THE EXACT MEASUREMENT OF PHENOMENA. 319 



on the process ad infinitum. Any finite space, then, must 

 be conceived as made up of an infinite number of parts, 

 each of which must consequently be infinitely small. We 

 cannot entertain some of the simplest geometrical notions 

 without allowing this. The conception of a square in- 

 volves the conception of a side and diagonal, which, as 

 Euclid admirably proves in the nyth proposition of his 

 tenth book, have no common measure f , meaning, as I 

 apprehend, no finite common measure. Incommensurable 

 quantities are, in fact, those which have for their 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- 

 posed 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 

 best measuring instruments, can appreciate. 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 approxi- 

 mations beyond quantities, which the senses can appreciate. 

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

 quantities to be equal, but only to be apparently equal. 

 Legendre really adopts this mode of treatment in the 

 twentieth proposition of the first book of his Geometry ; 

 and it is practically adopted throughout the physical 

 sciences, as we shall afterwards see. But though our 



f See De Morgan, 'Study of Mathematics/ in IT. K. S^ Library, p. 81. 



