158 LIMITS AND FLUXIONS 



The author speaks of *'that most accurate defìnì- 

 tion of the ultimate ratio's of vanìshing quantities ; 

 which we have at the latter end of Sch, Lemma XI 

 Princip. [see our §§ 10-15], ^^^ which is so plain, 

 that I wonder how our author [Berkeley] could help 

 understanding it ; which had he done, I am apt to 

 think that ali his Analyst says concerning the pro- 

 portion of quantities vanishing with the quantities 

 themselves, had never been heard : For according 

 to this definition, we are not obliged to consider the 

 last ratio as ever subsisting between the vanishing 

 quantities themselves. But between other quantities 

 it may subsist, not only after the vanishing 

 quantities are quite destroyed, but before when 

 they are as large as you please. And the reason 

 why we consider quantities as decreasing continually 

 till they vanish, is not in order to make, but to find 

 out, this last ratio. Sir Isaac Newton does not 

 indeed say that this last ratio is the ratio with which 

 the quantities themselves vanish ; but whether he 

 herein speaks with the utmost propriety or not, is 

 a mere nicety on which nothing at ali depends " 

 (p. 16, note). 



Velocity " signifies the degree of quickness with 

 which a body changes its situation in respect to 

 space"; the fluxion of a quantity ''signifies the 

 degree of quickness with which the quantity changes 

 its magnitude. " "And when our author asserts, that 

 in order to conceive of a second fluxion, we must 

 conceive of a velocity of velocity, and that this is 

 nonsense ; he plainly appeals to the sound and 



