i6o THE GRAMMAR OF SCIENCE 



by the aid of our most powerful microscopes. We have 

 then reached a Hmit to our mode of perceiving apart, — in 

 ordinary parlance, to the divisibility of space. We may 

 possibly conceive smaller divisions, but in doing this we 

 have passed from the sphere of the real to the ideal — 

 from the space of perception to the space of geometry. 

 It seems to me that this transition from perception 

 to conception, often made quite unconsciously, is the 

 basis of all the difficulties involved in the paradox as to 

 the infinite divisibility of space. The point has been 

 referred to by Hume in his Essay Concerning Human 

 Understanding} where he writes as follows : — 



" The chief objection against all abstract reasonings 

 is derived from the ideas of space and time — ideas which, 

 in common life and to a careless view, are very clear and 

 intelligible, but when they pass through the scrutiny of 

 the profound sciences (and they are the chief object of 

 those sciences) afford principles which seem full of ab- 

 surdity and contradiction. No priestly dogmas, invented 

 on purpose to tame and subdue the rebellious reason of 

 mankind, ever shocked common sense more than the 

 doctrine of the infinite divisibility of extension, with its 

 consequences, as they are pompously displayed by all 

 geometricians and metaphysicians with a kind of triumph 

 and exultation. A real quantity, infinitely less than any 

 finite quantity, containing quantities infinitely less than 

 itself, and so on in infitutum ; this is an edifice so bold 

 and prodigious that it is too weighty for any pretended 

 demonstration to support, because it shocks the clearest 

 and most natural principles of human reason. But what 

 renders the matter most extraordinary is that these 

 seemingly absurd opinions are supported by a chain of 

 reasoning, the clearest and most natural ; nor is it possible 

 for us to allow the premises without admitting the 

 consequences." 



Now the reader should carefully note the unconscious 

 transition in this passage from the ideas of space and time 

 to the infinite divisibility of real quantities. The transi- 

 1 Section xii. part ii. Green and Grose : Hume's Works, vol. iv. p. 128. 



