the probability that the
birth of a boy is less possible at London than at Paris, we have

{U (1 _ uy {u - xY (1-U + xY du dx

P =

\ie (1 - uY {u - xY (1 - M + xY die dx

LAPLACE. 483

Laplace says that the integral in the numerator is to be taken
from ic = 0 to u = x, and from a? = 0 to ic = 1, and that the integral
in the denominator is to be taken for all possible values of x and ii.
Thus putting u — x = s the denominator becomes

[ [ u'' (1 - u)' s^' (1 - sY du ds.

•J {^ ■J Q

Laplace's statement of the limits for the numerator is wrong ;
we should integrate for x from 0 to u, and then for u from 0 to 1.

There is also another mistake. Laplace has the equation

V_ £_ ,JP_ l__ = n

X l-X'^X-x l-X-^x

He finds correctly that when a? = 0 this gives

X =

jy+p

He says that when jr = 1 it gives X = l, which is wrong.

Laplace however really uses the right limits of integration in
his work. His solution is very obscure ; it is put in a much clearer
form in a subsequent memoir which we shall presently notice ; see
Art. 909. He uses the following values,

p = 251527, q = 24.1945,
;/= 737629, ^' = 698958,

and he obtains in the present memoir

410458 '
he obtains in the subsequent memoir

P =

410178*

The problem is also solved in the Theorie . . . des Proh. pages
381 — 384 ; the method there is different and free from the mis-
takes which occur in the memoir. Laplace there uses values of p
and q derived from longer observations, namely

. ^ = 393386, ^=377555;

31—2

484 LAPLACE.

he retains the same vakies oip and q as before, and he obtains

P =

328269 •

It will be seen that the new values of p and q make - a little

larger than the old values ; hence it is natural that P should be
increased.

903. Laplace gives in the memoir some important investiga-
tions on the probability of future events as deduced from ob-
served events; these are reproduced in the Theorie . . . des Proh.
pages 394—396.

904. Laplace devotes the last ten pages of his memoir to
the theory of errors ; he says that after his memoir in the sixth
volume of the Memoires...par divers Savans the subject had been
considered by Lagrange, Daniel Bernoulli and Euler. Since, how-
ever, their principles differed from his own he is induced to resume
the investigation, and to present his results in such a manner as to
leave no doubt of their exactness. Accordingly he gives, with
some extension, the same theory as before ; see Art. 874. The
theory does not seem, however, to have any great value.

905. The present memoir deserves to be regarded as very im-
portant in the history of the subject. The method of approxima-
tion to the values of definite integrals, which is here expounded,
must be esteemed a great contribution to mathematics in general
and to our special department in particular. The applications
made to the problems respecting births shew the power of the
method and its peculiar value in the- theory of probability.

906. Laplace's next memoir on our subject is entitled Memoir e
sur les Suites; it is published in the volume for 1779 of the
Histoire de l' A cad... Paris; the date of publication is 1782. The
memoir occupies pages 207 — 809 of the volume.

This memoir contains the theory of Generating Functions.
With the exception of pages 269 — 286 the whole memoir is
reproduced almost identically in the T}ieorie..,des Proh.; it forms
pages 9 — 80 of the work. The pages which are not reproduced

LAPLACE. 4S5

relate to tlie solution of partial differential equations of the
second order, and have no connexion with our subject.

The formulae which occur at the top of pages 18 and 19 of
the Theorie...des Proh. are stated in the memoir to agree with
those which had been given in Newton's Methodus differentialis ;
this reference is omitted in the TJieorie...des Proh.

907. Laplace's next memoir on our subject is entitled ^itr les
approximations des Formides qui sont fonctions de tres-grands nom-
hres; it is published in the volume for 1782 of the Histoire de
V Acad... Paris: the date of publication is 1785. The memoir
occupies pages 1 — 88 of the volume.

Laplace refers at the commencement to the evaluation of
the middle coefficient of a binomial raised to a high power by
the aid of Stirling's Theorem ; Laplace considers this to be one
of the most ingenious discoveries which had been made in the
theory of Series. His object in the memoir is to effect similar
transformations for other functions involving large numbers, in
order that it might be practicable to calculate the numerical
values of such functions.

The memoir is reproduced without any important change
in the T]ieorie...des Proh., in which it occupies pages 88 — 171-.
See Arts. 89-i, 899.

A mistake occurs at the beginning of page 29 of the memoir,
and extends its influence to the end of page 30. Suppose that a
function of two independent variables, 6 and 6', is to be exjDanded
in powers of these variables: we may denote the terms of the
second degree by Md^ + 2N96'+ P6"^ : Laplace's mistake amounts
to omitting the term 2X66'. The mistake does not occur in the
corresponding passage on page 108 of the Tlieorie...des Proh.

908. Laplace's next memoii* is the continuation of the pre-
ceding; it is entitled, Suite du Memoire sur les approximations
des Formides qui sont fonctions de tres-grands Komhres; it is pub-
lished in the volume for 1783 of the Histoire de V Acad... Paris:
the date of publication is 1786. The memoir occupies pages
423 — 467 of the volume.

909. Laplace gives here some matter which is reproduced in
the Theorie.., des Proh. pages 363—365, 394—396. Pages 440—444

486 LAPLACE.

of the memoir are not reproduced in the Theorie , . .des Proh.;
they depend partly on those pages of the memoir of 1782 which
are erroneous, as we saw in Art. 907.

Laplace in this memoir applies his formulae of approxima-
tion to the solution of questions in probability. See Arts. 767, 769.
He takes the problem which we have noticed in Art. 896, and
arrives at a result practically coincident with the former. He takes
the problem which we have noticed in Art. 902, gives a much
better investigation, and arrives at a result practically coincident
with the former. He solves the problem about the births during a
century to which we have referred in Art. 897, using the smaller
values of j9 and q which we have given in Art. 902; he finds
the required probability to be "664. In the Theorie...des Proh.
page 401 he uses the larger values of j) and q which we have
given in Art. 902, and obtains for the required probability "782.

910. This memoir also contains a calculation respecting a
lottery which is reproduced in the Theo7'ie...des Proh. page 195.
See Arts. 455, 864.

Laplace suggests on page 433 of the memoir that it would

be useful to form a table of the value of \e~^'dt for successive
limits of the integration : such a table we now possess.

911. In the same volume there is another memoir by La-
place which is entitled, Sur les naissances, les mcwiages et les
marts d Paris.... This memoir occupies pages 693 — 702 of the
volume.

The following problem is solved. Suppose we know for a
large country like France the number of births in a year ; and
suppose that for a certain district we know both the population
and the number of births. If we assume that the ratio of the
population to the number of births in a year is the same for the
whole country as it is for the district, we can determine the popu-
lation of the whole country. Laplace investigates the probability
that the error in the result will not exceed an assiofned amount.
He concludes from his result that the district ouo'ht to contain
not less than a million of people in order to obtain a sufficient
accuracy in the number of the population of France.

LAPLACE. 487

The problem is reproduced in tlie Theorie...des Proh. pages
391 — 39-i. The necessary observations were made by the Frencb
government at Laplace's request ; the population of the district
selected was a little more than two millions.

The solutions of the problem in the memoir and in the
Theorie. . .des Proh. are substantially the same.

912. In the Lecons de Mathematiques donnees a Tecole normale,
en 1795, par M. Laplace, we have one le^on devoted to the subject
of probabilities. The legons are given in the Journal de VEcole
Poly technique y vii^ et viii® cahiers, 1812; but we may infer from
page 164 that there had been an earlier publication. The lecon
on probabilities occupies pages 140 — 172. It is a popular state-
ment of some of the results which had been obtained in the
subject, and was expanded by Laplace into the Introduction
which appeared with the second edition of the Theorie.,. des Proh.,
as he himself states at the beginning of the Introduction.

913. With the exception of the unimportant matter noticed
in the preceding Ai'ticle, Laplace seems to have left the Theory
of Probability untouched for more than twenty-five'^years. His
attention was probably fully engaged in embodying his own re-
searches and those of other astronomers in his Mecanique Celeste,
the first four volumes of which appeared between 1798 and 1805.

914. Laplace's next memoir connected with the Theory of
Probability is entitled Memoire sur les approximations des for-
mules qui sont fonctions de tres-grands nomhres, et sur leur ap-
plication aux prohoMlites. This memoir is published in the
Memoires...de VInstitut for 1809; the date of publication is 1810 ;
the memoir occupies pages 353 — 415 of the volume, and a supj^le-
ment occupies pages 559 — oQo.

915. The first subject which is discussed is the problem re-
lating to the inclination of the orbits of the planets and comets
which is given in the Theorie... des Proh. pages 253 — 261; see
also Art. 888. The mode of discussion is nearly the same. [There
is however some difference in the jDrocess relating to i\iQ planets,
for in the memoir Laplace takes two right angles as the extreme

488 LAPLACE.

angle instead of one right angle whicli he takes in the Tlieorle...
des Prob. Laplace's words are, on page 362 of the memoir :

Si Ton fait varier les inclinaisons depuis zero jiisqu'a la demi-cir-
conference, on fait disparoitre la consideration des mouvemens retro-
grades j car le mouvement direct se change ^en retrograde, quand I'incli-
naison surpasse un angle droit.

Laplace obtains in the memoir the same numerical result as on
page 258 of the Theorie...des Proh. ; but in the latter place the
fact of the motions being all in the same direction is expressly
used, while in the former place Laplace implies that this fact still
remains to be considered.

The calculation for the comets, which follows some investiga-
tions noticed in the next Article, does not materially differ from
the corresponding calculation in the Theorie . . .des Proh.; 97 is
taken as the number of comets in the memoir, and 100 in the
Theorie . . . des Proh.

916. Laplace gives an investigation the object of which is
the approximate calculation of a formula which occurs in the
solution of the problem noticed in the preceding Article. The
formula is the series for Zl'^s', so far as the terms consist of
positive quantities raised to the power which i denotes. A large
part of the memoir bears on this subject, which is also treated
very fully in the Theorie... des Proh. pages 165 — 171, 475 — 482.
This memoir contains much that is not reproduced in the
Theorie... des Proh., being in fact superseded by better methods.

We may remark that Laplace gives two methods for finding the

value of I fe~'^^'' cos htdt, but he does not notice the simplest

'' 0

method, which would be to differentiate e-^^^ cos htdt four times

-J 0

with respect to h, or twice with respect to c ; see pages 368 — 370
of the memoir.

917. In pages 383 — 389 of the memoir we have an important
investigation resembling that given in pages 329 — 332 of the
TJieorie...des Proh., which amounts to finding the probability that
a linear function of a large number of errors shall have a certain

LAPLACE. 489

value, the law of facility of a single error being any what-
ever.

Pages 390 — 397 of the memoir are spent in demonstrat-
ing the formula marked [q] which occurs at the top of page 170
of the Theorie...des Prob. The remaining pages of the memoir
amount to demonstrating the formula marked fp) on page 168 of
the Theor{e...des Proh., which is again discussed in pages 475 — 482
of the Theorie...des Prob. The methods of the memoir are very
laborious and inferior to those of the Th^orie...des Prob.

918. The supplement to the memoir consists of the matter
Avhich is reproduced in pages 333 — 335 and 340 — 342 of the
TJieorie...des Prob. In his supplement Laplace refers to his
memoir of 1778; see Art. 901: the reference is not preserved
in the Theorie...des Prob. He names Daniel Bernoulli, Euler,
and Gauss; in the corresponding passage on page 335 of the
Theorie...des Prob., he simply says, des geomUres celebres.

919. Laplace's next memoir is entitled, Me moire sur les Inte-
grates Definies, et leur ajyj^Ucatiori aux Probabilites, et specialement
a la recherche dii milieu quil faut choisir entre les residtats des
observations. This memoir is published in the Memoires ...de
rinstitut for 1810; the date of publication is 1811 : the memoir
occupies pages 279 — 347 of the volume.

920. Laplace refers to his former memoirs on Generating
Functions and on Approximations ; he speaks of the approaching
publication of his work on Probabilities. In his former memoirs
he had obtained the values of some definite integrals by the
passage from real to imaginary values ; but he implies that such a
method should be considered one of invention rather than of
demonstration. Laplace says that Poisson had demonstrated several
of these results in the Bidletiii de la Societe Philoinatique for March
1811 ; Laplace now proposes to give direct investigations.

921. The first investigation is^that which is reproduced in
pages 482 — 484 of the Theorie...des Prob. Then follow those
which are reproduced in pages 97 — 99 of the Theorie...des Prob.
Next we have the problem of the Duration of Play, when the

490 LAPLACE.

players are of equal skill and one of them has an infinite capital ;
there is an approximate calculation which is reproduced in pages
235—238 of the Theorie...des Proh. Next we have the problem
about balls and the long dissertation on some integrals which we
find reproduced in pages 287 — 298 of the Theorie...des Proh.
Lastly we have the theory of errors substantially coincident with so
much of the same theory as we find in pages 314 — 328 and
340—342 of the Theorie...des Proh.

922. A theorem may be taken from page 327 of the memoir,
which is not reproduced in the Theorie...des Proh.

To shew that if ^|r (x) always decreases as x increases between
0 and 1 we shall have

I i/r [x) dx greater than S I x'^^jr (a?) dx.
Jo •'o

It is sufficient to shew that

x^ I i/r [x) dx is greater than S j x^yjr (x) dx,

or that 2x I -v^^ (x) dx is greater than 2x^ -^ {x)y

r X

or that I '^ (x) dx is greater than x -v/r (x),

"J 0
1 , r \ ' , i 1 I / \ d^ (x)

or that Y l*^) ^^ greater than y \F) + ^ — j — ?

but this is obviously true, for ^ is negative.

The result stated on page 321 of the Theorie...des Proh., that

k" . 1 .

under a certain condition -^ 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 qiiilfaut choisir entre les residtats d'tin grand
nomhre d'ohservations. The article contains the matter which is
reproduced in pages 322 — 329 of the Theorie...des Proh. Laplace
speaks of his work as soon about to appear.

LAPLACE. 491

924. In the Connaissance des Terns for 1815, which is dated
November 1812, there is an article on pages 215 — 221 relating to
Laplace's Theo7ne...des Proh. 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 Connaissance des Terns for 1816, which is dated
November 1813, there is an article by Laplace, on pages 213 — 220,
entitled, Sur les Cometes.

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 probability is very great that a comet would
move either in an ellipse 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 ?• 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 wdien 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, ■sr 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

r V sin OT = Va (1 — e ),
I) = a{l-e).

492 LAPLACE.

From these equations by eliminating a and e we have

sin^ OT =

r
7F

and from this we deduce

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 'sr. The values of the perihelion distance which
correspond to these limiting directions are 0 and D. Laplace
then proceeds thus;

...en supposant done toutes les valeurs de D egalement possibles, on
a pour la probabilite que la distance peiihelie sera comprise entre zero
et D,

yf'^(-?)--}

II faut multiplier cette valeur par dV ; en I'integrant ensuite dans
des limites determinees, et divisant I'integrale j^ar la plus grande valeur
de V, valeur que nous designerons par U ; on aura la probabihte que la
valeur de V sera comprise dans ces limites. Cela pose, la plus petite
valeur de V est celle qui rend nulle la quantite renfermee sous le radical
precedent ; ce qui donne

J2I)

rV =

1.*

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 'yjr ( V) stand for

.y/fp(l+f)-2Z)};

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 >/r ( F) that its perihelion distance will lie between 0 and D.
Now suppose we assume as a fact that the perihelion distance
does lie between 0 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

ff{V)
fylr{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 l'^{V)dV; for this pmpose he

assumes

For the assigned limits of V he takes ^ and —

The value of \^|r{V)dV between these limits he finds to be ap-
proximately

2r ir sjr '

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

■{it -2) si IB _ ^ \ ^ (tt - 2) V27>
2r ir^/r) ' 2r '

494 LAPLACE,

that is,

i (tt — 2)\lr'

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 - = -— V^ suppose a = — 100 ; then

^^, r + 200 ,, .., r + 200

^=T()or' '^'''' ^=-100--

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

V2^

i (tt — 2) ^r '

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 v^^tt •

Laplace then says that his analysis supposes that all values of
I) between 0 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 0 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 VajypUcation du Calcid des Prohabilites a la
Pkilosophie naturelle; the second is entitled, Sur le Calcul des
Prohabilites, applique a la Philosophie naturelle. The matter is
reproduced in the first Supplement to the Theorie...des Proh.
pages 1 — 25, except two pages, namely, 376, 377: these contain
an application of the formulge 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,
Ajyplication dii Calcul des ProhahiUtes, aux ojje rations geodesiqiies:
it is reproduced in the second Supplement to the Theorie...des
Proh. 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 dii Calcul des ProhahiUtes aux operations geodesiqnes
de la meridienne de France: it is reproduced in the 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 ProhahiUtes, This was published
in 1812, in quarto. There is a dedication to Napoleon-le-Grand ;
then follow 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 occujiies pages 485 — 506 : then two pages of errata are
given.

The pages 9 — 444 of the first edition luere not reprinted for
the second or third edition ; a few pages Avere 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 ivas not
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 Proh.; 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 will be found that in the text a page n of the editions pub-

lished by Laplace himself will correspond nearly to the page ^ + tt;

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 Napoleon-le-Grand. Sire, La bienveillance avec laquelle Yotre
Majeste a daigne accueillir rhommage de mon Traite de Mecanique
Celeste, m'a inspire le desir de Lui dedier cet Ouvrage sur le Calcul des
Prohabilites. Ce calcul delicat s'etend anx questions les plus impor-
taiites de la vie, qui ne sout en effet, pour la plupart, que des j^roblemes
de probabilite. 11 doit, sous ce rapport, interesser Yotre Majeste dont
le genie sait si bien apprecier et si dignement encourager tout ce qui
peut contribuer au progres des lumieres, et de la prosperite publique.
J'ose La supplier d'agreer ce nouvel hommage dicte par la plus vive
reconnaissance, et par les sentimens profonds d'adiniration et de respect,
avec lesquels je suis, Sire, de Votre Majeste, Le tres-liumble et tres-
obeissant serviteur et fidele 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 sup-
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 Proh. was pub-
lished separately in octavo under the title of Essai pkilosophique
sur les Prohahilites; we shall however refer to the introduction
by the pages of the third edition of the Theorie...des P7'ob.

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
mModes analytiques dii Calcul des Probahilites ; 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 symbolical
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 fi'om page
CI. of the second edition of the introduction : a student however

498 LAPLACE.

who wished to understand the treatment would have to consult
the original work, namely De Moivre's Miscellanea Analytical
pages 28 — 83. Also some slight historical reference to Wallis and
others is introduced on pages xxxv— xxxvn. ; this is merely an
abridgement of the pages 3 — 8 of the Theorie . . .des Proh.

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 events ; that is the
section is devoted to the consideration of James Bernoulli'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 XLiii.

Voyez au contraire, dans quel abime de mallieurs, les peuples ont
ete souvent precipites par I'ambition et par la perfidie de leurs chefs.
Toutes les fois qu'une grande puissance enivree de I'amonr des conquetes,
aspire h la dominatiou universelle; le sentiment de I'mdependance pro-
duit entre les nations menacees, une coalition dont elle devient presque
toiijours la victinie.

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 end 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 theorie des fonctions generatrices donne une expression tres
simple de cette probabilite, que Ton obtient en integrant le produit de
la differentielle de la quantite dont le resultat deduit d'un grand nombre
d' observations s'ecarte de la verite, par une coustante moindre que
I'unite, dependante de la nature du probleriie, et elevee a une puissance
dont I'exposant est le rapport du carre de cet ccart, au nombre des
observations. L'integrale prise entre des limites donnces, et divisee

LAPLACE. 499

par la meme integrale etendue a rinfini positif et negatif, exprimera la
probabilite que I'ecart de la veiite, est compris entre ces limites.

A student familiar with the Theorie...des Proh. itself might
not find it easy to say what formula Lajolace has in view ; it must
be that which is given on page 309 and elsewhere, namely

dre 4A".

7 "

/J TT

Other examples of the same absurdity will be found on page LL
of the introduction, and on page 5 of the first supplement.

938. A section occupies pages XLix — LXX. entitled Applica'
tion du Calcul des Probahilites, d la Pliilosophie naturelle. Tbe
principle which is here brought forward is simple ; we will take
one example which is discussed in the Theorie . . .des Prob. 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 irresistibly 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 sj)heroidal 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 tem23erature of the earth shewn
'to be constant for two thousand years : it does not appear that
Laplace himself here notices this result.

939. Li the second edition of the Theorie ...des Proh.

32—2

500 LAPLACE.

Laplace did not include the secular acceleration of the moon and
the theory of the tides in the list of his labours suggested by the
Theory of Probability. Also pages Li— LVL 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 Theorie...des Proh., 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 Theorie...desFroh.,
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 — of the whole mass. Nevertheless it

has since been recognised that the error was as large as — ; see

Poisson, Recherches sur la Proh..., page 816.

9-iO. Laplace devotes a page to the Application dii Calcul
des Prohabilites aux Sciences morales; he makes here some inter-
esting remarks on the opposing tendencies to change and to con-
servatism.

9-tl. The next section is entitled, De la Prohahilite des
temoignages; this section occupies pages Lxxi — Lxxxii : it is an
arithmetical reproduction of some of the algebraical investigations
of Chapter XL of the Theorie...des Proh. One of Laplace's discus-
sions has been criticised by John Stuart Mill in his Logic; see
Vol. 11. 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
Racine, Pascal and Locke. He examines with some detail a
famous argument by Pascal which he introduces thus :

Ici se presente naturellement la discussion d'un argument fameux
de Pascal, que Craig, mathematicieii anglais, a reproduit sous une forme

LAPLACE. 501

geomtStriqiie. Des t^moiiis attestent qu'ils tieunent de la Divinite meme,
qu'en se conformant k telle chose, on jouira, non pas d'une ou de deux,
mais d'une infinite de vies heureuses. Quelque faible que soit la proba-
bilite des temoignages, pourvu quelle ne soit pas infiniment petite; 11
est clair que I'avantage de ceux qui se conforment a la chose prescrite,
est infini, puisqu'il est le produit de cette probabilite par un bien
infiDi; on ne doit done point balancer a se procurer cet avantage.

See also the Athenceum for Jan. 14tli, 1865, page oo.

942. The next section is entitled, Des clioix et des decisions
des assemhUes; 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 probabilite des Jugemens
des tiibunaux; it occupies five pages: results are stated which
are obtained in the first supplement to the Theo7'ie...des Prob.
This section is nearly all new in the third edition of the
Theorie. . . des Prob.

The next section is entitled, Des Tables de mortalite, et des
durees moyennes de la vie, des mariages et des associations quel-
conques; it occupies six pages : results are stated which are ob-
tained in Chapter VIIL of the Theorie... des Prob.

The next section is entitled, Des benefices des etablissemens qui
dependent de la probabilite des evenemens; it occupies five pages.
This section relates to insurances : results are given which are ob-
tained in Chapter ix. of the Theorie... des Prob.

943. The next section is entitled, Des illusions dans Vesti-
mation des Probabilites ; this important section occupies pages
cii — cxxviii: in the second edition of the Theorie... des Prob. 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 ilkistratecl by the example which we have
noticed in Art. 85G, and by the case of the Chevaher de Mere:
see Art. 10. Laplace makes on his page cviii. 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 BernoulK. They estimated the

infinite series

1-1+1-1 + .. .

as equal to ^ ; because if we take an even number of terms we

Jit

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 Stewarfs Works
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 pliilosophe de Tantiqiiite, auqiiel on montrait dans un
temple, pour exalter la puissance du dieu qu'on y adorait, les ex voto
de tons cenx qui apres I'avoir invoqiie, s'etaient sauves du naufi^age, fit
une remarque conforme au calcul des prohabilites, en observant qu'il
ne voyait point inscrits, les noms de ceux qui, malgre cette invocation,
avaient peri.

944. A long discussion on what Laplace calls Psycliologie
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

94^5. The next section is entitled, Des divers moi/ens d'ap-
l^rocher de la certitude; it occupies six pages. Laplace says,

L'inductioii, Tanalogie, des hypotheses fondees siir les faits et recti-
fiees sans cesse par de nouvelles observations, un tact heureux donne
par la nature et fortifie par des comparaisons nonibreuses de ses indi-
cations avec I'experience; tels sont les priucipaux moyens de parvenir
a la verite.

A paragraph beginning on page cxxix. with the words Kous
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 donne pour la recherche de la verite, le precepte et non I'ex-
emple. Mais en insistant avec toute la force de la raison et de I'elo-
quence, sur la necessite d'abandonner les subtilites insignifiantes de
I'ecole, pour se livrer aux observations et aux experiences, et en indi-
quant la vraie niethode de s'elever aux causes general es des phenomenes;
ce grand philosophe a contribue aux progres immenses que I'esprit
humain a faits dans le beau siecle ou il a terniine sa carriere.

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. Notice
historique sur le Calcid des Prohabilites ; this is brief but very
good. The passage extending from the middle of page cxxxix.
to the end of page CXLI. 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 Theorie...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-

50^ LAPLACE.

place sums up the claims of the Theory of Probability well deserve
to be reproduced here:

On voit par cet Essai, que la tlieorie des i^robaLilites n'est an fond,
que le bon sens reduit au calcul : elle fait apprecier avec exactitude,
ce que les esprits justes sentent par une sorte d'instinct, sans qu'ils
puissent souvent s'en rendre compte. Si Ton considere les methodes
analytiques auxquelles cette theorie a donne naissance, la verite des
principes qui lui servent de base, la logique fine et delicate qu'exige
leur emploi dans la solution des problemes, les etablissemens d'utilite
publique qui s'appuient sur elle, et I'extension qu'elle a reque 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 clioses memes qui ne peuvent etre soumises
au calcul, elle donne les aperqus les plus surs qui puissent nous guider
dans nos jugemens, et qu'elle apprend a se garantir des illusions qui
souvent nous egarent; on verra qu'il n'est point de science plus digne
de nos meditations, et qu'il soit plus utile de faire entrer dans le systeme
de r instruction publique.

948. We now leave the introduction and pass to the Theorie...
des Proh. itself Laplace divides this into two books. Livre I. is
entitled Du Calcul des Fonctions Gene'ratrices: this occupies pages
1 — 177 ; Livre ii. is entitled Theoi^e generate des Prohahiliies;
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 oi Livre I. is entitled
Des Fonctions generati^ices, ct 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.

Pasres 39 — 49 contain various formulae of what we now call the
Calculus of Operations ; these formulse cannot be said to be cle-
monstrated by Laplace ; he is content to rely mainly on analogy.
LagTange had led the way here ; see the preceding Article.

One of the formulae may be reproduced ; see Laplace's page 41.
If we write Taylor's theorem symbolically we obtain

Ay.= v/^^-Vj/.,

where A indicates the difference in y^ arising from a difference h in
X. Then

Laplace transforms this into the following result,

/ hd_

The following is his method :

/d_ Y nh£/ hd_ h dY

W^"^ -l)y. = e ^.^'^•^ Ve2 ci^ _ ^'2 dx) y^^

506 LAPLACE.

Now let ^ ( -7- j denote any term arising from the development of

I g2 dx Q~ 2dxj

Then ,(^)....^^=,(^),,,

and the term on the right hand may be supposed to have arisen

/ hd_ _IlAY
Ve2 dx_

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

Ay,=\e "- -1).

He gives verbal directions as to the way in which the symbols
are to be treated, which of course make his formulae 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 h = l and change x into a? — - we

obtain

/ 1 £, 1 d\'

which Laplace obtains on his page 45 by another process.

952. The second Chapter of the first part of Livre 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 — (j^> a strange process for integrating the
following equation in Finite Differences,

LAPLACE. 507

We might suppose that z^,^ is the coefficient of fr' in the ex-
pansion of a function of t and r ; then it would easily follow that
this function must be of the form

^[t)^'^ (t)
nab \'

Tti C

\Tt T t J

where </> (t) is an arbitrary function of t, and yjr (r) an arbitrary
function of r.

Laplace, however, proceeds thus. He puts

1 a h ^
-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 t
which when expanded in powers of t and r has z^, y for the co-

efficient of fi^. Then in the expansion of -^-^ the coefficient of

^V^willbe^,,,.

Laplace then transforms —^ thus. By the equation generatrice
we have

1 T

' i-6

therefore,

u \T h h)

c + ah + a f h

(I _ ,)^

Develope the second member according to powers of 5 ;

thus

f~r

,-'l(H'-''M"-'-^'-M"-^-]

,ar-2

, _ X (c + ah) a""'^ x(x — l), , ,,2 ^'
X l^'+-^"J + 1.2 (^ + ^^) 7l ^

T Vt y

508 LAPLACE.

Multiply the two series together. Let

Fj = yld' + a; (c + al) a"'^,

J- ■ ^ 1 . ^

Tr_y(y-i)Cy-2)„ .

Then

^^=4^f--^y+F.f--^rv...-^^.

.T / VT

T \T J \T J

But the equation

gives

1 «j 5

7 --c = 0

CT T C

1 , c + a6 '
0

therefore

Now we pass from the generating functions to the coefficients,
and we pick out the coefficients of ^V° on both sides. This gives
z^y on the left-hand side, and on the right-hand side a series
which we shall now proceed to express.

Let A apply to x, and indicate a Finite Difference produced
by the change of x into ic + 1 ; and let 8 similarly apply to ?/, and
indicate a Finite Difference produced by the change of 7/ into
2/+1.

LAPLACE. 509

Now [ ^] ^^[1 ■"-) J hence in u( h) the coefficient

of fr^ will be h't'' (-jiA > provided we suppose that ?/ is made zero
after the operation denoted by 8'' has been performed on -^ .

Similarly in w (^ - aj the coefficient of fr' wiU be oTA' (^/] ,
provided we suppose that x is made zero after the operation de-
noted by A'" has been performed on -^ .

In this way we obtain

+ ^ V A (^] + — ^ F A'^ (^]

IT K f ( X A

+ • • • + 7 : 7T- r y^y. A I — ^

Thus we see that in order to obtain z^^y we must know
^0, 1 > ^0, 2' • • • ^P ^^ ^0, V > ^nd we must know z^^^, ^2, c • • ^p to z^^ „ .

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
generatrice 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

7/ into ?/ + 1 : then the fundamental equation we have to integrate

will be written

{EF- aF- bE- c) z,^, = 0,

or for abbreviation

EF-aF-bE-c = 0,

Then E'^F^ will be expanded in the way Laplace expands
and his result obtained from E^'F^z^^^. Thus we rely on the

foundations on which the Calculus of Operations is based.

510 LAPLACE.

We may notice tliat we have changed Laplace's notation in
order to avoid the dashes which are difficult in printing. La-
place uses X where we use y, t! where we use r, and 'A where we
use 8.

953. Laplace takes another equation in Finite Differences.
The equation we will denote thus

A"^,„+^A»-a^,.,+ J A-^ax, + ... = 0.

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

e-r*K'-r(i-)4.&-r&-r--=»-

He supposes that this equation is solved, and thus decomposed
into the following n equations :

t a\ W

t a V TV '

where q, q^, q^}"* ^^^ the n roots of the equation
Then, using the first root

u u f^ q

^' x.r'u.

<i\ 1

Then passing from the generating functions to the coefficients,
that is equating the coefficients of ^V°, we obtain

LAPLACE. 511

The second member may be put in tlie form

Denote the quantity [ — —Vz^^y by the arbitrary function
</) (?/). Thus

This value of Zx,y will then satisfy the equation in Finite Dif-
ferences.

Each of the n roots q, q^, q^, ... gives rise to a similar ex-
pression ; and the sum of the 7i particular values thus obtained for
Zx 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 dy. Then

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-

512 LAPLACE.

ferential equations; he concludes that such functions are ad-
missible under certain conditions. Professor Boole reofards the
argument 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 error
hereafter in Arts. 974, 980, 984.

956. We now arrive at the second part of Livre i., 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 Be
Tintegixttion jjar approximation, des differe^itielles qui renferment
des facteurs Sieves a de grandes puissances; this Chapter occupies
pages 88—109.

957. The method of approximation which Laplace gives 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

jydx=YJe-<^^dt.

Let y = (j) {x) ; suppose a the value of x which makes y have
the value Y : assume x= a+ 6.

Thus (^(a + (9)= Ye-'';

therefore f = log -7-7 ^r .

(p {a-^ o)

From this equation we may expand ^ in a series of ascending
powers of 6, and then by reversion of series we may obtain 6 in a
series of ascending powers of t. Suppose that thus we have

LAPLACE. 513

then J = J = ^, + 2i?,^ + 353^+...;

[ydx = YJe-^' {B^ + 2B,t + SBf + ...) 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 B^ inclusive will be
found in De Morgan's Differential and hitegral 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 will be — co and + oo .

Now if r be odd I e-^'fdt vanishes, and if r be even it is equal to

^ —00

(^_1)(^_3)... 3.1
Thus we have

3 5 3

ydx= Ys/ir\B^-]r-^B^-^-i^ B^-^.,.\.

Besides the transformation y = Ye~^^ Laplace also takes cases
in which the exponent of e instead of being — f has other values.
Thus on his page 88 the exponent is — t, and on his page 93
it is — f'' ; 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 useful to supply a few references.

The formula marked {T) on page 95 occurs in Laplace's memoir
of 1782, page 17.

COS rx e** ^ dx = -^r- e ia" .

Jo 2«

this was given by Laplace in the Memoires...de llnstitut for
1810, page 290 ; see also Tables dlntegrales Definies, 1858, by

D. Bierens de Haan, page 376.

514 LAPLACE.

dx-=^ ^

0 ^ ^

see D. Bierens de Haan, page 268.

where a is supposed positive ; these seem due to Laplace ; see
D. Bierens de Haan, page 282, TMorie...desProh., pages 99—134.
We may remark that these two results, together with

/■

J 0

sin ax dx it ,^ „•.

1 -i- X X Z

are referred by D. F. Gregory, in his Examples ofthe.,.Diffe7'ential
and Integral Calcidus, to Laplace's memoir of 1782 ; but they are
not explicitly given there : with respect to the last result see
D. Bierens de Haan, page 293.

959. Since the integral le~^' 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 :

t' 1 t' 1 t'
e-'V< = T-3+^g-^^ + ...;

j;>.,.„,-..(..K.jiS!.,5^*...)>

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

J T

a continued fraction, which he takes from his MScanique Celeste,
Livy^e x. See also De Morgan's Differential and Integral Calculus,
page 591, for this and some similar developments.

LAPLACE. 515

9 GO. 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 \\ydxdx' 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'.

Substitute these values of x and x in the function log — and

expand it in powers of 6 and 6' ; then since Y is by hypothesis the
maximum value of y the coefficients of 6 and 6' will vanish in this
expansion : hence we may write the result thus

that is J/ (" (9 + ^ ^'V j^(p-^\ d" =e^ t'\

Since we have made only one assumption respecting the inde-
pendent variables t and t' w^e are at liberty to make another ; we
will assume

and therefore 6' /(p-^\^t'.

Now by the ordinary theory for the transformation of double
integrals we have

\ydxdx = jj ^ ,

, T-K , 1 r dt dt' dt dt'

where x> stands lor -7-, ^7^, — -,^, -77: .

du do do do

Thus far the process is exact. For an approximation we may
suppose M, N, P to be functions of a and a only ; then we have

2r da" ' tY dada" 2Y da"'

3—2

516 LAPLACE.

Then we sliall find that

n //DiT A72N 1 /{d'^Yd'Y fd'Y\\

And the limits of t and t' will be — (X) and + co ; thus finally
we have approximately

I \y dx dx —

/{d'Y d'Y
y \dd' da!'

2irY^

( d'Y

jda da I
See Art. 907.

961. The second Chapter of the second part of Livre l. is
entitled De ^integration par approximation, des Equations lineaires
aux differences finies 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 L is
entitled App)lication des m^thodes prdc^dentes, a V ap>proximation
de diverses fonctions de tres-grands nonibi^es: this Chapter oc-
cupies pages 126 — 177.

The first example is the following. Suppose we have to in-
tegrate the equation in Finite Differences,

Assume y^— ix'ipdx, where ^ is a function of x at present

undetermined, and the limits of the integration are also unde-
termined.

Let By stand for x" ; then -~^ = sx^~'^. Hence the proposed

equation becomes

LAPLACE, 0I7

that is, by integrating by parts,

0 = [xhy<f>] +j |(1 -^) ^-^ (#)j Bi/dx.

Wliere by [xBi/^] we mean that xhy^ is to be taken between
limits.

Assume </> such that

and take the limits of integration such that [^ % </>] = 0 ; then
our proposed equation is satisfied.

From (1 — cc) (^ — y- (x(^) = 0, we obtain

where ^ is a constant. Then xBi/ (f> will vanish when x= 0 and
also when cc = 00 . Thus, finally

/•CO"

x^e'^dx.

Now we proceed to put this integral in the form of a series.
The maximum value of ic*e~* is easily found to be that which
corresponds to a; = 5. Assume, according to Art. 957,

23* e"* = s'e~^e~^\

and put x—s-\-Q\ thus

Take the logarithms of both sides ; thus

Hence by reversion of series we get

3 9 V2s

518 LAPLACE.

therefore dx^dQ^ di V25 1 1 + — r— + -^

\ 3V25 65

The limits of t corresponding to the limits 0 and go of a; will
be — 00 and + 00 . Therefore

/-«> /"» ( ^t ^ \

Jo J -00 I 8V2s 65 J

By integration we obtain

==As'^^e-'\^'27r \l+ Y^+ ...\ .

Laplace says we may determine the value of the factor

very simply thus.

7? O

Denote it by 1 ■] H -7 + ... so that

•^ s s

7/, = As'^'^e-W2^h + j + ~ + ...\
Substitute this value in the equation

thus

' ly^^ _, r B C ] , B 0 ,

therefore

n.?+f+...){;-H)-('n)_^}

8 S

B B-2G

--^ + — ^j— + ...

And

~ 126'^ "^12s^^ ••'
Thus

f, B G \ ( 1 1 I ^ , B-20

s ' ^^ ' J 12s' ' 12/ s' ' s

LAPLACE. 519

Hence, equating coefficients,

^ ~ 12 ' 288 '

The value of A in the expression for y^ must be determined
by some particular value of ?/g. Suppose that when s = yx we
have 2/^= Y.

Then Y^A\ x^e'^dx^

J a

thus A =

/•OO

Hence

Vs- p i^+ 12s'^288/"^*"'i*

I x^ e ax

^ 0

The original equation can be very easily integrated; and we

obtain

y,^ rOL6 + l)0+2) ...5.

Hence, by equating the two values of y^,
(/i + 1) (/I +2) ...s =

•^ 0

a;'^ e"'" cfa?

It will be observed that 5 — /z is assumed to be a positive
integer, but there is nothing to require that 5 itself should be an

integer.

963. One remark must be made on the process which we have
just given. Let 0 (s) denote

*■ "T T(r>_'ooo^2'"

will be denoted by <f>{—s).

Now Laplace does not shew that

520 LAPLACE.

although he assumes the truth of this on his page 134). It may
be shewn by adopting the usual mode of proving Stirling's Theo-
rem. For by using Euler's theorem for summation, given in
Art. 334, it will appear that

where t (^) = 9^ " ^l73 + '

26^ 3.4/ ' 5.Qs' '

the coefficients being the well-known numhers of Bernoidli.

Thus 'i/r(5)+i/r(-5)=0;

therefore e^^'^ x e^^"^ = e' ^1,

that is </) (5) <^ (— 5) = 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 = cos CT

(^^ + ^r

then

/<00

('00 •

J 0

A memoir by Cauchy on Definite Integrals is published in the
Journal de VEcole Folytechnique, 28^ Cahier ; this memoir was
presented to 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 a qiielques resultats nouveaux, ainsi qii'a la
demonstration directe de plusieurs formules, que M. Laplace a deduites

LAPLACE. 521

du passage du r^el a rimaginaire, dans le 3"^® chapitre du Calcul d'^s
P^'ohahilites, et qu'il vient de confirmer par des metliodes rigoureuses
dans quelques additions faites a cet ouvrage.

The additions to which Cauchy refers occupy pages 464 — 484
of the Theorie...des Froh., 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 coefficients of the
terms in the expansion of a high power of a certain polynomial.

Let the polynomial consist of 2n + l terms and be denoted
by

111 1

-^ + -iFT + -1F2 + — + - + 1 + «+••• + «""' + «*""' + «" ;

a a a ^ a

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 6 when

{

l + 2cos^ + 2cos2^+ ... + 2cos?i^y

is expanded and arranged according to cosines of multiples of 0.
This term will be found by integrating the above expression with
respect to 6 from 0 to tt, and dividing by tt. Sum the series of
cosines by the usual formula ; then the required term

. 2w + l^
t n^ 1 sm — 71 — u

ir\ J

TT j„ J • 1 /I

sm ^ o

2 C^' /sin m(l)
"ttJo Vsin^

where (f)=^-9, and 7n = 2n + 1.

Now the expression ( . ^j vanishes when

TT 27r Stt
0 = — or — or —
^ m m m

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
(j) = 0. Thus we may calculate by Art. 957 the value of the integral

'sm 7nd)\^ . . . . TT

— : — ~ d6 when the limits are consecutive multiples of — .
^ sni (p J ^ m

sm Tyicri
The equation which determines the maxima values of — ^ — ,-

sm<p

m, cos mcf) sin (/> — cos <j) sin mcf)

!•

sin^ (j)

It will be found that this is satisfied when <^ = 0 ; the situation
of the other values of cf) will be more easily discovered by putting
the equation in the form

tan m(j) — m tan (^ = 0 :
now we see that the next solution will lie between m(b = -— and

»">73" 7 77" 077"

m^— — , and then the next between m(p = -r- and m<^ = -^,
and so on.

We proceed then to find

''sin mcf)\

'711

d(f>.

The maximum value of the function which is to be integrated
occurs when 4> = 0, and is therefore m* ; assume

— 77i*e~^',

/sin m(f)^ *
V sin (/) ,

j = m*e~'';

take logarithms, thus we obtain

LAPLACE. 523

Therefore approximately

d<t> V6

dt ~^ls{m'-l)}'

The limits of ^ will be 0 and x . Hence approximately
2 r^ /sin 7?2(f)\* ,^ 2 w'' ^/6 r -t^ j.

m' V6 (2w + 1)' \/3

Laplace next considers the value of the integral with respect

to <h between the limits — and — , and then the value between
^ m m

the limits — • and ^ — , and so on ; he shews that when 5 is a very
771 m

large number these definite integrals diminish rapidly, and may

be neglected in comparison with the value obtained for the limits

0 and — . This result depends on the fact that the successive
m

Sm 'ITZCD

numerical maxima values of — -. — r^ diminish rapidly ; as we shall

sm ^

now shew. At a numerical maximum we have

sin vi(f) m cos m(f) m m

sin ^ cos ^ cos 0 V(l + ^^'^ tan^ </>) VCcos'^ ^ + m"^ sin^ 0) '

this is less than ^ — r , that is less than . . . — , and therefore

sm (p sm (p 9

a for^tiori less than — — , that is less than ^ — r .
•^ 2 <j) 2 incp

Hence at the second maximum —. — j- is less than - -— ,

sm 0 2 5

that is less than ^ , and therefore the ratio of the second nume-

524< LAPLACE.

rical maximum value of ( —. — ~] to the first is less than f J) .
Similarly the ratio of the third numerical maximum value to the
first is less than f^j . And so on.

Next suppose that we require the coefficient of a in the
expansion of

[4 + -4i + -^2 +... + - + 1 + « + •••+«""'+«""' + ^4'-
[a a a a )

The coefficient of a'" in this expansion will be the same as the
coefficient of a"*" ; denote the coefficient of a"" by A^.. Pat e^^
for a and suppose the expression to be arranged according to
cosines of the multiples of 6 ; then 2A^ cos r6 will be the term
corresponding to A^ (a*" + a"*"). If we multiply the expression by
cos W, and integrate between the limits 0 and tt, all the terms
will vanish except that for which r is equal to I; so that the

integral reduces to 2Ai j cos^ Wd6. Hence

cos WdO.

We put; as before, m — 2n + 1, and (f> = ^0; thus we have

As before assume

/sin m4i\^

sin md>Y „ _,,

then </) = .. , \ — TyT , approximately.

Hence the intoOTal becomes

to'

2 "^'^'5 f^-,^^^ 2?^y6 ^,

Vis K- 1)1 J" ^"V(«K-i)}

LAPLACE. 525

As before we take 0 and oo for the limits of t, and thus
neglect all that part of the integral with respect to 0 which is not

included between the limits 0 and — . Hence by Art. 958 we

have finally

2 77ZV6 Vtt -,-J^ (2/z + l)V3 -^-^,

;; sj[s {m' - 1)} 2 ^ ' ""^ ^/[n {n + 1) 2s7r}

Suppose now that we require the sum of the coefficients, from
that of a-^ to that of o} both inclusive ; we must find the sum of

2^j+2^z.i+2^i_2+... + 2^^ + ^,:

this is best effected by the aid of Euler's Theorem ; see Art. 384.
We have approximately

ri 11

r? 1 1

therefore ^^ u^ = I u^dx + « ^^a; + « ^o >

therefore 2^^^^ Ug; — u^ = 2 I ujix + u^,

J 0

Hence the required result is

We may observe that Laplace demonstrates Euler's Tlieorem
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

x^ ^ e^ dx

He demonstrates this in his own way ; it is sufficient to obseiTe
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

/• 00

I x^^ e-^ (e-^ - 1)" dx

^n - ^ -_o

6-*

:&-^ e~^ dx

Laplace calculates tlie approximate value of this expression,
supposing { 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
Tkeorie...des Proh. The demonstration involves much use of the
symbol \/(- !)• 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 t 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

Tl [71 — 1 )

(n-\-r sJnY - n (ii + r V?i - 2)^ + (n + r ^Jn- iy - ... ,

where the series is to extend only so long as the quantities raised
to the power yu, are positive, and />t is an integer a little greater or
a little less than n. See Arts. 916, 917.

The methods are of the kind already noticed ; that is they are
not demonstrative, but rest on a free use of the symbol a/ (— 1).

A point should be noticed with respect to Laplace's page 171.
He has to establish a certain formula; but the whole difficulty of
the process is passed over with the words determinant convenable-
ment la constante arhitraire. Laplace's formula is established by
Cauchy ; see page 240 of the memoir cited in Art. 964.

968. In conclusion we may observe that this Chapter contains
many important results, but it is to be regretted that the demon-

LAPLACE. 527

strations are very imperfect. The memoir of Cauchy to which we
have referred, is very laborious and difficult, so that this portion
of the Theorie...des Proh. remains in an unsatisfactory state.

969. We now arrive at Livre ii, which is entitled TMorie
Generate des Prohahilites.

It will be understood that when we speak of any Chapter in
Laplace's work without further specification, we always mean a
Chapter of Livre IL

The first Chapter is entitled Principes generaux de cette TMorie.
This occupies pages 179 — 188 ; it gives a brief statement, with
exemplification, of the first principles of the subject.

970. The second Chapter is entitled De la ProbcibiliU des
^venemens composes d'Svenemens simples dont les possihiliUs respec-
tives sont donnees. This occupies pages 189 — 27-i ; it contains the
solution of several problems in direct probability ; we will notice
them in order.

971. The first problem is one connected with a lottery ; see
Arts. 291, 44^8, 4<d5, 775, 864, 910.

The present discussion adds to what Laplace had formerly
given an approximate calculation. The French lottery was com-
posed of 90 numbers, 5 of which were drawn at a time. Laplace
shews that it is about an even chance that in 86 drawingfs all
the numbers will appear. This approximate calculation is an
example of the formula for AV given by Laplace on page 159 of
his work ; see Art. 966.

We may remark that Laplace also makes use of a rougher ap-
proximation originally given by De Moivre ; see Art. 292.

972. On his page 201 Laplace takes the problem of odd and
even; see Arts. 350, 865, 882.

Laplace adds the following problem. Suppose that an urn con-
tains X white balls, and the same number of black balls ; an even
number of balls is to be drawn out : required the probability that
as many white balls as black balls will be drawn out.

The whole number of cases is found to be 2"^"^ — 1, and the

528 LAPLACE.

whole number of favourable cases to be -^= 1 ; the required

X \x
probability therefore is the latter number divided by the former.

973, The next problem is the Problem of Points. Laplace
treats this very fully under its various modifications ; the dis-
cussion occupies his pages 203 — 217. See Arts. 872, 884.

We will exhibit in substance, Laplace's mode of investigation.
Two players A and B want respectively x and y points of winning
a set of games ; their chances of winning a single game are jp and
q resj^ectively, w^here the sum of "p and q is unity ; the stake is to
belong to the player who first makes up his set : determine the
probabilities in favour of each player.

Let </) {x^ y) denote ^'s probability. Then his chance of win-
ning the next game is p, and if he wins it his probability becomes
</) (a? — 1, y) ; and q is his chance of losing this game, and if he loses
it his probability becomes ^{x, y — 1) : thus

^ {x, y) =p ^ {x -1, y) -\- q (\> {x, y -1) (1).

Suppose that </> [x, y) is the coefficient of fr^ in the develop-
ment according to powers of t and r of a certain function u of
these variables. From (1) we shall obtain

u-t<i>(x, 0)f-t^{0,y) T^+^(0, 0)

==u(pt + qT)-pttcl>(x,0)f-qTtcl^{0,7j)T' (2),

where ^ cj) (x, 0) f denotes a summation with respect to x from
X = 0 inclusive to x= oo ; and X (/> (0, y) r'" denotes a summation
with respect to y from ?/ = 0 inclusive to y= cc . In order to shew
that (2) is true we have to observe two facts.

First, the coefficient of any such term as Tr", 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 7i 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

u — ,

1 —j)f — qr

LAPLACE. 529

wc may write this result thus,

Avhere i^(^) and /(r) are functions of t and r respectively, which

are at present undetermined. By supposing that the term in /(t)

vv'hich is independent of t is included in F{t), we may write the

result thus,

„ = %IO±ltM (4)_

1 — 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
sjDecial 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) = 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 (j) (x, 0) is the coefficient of f in the development

of ^^^ ; therefore v (t) = 0.

Again, A wins if he first makes up his set, so that cj) (0, y) = 1
for every value of i/ from unity upwards. But from (4) it follows

that (j) (0, y) is the coefficient of t^ in the development of r-^

so that

Tyjr (t) _ T

therefore

Thus finally

qr 1

-t'

T^fr (t)

t(1.
1-

•

- T

■t(1-

qr)

{l-T){l-pt-qT)

Now <^{x,y) is the coefficient of fr' in the development of w.
First expand according to powers of t ; thus we obtain for the

530 LAPLACE.

«.

p T

coefficient of f the expression _ J^ .^ _ y . Then expand

this expression according to powers of r, and we finally obtain for
the coefficient of fj^

This is therefore the probability in favour of A ; and that in

favour of JB may be obtained by interchanging p with q and x

with ?/.

The result is identical with the second of the two formuloe

which we have given in Art. 172.

97*i. The investigation just given is in substance Laplace's ;

he takes the particular case in which p = -^ and q = ^', but this

makes no difference in principle. But there is one important
difference. At the stage where we have

F{t) +/(t)

u =

1 —pt — qr

Laplace puts

fir)

u =

1 —jyi — 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-
lectinsc 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 x^ i^oints, the second x^ points, the third x^ points,
and so on. Let their respective chances of winning a single game
^^ 1\' 2\> 1\^ • ' • Let cj) (x^, x^, x^, . . .) denote the probability in
favour of the first player. Then, as in Art. 973, we obtain the
equation

LAPLACE. 531

Suppose that cj) {x^, x^, x^, ...) is the coefficient of ?f/i t^'^^ts^'s ...
in the development of a function u of these variables. Laplace
then proceeds thus. From (1) he passes to

?^ = w(M+M + M+---) (2)'

and then he deduces

i=M+M+M + O^)-

Hence

therefore

, ^1 (^1 + 1)

1.2

(M + M+--0'

cc, (a?, + 1) (x, + 2) , , . .

Now the coefficient of t^t^^^t^^... in — ^ is ^ (a^^, ajg, cCg, ...)•

Let 7i:w;:)^*i i^"* t^^ . . . denote any term of the right-hand member
of the last equation. Then the coefficient of t^ t,^2 t^a ... in this
term will be hp^i <f> (0, x^—m, x^—7i,...). But cj) (0, x^— m, x^—n,...)
is equal to unity, for if the first player wants no points he is en-
titled to the stake. Moreover we must reject all the values of
<f> if), x^ — m, x^ — n, ...) in which m is equal to or greater than x^^
in which n is equal to or greater than x^, and so on; for these
terms in fact do not exist, that is must be considered to be zero.
Hence finally

^ (a?j, OJg, a-g, . . .) =2\^' |l + a?, {p^ +793 + . . .)

* !■

34^—2

532 LAPLACE.

provided we reject all terms in wliich the power of p^ surpasses
x^ — 1, in which the power oi p^ surpasses x^ — 1, and so on.

Now on this process of Laplace's we remark : ' .

First, the equation (2) is not true ; as in Art. 973 we ought to
allow for terms in which one or more of the variables x^,x^,x^,...
is zero ; and therefore additional terms ought to be placed in each
member of equation (2) of the present Article, like those in equa-
tion (2) of Article 973.

Secondly, Laplace's treatment of his equation (3) is unintel-
ligible, as we have already remarked in a similar case ; see
Art. 952. By making use of the Calculus of Operations we might
however translate Laplace's process into another free from ob-
jection.

976. At this stage we shall find it convenient to introduce an
account of the fourth Supplement to the Theorie.,.des Prohahilites.
This supplement contains 28 pages. Laplace begins with a few
remarks on Generating Functions; he gives the correct formula
for the solution of an equation in Finite Differences for which he
had formerly given an incorrect formula: see Art. 955. He does
not refer to the Theorie...des Froh. nor take any notice of the
discrepancy of the two formulae. He says, on page 4 of the Sup-
plement,

Un des principaux avantages de cette maniere d'integrer les equa-
tions aux differences partielles, consiste en ce que I'analyse algebrique
fournissant divers moyens pour developper les fonctions, on peut choisir
celui qui convient le mieux a la question proposee. La solution des
]>roblemes suivans, par le Comte de Lai^lace, mon fils, et les considera-
tions qu'il y a jointes, repandront un nouveau jour sur le calcul des
fonctions generatrices.

We have therefore to ascribe all the rest of the fourth Sup-
plement to Laplace's son.

977. The main part of the fourth Supplement consists of the
solution of problems which may be considered as generalisations of
the Problem of Points. There are three of these problems ; we
will enunciate them.

LAPLACE. 533

I. A player A draws a ball from an urn containing white
balls and black balls ; his chance of drawing a white ball is p,
and his chance of drawing a black ball is q : after the ball has
been drawn it is replaced. Then a second player B draws a ball
from a second urn contairing white balls and black balls; his
chance of drawing a white ball is p, and his chance of drawing
a black ball is q : after the ball has been drawn it is replaced.
The two players continue thus to draw alternately a ball, each
from his own urn, and to replace the ball after it has been
drawn. If a player draws a white ball he counts a point ; if he
draws a black ball he counts nothing. Suppose that A wants x
points, and B wants x points to complete an assigned set, required
the probabilities in favour of each player.

II. Suppose A draws from an urn in which there are balls
of three kinds ; for a ball of the first kind he counts two points, for
a ball of the second kind he counts one point, and for a ball of the
third kind he counts no point: let his chances he p>,p)^, and q for
the three cases.

Similarly let B draw from a second urn containing similar
balls ; let^', j9j', and q be his chances for the three cases. Then,
as before, we require the probabilities for each player of his
making up an assigned set of points before his adversary makes
up an assigned set.

III. An urn contains a known number of black balls, and a
known number of white balls ; a ball is drawn and not replaced ;
then another ball, and so on : required the probability that a
given number of white balls will be drawn before another given
number of black balls.

These three problems are solved by the method of Generating
Functions used carefully and accurately ; that is, the terms which
are required to make the equations true are given, and not
omitted. See Art. 97^. After the problems are solved generally
particular cases are deduced.

The student of the fourth Supplement will have to bear in
mind that in the first problem p + q = l and p?' + 2=1* ^^^ ^^
the second problem p +p)^ + ^ = 1, p' +p^ -\- q =1,

534 LAPLACE.

978. After the solutions of these problems we have a few
pages headed Remarque sw les fonctions generatrices ; and this is
the part of the fourth Supplement with which we are chiefly
interested. It is here observed that in a case like that of our
Art. 975; the equation (2) is not an accurate deduction from equa-
tion (1) ; for additional terms ought to be added to each side, in
the manner of our Art. 973.

There is however a mistake at the top of page 24 of the fourth
Supplement : instead of adding a function of t, two functions must
be added, one of t and the other of t'.

The fourth Supplement then proceeds thus, on its page 24 :

Faute d' avoir egard a ces fonctions, on j^eut tomber dans des
erreurs graves, en se servant de ce moyen pour integrer les equations
aux differences partielles. Par cette meme raison, la marche suivie dans
la solution des problemes des n"^^ 8 et 10 du second livre de la Theorie
analytique des Probabilites n'est nullement rigoureuse, et semble impliquer
contradiction, en ce qu'elle etablit une liaison entre les variables qui
sont et doivent etre toujours independantes. Sans entrer dans les
considerations particulieres qui ont pu la faire reussir ici, et qu'il est
aise de saisir, nous allons faire voir que la metliode d'integration ex-
posee au commencement de ce Supplement s'applique egalement a ces
questions, et les resout avec non moins de simplicite.

The problem referred to as contained in No. 8 of the
TMorie...des Proh. is that which we have given in Art. 975;
the problem referred to as contained in No. 10 of the Theorie... des
Proh. is that which we shall notice in Art. 980. The fourth
Supplement gives solutions of these problems by the accurate use
of Generating Functions, in the manner of our Art. 973.

Thus as Laplace himself attached the fourth Supplement to
his work, we may conclude that he admitted the solutions in
question to be unsound. We consider that they are unsound, and
in fact unintelligible, as they are presented by Laplace ; but on
the other hand, we believe that they may be readily translated
into the language of the Calculus of Operations, and thus become
clear and satisfactory. See Art. 952.

979. We return from the fourth Supplement to the
Theorie... des Proh. itself. Laplace's next problem is that which

LAPLACE. 535

is connected with the game which is called Treize or Rencontre ;
see Arts. 162, 280, 286, 430, Q±Q.

Laplace devotes his pages 217 — 225 to this problem ; he gives
the solution, and then applies his method of approximation in
order to obtain numerical results when very high numbers are
involved.

980. Laplace takes next on his pages 225 — 238 the problem
of the Duration of Play. The results were enunciated by De
Moivre and demonstrated by Lagrange ; Laplace has made great
use of Lagrange's memoir on the subject ; see Arts. 311, 583,
588, 863, 885, 921. We may observe that before Laplace gives
his analytical solution he says, Ce probleme pent etre resolu
avec facilite par le precede suivant qui est en quelque sorte,
mecanique ; the process which he gives is due to De Moivre ;
it occurs on page 203 of the Doctrine of Chances. See also
Art. 303. In the course of the investigation, Laplace gives a
process of the kind we have already noticed, which is criticised in
the fourth Supplement ; see Art. 978.

981. Laplace takes next on his pages 238 — 2-17 the problem
which we have called Waldegrave's problem ; see Arts. 210, 249,
295, 348.

There are n-\-l players C^, G^, ... (7„^j. First C^ and Cg play
together ; the loser deposits a shilling in a common stock, and the
winner plays with C^ ; the loser again deposits a shilling, and the
winner plays with C^\ the process is continued until some one
player has beaten in succession all the rest, the turn of C^ coming
on again after that of (7„^j. The winner is to take all the money
in the common stock.

Laplace determines the probability that the play will terminate
precisely at the x^^'^ game, and also the probability that it will
terminate at or before the ic^'* game. He also determines the
probability that the r^^ player will win the money j)recisely at the
x^^ game ; that is to say, he exhibits a complex algebraical func-
tion of a variable t which must be expanded in powers of x
and the coefficient of t'' taken. He then deduces a general ex-
pression for the advantage of the r*^ player.

The part of the solution which is new in Laplace's discussion

536 LAPLACE.

is that which determines the probability that the r^^ player will-
win the money precisely at the x^^ game ; Nicolas Bernoulli had
confined himself to the probability which each player has of
winning the money on the whole.

982. We will give, after Laplace, the investigation of the
probability that the play will terminate precisely at the x^^
game.

Let z^ denote this probability. In order that the play may
terminate at the x^^ game, the player who enters into play at the
{x — n-\-Vf^ game must win this game and the n — 1 following
games.

Suppose that the winner of the money starts with a player
who has won only one game ; let P denote the probability of this

P .

event ; then — will be the corresponding probability that the

play will terminate at the x^^ game. But the probability that the
play will terminate at the {x — iy^ game, that is z^_^, is equal

P . .

to ^Tpi . For it is necessary to this end that a player who has

already won one game just before the {x — n + Vf^ game should
win this game and the n — ^ following games j and the probabilities

of these component events being respectively P and ^^^^i , the

. P

probability of the compound event is ^^^ . Thus

P 1

— V •

and therefore ^ z^_^ is the probability that the play will terminate

at the fl?*^ game, relative to this case.

Next suppose that the winner of the money starts with a player
who has won two games ; let P' denote the probability of this

event ; then -^ will be the corresponding probability that the play

will terminate at the x^"^ game. And -^^ = z^_^ : for in order that

the play should terminate at the (ic— 2)"* game it is necessary that
a player who has already won two games just before the (a; — 72+ 1)"'

LAPLACE. 537

game should win this, game and the n — 2 following games. Thus

F' 1

and therefore ^ z^_^ is the probability that the play will terminate

at the ic*^ sfame relative to this case.

By proceeding thus, and collecting all the partial probabilities
we obtain

1 1 )_ 1 n\

Suppose that z^ is the coefficient of f in the expansion accord-
ing to powers of ^ of a certain function u of this variable. Then
from (1) we have, as in Art. 937,

F(t)

u =

111 1

-*• 2 " 92 '' 2^ * " 2'*"'^

where -F (^) is a function of t which is at present undetermined.

Now if (1) were true for x = n SiS well as for higher values of
n, the function F(t) would be of the degree n — 1. But (1) does
not hold when x — n, for in forming (1) the player who wins the
money was supposed to start against an opponent who had won
one game at least ; so that in (1) we cannot suppose x to be less
than n-\-l. Hence the function F [t) will be of the degree n,
and we may put

a^ + ait + ctf + . • . + a.f'

u =

111 1

Now the play cannot terminate before the ?i*'' game, and the pro-
bability of its terminating at the n^^ game is ^^^ ; therefore a^

vanishes for values of x less than n, and a,, = — — • . Thus

538 LxlPLACE.

The coefficient of f in the expansion of u in powers of t gives
the probabiHty that the play will terminate at the x^^ game.

The probability that the play will terminate at or before the
x^^ game will be the sum of the coefficients of f and of the inferior
jDOwers of t in the expansion of u, which will be equal to the co-

efficient of f in the expansion of- ; that is, it will be the co-
efficient of f in the ex^Dansion of

1 f (2 - t)

This expression is equal to

1 r(2-o f f f r \

¥ {l-tf \ T (1 - 0 "^ 2'^' (1 - tf 2^''(l-/)-^'^ ""J '
The T^^ term of this development is

(- 1)*--^ (2 - 1) r

9r/i,

r+l >

that is

( 1) l^™-! (^1

ty^^ ^rn ^^ _ ^y^lj •

The expansion in powers of t of this r* term may now be
readily effected ; the coefficient of f will be

f 1 \ic-\-r — rii \ \x -\-r — rii — 1
(- ) I 2^^ \x-rn\r ~ W' \x-rn-\ [T j '

/ 2y-i \x -\-r — rn — 1

that is ^-TTri— , {x — rn + 2r).

^"^"^ x — rn\r ^ ^

The final result is that the probability that the play will termi-
nate at or before the x^^ game, is represented by as many terms
of the following series as there are units in the integer next

below - :
n

1 2 3~2^ ^ - o;i + uj — . . .

LAPLACE. 539

The sum of the coefficients of every power of t up to infinity
in the expansion of u will represent the probability that the play
will terminate if there be no limit assigned to the number of games.
But the sum of these coefficients will be equal to the value of i6
when t is made equal to unity ; and this value of u is unity. Hence
we infer that the probability of the termination of the play may
be made as near to unity as we please by allowing a sufficient
number of games.

983. In Laplace's own solution no notice is taken of the fact
that equation (1) does not hold for x = n. Professor De Morgan
remarks in a note to Art. 52 of the Theory of Probabilities in the
EnciJclopcBdia Metropolitana,

Laplace (p. 240) has omitted all allusion to this circumstance ; and
the omission is highly characteristic of his method of writing. ISTo one
was more sure of giving the result of an analytical process correctly, and
no one ever took so little care to point out the various small considera-
tions on which correctness depends. His Theorie cles Probabilites is by
very much the most difficult mathematical work we have ever met
with, and prmcipally from this circumstance : the Mecanique Celeste has
its full share of the same sort of difficulty; but the analysis is less intri-
cate.

984. We may observe that as Laplace continues his discussion
of Waldegrave's problem he arrives at the following equation in
Finite Differences,

i/r, X yr-\ , x-1 "f" ~e)n U i\ x-n ^ j

in integrating this, although his final result is correct, his process is
unsatisfactory, because it depends upon an error we have already
indicated. See Art. 955.

985. Laplace's next problem is that relating to a run of
events which was discussed by De Moivre and Condorcet ; see
Arts. 325, G77 : this problem occupies Laplace's pages 247 — 253.

Let 2^ denote the chance of the happening of the event in a
single trial ; let </> {x) denote the probability that in x trials the

540 LAPLACE.

event will happen i times in succession. Then from equation (1)
of Art. 678 by changing the notation we have

^ (x) ^p'+p'-' (1 -p) cj, (x-i) -\-p'-' (1 -p) ct>{x-i-^ 1) + ...

,..+p{l-p)c^{x-2) + (l^p)<t>(x-l) (1).

Laplace takes z^ to denote the probability that the run will
finish at the x^^ trial, and not before ; then he obtains

^x = (1 p) 1^.-1 + P^^, +P\-, + . . . + p"' ^x- j (2).

"VVe may deduce (2) thus ; it is obvious that

z^ = (f>{x)-(f){x-l);

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 x^^ trial the (x—iy^ trial must have been unfavourable, and
the following i trials favourable. Laplace then makes ^ distinct
cases.

I. The (x — i— \y^ trial unfavourable.

IL The {x-i-iy^ favourable; and the (a;-f-2y^ un-
favourable.

Ill The [x-i- \y^ and the {x - i- 2)"^ favourable, and the
{x — {— 3) ''^ unfavou rable.

IV. The {^x-i-Vf\ the {x-i-2f\ and the {x-i-Zf
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 [x — 4)*^ trial.

Now the part of z^ which arises from this case IV. is FJ(\. —p) j/\
for we require an unfavourable result at the {x — if^ trial, of

LAPIACE. 541

which the chance is 1—p, and then a run of t. Thus the part
of z^ is

^(1-i^)/^ or^/(l-^;)^,_,.

We have said that Laplace adopts nearly the method we have
given ; but he is rather obscure. In the method we have given
P^ denotes the probability of the following compound event : no
run of i before the (ic — /— 4)^^ trial, the (a?— ^— 4)^^ trial un-
favourable, and then the next three trials favourable. Similarly
our Pg would denote the probability of the following compound
event; no run of i before the (a? — ^ — 2)^^ trial, the {x—i—Tf^
trial unfavourable, and the next trial favourable. Laplace says,
Nommons P' la probability qu'il n'arrivera pas au coup x — i—%
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 Pj if we omit the last of the three clauses
which form our account of the probability represented by Pg ; so
that in fact pP' with Laplace denotes the same as F^ 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 run of i 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 d^^t^o, with what we have griven
in Art. 680, after Condorc3t.

Laplace then supplies some new matter, in which he considers
the expectation of each player supposing that after failiug 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 71 + 1 balls marked respectively 0, 1, ... w ; a ball is drawn
and replaced : required the probability that after i drawings the
sum of the numbers drawn will be s. This problem and applica-
tions of it occupy pages 253 — 261. See Arts. 888, 915.

The problem is due to De Moivre ; see Arts. 149, 364. La-
place's solution of the problem is very laborious. We will pass to

542 LAPLACE.

the application which Laphxce 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 x^ in the expansion of

— ^ (1 - ic^^y (1 - xY\

Thus we obtain for the required probability
1 {\i+s-l i U*+s-n-2

[n-^VfMi-l \s 1 [i:il Lf^i^^ij^l

i[i-\) h' + g-2;2~3

1.2 ^-l 5-2^

If the balls are marked respectively 0, Q, 2^, 8^, ,.,nd, this
expression gives the probability that after i drawings the sum of
the numbers drawn will be sQ.

Now suppose 0 to become indefinitely small, and n and 5 to
become indefinitely great. The above expression becomes ulti-
mately

i-\ \\nl \\n 1 ^ 1.2 \n j '" n

s 1

Let - be denoted by x, and - by dx, so that we obtain

n " ' n

this expression may be regarded as the conclusion of the follow-
ing problem. The numerical result at a single trial must lie
between 0 and 1, and all fractional values are equally probable :
determine the probability that after { trials the sum of the results
obtained will lie between x and x + dx^ where dx is indefinitely
small.

Hence if we require the probability that after i trials the sum
of the results obtained will lie between x^ and x^, we must inte-

LAPLACE. 543

grate the above expresBion between the limits x^ and 0*^, ; thus
we obtain

i|-.;-|K-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 { are positive.

We might have obtained this result more rapidly by using
Art. 3G4 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 914187
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 0 and "914187 of a right
angle. By the preceding expression we obtain for the result

-^ (-914187)'', that is about '00000011235.

Speaking of this probability, Laplace says :

... Elle est deja tres-petite ; mais il faut encore la combiner avec
la probability d'une circonstance tres-remarqiiable dans le systeme du
monde, et qui consiste en ce que toutes les planetes se meuvent dans le
meme sens que la terre. Si les mouvemens directs et retrogrades sont

supposes ^galement possibles, cette demiere probabilite est ( ) ; il

/IV"
faut done multiplier '00000011235 par f- ) , pour avoir la probabilite

que tous les mouvemens des planetes et de la terre seront diriges dans le

meme sens, et que la somme de leurs inclinaisons a I'orbite de la terre,

2-0972
sera comprise dans les limites zero et 91*''4187; on aurn ainsi jjt^i^

pour cette probabilite ; ce qui donne 1 — ^,io pour la probabilite que

cela n'a pas du avoir lieu, si toutes les inclinaisons, ainsi que les mouve-
mens directs et retrogrades, ont ete egalement faciles. Cette probabilite

544) LAPLACE.

approclie tellement de la certitude, que le resultat observe devient
invraisemblable dans cette liypothese ; ce resultat indique done avec
une tres-grande probabilite, I'existence d'une cause primitive qui a deter-
mine les mouvemens des planetes a se rapprocber du plan de I'ecliptique,
ou plus naturellement, du plan de I'equateur solaire, et a se mouvoir
dans le sens de la rotation du soleil...

Laplace then mentions other circumstances which 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 ground 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 Exposition de
la Theorie 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 Proh page 802.

But on the other hand two most distinguished philosophers have
recorded their dissatisfaction ; see Professor Boole's Laws of
Thought, page 864, and a note by K. L. Ellis in The Works of
Francis Bacon ... Vol L 1857, page 848.

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 quantites variables et positives t, t^,...ti_^ dont la somme soit
s, et dont la loi de possibilite soit connue ; on propose de trouver la
somme des produits de cliaque valeur que pent recevoir une fonction
donnee i(/(t, t^, t^, (fee.) de ces variables, multipliee par la probabilite
correspondante a 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 lie
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 him 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 z^ denote the first ordinate, let z^ + z,^ denote the second,
^1 + ^2+ ^3 ^^^ third, and so on : thus z^, z^, z^, ... z^ are all posi-
tive variables, and since the sum of the ordinates is s we have

nz^-^{n-l)z^-\- (n-2) z^+ ... +z,, = s (1).

The mean value of the r*^ ordinate will be

{z^-{-z^+ ... +Zr) dz^dz^,.. dz^

ctz^ (^z~ . . . CIZ^^

where the integrations are to be extended over all positive values
of the variables consistent with the limitation (1).

Put nz^ = x^, (ii — l)z^ = x^, and so on. Then our expression
becomes

... p + — ^H--^ + ... -\ ^^^--^]dx^dx.-^...dx,,

JJJ \n 71 — 1 71 — z 7i — r-\-l/

... CvX^ (JuJUn ... Ct/ll/j^

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

s (1 1 1 ,11

n \n 71 — 1 71- z 7i — 7'-\-l)

54i6 LAPLACE.

For let us suppose that instead of (2) we have the condition
that x^ + x^+ ... + Xn shall lie between s and s + As. Then by the
theorem to which we have just referred we have

m 1 2 n [72 + 1

and \\\ dx^dx^,»,dxn= ;

Hence by division we obtain

///

• • • Xnn CLX., (a/X„ • • • UitL',

jj^l^Oyj^U/tAy^ ••• ix/t*^„ f A Nn+1 W+1

(5 + AsY^"- - s'

• • • a*X/^ Ci/tO^ • • • Cvt^Aj
The limit of this expression when As is indefinitely diminished

///■

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

1 fl 1 1

n In n — 1 n — 2 n

-r + 1)

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 will 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 ^1, t^, ... tn 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 given ; 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 r*^ candidate will be

///••

vv CtL^ (^^n . • . Cvt',

• • • etc, ^^o • • • ^^n

The integrations are to be taken subject to the following con-
ditions : the variables are to be all positive, a variable ^^ 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 obtained. We may obtain it thus.

X Ut t,i = X^, fn_^ = ^n + ^n-i? ^n-2 ~ ^n-i + ^n-2> '" l tneU the

above expression for the mean value becomes

III • • • \p^n ~r" ^n—i i • • • "i~ "^r) ^«^i CiOS^ ... CIX^

with the condition that all the variables must be positive, and
that iCj + cCg + ••• + ^n i^^st not be greater than a. Then we may
shew in the manner of the preceding Ai'ticle that the result is

{n — r-VV) a

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—\ 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'election serait sans doute le meilleur, si des considerations
6ti'angeres au merite n'influaient point souvent sur le clioix des elec-
teurs, meme les plus honnetes, et ne les determinaient point a placer
aux derniers rangs, les candidats les plus redoutables a celui qu'ils pre-
ferent; ce qui donne un grand avantage aux candidats d'un merite
mediocre. Aussi I'experience I'a-t-elle fait abandonner aux etablissemens
qui I'avaient adopte.

It would be interesting to know where this mode of managing
elections bad 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-
hiliU, qui resultent de la midtiplication indefinie des evenemens : it
occupies pages 275 — 303.

993. The first problem is that which constitutes James Ber-
noulli's theorem. We will reproduce Laplace's investigation.

The probability of the hap]3ening 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 = 1 — p and yu, = m + r^ ; then the probability that the
event will happen m times and fail w times in fi trials is equal to
a certain term in the expansion of (/; + q^, namely

\m\n -^

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 yup and y^q
respectively. We say as nearly as possible, because jbLp is not

LAPLACE. 549

necessarily an integer, while m is. We may denote the value of
m by \xp + z, where z is some proper fraction, positive or negative ;
and then n = ^q — z.

The r^^ term, counting onwards, in the expansion of {p-\-q)'^

after -^p^q- is ^ , p"-y^

\m \n

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

m — r \/(27r) [ 12(/?i — r)

^ = (?i + r)-"-'-^' e"-''- -7^ [1 - ^

• t • I t

n + r ^ ' sl^'rr) \ 12 (?i + ?-)

We shall transform the term {in — r)*"""^"^. Its logarithm is

and wri-^U-'" '•' ''

We shall suppose that r^ does not surpass //, in order of mag-
nitude, and we shall neo^lect fractions of the order - ; we shall

thus neglect such a term as — ^, because vi is of the order //-.
Thus we have approximately

^r - m - 2) jlog w + log ^1 - ^J I

/ 1\ , r r^ r^

\^ 2/ ° , 2«i 2??i o»i

and then, passing from the logarithms to the numbers,

'-^ = m'--'"-^ 6*^2- ( 1 + -— _ — -J .

(.. - r)— ^ = m-- e-2-^ 1 + ^^ - ^.)

550 LAPLACE.

Similarly

Thus we have approximately

"7r-,+

,-r+.^^«+r+| JI9rrA 1 2ww 6m^ G^l'

Now suppose that the values of m and w are those which we
have already assigned as corresponding to the greatest term of
the expansion of (i? + ^)^ then

171 — 2 n + z

thus we have approximately

m-r n+r^ U + ^^ •

t ^ ^^ \ ran) ,

Therefore finally we have approximately for the r*^ term after
the greatest

tji^irmri) \ mn 2mn 6m^ 6n^

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

2^/ll .El

i^i^irmn)

If we take the sum of the values of this expression from r = 0
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 2^ + 1 terms of a binomial expansion which include
the greatest term as their middle term.

Now by Euler's theorem, given in Art. 33i,

LAPLACE. 551

2 J II -Ell

Here y = ——-!-— -e 2""» and the differential coefficients of v

with respect to r will introduce the factor -—- — , and its powers ;

and ^-— is of the order —j- 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,

where both the symbols S and I are supposed to indicate opera-

tions commencing with «• = 0, and ^ Y denotes the gi'eatest term

of the binomial expansion, that is the value oi ^y when r = 0.

The expression ^y 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

jydr-v^y + -^Y\

subtract the greatest term of the binomial, and thus we have

ydr-^^y.

/y kI 11,

Put T = ,.^ . ; thus we obtain finally

2 f ... 7, V/^

-2

This expression therefore is the apj^roximate value of the sum of
2r + 1 terms of the expansion of {p + qY, 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 — 'T and m + r, both inclusive, that is between

T \/(2mn) T , , r\/(2mn)
^^^.^^_V^__and/x^ + ^4--^^;

552 LAPLACE.

or, in other words, the 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

p^ ^^ ^. and p+ -+ -, — .

If fi be very large we may neglect z in comparison with fip or fj.q ;
and then onn = /Ji^pq approximately, so that we obtain the following
result ; If the number of trials, /jl, be very large, the probability
that the ratio of the number of times in which the event happens
to the whole number of trials will lie between

e-f^dt + ttJ^ ^e-^

\l'rr Jo Aj{27rfipq)

994. The result which has just been obtained is one of the
most important in the whole range of our subject. There are two
points to be noticed with respect to the result.

In the first place, it is obvious that supposing r to be constant
we may by sufficiently increasing jjl render the limits

as close as we please, while the corresponding probability is always
greater than —i— e *' dt

2 C"^ _
In the second place, it is known that the value of —r- j e^'' dt

^ NTT Jq

approaches very near to unity for even moderate values of r.
Tables of the value of this expression will be found in the works
of Professor De Morgan cited in Arts. 268 and 485, and in that of
Galloway cited in Art. 753. The following extract will sufficiently
illustrate the rapid approach to unity: the first column gives
values of t, and the second column the corresponding values of the

expression — — I e'^^ dt.

YttJo

LAPLACE.

•5

•5204999

•8427008

1-5

•9661052

2 0

•9953223

2-5

•9995930

3-0

•9999779

553

995. With respect to the history of the result obtained in
Art. 994, we have to remark that James Bernoulli began the
investigation ; then Stirling and De Moivre carried it on by the aid
of the theorem known by Stirling's name ; and lastly, the theorem
known by Euler's name gave the mode of expressing the finite
summation by means of an integral. See Arts. 123, 334, 335, 423.
But it will be seen that practically we use only the first term
of the series given in Euler's theorem, in fact no more than
amounts to evaluating an integral by a rough approximate quadra-
ture. Thus the result given by Laplace was within the power of
mathematicians as soon as Stirling's Theorem had been published.

Laplace, in his introduction, page XLII, speaking of James
Bernoulli's theorem says,

Ce theoreme indique par le bon sens, 6tait difficile a demontrer par
I'Analyse. Anssi I'illustre geometre Jacques Bernoulli qui s'en est
occup6 le premier, attachait-il une grande importance a la demonstra-
tion qu'il en a donnee. Le calcul des fonctions generatrices, applique
a cet objet, non-seulement d^montre avec facilite ce theoreme ; mais de
phis il donne la probabilite que le rapport des evenemens observes, ne
s'ecarte que dans certaines limites, du vrai rapport de leurs possibilites
respectives.

Laplace's words ascribe to the theory of generating functions
the merit which should be shared between the theorems known
by the names of Stirling and Euler.

We may remark that in one of his memoirs Laplace had used
a certain process of summation not connected with Euler's
theorem : see Art. 897.

996. Laplace gives the following example of the result ob-
tained in Art. 993.

554i LAPLACE.

Suppose that the probability of the birth of a boy to that of
the birth of a girl be as 18 to 17 : required the probability that
in 14000 births the number of boys will fall between 7363 and
7037.

Here

i?=^, q = ^, m=7200, n = 6S00, r=163:

the required probability is '994303.

The details of the calculation will be found in Art. 74 of the
TJieory of Prohdbilities in the EncyclopcEdia MetropoUtana,

997. We have now to notice a certain inverse application
which Laplace makes of James Bernoulli's theorem : this is a
point of considerable importance to which we have already alluded
in Art. 125, and which we must now carefully discuss.

In Art. 993 it is supposed that p is given, and we find the
probability that the ratio of the number of times in which the
event happens to the whole number of trials will lie between
assigned limits. Suppose however that p is not known a pi^oriy
but that we have observed the event to happen m times and to
fail n times in fi trials. Then Laplace assumes that the expression

given in Art. 993 will be the probability that p— — lies be-

tween

• ^^/(2mn) ^^^ ^ T>sJ{2mn) ^

that is, Laplace takes for this probability the expression

i-\\-i^di^ ,,y^ ,6-- (1).

He draws an inference from the formula, and then says, on
his page 282,

On parvient directement ^ ces resultats, en considerant p comme
une variable qui pent s'etendre depuis zero jusqu'a I'unite, et en deter-
minant, d'apres les evenemens observes, la probabilite de ses diverses
valeurs, comme on le verra lorsqne nous traiterons de la probabiUte des
causes, deduite des evenemens observes.

LAPLACE. 555

Accordingly we find that Laplace does in effect return to the
subject ; see his pages 363 — 366.

In the formula which we have given in Art. 697, suppose
a= 0, and 5 = 1; then if the event has been observed to happen
m times and to fail n times out oi m + n trials, the probability that
the chance at a single trial lies between a and yS is

J a

I x'^il-xydx
Jo

-f m T \/(2m7i) ^ m T \/(2mn)
Let a = , p = — 1 — ,

where fjL = m + n; then we shall shew, by using Laplace's method
of approximation, that the probability is nearly

[V«V^ (2).

VttJo

For with the notation of Art. 957 we have ?/ = £c"'(l -xY;
the value of x which makes ^ a maximum is found from the
equation

m n

= 0,
SO that a =

m + n
Then

e = log

^ {a-\-ey{i-a-ey

~2 K+(l-a)^J ~3 I? ■"(13^1 + -
Thus, approximately,

2 _ 1/ ) "t It, ] 6"

2 ja" ' (l-O 2»m

556 LAPLACE.

Therefore

x'^il -xydx Y e-^'df

jx'^il-xydx yJ e-t'dt
= -r- I e'^'dt= -J-- I V^ dt.

"We have thus two results, namely (1) and (2) : the former is
obtained by what we may call an assumed inversion of James
Bernoulli's theorem, and the latter we may say depends on Bayes's
theorem. It will be seen that the two results are not quite con-
sistent ; the difference is not practically very important, but it is
of interest theoretically.

The result (2) is in effect given by Laplace on his page 366 ;
he does not however make any remark on the difference between
this result and that which we find on his page 282.

On page 209 of his RecJier cites... sur la Proh. Poisson gives the
result (1) which he obtains by the same assumption as Laplace. But
on his page 213 Poisson gives a different result, for he finds in effect
that the probability that the chance at a single trial lies between

m V \/{2mn) , w [v -{■ dv) \J{2'mn)
— — - and '.

is Vdv,

where F=-r- e"^' ^7;:^ ^-e"^' (3).

This is inconsistent with Poisson's page 209 ; for if we take the
integral j Vdv between the limits — t and + t for v it reduces

to — / e~^ dt, so that we arrive at the result (2), and not at the

result (1). It is curious that Poisson makes no remark on the dif-
ference between his pages 209 and 213 ; perhaps he regarded his
page 209 as supplying a first approximation, and his page 213 as a
more correct investigation.

Poisson's result (3) is deduced by him in his RecheTches...sxir la
Proh. from the same kind of assumption as that by which he and

LAPLACE. 557

Laplace arrived at the result (1) ; but the assumption is used in
such a way as to diminish very decidedly the apprehension of any
erroneous consequences : the assumption, so to speak, is made to
extend over an indefinitely small interval instead of over a finite
interval.

Poisson had however previously considered the question in his
Memoire sur la proportion des naissances des deux sexes; this
memoir is published in the Memoires...de VInstitut, Yol. ix, 1830 ;
there he uses Bayes's theorem, and proceeds as we have done in
establishing (2), but he carries the approximation further: he
arrives at the result (3). See page 271 of the memoir.

Thus the result (3) is demonstrable in two ways, namely, by
the assumed inversion of James Bernoulli's theorem, and by
Bayes's theorem. As Poisson in his latest discussion of the ques-
tion adopted the inversion of James Bernoulli's theorem, we may
perhaps infer that he considered the amount of assumption thus
involved to be no greater than that which is required in the use of
Bayes's theorem. See Art. 552.

In a memoir published in the Cambridge Philosophical Trans-
actions, Vol. VI. 1837, Professor De Morgan drew attention to the
circumstance that Laplace and Poisson had arrived at the result (1)
by assuming what we have called an inversion of James Bernoulli's
theorem ; and he gave the investigation Avhich, as we have said,
depends on Ba3^es's theorem. Professor De Morgan however over-
looked the fact that Laplace had also implicitly given the result
(2), and that Poisson had arrived at the result (3) by both
methods. It will be found on examining page 428 of the volume
which contains Professor De Morgan's memoir, that his final
result amounts to changing v'"^ into v in the second term of the
value of V in Poisson's result (3). Poisson, however, is correct ;
the disagreement between the two mathematicians arises from the
fact that the approximations to the values of fi and v which Pro-
fessor De Morgan gives towards the top of the page under con-
sideration are not carried far enough for the object he has in
view.

In the Treatise on Probability by Galloway, which is con-
tained in the Encyclopcedia Britannica, reference is expressly made
to Professor De Morgan's memoir, without any qualifying remark ;

558 LAPLACE.

this is curious, for the Treatise may be described as an abridge-
ment of Poisson's Recherches . . .SUV la Prob., and Poisson himself
refers to his memoir of 1830 ; so that it might have been expected
that some, if not all, of our conclusions would have presented
themselves to Galloway's attention.

998. Laplace discusses in his pages 284 — 286 the following
problem. An urn contains a large number, n, of balls, some white,
and the rest black; at each drawing a ball is extracted and re-
placed by a black ball : required the probability that after r
drawings there will be x white balls in the urn.

999. The remainder of the Chapter, forming pages 287 — 303,
is devoted to investigations arising from the following problem.
There are two urns, A and B, each containing n balls, some white
and the rest black ; there are on the whole as many white balls as
black balls. A ball is drawn out from each urn and put into the
other urn; and this operation is repeated r times. Required the
probability that there will then be x white balls in the urn A.

This problem is formed on one which was originally given by
Daniel Bernoulli; see Arts. 417, 587, 807, 921.

Let ^^. ,. denote the required probability; then Laplace obtains
the following equation:

/ic + lV , 2x A, x\ , f^ x~ly

This equation however is too difficult for exact solution, and so
Laplace mutilates it most unsparingly. He supposes n to be very
large, and he says that we have then approximately

Q/Z^ If 1. CL Z rg ft

LAPLACE. 559

Let X = — ^ , r = nr, z^^^ = Z7; then lie saj^s that neglecting
terms of the order —^ the equation becomes

It is difficult to see how Laplace establishes this ; for if we adopt
his expressions for 2;j.^i,r, ^x-i,/? and z^^^^^-^, the equation becomes

dr \ n) \ n) dp

/, 11^ 4 4\ d^U

and thus the error seems to be of the order -, or even laro^er, since

n °

p^ 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 way which
is perhaps simpler than Laplace's.

To shew that, if i is a positive integer,

/» 00 /• 00

/ e-''-'-'{s-{-fiJ~^)'dsdtJL = 0.

*' -00 "^ -00

Transform the double integral by putting

s = r cos 6, fju = r^m.B\
we thus obtain

[ * ['" e-"' (cos ^ 6> + ^/^ sin i 6) r'^' dr dO.

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 t and q are positive integers and q less
than ij

560 LAPLACE.

»' —00 '' —00

Transforming as before we obtain

e-"' (cos 10+ sT^l sin 1 6) sin^ 6 r^^+^ Jr ^6^.

•'0 '' 0

Now sin^^ 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 qO. And we know that if m and n are
unequal integers we have

/•27r

I sin mO cos nO dO — 0,

0
r2rT

cos mO cos n6d6 = 0,

sin md sin nd dd = 0 ;

thus the required result is obtained.

Laplace finally takes the same problem as Daniel Bernoulli
had formerly given ; see Art. 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 prohahiUte des
erreurs des resultats nioyens d\in grand nomhre d observations, et
des resultats 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 Theorie des
ProhahiliUs, which is that in which the method of least squares
is proved." Camhriclge 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 pense que les remarques que j'ai eu I'occasion de faire en
1 etudiant, — that his memoirs form a kind of translation, Avhich he
made for his own satisfaction, of Laplace's investigations. Poisson
embodied a large part of his memoirs in the fourth Chapter of his
Recherches sur la Proh....

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

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 to. These values will be assumed to lie between aw
and /S&), both inclusive ; here a and (3 will be positive or negative
integers, or zero, and we shall suppose that a is algebraically

562 LAPLACE.

greater than p, so that a — ^ is positive. The chance of an as-
siofned error will not be assumed the same at each observation.
If 71 be any integer comprised between a and /3 we shall denote
the chance of an error 71(0 at the first observation by N^, at the
second observation by N^, at the third observation by N^, and
so on. Let -ot be a factor such that all the products -377^, -5772 ,
'S773, ... -5773 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 all values
from P to a, both inclusive ; and let

J- ^^ V1V2 * • * ^s *

then the probability that 'utE will be exactly equal to mo), where
m is a given integer, is the coefficient of t^'^ 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 f^ put e^^^ and denote by X what T becomes ; then the
required probability is equal to

Let X and fi be two given integers, such that \ — yit is positive ;
then the probability that tjjE will lie between \co and /xo), both
inclusive, may be derived from the last expression by putting for
m in succession the values jx, //- + !, ^6 + 2, ... X, and adding the re-
sults. Since the sum of the values of ^"'"^^"^ is

7~i

2sini6>

,-(x+i)0V-i _ p-{i^-i)esl-i

the required probability is equal to

sm - ^

we shall denote this probability by P.

LAPLACE. 563

Let us now suppose that w is indefinitely small, and that X
and jx are infinite ; and let

Xft) = (c + 7;) -ST, 11(0 = (c — rj) tar, 'urO =

cox.

The limits of the integration with respect to x will be + 00 .
Also we have

,^ ft) , • 1/1 «^
av = — ax, sm ^a = tt- .

Thus, neglecting ± - compared with X and /^, we obtain

P=- Xe-''^^-^ sin 77a'— (1).

This expression gives the probability that 'stE will lie between
(c + 77) t3- and (c — rj) 'ur, that is, the probability that E will lie
between c + rj and c — 17.

Since we suppose co indefinitely small we consider that the

error at each observation may have any one of an infinite number

of values ; the chance of each value will therefore be indefinitely

small. Let

aft) = a, /3ft) = h, noa — z\

then

ftayiUu

= ^^

YinflV-i _gy{i

ntax^~^-- gYiX^; V-i

Let

iv;=

= ^fi (^) ;

thus Qi

becomes

r^(^)

e-Zi^^^'^dz]

and for X in (1) we must put the new form which we thus obtain
for the product

Assume fi (z) cos 7"^ xz dz = pi cos ?\,

I fi (z) sin ji xz dz = pi sin r^ ;

then Qi- pi^

Let Y= p^p^ p^ ... p8>

y = rj + r^ + 7-3 + . . . + r, ;

86—2

564? LAPLACE.

then X=F6^^.

Substitute in (1) and we obtain

P= — j F cos (?/ — ex) sin vx —
+ / F sin (?/ — ex) sin 9;^ — .

TT J -00 ^

The elements in the second integral occur in pairs of equal
numerical value and of opposite signs, while the elements in the
first integral occur in pairs of equal numerical value and of the
same sign. Thus

2 f" dx
F = —j Y cos (2/ — ex) sin 7]x — (2).

Since each error is supposed to lie between a and h we have

I fi (^) ^^ = 1-

-^ b

Hence it follows that pi = l when x = 0; and we shall now
shew that when x has any other value pi is less than unity.

For pl = \j fi{z) cos jiXzdzY +\ fi(z)smyiXzdz[
fa ra

that is Pi^ = fi (^) COS 7^ xz dz fi (z) cos 7^ xz dz'

-lb J b

ra ra

+ 1 fi{z)^iriyixzdz j f {z') smji xz' dz

ra ra

""] J fi{^)fi(^^')^osry,x(z'-z')dzdz;
and this is less than

that is less than

J b J b

that is less than unity.

LAPLACE.

56o

Up to this point the investigation has been exact: we shall
now proceed to approximate. Suppose 5 to be a very large num-
ber; then Y is the product of a very large number of factors, each
of which is less than unity except when x =0. We may infer that
Fwill always be small except when x is very small; and we shall
find an a^Dproximate value of Y on the supposition that x is small.

Let

I zfi {z) dz = Ici,

J b

z'fi {z) dz = ki',

J b
fa

z^fi (z) dz = ki",

J b

z\fi {z) dz = kl",

Then we shall have in converging series

.4..4 7 '"

pi cos r^ = 1 —

^^ikl x^^ik,

7 ^V-^/'
pi sm Vi = X'^iki r^ f-

Let

{k- — kl) = hi ; then we obtain

Hence
therefore

pi=l —x^^m^- ,

Ti — X'^^^i-V

log /?i = - a^V V + ;

p. = e"^V/u2 approximately.

Let k'- stand for ^<yihi, and I for ^Yi^'i, each summation extend-
ing for the values of i from 1 to 5 inclusive. Then approximately

Thus (2) becomes

2 r o o . dx

P = — e"""^^ cos ilx — cx) sin rjx —
7rj„ ^ ^ ' x

(3).

The approximate values which have been given for Y and 2/
can only be considered to be near the truth when x is very small ;

566 LAPLACE.

but no serious error will arise from this circumstance, because
the true value of Y and the approximate value are both very-
small when X is sensibly different from zero. We may put (3) in
the form

F = — j < I cos (Ix — CX + XV) dv j- 6""^'^'^ dx ;

then by changing the order of integration, and using a result
given in Art. 958, we obtain

^=2^\f^^^' (^)-

This is therefore approximately the probability that E will lie
between c — rj and c-\-7j.

It is necessary to shew that the quantity which we have
denoted by k^ is really positive; this is the case since hi is really
positive, as we shall now shew. From the definition of hi in con-

junction with the equation I fi [z) dz =^ 1, we have

J 6

U^ = f " z'f, {z) dz ("f, (z) dz' - r zf, {z) dz ("z'f, (z') &■

J h J b J b J b

= jj\z''-zz')f{z)f,{z')dzdz'.

And so also

2A? = JJ 1^ (/2 - z^')f, {z)fi {£) dz dz'.
Hence, by addition,

^h^=n\z-z'ffi{z)f,{z')dzdz'.

J b J h

Thus ^hi 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 i^'^ observation ; this means that ^^ (2;) dz expresses the
chance that the error will lie between z 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
suffixes which are no longer necessary, we have

h = [ V(^) ^^' ^' = f"*^!/ (^) ^-^^

J b ■ J b

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 h^,
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 h^ and between a^ and a.

Thus in fact it is only necessary to suppose that a and h 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 — Ix \ we have more closely y—lx — l^ ,
where

Hence, approximately,

cos {y — ex) = cos {Ix — ex) + l^x^ sin {Ix — ex).

568 LAPLACE.

Therefore (2) becomes

P=— \ e cosilx — ex) sm rix —

TTJo ^

2L r^ -/c2^2 . \ 2 • 7

-\ \ e sm [Ix — ex) x^ sm 7)X ax.

TT J 0

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 / and
multiplying by l^ ; so that a transformation may be obtained 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 7i = 72= ••• =78= 1- 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

J —a

If C denote the constant value of f{z) we have then

2a (7=1.

Here Jc = 0, h' = -^— = — , 7^^ = 77 ,

Letc = 0; then by equation (4) of the preceding Article the
probability that the sum of the errors at the 5 observations will
lie between — 77 and 77

2a^[s'7r) J -ri ^VM Jo

Let —^^ = f; then the probability that the sum of the errors
will lie between — ra ^s and ra V^

This will be found to agree with Laplace's page 805.

LAPLACE. 569

100-i. We take Laplace's next case.

Let ry^ = 7^ = • • • = 7s = 1- ^^^ ^^^^ limits of error he ±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) =f{x).

Here h = 0, A^=|^', ^=0, K' = ~h'.

By equation (4) of Article 1002 the probability that the sum
of the errors at the s observations will lie between — rj and 77
is

dv.

2 [\-2sk'

This will be found to agree with Laplace's page 808.

'a r a

We have k' = z''f{z)dz=2 z'f{z) dz,

a ra

and 1 = f[z) dz=2 f{z) dz ;

-a

hence i^fiz) always decreases as z increases from 0 to a we see, as

in Art. 922, that h' is less than -^ .

1005. Laplace next considers the probability that the sum of
the errors in a large number of observations will lie between
certain limits, the sign of the error being disregarded, that is all
errors being treated as positive ; the function of the facility of
error is suj)posed 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 jDut 5 = 0 in
Poisson's problem.

Take 7^ = ^3 = • ■ • = 78 = 1. Then

Take c = Z; then, by equation (-i) of Art. 1002, the probability
that the sum of the errors will lie between l—r] and l-\-7] is

570 LAPLACE.

This will be found 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

[/(z) dz = 1,

•^ 0

we have aC =1.

2 _2

Thus ^' = 9 » ^' = -o ' ^*^ ~" ^"^ = 19 •

Therefore the probability that the sum of the errors will lie

T sa ^ sa .

between -^ — v and -^ + t; is

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 0i {z) denote any function of z : required the probability
that

will lie between the limits c — rj and c + 77. The investigation
will differ very slightly from that in Art. 1002. In that Article
we have

J b

g7,a;W-i^^

in the present case the exponent of e instead of being y^xzj—l,

will be x(pi {z) J— 1. The required probability will be found
to be

- — -r- e 4"' dv;

LAPLACE. 571

where Z = S I (j>ii^)fi (^) ^^'

and 2«=' = 2 j ^ U (2)| f, iz) rf.t! - 2 1 j^, [z)f, {z) rf.j .

The summations extend for all values of i from 1 to s, both
inclusive.

It is not necessary that </)i {£) should be restricted to denote
the same function of z for all the values of i\ Poisson however
finds it sufiicient for his purpose to allow this restriction.

Suppose now, for example, that cf)i{s) = z" for all the values

of ^; and let the function of the facility of error be the same

at every observation. Then, taking 6 = 0, as in the preceding

Article,

ra
1= s j z^f {z) dzj

'a { ra

2/c' = 5J z'f{z)dz-sU zy{z)dz)-.

Take c = l; then the probability that the sum of the squares
of the errors will lie between l — rj and I + 7] is

"~ dv.

This will be found to agree with Laplace's page 312.

1007. Laplace proceeds in his pages 313 — 321 to demonstrate
the advantage of the method of least squares in the simplest case,
that is when one unknown element is to be determined from
observations; see Art. 921. This leads him to make an investi-
gation similar to that which we have given in Art. 1002 from
Poisson : Laplace however assumes that the function of the facihty
of error is the same at every observation, and that positive and
negative errors are equally likely, and thus his investigation is
less general than Poisson's.

Laplace and Poisson agree closely in their application of the
investigation to the method of least squares : we will follow the
latter.

572 LAPLACE.

In a system of observations the quantity given by tbe observa-
tion is in general not the element which we wish to determine,
but some function of that element. We suppose that we already
know the approximate value of the element, and that the required
correction is so small that we may neglect its square and higher
powers. Let the correction be represented by u ; let A^ be the
approximate value of the function at the ^*^ observation, and
Ai 4- uq^i its corrected value. Let Bi be the value of the function
given by observation, e^ the unknown error of this observation.
Then we shall have

^i + 6i = u4, + uq^.

Put 8i for Bi — Ai, so that Bi is the excess of the observed
value above the approximate value of the function ; thus we
have

ei = uqi — 8i.

A similar equation will be furnished by each of the s observa-
tions. All the quantities of which qi and S^ are the types will
be known, and all those of which e^ is the type will be unknown.
We wish to obtain from the system of equations the best value
of u.

Form the sum of all such equations as the preceding, each
multiplied by a factor of which 7^ is the type. Thus we obtain

tyiei = u^yiqi-t%Si (1).

Then by equation (4) of Art. 1002 the probability that 2)7^6^
will lie between I— t] and Z -f ?; is

., _v^

^"^ dv,

where I and /c have the values assigned in that Article.

Put 27-5 = ^ ; thus the probability that Xyi€i will lie between
I — 2tk and I + 2tk is

~ I'e-^'dt (2).

LAPLACE. 573

If in (1) we put I for X^e^ we obtain

and there is therefore the probability assigned in (2) that the
error in the value of u will lie between

2tk , 2tk
and

Supposing then that r remains constant, the error to be ap-

prehended will be least when -^^ is least : and therefore the

factors of which 7^ is the type must be taken so as to make
this expression as small as possible. Put for k its value; and
then the expression becomes

We then make this expression a minimum by the rules of the
Differential Calculus, and we find that the factors must be deter-
mined by equations of which the type is

^' hi '
where z^ is a coefficient which is constant for all the factors.
With these values of the factors, equation (3) becomes

„ = _jf + _5 (4);

.hi ^ hi

and the limits of the error for which there is the probability
assigned in (2) become

If the function of the facility of error is the same at every

574 LAPLACE.

observation the quantities of which hi is the type are all equal,
and so are those of which Ic^ is the type. Thus (4) becomes

and the limits of error become

2t;i

V(2^,^) •

If we suppose also that positive and negative errors are
equally likely, we have ^ = 0, as in Art. 1004. Thus (5) be-
comes

u =

(6).

This agrees with Laplace's result.

Laplace also presents another view of the subject. Suppose
that i/r (x) dx represents the chance that an error will lie between

X and x-\-dx\ then I x^jr (x) dx may be called the mean value

of the positive error to be apprehended — la valeur moyenne de
Verreur a craindre en plus. Laplace compares an error with a
loss at play, and multiplies the amount of the error by the chance
of its happening, in the same way as we multiply a gain or loss
by the chance of its happening in order to obtain the advantage
or disadvantage of a player. Laplace then examines how the
mean value of the error to be apprehended may be made as small
as possible.

In equation (4) of Art. 1002 put c — rj', and suppose positive
and negative errors equally likely, so that 1=0: then the proba-
bility that 27^6^ will lie between 0 and 2?;

Thus the probability that Syi€i will lie between 0 and t is

e~^' dv,

■j^ z*^ «2

^k^JttJ^

LAPLACE. 575

and therefore the probabiHty that Xyid will lie between t and
T ■]- dr is

2iK sjlT

e ^^'dr.

This then is the probability that the error in u will lie

between =; • and -;=i ; and therefore the probability that

the error in u will lie between x and x-\-dx \^

P^e ^-^ dx.

This then is what we denoted above by i|r (x) dx ; and we
obtain therefore

/.oo

I x^lr (x) dx =

which is least when r^ is least. This leads to the same re-

suit as before. The mean value of the positive error to be ap-
prehended becomes — — ^^-^;r .

Since 6i = uqi — 8i we have

If we were to find u from the condition that the sum of the
squares of the errors shall be as small as possible, we should obtain
by the Differential Calculus

which coincides with (6) ; so that the result previously obtained
for u is the same as that assigned by the condition of making the
sum of the squares of the errors as small as possible. It will
be remembered that (6) was obtained by assuming that the
function of the facility of eri'or is the same at every observation,
and that positive and negative errors are equally likely. The
result in (4) does not involve these assumptions. It Tvdll be found

576 LAPLACE.

that the value of u in (4) is the same as we should obtain by
seeking the minimum value of

that is the minimum value of

1008. It is very important to observe how much is demon-
strated with respect to the results (4), (5), and (6) of the preceding
Article. There is nothing to assure us that we thus obtain the
most prohable value of u, in the strict sense of these words ; neither
Laplace nor Poisson makes such an assertion : they speak of the
method as the most advantageous method, as the method luhich
ought to he 'preferred.

Let us compare this method with another which would perhaps
appear the most natural, namely that in which each of the factors
7j, 72' ••• ^^ taken equal to unity.

In the preceding Ai'ticle we arrived at the following result,

Now suppose that instead of giving to the factors 7^, 73, ... the
values assigned in the preceding Article we take each of them
equal to unity ; then the quantity I of the preceding Article be-
comes SA^i, that is sh if we suppose the function of the facility of
error to be the same at each observation. Hence instead of (5) we
shall have

"=2^ + % • W-

Now (5) is preferable to (7) because it was shewn in the pre-
ceding Article that, corresponding to a given probability, the limits
of the error in (5) are less than the limits of the error in (7). In

2T7^
fact the limits of the error in (5) are + ..^ g, , and in (7) they

are + — ^ — - ; and the result that the former limits are less than
Mi

n^>7

LAPLACE. 577

the latter is equivalent to the known algebraical theorem that

{%q^^ is less than s'Sql.

Moreover suppose that we neglect the second temi on the right-
hand side of (5) and of (7); and thus arrive at

"=%/ (*5)' " = 2^, (^)'

then there is another reason why (6) is preferable to (8) ; for, by
virtue of the algebraical theorem just quoted, the term which is
neglected in arriving at (6), is less than the term which is neg-
lected in arriving at (8).

1009. It was shewn in Art. 1007 that there is the probability

(2) that the limits of the error in (6) are + ,,^ ^, . This involves

^ " V(S?i )
an unknown quantity h. Laplace proposes to obtain an approxi-
mate value of h from the observations themselves. It is shewn in
Art. 1006 that there is a certain probability that the sum of the
squares of the errors will lie between l—y and l-\-rj. Assume I
for the value of the sum of the squares of the errors ; thus

t€,' = l = s[ \y {2) dz = 2sh\

J 0

Therefore approximately

72 _ ^_ ^ (uqi - ^iY .

^ ~ 2s ~ 2s
and with the value of u from (6) of Art. 1007, we obtain

^^ 2sXqr

Thus the mean value of the positive error to be apprehended,

which was found in Art. 1007 to be ., ^ ^. , becomes

tqNi^'-rrs)

This agrees with Laplace's page 322.

578 LAPLACE.

1010. Laplace now proceeds in his pages 822 — 329 to the
case where two unknown elements are to be determined from a
large number of observations ; see Art. 923. Laplace arrives at
the conclusion that the method of least squares is advantageous
because the results which it gives coincide with those obtained by
making the mean values of the positive errors to be apprehended
as small as possible ; the investigation is very laborious. The
same assumptions are made as we have stated at the end of
Art. 1007.

Laplace considers that he has thus established the method of
least squares for any number of unknown quantities, for he asserts,
on his page 327, ... i7 est visible que V analyse precedente 2')eut seten-
dre a un nomhre quelconque d'elemens. This assertion, however,
seems very far from being obvious. '

Poisson has not considered this part of the subject ; on account
of its importance I shall now supply investigations by which the
conclusions obtained in Art. 1007 will be extended to the case of
more than one unknown element. I shall give, as in Art. 1007,
two modes of arriving at the result ; Laplace himself omits the
first, and he presents the second in a form extremely different from
that which will be here adopted. In drawing up the next Article
I have obtained great assistance from the memoir by R. L. Ellis
cited in Art. 1001.

1011. Suppose that instead of one element to be determined
by the aid of observations we have any number of elements ; sup-
pose that approximate values of these elements are known, and
that we have to find the small correction which each element
requires. Denote these corrections by x, 2/, z, ... Then the general
type of the equations furnished by the aid of observations will be

€i = aiX + hiy^CiZ+ ... -qi ( 1 ) .

Here 6^ is unknown, while a^, hi, c^, ... q^ are known. Multiply
(1) by 7i, and then form the sum of the products for all values of
^, which we suppose to be from 1 to s, both inclusive. And let the
factors 7^, ^y^, ... 7^ be taken subject to the conditions

27A=0, :^7A = 0, (2);

LAPLACE. 579

thus we obtain

«,=|m+|M 3^_

Now we know from equation (4) of Art. 1002 that there is the
probability

irl}"^' W.

that Syi^i will lie between Z — 2t/c and Z + ^tk, where, as before,
I = Syj^v Put Z for ^7^6^ ; thus (3) becomes

^=|Mt+ ^ ..(5);

and there is the probability (-i) that the error in the value of x,
when determined by (5), will Lie between

2T/g

We propose then to make ^ as small as possible, the fac-
tors being taken consistent with the limitations (2).

Since it is obvious that we want not the absolute values of
the factors 7^, 73, 73, •••, but only the ratio which they bear to
any arbitrary magnitude, we shall not really change the problem
if we impose the condition '^'yia^ = 1. Thus, since k^ = ^j^h^, we
require that '^jiV shall be a minimum consistent with the con-
ditions

Xyiai = l, tyihi=0, ^7^ = 0 (6).

Hence, by the Differential Calculus, we have

tyjitdyi=0,
^Gidyi = 0,
^hid^i = 0,

Therefore by the use of arbitrary multiphers \ fi, v, ... we
obtain a set of s equations of which the type is

ry.Jli^ = \a. -\- fJ^hi + VCi + ,....(/).

37—2

580

LAPLACE.

Let ji stand for y^ ; then from (7) we can deduce the follow-
ing system of equations :

1 = Xtaiji + fJ^afiiji + vtafiji + . . . ^

0 = XtaihJi+ fMtk^ji+ v%hiCiji-\- ... \^ (8).

0 = Xtafiji + tJ&hiCiji + vtciji + . . .

To obtain the first of equations (8) we multiply (7) by aji,
and then sum for all values of i paying regard to (6) ; to ob-
tain the second of equations (8) we multiply (7) by hji 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 \ fju, v, ... Thus equations
(8) will determine X, fi, v, ... ; and then from (5) we have

x = tyiqi + l (9).

We shall now shew how this value of x may practically
be best calculated.

Take s equations of which the type is

GiX + hi7/' + CiZ + =qi + h.

First multiply by aji and sum for all values of ^ ; then mul-
tiply by hji and sum ; then multiply by cj] and sum ; and so on :
thus we obtain the following system

xta.^ji + ytafiiji + zt^a^Cij^ + . . .
xtapji 4- ythlj\ + zthiCiji^ + ...
x'tafiji + y^hiCiji + ztclji + . . .

2 [qi -H h) aji

^ {qi + h) hji
2 {qt + h;) c,j,

(10).

Now we shall shew that if x be deduced from (10) we shall
have X = %%qi + I, and therefore x = x.

For multiply equations (10) in order by X, ^i, v, ... and add;
then by (8)

LAPLACE. 581

x' = Xt [qi + h) aji + ^lt (qi + k,) hij\ + vt {qt + h)CiJi + . . .
= 2 (^. + h^ji [Xtti -{■ lxhi + vCi+ ...]
=^tyi{qi + k) 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 z, 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 y^, %, y^^, "•
unchanged ; and thus we might have anticipated that a sym-
tnetrical 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 , z\ ...
which render the following expression a minimum,

'%f [ciiX + hiy + CiZ -i- ... — $'i - ^if;

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

^ji {€i — kiY that is Xf ^^-T—j '

If h^ be zero, and hi constant, for all values of t, 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.

5S2 LAPLACE.

Tims 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-\-dt\^

^e ^^dt (11).

For put c = ?; in equation (4) of Art. 1002. Then the proba-
bility that SYi^i will lie between 0 and 2?;

ZksIit J ^r, ZkisItt J 0

Thus the probability that S^ej will lie between t and r + dr is

e ^"^ dr,

and therefore the probability that ^Yi^i will lie between Z + t and
Z + T + dr is

1 _ j1

This is therefore the probability that the error in the value of
X when determined by (5) -will lie between

T 1 T + dr
;s— and -= .

And therefore the probability that the error in the value of x
when determined by (5) will lie betv/een t and t^dt\^ given by (11).

The mean value of the positive error to be apprehended in the
value of X will be obtained by multiplying the expression in (11)
by t and integrating between the limits 0 and oo for t. Thus, since

^Ji^i — Ij we obtain -^- for the result : and therefore if we pro-

ceed to make this mean error as small as possible we obtain the
same values as before for the factors y,^, %, %, -"

It will be interesting to develop the value of k. Multiply
equation (7) by 7^, and sum for all values of z; thus by (6) we
obtain

/c^ = X.

LAPLACE. 583

Suppose then we have two unknown quantities, x and y ; we
find from (8)

X =

and the mean error for x will be . .

The mean error to be apprehended for y may be deduced
from that for x by interchanging a^ with hi.

If there are three unknown quantities we may deduce the
mean error from that which has just been given in the case of
two unknown quantities by the following rule :

change ^a^ji into 'Za^^Ji ^o^'- >

{tkco\Y

cUi

change l^h^^i into %KJi ^

change Xajjiji into Zapij\ — ^^ ' ^ ^T ' ""^ .

To establish this rule we need only observe that if we have
three equations (8) we may begin the solution of them by ex-
pressing V from the last equation in terms of X and //., and sub-
stituting in the first and second.

By a similar rule we can deduce the mean en'or in the case of
four unknown quantities from that in the case of three unknown
quantities : and so on.

The rule is given by Laplace on his page 328, without any
demonstration. He assumes however the function of the facility
of error to be the same at every observation so that j\ is constant
for all values of i\ and he takes, as in Art. 1009,

li! =

2s '

1012. Laplace gives on his pages 329—332 an investigation
which approaches more nearly in generality to that which we
have sui3plied in Art. 1007 than those which we have hitherto
noticed in the fourth Chapter of the Theorie ...des Proh. ; see
Art. 917. Laplace takes the same function of the faciUty of error

58^ LAPLACE.

at every observation, but he does not assume that positive and
negative errors are equally likely, or have equal ranges.

1013. Laplace says, on his page 833, that hitherto he has
been considering observations not yet made ; but he "will now
consider observations that have been already made.

Suppose that observations assign values a^, a^, ^g, ... to an
unknown element ; let 0 (z) be the function of the facility of an
error z, the function being supposed the same at every observa-
tion. Let us now determine the probability that the true value
of the element is x, so that the errors are a^ — x, a^ — x, a^ — x,...
at the various observations.

Let P = (j) {a^ — x) . (j> (a^ — x) . <f) {a^ — x). ...

Then, by the ordinary principles of inverse probability, the pro-
ba^bility that the true value lies between x and x + dx is

Pdx

the integral in the denominator being supposed to extend over all
the values of which x is susceptible.

Let H be such that, with the proper limits of integration,

nlpdx^i.

and let y = Hcj) {a^ — x) . ^ [a^ — x) . cf) {a^ — x) . . . .

Laplace conceives that we draw the curve of which the ordi-
nate is 7/ 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 geometres celebres ont pris ponr le milieu qu'il faut choisir,
celui qui rend le resiiltat observe, le plus probable, et par consequent

LAPLACE. 585

I'abscisse qui i^epond k la plus grande ordonnee de la courbe ; mais le
milieu que nous adoptons, est evidemment iudique par la theorie des
probabilites.

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 7nost probable result but one which he con-
siders the most advantageous.

lOl-i. 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 2:)ro-
bable result, and assume that positive and negative errors are
equally hkely and that the function of the facility of error is the
same at every observation, what function of the facihty of error is
implicitly assumed ?

Let the function of the facility of an error z be denoted by
e-'/'(z')^ 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

He-^,

where (7 = '^{x — a^Y + y)r{x — a^Y + yjr {x — a^Y + • • •

To obtain the most probable result we must determine x so
that a shall be a minimum ; this gives the equation

{x — aj yjr' {x — ay + {x— aj i^' (^ ~ ^2)^

+ {x — a^) yjr' {x-ay+ ...^ 0.

Now let us assume that the average result is always the most
probable result ; suppose that out of s observations ^ coincide in
giving the result a^, and s — i coincide in giving the result a^ ; the
preceding equation becomes

I {x — aj -v/r' (x — a^ + (s — i) {x — a^ yfr' (x — a^ = 0.

The average value in this case is

ia^ + (g — i) a^
s

Substitute this value of x in the equation, and we obtain
(s-{ V {i ^*

586 LAPLACE.

■

This cannot hold for all values of - and a^ — a^^ unless i/r' {z) be

independent of z ; say i/r' (z) = c.

Hence '^ {z) = 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

CJ e-'''dz = l',

J —CO

therefore C = —r- •

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

{x-ay+ {x-a^y-\- ... + (x-a^y

we obtain

a +a +... + as

x = — .

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 310 — 312 to shewing, as he
says, that in a certain case the method of least squares becomes
necessary. The investigation is very simj)le 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 a^ 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 Art. 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.i^ ^ e-^i'^C^- a)'' dx.

LAPLACE. 587

Here SJ^ stands for ; J.X., , and tlie value of B, will therefore

depend on the values of the factors y^^, %, ... 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
called the most advantageous system : if we adopt the latter we

Similarly the probability that the error of the result deduced
from the second set of observations will lie between x — a^ and

And so on for the other sets of observations.

Hence we shall find, in the manner of Art. 1013, that the pro-
bability that X is the true value of the unknown quantity is pro-
portional to

x-\-dx-a^ is -^ e-^2Mx-a2)Vic,

e-^

where (t = ^^ (x-a^^ + ^^ (x- a^"" + I3^\x- a^' + ...

Now determine x so that this probability shall have its
greatest value ; a must be a minimum, and we find that

^ /3>^ + ff /g, + /g>., + . . .

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 p7'ohahle inference. If a^, 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 S-y^, and the

denominator with the Sti> 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 errors 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 IIL
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 Proh.,
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

7i = 72 = "- =7s = l
in Art. 1007, so that

S3,

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, 5oQ. Lambert and Bos-
covich also suggested rules on the subject ; see the article Milieu of
the Encyclopedie Methodique and Dr Bowditch's translation of the
Mecanique Celeste, Yol. 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 Britamiica ; we would also refer the student to the
work by the Astronomer Eoyal On the A Igebraical and Numerical
Theory of Errors of Observations and the combination of Observa-
tions.

1018. Laplace's fifth Chapter is entitled Application du Calcid
des Probabilites, a la recherche des phenomenes et de leurs caicses :
it occupies pages 31;9 — 3G2.

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 will be positive or negative, the probability
that the sum of s excesses will exceed the positive quantity c

,00 V

2sk' ^^

^^{2k's7^) J,

= —r- I e~^' dt, where t = ,,^ ,,, .
VttJt f^{zsk)

Hence the probability that the sum will be algebraically less
than c is

1 r

590 LAPLACE.

Now, as in Art. 1004, we may take -^ as the greatest value

c \/3
of 1c y so that the least value of r is — 77^ ; also a = 4, c = 400,

a \l{^s) ' '

5\/ 3
s = 400 : thus the least value of r is —j^ , that is \/(S^'5).

1 r

Hence 1 j- \ e~^'^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 probabilities 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 will fall

across aline 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 will be very nearly — : we might

therefore determine by experiment an approximate value of tt.

8r
Laplace adds . . . et il est facile de voir que le rapport — qui,

pour un nombre donn^ de projections, rend I'erreur a craindre la
plus petite, est I'unite... Laplace seems to have proceeded thus.
Suppose^ the chance of the event in one trial; then, by Art. 993,

I^APLACE. 591

the probability that in fjb trials the number of times in which the
event happens will lie between

jpiJb — T sl'lfii? (1 —p) and p>iJi + T V2/x^j> (1-2^)

2 f"^
is approximately -j- e'^^ dt

Hence to make the limits as close as possible we must have
p [^-p) as small as possible, and thus p = -^. This, we say, ap-
pears to have been Laplace's process. It is however wrong ; for
p (1 — ^;) is a maximum and not a minimum when p=-^. More-

over we have not to make r V2yC6p (1 - p) as small as possible,
but the ratio of this expression to p\x. Hence we have to make

sj p n _ 27) \

—^—^ ^ 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 ^ = — ; we must therefore make this as great

as j)ossible : 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 Encyclopcedia

Metro])olitana. 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 facile de voir

2r .
que le rapport — qui, pour un nombre donne de projections,

rend I'erreur a craindre la plus petite, est I'unite . . . 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 Buffon himself had failed; Lai^lace'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 prohahilite des
causes et des Mnemens futurs, tiree des evenemens observes: 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 observed ; then the probability that the
chance of the simple event lies between a and ^ is

I y dx

\ y dx

Jo '

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 fi and the chance of the other between a! and ^' is

y dx dx

ny dx dx
0

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 I e-^' dt between the limits — r and r we may for an ap-

LAPLACE. 593

proximation take the integral between the limits 0 and

2 /*

and double the result : he says this amounts to neglecting the
square of t'^ — rl We may put the matter in the following form :
suppose that a and h are positive, and we require x such that^

"'e-''dt+{\-^dt = 2\ ""e-^' dt

Suppose a less than h ; then in fact we require that

a J X

'a' + 1'

Laplace, in effect, tells us that we should take x =

2 ;

as an approximation. He gives no reason however, and the more

natural approximation would be to take x = ^ (a + 5), 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

1 . , . . . 1

^ (a + 5), while Laplace's approximation is greater than ^ (a + V).

1024. Laplace discusses on his pages 869 — 376 a problem re-
lating to play ; see Art. 868. A and B 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 -4'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/i and 3?i : 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'i^-xYdx

[ x^ii-xydx

■^ 0

where p and q have the values derived from observations during

594 LAPLACE.

40 years ; these values are given in Art. 902. Laplace finds that
the probability is approximately

1 - -0030761

where /a 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 life 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 u^ =

let e~^^ be integrated with respect to each of the n — 1 variables
z^, z^, ... Zn_^, between the limits — oo and oo : then the result
will be

n-l

Cf|i*2 • • • ^'n_i^7i

e-Y^^„',

where -^ = — ^^ + ~^ + —H-fj}^ + . . . + -i-^ — 2—^

LAPLACE. 595

Let us consider first the integration with respect to z^ ; we have

= (a^+a%') (z ^2'^i^2 Y , 2 2 <^xX'

2 2 2

a. a„ ^.

a^ + a^ ^1

where ^ = 2;, 2"^— ^~2^~2 •

1 a/ + a^b^

The limits of ^ will be — x and 00 ; integrate with respect to
t : thus we remove z^ entirely, and obtain the factor

and instead of the first two terms in w^ we have the single term

ttj ci^ z^

We integrate next with respect to z^ ; thus we shall remove
^2 entirely, and introduce the factor

and instead of the first three terms in ic^ we shall have the single
term

2 2 2„ 2 /» /v 2 2 ■N -1

"'1 ^'2 ^3 ^3 j ^1 S I ^ 2X 2I

< + <^"' l< + <^l' "^ ^3 ^2 I •

Thus we have now on the whole the factor
where -rs = — 2 + -^ + ^ ^ '

- «3 «2 «i

2 ?

and the first three terms in u^ are replaced by the single term \\^.

38—2

596 LAPLACE.

We integrate next with respect to z^ ; thus we shall remove z^
entirely, and introduce the factor

that IS, ^-

LLiJlT
^ 1 1 h

where -^ = —2 +

and the first four terms in u^ are replaced by the single term

\^a 2 2
^2 , ' 2^7 2 J that is, by /x'2;/.

A + «4 t>3

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 h.2iYQ followed
the discussion which we are about to analyse in our next Article,
and which begins thus, Considerons maintenant la prohahilite des
ivenemens futurs, tir^e des evenemens observes.

1030. Laplace considers in his pages 394 — 396 the impor-
tant subject of the probability of future events deduced from
observed events : see Ai'ts. 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 probability, P, of this future event
will be given by

I yzdx

0 _

j y dx

J 0

Laplace then suggests approximations for the integrals in the

LAPLACE. 597

above expression. We will reproduce the substance of his remarks.
In Art. 957 we have

^2 = log r-log^(a+(9)

= log Y- log !</, (a) + ecp' (a) + | f [a) + .. j

~ 2 (/)(«) '^"''
for Y= (j) (a), and ^' (a) = 0, by hypothesis.
Thus approximately

Hence if i/ vanishes when a; = 0 and when x=l, we have
approximately

^/(-

Similarly if we suppose that yz is a maximum when x = a,
and that then yz = Y'Z', we have

j/'^'^^ // d'Y'Z'\'

Suppose that 0 is a function of 3/, say s = (j) (3/), then 3/2 is
a maximum when ?/ is a maximum, so that a =a\ and since

--— = 0, we find that
da

d'Y'Z' ( . ,_ . ,..,,_) ^^F

J^ = {<^(r) + rf(r)|

Hence we have approximately

■<^(n

a/{

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 ^ be the observed number of baptisms of boys during a
certain number of years, q^ the observed number of baptisms of
girls, 2w the annual number of baptisms. Let x represent the
chance that an infant about to be born and baptised will be a
boy.

Let (a? + 1 — xf" be expanded in a series

x^^ + ^^nx'^-' (1 - ^) + ^"^ ^f ' ~ ^^ x''^'' {I -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 ^; then f^ will be the probability that
the superiority will be maintained during i years.

Hence we put x^ {1 — xY for y and f ^ for z in the formula of
the preceding Article, and obtain

P =

I x"" [1 - x)" t,' dx
1x^(1- xy dx

Laplace applies his method of approximation with great success
to evaluate the integrals. He uses the larger values of p and q^
given in Art. 902 ; and he finds that P= '782 approximately.

1032. Laplace's seventh Chapter is entitled De V influence des
ineg allies inconnues qui peuvent exister entre des chances que Von
suppose parfaitement egales : 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

, . 1 r 72 (n — 1) 2 _ 71 (w — 1) (^ — 2) (7^ — 8) 4
that IS, ^ jl + ^^ ^ ^a^-f— ^ ^ I ^ ^ ^ ^a*+...

LAPLACE. 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 B', let the chances of head
and tail in ^ be ^ 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 + qqY.

Suppose that

1 +a 1-a

, 1 + a' , 1-a'

then {pp + qqf = ^. (1 + «« )" •

But as we do not know to which faces the want of S3nTimetry
is favourable, the preceding expression might also be »^ (1 — aa')"

by interchanging the forms of p and q or of p and q. Thus
the true value will be

iji(l + aar+i(l-«a')''},

that is

1 f n{n-l) , n{n-\)[n-^){n-Z) ,„ {
2-„|l+ j_2 aa + g aa +...^.

It is obvious that this expression is nearer to — than that

which was found for the probability of securing n heads in 7i
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 b
counters. Then A's chance of ruining B is

600 LAPLACE.

Laplace puts for p in succession ^ (1 + a) and ^ (1 — a), and
takes half the sum. Thus he obtains for ^'s chance

1 {(1 + g)"- (1 -^■)"1 [il + oy+{l-of]

2 (1 + a)"+' - (1 - af^""

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. 891, but the proof will be more easy, because the trans-
formation there adopted is not reproduced.

Put -z. = X,

1 — a

and 'ii = . „+6

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

1 du _ ax" [x^ -l)-ha^ [x"^ - 1)
udx" x[x''-l){x'' + l){x''^-l)'

We shall shew that this expression cannot be negative.
We have to shew that

h a

cannot be negative. •

This expression vanishes when a; = 1, and its differential coeffi-
cient is {o^~^ — x°-~^) (1 — ic~"~*), which is positive if x lie between
1 and oc ; 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 lea and h into hh : we have then to shew that

(a^" - 1) (a^^ + 1)

LAPLACE. 601

continually increases with h. Let x^ —y\ then we have to shew

that

(/-1)(/+1)

2/"+^ - 1

continually increases as y increases from unity : and this is what
we have already shewn.

1034. Laplace's eighth Chapter is entitled Des durees moyennes
de la vie, des mariages et des associations quelcoiiques: 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

dx n mn '

fc . 1 1

where o' = - . Put i for - , and r for — : and let i and r not be
^ s n m

assumed constant. Thus we have

-y- =iq — ir.

ax ^

Let V denote e"/^^; thus

-gv = -{rv;

therefore qv— constant — Itrv dx.

602 LAPLACE.

The constant is unity, if we suppose the lower limit of the
integral to be 0, for q and v are each unity when a; = 0 ; thus

gy = 1— I irv dx.

The differential equation obtained in Art. 405 becomes when
expressed in our present notation

1 dz 1 d^ _ ir _ irv

z dx ^ dx q - r . , '

1 — \irv dx

therefore, by integration,

z _ constant

1—1 irv dx
As before the constant is unity ; thus

z =

1 — I irx) dx

This result agrees with that on Laplace's page 414.

Laplace intimates that this would be an advantageous formula
if i and r were constants ; but as these quantities may vary, he
prefers another formula which he had previously investigated, and
which we have given from D'Alembert in Art. 483. He says that
by using the data furnished by observation, it appears that the
extinction of the small-pox would increase by three years the
mean duration of life, provided this duration be not affected by
a diminution of food owing to the increase of population.

1036. Laplace discusses in his pages 415 — 418 the problem
of the mean duration of marriages which had been originally
started by Daniel Bernoulli ; see Arts. 412, 790.

Laplace's investigation is very obscure : we will examine various
ways in which the problem may be treated.

Suppose yu, men aged A years to marry yu, women of the same
age, /I, being a large number : determine the probability that at
the end of T years there Avill remain an assigned number of un-

LAPLACE. 603

broken couples. The law of mortality is assumed to be the same
for men as for women ; and we suppose that the tables shew that
out of 7)1^ + 71^ persons aged A years, 77i^ were alive at the end of
T years, tti^ and 7ij^ being large.

One mode of solving the proposed problem would be as follows.

Take — as the chance that a specified individual will be alive

771^ 4- Wj

/ m \^
at the end of T years ; then — will be the chance that a

specified pair will be alive, and we shall denote this by 2?. There-
fore the chance that at the end of T years there will be v un-
broken couples, out of the original jii couples, is

la
— — p^'Cl-pY-".

fjb— V \v

This is rioforous on the assumption that ^ — is exactly the

^ ^ 711^ + 71^ -^

chance that a specified individual will be alive at the end of
T years : the assumption is analogous to what we have called an
inverse use of James Bernoulli's theorem ; see Art. 997.

Or we may solve the problem according to the usual principles
of inverse probability as given by Bayes and Laplace. Let x
denote the chance, supposed unknown, that an individual aged
A years will be alive at the end of T years. We have the ob-
served event recorded in the tables of mortality, that out of m^-\-n^
persons aged A years, m^ were alive at the end of J' years. Hence
the quantity denoted by y in Art. 1030 is

m^ \7i^

and the quantity denoted by z is

{xy{i-xy-^;

therefore P =

fjL— v\v

, f V^ (1 - xY' (x'Y (1 - x'y-^ dx

604? LAPLACE.

Laplace however adopts neither of the above methods ; but
forms a mixture of them. His process may be described thus :
Take the first form of solution, but use Bayes's theorem to deter-

m, \^

mine the value of ^, instead of putting^ equal to i

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

^ — (x'Y (1 - xy-- = - ,

where r is supposed to be not large, and to be such that nearly

V = x^fjb — r, fjb — V = {1 — x^) [JL -^ r.

Thus p= ^o^2^/;fa-^-'0 .

I x'^^ii -xy^dx

J 0

Then, as in Arts. 957, 997, we put

x^^ (1 - xY^ = Ye-^\

x = a+ ~-^^l , nearly,
where a =

And finally we have approximately

e ifia^il-a-i)

^2'7rfid' (1 - a'O ■
Then we have to effect a s-ummation for different values of r,

LAPLACE. 60o

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\f2fia'{l-d') and /J^d" + t ^/2y.a'' {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

271 f (1 - (j^') '

See Art. 148 of the Theory of Prohahilities in the Encyclopcedia
Metropolitana.

There is still another way in which the problem may be solved.
We may take it as a result of observation that out of fi^ man-iages
of persons aged A years there remained v^ unbroken couples at
the end of T years, and we require the consequent probability
that out of fji marriages now contracted between persons aged
A years v unbroken couples will remain at the end of T years.
Then as in Art. 1030 we obtain

P =

\1 \f: — ^ I ic"! (1 - a?)'*i-»'i dx

J 0

The result will be like that which we have found by the

second method, having — instead of al Practically -^ may be

nearly equal to a^, but tliey must not be confounded in theory,
being obtained from ^Ufferent data. The last mode is simpler in
theory than the second, but it assumes that we have from observa-
tion data which bear more immediately on the problem.

1037. Laplace's ninth Chapter is entitled Des benefices depen-
dans de la prohabiliU des Svenemens futiirs: it occupies pages
419—481.

606 LAPLACE.

Suppose that a large number of trials, s, is to be made, and
that at each trial one of two cases will happen ; suppose that in
one case a certain sum of money is to be received, and in the
other case a certain other sum : determine the expectation.

Laplace applies an analysis of the same kind as in his fourth
Chapter ; we shall deduce the required result from the investiga-
tion in Art. 1002. We supposed in Art. 1002 that all values of
a certain variable z were possible, and that fi {£) denoted the
chance at the ^"^ trial that the value would lie between z and
z + hz. Suppose however that only two values are possible which
we may denote by £ and ^^ ; then we must suppose that f^ {z)
vanishes for all values of z except when z is very nearly equal
to fi or to ^i, and we may put

ra
J b

where ^^ stands for the part of the integral arising from values
of z nearly equal to ^^ and qi stands for the part of the integral
arising from values of z nearly equal to f ^ ; and thus

Pi+qi = l.

fa ^

Again, zf^ (z) dz will reduce to two terms arising from values

of z nearly equal to fj and ^^ respectively, so that we shall have

I ^^(2;) J^=£i?i + ft2'^•

Similarly,

Suppose now in Art. 1002 that 7i = 72 = ••• = 7s = 1 ; then
= X [iSlp, + ^^q^) (p, + q^) - (£p, + ^,qy]

= ^ptqi (S - ft)'.

LAPLACE. 607

And there is, by Art. 1002, tlie probability —r- I e'^"^ dt that
Se will lie between

t (5i?i + ?i^i) - 2t/c and t {t^, + ft^^) + 2T/t.
There has been no limitation as to the sign of f^ or l^i-

■ This result will be found to agree with that given by Laplace
on his page 423 ; he had previously, on his page 420, treated the
particular case in which the function /^ {£) is supposed the same at
every trial, so that the suffix i becomes unnecessary, and the result
simplifies in the manner which we have explained towards the
end of Art. 1002.

1038. An important consequence follows so naturally from the
investigation in the preceding Article, that in order to explain it we
will interrupt our analysis of Laplace. Suppose that fi = 1 and
fi = 0, for all values of i : thus

and Sfi becomes equal to the number of times in which an event

happens ovit of s trials, the chance of the happening of the event

2 /''■
being pi at i}^ trial. Thus we have the probability -r- e'^ dt

that the number of times will lie between

%Pi — T^/'2Spiqi and S^i + r V22^i^i.

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 Recherches
sur la Prob. ..., page 246.

1039. If instead of two values at the i*^ trial as in Art. 1037,
we suppose a larger number, the investigation will be similar to

608 LAPLACE.

that already given. Denote these values by fj, ft, %» ... ; we shall
have

-where pi + qi + Wi+ ... = !';

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 /x^
trials a certain result has been obtained v^^ times : if /jl 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- e'^"^ dt that the number

of times the result is obtained will lie between

But if the result is obtained a times in fx trials the advan-
tage is

o-f- {fi - a) f, that is, c (f + f) - yt^f.

Hence there is the probability above assigned that the advan-
tage will lie between

This will be found to agree substantially with Laplace's
page 425.

1041. Laplace passes on to questions connected with life in-
surances : he shews that the stability of insurance companies
depends on their obtaining a very large amount of business. It
has been pointed out by Bienaym^, that if the consideration of

LAPLACE. 609

compound interest is neglected we shall form too high an estimate
of the stability of insurance companies ; see Cournot's Exposition
de la Theorie des Chances... ^Rge 333: se^ also page 143 of the
same work for a formula by Bienayme connected with the result
given in Ai't. 1038.

1042. Laplace's tenth Chapter is entitled De Tesperance morale:
it occupies pages 432 — 445. This Chapter may be described as
mainly a reproduction of the memoir by Daniel Bernoulli, which
we have analysed in Arts. 377 — 393 ; Laplace himself names his
predecessor. Laplace adds the demonstration to which we have
referred in Art. 388 ; see his pages 436, 437. Laplace also applies
the theory of moral expectation to an example connected with life
annuities ; see his pages 442 — 444.

The following example in inequalities is involved in Laplace's
page 444. If a^, a^, a^, ... and h^, h^, h^, ... are series both in in-
creasing or both in decreasing order of magnitude

is greater than

a^\ + a^\ + o^\ + . . . + OnK

afi^ + <^2^2 + ^3^3 + • • • + <^n^n

a^ + a^ + a^+ ... + Gn '

for if we multiiDly up and bring all the terms together, we find
that the result follows from the fact that ara^ {cij. — a^ {hr — h^ is
positive.

Hence too if one of the two series is in increasing and one in
decreasing order of magnitude the inequality becomes inverted.

1043. Laplace's eleventh Chapter is entitled De la Prohahilite
des Umoig^iages: it occupies pages 446 — 461.

We have given sufficient indication of the main principle of
the Chapter in Art. 735 ; see also Art. 941.

Laplace's process on his page 457, although it leads to no eiTor
in the case he considers, involves an unjustifiable assumption ; see
Poisson, Eecherches siir la Pro6. ...page 112. See also pages
3 and 364 of Poisson's work for criticisms bearing on Laplace's
eleventh Chapter.

610 LAPLACF.

1044. Laplace's pages 464 — 484 are headed Additions; see
Arts. 916, 921. There are three subjects discussed.

I. Laplace demonstrates Wallis's theorem, and he gives an
account of the curious way in which the theorem was discovered,
although it cannot be said to have been demonstrated by its dis-
coverer.

II. Laplace demonstrates a formula for A"s' which he had
formerly obtained by a bold assumption ; see Arts. 916, 966.

III. Laplace demonstrates the formula marked (p) on page 168
of the Theorie...des Prob.; see Art. 917.

1045. The first Supplement to the Theorie...des Proh. is en-
titled Sur V application du Calcul des Prohabilites d la Philosophie
Naturelle ; it occupies 34 pages: see Art. 926. The title of the
Supplement does not seem adapted to give any notion of the
contents.

1046. We have seen in Art. 1009 that in Laplace's theory of
the errors of observations a certain quantity occurs the value of
which is not known a priori, but which may be approximately
determined from the observations themselves. Laplace proposes
to illustrate this point, and to shew that this approximation is one
which we need not hesitate to adopt : see pages 7 — 11 of the first
Supplement. It does not appear to me however that much con-
viction could be gained from Laplace's investigation.

A very remarkable theorem is enunciated by Laplace on page 8
of the first Supplement. He gives no demonstration, but says
in his characteristic way, L'analyse du n° 21 du seconde Livre
conduit a ce theoreme general.... The theorem is as follows:
SupjDose, as in Art. 1011, that certain quantities are to be deter-
mined by the aid of observations ; for simplicity we will assume
that there are three quantities x, y, z. Let values be found for
these quantities by the most advantageous method, and denote
these values by o;^, y^, z^, respectively. Put

aj = a?j + f , y = y^ + V, z = z^+^.
Then Laplace's theorem asserts that the probability of the simul-

LAPLACE. 611

taneoiis existence of f, 77, f, as values of the errors of the quantities
to be determined, is 23roportional to e~'^, where

I am compelled to omit the demonstration of this theorem for want
of space ; but I shall endeavour to publish it on some other
occasion.

1047. Laplace next supposes that six elements are to be
determined from a large number of observations by the most ad-
vantageous method. He arranges the algebraical work in what
he considers a convenient form, supposing that we wish to de-
termine for each variable the mean value of the error to be appre-
hended, or to determine the probability that the error wdll lie
within assigned limits ; see pages 11 — 19 of the first Supplement.
He then, on his pages 21 — 26, makes a numerical application, and
arrives at the result to which we have alreadv referred in Art. 939.

1048. Laplace observes that all his analysis rests on the as-
sumption that positive and negative errors are equally likely, and
he now proposes to shew that this limitation does not practically
affect the value of his results: see his pages 19 — 21. Here again
however it does not appear to me that much conviction would be
gained from Laplace's investigation.

1049. The first Supplement closes with a section on the Pro-
bability of judgments; it is connected with the eleventh ChajDter :
see Art. 1043.

1050. The second Supplement is entitled Application du
Calcul des Prohahilites aux operations geodesiques : it occupies 50
pages: see Art. 927. This Supplement is dated February 1818.

This Sup23lement is very interesting, and considering the sub-
ject and the author it cannot be called difficult. Laplace shews
how the knowledge obtained from measuring a base of verification
may be used to correct the values of the elements of the triangles
of a survey. He speaks favourably of the use of repeating circles;
see his pages 5, 8, 20. He devotes more space than the subject
seems to deserve to discuss an arbitrary method proj^osed by

39—2

(112 LAPLACE.

Svanberg for deducing a result from observations made with a re-
peating circle : see Laplace's pages 82 — 35.

Laplace explains a method of treating observations which he
calls the method of situation, and which he considers may in
some cases claim to be preferable to the "inost advantageous method
explained in his fourth Chapter. This method of situation had
been given in the Mecanique Celeste, Livre III., but without re-
ceiving a special name: see Art. 1016. Laplace gives an investi-
gation to determine when the inethod of situation should be pre-
ferred to the most advantageous method, and an investigation of the
value of a combination of the two methods.

1051. The third Supplement is entitled A^yplication des
formules geodesiques de py^ohahilite, d la meridienne de France;
it occupies 36 pages: see Art. 928.

Laplace begins by giving a numerical example of some of the
formulae in the second Supplement. In his pages 7 — 15 he gives
what he calls a simple example of the application of the geodesic
formulae. He takes a system of isosceles triangles, having their
bases all parallel to a given line, and he finds the errors in lengths
arising from errors in the angles. The investigation is like that in
the second Supplement.

Laplace devotes his pages 16 — 28 to discussions respecting the
error in level in large trigonometrical surveys.

Pages 29 — 36 contain what Laplace calls Methods generale du
calcul des prohabilites, lorsquil y a plusieurs sources d'erreurs.

1052. Here we close our account of the Theorie Analytique
des Prohahilites. After every allowance has been made for the aid
which Laplace obtained from his predecessors there will remain
enough of his own to justify us in borrowing the words applied to
his Theory of the Tides by a most distinguished writer, and pro-
nouncing this also " to be one of the most splendid works of the
greatest mathematician of the past age."

For remarks which will interest a student of Laplace's work I
may refer to the first page in the Appendix to De Morgan's Essay
on Probabilities .. .'va the Cabinet Cyclopaedia; to the History of the
Science which forms the introduction to Galloway's Treatise pub-

LAPLACE. C13

lished in tlie Encyclopcedia Britannica; to the work of Gouraud,
pages 107 — 128 ; and to various passages in Dugald Stewart's Works
edited hy Hamilton, which will be found by consulting the General
Index in the Supplementary volume.

Some observations by Poisson will find an appropriate place
here: they occur in the Comptes Rendus... Yol. IL page 396.

Sans doute LajDlace s'est montre un homme de genie dans la meca-
nique celeste ; c'est lui qui a fait preuve de la sagacite la plus penetrante
pour decouvrir les causes des phenomenes ; et c'est ainsi qu'il a trouve la
cause de Tacceleration du mouvement de la Lune et celle des grandes
inegalites de Saturn e et de Jupiter, qu'Euler et Lagrange avaient cher-
chees infructueusement. Mais on pent dire que c'est encore plutot dans
le calcul des probabilites qu'il a ete un grand geometre; car ce sent les
nombreuses ap[)lications qu'il a faites de ce calcul qui out donne naissauce
au calcul aux differences fiuies partielles, a sa methode pour la reduction
de certaines integrales en series, et a ce qu'il a nomme la theorie des /onc-
tions generatrices. Un des plus beaux ouvrages de Lagrange, son Me-
moire de 1775, a aussi pour occasion, et en partie pour objet, le calcul
des probabilites. Croyons done qu'un sujet qui a fixe 1' attention de
pareils hommes est digne de la notre; et tachons, si cela nous est pos-
sible, d'aj outer quelque chose a ce qu'ils ont trouve dans une matiere
aussi difficile et aussi interessante.

APPENDIX.

1053. This Appendix gives a notice of some writings which
came under my attention during the printing of the book, too
late to be referred to their proper 2:)laces.

105-i. John de Witt's tract which was mentioned in the fifth
Chapter has been recovered in modern times, and printed in an
Enghsh translation. See Contributions to the History of Insur-
ance... by Frederick Hendriks, Esq. in the Assurance Magazine,
Yol. II. 1852, page 231. For some remarks on John de Witt's hypo-
thesis as to the rate of mortality, see page 393 of the same
volume.

Many interesting and valuable memoirs connected with the
history of Insurance and kindred subjects will be found in the
volumes of the Assurance Magazine.

1055. A memoir on our subject occurs in the Actoruni Eru-
ditorum. . .Supplemenia. Tomus ix. Lipsise, 1729. The memoir is
entitled, Johannis Rizzetti Ludorunn Scientia, sive Artis conjectandi
elementa ad alias appUcata: it occupies pages 215 — 229 and
295—307 of the volume.

It appears from page 297 of the memoir that Daniel Ber-
noulli had a controversy with Rizzetti and Riccati relating to
some problems in chances ; I have found no other reference to
this controversy. Rizzetti cites the Exercitationes Mathematicce
of Daniel Bernoulli ; I have not seen this book myself, which
ajDpears to have been published in 1721^.

The chief point in dispute may be said to be the proper defi-
nition of expectation. Suppose that A and B play together ; let
A stake the sum a, and B stake the sum h ; suppose that there
are m + n + p equally likely cases, in m of them A is to take both
the stakes, in 7i of them B is to take both the stakes, and in p of

APPENDIX. 615

them each takes his own stake. Then according to the ordinary-
principles we estimate the expectation of A at

m {a + i) +2^^
m + n +p '

so far as it depends upon the game which is to be played. Or if
we wish to take account of the fact that ^ has already paid down
the sum a, w^e may take for the expectation

mia +h) ■\- pa ,. , . mh — na
— a, that is, .

Eizzetti however prefers another definition ; he says that A has

m chances out of 7n -{-n + p of gaining the sum h ; so that his

7726

expectation is . Rizzetti tries to she^v that the ordinary

m + 71 -\-p

definition employed by Montmort and Daniel Bernoulli leads to
confusion and error ; but these consequences do not really follow
from the ordinary definition but from the mistakes and unskil-
fulness of Rizzetti himself

The memoir does not give evidence of any power in the sub-
ject. Rizzetti considers that he demonstrates James Bernoulli's
famous theorem by some general reasoning which mainly rests
on the axiom, Effectus constans et immutabilis pendet a causa
constante, et immutabili. On his page 224 he gives wdiat he con-
siders a short investigation of a problem discussed by Huygens
and James Bernoulli ; see Arts. 33, 103 : but the investigation is
unsatisfactory, and shews that Rizzetti did not clearly understand
the j)roblem.

1056. I am indebted for a reference to the memoir noticed
in the preceding Article to Professor De Morgan who derived it
from Kahle, Bihliothecce Philosopliice Struviance...GoiimgQn, 1740.
2 Vols. 8vo. Vol. I. p. 295. Professor De Morgan supplied me
from the same place with references to the following works which
I have not been so fortunate as to obtain.

Andrew Budiger, De sensu falsi et veri, lib. I. cap. xii. et
lib. III. : no further description given.

Kahle himself Elementa logicce probabilium, methodo mathe-
matica...'RRlse MagdeburgiccT, 1735, 8vo.

616 APPENDIX.

1057. The work which we have quoted at the beginning of
Art. 347 contains some remarks on onr subject; they form part
of the Introduction a la Philosopliie, and occur on pages 82 — 93 of
the second volume. It appears from page XLVII of the first volume
that this work was first published by 's Gravesande in 1736. The
remarks amount to an outline of the mathematical Theory of Pro-
bability. It is interesting to observe that 's Gravesande gives in
effect an example of the inverse use of James Bernoulli's theorem;
see his page 85 : the example is of the kind which we have used
for illustration in Art. 125.

1058. The result attributed to Euler in Art. 131 is I find
really due to John Bernoulli. See Johannis Bernoulli... Opera
Omnia, Tomus Quartus, 1742, p. 22. He says,

Atque ita satisfactum est ardenti desiclerio Fratris mei, qui agnoscens
summse liujus pervestigationem difficiliorem esse qiiam quis putaverit,
ingenue fassus est, omnem suam industriani fuisse elusam : Si quis in-
veniat, inquit, tiohisque commu7iicet, quod i7idustriam nostram elusit
hactenus, magna s de nobis gratias ft ret. Yid. Tractat. de Seriehus inji-
nitis, jD. 254:. Utinam Frater superstes esset.

1059. An essay on Probability was written by the celebrated
Moses Mendelsohn ; it seems to have been published in his Phi-
losophische Schriften in 1761. I have read it in the edition of the
Philosophische Schriften which appeared at Berlin in 1771, in two
small volumes. The essay occupies pages 243 — 283 of the second
volume.

Mendelsohn names as writers on the subject, Pascal, Format,
Huygens, Halley, Craig, Petty, Montmort, and De Moivre. Men-
delsohn cites a passage from the work of 's Gravesande, which
amounts to an example of James Bernoulli's theorem ; and Men-
delsohn gives what he -considers to be a demonstration of the
theorem, but it is merely brief general reasoning.

The only point of interest in the memoir is the following.
Suppose an event A has happened simultaneously, or nearly so,
with an event B; we are then led to enquire whether the con-
currence is accidental or due to some causal connexion. Men-

APPENDIX. 617

delsohn says that if the concurrence has happened n times the

probability that there is a causal connexion is ; but he gives

no intimation of the way in which he obtains this result. He
takes the following illustration : suppose a person to drink coffee,
and to be attacked with giddiness ; the concurrence may be acci-
dental or there may be some causal connexion : if the concurrence

has been observed n times the probability is that the o-id-

diness will follow the drinking of coffee.

If we apply the theorem of Bayes and Laplace, and suppose

that an event has happened n times, the probability that it will

?i + 1
happen at the next trial is ^ ; see Art. 848. It is certainly

curious that Mendelsohn's rule should agi-ee so nearly with this
result when n is large, but it is apparently only an accidental
coincidence, for there is nothing in Mendelsohn's essay which
suggests that he had much knowledge of the subject or any great
mathematical power : we cannot therefore consider that he in any
way anticipated Bayes.

Mendelsohn makes his rule serve as the foundation of some
remarks on the confidence which we repose on the testimony of
our senses, referring especially to the scepticism of Hume. Men-
delsohn also touches on the subjects of Free Will and the Divine
Foreknowledge ; but as it appears to me without throwing any
light on these difficult problems.

I was aware that Mendelsohn had written on Probability from
the occurrence of his name in Art. 840, but I assumed that his
essay would not contain any matter bearing on the mathematical
theory, and so I omitted to examine it. I supply the omission
at the request of the late Professor Boole ; he had seen a reference
to Mendelsohn in some manuscripts left by Dr Bernard, formerly
teacher of Hebrew in the University of Cambridge, and, in con-
sequence of this reference, expressed a wish that I would report
on the character of the essay.

1060. I take from Booksellers' Catalogues the titles of four
works which T have never seen.

018 APPENDIX.

Thubeuf. EMmens et principes de la royale Arithm^tique
aux jettons, etc. 12mo. Paris, 1661.

Marpurg, F. W., Die Kunst, sein Gliick spielend zu maclien.
Hamburg, 1765. 4to.

Fenn, (I.) Calculations and formulae for determining the Ad-
vantages or Disadvantages of Gamesters,... 1772. 4to.

Frommichen Ueber Lelire d. Wahrsclieinl. Braunscbw. 1 773 . 4 to.

1061. I had overlooked a passage in Montucla which bears
on the point noticed in Art. 990; see Montucla, page 421. It
seems that a mode of election suggested by Condorcet was for
some time adopted at Geneva. The defects of the mode were
indicated in a work by Lhuilier entitled, Examen dii mode d'elec-
tion propose en fevrier 1793, a la Convention nationale de France^
et adopts d Geneve (1794, en 8^).

1062. A very curious application of the Theory of Proba-
bility was stated by Waring; see his Meditotiones Algehraicce,
3rd Edition, 1782, pages xi, 69, 73. For example, he gives a rule for
ascertaining the number of imaginary roots in an equation, and
says : Haec methodus in quadraticis sequationibus verum prgebet
numerum impossibilium radicum : in cubicis autem ejus proba-
bilitas inveniendi impossibiles radices non videtur majorem habere
rationem ad probabilitatem fallendi quam 2:1.

I owe this reference to the kindness of Professor Sylvester in
sending me a copy of his remarkable memoir in the Philosophical
Transactions for 1864, on the Real and Imaginary roots of Alge-
hi'aical Equations. Professor Sylvester had independently made
the same kind of application ; see page 580 of the volume, where
he says : " Like myself, too, in the body of the memoir Waring
has given theorems of probability in connexion with rules of this
kind, but without any clue to his method of arriving at them.
Their correctness may legitimately be doubted."

CHRONOLOGICAL LIST OF AUTHORS.

The figures refer to the pages of the Volume.

1606

1617

1654

1657
1657

1662
1666
1670
1671
1679
16S5
1685
1690
1690
1692
1693

i(>93
1699

1708

1709

1710

1711

1712

1713
1714

1714

1718

PAGE

Cardan i

Buckley 26

Kepler 4

G ALTLEO 4

puteanus 27

Pascal 7, 28

Fehm AT 7

HUTGENS 22

schooten 30

Graunt 38

Leibnitz 31

Car AMUEL 44

John DE Witt 38,614

Sauveur 46

Wallis .. 34

James Bernoulli 47

Petty 39

James Bernoulli 47

Arbuthnot 49

H ALLEY 41

Roberts 53

Craig 54

MoNTMORT 79

Nicolas Bernoulli 194

Barbeyrac 196

Arbuthnot 196

De Moivre 136

's Gravesande 197

James Bernoulli 58

Montmort 79

Nicolas Bernoulli 199

De Moivre 199

Browne 199

De Moivre 136

728
729

730
730
730

733

734
736
738
738
739
740
742

751
751

753
754
754
756

757
757
758
760
760
761
761
762

763
764
765
765
766

767
767

PAGE

Mairan 200

Rizzetti 614

Nicole 201

Daniel Bernoulli 213

Cramer 221

BuFFON 203

Daniel Bernoulli 222

's Gravesande 616

De Moivre 1 36

Ham 203

D.M 205

Simpson 206

John Bernoulli 212

EuLER 239

K aestner 321

DODSON 322

Hoyle 322

D'Alembert 258

De Moivre 1 36

Simpson 211

D'Alembert 259

Clark 323

Daniel Bernoulli 224

EULER 240

D'Alembert 261

Mendelsohn 616

Mallet 325

Bayes and Price 294

EULF.R 243

EULER 245

Beguelin 328

Daniel Bernoulli 228,229

Beguelin 331

Michell 332

620

CHRONOLOGICAL LIST OF AUTHORS.

PAGE

768 John Bernoulli 335

768 D'Alembert -272, 273, 285

769 John Bernoulli 325

769 EuLER •. 247

769 Daniel Bernoulli 231, 235

770 Lagrange 301

771 Lambert 335

772 Mallet 337

773 Laplace 473

774 Laplace 464,465

775 Lagrange 313

776 Emerson 343

777 Daniel Bernoulli 236

777 EULER 237

777 BUFFON 344

778 Laplace 476

779 ^uss 349

779 Laplace 484

780 Fuss 350

780 Prevost 432

781 Prevost 432

781 BoRDA 432

781 condorcet 392

782 condorcet 395

782 Malfatti 434

782 Laplace 485

782 Waring 618

783 Laplace 485, 486

783 BiCQUILLEY 438

783 CONDORCET 398

784 CONDORCET 401

PAGE

784 D'Anieres 445

784 Encyclopedie Methodique 44 t

785 Euler 256

785 condorcet 351

786 D'Anieres 445

792 Lagrange 320

792 Waring 446

793 Trembley 411

794 Ancillon 453

794 Trembley 420

794 Waring 452

795 Trembley 413

795 Laplace 487

796 Trembley 423

796 Prevost and Lhuilier 453

797 Prevost and Lhuilier 461

797 Trembley 426

799 Trembley 426

8or Trembley 428

802 Trembley 429

809 Laplace 487

810 Laplace 489

812 Laplace 495

813 Laplace 490

814 Laplace 495

815 Laplace 491

816 Laplace 491

818 Laplace 494

820 Laplace 495

822 Laplace 495

INDEX.

The figures refer to the pages of the Volume.

Absent, 260, 441.

Accius Nsevius, 409.

Airy, 589, 612.

Ancillon, 453.

Annuities, 4-2, 742, 320, 447.

Arbuthnot, 49 to 53, 130, 193, 197, '200.

Arguments, Probability of, 70, 462.

Arithmetical Triangle, 17, 28 to 30, 64, 82.

Assurance, 256, 441.

Athenaeum, 38, 55, 501.

Bacon, 503.

Baily, 452.
Barbeyrac, 196.
Barrow, 2 1 .
Bauhusius, 27.

Bayes, 294 to 300, 378, 466, 476, 592.
Bayle's Dictionary, 12.
Bernard, 617.

Bernoulli, James, 22, 24, 28, 39, 46, 47,
56 to 77, 80, 93, 98, 122, 124, 147,

279» 378, 46^, 553.
Bernoulli, Jolin, 40, 45, 57, 88, 91, 96, 98,

113 to 116, 121, 135, 139, 210, 2T2,

222, 616,
Bernoulli, Nicolas, 45, 58, 90, 91, loi,

105 to 134, 148, 157, 162, 193 to 199,

210, 220, 243, 429, 536.
Bernoulli, Daniel, 42, 213 to 238, 265 to

285, 293, 319, 377, 393, 423 to 428,

434, 442, 469, 475, 489, 502, 601,609,

614.
Bernoulli, John, 325 to 328, 442, 469.
Bicquilley, 55, 438 to 441.
Bienayme, 608.
Binet, 292.

Boole, 7, 505, 512, 544, 617.
Borda, 391, 432 to 434, 548.
Boscovich, 588.
Bowditch, 478, 588.
Breslau Registers, 41, 226, 322.
Browne, 23, 49, 199.
Buckley, 26.

BufFon, 203, 262, 275, 277, 285, 344 to
349; 376, 386, 440.

Bullialdus, 65.
Buteo, 33.

Calandrin, 149.

Calculus of Operations, 505 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.

Commercium Epistolicum, 131.

Condorcet, 41, 186, 261, 292, 351 to 410,

432, 441, 456, 458, 539' 541. 548, 618,
Cotes, 143, 588.
Cournot, 222, 383, 544, 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'Anieres, 445.

Dante, i, 323.

De Beaune, 127, 130.

De Ganiferes, 29.

De Haan, 513, 514.

De la Hontan, 95.

De la Roche, 459.

De M^re, 7, 8, 11, 63, 145, 502.

De Moivre, 43, 52, 54, 63, 78 to 94,
100 to 105, 128, T34 to 193, 199 to

211, 250 to 253, 305, 315 to 325, 340,

361, 378, 412, 421, 466, 468, 497, 527,

535, 539^ 541, 553-
De Morgan, 26, 49, 147, 346, 379, 400,

452, 513. 514, 539» 552, 557: 591, 605,
612, 615.
Descartes, 21, 59, 132, 505.

622

INDEX.

De Witt, 37 to 41, 614.
Diderot, 55, 260.
Dodson, 322,

Election, modes of, 374, 547, 618.

Ellis, 544, 561, 578.

Emerson, 343.

EncydopecUe, 39, 55, 201, 258 to 265,

286, 290, 441 to 445.
Errors, Theory of, 236 to 238, 301 to

309, 428, 442, 468 to 470, 484, 488,

490, 561 to 589.
Euler, 90, 237 to 257, 325 to 328, 422,,

443, 489. 553, 588.
Expectation, 213, 261, 392, 609, 614.

Faulhaberus, 65.

Fenn, 618.

Eermat, 7 to 21, 35, 97j M^-

Eontaitie, 222, 261, 346.

Eontana, 186.

Foiitenelle, 46, 57, 78, 188.

Forbes, 334.

Ereret, 407.

Frommichen, 618.

Fuss, 69, 349.

Gaeta, 186.

Galileo, 4 to 6.

Galloway, 48, 409, 552, 557, 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.

Eerme, 106.

Gageure, 259, 263, 264.

Hazard, 48, 53, 94, 163, 164, 205.

Her, 106, 133, 429.

Krabs, 444.
Lansquenet, 91.

Lotteries, 48, 53, roo, 151, 203,206,
245 to 256, 260, 325, 338, 421, 465,

527-
Noyau X, 95.

Odd and Even, 200, 465, 473, 527-
Oublieux, 100.
Pari, 260.

Passe-Dix, 45, 94, 444.
Paume, 75, 125.
Pharaon, 48, 80, 87, 116, 150, 152, 203,

243» 345-
Piquet, 94, 166.
Quadrille, 152, 201.
Qninquenove, 94.
Rafle, Raffling, 94, 164.
Royal Oak, 52.
Tas, 106, 110, 124.
Treize or Rencontre, 91, 105, 115, 120,

15^5 239, 335, 452, 535-

Trente et Quarante, 205, 444.

Tiijaques, 69.

Trois Dez, 94.

Whist, 52, 164, 445.
Garve, 453.
Gauss, 489, 588.
Generating Functions, 484, 497, 504,

53O; 534.
Gouraud, i, 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, 8r, 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,

£^8 to 62, 81, 138 to 141, 143, 199,

432, 444-

Izquierdus, 44.

INDEX.

G23

Jones, 82.
Justell, 41.

Kaestner, 321.
Kahle, 615.
Karstens, 42 7.
Kepler, 4,
Kersebooni^ 241.

Lacroix, 377.

Lagrange, 178, 211, 249, 301 to 320, 428,
466, 469, 478, 484, 505, 535. 613.

Lambert, 71, 93, 335 to 337, 428, 459,
462, 588.

Laplace, 7, 123, T57 to 163, 169 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 to 613.

Laplace, Comte de, 532 to 534.

Laplace's Functions, 559.

Least Squares, 560, 575.

Leibnitz, 12, r4, 21, 22, 31 to 33,39, 47>

56, 73» 79. 502, 505.
Leslie, 26.

Lhuilier, 618. See Prevost and Lhuilier.
Libri, I, 2, 5, 6.
Loc ,50.
Longomontanus, 45.
Lubbock aud Drinkwater, 11, 23, 33, 37,

4h 48, 50; 54; 55, 299.
Lully, 44.

M., 205.

Maclaurin, 192.

Mairan, 200.

Malebranche, 78, 126.

Malfatti, 235, 434 to 438.

MaUet, 325, 337 to 343, 350.

Mar pur g, 618.

Maseres, 34, 59, 65.

Mayer, 588.

Mead, 199.

Mecanique Celeste, 478, 487, 514, 588.

Mechanique du Feu, 131.

T^Iend 1 hn, 453, uiO.

Mercator, 65.

Merian, -.6.

Michaelis, 93.

IVIicheU, 332 to 335, 393, 491.

Miclielsen, 427.

Mill, 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, II, 12, 16, 22, 26, 38, 39, 42,

46, 48, 79, 133, 222, 224, 261, 292,

^93> 316, 332, 618.
Mortality, 37 to 43, 240, 268, 285.
Motte, 23, 48.

Napoleon, 495 to 497.

Necker, 259, 264.

Neiimann, 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, 500, 502.
Payne, 324.
Peacock, 26,

Permutations, 34, 64, 67, 150.
Peterson, 54.
Petty, 39, 81.
Peverone, i.
Poisson, 7, 206, 222, 410, 489, 544, 556

to 561, 567, 571, 576, 607, 609, 613.
Pre motion Ph tjsiq ue, 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, 558, 560.

Births of boys and girls, 130, 193, 196
to 198, 235, 415 to 420, 480 to 484,

593. 597.

G24<

INDEX.

Problems :

Buffon's, 260, 347, 59°'

Cuming's, 18-2.

Duration of Marriages, 229, 335, 426,

602.
Duration of Play, 61, loi to 105, 147,
167 to 183, 209, 317 to 320, 448,

465. 474, 476, 489, 535-
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,

275, 280, 286 to 289, 332, 345, 393>

470.
Points, 8 to 19, 59, 96 to 99, 137^

146, 201 to 203, 316, 412, 468, 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, 3^5j 535-
Woodcock's, 147 to 149.

Puteanus, 27.

Kacine, 500.

Riccati, 614.

Kizzetti, 614.

Eoberts, 53, 136, 137, 159, 1^4.

Roberval, 8, 12 to 15.

Rudiger, 615.

Saurin, 58.

Sauveur, 46, 201.

Schooten, 22, 26, 30, 64, 67.

Schwenter, 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.
Stewart, Dugald, 4, 349> 409» 45 3> 458,

502, 503,613.
Stifel, 33.

Struve, 334.
Sussmilch, 320.
Svanberg, 612,
Sylvester, 618.

Tacquet, 36.
Tartaglia, i.
Taylor, 162.
Terrot, 457.
Tetens, 427.
Theorems :

Eayes's, 73, 294 to 300, 398, 410, 557,

603.
Bernoulli's, James, 71 to 73, i3i> 183,
198, 360, 393, 548, 554, 556, 607.
Binomial, 65, 82.
De Moivre's on Dice, 84, 138, 146,

189, 208, 305, 350, 4^8, 542.
De Moivre's approximation, 138, 144,

207.
Euler's, 192.
Stirling's 72, t88, 235, 467, 485, 520,

549. 553-
Van derm onde's, 451.

Wallis's, 610.
Thomson, 49.
Thubeuf, 618.
Tltius, 54.

Trembley, no, 160, 230, 250, 411 to 431.
Trial by Jury, 388.
Turgot, 352.

Van Hudden, 38.
Varignon, 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.

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