494 MR W. ROBERTSON SMITH ON HEGEL 



determining it." What follows will, I hope, serve to show that these facts imply 

 that Newton had all along a firm grasp of the principle of his method, and that 

 his frequent employment of abbreviated practical processes was really based on 

 a consciousness of the strength of his method, according to the general principle 

 of mathematicians, who never hesitate to apply the boldest symbolical methods 

 in detail, when they feel confident of the starting-point in the use of these 

 symbols. This, in fact, is a point that metaphysicians have never properly 

 attended to. One is disposed to cap Dr Stirling's wish that some great analyst 

 would study Hegel, by expressing a hope that some metaphysician of real ability 

 may pay sufficient attention to what are technically called the " Symbolical 

 Methods" of mathematics, to enable him to appreciate Boole's profound preface 

 to his treatise on " Differential Equations." This exercise would at least make 

 it clear that metaphysical criticism on mathematics is still — I speak without any 

 desire to be disrespectful — in the circle- squaring stage, i.e., still treats as the real 

 questions for discussion points that mathematicians have long seen to be merely 

 special cases of general principles, and therefore to be no longer possessed of 

 independent interest. 



To return from this digression. Newton saw that there were two ways in 

 which quantities might be conceived as generated. The first of these is that 

 which the usual processes of arithmetic have made familiar to everybody, viz., 

 the addition of discrete units. The theory of numbers thus viewed is contained 

 in the arithmetic of integers, to which may be added the doctrine of arithmetical 

 fractions as an extension of the method, reached by supposing the unit itself to 

 change in value. Newton was especially attentive to the importance of the 

 doctrine of decimal fractions, in which the change of unit is so regulated as to 

 give the greatest possible increase of power that the arithmetical conception of 

 quantity admits of ; and the opening pages of his " Geometria Analytica" are 

 expressly directed to show that these advantages may be made available in literal 

 as well as in numerical calculations. [See also the treatise " De Analysi per 

 Equationes Numero Terminorum Infinitas."] 



Newton saw, however, that arithmetic in its most perfect form could give full 

 mastery over quantity, only on the supposition that quantity, as it comes before 

 us in the universe, is always produced by the synthesis of ultimate units, or, in 

 other words, of indivisibles. And this, says Newton, is contrary to what Euclid 

 has proved concerning incommensurables in the tenth book of the Elements 

 (Princ. lib. i. sec. i. schol.) 



Instead, therefore, of endeavouring to eke out this view of quantity by arbi- 

 trary assumptions, Newton resolved to turn to Nature herself, and inquire how 

 quantity is really generated in the objective universe. " Lineae," he writes 

 " describuntur ac describendo generantur non per appositiones partium sed per 

 motum continuum punctorum ; superficies per motum linearum ; solida per 



