the " Method of Least Squares" 325 



but that we have exceedingly good reason to believe that they 

 operate differently in different classes of observations ; and in 

 the second, that mere ignorance is no ground for any infer- 

 ence whatever. Ex nihilo nihil. It cannot be that because 

 we are ignorant of the matter we know something about it. 

 Or are we to believe that the assumption is legitimate, inas- 

 much as it in a manner corresponds to and represents our 

 ignorance? But then what reason have we for believing that 

 it can lead us to conclusions which correspond to and repre- 

 sent outward realities? And yet the reviewer at the conclu- 

 sion of his proof asserts, that, on the long run, and exceptis 

 excipiendis, the results of observation " will be found to group 



themselves according to one invariable law." Thus the 



assumption, though "it is nothing more than the expression 

 of our state of complete ignorance of the causes of error and 

 their mode of action," leads us by a few steps of reasoning to 

 the knowledge of a positive fact, and makes us acquainted 

 with a general law, which is as independent of our knowledge 

 or our ignorance as the law of gravitation. 



Let us, however, suppose it to be true that the law of error 

 is always the same, and that equal positive and negative errors 

 are equally probable. To determine the special form of the 

 law, the reviewer employs a particular case — he supposes a 

 stone to be dropt with the intention that it shall fall on a 

 given mark. Deviation from this mark is error; and the pro- 

 bability of an error r may be expressed by the function f(r q ) 

 or/(# 2 + ?/ 2 ), the origin of coordinates being placed at the mark. 

 It is of course supposed that equal errors in all directions are 

 equally probable. We have now only to determine the form 

 of/. This the reviewer accomplishes in virtue of a new as- 

 sumption, namely, that the observed deviation is equivalent 

 to two deviations parallel respectively to the coordinate axes, 

 " and is therefore a compound event of which they are the sim- 

 ple constituents, therefore its probability will be the product 

 of their separate probabilities. Thus the form of our unknown 

 function comes to be determined from this condition, viz. that 

 the product of such functions of two independent elements is 

 equal to the same function of their sum." Or in other words, 

 we have to solve the functional equation 



But it is not true that the probability of a compound event is 

 the product of those of its constituents, unless the simple 

 events into which we resolve it are independent of each other; 

 and there is no shadow of reason for supposing that the oc- 

 currence of a deviation in one direction is independent of that 



