

NOTION AND DEFINITION OF NL'MUKK. 3 



Minerva, the ' ' inventrix " of counting, for the purpose of showing the 

 number of years which had elapsed since the building of the edifice. 

 We learn from the same source that also in the temple at Volsinii 

 nails were shown which the Etruscans had placed there as marks for 

 the number of years. 



Also recent researches in the civilisation of ancient Mexico show 

 that natural number-pictures were the first stage of numeral nota- 

 tion. Whosoever has carefully studied in any large ethnographical 

 collection the monuments of ancient Mexico, will surely have re- 

 marked that the nations which inhabited Mexico before its conquest 

 by the Spaniards, possessed natural number-signs for all numbers 

 from one to nineteen, which they formed by combinations of circles. 

 If in our studies of the past of modern civilised peoples, we meet 

 with natural number-pictures only among the Greeks or Romans, 

 and some Oriental nations, the reason is that the other nations, as the 

 Germans, before they came into contact with the Romans and adopted 

 the more highly developed notation of the latter, were not yet suffi- 

 ciently advanced in civilisation to feel any need of expressing num- 

 bers symbolically. But since the most perfect of all systems of nu- 

 meration, the Hindu system of "local value," was introduced and 

 adopted in Europe in the twelfth century, the Roman numeral sys- 

 tem gradually disappeared, at least from practical computation, and 

 at present we are only reminded by the Roman characters of inscrip- 

 tions of the first and primitive stage of all numeral notation. To- 

 day we see natural number-pictures, except in the above-mentioned 

 games, only very rarely, as where the tally-men of wharves or ware- 

 houses make single strokes with a pencil or a piece of chalk, one for 

 each bale or sack which is counted. 



As in writing it is of consequence to associate with each of tin- 

 things to be counted some simple sign, so in speaking it is of con 

 sequence to utter for each single thing counted some short sound. 

 It is quite indifferent here what this sound is called, also whether 

 the sounds which are associated with the things to be counted arc 

 the same in kind or not, and finally, whether they are uttered at 

 equal or unequal intervals of time. Yet it is more convenient and 

 simpler to employ the same sound and to observe equal intervals in 



