THE ARABS. 225 



Arabs to a knowledge of Indian algebra, enabled them also to 

 obtain, iu 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 

 of Sicily. Nevertheless, the important 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 1819 and 1829), and presented to 

 the Academic des Inscriptions at Paris, and the Academy of 

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



* Chasles, Apcrcu Histonque des Mcthodes en Geomitrie, 1837, p. 

 464-472 ; also in the Comptes Rendus de I'Acad. des Sciences, t. viii., 

 1839, p. 78; t. ix., 1839, p. 449 ; t. xvi., 1843, p. 156-173, and 218-246; 

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



t Hum bold t, Ueber die lei verschiedenen Vdlkem ublichen Systeme von 

 Zahlezeichen und uber den Urspnmg des Stellenwerlhes in den Indischen 

 Znhlcn, in CrelCs Journal fur die reine und angetvandte Mathematik, btl. 

 iv. (1829), s. 205-231. Compare, also, my Examen Crit. de VHist. de 

 la Geographic, 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 

 perpendicularly or horizontally, by means of " indicators," correspond- 

 ing to the different divisions of the abacus (thus, M 3 C 5 X 6 I 8 ), we shall 

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

 our Indian numbers are, however, 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- 

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

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

 3, 5, 6, and 8 balls. In the Suanpan there is no apparent group-sign ; 

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

 are occupied by units (3, 5, 6, and 8) as multipliers or indicators. la 

 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 writing If a group (a member of the progression) be want 



K 2 



