[262] Gerhardt, _Entstehung _etc., p. 20.

[263] H. Suter, "Das Rechenbuch des Ab[=u] Zakar[=i]j[=a]

el-[H.]a[s.][s.][=a]r," _Bibliotheca Mathematica_, Vol. II (3), p. 15.

[264] A. Devoulx, "Les chiffres arabes," _Revue Africaine_, Vol. XVI, pp.

455-458.

[265] _Kit[=a]b al-Fihrist_, G. Flugel, Leipzig, Vol. I, 1871, and Vol. II, 1872. This work was published after Professor Flugel's death by J. Roediger and A. Mueller. The first volume contains the Arabic text and the second volume contains critical notes upon it.

[266] Like those of line 5 in the ill.u.s.tration on page 69.

[267] Woepcke, _Recherches sur l'histoire des sciences mathematiques chez les orientaux_, loc. cit.; _Propagation, _p. 57.

[268] Al-[H.]a[s.][s.][=a]r's forms, Suter, _Bibliotheca Mathematica_, Vol.

II (3), p. 15.

[269] Woepcke, _Sur une donnee historique_, etc., loc. cit. The name _[.g]ob[=a]r_ is not used in the text. The ma.n.u.script from which these are taken is the oldest (970 A.D.) Arabic doc.u.ment known to contain all of the numerals.

[270] Silvestre de Sacy, loc. cit. He gives the ordinary modern Arabic forms, calling them _Indien_.

[271] Woepcke, "Introduction au calcul Gob[=a]r[=i] et Haw[=a][=i]," _Atti dell' accademia pontificia dei nuovi Lincei_, Vol. XIX. The adjective applied to the forms in 5 is _gob[=a]r[=i]_ and to those in 6 _indienne_.

This is the direct opposite of Woepcke's use of these adjectives in the _Recherches sur l'histoire_ cited above, in which the ordinary Arabic forms (like those in row 5) are called _indiens_.

These forms are usually written from right to left.

[272] J. G. Wilkinson, _The Manners and Customs of the Ancient Egyptians_, revised by S. Birch, London, 1878, Vol. II, p. 493, plate XVI.

[273] There is an extensive literature on this "Boethius-Frage." The reader who cares to go fully into it should consult the various volumes of the _Jahrbuch uber die Fortschritte der Mathematik_.

[274] This t.i.tle was first applied to Roman emperors in posthumous coins of Julius Caesar. Subsequently the emperors a.s.sumed it during their own lifetimes, thus deifying themselves. See F. Gnecchi, _Monete romane_, 2d ed., Milan, 1900, p. 299.

[275] This is the common spelling of the name, although the more correct Latin form is Boetius. See Harper's _Dict. of Cla.s.s. Lit. and Antiq._, New York, 1897, Vol. I, p. 213. There is much uncertainty as to his life. A good summary of the evidence is given in the last two editions of the _Encyclopaedia Britannica_.

[276] His father, Flavius Manlius Boethius, was consul in 487.

[277] There is, however, no good historic evidence of this sojourn in Athens.

[278] His arithmetic is dedicated to Symmachus: "Domino suo patricio Symmacho Boetius." [Friedlein ed., p. 3.]

[279] It was while here that he wrote _De consolatione philosophiae_.

[280] It is sometimes given as 525.

[281] There was a medieval tradition that he was executed because of a work on the Trinity.

[282] Hence the _Divus_ in his name.

[283] Thus Dante, speaking of his burial place in the monastery of St.

Pietro in Ciel d'Oro, at Pavia, says:

"The saintly soul, that shows The world's deceitfulness, to all who hear him, Is, with the sight of all the good that is, Blest there. The limbs, whence it was driven, lie Down in Cieldauro; and from martyrdom And exile came it here."--_Paradiso_, Canto X.

[284] Not, however, in the mercantile schools. The arithmetic of Boethius would have been about the last book to be thought of in such inst.i.tutions.

While referred to by Baeda (672-735) and Hraba.n.u.s Maurus (c. 776-856), it was only after Gerbert's time that the _Boetii de inst.i.tutione arithmetica libri duo_ was really a common work.

[285] Also spelled Ca.s.siodorius.

[286] As a matter of fact, Boethius could not have translated any work by Pythagoras on music, because there was no such work, but he did make the theories of the Pythagoreans known. Neither did he translate Nicomachus, although he embodied many of the ideas of the Greek writer in his own arithmetic. Gibbon follows Ca.s.siodorus in these statements in his _Decline and Fall of the Roman Empire_, chap. x.x.xix. Martin pointed out with positiveness the similarity of the first book of Boethius to the first five books of Nicomachus. [_Les signes numeraux_ etc., reprint, p. 4.]

[287] The general idea goes back to Pythagoras, however.

[288] J. C. Scaliger in his _Poetice_ also said of him: "Boethii Severini ingenium, eruditio, ars, sapientia facile provocat omnes auctores, sive illi Graeci sint, sive Latini" [Heilbronner, _Hist. math. univ._, p. 387].

Libri, speaking of the time of Boethius, remarks: "Nous voyons du temps de Theodoric, les lettres reprendre une nouvelle vie en Italie, les ecoles florissantes et les savans honores. Et certes les ouvrages de Boece, de Ca.s.siodore, de Symmaque, surpa.s.sent de beaucoup toutes les productions du siecle precedent." [_Histoire des mathematiques_, Vol. I, p. 78.]

[289] Carra de Vaux, _Avicenne_, Paris, 1900; Woepcke, _Sur l'introduction_, etc.; Gerhardt, _Entstehung_ etc., p. 20. Avicenna is a corruption from Ibn S[=i]n[=a], as pointed out by Wustenfeld, _Geschichte der arabischen Aerzte und Naturforscher_, Gottingen, 1840. His full name is Ab[=u] 'Al[=i] al-[H.]osein ibn S[=i]n[=a]. For notes on Avicenna's arithmetic, see Woepcke, _Propagation_, p. 502.

[290] On the early travel between the East and the West the following works may be consulted: A. Hillebrandt, _Alt-Indien_, containing "Chinesische Reisende in Indien," Breslau, 1899, p. 179; C. A. Skeel, _Travel in the First Century after Christ_, Cambridge, 1901, p. 142; M. Reinaud, "Relations politiques et commerciales de l'empire romain avec l'Asie orientale," in the _Journal Asiatique_, Mars-Avril, 1863, Vol. I (6), p.

93; Beazley, _Dawn of Modern Geography, a History of Exploration and Geographical Science from the Conversion of the Roman Empire to A.D. 1420_, London, 1897-1906, 3 vols.; Heyd, _Geschichte des Levanthandels im Mittelalter_, Stuttgart, 1897; J. Keane, _The Evolution of Geography_, London, 1899, p. 38; A. Cunningham, _Corpus inscriptionum Indicarum_, Calcutta, 1877, Vol. I; A. Neander, _General History of the Christian Religion and Church_, 5th American ed., Boston, 1855, Vol. III, p. 89; R.

C. Dutt, _A History of Civilization in Ancient India_, Vol. II, Bk. V, chap, ii; E. C. Bayley, loc. cit., p. 28 et seq.; A. C. Burnell, loc. cit., p. 3; J. E. Tennent, _Ceylon_, London, 1859, Vol. I, p. 159; Geo. Turnour, _Epitome of the History of Ceylon_, London, n.d., preface; "Philalethes,"

_History of Ceylon_, London, 1816, chap, i; H. C. Sirr, _Ceylon and the Cingalese_, London, 1850, Vol. I, chap. ix. On the Hindu knowledge of the Nile see F. Wilford, _Asiatick Researches_, Vol. III, p. 295, Calcutta, 1792.

[291] G. Oppert, _On the Ancient Commerce of India_, Madras, 1879, p. 8.

[292] Gerhardt, _etudes_ etc., pp. 8, 11.

[293] See Smith's _Dictionary of Greek and Roman Biography and Mythology_.

[294] P. M. Sykes, _Ten Thousand Miles in Persia, or Eight Years in Iran_, London, 1902, p. 167. Sykes was the first European to follow the course of Alexander's army across eastern Persia.

[295] Buhler, _Indian Br[=a]hma Alphabet_, note, p. 27; _Palaeographie_, p.

2; _Herodoti Halicarna.s.sei historia_, Amsterdam, 1763, Bk. IV, p. 300; Isaac Vossius, _Periplus Scylacis Caryandensis_, 1639. It is doubtful whether the work attributed to Scylax was written by him, but in any case the work dates back to the fourth century B.C. See Smith's _Dictionary of Greek and Roman Biography_.

[296] Herodotus, Bk. III.

[297] Rameses II(?), the _Sesoosis_ of Diodorus Siculus.

[298] _Indian Antiquary_, Vol. I, p. 229; F. B. Jevons, _Manual of Greek Antiquities_, London, 1895, p. 386. On the relations, political and commercial, between India and Egypt c. 72 B.C., under Ptolemy Auletes, see the _Journal Asiatique_, 1863, p. 297.

[299] Sikandar, as the name still remains in northern India.

[300] _Harper's Cla.s.sical Dict._, New York, 1897, Vol. I, p. 724; F. B.

Jevons, loc. cit., p. 389; J. C. Marshman, _Abridgment of the History of India_, chaps. i and ii.

[301] Oppert, loc. cit., p. 11. It was at or near this place that the first great Indian mathematician, [=A]ryabha[t.]a, was born in 476 A.D.

[302] Buhler, _Palaeographie_, p. 2, speaks of Greek coins of a period anterior to Alexander, found in northern India. More complete information may be found in _Indian Coins_, by E. J. Rapson, Stra.s.sburg, 1898, pp. 3-7.

[303] Oppert, loc. cit., p. 14; and to him is due other similar information.