324 On the Systems of Numerical Signs 



of expressing every number only by unities (multiplicators) is 

 moreovier suggested by the suanpan, of which the strings indi- 

 'cated the thousands, hundreds, tenths and unities in fixed 

 order. To express the above-mentioned number the strings 

 contained 3, 5, 6 and 8 balls. Ko sign of groups is here to be 

 observed. The places themselves supply the signs of groups, 

 and these places (the strings) are filled up with the unities 

 (multiplicators). Thus the Indian system may have been in- 

 vented in either way, by the figurative as well as by the pal- 

 pable arithmetic. If a string was empty, or in writing a layer 

 was not occupied by a figure, if consequently a group (a mem- 

 ber of the progression) was wanting, the empty place in writing 

 was filled up by the hieroglyphic of emptiness, a circle open in 

 the middle, sunya, sifron, cyphra*. 



That the notation of numerical quantities has been improved 

 and brought to perfection in India only by degrees, is proved 

 by the numerical figures of the Tamul language, in which 

 every numerical quantity is expressed by the nine figures of 

 the unities, by distinct figures for the groups of 10, 100, and 

 1000, and by multiplicators added to the left hand. The same 

 is also proved by the remarkable arithmoi indikoi, in the 

 scholion of the monk, Neophytus, which is preserved in the 

 library of Paris (Cod. Reg. fol. 15), and was communicated 

 to me by the kindness of Professor Brandes. The nine figures* 

 of Neophytus are, the four excepted, all of them like those of 

 the Persians. The figures of 1, 2, 3, and 9 are even found in 

 some Egyptian inscriptions containing numbers f . The unities 

 are multiphcated, a ten, a hundred, and a thousand times, 



o 



by writing over them one, two, or three cyphers, as Sz=:20, 



o 

 o oo oo 



24=240, 5:^500, 6=6000. If, instead of cyphers, points are 

 used, the Arabic Gobar figures are obtained. I shall give 

 here a literal translation in Latin of the scholion itself, and 

 observe only, that the monk erroneously calls the expression 

 tzUphron an Indian word. 



* In English, the expression cypher for nullity has heen preserved ; whilst, in 

 the other languz^ges of the west, this word is used to indicate the numerical figures 

 in general ; in them the cypher is called zero (sifron, siron). According to WilsQA 

 ^ny numerical quantity is called in Sanscrit sambhara, 

 ■{- Kosegaxteu; p, 54. 



