6o6 THE POPULAR SCIENCE MONTHLY 



Here the characters for 1, 4, 6, and 7 are easily recognizable, while the 

 2, 3, 5, and 9 can be developed without difficulty. 



This system is neither different from that of the Greeks nor 

 superior to it. Neither has a zero, and in neither have the characters 

 a value of place. That is, all of the important numbers must be 

 designated by signs of their own. In our system 2 moved one place to 

 the left becomes 20, but in Greek ft= 2 and k'= 20. So, in the Nasik 

 inscription 2 and 20 are designated by characters separate and distinct. 



In this far there is similarity; but while the Greeks used the letters 

 of their alphabet, the Hindus did not. The meaning of some of these 

 signs, such as the strokes for 1, 2, and 3, is apparent, but the origin 

 and meaning of others are not known. They may have been made 

 from alphabetic characters, but as yet no theory has been substantiated. 



It may be seen that all of the different systems of numerical nota- 

 tion which had been developed, whether in China, in India, or in 

 Greece, had the same general characteristics : there was no zero, and the 

 symbols had no place value. Because of this it was necessary to employ 

 a great many different symbols. Such a system might be used for 

 mathematical calculation, but it was bound to be complicated and 

 intricate. What was needed was a system with fewer signs; but when 

 this was constructed, as it was among the Romans, and the Greeks of 

 Solon's time, it was so rigid and inflexible as to admit of no progress 

 in mathematics. Something altogether different was needed. 



Gradually, by processes of which we know little, a revolution was 

 wrought in all mathematical work and new numeral systems were 

 developed. This revolution was effected by the use of the counting- 

 board, or, as we now call it, the abacus, from a Greek word the mean- 

 ing of which is in dispute. At first all calculation was probably mental 

 or performed upon the fingers, but as time went by, a mechanical 

 device was perfected wherever men strove to make readier and more 

 elaborate computations. This device is said to have been invented by 

 the Chinese, though of such tradition there is no certain proof At all 

 events it was used by the Chinese, the Babylonians, and the Hindus in 

 immemorial times. The Greeks and the Eomans had it; and it con- 

 tinued to be used in Europe throughout the middle ages. According 

 to the " Dialogus de Scaccario " of the twelfth century, the officials of 

 the exchequer reckoned the king of England's revenue by means of it. 

 To this day it is employed generally in Eussia, and in schools wherever 

 children are learning to count. 



Fundamentally the abacus consists of a board or table marked off 

 in parallel columns within which counters can be placed. The prin- 

 ciple is the same if the counters are strung along parallel wires. The 

 important thing is that on the abacus each column has a value of its 

 own, a value of place. Thus, several numerals may be employed with 



