466 Theory of the Average. [CHAP. xix. 



logical principles underlying the methods and processes in 

 question are not, I apprehend, particularly difficult to grasp : 

 though, owing to the extremely technical style of treatment 

 adopted even in comparatively elementary discussions of the 

 subject, it is far from easy for those who have but a moderate 

 command of mathematical resources to disentangle these 

 principles from the symbols in which they are clothed. The 

 present chapter contains an attempt to remove these difficul 

 ties, so far as a general comprehension of the subject is con 

 cerned. As the treatment thus adopted involves a con 

 siderable number of subdivisions, the reader will probably 

 find it convenient to refer back occasionally to the table of 

 contents at the commencement of this volume. 



2. The subject, in the form in which we shall discuss 

 it, will be narrowed to the consideration of the average, on 

 account of the comparative simplicity and very wide preva 

 lence of this aspect of the problem. The problem is however 

 very commonly referred to, even in non-mathematical treatises, 

 as the Rule or Method of Least Squares ; the fact being 

 that, in such cases as we shall be concerned with, the Rule 

 of Least Squares resolves itself into the simpler and more 

 familiar process of taking the arithmetical average. A very 

 simple example, one given by Herschel, will explain the 

 general nature of the task under a slightly wider treatment, 

 and will serve to justify the familiar designation. 



Suppose that a man had been firing for some time with a 

 pistol at a small mark, say a wafer on a wall. We may take 

 it for granted that the shot-marks would tend to group 

 themselves about the wafer as a centre, with a density vary 

 ing in some way inversely with the distance from the centre. 

 But now suppose that the wafer which marked the centre 

 was removed, so that we could see nothing but the surface of 

 the wall spotted with the shot-marks ; and that we were 



