42 ON PROBABILITY: 
has become unusual to mention, without adding that he was a man of talent, u 
but no mathematician) was thas comparing events which have no connec. ca 
tion with each other. We shall have occasion presently to mention errors ru 
in the principles of this science committed under the sanction of a name of bd 
greater influence and authority among mathematicians. (a 
66. The other problem (which afterwards obtained the name of the Pro- fall 
blem of Points) appeared to Pascal of greater interest ; he communicated it ous 
to Fermat; Roberval, and others; none of whom, but Fermat, returned him the 
a satisfactory solution. The correspondence which passed on this subject tas 
between Fermat and Pascal appeared in 1679, in the posthumous edition of ! 
Fermat’s works published at Toulouse, and is now also to be found in the 
complete edition of Pascal’s works. Pascal began by considering the 
simplest case, in which one of two players, whom we will call A, wants one, 
and B, the other, two points of winning the game. He determined the si 
required proportion from the consideration that if B win the next point, of ins! 
which his chance is only 1, these players would be in a condition of equality ; che 
and if they were then to separate, the stake ought to be equally divided " 
between them; so that A’s present share should be made up of half the 0 
stake corresponding to his equal chance of winning the next point, and one th 
quarter corresponding to the present chance of his share, if B were to win 
the next ; making 3 in all. This mode of solution is very elegant, but there 
is some difficulty in applying the same principle to more complicated ques- 
tions. It has been adopted by subsequent writers in examining the most 
difficult parts of the theory, but aided by a method of analysis very far 
superior to anything with which Pascal or his contemporaries were acquainted. 
In fact, Pascal was led into an error when he attempted to extend his 
method to the general problem, which occasioned a short controversy between 5 
him and Fermat, who had preferred the more laborious, but also more direct, 
method of enumerating all the possible ways in which the game might be 
terminated, and proportioning the division of the stakes according to the 
numbers which appear favourable to either party. Pascal found some 
difficulty in admitting that this method of Fermat's is good in every case, 
and confirmed himself in his mistake in consequence of an erroneous distri- 
bution which he made of the permutations of three letters, when he attempted 
to apply Fermat’s method to the case of three players. Fermat pointed out 
the nature of his error, at least we may presume so from a letter of Pascal’s, 
in which he retracts his former objection, saying, that Fermat's last remarks 
had been entirely satisfactory. 
67. This correspondence was still unpublished, when Huyghens turned his 
thoughts to the theory of probability, and composed a short Latin treatise, 
‘ De Ratiociniis in Ludo Alea,” first printed by Schooten in 1658 at the 
end of his “ Exercitationes Geometrice.” This is the earliest regular 
treatise on the subject, which thenceforward continued to draw more and 
more the attention of mathematicians. Besides an examination in detail of 
Meré’s questions, Huyghens’ treatise contains the enunciation of the general 
and fundamental theorem of this branch of the science, that if any player 
have p chances of gaining a sum represented by a and ¢ chances of gaining 
b, his expectation (a term then first introduced) will be rightly represented 
by peta? Elementary as this truth may now appear, it was not re- 
ptyq 
ceived altogether without opposition. 
68. In the year 1670, a Jesuit named Caramuel published the two first 
volumes of his course of mathematics, under the title of “ Mathesis Biceps,”