552 THE ROYAL SOCIETY OF CANADA 



ancy show that it is due to incidental error, the statement of the two 

 in agreement is accepted; and this course is justified on the following 

 purely scientific grounds. If we assume, for the purpose of this illus- 

 tration, that in history as a science, and therefore in any of its trust- 

 worthy records, the proportion of incidental error and the average of 

 essential correctness are respectively 1/50 and 49/50, then the pure 

 probability that the single record is in incidental error on the point 

 in contradiction will be 1 /50, and the probability that it is correct 

 will be 49/50 or 49:1. On the other hand, the pure probability that 

 the two records in agreement are both in incidental error on the point 

 ( = joint occurrence of two independent events) will be the product of 

 their separate probabilities (1/50 X 1 /50) = 1 /2,500, and the proba- 

 bility that they are correct will be 2,499 /2,500 or 2,499 :1. The chances 

 that the two records are right, rather than the one, will be therefore as 

 2,499:49 = 51:1, and the pure probability in favor of their correctness 

 will be 51 /52. In the formulation of an historical narrative or in any 

 other considerable historical research, there will be a series of occa- 

 sions where the above rule is applicable, and the degree of probability 

 involved (51 /52) will develop in this series an average of correct results 

 corresponding approximately to the average of essential correctness 

 (49/50) required by history as a science in such a narrative or research. 



(c). Formal Probability (the Product of Correct Processes). 



In the introduction to the present part of this paper (p. 492), it 

 was noted that, under accredited (competent) operation, correct 

 processes cannot ordinarily prevent incidental error entirely, but they 

 guarantee in a series of results an average of essential correctness 

 corresponding in height to the exactness of the branch of science in- 

 volved. By setting the proportion of correct results in such a series, 

 e.g., in Chambers' Mathematical Tables, against the proportion of 

 unlocated, incidental errors, one can strike a ratio between the re- 

 spective proportions, and so reduce to a form of probability the ques- 

 tion of the correctness of any individual result in the table or series. 

 Thus, in the mathematical tables in question, containing approximately 

 200,000 numerical quantities, if the number of correct quantities 

 be taken as 199,990 and the number of unlocated, incidental errors 

 as 10, an approximate probability of 199,990/200,000 or 199,990:10 

 may be deduced, that any individual quantity in the work is correct; 

 and thus also, in the foregoing illustration of the function of pure 

 probability, assuming that the average of essential correctness in a 

 trustworthy record is 49 /50, or 49 correct statements to every unlocated 

 incidental error, a probability of 49 /50or 49 :1 can be deduced (apart from 

 contradictions or confirmations by other records) that any individual 



