462 PREVOST AND LHUILIER. 



The memoir first notices tlie criticism given in Lambert's Orga- 

 non of James Bernoulli's formula winch we have already given in 

 Art. 122. 



It then passes on to the theory of concurrent testimony now 

 commonly received. Suppose a witness to speak truth m times and 

 falsehood n times out oi m + n times ; let m and ?i' have similar 

 meanings for a second witness. Then if they agree in an assertion 



the probability of its truth is -, r • 



mm ■\-n7i 



The ordinary theory of traditional testimony is also given. 

 Using the same notation as before if one witness reports a state- 

 ment from the report of another the probability of its truth is 



mm' + nn 



J\ 5 



{m + m') {71 + n) 



for the statement is true if they both tell the truth or if they both 

 tell a falsehood. If there be two witnesses in succession each of 

 whom reverses the statement he ought to give, the result is true ; 

 that is a double falsehood gives a truth. It is stated that this con- 

 sequence was first indicated in 1794 by Prevost. 



The hypothesis of Craig is noticed ; see Art. 91. 



The only new point in the memoir is an hypothesis which is 

 proposed relating to traditional testimony, and which is admitted 

 to be arbitrary, but of which the consequences are examined. The 

 hypothesis is that no testimony founded on falsehood can give the 

 truth. The meaning of this hypothesis is best seen by an example: 



suppose the two witnesses precisely alike, then instead of taking 



2 I 2 

 071 "t" n 



-7 rr as the probability of the truth in the case above considered 



{m -\-n) 



m^ 



we should take rj : that is we reject the term n^ in the 



[m + n) ** 



numerator which arises from the agreement of the witnesses in a 



falsehood. 



rrn J 1 ^^ 1 ^nm -\-n^ , , . , 



Ihus we take 7 V2 ^'^^ -, \¥ to represent respectively 



[in + n) {in + ^) ^ ^ -^ 



the probabilities of the truth and falsehood of the statement on 

 which the witnesses agree. 



Suppose now that there is a second pair of witnesses inde- 

 pendent of the former, of the same character, and that the same 



