496 



THE POPULAR SCIENCE MONTHLY 



denoted 24ths, 48ths and 36ths respectively. On the Greek abacus also 

 the right-hand columns were of mixed systems which serve to make the 

 calculating more difficult. 



A late development of the same nature was the reckoning on lines 

 which continued into the sixteenth century. The essentials are similar. 



A glance at the accompanying diagram explains the connection be- 

 tween this system and the decimal place system. The upper part rep- 

 resents the number 4,063, the lower part the number 3,251. It seems 

 such a slight step, after acquiring special symbols for the groups of one 



M 



c 



X 



I 





















































 

 





 















































to nine to construct a symbol to indicate a blank space, but that step 

 took centuries to achieve. 



Of all ancient peoples the Hindus occupied themselves most deeply 

 with numbers. To some of their scholars came the conception of a 

 connection between the infinite of the universe and the infinite of num- 

 bers. This longing for the infinite found expression in the construc- 

 tion of ever increasing numbers. Buddha calculates the number of 

 grains of sand in a mile and shows how to compute the number in a 

 sphere whose radius is the distance to one of the fixed stars. Not con- 

 tent with this, the Buddha goes on to show how even greater numbers 

 may be expressed, arriving at the equivalent in modern exponential no- 

 tation of 10 (7+9 - 46) = 10 421 . The numerals of the ancient Tamils, who, 

 like the mountaineers of Appalachian America, conserve the traditions 

 of a more remote civilization, show us that the Hindu peoples origi- 

 nally had special symbols not only for the first nine units, but also for 

 the nine tens, the nine hundreds and even the nine thousands. The 

 formation of the sequences of large numbers revealed the futility of hav- 

 ing separate signs for the mixed tens and hundreds, with the conse- 

 quent result that they dropped the separate symbols for 20 to 90, 200 

 to 900, and used the pure decimal units in connection with the symbols 

 for one to nine. A similar development took place in quite early times 

 — pre-Christian — among the Chinese but their clumsy notation ob- 

 scured the realization of the possibility of a simpler place system. 



The reading of a large number in Hindu style reveals how close 

 their nomenclature brought them to the place system. 



