440 



THE PRINCIPLES OF SCIENCE. 



increased and the amount of each error decreased, and the 

 arithmetical triangle will always give us the proportional 

 frequency of the resulting errors. Thus if there be five 

 positive causes of error and five negative causes, the fol 

 lowing table shows the comparative numbers of aggregate 

 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. Thus the probability of a positive 



error of exactly one inch is 



1024 



in which fraction the 



numerator is the exact 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 + 210+252 + 210+120 + 45, and the denomi 

 nator, 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: thus the odds are 1002 to 22, or more than 45 to i, 

 that the error will not exceed three inches ; and the odds 

 are 1022 to 2 against the occurrence of the greatest pos 

 sible error of five inches. The existence of no error at all 

 is the most likely event ; but a small error, such as that of 

 one inch positive, is little less likely. 



