THE LA W OF ERROR. 443 



being both assumed equal to unity, in order to simplify 

 the calculations. In the same figure are inserted eleven 

 dots, whose heights above the base line are proportional 

 to the numbers in the eleventh line of the Arithmetical 

 Triangle, thus representing the comparative probabilities 

 of errors of various amounts arising from ten equal causes 

 of error. It is apparent that 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, for instance 

 NM, nm, n'm f , represent values of y in the equation ex- 

 pressing 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 

 definite 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 exceeding unity will occur, 

 is represented by the area MmWm', in short, by the area 

 standing upon the line nri. Since every observation 

 must either have some definite error or none at all, it 

 follows that the whole area of the curve should be con- 

 sidered as the unit expressing certainty, and the proba- 

 bility 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. 



Derivation of the Law of Error from Simple 

 Logical Principles. 



It is worthy of notice that this Law of Error, abstruse 

 though the subject may seem, is really founded upon the 

 simplest principles. It arises entirely out of the difference 



