46 ON PROBABILITY. 
a work entitled “Du Calcul des Probabilités,” and by Condorcet in the 
article “Probabilité,” in the French Encyclopadia. 
78. James Bernoulli was employed in preparing a copious work on the the 
science of chances, till his death in 17035, by which its appearance was delayed less 
during ten years, after which his nephew, Nicolas Bernoulli, found leisure nut 
to superintend its publication, though in an unfinished state. In the meantime expe 
Montmort published his celebrated work, * Essai d’Analyse sur les Jeux de his @ 
Hazard,” the most extensive of the sort which had till then appeared, in and 
which the conditions of all the principal games then in vogue are discussed <0 4 
at considerable length, and the theory of combinations extended and enriched 7] 
with several new theorems. Immediately after the publication of Montmort’s in 
book, Demoivre, a Frenchman naturalized in England after the revocation of 0 
the Edict of Nantes, inserted, in the Philosophical Transactions for 1711, a ect] 
short essay entitled * De Mensura Sortis,” which, in 1716, he published in ho ne 
a greatly enlarged form, under the title of ““ The Doctrine of Chances.” This bv il 
work is far superior, both in research and elegance, to all which had preceded ool 
it; the collection of problems which it contains is far too extensive to admit ae 
any complete notice of it being given in this place: it will be sufficient to in- 2 
stance the doctrine of recurring series, and the theorems on the duration of oh 
play, which are to be met with in it for the first time, to show how much A, 
farther Demoivre carried his inquiries than those who had written on the sub- Fe 
ject before him. Montmort, who had been a personal friend of Demoivre, 
thought he had some reason to complain of the' manner in which Demoivre 5 
spoke of his methods, and a coolness existed in consequence for some time = 
between them. pe 
79. The treatise by James Bernoulli, mentioned above, which was pub- her 
lished in 1715, by his nephew Nicolas, and entitled * Ars Conjectandi,” may nd 
be considered as belonging to the earlier period, at which unquestionably it F-1 
was written. It is divided into four parts; the first consisting of Huyghens’s -, 
treatise, with a commentary on most of the propositions. The second con- x 
tains the abstract theory of combinations, in which are many new and elegant pe 
results ; amongst others, the expression for the sum of the a powers of the he} 
natural numbers, in which series occur for the first time those remarkable 4 
coefficients since become so famous, under the name of Bernoulli’s Num- 3 
bers. A less profitable labour, which is also to be met with in this part of PY 
the work, is the curious analysis of the permutations of the celebrated Latin EK 
verse, Tot tibi sunt dotes, Virgo, quot sidera ceelo, which are determined to pug 
be 3312 in number, without transgressing the laws of Latin metre. It does uu 
not appear that James Bernoulli intended to publish this analysis, which pro 
was found by his nephew among his loose papers. The third part gives the a 
application of the preceding principles to a variety of questions. The follow- a 
ing problem deserves notice, because Bernoulli has given a false, though 2 
plausible solution of it, together with the true one, in order, as he says, to . 
show what care is necessary to avoid error in the discussion of these ques- : 
tions. A isto throw a die, and to repeat his throw as many times as the Je 
number thrown the first time. If the sum of the points given by the latter jai 
set of throws be less than 12, A loses; if more, he wins; if they equal 12, acy 
he takes half the stake, His expectation is required... The true value of his Pe 
-— 15295 _ 
expectation is found to be = STToL rather less than 1, the false solution is > 
as follows :—A has I chance of throwing an ace at the first trial, in which case uy 
he will have but one throw to reckon upon, and as this may equally give 
him any - number from 1 to 6, his chance from it may be reckoned at