*THE HINDU-ARABIC NUMERALS*

reference to the counters in the first column, for example, our 1, 2, 3, 4, 5, 6, 7, 8, 9. Then these same numerals may be employed for the second, but they will now have in their new place a new value, let us say, ten times as great, so that a 2 in the second column will denote 2 tens, or 20. And so in other columns, which give the value of 100, or 1000. By this means the entire number of numeral signs may at once be reduced to the number of signs used in the first column. Without the abacus the Greeks must have nineteen signs in counting from one to a hundred: on the abacus there is need only of nine.

The place value assigned on such an instrument would depend upon the practical system of counting which had been developed. Just as nowadays the workman, or the boy playing a game, will tally five, and then begin another five, and then another, thus making a group of fives which he can handle easily, rather than one longer series, so the practical calculators of bygone days worked with fives, or sixes, or tens, or twenties. Various systems have been used. The Babylonians employed the sexagesimal, reckoning by sixties. Some of the African tribes count by sixes, and some of the New Zealanders are said to use elevens The duodecimal or twelve system has passed away, but in the dozen we still preserve traces of it ourselves.

As a rule, however, the system has been none of these, but counting has been done by fives, by tens, or twenties, and this simply because it has been based upon the antique but persistent habit which man has of counting upon his fingers. To this day there is a widespread custom of reckoning roughly by fives. The Mayas of Yucatan used the vigesimal or twenty system; so did the men of Palmyra in Zenobia's time, and the Syrians before the days of Mohammed. The same is said to have been true of the Celts, and the French seem to preserve traces of it now when they say quatre-vingts, four twenties, for 80. But after all that system which has been most widely adopted is the decimal, based upon all the fingers of the two hands.

The Hindus came to employ the decimal system, as did the Greeks and the Chinese. It found place in the written language as well as in the numeral notation; but, as has been said, it existed only in complicated form. Thus, the Greeks and the Romans had words to express numbers from one to ten, as they had signs. In Greek **εἲς** and d denoted one; **δέκα** and **ί,** ten; after which there were words and signs for twenty, thirty, and so on, at intervals of ten up to one hundred; followed by words and signs at intervals of a hundred. In between, the numbers were expressed by combination: **ἕνδεκα** (one-ten), eleven; **δίο** **καῚ τριάκοτα** (two and thirty), thirty-two. The Roman system was entirely similar, except that it employed fewer signs. The Hindus used the decimal system even more consistently, since they preserved it in counting beyond thousands indefinitely.