322 On the Systems of Numerical Signs 



consequently, was known in the west long before the invasion 

 of the Arabs *. 



We have seen, that the Chinese, by placing indicators per- 

 pendicularly over or under the groups, indicate the difference 



s 



between X=12, and X=20. The same effect is produced 



among the Greeks, Armenians, and those tribes of the Hin* 

 doos, who speak the Tamul language, by adding figures in 

 horizontal direction. Diophantus and Pappus wrote (SMu 

 for twice ten thousands, whilst ocMu^ (where /3 is placed to the 

 right of the initials of the myriad) signifies ten thousands, plus 

 two or 10,002. In the same manner, the Tamul figures are 

 used, as, for instance : 4X=:40, and X4=14. In the ancient 

 Persian Pehlwi, according to Anquetil, and among the Arme- 

 nians, according to Cerbied f , multiplicators employed to ex- 

 press the multipla of the hundred, are found placed to the 

 left. To these examples may yet be added, the above-men- 

 tioned point of Diophantus, which is used instead of My, and 

 multiplies the figures to the left a thousand times. For, as iw£ 

 as regards the method, it belongs to this class J. 



Fourth Method. — Multiplication and diminution^ in 

 ascending and descending direction, brought about by dividing 

 the figures in layers^ of which the value decreases in geome- 

 trical proportion. 



Both Archimedes in the Octades, and Apollonius in the 

 Tetrades, make use of this notation, but only in numbers above 

 (10,000)®, and in a hundred of millions, or a myriad of my- 

 riads J. Here, as well as in the descending sexagesimal scale 



sometimes to a peculiar relation in which the letter omicron stands to the sexage- 

 simal system, 1. c, t. ii., p. 14, and Journal des Savans, 1817, p. 539. It is strange, 

 that in the old Indian arithmetic of the Lilawati, the figure of the cypher, placed 

 aside a number, indicates, that such a number is to be subtracted. Delambre, vol.i., 

 p. 540. What does the Ling (the true cypher) signify, which in Chinese figures is 

 written imder 12, 13, 22, 132? In Greek inscriptions the signs of cyphers indi- 

 cate oboles. (Bdckh, Staatshaushaltung der Athener, b.ii., p. 379.) 



* Planudes, Treatise on the Arithmoi Indikoi, third method, Multiplication of the 

 Value by Coefficients. 



f Grammaire Armenienne, 1825, p. 25. 



\ This mode of separating numbers by points, though it is otherwise used in a 

 very inconsistent manner, expresses properly the value by position. It is also 

 used in three passages of the elder Pliny (vi., 24, 33 j xxx., 3), which have given 

 rise to many controversies. 



§ Delambre, Hist, de I'Astron. Ancieuue, t. i, p 105 ; t. ii., p. 9. 



