ON THE METHOD OF LEAST SQUARES. 15 



not only is Se = ultimately, but S./e = 0, where /e-is any func- 

 tion such that / = / ( e) ; and therefore we should have 



as an equation which ultimately would give the true value of x 

 when the number of observations increases 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 causes had 

 disappeared from a given series of observations by assigning 

 different forms to f, and comparing the different values thus 

 found for a. 



No satisfactory reason can be assigned why, setting aside 

 mere convenience, the rule of the arithmetical mean should be 

 singled out from the other rules which are included in the general 

 equation S/ (x a) = 0. 



Let us enquire, therefore, whether there is any sufficient 

 reason for saying that the rule of the arithmetical mean gives the 

 most probable value of the unknown magnitude. In the first 

 place, it is only one rule out of many among which it has no 

 prerogative but that of being in practice more convenient than 

 any other: in the second place, if this were not so, it would not 

 follow that in the accurate sense of the words it gave the most 

 probable result. This objection I shall defer for a moment, and 

 proceed to consider the manner in which Gauss makes use of the 

 postulate on which his method is founded. 



From the first principles of what is called the theory of 

 probabilities a posteriori, it appears that the most probable value 

 which can be assigned to the magnitude which our observations 

 are intended to determine, is that which shall make the a priori 

 probability of the observed phenomena a maximum. That is to 

 say, if a be the true value sought, a? x being the value observed 

 at the first observation, x z the corresponding quantity for the 

 second, and so on, the errors at the first, second, &c. observation 

 must be x v a, x 2 a, &c., respectively; and if <fte . de be the 

 probability of an error e in any observation of the 



