326 COSMOS. 



problem, concerning wkich much yet remains to be elucida- 

 ted, the question arises, whether position-value- -the ingenious 



ing, the vacuum is graphically filled by the symbol of a vacuum {sunyn, 

 sifron, tzuphra). lu the " Method of Eutocius,'^ I find in the group of 

 the myriads the first trace of the exponential or indicational system of 



the Greeks, which was so influential in the East: M'', M^, M'^, desig- 

 nate 10,000, 20,000, 30,000. That which is here alone applied to the 

 myriads, passes amoug the Chinese and the Japanese, who derived 

 tlieir knowledge from the Chinese two hundred years before the Chns- 

 dau era, through all the multiples of the groups. In the Gobar, the 

 \rabiau "dust-writing" (discovered by my deceased friend and teacher 

 Silvestre de Sacy, m a manuscript in the libraiy of the old Abbey of 

 St. Germain des Pres), the group-signs are points — therefore zeros oi 

 ciphers ; for in India, Thibet, and Persia, zeros and points are identical. 



[u the Gobar, 3 • is written for 30 ; 4 • • for 400 ; and G •*• for GOOO. 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 Pehhvi numbers. The 

 conjecture of the successive improvements that have been made in the 

 Indian notation appears to me to be supported by the Tamul system, 

 which expresses units by nine characters, and all other values by group- 

 signs for 10, 100, and 1000, with multipliers added to the left. The 

 singular upLdfcol 'Iv6lkoI, in a scholium of the monk Neophytos, discov- 

 ered by Prof. Brandis in the library of Paris, and kindly communicated 

 to me for publication, appear to corroborate the opinion of such a grad- 

 ual process of improvement. The nine characters of Neophytos are, 

 with the exception of the 4, quite similar to the present Persian; but 

 the value of these nine units is raised to 10, 100, 1000 fold by writing 



one, two, or three ciphers or zero-signs above them; as 2 for 20, 2 4 



for 24, 5 for 500, and 3 G for 30G. If we suppose points to be used 

 instead of zeros, we have the Arabic dust-writing, Gobar. As my 

 bi-other, Wilhelm von Humboldt, has often remarked of the Sanscrit, 

 that it is very inappropriately designated by the terms "Indian" and 

 " ancient Indian" language, since there are in the Indian peninsula 

 several very ancient languages not at all derived from the Sanscrit, so 

 the expression Indian or ancient Indian arithmetical characters is also 

 very vague, and this vagueness applies both to the form of the charac- 

 ters and to the spirit of the methods, which sometimes consist in mere 

 juxtaposition, sometimes in the employment of coefficients and indica- 

 tors, and sometimes in the actual value of position. Even the existence 

 of the cipher or zero is, as the scholium of Necphytos shows, not a 

 necessary condition of the simple position-value in Indian numerical 

 characters. The Indians who speak the Tamul language have arith- 

 metical symbols which differ from their alphabetic-al characters, and of 

 which the 2 and the 8 have a faint resemblance to the 2 and the 5 of 

 the Devanagari figures (Rob. Anderson, Rudiments of Tamul Grammar, 

 1821, p. 135) ; and yet an accurate comparison proves that the Tamul 

 arithmetical characters are derived from the Tamul alphabetical writing. 

 According to Carey, the Cingalese are still more different from the 

 Devanagari characters. lu the Cingalese and in the Tamul, there is 

 no position-value or zero-sign, but symbols for the groups of tens, hund* 

 reds, and thousands. The Cingalese work, like the Romans, by juxta 

 position, the Tamuls by coefficients. Ptolemy uses the present zero 



