SECT. 2.] Theory of the Average. 467 



asked to guess the position of the wafer. Had there been 

 only one shot, common sense would suggest our assuming 

 (of course very precariously) that this marked the real centre. 

 Had there been two, common sense would suggest our taking 

 the mid-point between them. But if three or more were 

 involved, common sense would be at a loss. It would feel 

 that some intermediate point ought to be selected, but 

 would not see its way to a more precise determination, be- 

 :ause its familiar reliance, the arithmetical average, does 

 not seem at hand here. The rule in question tells us how to 

 proceed. It directs us to select that point which will render 

 the sum of the squares of all the distances of the various 

 shot-marks from it the least possible \ 



This is merely by way of illustration, and to justify the 

 familiar designation of the rule. The sort of cases with 

 which we shall be exclusively occupied are those compara 

 tively simple ones in which only linear magnitude, or some 

 quality which can be adequately represented by linear mag 

 nitude, is the object under consideration. In respect of these 

 the Rule of Least Squares reduces itself to the process of 

 taking the average, in the most familiar sense of that term, 

 viz. the arithmetical mean ; and a single Law of Error, or its 

 graphical equivalent, a Curve of Facility, will suffice accu 

 rately to indicate the comparative frequency of the different 

 amounts of the one variable magnitude involved. 



1 In other words, we are to take when we are dealing with such cases 



the &quot; centre of gravity &quot; of the shot- as occur in Mensuration, where we 



marks, regarding them all as of equal have to combine or reconcile three or 



weight. This is, in reality, the aver- more inconsistent equations, some 



ige of all the marks, as the elemen- such rule as that of Least Squares 



;ary geometrical construction for becomes imperative. No taking of 



obtaining the centre of gravity of a an average will get us out of the 



system of points will show ; but it is difficulty, 

 lot familiarly so regarded. Of course, 



302 



