6o LIMITS AND FLUXIONS 



a certain point is supposed, by virtue of which 

 certain other points are attained ; and such supposed 

 point be itself afterwards destroyed or rejected by 

 a contrary supposition ; in that case, ali the other 

 points attained thereby, and consequent thereupon, 

 must also be destroyed and rejected, so as from 

 thenceforward to be no more supposed or applied in 

 the demonstration.' This is so plain as to need no 

 proof"(§ 12). 



79. Berkeley examines now the method of obtain- 

 ing the fluxion oi x*^ by writing x-\-o in the place 

 of ;r, expanding by the binomial formula, writing 

 down the increments of x and x"", which are in the 

 ratio of 



I to «;ir«-i + ^-^^^?-Z^W-2+ etc, 

 2 



or, when the increment is made to vanish, in the 

 ratio of I to nx*"'^. Berkeley argues : 



" But it should seem that this reasoning is 

 not fair or conclusive. For when it is said, let 

 the increments vanish, i.e. let the increments be 

 nothing, or let there be no increments, the former 

 supposition that the increments were something, or 

 that there were increments, is destroyed, and yet a 

 consequence of that supposition, i.e. an expression 

 got by virtue thereof, is retained. Which, by 

 the foregoing lemma, is a false way of reasoning. 

 Certainly when we suppose the increments to vanish, 

 we must suppose their proportions, their expres- 

 sions, and everything else derived from the supposi- 

 tion of their existence, to vanish with them (§ 13). 



