82 INTRODUCTION 



Accordingly, for instance, Newton regarded all geo- 

 metrical magnitudes as capable of generation by con- 

 tinuous motion. Lines may be regarded as generated by 

 the motion of points, surfaces by that of lines, and solids 

 by that of surfaces. That is to say, these figures are dis- 

 tinct from one another, not absolutely, but merely in the 

 degree in which they possess certain characteristics. The 

 difference between the point and the line is an infinitely 

 small degree of length, the difference between the line 

 and the surface is an infinitely small degree of breadth, 

 the difference between the surface and the solid is an in- 

 finitely small degree of depth. * Motion, ' in Newton's way 

 of putting it, is in this connexion merely a metaphor for 

 continuity. Again, in physical science we have to deal 

 with phenomena which not merely are variable but are 

 continually varying, and the Infinitesimal Calculus is of 

 the utmost value in enabling us to state the laws of these 

 variations, that is to say, to establish proportions between 

 different sets of constantly changing phenomena. 



The value of the Infinitesimal Calculus in the interpre- 

 tation of nature rests ultimately on this, that the con- 

 ception of ' infinitesimals ' which it employs is a virtual 

 recognition of System in knowledge or of the principle of 

 Becoming as distinct from that of abstract Being. When 

 we say that a thing (a geometrical figure, for instance) has 

 a certain quality or characteristic in an infinitely small 

 amount, we mean that it both has and has not that quality 

 or characteristic, or (to use another metaphor made familiar 

 by Psycho-physics) that it is on the ' threshold ' of having 

 it. The identity of the thing is not merely superficial, 

 of such a kind that when a quality seems to pass away 

 from it the thing ceases to exist and another thing ap- 

 pears ; the identity of the thing is maintained through an 

 indefinite amount of difference. Thus, as we have seen, 

 the point, the line, the surface, and the solid are all recog- 

 nized as differences or relations within one system. So in 

 general, when we have shown that the difference between 



