56 REMARKS ON AN ALLEGED PROOF OF 



observations is very large, this mean value approximates to a 

 certain limit, the form of which is independent of the law of 

 probability. The essence of Laplace's demonstration consists 

 in its enabling us to determine this limi't. When this is done, 

 it may easily be shown that the most advantageous system of 

 factors, those, namely, which make this limiting mean value 

 of the error a minimum, will give the same value to the 

 element to be determined as the system of final equations 

 obtained by employing the method of least squares, provided 

 equal positive and negative errors are equally probable. And 

 the same is of course true with respect to the remaining 

 elements. Thus this system of final equations gives to each 

 element a value affected by a smaller average error than any 

 other linear system, if the number of observations is sufficiently 

 large. It nowise follows that these values are the most pro- 

 bable ; that is, that the errors which must have been com- 

 mitted if these are the true values, form a combination a priori 

 more probable than the errors which in like manner have been 

 committed if any other set of values are the true ones. The 

 most advantageous set of factors for determining any element 

 depends only on the coefficients of the equations to be dis- 

 cussed, and not on their constant terms, which are the direct 

 result of observation. Thus these factors are determinable, ^ 

 priori, before the observations are made. But it is only after 

 the observations have been made that the most probable values 

 of the elements can be found, and then only if we know the 

 law of probability of error. Laplace has pointed out the differ- 

 ence between the two investigations. 



This difference, however, the reviewer does not seem to 

 have apprehended. He plainly supposes that Laplace proves 

 the results of the method of least squares to be the most pro- 

 bable results, which can only be the case, as Gauss had in 

 effect shown, if a special law of error obtains. He therefore 

 undertakes to prove, that for all kinds of observations this is 

 actually the only possible law. 



But for the supposed authority of Laplace, he would pro- 

 bably have perceived that nothing can be more unlikely than 

 that the errors committed in all classes of observations should 

 follow the same law; and that at any rate this proposition, if 

 true, could only be proved inductively, and not by an d priori 



