On Infinites. 831 



quantity, it is called an infinitesimal." These definitions 

 arc perfectly correct, taking in the two last "any determi- 

 nate limit," and " any determinate quantities," for all deter- 

 minate limits, and all determinate quantities whatever. The 

 idea of mathematical infinity can be clearly obtained only 

 from space and duration. In abstract number if we add 

 millions to millions for ever so long a period, we see with 

 certainty that the numbers thus obtained are in all cases 

 finite. And if we suppose an abstract number infinite in 

 the highest sense, that is, so great that nothing can be added 

 to it, or supposed to be added, the mind sees at once that 

 the supposition is an absurdity. It even appears while we 

 deal only with abstract numbers, as if there could not be 

 such a number really infinite, even in a mathematical sense ; 

 that IS, so great, that though it is capable of increase, it is 

 notwithstanding incalculable by every finite mind. I5ut if 

 we suppose a line iijfinite in one d-rection, and terminating 

 in the other at a given point, this line may properly be 

 said to be mathematically infinite: for though it is evident- 

 ly capable of increase, yet it is measureable by no finite 

 mind. In the same sense past duration has been infinite : 

 it is capable of increase, but the repetition of no finite du- 

 ration can measure it. But in this sense future duration 

 never will and never can be infinite. The infinity which is 

 usually applied to it is the infinity of a mere abstraction of 

 the mind. We see, it is true, that future duration will never 

 terminate, but we see with equal certainty that it will never 

 arrive at infinity. We will at present take it for granted 

 that the infinite line supposed above, and infinite past dura- 

 tion, may be divided into finite parts. Then however far 

 the idea of the finite part which we call uhity is extended, 

 we shall have in each case a number mathematically infi- 

 nite. It is usually said, that we obtain the idea of an infini- 

 tesmal by dividing a given space, or numerical unit into a 

 certain number of parts, then into a greater number, and so 

 on, increasing at each step, till the mind is wearied, and 

 then because we see that the number may be still increased, 

 and the quantity of each part diminished, we conclude 

 there may be a part so small that no finite mind can meas- 

 ure it. It is obvious that these steps are exactly analogous 

 to those which are first taken in quest of the idea of infi- 

 nite space. How the mind draws the conclusion that the 



