

On the Fundamental Principles of Mathematics. 181 



s 



Thus a straight line and a curve are of the same species ; since 

 each exceeds its less in length — in which respect alone a line can 



be either great or small. But, a straight line and a square are of 



different species ; since the one exceeds its less in length, while 

 the other exceeds its less in surface ; and this, although the boun- 

 danes or limits of the square are, themselves, straight lines. 



A straight line and an hour are quantities of a different species, 

 since the one exceeds its less in length, but the other exceeds its 

 less (e. g., a minute) in duration. 



(7.) So fundamental and inherent is the distinction between 

 quantities of different species, that the combination of them by- 

 addition, or the attempt to subtract one from the other, or to com- 

 pare what constitutes greatness in the one species with that which 

 constitutes it in the other, will all be found to be impracticable, 

 and even manifestly absurd. 



Thus a straight line cannot be added to a day, nor a pound in 

 weight be subtracted from the surface of a triangle; nor can we 

 say of an hour and a square that one is larger than the other, or 

 even compare them at all as to greatness. 



The single point of resemblance between quantities of different 

 species, is that indicated (5.) in the definition already given of 

 quantity in general ; viz., that the distinction of greater and less 

 in some seiise, is every where admissible. Hence it is possible to 



| compare the ratio of two quantities of one species with that o( 



tWo other quau ti ties of another species : or even that an equality 



J of such ratios should exist; one of the first pair being precisely 



as great or small in comparison with the other, in the peculiar 

 sense of great or small which belongs to that species, as one of 

 the other pair of quantities is great or small in comparison with 

 |he other, though in the peculiar sense of great or small which 

 belongs to that species. Thus, 2 feet : 1 foot : : 2 hours : 1 hour. 



Of the Limits of Various Quantities, and the Nature of Zero. 



(&) The nature of the boundaries or limits of the quantities 

 of various species will next be considered ; and this will naturally 

 lead to an examination of the nature of zero. 



The limits of bounded space being the most obvious, and also 

 those with which we are most familiar, may well claim our 

 attention first. 



Solids, or rather volumes, occupy space; and their limits are 

 surfaces. In accordance with what has already been advanced, 

 (2.) it will be observed, that it is with the form, capacity, &c, 

 of the space thus occupied, that the mathematician, as such, has 

 t0 do, and not with the nature of the substance to which they 



m *y appertain. 



A. surface (i. e., the very outside) of a solid, although it bounds 

 that solid, is yet no part of the solid itself. To remove or even 



