[102] Chalfont, F. H., _Memoirs of the Carnegie Museum_, Vol. IV, no. 1; J.

Hager, _An Explanation of the Elementary Characters of the Chinese_, London, 1801.

[103] H. V. Hilprecht, _Mathematical, Metrological and Chronological Tablets from the Temple Library at Nippur_, Vol. XX, part I, of Series A, Cuneiform Texts Published by the Babylonian Expedition of the University of Pennsylvania, 1906; A. Eisenlohr, _Ein altbabylonischer Felderplan_, Leipzig, 1906; Maspero, _Dawn of Civilization_, p. 773.

[104] Sir H. H. Howard, "On the Earliest Inscriptions from Chaldea,"

_Proceedings of the Society of Biblical Archaeology_, XXI, p. 301, London, 1899.

[105] For a bibliography of the princ.i.p.al hypotheses of this nature see Buhler, loc. cit., p. 77. Buhler (p. 78) feels that of all these hypotheses that which connects the Br[=a]hm[=i] with the Egyptian numerals is the most plausible, although he does not adduce any convincing proof. Th. Henri Martin, "Les signes numeraux et l'arithmetique chez les peuples de l'antiquite et du moyen age" (being an examination of Cantor's _Mathematische Beitrage zum Culturleben der Volker_), _Annali di matematica pura ed applicata_, Vol. V, Rome, 1864, pp. 8, 70. Also, same author, "Recherches nouvelles sur l'origine de notre systeme de numeration ecrite,"

_Revue Archeologique_, 1857, pp. 36, 55. See also the tables given later in this work.

[106] _Journal of the Royal Asiatic Society, Bombay Branch_, Vol. XXIII.

[107] Loc. cit., reprint, Part I, pp. 12, 17. Bayley's deductions are generally regarded as unwarranted.

[108] _The Alphabet_; London, 1883, Vol. II, pp. 265, 266, and _The Academy_ of Jan. 28, 1882.

[109] Taylor, _The Alphabet_, loc. cit., table on p. 266.

[110] Buhler, _On the Origin of the Indian Br[=a]hma Alphabet_, Stra.s.sburg, 1898, footnote, pp. 52, 53.

[111] Albrecht Weber, _History of Indian Literature_, English ed., Boston, 1878, p. 256: "The Indian figures from 1-9 are abbreviated forms of the initial letters of the numerals themselves...: the zero, too, has arisen out of the first letter of the word _[s.]unya_ (empty) (it occurs even in Pingala). It is the decimal place value of these figures which gives them significance." C. Henry, "Sur l'origine de quelques notations mathematiques," _Revue Archeologique_, June and July, 1879, attempts to derive the Boethian forms from the initials of Latin words. See also J.

Prinsep, "Examination of the Inscriptions from Girnar in Gujerat, and Dhauli in Cuttach," _Journal of the Asiatic Society of Bengal_, 1838, especially Plate XX, p. 348; this was the first work on the subject.

[112] Buhler, _Palaeographie_, p. 75, gives the list, with the list of letters (p. 76) corresponding to the number symbols.

[113] For a general discussion of the connection between the numerals and the different kinds of alphabets, see the articles by U. Ceretti, "Sulla origine delle cifre numerali moderne," _Rivista di fisica, matematica e scienze naturali_, Pisa and Pavia, 1909, anno X, numbers 114, 118, 119, and 120, and continuation in 1910.

[114] This is one of Buhler's hypotheses. See Bayley, loc. cit., reprint p.

4; a good bibliography of original sources is given in this work, p. 38.

[115] Loc. cit., reprint, part I, pp. 12, 17. See also Burnell, loc. cit., p. 64, and tables in plate XXIII.

[116] This was a.s.serted by G. Hager (_Memoria sulle cifre arabiche_, Milan, 1813, also published in _Fundgruben des Orients_, Vienna, 1811, and in _Bibliotheque Britannique_, Geneva, 1812). See also the recent article by Major Charles E. Woodruff, "The Evolution of Modern Numerals from Tally Marks," _American Mathematical Monthly_, August-September, 1909.

Biernatzki, "Die Arithmetik der Chinesen," _Crelle's Journal fur die reine und angewandte Mathematik_, Vol. LII, 1857, pp. 59-96, also a.s.serts the priority of the Chinese claim for a place system and the zero, but upon the flimsiest authority. Ch. de Paravey, _Essai sur l'origine unique et hieroglyphique des chiffres et des lettres de tous les peuples_, Paris, 1826; G. Kleinwachter, "The Origin of the Arabic Numerals," _China Review_, Vol. XI, 1882-1883, pp. 379-381, Vol. XII, pp. 28-30; Biot, "Note sur la connaissance que les Chinois ont eue de la valeur de position des chiffres," _Journal Asiatique_, 1839, pp. 497-502. A. Terrien de Lacouperie, "The Old Numerals, the Counting-Rods and the Swan-Pan in China," _Numismatic Chronicle_, Vol. III (3), pp. 297-340, and Crowder B.

Moseley, "Numeral Characters: Theory of Origin and Development," _American Antiquarian_, Vol. XXII, pp. 279-284, both propose to derive our numerals from Chinese characters, in much the same way as is done by Major Woodruff, in the article above cited.

[117] The Greeks, probably following the Semitic custom, used nine letters of the alphabet for the numerals from 1 to 9, then nine others for 10 to 90, and further letters to represent 100 to 900. As the ordinary Greek alphabet was insufficient, containing only twenty-four letters, an alphabet of twenty-seven letters was used.

[118] _Inst.i.tutiones mathematicae_, 2 vols., Stra.s.sburg, 1593-1596, a somewhat rare work from which the following quotation is taken:

"_Quis est harum Cyphrarum autor?_

"A quibus hae usitatae syphrarum notae sint inventae: hactenus incertum fuit: meo tamen iudicio, quod exiguum esse fateor: a graecis librarijs (quorum olim magna fuit copia) literae Graecorum quibus veteres Graeci tamquam numerorum notis sunt usi: fuerunt corruptae. vt ex his licet videre.

"Graecorum Literae corruptae.

[Ill.u.s.tration]

_"Sed qua ratione graecorum literae ita fuerunt corruptae?_

"Finxerunt has corruptas Graecorum literarum notas: vel abiectione vt in nota binarij numeri, vel additione vt in ternarij, vel inuersione vt in septenarij, numeri nota, nostrae notae, quibus hodie utimur: ab his sola differunt elegantia, vt apparet."

See also Bayer, _Historia regni Graecorum Bactriani_, St. Petersburg, 1788, pp. 129-130, quoted by Martin, _Recherches nouvelles_, etc., loc. cit.

[119] P. D. Huet, _Demonstratio evangelica_, Paris, 1769, note to p. 139 on p. 647: "Ab Arabibus vel ab Indis inventas esse, non vulgus eruditorum modo, sed doctissimi quique ad hanc diem arbitrati sunt. Ego vero falsum id esse, merosque esse Graecorum characteres aio; a librariis Graecae linguae ignaris interpolatos, et diuturna scribendi consuetudine corruptos. Nam primum 1 apex fuit, seu virgula, nota [Greek: monados]. 2, est ipsum [beta]

extremis suis truncatum. [gamma], si in sinistram partem inclinaveris & cauda mutilaveris & sinistrum cornu sinistrorsum flexeris, fiet 3. Res ipsa loquitur 4 ipsissimum esse [Delta], cujus crus sinistrum erigitur [Greek: kata katheton], & infra basim descendit; basis vero ipsa ultra crus producta eminet. Vides quam 5 simile sit [Greek: toi] [epsilon]; infimo tantum semicirculo, qui sinistrorsum patebat, dextrorsum converso. [Greek: episemon bau] quod ita notabatur [digamma], rotundato ventre, pede detracto, peperit [Greek: to] 6. Ex [Zeta] basi sua mutilato, ortum est [Greek: to] 7. Si [Eta] inflexis introrsum apicibus in rotundiorem & commodiorem formam mutaveris, exurget [Greek: to] 8. At 9 ipsissimum est [alt theta]."

I. Weidler, _Spicilegium observationum ad historiam notarum numeralium_, Wittenberg, 1755, derives them from the Hebrew letters; Dom Augustin Calmet, "Recherches sur l'origine des chiffres d'arithmetique," _Memoires pour l'histoire des sciences et des beaux arts_, Trevoux, 1707 (pp.

1620-1635, with two plates), derives the current symbols from the Romans, stating that they are relics of the ancient "Notae Tironianae." These "notes" were part of a system of shorthand invented, or at least perfected, by Tiro, a slave who was freed by Cicero. L. A. Sedillot, "Sur l'origine de nos chiffres," _Atti dell' Accademia pontificia dei nuovi Lincei_, Vol.

XVIII, 1864-1865, pp. 316-322, derives the Arabic forms from the Roman numerals.

[120] Athanasius Kircher, _Arithmologia sive De abditis Numerorum, mysterijs qua origo, antiquitas & fabrica Numerorum exponitur_, Rome, 1665.

[121] See Suter, _Die Mathematiker und Astronomen der Araber_, p. 100.

[122] "Et hi numeri sunt numeri Indiani, a Brachmanis Indiae Sapientibus ex figura circuli secti inuenti."

[123] V. A. Smith, _The Early History of India_, Oxford, 2d ed., 1908, p.

333.

[124] C. J. Ball, "An Inscribed Limestone Tablet from Sippara,"

_Proceedings of the Society of Biblical Archaeology_, Vol. XX, p. 25 (London, 1898). Terrien de Lacouperie states that the Chinese used the circle for 10 before the beginning of the Christian era. [_Catalogue of Chinese Coins_, London, 1892, p. xl.]

[125] For a purely fanciful derivation from the corresponding number of strokes, see W. W. R. Ball, _A Short Account of the History of Mathematics_, 1st ed., London, 1888, p. 147; similarly J. B. Reveillaud, _Essai sur les chiffres arabes_, Paris, 1883; P. Voizot, "Les chiffres arabes et leur origine," _La Nature_, 1899, p. 222; G. Dumesnil, "De la forme des chiffres usuels," _Annales de l'universite de Gren.o.ble_, 1907, Vol. XIX, pp. 657-674, also a note in _Revue Archeologique_, 1890, Vol. XVI (3), pp. 342-348; one of the earliest references to a possible derivation from points is in a work by Bettino ent.i.tled _Apiaria universae philosophiae mathematicae in quibus paradoxa et noua machinamenta ad usus eximios traducta, et facillimis demonstrationibus confirmata_, Bologna, 1545, Vol. II, Apiarium XI, p. 5.

[126] _Alphabetum Barmanum_, Romae, MDCCLXXVI, p. 50. The 1 is evidently Sanskrit, and the 4, 7, and possibly 9 are from India.

[127] _Alphabetum Grandonico-Malabaric.u.m_, Romae, MDCCLXXII, p. 90. The zero is not used, but the symbols for 10, 100, and so on, are joined to the units to make the higher numbers.

[128] _Alphabetum Tangutanum_, Romae, MDCCLXXIII, p. 107. In a Tibetan MS.

in the library of Professor Smith, probably of the eighteenth century, substantially these forms are given.

[129] Bayley, loc. cit., plate II. Similar forms to these here shown, and numerous other forms found in India, as well as those of other oriental countries, are given by A. P. Pihan, _Expose des signes de numeration usites chez les peuples orientaux anciens et modernes_, Paris, 1860.

[130] Buhler, loc. cit., p. 80; J. F. Fleet, _Corpus inscriptionum Indicarum_, Vol. III, Calcutta, 1888. Lists of such words are given also by Al-B[=i]r[=u]n[=i] in his work _India_; by Burnell, loc. cit.; by E.

Jacquet, "Mode d'expression symbolique des nombres employe par les Indiens, les Tibetains et les Javanais," _Journal Asiatique_, Vol. XVI, Paris, 1835.

[131] This date is given by Fleet, loc. cit., Vol. III, p. 73, as the earliest epigraphical instance of this usage in India proper.

[132] Weber, _Indische Studien_, Vol. VIII, p. 166 seq.

[133] _Journal of the Royal Asiatic Society_, Vol. I (N.S.), p. 407.

[134] VIII, 20, 21.

[135] Th. H. Martin, _Les signes numeraux_ ..., Rome, 1864; La.s.sen, _Indische Alterthumskunde_, Vol. II, 2d ed., Leipzig and London, 1874, p.

1153.