Mr. ELLIS, ON THE METHOD OF LEAST SQUARES. 205 



From this imperfect sketch of the history of the subject, we perceive that the methods which 

 have been pursued may be thus classified. 



(l). Gauss's method in the Theoria Motiis, which is followed and developed by Encke and 

 other German writers. 



(2). That of Laplace and Poisson. 

 (3). Gauss's second method. 

 (4.). Those of Mr. Ivory. 

 I proceed to consider these separately, and in detail. 



For the analysis of Laplace and Poisson, I have substituted another, founded on what is 

 generally known as Fouriei-'s theorem, having been first given by him in the Thcorie de la Chaleur. 

 It will be seen that the mathematical difficulty is greatly diminished by the change. 



GAUSS'S FIRST METHOD. 



This method is founded on the assumption that in a series of direct observations, of tiie 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 observations, 

 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 imme- 

 diate 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 



.r, - a = e, 



w^ — a — fia 



&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 



S (.??i - a) = 0\ 



1 

 or, a — - 2i?'i ; 



n 



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 a priori considerations. But 



it is to be remarked, that it is not the only rule to which these considerations might lead ns. 



For not only is 2:e = ultimately, but 2/e = o, where fe is any function sucii that fc = -/( - e); 



and therefore we should have 



2/(.^ - a) = 0, 



as an equation which ultimately would give the true value of .r when the number of observations in- 

 creases sine limite, and which therefore for a finite number of observations may be looked on in 

 precisely the same way as the equation which expresses the rule of the arithmetical mean. There is 

 no discrepancy between these two results. At the limit they coincide: short of the limit both are 

 approximations to the truth. Indeed, we might form some idea how far the action of fortuitous 



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