140 Mr. Moon on the Evaluation 



Prof. Young appears to be haunted by the ghost of an ar- 

 gument, and it is somewhat difficult for others whose rest is 

 not scared by the same phantom, to tell what he is driving at ; 

 but it appears to me that his difficulties, whatever they may 

 be, arise from an inaccurate mode of expression which has 

 crept into use, and of which he has not perceived the impro- 

 priety. Thus later on, we find him saying, " The real error, 

 so frequently committed in analysis, consists in confounding 



— + cos 9 + cos 20 + &c. in inf. 



with the limit of 



— - -f A cos 9 + A 2 cos20 -f &c. in inf. 



38 



and calling [1.], when A = 1, the sum of the former." Now 

 I would observe that the expression, the limit of the series 



i+ A cos + A 2 cos 2 0, 

 2 



is a relative term, and in this case means the value to which 

 the series tends, when the difference between A and 1 gra- 

 dually diminishes; and from which value (or rather quantity) 

 it can be made to differ by a quantity less than any that can 

 be assigned. But there is no propriety in the expression, " the 

 limit of the series when A= 1." The phrase should be, the 

 particular value of the series when A= 1. It is perfectly true 

 that so long as the difference between A and 1 is an actual 

 magnitude (which phrase I use advisedly, as the abuse of the 

 expressions finite quantities and indefinitely small quantities 

 has led many people to believe that there is after all no essen- 

 tial difference between entity and non-entity), the series by di- 

 minishing that difference can be made to differ from 

 {1 — A 2 "1 1 



2(l-2Acos0 + A 2 ) fU L 6 ' fl '° m ^1 



2 



by a quantity less than any that can be assigned ; but it does 

 not thence follow, nor is it the fact, that when A = 1 the 



series becomes = t- ; for from the same evidence as 



2sin^— - 

 2 



that by which we are led to the conclusion that when A < 1 



limit, 



1- A cos + A 2 cos 2 + &c. in inf. = r-, 



2 sin 2 — 

 2 



