i8 LECTURES AND ESSAYS [1869- 



Methods &quot; of mathematics, to enable him to appreciate 

 Boole s profound preface to his treatise on &quot; Differential 

 Equations.&quot; 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 calcula 

 tions. [See also the treatise De Analysi per Equationes 

 Numero Terminomm Infinites.] 



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 arbitrary assumptions, Newton 



