NOTES. 

 / 



Systeme von Zahlzeichen und iiber den Ursprung des Stellenwerthes in dea 

 indischen Zahlen, in Crelle's Journal f iir die reine und angewandte MatLematik, 

 Bd. iv. (1829), S. 205231 ; compare also my Examen crit. de 1'Hist. de la 

 Geographic, T. iv. p. 275. The simple relation of the different methods 

 which nations, to whom the Indian arithmetic by position was unknown, em- 

 ployed for expressing the multiplier of the fundamental group, contains, I 

 believe, the explanation of the gradual rise or origin of the Indian system. 

 If we express the number 3568, either perpendicularly or horizontally, by means 

 ef "indicators," which correspond to the different divisions of the Abacus, 

 (thus, M c X l) we staU easilv P erceive tliat the group-signs (M C X I) 

 could be left out. But our Indian numbers are no other than these indicators ; 

 they are the multipliers of the different groups. We are also reminded of this 

 designation (solely by means of indicators) by the ancient Indian Suanpan (the 

 reckoning machine which the Moguls introduced into Russia), which has 

 successive rows or wires representing the thousands, hundreds, tens, and units. 

 These rows would present, in the numerical example just cited, 3, 5, 6, and 8 

 balls. In the Suanpan, no group-sign is visible : the group-signs are the 

 positions themselves; and these positions (rows or wires) are occupied by 

 units (3, 5, 6, and 8) as multipliers or indicators. In both ways, whether 

 by the written or by the palpable arithmetic, we arrive at position-value, and 

 at the simple use of nine numbers. If a row is empty, the place will be un- 

 filled in writing. If a group (a member of the progression) is wanting, the 

 vacuity is graphically filled by the symbol of vacuity (sunya, sifron, tziiphra). 

 In the " Method of Eutocius," I find, in the group of the myriads, the first 

 trace of the exponential system of the Greeks so important for the East : 

 M a , M*, M Y , designate 10000, 20000, 30000. That which is here applied 

 only to the myriads extends among the Chinese and Japanese, who derived 

 their instruction from the Chinese 200 years before the Christian Era, to all 

 the multipliers of the groups. In the Gobar, the Arabian " dust writing," (dis- 

 covered by my deceased friend and teacher, Silvestre de Sacy, in a manuscript 

 in the library of the old Abbey of St. Germain des Pres,) the group-signs are 

 points therefore, noughts or ciphers; for in India, Thibet, and Persia, 

 noughts and points are identical. In the Gobar, 3" is 30 ; 4" is 400; and 

 6v is 6000. The Indian numbers, and the knowledge of the value of position, 

 must be more modern than the separation of the Indians and the Arians ; for 

 the Zend nation only used the far less convenient Pehlvi numbers. The 

 opinion that the Indian notation has undergone successive improvements 

 appears to me to derive particular support from the Tamul system, which ex- 



