xvi.j THE METHOD OF MEANS. 359 



more likely than any other, as a general rule, to Taring us 

 near the truth. The aptarov ^rpov, or the aurea mediocritas, 

 was highly esteemed in the ancient philosophy of Greece 

 and Eome ; but it is not probable that any of the ancients 

 should have been able clearly to analyse and express the 

 reasons why they advocated the mean as the safest course. 

 But in the last two centuries this apparently simple 

 question of the mean has been found to afford a field for 

 the exercise of the utmost mathematical skill, Roger 

 Cotes, the editor of the Principia, appears to have had 

 some insight into the value of the mean; but profound 

 mathematicians such as De Moivre, Daniel Bernoulli, 

 Laplace, Lagrange, Gauss, Quetelet, De Morgan, Airy, 

 Leslie Ellis, Boole, Glaisher, and others, have hardly ex- 

 hausted the subject. 



Several uses of the Mean Result. 



The elimination of errors of unknown sources, is almost 

 always accomplished by the simple arithmetical process 

 of taking the mean, or, as it is often called, the average 

 of several discrepant numbers. To take an average is to 

 add the several quantities together, and divide by the 

 number of quantities thus added, which gives a quotient 

 lying among, or in the middle of, the several quantities. 

 Before however inquiring fully into the grounds of this 

 procedure, it is essential to observe that this one arith- 

 metical process is really applied in at least three different 

 cases, for different purposes, and upon different principles, 

 and we must take great care not to confuse one applica- 

 tion of the process with another. A mean result, then, 

 may have any one of the following significations. 



(1) It may give a merely representative number, 

 expressing the general magnitude of a series of quantities, 

 and serving as a convenient mode of comparing them 

 with other series of quantities. Such a number is properly 

 called The fictitious mean or The average result. 



(2) It may give a result approximately free from 

 disturbing quantities, which are known to affect some 

 results in one direction, and other results equally in the 

 opposite direction. We may say that in this case we get 

 a Precise mean result, 



