TEE ORIGIN OF NUMBER SYSTEMS. 527 



nitely as many. 3. The names of the six numerals, from 5 to 10, 

 may possibly be derived etymologically from a hand or finger 

 source. 



Mr. Gow's statements with regard to the grammatical char- 

 acter of the words is easily seen to be quite what we should 

 expect. When man first begins to count, the numerals do not 

 represent to him abstract ideas. He does not think of " 3," for 

 example, as a mere number, as we do. It represents nothing to 

 him if separated from the group of objects which it is used to 

 describe. The numeral words, therefore, naturally take on the 

 form of adjectives. Later, when his ideas are further developed 

 and he begins to use his fingers in counting, the real idea of 

 number begins to assert itself, and the words used to designate 

 different numbers appear in a more abstract form. The fingers 

 become, in fact, when used thus, real numerical symbols, as much 

 so as written ones, and the mind gradually becomes accustomed 

 to thinking of the number of a set of objects as something which 

 can be considered apart from the objects themselves, and which 

 can be represented just as well by an equal number of other 

 objects differing from the first set in all respects except that of 

 number. Thus the abstract idea of number is formed. 



Mr. Gow's ideas upon grammatical number seem to me, if 

 they have the significance he claims, to point back still further. 

 As soon as the idea of two or more as distinct from one is con- 

 ceived, the necessity for a new grammatical form arises. Now, 

 if the number sense were at all developed, the formation of gram- 

 matical number ought to stop here ; for it would be apparent at 

 once that to have a different grammatical form for every num- 

 ber is impracticable. But we find a distinct inflection set apart 

 to express two, and a new inflection to designate three or more. 

 The existence of this third or plural number would then indicate 

 that the idea of three, or many, as distinct from two corresponds 

 to another step in the development of the number sense. That 

 the process of forming new grammatical numbers went no fur- 

 ther then becomes an argument to show that the subsequent 

 number development was more rapid, and the impossibility of 

 making the former keep pace with the latter was realized. 



The etymology of the first three or four of our numerals is 

 probably quite beyond our reach. It has already been pointed 

 out that in the selection of words to represent 1 or 2 the 

 savage has such a wide range of objects to choose from that it is 

 very much a matter of chance what he will select. Any concrete 

 object that possesses the essential quality of unity or duality 

 may be impressed into the service. For 3 and 4 the range of 

 objects that will serve his purpose is more limited, but it is 

 still sufiiciently large to make it a mere accident what he will 



