BERKELEY'S ANALYST (1734) 71 



Magnitudes of synchronal or isochronal Increments 

 must be infinitely diminished and become evan- 

 escent, in order to obtain their first or last Ratios, 

 to which Ratios the Ratios of their corresponding 

 Fluxions are equal. " The moment of the rectangle 

 AB is Ab+Ba, for consider Ab-\-Ba-\-ab and Ab + 

 Ba, "under a Constant Diminution of the Incre- 

 ments a and b . . . [they] constantly tend to an 

 Equality . . . [and] they become equal, and their 

 Ratio becomes a Ratio of Equality. ..." Hence 

 Ab-\-Ba-\-ab " is not the Moment or Fluxion of 

 the Rectangle AB, except in the very Instant 

 vvhen it begins or ceases to exist." Here fluxions 

 appear to be no longer velocities (finite magnitudes) 

 but moments. Walton next quotes a Latin passage 

 from the Quadratura Curvai um. He says that 

 Berkeley seems ''to have been deceived by an 

 Opinion that there can be no first or last Ratios 

 of mathematical Ouantities," but Walton insists 

 that if quantities are generated together, or if they 

 vanish together, they will do so "under certain 

 Ratios, which are their first or last Ratios." 

 Walton claims that Berkeley's lemma " is in no 

 Way pertinent to the Case for which it was in- 

 tended " ; he explains the Newtonian process of 

 finding the fluxion of x*^, supposing x to increase 

 uniformly, and points out that this is done without 

 rejecting quantities " on account of their exceeding 

 smallness." Commenting on Berkeley's contention 

 that "no geometrie Ouantity, by being infinitely 

 diminished, can ever be exhausted or become 



