498 MR W. ROBERTSON SMITH ON HEGEL 



got by allowing the motion at any point to become uniform for a unit of time. 

 But if one wishes, as Hegel would say, to substitute for this notion a convenient 

 " Vorstellung" to assist the imagination, Newton is ready, by means of the 

 doctrine of prime and ultimate ratios, to point out a way in which we may avail 

 ourselves of the method of indivisibles, always remembering that this method 

 shall have merely a symbolic value, and so must be used with caution. 



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 increments 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 ultimately be unity— i.e., that quantity which, varying 

 according 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, which 

 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 reason- 

 ing, 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 in geometry. Everything is as plainly and undeniably reduced to 

 ordinary geometrical intuition as anything in Euclid, when we once bring with 

 us the fundamental kinematical ideas of velocity and acceleration. It is obvious, 



moreover, that to Newton the fraction ~, as above explained, means simply the 



ratio of the rates at which two quantities are flowing at the moment at which 

 they pass together through the point from which we have agreed to reckon their 

 magnitude backwards and forwards. Except where such rates can be assigned 



