58 Prof. Bonkin on certain Questionji relating to 



for May, §§ 17, 18.) The reader who is acquainted with 

 Mr. Ellis's paper, will see that I have been here referring to 

 some parts of his reasoning at pp. 324, 325 ; and is requested 

 also to observe, that in pointing out the invalidity of a particular 

 objection against the Edinburgh Reviewer's result, I am not de- 

 fending his argument, about which I shall say something here- 

 after. 



If it be now asked what positive grounds there are for using 

 the method of least squares in the case of a moderate number of 

 observations, beyond motives of mere convenience, I think it 

 may be answered that the method has been proved by Gauss (in 

 the Theoria Combinationis Ohsei'vationum, ^c.) to be a very good 

 method, though it has not been proved to be the best method. 

 He has not shown that it gives the most probable result ; but he 

 has shown that it gives a result such, that if the whole system 

 of obsen atious were repeated an infinite number of times, the 

 average value of the square of the error would be a minimum. 

 I presume that Mr. Ellis does not mean to imply more than this 

 when he says (p. 321) that "Gauss afterwards gave another 

 demonstration which is perfectly rigorous.'' In fact, Gauss him- 

 self expressly points out that there is something arbitrary in 

 assuming the square of the eiTor as the function whose average 

 value is to be a minimum. (Theor. Comb. § 6.) Perhaps he 

 might have added, that the assumption is less arbitrary than any 

 other which could have been made ; but I shall not attempt to 

 discuss the question how far this fact, supposing it admitted, 

 would tend to give a demonstrative character to the reasoning, 

 considered as an attempt to establish the method of least squares 

 as the best method. The point to be observed is, that though 

 Gauss rigorously demonstrates what he professes to demonstrate, 

 he does not profess to demonstrate the method of least squares, in 

 the sense in which these words would be commonly understood 

 without explanation. 



- I shall conclude with a few general remarks on the other 

 proofs which have been, or may be offered, of this remarkable 

 method. And I must remind the reader, that everything which 

 is here said applies only to the case in which the actual law of 

 facility of errors is not known. 



Since the rigorous solution of the problem is unattainable, 

 every professed solution which puts on an appearance of demon- 

 stration must involve an assumption, leading more or less di- 

 rectly to the employment of a particular law of facility as if it 

 were known to be the actual law. And it would appear natm*al 

 to prefer that solution in which the assumed condition should 

 be most simple, least arbitrary, and most in accordance with 

 common notions and experience. That all solutions which 



