338 



On the Fundamental Principles of Mathematics. 



If a point (P) be assumed in a straight line, (19.), specifically 



Fig. 3. 



™ 



P 



p 





infinite, the line in either direction from that point will be inter- 

 minable; and the two portions (if they may so be called), one 

 on each side of the point, may be regarded as being, in effect, 

 equal. If, again, another point (P') be assumed in the same line, 

 however remote from the former, the two portions, one on each 

 side of it, may again be regarded as being, in effect, equal ; though 

 the whole intervening distance (PP') will have been added to one of 

 the portions into which the line was divided at the first point (P) 

 and subtracted from the other portion. It appears therefore that 

 any such distance, however great, must be regarded as nothing in 

 comparison with a straight line interminable in only one direction. 



Similar reasoning applied to the case of any other of the spe- 

 cifically infinite quantities described in (19.) ; (or rather to what 

 in like manner may be regarded as being in effect their halves,) 

 would lead to a similar conclusion with respect to them. 



If moreover a plane without border be extended any where in 

 space, all that region of absolute space on the one side of the 

 plane must be regarded as being, in effect, the half of all space 

 and all that region on the other side of the plane, as being, in effect, 

 the other half of the same. But the like will be true of the 

 regions found, the one on one side, and the other, on the other 

 side of a plane of the same description, parallel, it may be, to the 

 former, but at any distance from it, however remote ; the one 



Fig. 4. 



7 



being without border and coinciding, in direction in space, with 

 AB; the other alike without border, but coinciding, indirection 

 in space, with CD. But if the dividing or separating limit be at 

 one of these planes, instead of the other, all the intervening space 

 will, as it were, have been added to the one half of all space, and 

 taken from the other. Yet the two regions which are separated 

 by the second plane of which CD is a part, are still to be regard- 

 ed as being, in effect, equal, or each as still, in effect, the half of 

 all space.* Hence, all the intervening space separated from the 



* In this as in other instance.- it will be observed, that the truth arrived at, in so 

 far as the so-called halves are concerned, admits of being otherwise illustrated. 

 When a straight lin< such as PP' in the figure at the run acement of this article 

 (22.) is finite, the middle is at an equal distance from each end. 



^ But when the line is interminable in both directions, there is no extremity in 

 either direction to measure from, and thus determine the middle. The middle 



