TIIK GERM OF NUMERICAL IDEAS. 155 



ty. And this notion of duality must necessarily have grown 

 up side by side with those of likeness and equality ; seeing 

 that it is impossible to recognise the likeness of tAvo things 

 without also perceiving that there are two. From the 

 very beginning the conception of number must have been, 

 as it is still, associated with the likeness or equality of 

 the things numbered. If we analyze it, we find that sim 

 ple enumeration is a registration of repeated impres 

 sions of any kind. That these may be capable of enu 

 meration it is needful that they be more or less alike ; and 

 before any absolutely true numerical results can be reach 

 ed, it is requisite that the units be absolutely equal. The 

 only way in which we can establish a numerical relation 

 ship between things that do not yield us like impressions, 

 is to divide them into parts that do yield us like impres 

 sions. Two unlike magnitudes of extension , force, time, 

 weight, or what not, can have their relative amounts esti 

 mated, only by means of some small unit that is contained 

 many times in both ; and even if we finally write down the 

 greater one as a unit and the other as a fraction of it, we 

 state, in the denominator of the fraction, the number of 

 parts into which the unit must be divided to be compara 

 ble with the fraction. 



It is, indeed, true, that by an evidently modern process of 

 abstraction, we occasionally apply numbers to unequal units, 

 as the furniture at a sale or the various animals on a farm, 

 simply as so many separate entities ; but no true result can 

 be brought out by calculation with units of this order. 

 And, indeed, it is the distinctive peculiarity of the calculus 

 in general, that it proceeds on the hypothesis of that abso 

 lute equality of its abstract units, which no real units pos 

 sess ; and that the exactness of its results holds only in 

 virtue of this hypothesis. The first ideas of number must 

 necessarily then have been derived from like or equal mag 

 nitudes as seen chiefly in organic objects ; and as the like 



