XVII.] THE LAW OF ERROR. 379 



rare occurrence. If we now suppose the errors to act as 

 often in one direction as the other, the effect will l3e to 

 alter the average error by the amount of two inches, and 

 we shall have the following results : — 



Negative error of 2 inches i way. 



Negative error of i inch 4 ways. 



No error at all 6 ways. 



Positive error of i inch 4 ways. 



Positive error of 2 inches i way. 



We may now imagine the number of causes of error 

 increased and the amount of each error decreased, and the 

 arithmetical triangle will give us the frequency of the re- 

 sulting errors. Thus if there be live positive causes of 

 error and five negative causes, the following table shows 

 the numbers of errors of various amount which will be the 

 result : — 



It is plain that from such numbers I can ascertain 

 the probability of any particular amount of error under 

 the conditions supposed. The probability of a positive 



error of exactly one inch is > in which fraction the 



*' 1024 



numerator is the number of combinations giving one 

 inch positive error, and the denominator the whole 

 number of possible errors of all magnitudes. I can also, 

 by adding together the appropriate numbers get the pro- 

 bability of an error not exceeding a certain amount. Thus 

 the probability of an error of three inches or less, positive 

 or negative, is a fraction whose numerator is the sum of 

 45 + 120 -t- 210 -I- 252 + 210 + 120-1-45, and the deno- 

 minator, as before, giving the result — ^. We may see at 



once that, according to these principles, the probability of 

 small errors is far greater than of large ones : the odds are 

 1002 to 22, or more than 45 to i, that the error will not 



