XVI I.] THE LAW OF ERROK. ?'P3 



Logical Origin of the Law of Error, 



It is worthy of notice that this Law of Error, abstruse 

 though the suhject may seem, is really founded upon the 

 simplest principles. It arises entirely out of the difference 

 "between permutations and combinations, a subject upon 

 "which T may seem to have dwelt with unnecessary prolixity 

 in previous pages (pp. 170, 189). The order in which we 

 add quantities together does not affect the amount of the 

 sum, so that if there be three positive and five negative 

 causes of error in operation, it does not matter in which 

 order they are considered as acting. They may be inter- 

 mixed in any arrangement, and yet the result will be the 

 same. The reader should not fail to notice how laws or 

 principles which appeared to be absurdly simple and 

 evident when tirst noticed, reappear in the most complicated 

 and mysterious processes of scientific method. The funda- 

 mental Laws of Identity and Difference gave rise to the 

 Logical Alphabet which, after abstracting the character of 

 the differences, led to the Arithmetical Triangle. The 

 Law of Error is defined by an infinitely high line of that 

 triangle, and the law proves that the mean is the most pro- 

 bable result, and that divergencies from the mean become 

 much less probable as they increase in amount. Now the 

 comparative greatness of the numbers towards the middle 

 of each line of the Arithmetical Triangle is entirely due 

 to the indifference of order in space or time, which was 

 iir-st prominently pointed out as a condition of logical re- 

 lations, and the symbols indicating them (pp. 32-35), and 

 which was afterwards shown to attach equally to numerical 

 symbols, the derivatives of logical terms (p. 160). 



Verification of the Law of Error. 



The theory of error which we have been considering 

 rests entirely upon an assumption, namely that when 

 Known sources of distui-bances are allowed ibr, there yet 

 remain an indefinite, possibly an infinite number of other 

 minute sources of error, which vvill as often produce ex- 

 cess a.s deficiency. Granting this as.sumption, tlie Law of 

 Error must be as it is usually taken to be, and there is 

 no more need to verify it empirically than to test the truth 



