138 Mr. Moon on the Evaluation 



than 0, is — , but let no one thence attempt to draw the in- 



ference that 1 — 1 + 1 — &c. = — - ; for 1 — 1 + 1 — &c. is not the 



2 



limit of either of the series, it is simply the form they respect- 

 ively assume when the variable has its limiting value, which 

 is a very different thing, as we have seen. 



The broad fact which, although as clear as the sun at noon 

 day, so many seem to hesitate to admit, is, that when x is very 

 small, if it be an actual magnitude, . 



l-(l-x)-(l-x)* + &c. 



differs very little from — , but that when x vanishes, it assumes 



two values, I and 0. There is in this case no middle term 

 between entity and non-entity. The idea is simple and the 

 fact certain. 



Prof. Young holds it to be an axiom, that " the value which 

 suffices for all cases except the extreme case, will suffice for 

 that too," or uses words to that effect. This is a most un- 

 warrantable and false assumption. Take the case of the same 

 series, 



1 — X + x 2 — X 3 + &c, 



where x is greater than 1. The value in this case, as is easily 

 seen from equation (</.), is + oo indifferently; and this holding 

 always, except in the extreme case, when x= 1, it would follow, 

 on Prof. Young's principle, that it holds in that too, and 

 therefore that 1 — 1 + 1 — &c. = + co ; and he has before sup- 

 posed it to be equal to — , which is absurd. 



We may hence see the absurdity of any attempt to prove 

 that 1 — 1 + 1 — &c., considered as the limit of 1 — x + # 2 — &c, 



where x is less than 1, is equal to — ; for by this nothing else 



25 



can be meant than to prove that the one series is the limit of 

 the other, which is contrary to the fact. 



Prof. Young's attempt in this respect depends on the as- 

 sumption that ( 1 ) = e y the base of the Napierian sy- 

 stem, which is untrue, and which at any rate I challenge him 



i_ 



to prove. Does he consider that (1 — 0) ° = e? 

 Liverpool, November 10, 1845. 



