ri 



On Infinites. 33 1 



quantity, it is called an infinitesimal," These definitions 



idea of mathematical infinity can he clearly obtained on!) 

 from space and duration. In abstract number if we add 

 miljions to millions for ever so long a period, we see nitli 

 certainty that the numbers thus obtained are in all cases 

 finite* And if we suppose an abstract numher infinite in 

 I the highest sense, that isj 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 i 

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

 Notwithstanding incalculable by every finite mind. Hut if 

 we suppose a line infinite in one direction, and terminating 

 in the other at a given pointy this line may properly be 

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

 ly capable of increase, yet it is measure.ible by no finite 

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

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

 ration can mf;asure it. But in this sense future duration 

 never will and never can be infioUe, The infinity which is 

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

 the mind* We see, it is tru^^, that future duration will never 

 terminate, but we see with equal certainly 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 unity is extended, 

 We shall have in each case a number mathematically infi- 

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

 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 st- ps are exactly analogous 

 to those w^hich are first taken in quest of ihe idea of infi- 

 ^Jte space. How the mind draws the conclusion that the 



