[14] See also Woepcke, _Propagation_, p. 505. The origin is discussed at much length by G. R. Kaye, "Notes on Indian Mathematics.--Arithmetical Notation," _Journ. and Proc. of the Asiatic Soc. of Bengal_, Vol. III, 1907, p. 489.

[15] _Alberuni's India_, Arabic version, London, 1887; English translation, ibid., 1888.

[16] _Chronology of Ancient Nations_, London, 1879. Arabic and English versions, by C. E. Sachau.

[17] _India_, Vol. I, chap. xvi.

[18] The Hindu name for the symbols of the decimal place system.

[19] Sachau's English edition of the _Chronology_, p. 64.

[20] _Litterature arabe_, Cl. Huart, Paris, 1902.

[21] Huart, _History of Arabic Literature_, English ed., New York, 1903, p.

182 seq.

[22] Al-Mas'[=u]d[=i]'s _Meadows of Gold_, translated in part by Aloys Sprenger, London, 1841; _Les prairies d'or_, trad. par C. Barbier de Meynard et Pavet de Courteille, Vols. I to IX, Paris, 1861-1877.

[23] _Les prairies d'or_, Vol. VIII, p. 289 seq.

[24] _Essays_, Vol. II, p. 428.

[25] Loc. cit., p. 504.

[26] _Materiaux pour servir a l'histoire comparee des sciences mathematiques chez les Grecs et les Orientaux_, 2 vols., Paris, 1845-1849, pp. 438-439.

[27] He made an exception, however, in favor of the numerals, loc. cit., Vol. II, p. 503.

[28] _Bibliotheca Arabico-Hispana Escurialensis_, Madrid, 1760-1770, pp.

426-427.

[29] The author, Ibn al-Qif[t.][=i], flourished A.D. 1198 [Colebrooke, loc.

cit., note Vol. II, p. 510].

[30] "Liber Artis Logisticae a Mohamado Ben Musa _Alkhuarezmita_ exornatus, qui ceteros omnes brevitate methodi ac facilitate praestat, Indorum que in praeclarissimis inventis ingenium & ac.u.men ostendit." [Casiri, loc. cit., p. 427.]

[31] Macoudi, _Le livre de l'avertiss.e.m.e.nt et de la revision_. Translation by B. Carra de Vaux, Paris, 1896.

[32] Verifying the hypothesis of Woepcke, _Propagation_, that the Sindhind included a treatment of arithmetic.

[33] A[h.]med ibn 'Abdall[=a]h, Suter, _Die Mathematiker_, etc., p. 12.

[34] _India_, Vol. II, p. 15.

[35] See H. Suter, "Das Mathematiker-Verzeichniss im Fihrist,"

_Abhandlungen zur Geschichte der Mathematik_, Vol. VI, Leipzig, 1892. For further references to early Arabic writers the reader is referred to H.

Suter, _Die Mathematiker und Astronomen der Araber und ihre Werke_. Also "Nachtrage und Berichtigungen" to the same (_Abhandlungen_, Vol. XIV, 1902, pp. 155-186).

[36] Suter, loc. cit., note 165, pp. 62-63.

[37] "Send Ben Ali,... tum arithmetica scripta maxime celebrata, quae publici juris fecit." [Loc. cit., p. 440.]

[38] _Scritti di Leonardo Pisano_, Vol. I, _Liber Abbaci_ (1857); Vol. II, _Scritti_ (1862); published by Balda.s.sarre Boncompagni, Rome. Also _Tre Scritti Inediti_, and _Intorno ad Opere di Leonardo Pisano_, Rome, 1854.

[39] "Ubi ex mirabili magisterio in arte per novem figuras indorum introductus" etc. In another place, as a heading to a separate division, he writes, "De cognitione novem figurarum yndorum" etc. "Novem figure indorum he sunt 9 8 7 6 5 4 3 2 1."

[40] See _An Ancient English Algorism_, by David Eugene Smith, in _Festschrift Moritz Cantor_, Leipzig, 1909. See also Victor Mortet, "Le plus ancien traite francais d'algorisme," _Bibliotheca Mathematica_, Vol.

IX (3), pp. 55-64.

[41] These are the two opening lines of the _Carmen de Algorismo_ that the anonymous author is explaining. They should read as follows:

Haec algorismus ars praesens dicitur, in qua Talibus Indorum fruimur bis quinque figuris.

What follows is the translation.

[42] Thibaut, _Astronomie, Astrologie und Mathematik_, Stra.s.sburg, 1899.

[43] Gustave Schlegel, _Uranographie chinoise ou preuves directes que l'astronomie primitive est originaire de la Chine, et qu'elle a ete empruntee par les anciens peuples occidentaux a la sphere chinoise; ouvrage accompagne d'un atlas celeste chinois et grec_, The Hague and Leyden, 1875.

[44] E. W. Hopkins, _The Religions of India_, Boston, 1898, p. 7.

[45] R. C. Dutt, _History of India_, London, 1906.

[46] W. D. Whitney, _Sanskrit Grammar_, 3d ed., Leipzig, 1896.

[47] "Das [=A]pastamba-['S]ulba-S[=u]tra," _Zeitschrift der deutschen Morgenlandischen Gesellschaft_, Vol. LV, p. 543, and Vol. LVI, p. 327.

[48] _Geschichte der Math._, Vol. I, 2d ed., p. 595.

[49] L. von Schroeder, _Pythagoras und die Inder_, Leipzig, 1884; H. Vogt, "Haben die alten Inder den Pythagoreischen Lehrsatz und das Irrationale gekannt?" _Bibliotheca Mathematica_, Vol. VII (3), pp. 6-20; A. Burk, loc.

cit.; Max Simon, _Geschichte der Mathematik im Altertum_, Berlin, 1909, pp.

137-165; three S[=u]tras are translated in part by Thibaut, _Journal of the Asiatic Society of Bengal_, 1875, and one appeared in _The Pandit_, 1875; Beppo Levi, "Osservazioni e congetture sopra la geometria degli indiani,"

_Bibliotheca Mathematica_, Vol. IX (3), 1908, pp. 97-105.

[50] Loc. cit.; also _Indiens Literatur und Cultur_, Leipzig, 1887.

[51] It is generally agreed that the name of the river Sindhu, corrupted by western peoples to Hindhu, Indos, Indus, is the root of Hindustan and of India. Reclus, _Asia_, English ed., Vol. III, p. 14.

[52] See the comments of Oppert, _On the Original Inhabitants of Bharatavar[s.]a or India_, London, 1893, p. 1.

[53] A. Hillebrandt, _Alt-Indien_, Breslau, 1899, p. 111. Fragmentary records relate that Kh[=a]ravela, king of Kali[.n]ga, learned as a boy _lekh[=a]_ (writing), _ga[n.]an[=a]_ (reckoning), and _r[=u]pa_ (arithmetic applied to monetary affairs and mensuration), probably in the 5th century B.C. [Buhler, _Indische Palaeographie_, Stra.s.sburg, 1896, p. 5.]

[54] R. C. Dutt, _A History of Civilization in Ancient India_, London, 1893, Vol. I, p. 174.

[55] The Buddha. The date of his birth is uncertain. Sir Edwin Arnold put it c. 620 B.C.

[56] I.e. 10010^7.

[57] There is some uncertainty about this limit.