436 THE PRINCIPLES OF SCIENCE. 



tion; Herschel rests upon geometrical considerations; while 

 Laplace and Quetelet regard the Law of Error as a de- 

 velopment of the doctrine of combinations ; that of Gauss 

 may be first noticed. 



The Law of Error expresses the comparative probability 

 of errors of various magnitude, and partly from ex- 

 perience, partly from a priori considerations, 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 im- 

 possible, 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 pro- 

 bable, 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 probability 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 accordingly as the error was 

 positive or negative. The even powers x 2 , a? 4 , a; 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 powers would 

 fulfil the conditions as well as the second 6 , but in the 

 absence of any theoretical reasons we should prefer the 

 second power, because it leads to formulae of great com- 

 parative simplicity. Did the Law of Error necessitate the 

 use of the higher powers of the error, the complexity of 



c ' Me'thode des Moindres Carres.' ' Me'moires sur la Combinaison des 

 Observations, par Ch. Fr. Gauss. Traduit en Fran^ais par J. Bertram!, 

 Paris, 1855, PP- 6, 133, &c. 



