METHOD OF DISCOVERING CAUSAL LAWS 205 



mean is taken. &quot; When, however, as happens in a few cases, some condition 

 is known to be operative which varies as the square of the distance, the 

 Geometrical Mean 1 i.e. \/o gives the more accurate result. For ex 

 ample, if the true weight of an object is sought by weighing it successively in 

 the two scales of an imperfect balance, as gravity is an operative condition, the 

 true result will be the geometrical mean . . . though as, in small numbers, 

 this differs but little from the arithmetical mean, the latter is generally taken 

 as more easily calculated.&quot; a 



The second method, called the &quot; METHOD OF LEAST SQUARES,&quot; is applic 

 able when, as is usually the case, the magnitude we are measuring is com 

 pound, or, in other words, involves measurements of two or more separate 

 quantities one of which is some function of another. Our total measurement 

 will, in such cases, be a combination of two or more distinct series, such as 

 [Pj + P 2 + P 3 + P 4 + ] + [Qi + Q 2 + 0.3 + Q 4 + . . . ] + [R, + R, + 

 R 3 -t- R 4 +...], where P is perhaps some multiple of Q, and Q perhaps 

 some function, such as the square root, of R. Now, we cannot reduce these 

 three series to one, so as to take the arithmetical or geometrical mean of the 

 whole ; because there is already a parity between the errors of the three 

 series, and hence by squaring or otherwise altering the values of any single 

 series, for the purpose of reducing it to terms of another, we alter the value of 

 its errors as compared with those of the other series. We therefore have re 

 course to the Method of Least Squares, which rests upon this theorem : That 

 magnitude is most probably the true magnitude, which makes the sum of the 

 squares of the errors in the actual measurements the LEAST POSSIBLE. Such a 

 magnitude can always be discovered, from the actual values, by an algebraic 

 process. This method is really &quot; an extension of the method of means, in that 

 it indicates the most probable mean in cases which involve a plurality of arith 

 metical means &quot;. 3 The property of having the sum of the squares of the 

 residual errors the least possible is always true of the arithmetical mean 4 ; 

 but it is also true of the mean magnitude of a compound series, a magnitude 

 which cannot be obtained by the simple process of finding the arithmetical 

 mean. The present method is, therefore, &quot; the most general mode of finding 

 the true magnitude from a number of divergent measurements ; but when 

 these measurements involve one magnitude only, the simplest mode of applying 

 the method is to take the arithmetical mean &quot;, 5 



247. &quot; EMPIRICAL LAWS &quot; AND THEIR EXPLANATION : 

 TRANSITION TO PART V. In the present chapter an account 

 has been given of the analytical process by which we seek to 

 discover and establish laws by way of hypothesis ; and the con 

 ditions requisite for the latter were outlined in the last preceding 



1 Rather than the Arithmetical Mean, iJL_. a WELTON op. cit., pp. 184-5. 



3 ibid., p. 186. 



4 For example, the sum of the squares on the residuals in the series 2, 5, 8 [(5 

 2)2 + (8 - 5) 2 = 18] where the middie value is the arithmetical mean, is less than in 

 the series 2, 4, 8 [(4 - 2 ) J + (8 - 4)* =* 20) where the intermediate value is not the 

 arithmetical mean. d WELTON, op. cit., p. 187. 



