TMK Al!M:S. 227 



application of position — which occurs in the Tuscan abacus, 

 and in the Suampan of Inner Asia, has been tvv^ice independ- 

 ently invented, in the East under the Ptolemies, and in the 

 West ? or whether the system of position-value may not have 

 been transferred by the direction of universal traffic from the 

 Indian w^estern peninsula to Alexandria, and subsequently 

 have been given out amid the renewed dreams of the Pytha- 

 goreans as an invention of the founder of their sect ? The 

 bare possibility 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 difierent 

 origin, to combinations of the same ideas ? • 



While 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 bene- 

 ficially on the brilliant epoch of the Italian mathematicians 

 of the Middle Ages, notwithstanding a great deficiency in 

 symbolical 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 si- 

 multaneous diffusion of the knowledge of the science of num- 

 bers and of numerical symbols with value by position — have 

 variously, but powerfully, favored the advance of the mathe- 

 matical portion of natural science, and facilitated access to the 

 more abstruse departments of astronomy, optics, physical geog- 

 raphy, and the theories of heat and magnetism, which, with- 

 out such aids, would have remained unopened. 



The question has often been asked, in the history of nations, 

 v/hat would have been the course of events if Carthage had 

 conquered Rome and subdued the West ? " We may ask 

 with equal justice," as Wilhelm von Humboldt* observes, 

 " what would be the condition of our civilization at the pres- 

 ent day if the Arabs had remained, as they long did, the sole 

 possessors of scientific knowledge, and had spread themselves 

 permanently over the West ? A less favorable result would 



sign to i-epresent the descending negative scale for degrees and minutes 

 both in his Ahnagest and in his Geography. The zero-sign was conse- 

 quently 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, p. 215, 219, 223, and 227.) 



* Wilhelm von Humboldt, Ueber 4ie Kawi-Sprache, bd. i., s. cclxii. 

 Compare, also, the excellent desciiption of the Arabs in Herder's Ideen 

 zur Gesch. d-er Maischeit, book xix., 4 and 5. 



