318 On the Systems of Numerical Signs 



the scarcity of the materials, if we do not wish to content our- 

 selves with a few negative results. 



We have seen that some nations, in expressing numbers by 

 writing, mix together alphabetical letters and ideographic 

 figures arbitrarily chosen. Likewise we find, that respecting 

 the mode of expressing the muUipla of the fundamental groups 

 the most heterogeneous methods are used. We discover even 

 that one system completely developes what, in another, is only 

 slightly indicated. The same incongruity obtains in lan- 

 guages. In one language, some grammatical forms do appear 

 only in a few instances, and are slightly expressed, whilst 

 another has developed them with a peculiar predilection, and 

 with every effort of mental power. Should I, therefore, ex- 

 plain the numerical systems singly, as they are used by dif- 

 ferent nations, the similarity of their methods would be ren- 

 dered obscure, and we should lose the track on which the 

 human mind proceeding, at last arrived to discover the master- 

 piece of the Indian arithmetic, in which every figure has a 

 double value, an absolute and a relative, of which the last is 

 increasing, in a geometrical progression, from the right to the 

 left. In my following observations I shall therefore abandon 

 the ethnographical order, and only consider the diff^erent means 

 employed by nations to express in writing the groups of the 

 units. 



First method. Juxtaposition is effected by simple addi- 

 tion in numerical figures as well as in alphabetical signs. It 

 was in use among the ancient Tuscans, the Romans, among the 

 Greeks only up to a myriad, among the Semitic tribes, the 

 Mexicans, and also in the greatest part of the Pehlwi calculations. 

 This method renders the computation extremely difficult, when 

 the multipla of the groups (2n, 3n, 2n* . . . . ) are not expressed 

 by distinct signs. The Tuscans and the Romans repeated the 

 figure of ten as far as fifty ; the Mexicans, whose first figure 

 of a group was 20 (a flag), repeated this hieroglyphic up 

 to 400. The Greeks, however, have in the rows of the tenths 

 and hundreds, which begin with iota and rho^ distinct figures 

 for 20, 30, 400 and 600. The three episemes (letters of an 

 obsolete alphabet), bau, koppa, and sanpi, serve to express 6, 

 90, and 900. The two last terminate the rows of the tenths 



