468 Theory of the Average. [CHAP. xix. 



3. We may conveniently here again call attention to a 

 misconception or confusion which has been already noticed 

 in a former chapter. It is that of confounding the Law of; 

 Error with the Method of Least Squares. These are things 

 of an entirely distinct kind. The former is of the nature of 

 a physical fact, and its production is one which in many 

 cases is entirely beyond our control. The latter, or any 

 simplified application of it, such as the arithmetical average, 

 is no law whatever in the physical sense. It is rather a 

 precept or rule for our guidance. The Law states, in any 

 given case, how the errors tend to occur in respect of their 

 magnitude and frequency. The Method directs us how to 

 treat these errors when any number of them are presented \ 

 to us. No doubt there is a relation between the two, as will j 

 be pointed out in the course of the following pages ; but 

 there is nothing really to prevent us from using the same 

 method for different laws of error, or different methods for] 

 the same law. In so doing, the question of distinct right 

 and wrong would seldom be involved, but rather one of morel 

 or less propriety. 



4. The reader must understand, as was implied in 

 the illustration about the pistol shots, that the ultimate 

 problem before us is an inverse one. That is, we are sup-| 

 posed to have a moderate number of errors before us and 

 we are to undertake to say whereabouts is the centre from 

 which they diverge. This resembles the determination of a 

 cause from the observation of an effect. But, as mostly 

 happens in inverse problems, we must commence with the 

 consideration of the direct problem. In other words, so far 

 as concerns the case before us, we shall have to begin by 

 supposing that the ultimate object of our aim, that is, the 

 true centre of our curve of frequency, is already known to 

 us : in which case all that remains to be done is to study the 



