THE AEABS. 599 



dently invented in the east under the Ptolemies and in the 

 or, whether the system of position-value may not have 

 been transferred by the direction of universal traffic, from the 

 Indian western peninsula to Alexandria, and subsequently have 

 been given out amid the renewed dreams of the Pythagoreans, 

 as an invention of the founder of their sect ? The bare possi- 

 bility of ancient and wholly unknown combinations anterior to 

 the sixtieth Olympiad, is scarcely worthy of notice. Wherefore 

 should a feeling of similar requirements not have severally 

 given rise, among highly-gifted nations of different origin, to 

 combinations of the same ideas ? 



Whilst the algebra of the Arabs, by means of that which 

 they had acquired from the Greeks and Indians, combined with 

 the* portions due to their own invention, acted so beneficially 

 on the brilliant epoch of the Italian mathematicians of the 

 middle ages, notwithstanding a great deficiency in sym- 

 bolical designations, we likewise owe to the same people the 

 merit of having furthered the use of the Indian numerical 

 system from Bagdad to Cordova by their writings and their 

 extended commercial relations. Both these effects the 

 simultaneous diffusion of the knowledge of the science of num- 

 bers and of numerical symbols with value by position have 

 variously, but powerfully favoured the advance of the mathe- 

 matical portion of natural science, and facilitated access to the 



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

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

 condition of the simple position-value in Indian numerical characters. 

 The Indians who speak the Tamul language, have arithmetical symbols 

 which differ from their alphabetical 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. In the Cingalese and in the Tamul, there is no position- 

 value or zero sign, 'but symbols for the groups of tens, hundreds, and 

 thousands. The Cingalese work, like the Romans, by juxta-position, 

 the Tamuls by coefficients. Ptolemy uses the present zero sign to 

 represent the descending negative scale for degrees and minutes, both 

 in his Almagest and in his Geography. The zero sign was consequently 

 in use in the west much earlier than the epoch of the invasion of the 

 Arabs. (See my work above cited, and the memoir printed in Crell's 

 Mathematical" Journal, pp. 215, 219, 223, and 227.) 



