the same letter (_ka_) appearing in the successive consonant forms, _ka_, _kha_, _ga_, _gha_, etc. See C. I. Gerhardt, _Ueber die Entstehung und Ausbreitung des dekadischen Zahlensystems_, Programm, p. 17, Salzwedel, 1853, and _Etudes historiques sur l'arithmetique de position_, Programm, p. 24, Berlin, 1856; E. Jacquet, _Mode d'expression symbolique des nombres_, loc. cit., p. 97; L. Rodet, "Sur la veritable signification de la notation numerique inventee par [=A]ryabhata," _Journal Asiatique_, Vol. XVI (7), pp. 440-485. On the two [=A]ryabha[t.]as see Kaye, _Bibl. Math._, Vol. X (3), p. 289. [150] Using _kha_, a synonym of _['s][=u]nya_. [Bayley, loc. cit., p. 22, and L. Rodet, _Journal Asiatique_, Vol. XVI (7), p. 443.] [151] Var[=a]ha-Mihira, _Pancasiddh[=a]ntik[=a]_, translated by G. Thibaut and M. S. Dvived[=i], Benares, 1889; see Buehler, loc. cit., p. 78; Bayley, loc. cit., p. 23. [152] _B[r.]hat Sa[m.]hit[=a]_, translated by Kern, _Journal of the Royal Asiatic Society_, 1870-1875. [153] It is stated by Buehler in a personal letter to Bayley (loc. cit., p. 65) that there are hundreds of instances of this usage in the _B[r.]hat Sa[m.]hit[=a]_. The system was also used in the _Pancasiddh[=a]ntik[=a]_ as early as 505 A.D. [Buehler, _Palaeographie_, p. 80, and Fleet, _Journal of the Royal Asiatic Society_, 1910, p. 819.] [154] Cantor, _Geschichte der Mathematik_, Vol. I (3), p. 608. [155] Buehler, loc. cit., p. 78. [156] Bayley, p. 38. [157] Noviomagus, in his _De numeris libri duo_, Paris, 1539, confesses his ignorance as to the origin of the zero, but says: "D. Henricus Grauius, vir Graece & Hebraice exime doctus, Hebraicam originem ostendit," adding that Valla "Indis Orientalibus gentibus inventionem tribuit." [158] See _Essays_, Vol. II, pp. 287 and 288. [159] Vol. XXX, p. 205 seqq. [160] Loc. cit., p. 284 seqq. [161] Colebrooke, loc. cit., p. 288. [162] Loc. cit., p. 78. [163] Hereafter, unless expressly stated to the contrary, we shall use the word "numerals" to mean numerals with place value. [164] "The Gurjaras of R[=a]jput[=a]na and Kanauj," in _Journal of the Royal Asiatic Society_, January and April, 1909. [165] Vol. IX, 1908, p. 248. [166] _Epigraphia Indica_, Vol. IX, pp. 193 and 198. [167] _Epigraphia Indica_, Vol. IX, p. 1. [168] Loc. cit., p. 71. [169] Thibaut, p. 71. [170] "Est autem in aliquibus figurarum istaram apud multos diuersitas. Quidam enim septimam hanc figuram representant," etc. [Boncompagni, _Trattati_, p. 28.] Enestroem has shown that very likely this work is incorrectly attributed to Johannes Hispalensis. [_Bibliotheca Mathematica_, Vol. IX (3), p. 2.] [171] _Indische Palaeographie_, Tafel IX. [172] Edited by Bloomfield and Garbe, Baltimore, 1901, containing photographic reproductions of the manuscript.