[136] But see Burnell, loc. cit., and Thibaut, _Astronomie, Astrologie und Mathematik_, p. 71.

[137] A. Barth, "Inscriptions Sanscrites du Cambodge," in the _Notices et extraits des Mss. de la Bibliotheque nationale_, Vol. XXVII, Part I, pp.

1-180, 1885; see also numerous articles in _Journal Asiatique_, by Aymonier.

[138] Buhler, loc. cit., p. 82.

[139] Loc. cit., p. 79.

[140] Buhler, loc. cit., p. 83. The Hindu astrologers still use an alphabetical system of numerals. [Burnell, loc. cit., p. 79.]

[141] Well could Ramus say, "Quicunq; autem fuerit inventor decem notarum laudem magnam meruit."

[142] Al-B[=i]r[=u]n[=i] gives lists.

[143] _Propagation_, loc. cit., p. 443.

[144] See the quotation from _The Light of Asia_ in Chapter II, p. 16.

[145] The nine ciphers were called _a[.n]ka_.

[146] "Zur Geschichte des indischen Ziffernsystems," _Zeitschrift fur die Kunde des Morgenlandes_, Vol. IV, 1842, pp. 74-83.

[147] It is found in the Bakh[s.][=a]l[=i] MS. of an elementary arithmetic which h.o.e.rnle 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.--[=A]ryabha[t.]a," _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 10^{18} and was not difficult, beginning as follows:

[Ill.u.s.tration]

the same letter (_ka_) appearing in the successive consonant forms, _ka_, _kha_, _ga_, _gha_, etc. See C. I. Gerhardt, _uber 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 Buhler, 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 Buhler 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. [Buhler, _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] Buhler, 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. x.x.x, 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.] Enestrom 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 ma.n.u.script.

[173] Bakh[s.][=a]l[=i] MS. See page 43; h.o.e.rnle, R., _The Indian Antiquary_, Vol. XVII, pp. 33-48, 1 plate; h.o.e.rnle, _Verhandlungen des VII.

Internationalen Orientalisten-Congresses, Arische Section_, Vienna, 1888, "On the Baksh[=a]l[=i] Ma.n.u.script," pp. 127-147, 3 plates; Buhler, loc.

cit.

[174] 3, 4, 6, from H. H. Dhruva, "Three Land-Grants from Sankheda,"

_Epigraphia Indica_, Vol. II, pp. 19-24 with plates; date 595 A.D. 7, 1, 5, from Bhandarkar, "Daulatabad Plates," _Epigraphia Indica_, Vol. IX, part V; date c. 798 A.D.

[175] 8, 7, 2, from "Buckhala Inscription of Nagabhatta," Bhandarkar, _Epigraphia Indica_, Vol. IX, part V; date 815 A.D. 5 from "The Morbi Copper-Plate," Bhandarkar, _The Indian Antiquary_, Vol. II, pp. 257-258, with plate; date 804 A.D. See Buhler, loc. cit.

[176] 8 from the above Morbi Copper-Plate. 4, 5, 7, 9, and 0, from "Asni Inscription of Mahipala," _The Indian Antiquary_, Vol. XVI, pp. 174-175; inscription is on red sandstone, date 917 A.D. See Buhler.

[177] 8, 9, 4, from "Rashtrakuta Grant of Amoghavarsha," J. F. Fleet, _The Indian Antiquary_, Vol. XII, pp. 263-272; copper-plate grant of date c. 972 A.D. See Buhler. 7, 3, 5, from "Torkhede Copper-Plate Grant of the Time of Govindaraja of Gujerat," Fleet, _Epigraphia Indica_, Vol. III, pp. 53-58.

See Buhler.

[178] From "A Copper-Plate Grant of King Tritochanapala Chanlukya of L[=a][t.]ade['s]a," H.H. Dhruva, _Indian Antiquary_, Vol. XII, pp. 196-205; date 1050 A.D. See Buhler.

[179] Burnell, A. C., _South Indian Palaeography_, plate XXIII, Telugu-Canarese numerals of the eleventh century. See Buhler.