450 THE PRINCIPLES OF SCIENCE. 



of recording with an observation the value or degree of 

 confidence with which they regard it, freely estimated 

 according to the impression of success or failure in accuracy 

 immediately after the observation. This value is usually 

 expressed in a decimal scale, so that 10 denotes the 

 highest degree of satisfaction with the result, and I the 

 least degree. Before taking the mean of the observations 

 each number is multiplied by its weight or value, and the 

 sum of the products is divided by the sum of the weights. 

 Thus if a, b, c, &c., be the observed numbers, arid w, w , iv&quot;, 

 &c., the weights, then the most probable mean is 



cw&quot; + . . . . ,, . , . .,, , , . 



71 TT-T -Lhis formula, it will be observed, is 

 +w + . . . . 



identical in form with that for finding the centre of 

 gravity of particles of different weights arranged in a 

 straight line. When we regard w, w , ID&quot; , &c., as all equal, 

 it becomes identical with the formula for the ordinary 

 mean. This method of weighting observations, now of 

 much importance in astronomical and other very exactly 

 quantitative investigations, appears to have been first pro 

 posed by Roger Cotes, the editor of the Principia/ as 

 pointed out by De Morgan 11 . 



The practice of giving weights would open the way to 

 much error and abuse, if the weights were assigned when 

 the mean was being drawn, and when the divergence of 

 some results from the others would be likely to become 

 the guide. As a general rule the weights must be as 

 signed at the moment of observation, and afterwards 

 rigidly maintained, and they must be assigned not from 

 regard to the apparent intrinsic accuracy of the result, 

 but the extrinsic circumstances which seem to render it 

 valuable. . An observed result, in short, must be discre 

 dited, not because it is divergent, but because there were 

 other reasons to suppose that it would be divergent. 



n Penny Cyclopsedia, art. Least Squares, 



