324 Mr. R. L. Ellis on an alleged proof 'of 



equal positive and negative errors are equally probable. And 

 the same is of course true with respect to the remaining ele- 

 ments. 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 suffi- 

 ciently large. It nowise follows that these values are the most 

 probable; 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, 

 a 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 

 difference 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 rale this proposition, if 

 true, could only be proved inductively, and not by andpriori 

 demonstration. For it is beyond question distinctly concei- 

 vable, that different laws may exist in different classes of ob- 

 servation; and that which is distinctly conceivable is a priori 

 possible. So that we cannot prove it to be impossible, though 

 we may be able to show empirically that it is not true. 



You will probably agree with me in thinking that a wrong 

 notion of Laplace's reasoning lies at the root of the reviewer's 

 new demonstration. But we now come to the demonstration 

 itself. The assumption that the law of error is in all cases 

 the same, is, we are told, justified by our ignorance of the 

 causes on which errors of observation depend. The law 

 " must necessarily be general, and apply alike to all cases, 

 since the causes of error are supposed alike unknown in all." 

 Two remarks are suggested by this statement : in the first 

 place, that our ignorance of the causes of error is not so great 



