376 THE PRINCIPLES OF SCIENCE. [CHAP. 



According to Gauss the Law of Error expresses the 

 comparative probability of errors of various magnitude, and 

 partly from experience, partly from a priori considera- 

 tions, we may readily lay down certain conditions to which 

 the law will certainly conform.. It may fairly be assumed 

 as a first principle to guide us in the selection of the 

 law, that large errors will be far less frequent and probable 

 than small ones. We know that very large errors are 

 almost impossible, so that the probability must rapidly 

 decrease as the amount of the error increases. A second 

 principle is that positive and negative errors shall be 

 equally probable, which may certainly be assumed, because 

 we are supposed to be devoid of any knowledge as to the 

 causes of the residual errors. It follows that the proba- 

 bility of the error must be a function of an even power of 

 the magnitude, that is of the square, or the fourth power, 

 or the sixth power, otherwise the probability of the same 

 amount of error would vary according as the error was 

 positive or negative. The even powers re 8 , x*, re 6 , &c., are 

 always intrinsically positive, whether x be positive or 

 negative. There is no d priori reason why one rather than 

 another of these even powers should be selected. Gauss 

 himself allows that the fourth or sixth power would fulfil 

 the conditions as well as the second ; l but in the absence 

 of any theoretical reasons we should prefer the second 

 power, because it leads to formulae of great comparative 

 simplicity. Did the Law of Error necessitate the use of 

 the higher powers of the error, the complexity of the 

 necessary calculations would much reduce the utility of 

 the theory. 



By mathematical reasoning which it would be unde- 

 sirable to attempt to follow in this book, it is shown 

 that under these conditions, the facility of occurrence. 

 or in other words, the probability of error is expressed 

 by a function of the general form e~* s **, in which x repre- 

 sents the variable amount of errors. From this law, 

 to be inoj-e fully described in the following sections, it at 

 once follows that the most probable result of any observa- 



1 Methode des Moindres Carres. Memoires sur la Combinaison des 

 Observations, par Ch. Fr. Gauss. Traduit en Franqais par J, 

 Bertrand, Paris, 1855, pp. 6, 133, &c. 



