78 INTRODUCTION 



relations of quantity (that is to say, of mere aggregation) 

 may become signs or symbols of relations which are 

 more than quantitative, relations in which the part is 

 not indifferent to the whole but characteristic of it. All 

 the processes of Algebra, however complex and elaborate, 

 are forms of the addition and the subtraction (or separa- 

 tion) of abstract units. Thus the abstract number i 

 remains the same, into whatever algebraic combination 

 it may enter as a part. But the conception of a straight 

 line, for instance, varies (the line has various functions) 

 according to the nature of the whole into which it enters 

 as a part, and according to the special way in which it is 

 related to the whole. Thus in relation to different kinds 

 of figures (rectilineal, curved, &c.), or on account of the 

 various forms of its relation to one and the same figure, 

 a straight line is a side, a tangent, a raclius, a directrix, 

 an axis, a sine, &c. There is a closer, more real unity 

 between the part and the whole than in the relation of 

 mere quantity, where the part is indifferent to the special 

 character of the whole. 



Relations of purely quantitative Unity and geometrical 

 Unity. Infinite Series and the infinitely little. 



But there is no absolute gulf fixed between quantita- 

 tive unity and geometrical unity. The difference is, 

 that geometrical unity, while abstract in comparison 

 with organic unity or with the real concrete unity of all 

 existence, is less abstract than merely quantitative unity 1 . 

 And the bridge between the unity which is expressed in 

 the Algebra of finite quantities and that which is expressed 

 in the Geometry of finite space-relations is to be found in 

 the analysis of a finite quantity into an infinite series. 

 No finite quantity can be resolved into an infinite series 



1 Strictly speaking, a merely quantitative unity is a contradiction 

 in terms, for mere quantity is pure difference, the absence of unity. 

 But what I mean here is unity of the lowest degree, unity on the 

 point of vanishing, or the most indeterminate unity. 



