TI1E LA W OF ERROR. 439 



arguments are satisfactory and conclusive. The law 

 adopted is chosen rather on the grounds of convenience 

 and plausibility, than because it can be seen to be the 

 true and necessary law. We can however approach the 

 subject from an entirely different point of view, and yet 

 get to the same result. 



Let us assume that a particular observation is subject 

 to four chances of error, each of which will" increase the 

 result one inch if it occurs. Each of these errors is to be 

 regarded as an event independent of the rest and we can 

 therefore assign, by the theory of probability, the com- 

 parative probability and frequency of each conjunction of 

 errors. From the Arithmetical Triangle (pp. 208, 213) we 

 learn that the ways of happening are as follows : 



No error at all . . . i way. 



Error of i inch . ... 4 ways. 



Error of 2 inches . . .6 ways. 



Error of 3 inches . . .4 ways. 



Error of 4 inches . . i way. 



We may infer that the error of two inches is the most 

 likely to occur, and will occur in the long run in six cases 

 out of sixteen. Errors of one and three inches will be 

 equally likely, but will occur less frequently ; while no 

 error at all, or one of four inches will be a comparatively 

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

 often in one direction as the other, the effect will be 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 



