THE ARAB3. 225 



Arabs to a knowleds'e of Indian alsfebra, enabled thein also to 

 obtain, in the ninth century, Indian numerals from Per&ia and 

 the shores of the Euphrates. Persians were established at 

 that period as revenue collectors on the Indus, and the use of 

 Indian numerals was gradually transmitted to the revenue' 

 officers of the Arabs in Northern Africa, opposite the shores 

 cf Sicily. Nevertheless, the imjDorlant historical investiga- 

 tions of the distinguished mathematician Chasles* have ren- 

 dered it more than probable, according to his correct interpre- 

 tation of the so-called Pythagorean table in the Geometry of 

 Boethius, that the Christians in the West were familiar with 

 Indian numerals even earlier than the Arabs, and that they 

 were acquainted with the use of nine figures or characters, 

 according to their position value, under the name of the system 

 of the abacus. 



The present is not a fitting place to enter more fully into 

 the consideration of this subject, which I have already treated 

 of in two papers (written in 1S19 and 1829), and presented to 

 the Academie des Inscrijitions at Paris, and the Academy of 

 Sciences at Berlin ;t but, in our attempts to solve a historical 



* Chasles, Apcrcu Hislonqne des Mcthodcs en Geometrle, 1837, p. 

 4G4-472 ; also in the Comples Rendus de VAcnd. des Sciences, t. viii., 

 1839, p. 78; t. ix., 1839, p. 449 ; t. xvi., 1843, p. 156-173, and 218-24G; 

 t. xvii., 1843, p. 143-154. 



t Humboldt, Ueher die hei versckiedenen Vdlkern uhlichen Systems von 

 Zahlez eichen und uber den Ur sprung des Stellentoerthes in den Indischen 

 Zahlen, in CrelVs Journal fur die reine und angewandte Mathematik, bd. 

 iv. (18-29), s. 205-231. Compare, also, my Examen Crit. de VHist. de 

 la Geographie, t. iv., p. 275. The simple enumeration of the different 

 methods which nations, to whom the Indian arithmetic by position was 

 unknown, employed for expressing the multiplier of the fundamental 

 groups, furnishes, in my opinion, an explanation of the gradual rise or 

 origin of the Indian system. If we express the number 3568, either 

 pei'pendicularly or horizontally, by means of '• indicators," correspond- 

 ing to the different divisions of the abacus (thus, M^C^X^F), we shall 

 easily perceive that the group-signs (MCXI) might be omitted. But 

 our Indian numbers are, hov/ever, nothing more than these indicators 

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

 designation by indicators by the ancient Asiatic Suanpan (the reckon- 

 ing machine which the Moguls introduced into Russia), which has suc- 

 ressive rows of strings, to represent thousands, hundreds, tens, and 

 mits. These strings would bear in the numerical example just cited, 

 I. 5, 6, and 8 balls. In the Suanpan there is no apparent group-sign ; 

 the group-signs are the positions themselves; and these positions (strmgs) 

 are occupied by units (3, 5, G, and 8) as multipliers or indicators. In 

 both ways, whether by the figurative (the written) or by the palpable 

 arithmetic, we arrive at the value of position and at the simple use of 

 nine numbers. If a string be without any ball, the place will be left 

 blank in wiiting If a crroup (a member of the progression) be want 



K ? 



