33G 



On the Fundamental Principles of Mathematics. 



And no velocity could be so great, that the body moved would 

 be in two places, at the same instant; for that would contradict what 

 all experience has shown to be true of the nature of body. Hence 

 the transfer of a body from one place to another, however rapid, 

 must occupy some time : and it is mathematically supposable, 

 and, for aught that can be discerned, physically possible, that the 



body might be made to pass through or over a greater distance in 



the same time ; i. e., any velocity, even that which is too great 

 to be measured, might admit, it would seem, of an increase. 



So also, however great a mechanical force may be applied in 



any case ; another might (for aught that can be discerned) be 

 added to and combined with that force. 



(20. ) It may be observed in brief that the three descriptions 

 of infinity obtain respectively, thus: 



1st. Absolute infinity, when the quantity is so great that there 

 is no limit to it. 



2d. Specific infinity, when this boundlessness exists in certain 

 respects only. 



3d. Relative infinity, when the one of two quantities of the 

 same species is too great to be measured by the other. 



Of Finite Quantities which are specifically infinite in one 



Dimension. 



(21.) The results of the Integral Calculus have long since in- 

 dicated that certain areas whose limits in part are lines intermin- 

 able in one direction, may yet themselves be finite. Such are 

 the areas bounded, in part, by certain curves and their asymptotes. 

 An area may also exist having for its partial 

 limits straight lines, and on one side a line 

 interminable in one direction, or even in 

 both directions, and thus, (19.), specifically 

 infinite ; and yet, as it would seem, be finite 

 in surface. Such an area will exist, if the 

 arrangement of its portions be that repre- 

 sented in the figure; each parallelogram hav- 

 ing its sides in one direction equal, each to 

 each, to those of any other which are situated 

 in the same direction ; but each having its 

 sides, respectively, in the other direction, but 

 one half of the length of those which imme- 

 diately precede them, in the series. Then 

 if Q be the area of the first parallelogram, 

 or that represented as lowest in the figure, 

 the sum of the " infinite series" or 



Fig. 2. 



*7 



S 



a-HQ + i 



the line OY being supposed to be interminable in 



the direction of 



> 



