ON ERRORS OP OBSERVATION. 133 



to express that errors of all magnitudes are very nearly 

 equally likely, we draw such a curve as the upper 

 dotted line : while, if we wish to express that the pre- 

 sumptions are very strong for each measure being very 

 nearly true/ we describe the lower dotted line. We 

 can thus figure to the eye a representation of the law of 

 errors, be it what it may ; and the description of a law of 

 error is that of a curve. 



Generally speaking, it is impossible to commit an 

 error of more than a certain magnitude : but this cir- 

 cumstance is one which embarrasses the mathematical 

 treatment of the subject exceedingly. It is practically 

 the same thing to consider any error, however great, as 

 possible, but errors of more than a certain magnitude 

 as extremely improbable. If, for instance, a case should 

 arise in which an error of more than an inch is impos- 

 sible, let it be agreed to consider such an error as not 

 impossible, but so improbable that there shall not be 

 an even chance of its happening once in a million of 

 times. The curve which exhibits the law of error 

 must then be of the following kind, never meeting the 

 axis, but continually approaching towards it, so that the 

 whole of its area from and after L, is incomparably 

 small by the side of the area C B L. 



The curve here drawn is something like that in 

 p. 17., and we might suspect from the utility of the 

 latter and of the tables derived from it, that it should 

 play an important part in the present subject. This 

 we shall presently see : in the mean while I proceed to 

 explain the sense in which I use the terms average 

 balance, average error, mean risk of error and probable 

 error, to which I direct the reader's particular at- 

 K 3 



