24 LECTURES AND ESSAYS [1869- 



If two quantities have the same fluxion at any moment, 

 they begin at that moment to increase at the same rate. 

 It does not follow from this that the two quantities shall 

 receive equal increments in any space of time however 

 small, unless during that time the rates of flow remain 

 constant. But Newton shows that in a very large class 

 of cases, which he takes up one by one in the first section 

 of the Principia, not only may we, by taking the time of 

 flow small enough, make the difference of the increments 

 generated in that time as small as we please, but if we 

 enlarge both increments on the same scale up to any given 

 size, we may make the differences of the increased incre 

 ments as small as we please, while the time of flow has still 

 a definite value. Since, then, the ratio of the increments 

 is always nearer to unity the less the time of flow, and may 

 be brought as near to unity as we please by taking the 

 time short enough, but still finite, the ratio must ultim 

 ately be unity i.e. that quantity which, varying accord 

 ing to a definite rule, always represents at any given time 

 the ratio of the increments, may still be constructed when 

 the time is made zero, and is now equal to unity, or is 

 equal to the ratio at which the increments start, \vhich 

 Newton calls their prime or ultimate ratio. 



The practical application of this reasoning is, of course, 

 that in virtue of it, we may in certain cases with strict 

 accuracy treat the increments of two variables (of a curve, 

 for example, and its tangent) as equal, if, before closing 

 our reasoning, we proceed to take the limit. Thus, if any 

 one finds that it assists his imagination to deal with 

 magnitudes as if they were composed of indivisibles, 

 instead of confining himself to fluxions, Newton provides 

 in the method of prime ratios a criterion by which the 

 applicability of the process may be judged. The details 

 by which it is shown in the Principia, that the limit of the 

 ratio of the increments is equal to the ratio of the fluxions 

 whenever the fluents may be geometrically represented as 

 curves of continuous curvature, involve no new principle 



