[304] J. Beloch, _Griechische Geschichte_, Vol. III, Stra.s.sburg, 1904, pp.

30-31.

[305] E.g., the denarius, the words for hour and minute ([Greek: hora, lepton]), and possibly the signs of the zodiac. [R. Caldwell, _Comparative Grammar of the Dravidian Languages_, London, 1856, p. 438.] On the probable Chinese origin of the zodiac see Schlegel, loc. cit.

[306] Marie, Vol. II, p. 73; R. Caldwell, loc. cit.

[307] A. Cunningham, loc. cit., p. 50.

[308] C. A. J. Skeel, _Travel_, loc. cit., p. 14.

[309] _Inchiver_, from _inchi_, "the green root." [_Indian Antiquary_, Vol.

I, p. 352.]

[310] In China dating only from the second century A.D., however.

[311] The Italian _morra_.

[312] J. Bowring, _The Decimal System_, London, 1854, p. 2.

[313] H. A. Giles, lecture at Columbia University, March 12, 1902, on "China and Ancient Greece."

[314] Giles, loc. cit.

[315] E.g., the names for grape, radish (_la-po_, [Greek: rhaphe]), water-lily (_si-kua_, "west gourds"; [Greek: sikua], "gourds"), are much alike. [Giles, loc. cit.]

[316] _Epistles_, I, 1, 45-46. On the Roman trade routes, see Beazley, loc.

cit., Vol. I, p. 179.

[317] _Am. Journ. of Archeol._, Vol. IV, p. 366.

[318] M. Perrot gives this conjectural restoration of his words: "Ad me ex India regum legationes saepe missi sunt numquam antea visae apud quemquam principem Romanorum." [M. Reinaud, "Relations politiques et commerciales de l'empire romain avec l'Asie orientale," _Journ. Asiat._, Vol. I (6), p.

93.]

[319] Reinaud, loc. cit., p. 189. Florus, II, 34 (IV, 12), refers to it: "Seres etiam habitantesque sub ipso sole Indi, c.u.m gemmis et margaritis elephantes quoque inter munera trahentes nihil magis quam longinquitatem viae imputabant." Horace shows his geographical knowledge by saying: "Not those who drink of the deep Danube shall now break the Julian edicts; not the Getae, not the Seres, nor the perfidious Persians, nor those born on the river Tanas." [_Odes_, Bk. IV, Ode 15, 21-24.]

[320] "Qua virtutis moderationisque fama Indos etiam ac Scythas auditu modo cognitos pellexit ad amicitiam suam populique Romani ultro per legatos petendam." [Reinaud, loc. cit., p. 180.]

[321] Reinaud, loc. cit., p. 180.

[322] _Georgics_, II, 170-172. So Propertius (_Elegies_, III, 4):

Arma deus Caesar dites meditatur ad Indos Et freta gemmiferi findere cla.s.se maris.

"The divine Caesar meditated carrying arms against opulent India, and with his ships to cut the gem-bearing seas."

[323] Heyd, loc. cit., Vol. I, p. 4.

[324] Reinaud, loc. cit., p. 393.

[325] The t.i.tle page of Calandri (1491), for example, represents Pythagoras with these numerals before him. [Smith, _Rara Arithmetica_, p. 46.] Isaacus Vossius, _Observationes ad Pomponium Melam de situ orbis_, 1658, maintained that the Arabs derived these numerals from the west. A learned dissertation to this effect, but deriving them from the Romans instead of the Greeks, was written by Ginanni in 1753 (_Dissertatio mathematica critica de numeralium notarum minuscularum origine_, Venice, 1753). See also Mannert, _De numerorum quos arabicos vocant vera origine Pythagorica_, Nurnberg, 1801. Even as late as 1827 Romagnosi (in his supplement to _Ricerche storiche sull' India_ etc., by Robertson, Vol. II, p. 580, 1827) a.s.serted that Pythagoras originated them. [R. Bombelli, _L'antica numerazione italica_, Rome, 1876, p. 59.] Gow (_Hist. of Greek Math._, p. 98) thinks that Iamblichus must have known a similar system in order to have worked out certain of his theorems, but this is an unwarranted deduction from the pa.s.sage given.

[326] A. Hillebrandt, _Alt-Indien_, p. 179.

[327] J. C. Marshman, loc. cit., chaps. i and ii.

[328] He reigned 631-579 A.D.; called Nu['s][=i]rw[=a]n, _the holy one_.

[329] J. Keane, _The Evolution of Geography_, London, 1899, p. 38.

[330] The Arabs who lived in and about Mecca.

[331] S. Guyard, in _Encyc. Brit._, 9th ed., Vol. XVI, p. 597.

[332] Oppert, loc. cit., p. 29.

[333] "At non credendum est id in Autographis contigisse, aut vetustioribus Codd. MSS." [Wallis, _Opera omnia_, Vol. II, p. 11.]

[334] In _Observationes ad Pomponium Melam de situ orbis_. The question was next taken up in a large way by Weidler, loc. cit., _De characteribus_ etc., 1727, and in _Spicilegium_ etc., 1755.

[335] The best edition of these works is that of G. Friedlein, _Anicii Manlii Torquati Severini Boetii de inst.i.tutione arithmetica libri duo, de inst.i.tutione musica libri quinque. Accedit geometria quae fertur Boetii_.... Leipzig.... MDCCCLXVII.

[336] See also P. Tannery, "Notes sur la pseudo-geometrie de Boece," in _Bibliotheca Mathematica_, Vol. I (3), p. 39. This is not the geometry in two books in which are mentioned the numerals. There is a ma.n.u.script of this pseudo-geometry of the ninth century, but the earliest one of the other work is of the eleventh century (Tannery), unless the Vatican codex is of the tenth century as Friedlein (p. 372) a.s.serts.

[337] Friedlein feels that it is partly spurious, but he says: "Eorum librorum, quos Boetius de geometria scripsisse dicitur, investigare veram inscriptionem nihil aliud esset nisi operam et tempus perdere." [Preface, p. v.] N. Bubnov in the Russian _Journal of the Ministry of Public Instruction_, 1907, in an article of which a synopsis is given in the _Jahrbuch uber die Fortschritte der Mathematik_ for 1907, a.s.serts that the geometry was written in the eleventh century.

[338] The most noteworthy of these was for a long time Cantor (_Geschichte_, Vol. I., 3d ed., pp. 587-588), who in his earlier days even believed that Pythagoras had known them. Cantor says (_Die romischen Agrimensoren_, Leipzig, 1875, p. 130): "Uns also, wir wiederholen es, ist die Geometrie des Boetius echt, dieselbe Schrift, welche er nach Euklid bearbeitete, von welcher ein Codex bereits in Jahre 821 im Kloster Reichenau vorhanden war, von welcher ein anderes Exemplar im Jahre 982 zu Mantua in die Hande Gerbert's gelangte, von welcher mannigfache Handschriften noch heute vorhanden sind." But against this opinion of the antiquity of MSS. containing these numerals is the important statement of P. Tannery, perhaps the most critical of modern historians of mathematics, that none exists earlier than the eleventh century. See also J. L. Heiberg in _Philologus, Zeitschrift f. d. kla.s.s. Altertum_, Vol. XLIII, p. 508.

Of Cantor's predecessors, Th. H. Martin was one of the most prominent, his argument for authenticity appearing in the _Revue Archeologique_ for 1856-1857, and in his treatise _Les signes numeraux_ etc. See also M.

Chasles, "De la connaissance qu'ont eu les anciens d'une numeration decimale ecrite qui fait usage de neuf chiffres prenant les valeurs de position," _Comptes rendus_, Vol. VI, pp. 678-680; "Sur l'origine de notre systeme de numeration," _Comptes rendus_, Vol. VIII, pp. 72-81; and note "Sur le pa.s.sage du premier livre de la geometrie de Boece, relatif a un nouveau systeme de numeration," in his work _Apercu historique sur l'origine et le developpement des methodes en geometrie_, of which the first edition appeared in 1837.

[339] J. L. Heiberg places the book in the eleventh century on philological grounds, _Philologus_, loc. cit.; Woepcke, in _Propagation_, p. 44; Blume, Lachmann, and Rudorff, _Die Schriften der romischen Feldmesser_, Berlin, 1848; Boeckh, _De abaco graecorum_, Berlin, 1841; Friedlein, in his Leipzig edition of 1867; Weissenborn, _Abhandlungen_, Vol. II, p. 185, his _Gerbert_, pp. 1, 247, and his _Geschichte der Einfuhrung der jetzigen Ziffern in Europa durch Gerbert_, Berlin, 1892, p. 11; Bayley, loc. cit., p. 59; Gerhardt, _etudes_, p. 17, _Entstehung und Ausbreitung_, p. 14; Nagl, _Gerbert_, p. 57; Bubnov, loc. cit. See also the discussion by Chasles, Halliwell, and Libri, in the _Comptes rendus_, 1839, Vol. IX, p.

447, and in Vols. VIII, XVI, XVII of the same journal.

[340] J. Marquardt, _La vie privee des Romains_, Vol. II (French trans.), p. 505, Paris, 1893.

[341] In a Plimpton ma.n.u.script of the arithmetic of Boethius of the thirteenth century, for example, the Roman numerals are all replaced by the Arabic, and the same is true in the first printed edition of the book. (See Smith's _Rara Arithmetica_, pp. 434, 25-27.) D. E. Smith also copied from a ma.n.u.script of the arithmetic in the Laurentian library at Florence, of 1370, the following forms, [Forged numerals

[342] Halliwell, in his _Rara Mathematica, _p. 107, states that the disputed pa.s.sage is not in a ma.n.u.script belonging to Mr. Ames, nor in one at Trinity College. See also Woepcke, in _Propagation_, pp. 37 and 42. It was the evident corruption of the texts in such editions of Boethius as those of Venice, 1499, Basel, 1546 and 1570, that led Woepcke to publish his work _Sur l'introduction de l'arithmetique indienne en Occident_.

[343] They are found in none of the very ancient ma.n.u.scripts, as, for example, in the ninth-century (?) codex in the Laurentian library which one of the authors has examined. It should be said, however, that the disputed pa.s.sage was written after the arithmetic, for it contains a reference to that work. See the Friedlein ed., p. 397.

[344] Smith, _Rara Arithmetica_, p. 66.

[345] J. L. Heiberg, _Philologus_, Vol. XLIII, p. 507.

[346] "Nosse autem huius artis dispicientem, quid sint digiti, quid articuli, quid compositi, quid incompositi numeri." [Friedlein ed., p.

395.]

[347] _De ratione abaci._ In this he describes "quandam formulam, quam ob honorem sui praeceptoris mensam Pythagoream nominabant ... a posterioribus appellabatur abacus." This, as pictured in the text, is the common Gerbert abacus. In the edition in Migne's _Patrologia Latina_, Vol. LXIII, an ordinary multiplication table (sometimes called Pythagorean abacus) is given in the ill.u.s.tration.