ON PROBABILITY. 41 
Vai in dM dX) eC ; 
| do pot (F) and (2) may be calculated directly, but it is better to infer them by 
 %: 1 
lope assuming an arbitrary error de, and by finding the corresponding error 6 \, 
Sof da dX ; 
If @ and b be put for F) and (2) the n” observation furnishes the 
fiom oe equation ON, =a, de} 5,3, 
and the two equations which serve to determine §e and 3 = by the method 
of least syuares are 
a 20ON,a,=3%e3 (a) +-w2a,b, 
quant 
By ZEN) b=3eSa,b,+3m3 5) 
i (he ques. If only one quantity has to be determined, this method evidently resolves 
iat sytem itself into taking the mean of all the values given by observation. 
8 39,2, is 64. We shall now, in conclusion, trace the theory of probability through 
the different stages of its progress, and mention the principal writers who 
be the mst have assisted in establishing its principles. The estimation of the probability 
0 that the of a future event, by enumeration of the cases supposed to be similarly circum- 
stanced, does not appear to have been attempted until the early part of the 
seventeenth century ; and the very elementary nature of the first problem of 
which the solution is on record, serves to show that the subject was then 
altogether new. It is contained in a fragment of uncertain date, written by 
the celebrated Galileo, who died in 1642. It was addressed to a friend who 
thought the fact that the points 9 and 10 can both be produced by six 
different combinations of numbers on three dice difficult to reconcile with the 
separately, notorious preference given by gamesters to the latter number beyond the 
alls sena. former. The difficulty is explained by Galileo, by taking into account the 
permutations of the component numbers, and the respective chances of these 
two numbers are thus shown to be as 25 to 27. A correct table is subjoined 
of the permutations of all numbers which can be thrown on three dice ; and 
it is added, that the consideration of this table will serve for the solution of 
other problems of the same nature. All this must be admitted to belong to 
the infancy of the science, nor does it appear that Galileo thought the subject 
of sufficient interest to call for further inquiry. 
65. The history of the theory of probability is generally made to begin 
several years later, when, in the year 1654, the two following problems were 
proposed by the Chevalier de Meré to Blaise Pascal. 
ad hence a Ist. Two players want each a given number of points towards winning. 
a White If they separate without playing out the game, how should the stakes be 
thousand divided between them ? 
Heme. 2d. Inhow many trials is it an even wager to throw sixes upon two dice ? 
1 2CCNTacy We are told in one of Pascal’s letters to Fermat, that his answer to the 
seals by latter question, that the odds are against twenty-four trials, and in favour 
of twenty-five, though undoubtedly correct, scandalized Mr. de Meré, and 
om made him declare loudly that the science of arithmetic is inconsistent with 
heen itself.”* The Chevalier thought that the chances being in favour of throw- 
x abv ing six in four trials with one die, on which are six different numbers, they 
hobs 4] ought also to be in favour of throwing two sixes in six times as many, or 
Wi twenty-four trials with two dice, on which are six times as many, or thirty- 
¢ Tar} six different numbers. Those who have read the preceding pages with any 
degree of attention, will readily perceive that the Chevalier (whose name it 
#¥ Opera Petri de Fermat, Tolose, 1679,