Chapter III

.

[94] From a copper plate of 510 A.D., found at Majhgaw[=a]in, Central India. [Fleet, loc. cit., Plate XIV.]

[95] From an inscription of 588 A.D., found at B[=o]dh-Gay[=a], Bengal Presidency. [Fleet, loc. cit., Plate XXIV.]

[96] From a copper plate of 571 A.D., found at M[=a]liy[=a], Bombay Presidency. [Fleet, loc. cit., Plate XXIV.]

[97] From a Bijayaga[d.]h pillar inscription of 372 A.D. [Fleet, loc. cit., Plate XXXVI, C.]

[98] From a copper plate of 434 A.D. [_Indian Antiquary_, Vol. I, p. 60.]

[99] Gadhwa inscription, c. 417 A.D. [Fleet, loc. cit., Plate IV, D.]

[100] K[=a]r[=i]tal[=a][=i] plate of 493 A.D., referred to above.

[101] It seems evident that the Chinese four, curiously enough called "eight in the mouth," is only a cursive [4 vertical strokes].

[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 Archæology_, XXI, p. 301, London, 1899.

[105] For a bibliography of the principal hypotheses of this nature see Bühler, loc. cit., p. 77. Bühler (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 numéraux et l'arithmétique chez les peuples de l'antiquité et du moyen âge" (being an examination of Cantor's _Mathematische Beiträge zum Culturleben der Völker_), _Annali di matematica pura ed applicata_, Vol. V, Rome, 1864, pp. 8, 70. Also, same author, "Recherches nouvelles sur l'origine de notre système de numération écrite," _Revue Archéologique_, 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] Bühler, _On the Origin of the Indian Br[=a]hma Alphabet_, Strassburg, 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 Piñgala). It is the decimal place value of these figures which gives them significance." C. Henry, "Sur l'origine de quelques notations mathématiques," _Revue Archéologique_, 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] Bühler, _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 Bühler'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 asserted by G. Hager (_Memoria sulle cifre arabiche_, Milan, 1813, also published in _Fundgruben des Orients_, Vienna, 1811, and in _Bibliothèque 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 für die reine und angewandte Mathematik_, Vol. LII, 1857, pp. 59-96, also asserts 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 hiéroglyphique des chiffres et des lettres de tous les peuples_, Paris, 1826; G. Kleinwächter, "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] _Institutiones mathematicae_, 2 vols., Strassburg, 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.

[Illustration]

_"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; à 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: tôi] [epsilon]; infimo tantum semicirculo, qui sinistrorsum patebat, dextrorsum converso. [Greek: episêmon 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'arithmétique," _Mémoires pour l'histoire des sciences et des beaux arts_, Trévoux, 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 Archæology_, 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'université de Grenoble_, 1907, Vol. XIX, pp. 657-674, also a note in _Revue Archéologique_, 1890, Vol. XVI (3), pp. 342-348; one of the earliest references to a possible derivation from points is in a work by Bettino entitled _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-Malabaricum_, 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, _Exposé des signes de numération usités chez les peuples orientaux anciens et modernes_, Paris, 1860.

[130] Bühler, 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 employé par les Indiens, les Tibétains 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 numéraux_ ..., Rome, 1864; Lassen, _Indische Alterthumskunde_, Vol. II, 2d ed., Leipzig and London, 1874, p. 1153.

[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 Bibliothèque nationale_, Vol. XXVII, Part I, pp. 1-180, 1885; see also numerous articles in _Journal Asiatique_, by Aymonier.

[138] Bühler, loc. cit., p. 82.

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

[140] Bühler, 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