fication 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.

[173] Bakh[s.][=a]l[=i] MS. See page