2 NOTION AND DEFINITION OF NUMBER. 



have seen, is the exact, individual conjunction or association of other 

 things with the things to be counted, is to be mentioned a second 

 important element, which in some respects perhaps is not so abso- 

 lutely essential; namely, that the things to be counted shall be re- 

 garded as of the same kind. Thus, any one who subjects apples and 

 nuts collectively to a process of numeration will regard them for the 

 time being as objects of the same kind, perhaps by subsuming them 

 under the common notion of fruit. We may therefore lay down pro- 

 visionally the following as a definition of counting : to count a group 

 of things is to regard the things as the same in kind and to associate 

 ordinally, accurately, and singly with them other things. In writing, 

 we associate with the things to be counted simple signs, like points, 

 strokes, or circles. The form of the symbols we use is indifferent. 

 Neither need they be uniform. It is also indifferent what the spa- 

 tial relations or dispositions of these symbols are. Although, of 

 course, it is much more convenient and simpler to fashion symbols 

 growing out of operations of counting on principles of uniformity 

 and to place them spatially near each other. In this manner are 

 produced what I have called * natural number-pictures ; for ex- 

 ample, 



etc. 



Now-a-days such natural number-pictures are rarely employed, and 

 are to be seen only on dominoes, dice, and sometimes, also, on play- 

 ing-cards. 



It can be shown by archaeological evidence that originally nu- 

 meral writing was made up wholly of natural number-pictures. For 

 example, the Romans in early times represented all numbers, which 

 were written at all, by assemblages of strokes. We have remnants 

 of this writing in the first three numerals of the modern Roman sys- 

 tem. If we needed additional evidence that the Romans originally 

 employed natural number-signs, we might cite the passage in Livy, 

 VII. 3, where we are told, that, in accordance with a very ancient 

 law, a nail was annually driven into a certain spot in the sanctuary of 



* System der Arithmetik. (Potsdam : Aug. Stein. 1885.) 



