382 THE PRINCIPLES OF SCIENCE. [CHAP. 



of error. The correspondence of the general and the 

 special Law of Error is almost as close as can be exhibited 

 in the figure, and the assumption of a greater number of 

 equal causes of error would render the correspondence far 

 more close. 



It may be explained that the ordinates NM, nm, n'm', 

 represent values of y in the equation expressing the Law 

 of Error. The occurrence of any one definite amount of 

 error is infinitely improbable, because an infinite number 

 of such ordinates might be drawn. But the probability of 

 an error occurring between certain limits is finite, and is 

 represented by a portion of the area of the curve. Thus the 

 probability that an error, positive or negative, not exceed- 

 ing unity will occur, is represented by the area Mmnn'mf, 

 in short, by the area standing upon the line nn. 

 Since every observation must either have some definite 

 error or none at all, it follows that the whole area of the 

 curve should be considered as the unit expressing certainty, 

 and the probability of an error falling between particular 

 limits will then be expressed by the ratio which the area 

 of the curve between those limits bears to the whole area 

 of the curve. 



The mere fact that the Law of Error allo\vs of the possi- 

 ble existence of errors of every assignable amount shows 

 that it is only approximately true. We may fairly say 

 that in measuring a mile it would be impossible to commit 

 an error of a hundred miles, and the length of life would 

 never allow of our committing an error of one million 

 miles. Nevertheless the general Law of Error would assign 

 a probability for an error of that amount or more, but so 

 small a probability as to be utterly inconsiderable and 

 almost inconceivable. All that can, or in fact need, be 

 said in defence of the law is, that it may be made to re- 

 present the errors in any special case to a very close 

 approximation, and that the probability of large and prac- 

 tically impossible errors, as given by the law, will be so 

 small as to be entirely inconsiderable. And as we are 

 dealing with error itself, and our results pretend to nothing 

 more than approximation and probability, an indefinitely 

 small error in our process of approximation is of no import- 

 ance whatever. 



