526 POPULAR SCIENCE MONTHLY. 



" 10 = One (group), two fives (hands), half-a-man, one man. 



" 15 = Ten-five, one foot, three fives. 



" 20 = Two tens, one man, two feet." 



One of the most significant things to be observed in this table 

 is the absence of any reference to the figures in the numerals for 

 1, 2, 3, and 4. This strongly confirms the view already expressed, 

 that counting began before the use of the fingers as an aid was 

 thought of. The higher numerals, on the contrary, are made up 

 almost entirely of finger- words and their adjuncts. This does not 

 appear in the English translations, but in the original words it is 

 seen at once. 



We may say, therefore, that the human race in learning to 

 count passes through three stages. In the first stage the fingers 

 are not used; progress is very slow; no distinct conception of 

 numbers greater than two or three is formed ; all beyond this is 

 "many." Indeed, in this stage it is altogether probable that no 

 conception of number, properly so called, is formed at all ; that 

 is, the idea of the number of things in a group is not distinctly 

 abstracted from the objects themselves. In the second stage the 

 fingers and toes are used, and counting can be carried as far as 

 ten or twenty, or perhaps, by the use of more than one man, even 

 a little further ; but corresponding numeral words are not yet 

 invented, so that counting is by gestures. In the third stage, 

 words or expressions describing the gestures used in the second 

 stage are assigned to do duty as numerals, and in the course of 

 time they become pure numeral words that is, they are used 

 merely to indicate numbers, the mind no longer thinking of them 

 as describing gestures that once served the same purpose. 



The question now arises whether we can find any trace of 

 finger-counting in our own numerals, and whether we can trace 

 the origin of the lower numerals those in which we should not 

 naturally expect to find a finger origin. Mr. James Gow, of 

 Cambridge, in his Short History of Greek Mathematics, Chapter 

 I, gives some reasons that seem to show that our own Aryan 

 ancestors, like other races, could not at first count beyond three 

 or four, and afterward learned to count on their fingers. His 

 reasons are three, as follows: 1. The words for 1, 2, 3, and 4 

 show a different grammatical character from the next six. He 

 says (page 2) : " The first three are adjectives, agreeing with only 

 casual and partial exceptions (e. g., 8vo) in gender and case with 

 the substantives which they qualify. The same might be said of 

 the fourth, but that in Latin quattuor is wholly indeclinable. 

 The rest, from 5 to 10, are generally uninflected, and have, or 

 had originally, the form of a neuter singular." 2. The existence 

 of three grammatical numbers singular, dual, and plural prob- 

 ably points to a time when more than two was regarded indefi- 



