30 Colorado College Studies. 



"But men did not arrive at this use of the fingers till they had 

 already made some little progress in calculation without them. That 

 this is the true history of the art of counting is evident, if we consider 

 the following facts in order: 



" Fir.st, there is hardly any language in the world in which the first 

 three or four numerals bear, on the face of them, any reference to the 

 fingers. Secondly, there are many savage languages in which these 

 numerals are obviously taken (not from the fingers, but) from small 

 symmetrical groups of common objects. Thus, ' two ' is, among the 

 Chinese, ny and cetd, which also mean 'ears'; in Thibet, paksha, 

 ' wing ' ; in Hottentot, VKoam, ' hand ' ; and so, also, among the Java- 

 nese, Samoyeds, Sioux and other peoples. So, again, with the Abipones, 

 ' four ' is geyenknat^, ' ostrich toes ' : ' five ' is neenhalek, ' a hide spotted 

 with five colours " ; with the Marquesans 'four' is pona, a. 'bunch of 

 four fruits,' etc. Thirdly, there are also many savages who, having 

 only a very few low numerals, count to much higher numbers dumbly 

 by means of the fingers. 



" But just as, in the examples quoted above, the name of the pattern 

 group (e. g. ears or hands) becomes the name of the number which that 

 group contains, so with finger-counting, the savage, advancing in intel- 

 ligence, begins to name the gesture with or without performing it, and 

 this name becomes the symbol of the number, which the gesture is meant 

 to indicate. Hence, all the world over, in nearly every language under 

 the sun where names for the higher units exist and show a clear 

 etymology, the word for 'five' means 'hand,' and the other numbers, 

 up to ten or twenty, as the case may be, are merely descriptive of finger- 

 and-toe-counting." — J. Gow, History of Greek MatJiematics. 188i, 

 pp. G and 7. 



Kemarks. 



(1) From the above citations, copied from representative 

 books of our time, it appears that the mathematicians of the 

 present day are unanimous in describing the earliest notions 

 (jf number as being free from ratio and measurement. It is 

 worthy of notice, moreover, tliatthe great historians of mathe- 

 matics are led from arclneological and ethnological study to 

 results in agreement with those of the mathematicians. 



(2) The only data which must be sujjpliod to the mind 

 through the senses for the cognition of number are the sepa- 

 rdloiess or distinctness of objects. For the earliest cogni- 



