1S2 On the Fundamental Principles of Mathematics. 



separately mark out any portion of the solid, the region at which 

 the division is made or indicated, must lie beneath the surface. 

 The surface exists only where the solid, or the space occupied by 



the solid, ends and other space begins. The surface itself occu- 



pies space not at all ; it only divides space. It is not somewhat, 

 in the same sense in which the solid is somewhat, but only some- 

 where: viz., where, as already stated, the solid ends, and space 



exterior to it begins. 



The surface, then, having no capacity, is in that respect a zero 



of solidity ; and we may with propriety say, when a solid such 

 as a cube or a parallelopiped is reduced to its base (its altitude 

 being reduced to zero), that the solid (as such) is reduced to zero. 



The base or other surface, though thus a zero of capacity, is 

 yet somewhat in its own sense — in the sense peculiar (6.) to that 

 species of quantity — viz., in superficial extent ; i. e., it still pos- 

 sesses, as it were, the property of covering or extending over, as 

 well as limiting, a portion of the solid, and also that of divi- 

 ding space. 



But a line existing at the edge of such a surface, or any other, 

 is not somewhat, even in the sense last mentioned, but only some- 

 where: viz., at the very edge of the surface. It does not divide, 

 but only penetrates space. 



If then a figure, such as a parallelogram or triangle, be reduced 

 to its base (its altitude being reduced to zero), the surface of that 

 figure will be reduced lo zero; or the base having no surface, 

 will be in that respect zero; i.e., zero of surface, or of area, 

 which is measured surface. 



A straight line whether it thus exist as the edge of a surface or be 

 otherwise defined (e. g., the axis of a sphere), is yet somewhat in 

 its own sense — in the sense peculiar to all lines — viz., in length; 

 whereby, though it does not divide, it penetrates space. 



A point, at the extremity of such a line, is not someichat in 

 any sense, but only somewhere; viz., at the very end of the line. 



The like is true of a point, though otherwise situated ; e.g, at 

 the centre of a sphere ; where it is precisely at an equal distance 

 from any and every point in the surface. This would cease to 

 be true at any other position ; though it were even at the small- 

 est distance from the centre : so that this last cannot extend some- 

 what in any direction, nor yet be situated anywhere else, than in 

 the position which has been already defined. 



A point is thus the absolute zero of space; having "position, 

 but not magnitude." 



(9.) As in space there is room for the separate existence of all 

 the material substances with which we are conversant, so, in du- 

 ration, there is room, in a metaphorical sense, for the (successive) 

 occurrence of events ; and time is separated into portions, or has 

 their termination marked by the limits of duration, as space is 





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