| "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., δύο) 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—probably points to a time when more than two was regarded indefi-
