[146] "Zur Geschichte des indischen Ziffernsystems," Zeitschrift für die Kunde des Morgenlandes, Vol. IV, 1842, pp. 74-83.
[147] It is found in the Bakhṣālī MS. of an elementary arithmetic which Hoernle placed, at first, about the beginning of our era, but the date is much in question. G. Thibaut, loc. cit., places it between 700 and 900 A.D.; Cantor places the body of the work about the third or fourth century A.D., Geschichte der Mathematik, Vol. I (3), p. 598.
[148] For the opposite side of the case see G. R. Kaye, "Notes on Indian Mathematics, No. 2.—Āryabhaṭa," Journ. and Proc. of the Asiatic Soc. of Bengal, Vol. IV, 1908, pp. 111-141.
[149] He used one of the alphabetic systems explained above. This ran up to 1018 and was not difficult, beginning as follows:
the same letter (ka) appearing in the successive consonant forms, ka, kha, ga, gha, etc. See C. I. Gerhardt, Über die Entstehung und Ausbreitung des dekadischen Zahlensystems, Programm, p. 17, Salzwedel, 1853, and Études historiques sur l'arithmétique de position, Programm, p. 24, Berlin, 1856; E. Jacquet, Mode d'expression symbolique des nombres, loc. cit., p. 97; L. Rodet, "Sur la véritable signification de la notation numérique inventée par Āryabhata," Journal Asiatique, Vol. XVI (7), pp. 440-485. On the two Āryabhaṭas see Kaye, Bibl. Math., Vol. X (3), p. 289.
[150] Using kha, a synonym of śūnya. [Bayley, loc. cit., p. 22, and L. Rodet, Journal Asiatique, Vol. XVI (7), p. 443.]
[151] Varāha-Mihira, Pañcasiddhāntikā, translated by G. Thibaut and M. S. Dvivedī, Benares, 1889; see Bühler, loc. cit., p. 78; Bayley, loc. cit., p. 23.
[152] Bṛhat Saṃhitā, translated by Kern, Journal of the Royal Asiatic Society, 1870-1875.
[153] It is stated by Bühler in a personal letter to Bayley (loc. cit., p. 65) that there are hundreds of instances of this usage in the Bṛhat Saṃhitā. The system was also used in the Pañcasiddhāntikā as early as 505 A.D. [Bühler, Palaeographie, p. 80, and Fleet, Journal of the Royal Asiatic Society, 1910, p. 819.]