XIII.] MEASUREMENT OF PHENOMENA. 275 



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as that which is divisible without limit. We can divide 

 a millimetre into ten, or a hundred, or a thousand, or tea 

 1 iiousand parts, and mentally at any rate we can carry 

 on the division ad infinitum. Any finite space, then, 

 must be conceived as made up of an infinite number of 

 }javts each infinitely small. We cannot entertain the 

 simplest geometrical notions without allowing this. The 

 conception of a square involves the conception of a side 

 and diagonal, which^ as Euclid beautifully proves in the 

 117th proposition of his tenth book, have no common 

 measure,^ meaning no finite common measure. Incom- 

 mensurable quantities are, in fact, those which have for tlieir 

 only common measure an infinitely small quantity. It is 

 somewhat startling to find, too, that in theory incommen- 

 surable quantities will be infinitely more frequent than 

 commensurable. Let any two lines be drawn haphazard ; 

 it is infinitely unlikely that they will be commensurable, 

 so that the commensurable quantities, which we are sup- 

 ])osed to deal with in practice, are but singular cases 

 among an infinitely greater number of incommensurable 

 cases. 



Practically, however, we treat all quantities as made up 

 of the least quantities which our senses, assisted by the 

 b(;st measuring instruments, can perceive. So long as 

 microscopes were uninvented, it was sufficient to regard 

 an inch as made up of a thousand thousandths of an 

 inch; now we must treat it as composed of a million 

 millionths. We might apparently avoid all mention of 

 infinitely small quantities, by never carrying our appro.K:i- 

 inations beyond quantities which the senses can appreciate. 

 In geometry, as thus treated, we should never assert two 

 (piantities to be equal, but only to be cqji'arently equal, 

 l^egendre really adopts this mode of tj'eatment in the 

 twentieth proposition of the first book of his Geometry ; 

 and it is practically adopted throughout the ])hysical 

 sciences, as we shall afterwards see. l>ut though our 

 fingers, and senses, and instruments must stop somewhere, 

 there is no reason why the mind should not go on. We 

 can see that a proof which is only carried through a few 

 steps in I'act, might be carried on without limit, and it is 



* See Df M(iiyan, Study of Mathcmalics, in U.K.S. LilnMiv, p. 81. 



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