THE HINDU-ARABIC NUMERALS 607 



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 

 Dowadays 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 prac- 

 tical 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 compli- 

 cated form. Thus, the Greeks and the Bomans had words to express 

 numbers from one to ten, as they had signs. In Greek zU and d de- 

 noted one ; Scko. and t, ten ; after which there were words and signs for 

 twenty, thirty, and so on, at intervals of ten up to one hundred; fol- 

 lowed by words and signs at intervals of a hundred. In between, the 

 numbers were expressed by combination: evSa<a (one-ten), eleven; 8vo 

 Kal TpiaxovTa (two and thirty), thirty- two. The Boman 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. 



