SECT. 33.] Theory of the Average. 497 



32. Suppose then that two scouts were sent to take 

 the calibre of a gun in a hostile fort, we may conceive that 

 the fort was to be occupied next day, and used against the 

 enemy, and that it was important to have a supply of shot or 

 shell, and that the result is that one of them reports the 

 calibre to be 8 inches and the other 9. Would it be wise to 

 assume that the mean of these two, viz. 8J inches, was a 

 likelier value than either separately? 



The answer seems to be this. If we have reason to 

 suppose that the possible calibres partake of the nature of a 

 continuous magnitude, i.e. that all values, with certain 

 limits, are to be considered as admissible, (an assumption 

 which we always make in our ordinary inverse step from an 

 observation or magnitude to the thing observed or measured) 

 then we should be justified in selecting the average as the 

 likelier value. But if, on the other hand, we had reason to 

 suppose that whole inches are always or generally preferred, 

 as is in fact the case now with heavy guns, we should do 

 better to take, even at hazard, one of the two estimates set 

 before us, and trust this alone instead of taking an average 

 of the two. 



33. The principle upon which we act here may be 

 stated thus. Just as in the direct process of calculating or 

 displaying the errors , whether in an algebraic formula or in 

 a diagram, we generally assume that their possibility is 

 continuous, i.e. that all intermediate values are possible; so, 

 in the inverse process of determining the probable position of 

 the original from the known value of two or more errors, we 

 assume that that position is capable of falling at any point 

 whatever between certain limits. In such an example as the 

 above, where we know or suspect a discontinuity of that 

 possibility of position, the value of the average may be 

 entirely destroyed. 



v. 32 



