MATHEMATICAL IDEAS. 73 



compared and the relations between them established, 

 only by some direct or indirect enumeration of their 

 component units; and the ultimate units into which 

 all others are decomposable, are such occupied posi 

 tions in Space as can, by making impressions on 

 consciousness, produce occupied positions in Time. 

 Among units that are unspecified in their natures 

 (extensive, protensive, or intensive), but are ideally 

 endowed with existence considered apart from attri 

 butes, the quantitative relations that arise, are those 

 most general relations expressed by numbers. Such 

 relations fall into either of two orders, according as 



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the units are considered simply as capable of filling 

 separate places in consciousness, or according as they 

 are considered as filling places that are not only sepa 

 rate, but equal. In the one case, we have that inde 

 finite calculus by which, numbers of abstract existences, 

 but not sums of abstract existence, are predicable. In 

 the other case, we have that definite calculus by which 

 both numbers of abstract existences and sums of 

 abstract existence are predicable. Next comes that 

 division of Mathematics which deals with the quanti 

 tative relations of magnitudes (or aggregates of units) 

 considered as coexistent, or as occupying Space the 

 division called Geometry. And then we arrive at 

 relations, the terms of which include both quantities 

 of Time and quantities of Space those in which 

 times are estimated by the units of space traversed 

 at a uniform velocity, and those in which equal 



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