328 On the Si/stems of Numerical Signs 



existed others, which did not express value by the same method. 

 Now it may be the case, that Alexander and his successors in 

 Bactria, in their temporary incursions, did not have intercourse 

 with any tribes, among whom the knowledge of the system ex- 

 pressing value by position had then become prevalent. 



I could wish that the traces of what is still to be discovered 

 (and that is yet very much), might soon be pursued with in- 

 creasing zeal, by philologists, who have opportunity of examin- 

 ing either Greek, Persian, or Arabian manuscripts*. Th6 

 manner in which old manuscripts of the Sanscrit literature 

 are paged, can sometimes bring ns to important observa- 

 tions and discoveries. To give £ln instance, hardly any 

 person would have expected to find in India, besides the 

 decimal system with position, a sedecimal system without 

 position. It seems, however, that some Indian tribes had 

 adopted in their calculations groups of sixteen, as the natives 

 of America, and the Gauls and Biscayans, those of twenty. 

 For such a remarkable numeration has been discovered, more 

 than ten years ago, by M. Bopp, in a manuscript of the old 

 Indian poem, Mahabharata (Cod. Reg., Paris, p . 178). He 

 had the kindness to communicate it to me for publication, when 

 I laid before the Academie des Inscriptions et Belles Lettres 

 my first essay on the numerical signs of the different nations. 

 The first sixty-five pages are paged with Indian alphabetic let- 

 ters, but only the consonants of the Sanscrit alphabet are used 

 (A: for 1, M for 2 ). This refutes the opinion till now ge- 

 nerally prevailing, that the Hindoos always used ideographic? 

 figures to express numbers, and never alphabetical letters, as 

 the Semitic tribes and the Greeksf . 



On the sixtieth page begins the extraordinary sedecimal 



* Among the Arabian manuscripts, those especially are to be recommended to 

 peculiar attention, which treat of custom-house or financial affairs, or of arithmetic 

 in general ; as, for instance, Abu Jose Alchindus de Arithmetica Indica; Abdel- 

 hamid Ben Vasee Abulphadl de Numerorum Proprietatibus; Amad Ben Omar 

 Alkarabisi Liber de Indica Numerondi Ratione ; the Indian Algebra by Katka ; 

 Mohammed Ben Lara de Numerorum Disciplina (Casari Biblioth. Arabico-His- 

 pana, t. i. p. 353, 405, 410, 426, 433.) 



f Si I'arithmetique de position n'est pas originaire de I'Inde, elle doit au moins y 

 avoir existe de temps immemorial ; car on ne trouve chez les Indiens aucune trace 

 d'une notation alphabetique telle que la notation des Hebreux, des Grecs et des 

 Arabes. — (Delambre, Histoire de rAstronomie Ancienpe, t. i., p. 543.) 



