14 ON THE METHOD OF LEAST SQUARES. 



rem, having been first given by him in the Theorie de la Chaleur. 

 It will be seen that the mathematical difficulty is greatly dimin- 

 ished by the change. 



GAUSS'S FIEST METHOD. 



This method is founded on the assumption that in a series of 

 direct observations, of the same quantity or magnitude, the 

 arithmetical mean gives the most probable result. This seems so 

 natural a postulate that no one would at first refuse to assent to 

 it. For it has been the universal practice of mankind to take 

 the arithmetical mean of any series of equally good direct ob- 

 servations, and to employ the result as the approximately true 

 value of the magnitude observed. 



The principle of the arithmetical mean seems therefore to be 

 true a priori. Undoubtedly the conviction that the effect of 

 fortuitous causes will disappear on a long series of trials, is an 

 immediate consequence of our confidence in the permanence of 

 nature. And this conviction leads to the rule of the arithmetical 

 mean, as giving a result which as the number of observations 

 increases sine limite, tends to coincide with the true value of the 

 magnitude observed. For let a be this value, x the observed 

 value, e the error, then we have 



x l -a=e l 



x 2 -a = e 2 



&c. = &c. 



And as on the long run the action of fortuitous causes disappears, 

 and there is no permanent cause tending to make the sum of the 

 positive differ from that of the negative errors, Se = 0, and 

 therefore 



or, a- 5 



which expresses the rule of the arithmetical mean, and which is 

 thus seen to be absolutely true ultimately when n increases sine 

 limite. 



In this sense therefore the rule in question is deducible from 

 & priori considerations. But it is to be remarked, that it is not 

 the only rule to which these considerations might lead us. For 



