V. Aryabhata's System of expressing Numbers

SEVERAL systems of expressing numbers, used in India, have been explained by Professor Bühler in §§ 33 to 35 of his work on Indian palaeography. There is a system, a highly interesting one, which was not noticed by him, because it has not been found used in inscriptions or in the pagination of literary works; namely, that of the astronomer Āryabhaṭa. It has been mentioned briefly by various other writers. And it was considered in some detail by Mr. C. M. Whish in 1820, and at more length by M. Léon Rodet in 1880. Those two treatments of it, however, scarcely suffice to do justice to it; particularly from lacking any table to make its details clear. And it deserves a full exposition, because it is of special interest in connection with two topics which have been reopened lately by Mr. G. R. Kaye; namely, the early use of the abacus in India, and the development of the decimal notation, that is, of the system of the nine significant digits 1 to 9, with the zero, cipher, or naught, used with place-value so that any particular sign denotes units, tens, thousands, etc., or the absence of them, according to its position as written in a row of figures. I propose, therefore, to consider it exhaustively here, but without venturing at present to offer any opinion on the two topics which Mr. Kaye has reopened: I only seek to exhibit fully, with a few introductory remarks about Āryabhaṭa himself, a system of numeration which must certainly be regarded as an important factor in considering them.