TEXT-BOOKS OF MIDDLE OFCENTUR F 191 



any Thing assignable, is the same Thing as no 

 Difference at ali : for repeat it as often as you 

 please, it can never be equal to any finite Ouantity : 

 and therefore can bear no Ratio to it, by Def. 4, 

 Bk. 5 [of Euclid's] Elements" (p. 37). Stewart 

 gives definitions of ultimate ratio of quantities and 

 of evanescent quantities, also definitions of prime 

 ratio of quantities and of nascent quantities. The 

 foUowing is a specimen: ''The ultimate Ratio of 

 evanescent Quantities is the Limit to which the 

 Ratio of variable Quantities diminishing without 

 Bound, continually approaches, to come nearer to 

 it than by any given Difference ; but which never 

 goes beyond ; yet no sooner attains to, than the 

 Quantities being diminished infinitely, vanish." The 

 following additional statement follows closely the 

 language of Newton (p. 39): '' If any one should 

 object that there can be no ultimate Ratio of 

 continually diminishing and at last evanescent 

 Quantities : because before they vanish it is not the 

 last ; and after they vanish, they have no Ratio. 

 The Answer is, that the ultimate Ratio is neither 

 the Ratio of them before they vanish ; nor after 

 they vanish ; but the Ratio wherewith they vanish, 

 or the Limit to which their varying Ratio no sooner 

 arrives, than they vanish ; . . . that Ratio they 

 have that very Instant they vanish. . . . It signi- 

 fies nothing to say ultimate Quantities cannot be 

 assigned, in regard Quantity is divisible without 

 End : for it is not the Quantities themselves that 

 are hereby determined, but only their Ratio : 



