34 ON THE METHOD OF LEAST SQUARES. 



bilities, which appeared to him to be foreign to the question. 

 The grounds of this opinion he has not clearly developed : 

 perhaps the best refutation of it will be found in the unsatis- 

 factory character of the demonstrations which he proposed to 

 substitute for the methods of Laplace and Poisson. In common 

 with many others, Mr Ivory appears to have looked with some 

 distrust on the results obtained by means of this theory : a not 

 unnatural consequence of the extravagant pretensions sometimes 

 advanced on its behalf. 



The first of his demonstrations rests upon what I cannot help 

 considering a vague analogy. In the equation of condition 



e = ax V, 



lie remarks that the influence of the error e on the value of x 

 increases as a decreases, and versd vice : that consequently the 

 case is precisely similar to that of a lever which is to produce a 

 given effect, as of course the length of the arm must vary in- 

 versely as the weight which it supports. 



Consequently, he argues, the condition to be fulfilled, in 

 order that the equations of condition may be combined in the 

 most advantageous manner, is the same as what would be the 

 condition of equilibrium, were a a a" &c. weights on a lever, 

 acting at arms e e e' &c. This condition is of course 



%ae = 0, whence S(a# F) a = 0, 

 the result given by the method of least squares. 



But, granting that the influence of an error e, ought to be 

 greater when a is less, and versd vice, how are we entitled to 

 assume that the case is precisely similar to that of equilibrium 

 on a lever ? Apart from this assumption, there seems to be no 

 reason for inferring that because this influence increases as a 

 decreases, it must therefore vary inversely as a. By what 

 function of a the influence of e ought to be represented, is the 

 very essence of the question ; to determine, by introducing the 



extraneous idea of equilibrium on a lever, that - is the function 



a 



required, seems to be little else than a petitio principii, concealed 

 by a metaphor*. 



* I have omitted to notice some remarks which Mr Ivory appends to this 

 demonstration, as they do not appear to affect the view taken in the text. 



