io8 LIMirS AND FLUXIONS 



less than the assign'd difìference. " Mere Robins 

 expresses the idea that the rapidity of approach 

 toward the limit can be arbitrarily altered, yet he 

 does not apparently perceive — certainly he does not 

 admit — that this rapidity may be altered in such a 

 way that the variable actually does reach its limit. 

 On the contrary, he maintains that *'where the 

 approach is determin'd by a subdivision into parts," 

 **it is obvious, that no coincidence can ever be 

 obtain'd." A coincidence such as Philalethes 

 explains in the case of rectangles circumscribed 

 and inscribed in a curve, if it could take place, is 

 not a coincidence such as Newton intended, for 

 Newton did not in this instance use motion, but 

 continuai subdivision. Robins tries to establish 

 his view of the matter by giving an instance of 

 erroneous results being deduced by letting the 

 variable reach its limit. He takes a hyperbola 

 and revolves its principal axis in the piane of the 

 curve, around the point of intersection of this axis 

 and an asymptote, until the two lines coincide. 

 At the end '*the hyperbola coincides with the 

 asymptote," which is "absurd." As a matter of 

 fact there is no absurdity. In lAx^ — a?'y'^ — a^U^ ^ 

 the slope of the asymptote is m = b j a. Robins's 

 process amounts to making in = o, which gives a 

 real locus when b = o, namely the \oc\xs y^ = o. The 

 only objection lies in stili calling the final curve a 

 " hyperbola." 



