334 On the Fundamental Principles of Mathematics. 



of absolutely infinite space, as well as extend through what, for 

 want of a better term, must be called its length.] 



A surface without border would be specifically infinite in 

 length, breadth, and a contiguous and plane superficial area. 



[A surface without border and altogether plane, must, notwith- 

 standing, be regarded as less than another surface, which though 

 alike without border, yet deviates any where from a plane ; since 

 the latter not merely extends through all space in every direction 

 which can be called length and breadth, such as exist upon a 

 plane, but also encroaches upon what we may, in such a compar- 

 ison, term the thickness of space.] 



The quantities here characterized as specifically infinite are 

 innumerable; and some maybe parallel to one another: while, 

 in so far as we can discern, there is but one absolute space ; i. e., 

 one absolute infinite of extension. So also there is but one ab- 



finite of 



We 



ifi 



other be too great to be expressed by any assignable number, 

 however large. 



Any assignable number, however large, may be exceeded by 

 the continued addition of the number 1 to itself; and then again 

 to the sum, &c. &c. ; and the like must be true with regard to 

 any series or aggregate of the units of any species; when the 

 number of units is assignable. But if, instead of continually 

 adding the original unit, or its equivalent, we take its double, 

 and of that product its double, &c f &c, and continue the pro- 

 cess of successive doubling, until the number of such individual 

 processes is as large as any number which we can assign ; the 

 aggregate will far exceed that obtained by successive additions, 

 repeated as often. 



For in the one series, the quantity to be added, at each succes- 

 sive step, is constant ; so that if Q, denote the original unit, the 

 aggregate of the series or 



s = a + a + a + &c ; 



but in the process of continued doubling, each term consists of 

 the aggregate of all that preceded, added to as much as itself; 

 and therefore the sum of such a series, or rather the resulting 

 aggregate, 



S' = Q + Q + (2Q)-f (4Q) 



the terms after the second continually increasing. If then the 

 number of terms in each series be as great as any that we can 

 assign, the number of times the original quantity Q, contained in 

 the aggregate of the second series will be too great to be assigned; 

 and will in any case exceed the number of units such as U, 

 which we may assign to the first series, however great that nimi- 



