332 ■ On Infinites. 



spaces or expressions may be infinitely small when it knows 

 with certainty that their becoming so by this process is 

 wholly impossible, may seem mysterious. But the myste- 

 ry lies in another step of which perhaps we are not always 

 conscious. We suppose a space between two limits, and 

 substitute it for a!l measureable spaces whatever. We 

 know that the substitution is perfectly correct. Then the 

 space not being in any case absolutely nothing, and the lim- 

 its being of no extent, we see with certaii'ty that another 

 separating limit of no extent can be crowded in between 

 them. It would seem as if no judicious mind could feel a 

 conviction of the possibility of a quantity infinitely small 

 merely from the first similar steps of the process : and even 

 with the last additional one the subject remains considera- 

 bly obscure : for notwithstanding the conditions of the prob- 

 lem, the mind still seizes on the ultimate portions, as measur- 

 able by finite quantities. But suppose a fractional ex- 

 pression, the numerator of which is one, and the denomi- 

 nator a series of figures infinitely extended. That the sup- 

 position of such a series is admissible, we have already 

 taken for granted. The proof will come hereafter. We 

 should then obviously have the expression for an extent so 

 small that no finite mind could measure it. Taking this 

 supposed extent as in the case above, and placing it be- 

 tween two limits, we see with certainty, that it is not abso- 

 lutely nothing: and the suppose d limits having no extent, 

 we see with the same certainty, that an intervening limit of 

 no extent can be pushed in between them. Here then, af- 

 ter the space is infinitely small, we see that it may still be 

 divided and diminished. 



On the subject of an infinite series of units, I am happy 

 to adduce the opinion of the late Professor Fisher, in his 

 own lant(uage. '" If you say it is metaphysically impossi- 

 ble that the earth should have performed infinite revolu- 

 tions about the sun, you maintain that there was a certain 

 revolution and a certain point in the orbit, suppose A, at 

 which it must have begun to move. In other words, it is 

 seen by the mind to be impossible, that the earth should 

 have described an arc of a foot before the point A. But 

 the mind Hoes not in fact perceive any such impossibility : 

 on the other hand, it appears just as easy, that the foot 

 which precedes this point should have been described, a? 



