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



But, granting that the influence of an error e, ought to be greater when a is less, and 

 versa 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 deter- 

 mine bv introducing the extraneous idea of equilibrium on a lever, that - is the function re- 

 -' ° a 



quired, seems to be little else than a petitio principii, concealed by a metaphor*. 



The second demonstration may be thus briefly stated. 



The values of different sets of observations might be compared if we knew the average error in 

 each set, or if we knew the average value of the squares of the errors in each. In either case that 

 would be the best set of observations in whicli the quantity taken as the measure of precision was 

 the smallest. 



Similarly, by assigning diff'erent values to the unknown quantities >v,y, he. involved in a system 

 of equations of condition, we can make it appear that the mean of the squares of the errors has a 

 m-eater or less value. Therefore as of sets of observations, that is the best in which this quantity 

 is least ; so of different sets of results deduced from one set of observations, the same is also true ; 

 and therefore the sum of the squares of the apparent errors is to be made a minimum. 



There seems to be involved in this reasoning a confusion of two distinct ideas; the precision of 

 a .set of observations is undoubtedly measured by the average of the errors actually committed, and 

 if we knew this average, we should be able to compare the values of diff'erent sets of observations. 

 But it is not measured by the average of tiie calculated errors, namely, those which are determined 

 from the equations of condition when particular values have been assigned to w, y, &c. 



The problem to be solved may be stated thus. Given that the single observations of which the 

 set is composed are liable to a certain average of error, to combine them so that the resulting values 

 of the unknown quantities may be liable to the smallest average of error. 



This problem Laplace and Gauss have both solved. Their solutions diff'er, because they 

 estimated the average error in different manners. 



But how are we justified in assuming that to be the best mode of combining the observations 

 wliicii merely gives the appearance of precision 7iot to the final results, but only to the individual 

 observations, and which, with reference to them, gives no estimation of the probability that this 

 appearance of accuracy is not altogether illusory ? 



The third of Mr. Ivory's demonstrations is not, I think, more satisfactory than the other two. 



The kind of observations to which the method of least squares is applicable, are such, Mr. Ivory 

 observes, that there exists no bias tending regularly to produce error in one direction, and that the 

 error in one case is supposed to have no influence whatever on the error in any other case. 



From this principle he attempts to show that the method of least squares is the only one 

 which is consistent with the independence of the errors. 



When, however, we speak of the errors as being independent of one another, only this can 

 be meant, that tlie circumstances under which one observation takes place do not aff"ect the others. 

 In rerum naturd the errors are independent of one another. Nevertheless, with reference to our 

 knowledge they are not so, that is to say, if we know one error we know all, at least in the case 

 in which the equations of condition involve only one unknown quantity, which is that considered 

 by Mr. Ivory. For the knowledge of one error would imply the knowledge of the true value of 

 the unknown quantity, and thence that of all the other errors. 



• I have omitted to notice some remarks wiiicli Mr. Ivory appends to this demonstration, as they do not appear to affect the view 

 taken in the text. 



