Chapter VIII: The Spread of the Numerals in Europe (2)

[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_, Strassburg, 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 Morgenlaendischen 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. Buerk, 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. [Buehler, _Indische Palaeographie_, Strassburg, 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. 100.10^7.

[57] There is some uncertainty about this limit.

[58] This problem deserves more study than has yet been given it. A beginning may be made with Comte Goblet d'Alviella, _Ce que l'Inde doit a la Grece_, Paris, 1897, and H. G. Keene's review, "The Greeks in India," in the _Calcutta Review_, Vol. CXIV, 1902, p. 1. See also F. Woepeke, _Propagation_, p. 253; G. R. Kaye, loc. cit., p. 475 seq., and "The Source of Hindu Mathematics," _Journal of the Royal Asiatic Society_, July, 1910, pp. 749-760; G. Thibaut, _Astronomie, Astrologie und Mathematik_, pp. 43-50 and 76-79. It will be discussed more fully in Chapter VI.

[59] I.e. to 100,000. The lakh is still the common large unit in India, like the myriad in ancient Greece and the million in the West.

[60] This again suggests the _Psammites_, or _De harenae numero_ as it is called in the 1544 edition of the _Opera_ of Archimedes, a work in which the great Syracusan proposes to show to the king "by geometric proofs which you can follow, that the numbers which have been named by us ... are sufficient to exceed not only the number of a sand-heap as large as the whole earth, but one as large as the universe." For a list of early editions of this work see D. E. Smith, _Rara Arithmetica_, Boston, 1909, p. 227.

[61] I.e. the Wise.

[62] Sir Monier Monier-Williams, _Indian Wisdom_, 4th ed., London, 1893, pp. 144, 177. See also J. C. Marshman, _Abridgment of the History of India_, London, 1893, p. 2.

[63] For a list and for some description of these works see R. C. Dutt, _A History of Civilization in Ancient India_, Vol. II, p. 121.

[64] Professor Ramkrishna Gopal Bhandarkar fixes the date as the fifth century B.C. ["Consideration of the Date of the Mah[=a]bh[=a]rata," in the _Journal of the Bombay Branch of the R. A. Soc._, Bombay, 1873, Vol. X, p. 2.].

[65] Marshman, loc. cit., p. 2.

[66] A. C. Burnell, _South Indian Palaeography_, 2d ed., London, 1878, p. 1, seq.

[67] This extensive subject of palpable arithmetic, essentially the history of the abacus, deserves to be treated in a work by itself.

[68] The following are the leading sources of information upon this subject: G. Buehler, _Indische Palaeographie_, particularly chap. vi; A. C. Burnell, _South Indian Palaeography_, 2d ed., London, 1878, where tables of the various Indian numerals are given in Plate XXIII; E. C. Bayley, "On the Genealogy of Modern Numerals," _Journal of the Royal Asiatic Society_, Vol. XIV, part 3, and Vol. XV, part 1, and reprint, London, 1882; I. Taylor, in _The Academy_, January 28, 1882, with a repetition of his argument in his work _The Alphabet_, London, 1883, Vol. II, p. 265, based on Bayley; G. R. Kaye, loc. cit., in some respects one of the most critical articles thus far published; J. C. Fleet, _Corpus inscriptionum Indicarum_, London, 1888, Vol. III, with facsimiles of many Indian inscriptions, and _Indian Epigraphy_, Oxford, 1907, reprinted from the _Imperial Gazetteer of India_, Vol. II, pp. 1-88, 1907; G. Thibaut, loc. cit., _Astronomie_ etc.; R. Caldwell, _Comparative Grammar of the Dravidian Languages_, London, 1856, p. 262 seq.; and _Epigraphia Indica_ (official publication of the government of India), Vols. I-IX. Another work of Buehler's, _On the Origin of the Indian Br[=a]hma Alphabet_, is also of value.

[69] The earliest work on the subject was by James Prinsep, "On the Inscriptions of Piyadasi or A['s]oka," etc., _Journal of the Asiatic Society of Bengal_, 1838, following a preliminary suggestion in the same journal in 1837. See also "A['s]oka Notes," by V. A. Smith, _The Indian Antiquary_, Vol. XXXVII, 1908, p. 24 seq., Vol. XXXVIII, pp. 151-159, June, 1909; _The Early History of India_, 2d ed., Oxford, 1908, p. 154; J. F. Fleet, "The Last Words of A['s]oka," _Journal of the Royal Asiatic Society_, October, 1909, pp. 981-1016; E. Senart, _Les inscriptions de Piyadasi_, 2 vols., Paris, 1887.

[70] For a discussion of the minor details of this system, see Buehler, loc. cit., p. 73.

[71] Julius Euting, _Nabataeische Inschriften aus Arabien_, Berlin, 1885, pp. 96-97, with a table of numerals.

[72] For the five principal theories see Buehler, loc. cit., p. 10.

[73] Bayley, loc. cit., reprint p. 3.

[74] Buehler, loc. cit.; _Epigraphia Indica_, Vol. III, p. 134; _Indian Antiquary_, Vol. VI, p. 155 seq., and Vol. X, p. 107.

[75] Pandit Bhagav[=a]nl[=a]l Indr[=a]j[=i], "On Ancient N[=a]g[=a]ri Numeration; from an Inscription at N[=a]negh[=a]t," _Journal of the Bombay Branch of the Royal Asiatic Society_, 1876, Vol. XII, p. 404.

[76] Ib., p. 405. He gives also a plate and an interpretation of each numeral.

[77] These may be compared with Buehler's drawings, loc. cit.; with Bayley, loc. cit., p. 337 and plates; and with Bayley's article in the _Encyclopaedia Britannica_, 9th ed., art. "Numerals."

[78] E. Senart, "The Inscriptions in the Caves at Nasik," _Epigraphia Indica_, Vol. VIII, pp. 59-96; "The Inscriptions in the Cave at Karle," _Epigraphia Indica_, Vol. VII, pp. 47-74; Buehler, _Palaeographie_, Tafel IX.

[79] See Fleet, loc. cit. See also T. Benfey, _Sanskrit Grammar_, London, 1863, p. 217; M. R. Kale, _Higher Sanskrit Grammar_, 2d ed., Bombay, 1898, p. 110, and other authorities as cited.

[80] Kharo[s.][t.]h[=i] numerals, A['s]oka inscriptions, c. 250 B.C. Senart, _Notes d'epigraphie indienne_. Given by Buehler, loc. cit., Tafel I.

[81] Same, ['S]aka inscriptions, probably of the first century B.C. Senart, loc. cit.; Buehler, loc. cit.

[82] Br[=a]hm[=i] numerals, A['s]oka inscriptions, c. 250 B.C. _Indian Antiquary_, Vol. VI, p. 155 seq.

[83] Same, N[=a]n[=a] Gh[=a]t inscriptions, c. 150 B.C. Bhagav[=a]nl[=a]l Indr[=a]j[=i], _On Ancient N[=a]gar[=i] Numeration_, loc. cit. Copied from a squeeze of the original.

[84] Same, Nasik inscription, c. 100 B.C. Burgess, _Archeological Survey Report, Western India_; Senart, _Epigraphia Indica_, Vol. VII, pp. 47-79, and Vol. VIII, pp. 59-96.

[85] K[s.]atrapa coins, c. 200 A.D. _Journal of the Royal Asiatic Society_, 1890, p. 639.

[86] Ku[s.]ana inscriptions, c. 150 A.D. _Epigraphia Indica_, Vol. I, p. 381, and Vol. II, p. 201.

[87] Gupta Inscriptions, c. 300 A.D. to 450 A.D. Fleet, loc. cit., Vol. III.

[88] Valhab[=i], c. 600 A.D. _Corpus_, Vol. III.

[89] Bendall's Table of Numerals, in _Cat. Sansk. Budd. MSS._, British Museum.

[90] _Indian Antiquary_, Vol. XIII, 120; _Epigraphia Indica_, Vol. III, 127 ff.

[91] Fleet, loc. cit.

[92] Bayley, loc. cit., p. 335.

[93] From a copper plate of 493 A.D., found at K[=a]r[=i]tal[=a][=i], Central India. [Fleet, loc. cit., Plate XVI.] It should be stated, however, that many of these copper plates, being deeds of property, have forged dates so as to give the appearance of antiquity of title. On the other hand, as Colebrooke long ago pointed out, a successful forgery has to imitate the writing of the period in question, so that it becomes evidence well worth considering, as shown in 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 Archaeology_, XXI, p. 301, London, 1899.

[105] For a bibliography of the principal hypotheses of this nature see Buehler, loc. cit., p. 77. Buehler (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 Beitraege zum Culturleben der Voelker_), _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] Buehler, _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 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] Buehler, _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 Buehler'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 _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 fuer 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 hieroglyphique des chiffres et des lettres de tous les peuples_, Paris, 1826; G. Kleinwaechter, "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.

_"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 Grenoble_, 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 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, _Expose des signes de numeration usites chez les peuples orientaux anciens et modernes_, Paris, 1860.

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

[138] Buehler, loc. cit., p. 82.

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

[140] Buehler, 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 Chapter II, p. 16.

[145] The nine ciphers were called _a[.n]ka_.

[146] "Zur Geschichte des indischen Ziffernsystems," _Zeitschrift fuer die Kunde des Morgenlandes_, Vol. IV, 1842, pp. 74-83.

[147] It is found in the Bakh[s.][=a]l[=i] MS. of an elementary arithmetic which Hoernle placed, at first, about the beginning of our era, but the date is much in question. G. Thibaut, loc. cit., places it between 700 and 900 A.D.; Cantor places the body of the work about the third or fourth century A.D., _Geschichte der Mathematik_, Vol. I (3), p. 598.

[148] For the opposite side of the case see G. R. Kaye, "Notes on Indian Mathematics, No. 2.--[=A]ryabha[t.]a," _Journ. and Proc. of the Asiatic Soc. of Bengal_, Vol. IV, 1908, pp. 111-141.

[149] He used one of the alphabetic systems explained above. This ran up to 10^{18} and was not difficult, beginning as follows:

the same letter (_ka_) appearing in the successive consonant forms, _ka_, _kha_, _ga_, _gha_, etc. See C. I. Gerhardt, _Ueber die Entstehung und Ausbreitung des dekadischen Zahlensystems_, Programm, p. 17, Salzwedel, 1853, and _Etudes historiques sur l'arithmetique de position_, Programm, p. 24, Berlin, 1856; E. Jacquet, _Mode d'expression symbolique des nombres_, loc. cit., p. 97; L. Rodet, "Sur la veritable signification de la notation numerique inventee par [=A]ryabhata," _Journal Asiatique_, Vol. XVI (7), pp. 440-485. On the two [=A]ryabha[t.]as see Kaye, _Bibl. Math._, Vol. X (3), p. 289.

[150] Using _kha_, a synonym of _['s][=u]nya_. [Bayley, loc. cit., p. 22, and L. Rodet, _Journal Asiatique_, Vol. XVI (7), p. 443.]

[151] Var[=a]ha-Mihira, _Pancasiddh[=a]ntik[=a]_, translated by G. Thibaut and M. S. Dvived[=i], Benares, 1889; see Buehler, loc. cit., p. 78; Bayley, loc. cit., p. 23.

[152] _B[r.]hat Sa[m.]hit[=a]_, translated by Kern, _Journal of the Royal Asiatic Society_, 1870-1875.

[153] It is stated by Buehler in a personal letter to Bayley (loc. cit., p. 65) that there are hundreds of instances of this usage in the _B[r.]hat Sa[m.]hit[=a]_. The system was also used in the _Pancasiddh[=a]ntik[=a]_ as early as 505 A.D. [Buehler, _Palaeographie_, p. 80, and Fleet, _Journal of the Royal Asiatic Society_, 1910, p. 819.]

[154] Cantor, _Geschichte der Mathematik_, Vol. I (3), p. 608.

[155] Buehler, loc. cit., p. 78.

[156] Bayley, p. 38.

[157] Noviomagus, in his _De numeris libri duo_, Paris, 1539, confesses his ignorance as to the origin of the zero, but says: "D. Henricus Grauius, vir Graece & Hebraice exime doctus, Hebraicam originem ostendit," adding that Valla "Indis Orientalibus gentibus inventionem tribuit."

[158] See _Essays_, Vol. II, pp. 287 and 288.

[159] Vol. XXX, p. 205 seqq.

[160] Loc. cit., p. 284 seqq.

[161] Colebrooke, loc. cit., p. 288.

[162] Loc. cit., p. 78.

[163] Hereafter, unless expressly stated to the contrary, we shall use the word "numerals" to mean numerals with place value.

[164] "The Gurjaras of R[=a]jput[=a]na and Kanauj," in _Journal of the Royal Asiatic Society_, January and April, 1909.

[165] Vol. IX, 1908, p. 248.

[166] _Epigraphia Indica_, Vol. IX, pp. 193 and 198.

[167] _Epigraphia Indica_, Vol. IX, p. 1.

[168] Loc. cit., p. 71.

[169] Thibaut, p. 71.

[170] "Est autem in aliquibus figurarum istaram apud multos diuersitas. Quidam enim septimam hanc figuram representant," etc. [Boncompagni, _Trattati_, p. 28.] Enestroem has shown that very likely this work is incorrectly attributed to Johannes Hispalensis. [_Bibliotheca Mathematica_, Vol. IX (3), p. 2.]

[171] _Indische Palaeographie_, Tafel IX.

[172] Edited by Bloomfield and Garbe, Baltimore, 1901, containing photographic reproductions of the manuscript.

[173] Bakh[s.][=a]l[=i] MS. See page 43; Hoernle, R., _The Indian Antiquary_, Vol. XVII, pp. 33-48, 1 plate; Hoernle, _Verhandlungen des VII. Internationalen Orientalisten-Congresses, Arische Section_, Vienna, 1888, "On the Baksh[=a]l[=i] Manuscript," pp. 127-147, 3 plates; Buehler, loc. cit.

[174] 3, 4, 6, from H. H. Dhruva, "Three Land-Grants from Sankheda," _Epigraphia Indica_, Vol. II, pp. 19-24 with plates; date 595 A.D. 7, 1, 5, from Bhandarkar, "Daulatabad Plates," _Epigraphia Indica_, Vol. IX, part V; date c. 798 A.D.

[175] 8, 7, 2, from "Buckhala Inscription of Nagabhatta," Bhandarkar, _Epigraphia Indica_, Vol. IX, part V; date 815 A.D. 5 from "The Morbi Copper-Plate," Bhandarkar, _The Indian Antiquary_, Vol. II, pp. 257-258, with plate; date 804 A.D. See Buehler, loc. cit.

[176] 8 from the above Morbi Copper-Plate. 4, 5, 7, 9, and 0, from "Asni Inscription of Mahipala," _The Indian Antiquary_, Vol. XVI, pp. 174-175; inscription is on red sandstone, date 917 A.D. See Buehler.

[177] 8, 9, 4, from "Rashtrakuta Grant of Amoghavarsha," J. F. Fleet, _The Indian Antiquary_, Vol. XII, pp. 263-272; copper-plate grant of date c. 972 A.D. See Buehler. 7, 3, 5, from "Torkhede Copper-Plate Grant of the Time of Govindaraja of Gujerat," Fleet, _Epigraphia Indica_, Vol. III, pp. 53-58. See Buehler.

[178] From "A Copper-Plate Grant of King Tritochanapala Chanlukya of L[=a][t.]ade['s]a," H.H. Dhruva, _Indian Antiquary_, Vol. XII, pp. 196-205; date 1050 A.D. See Buehler.

[179] Burnell, A. C., _South Indian Palaeography_, plate XXIII, Telugu-Canarese numerals of the eleventh century. See Buehler.

[180] From a manuscript of the second half of the thirteenth century, reproduced in "Della vita e delle opere di Leonardo Pisano," Baldassare Boncompagni, Rome, 1852, in _Atti dell' Accademia Pontificia dei nuovi Lincei_, anno V.

[181] From a fourteenth-century manuscript, as reproduced in _Della vita_ etc., Boncompagni, loc. cit.

[182] From a Tibetan MS. in the library of D. E. Smith.

[183] From a Tibetan block-book in the library of D. E. Smith.

[184] ['S][=a]rad[=a] numerals from _The Kashmirian Atharva-Veda, reproduced by chromophotography from the manuscript in the University Library at Tuebingen_, Bloomfield and Garbe, Baltimore, 1901. Somewhat similar forms are given under "Numeration Cachemirienne," by Pihan, _Expose_ etc., p. 84.

[185] Franz X. Kugler, _Die Babylonische Mondrechnung_, Freiburg i. Br., 1900, in the numerous plates at the end of the book; practically all of these contain the symbol to which reference is made. Cantor, _Geschichte_, Vol. I, p. 31.

[186] F. X. Kugler, _Sternkunde und Sterndienst in Babel_, I. Buch, from the beginnings to the time of Christ, Muenster i. Westfalen, 1907. It also has numerous tables containing the above zero.

[187] From a letter to D. E. Smith, from G. F. Hill of the British Museum. See also his monograph "On the Early Use of Arabic Numerals in Europe," in _Archaeologia_, Vol. LXII (1910), p. 137.

[188] R. Hoernle, "The Baksh[=a]l[=i] Manuscript," _Indian Antiquary_, Vol. XVII, pp. 33-48 and 275-279, 1888; Thibaut, _Astronomie, Astrologie und Mathematik_, p. 75; Hoernle, _Verhandlungen_, loc. cit., p. 132.

[189] Bayley, loc. cit., Vol. XV, p. 29. Also Bendall, "On a System of Numerals used in South India," _Journal of the Royal Asiatic Society_, 1896, pp. 789-792.

[190] V. A. Smith, _The Early History of India_, 2d ed., Oxford, 1908, p. 14.

[191] Colebrooke, _Algebra, with Arithmetic and Mensuration, from the Sanskrit of Brahmegupta and Bhascara_, London, 1817, pp. 339-340.

[192] Ibid., p. 138.

[193] D. E. Smith, in the _Bibliotheca Mathematica_, Vol. IX (3), pp. 106-110.

[194] As when we use three dots (...).

[195] "The Hindus call the nought explicitly _['s][=u]nyabindu_ 'the dot marking a blank,' and about 500 A.D. they marked it by a simple dot, which latter is commonly used in inscriptions and MSS. in order to mark a blank, and which was later converted into a small circle." [Buehler, _On the Origin of the Indian Alphabet_, p. 53, note.]

[196] Fazzari, _Dell' origine delle parole zero e cifra_, Naples, 1903.

[197] E. Wappler, "Zur Geschichte der Mathematik im 15. Jahrhundert," in the _Zeitschrift fuer Mathematik und Physik_, Vol. XLV, _Hist.-lit. Abt._, p. 47. The manuscript is No. C. 80, in the Dresden library.

[198] J. G. Praendel, _Algebra nebst ihrer literarischen Geschichte_, p. 572, Munich, 1795.

[199] See the table, p. 23. Does the fact that the early European arithmetics, following the Arab custom, always put the 0 after the 9, suggest that the 0 was derived from the old Hindu symbol for 10?

[200] Bayley, loc. cit., p. 48. From this fact Delambre (_Histoire de l'astronomie ancienne_) inferred that Ptolemy knew the zero, a theory accepted by Chasles, _Apercu historique sur l'origine et le developpement des methodes en geometrie_, 1875 ed., p. 476; Nesselmann, however, showed (_Algebra der Griechen_, 1842, p. 138), that Ptolemy merely used [Greek: o] for [Greek: ouden], with no notion of zero. See also G. Fazzari, "Dell' origine delle parole zero e cifra," _Ateneo_, Anno I, No. 11, reprinted at Naples in 1903, where the use of the point and the small cross for zero is also mentioned. Th. H. Martin, _Les signes numeraux_ etc., reprint p. 30, and J. Brandis, _Das Muenz-, Mass- und Gewichtswesen in Vorderasien bis auf Alexander den Grossen_, Berlin, 1866, p. 10, also discuss this usage of [Greek: o], without the notion of place value, by the Greeks.

[201] _Al-Batt[=a]n[=i] sive Albatenii opus astronomicum_. Ad fidem codicis escurialensis arabice editum, latine versum, adnotationibus instructum a Carolo Alphonso Nallino, 1899-1907. Publicazioni del R. Osservatorio di Brera in Milano, No. XL.

[202] Loc. cit., Vol. II, p. 271.

[203] C. Henry, "Prologus N. Ocreati in Helceph ad Adelardum Batensem magistrum suum," _Abhandlungen zur Geschichte der Mathematik_, Vol. III, 1880.

[204] Max. Curtze, "Ueber eine Algorismus-Schrift des XII. Jahrhunderts," _Abhandlungen zur Geschichte der Mathematik_, Vol. VIII, 1898, pp. 1-27; Alfred Nagl, "Ueber eine Algorismus-Schrift des XII. Jahrhunderts und ueber die Verbreitung der indisch-arabischen Rechenkunst und Zahlzeichen im christl. Abendlande," _Zeitschrift fuer Mathematik und Physik, Hist.-lit. Abth._, Vol. XXXIV, pp. 129-146 and 161-170, with one plate.

[205] "Byzantinische Analekten," _Abhandlungen zur Geschichte der Mathematik_, Vol. IX, pp. 161-189.

[206] [symbol] or [symbol] for 0. [symbol] also used for 5. [symbols] for 13. [Heiberg, loc. cit.]

[207] Gerhardt, _Etudes historiques sur l'arithmetique de position_, Berlin, 1856, p. 12; J. Bowring, _The Decimal System in Numbers, Coins, & Accounts_, London, 1854, p. 33.

[208] Karabacek, _Wiener Zeitschrift fuer die Kunde des Morgenlandes_, Vol. XI, p. 13; _Fuehrer durch die Papyrus-Ausstellung Erzherzog Rainer_, Vienna, 1894, p. 216.

[209] In the library of G. A. Plimpton, Esq.

[210] Cantor, _Geschichte_, Vol. I (3), p. 674; Y. Mikami, "A Remark on the Chinese Mathematics in Cantor's Geschichte der Mathematik," _Archiv der Mathematik und Physik_, Vol. XV (3), pp. 68-70.

[211] Of course the earlier historians made innumerable guesses as to the origin of the word _cipher_. E.g. Matthew Hostus, _De numeratione emendata_, Antwerp, 1582, p. 10, says: "Siphra vox Hebraeam originem sapit refertque: & ut docti arbitrantur, a verbo saphar, quod Ordine numerauit significat. Unde Sephar numerus est: hinc Siphra (vulgo corruptius). Etsi vero gens Iudaica his notis, quae hodie Siphrae vocantur, usa non fuit: mansit tamen rei appellatio apud multas gentes." Dasypodius, _Institutiones mathematicae_, Vol. I, 1593, gives a large part of this quotation word for word, without any mention of the source. Hermannus Hugo, _De prima scribendi origine_, Trajecti ad Rhenum, 1738, pp. 304-305, and note, p. 305; Karl Krumbacher, "Woher stammt das Wort Ziffer (Chiffre)?", _Etudes de philologie neo-grecque_, Paris, 1892.

[212] Buehler, loc. cit., p. 78 and p. 86.

[213] Fazzari, loc. cit., p. 4. So Elia Misrachi (1455-1526) in his posthumous _Book of Number_, Constantinople, 1534, explains _sifra_ as being Arabic. See also Steinschneider, _Bibliotheca Mathematica_, 1893, p. 69, and G. Wertheim, _Die Arithmetik des Elia Misrachi_, Programm, Frankfurt, 1893.

[214] "Cum his novem figuris, et cum hoc signo 0, quod arabice zephirum appellatur, scribitur quilibet numerus."

[215] [Greek: tziphra], a form also used by Neophytos (date unknown, probably c. 1330). It is curious that Finaeus (1555 ed., f. 2) used the form _tziphra_ throughout. A. J. H. Vincent ["Sur l'origine de nos chiffres," _Notices et Extraits des MSS._, Paris, 1847, pp. 143-150] says: "Ce cercle fut nomme par les uns, _sipos, rota, galgal_ ...; par les autres _tsiphra_ (de [Hebrew: TSPR], _couronne_ ou _diademe_) ou _ciphra_ (de [Hebrew: SPR], _numeration_)." Ch. de Paravey, _Essai sur l'origine unique et hieroglyphique des chiffres et des lettres de tous les peuples_, Paris, 1826, p. 165, a rather fanciful work, gives "vase, vase arrondi et ferme par un couvercle, qui est le symbole de la 10^e Heure, [symbol]," among the Chinese; also "Tsiphron Zeron, ou tout a fait vide en arabe, [Greek: tziphra] en grec ... d'ou chiffre (qui derive plutot, suivant nous, de l'Hebreu _Sepher_, compter.")

[216] "Compilatus a Magistro Jacobo de Florentia apud montem pesalanum," and described by G. Lami in his _Catalogus codicum manuscriptorum qui in bibliotheca Riccardiana Florentiae adservantur_. See Fazzari, loc. cit., p. 5.

[217] "Et doveto sapere chel zeuero per se solo non significa nulla ma e potentia di fare significare, ... Et decina o centinaia o migliaia non si puote scrivere senza questo segno 0. la quale si chiama zeuero." [Fazzari, loc. cit., p. 5.]

[218] Ibid., p. 6.

[219] Avicenna (980-1036), translation by Gasbarri et Francois, "piu il punto (gli Arabi adoperavano il punto in vece dello zero il cui segno 0 in arabo si chiama _zepiro_ donde il vocabolo zero), che per se stesso non esprime nessun numero." This quotation is taken from D. C. Martines, _Origine e progressi dell' aritmetica_, Messina, 1865.

[220] Leo Jordan, "Materialien zur Geschichte der arabischen Zahlzeichen in Frankreich," _Archiv fuer Kulturgeschichte_, Berlin, 1905, pp. 155-195, gives the following two schemes of derivation, (1) "zefiro, zeviro, zeiro, zero," (2) "zefiro, zefro, zevro, zero."

[221] Koebel (1518 ed., f. A_4) speaks of the numerals in general as "die der gemain man Zyfer nendt." Recorde (_Grounde of Artes_, 1558 ed., f. B_6) says that the zero is "called priuatly a Cyphar, though all the other sometimes be likewise named."

[222] "Decimo X 0 theca, circul[us] cifra sive figura nihili appelat'." [_Enchiridion Algorismi_, Cologne, 1501.] Later, "quoniam de integris tam in cifris quam in proiectilibus,"--the word _proiectilibus_ referring to markers "thrown" and used on an abacus, whence the French _jetons_ and the English expression "to _cast_ an account."

[223] "Decima vero o dicitur teca, circulus, vel cyfra vel figura nichili." [Maximilian Curtze, _Petri Philomeni de Dacia in Algorismum Vulgarem Johannis de Sacrobosco commentarius, una cum Algorismo ipso_, Copenhagen, 1897, p. 2.] Curtze cites five manuscripts (fourteenth and fifteenth centuries) of Dacia's commentary in the libraries at Erfurt, Leipzig, and Salzburg, in addition to those given by Enestroem, _Oefversigt af Kongl. Vetenskaps-Akademiens Foerhandlingar_, 1885, pp. 15-27, 65-70; 1886, pp. 57-60.

[224] Curtze, loc. cit., p. VI.

[225] _Rara Mathematica_, London, 1841, chap, i, "Joannis de Sacro-Bosco Tractatus de Arte Numerandi."

[226] Smith, _Rara Arithmetica_, Boston, 1909.

[227] In the 1484 edition, Borghi uses the form "cefiro: ouero nulla:" while in the 1488 edition he uses "zefiro: ouero nulla," and in the 1540 edition, f. 3, appears "Chiamata zero, ouero nulla." Woepcke asserted that it first appeared in Calandri (1491) in this sentence: "Sono dieci le figure con le quali ciascuno numero si puo significare: delle quali n'e una che si chiama zero: et per se sola nulla significa." (f. 4). [See _Propagation_, p. 522.]

[228] Boncompagni _Bulletino_, Vol. XVI, pp. 673-685.

[229] Leo Jordan, loc. cit. In the _Catalogue of MSS., Bibl. de l'Arsenal_, Vol. III, pp. 154-156, this work is No. 2904 (184 S.A.F.), Bibl. Nat., and is also called _Petit traicte de algorisme_.

[230] Texada (1546) says that there are "nueue letros yvn zero o cifra" (f. 3).

[231] Savonne (1563, 1751 ed., f. 1): "Vne ansi formee (o) qui s'appelle nulle, & entre marchans zero," showing the influence of Italian names on French mercantile customs. Trenchant (Lyons, 1566, 1578 ed., p. 12) also says: "La derniere qui s'apele nulle, ou zero;" but Champenois, his contemporary, writing in Paris in 1577 (although the work was not published until 1578), uses "cipher," the Italian influence showing itself less in this center of university culture than in the commercial atmosphere of Lyons.

[232] Thus Radulph of Laon (c. 1100): "Inscribitur in ultimo ordine et figura [symbol] sipos nomine, quae, licet numerum nullum signitet, tantum ad alia quaedam utilis, ut insequentibus declarabitur." ["Der Arithmetische Tractat des Radulph von Laon," _Abhandlungen zur Geschichte der Mathematik_, Vol. V, p. 97, from a manuscript of the thirteenth century.] Chasles (_Comptes rendus_, t. 16, 1843, pp. 1393, 1408) calls attention to the fact that Radulph did not know how to use the zero, and he doubts if the sipos was really identical with it. Radulph says: "... figuram, cui sipos nomen est [symbol] in motum rotulae formatam nullius numeri significatione inscribi solere praediximus," and thereafter uses _rotula_. He uses the sipos simply as a kind of marker on the abacus.

[233] Rabbi ben Ezra (1092-1168) used both [Hebrew: GLGL], _galgal_ (the Hebrew for _wheel_), and [Hebrew: SPR'], _sifra_. See M. Steinschneider, "Die Mathematik bei den Juden," in _Bibliotheca Mathematica_, 1893, p. 69, and Silberberg, _Das Buch der Zahl des R. Abraham ibn Esra_, Frankfurt a. M., 1895, p. 96, note 23; in this work the Hebrew letters are used for numerals with place value, having the zero.

[234] E.g., in the twelfth-century _Liber aligorismi_ (see Boncompagni's _Trattati_, II, p. 28). So Ramus (_Libri II_, 1569 ed., p. 1) says: "Circulus quae nota est ultima: nil per se significat." (See also the Schonerus ed. of Ramus, 1586, p. 1.)

[235] "Und wirt das ringlein o. die Ziffer genant die nichts bedeut." [Koebel's _Rechenbuch_, 1549 ed., f. 10, and other editions.]

[236] I.e. "circular figure," our word _notation_ having come from the medieval _nota_. Thus Tzwivel (1507, f. 2) says: "Nota autem circularis .o. per se sumpta nihil vsus habet. alijs tamen adiuncta earum significantiam et auget et ordinem permutat quantum quo ponit ordinem. vt adiuncta note binarij hoc modo 20 facit eam significare bis decem etc." Also (ibid., f. 4), "figura circularis," "circularis nota." Clichtoveus (1503 ed., f. XXXVII) calls it "nota aut circularis o," "circularis nota," and "figura circularis." Tonstall (1522, f. B_3) says of it: "Decimo uero nota ad formam [symbol] litterae circulari figura est: quam alij circulum, uulgus cyphram uocat," and later (f. C_4) speaks of the "circulos." Grammateus, in his _Algorismus de integris_ (Erfurt, 1523, f. A_2), speaking of the nine significant figures, remarks: "His autem superadditur decima figura circularis ut 0 existens que ratione sua nihil significat." Noviomagus (_De Numeris libri II_, Paris, 1539, chap. xvi, "De notis numerorum, quas zyphras vocant") calls it "circularis nota, quam ex his solam, alij sipheram, Georgius Valla zyphram."

[237] Huswirt, as above. Ramus (_Scholae mathematicae_, 1569 ed., p. 112) discusses the name interestingly, saying: "Circulum appellamus cum multis, quam alii thecam, alii figuram nihili, alii figuram privationis, seu figuram nullam vocant, alii ciphram, cum tamen hodie omnes hae notae vulgo ciphrae nominentur, & his notis numerare idem sit quod ciphrare." Tartaglia (1592 ed., f. 9) says: "si chiama da alcuni tecca, da alcuni circolo, da altri cifra, da altri zero, & da alcuni altri nulla."

[238] "Quare autem aliis nominibus vocetur, non dicit auctor, quia omnia alia nomina habent rationem suae lineationis sive figurationis. Quia rotunda est, dicitur haec figura teca ad similitudinem tecae. Teca enim est ferrum figurae rotundae, quod ignitum solet in quibusdam regionibus imprimi fronti vel maxillae furis seu latronum." [Loc. cit., p. 26.] But in Greek _theca_ ([THEKE], [Greek: theke]) is a place to put something, a receptacle. If a vacant column, e.g. in the abacus, was so called, the initial might have given the early forms [symbol] and [symbol] for the zero.

[239] Buteo, _Logistica_, Lyons, 1559. See also Wertheim in the _Bibliotheca Mathematica_, 1901, p. 214.

[240] "0 est appellee chiffre ou nulle ou figure de nulle valeur." [La Roche, _L'arithmetique_, Lyons, 1520.]

[241] "Decima autem figura nihil uocata," "figura nihili (quam etiam cifram uocant)." [Stifel, _Arithmetica integra_, 1544, f. 1.]

[242] "Zifra, & Nulla uel figura Nihili." [Scheubel, 1545, p. 1 of ch. 1.] _Nulla_ is also used by Italian writers. Thus Sfortunati (1545 ed., f. 4) says: "et la decima nulla & e chiamata questa decima zero;" Cataldi (1602, p. 1): "La prima, che e o, si chiama nulla, ouero zero, ouero niente." It also found its way into the Dutch arithmetics, e.g. Raets (1576, 1580 ed., f. A_3): "Nullo dat ist niet;" Van der Schuere (1600, 1624 ed., f. 7); Wilkens (1669 ed., p. 1). In Germany Johann Albert (Wittenberg, 1534) and Rudolff (1526) both adopted the Italian _nulla_ and popularized it. (See also Kuckuck, _Die Rechenkunst im sechzehnten Jahrhundert_, Berlin, 1874, p. 7; Guenther, _Geschichte_, p. 316.)

[243] "La dixieme s'appelle chifre vulgairement: les vns l'appellant zero: nous la pourrons appeller vn Rien." [Peletier, 1607 ed., p. 14.]

[244] It appears in the Polish arithmetic of K[=l]os (1538) as _cyfra_. "The Ciphra 0 augmenteth places, but of himselfe signifieth not," Digges, 1579, p. 1. Hodder (10th ed., 1672, p. 2) uses only this word (cypher or cipher), and the same is true of the first native American arithmetic, written by Isaac Greenwood (1729, p. 1). Petrus de Dacia derives _cyfra_ from circumference. "Vocatur etiam cyfra, quasi circumfacta vel circumferenda, quod idem est, quod circulus non habito respectu ad centrum." [Loc. cit., p. 26.]

[245] _Opera mathematica_, 1695, Oxford, Vol. I, chap. ix, _Mathesis universalis_, "De figuris numeralibus," pp. 46-49; Vol. II, _Algebra_, p. 10.

[246] Martin, _Origine de notre systeme de numeration ecrite_, note 149, p. 36 of reprint, spells [Greek: tsiphra] from Maximus Planudes, citing Wallis as an authority. This is an error, for Wallis gives the correct form as above.

Alexander von Humboldt, "Ueber die bei verschiedenen Voelkern ueblichen Systeme von Zahlzeichen und ueber den Ursprung des Stellenwerthes in den indischen Zahlen," Crelle's _Journal fuer reine und angewandte Mathematik_, Vol. IV, 1829, called attention to the work [Greek: arithmoi Indikoi] of the monk Neophytos, supposed to be of the fourteenth century. In this work the forms [Greek: tzuphra] and [Greek: tzumphra] appear. See also Boeckh, _De abaco Graecorum_, Berlin, 1841, and Tannery, "Le Scholie du moine Neophytos," _Revue Archeologique_, 1885, pp. 99-102. Jordan, loc. cit., gives from twelfth and thirteenth century manuscripts the forms _cifra_, _ciffre_, _chifras_, and _cifrus_. Du Cange, _Glossarium mediae et infimae Latinitatis_, Paris, 1842, gives also _chilerae_. Dasypodius, _Institutiones Mathematicae_, Strassburg, 1593-1596, adds the forms _zyphra_ and _syphra_. Boissiere, _L'art d'arythmetique contenant toute dimention, tres-singulier et commode, tant pour l'art militaire que autres calculations_, Paris, 1554: "Puis y en a vn autre dict zero lequel ne designe nulle quantite par soy, ains seulement les loges vuides."

[247] _Propagation_, pp. 27, 234, 442. Treutlein, "Das Rechnen im 16. Jahrhundert," _Abhandlungen zur Geschichte der Mathematik_, Vol. I, p. 5, favors the same view. It is combated by many writers, e.g. A. C. Burnell, loc. cit., p. 59. Long before Woepcke, I. F. and G. I. Weidler, _De characteribus numerorum vulgaribus et eorum aetatibus_, Wittenberg, 1727, asserted the possibility of their introduction into Greece by Pythagoras or one of his followers: "Potuerunt autem ex oriente, uel ex phoenicia, ad graecos traduci, uel Pythagorae, uel eius discipulorum auxilio, cum aliquis eo, proficiendi in literis causa, iter faceret, et hoc quoque inuentum addisceret."

[248] E.g., they adopted the Greek numerals in use in Damascus and Syria, and the Coptic in Egypt. Theophanes (758-818 A.D.), _Chronographia_, Scriptores Historiae Byzantinae, Vol. XXXIX, Bonnae, 1839, p. 575, relates that in 699 A.D. the caliph Wal[=i]d forbade the use of the Greek language in the bookkeeping of the treasury of the caliphate, but permitted the use of the Greek alphabetic numerals, since the Arabs had no convenient number notation: [Greek: kai ekoluse graphesthai Hellenisti tous demosious ton logothesion kodikas, all' Arabiois auta parasemainesthai, choris ton psephon, epeide adunaton tei ekeinon glossei monada e duada e triada e okto hemisu e tria graphesthai; dio kai heos semeron eisin sun autois notarioi Christianoi.] The importance of this contemporaneous document was pointed out by Martin, loc. cit. Karabacek, "Die Involutio im arabischen Schriftwesen," Vol. CXXXV of _Sitzungsberichte d. phil.-hist. Classe d. k. Akad. d. Wiss._, Vienna, 1896, p. 25, gives an Arabic date of 868 A.D. in Greek letters.

[249] _The Origin and History of Our Numerals_ (in Russian), Kiev, 1908; _The Independence of European Arithmetic_ (in Russian), Kiev.

[250] Woepcke, loc. cit., pp. 462, 262.

[251] Woepcke, loc. cit., p. 240. _[H.]is[=a]b-al-[.G]ob[=a]r_, by an anonymous author, probably Ab[=u] Sahl Dunash ibn Tamim, is given by Steinschneider, "Die Mathematik bei den Juden," _Bibliotheca Mathematica_, 1896, p. 26.

[252] Steinschneider in the _Abhandlungen_, Vol. III, p. 110.

[253] See his _Grammaire arabe_, Vol. I, Paris, 1810, plate VIII; Gerhardt, _Etudes_, pp. 9-11, and _Entstehung_ etc., p. 8; I. F. Weidler, _Spicilegium observationum ad historiam notarum numeralium pertinentium_, Wittenberg, 1755, speaks of the "figura cifrarum Saracenicarum" as being different from that of the "characterum Boethianorum," which are similar to the "vulgar" or common numerals; see also Humboldt, loc. cit.

[254] Gerhardt mentions it in his _Entstehung_ etc., p. 8; Woepcke, _Propagation_, states that these numerals were used not for calculation, but very much as we use Roman numerals. These superposed dots are found with both forms of numerals (_Propagation_, pp. 244-246).

[255] Gerhardt (_Etudes_, p. 9) from a manuscript in the Bibliotheque Nationale. The numeral forms are [symbols], 20 being indicated by [symbol with dot] and 200 by [symbol with 2 dots]. This scheme of zero dots was also adopted by the Byzantine Greeks, for a manuscript of Planudes in the Bibliotheque Nationale has numbers like [pi alpha with 4 dots] for 8,100,000,000. See Gerhardt, _Etudes_, p. 19. Pihan, _Expose_ etc., p. 208, gives two forms, Asiatic and Maghrebian, of "Ghob[=a]r" numerals.

[256] See Chap. IV.

[257] Possibly as early as the third century A.D., but probably of the eighth or ninth. See Cantor, I (3), p. 598.

[258] Ascribed by the Arabic writer to India.

[259] See Woepcke's description of a manuscript in the Chasles library, "Recherches sur l'histoire des sciences mathematiques chez les orientaux," _Journal Asiatique_, IV (5), 1859, p. 358, note.

[260] P. 56.

[261] Reinaud, _Memoire sur l'Inde_, p. 399. In the fourteenth century one Sih[=a]b al-D[=i]n wrote a work on which, a scholiast to the Bodleian manuscript remarks: "The science is called Algobar because the inventor had the habit of writing the figures on a tablet covered with sand." [Gerhardt, _Etudes, _p. 11, note.]

[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. Fluegel, Leipzig, Vol. I, 1871, and Vol. II, 1872. This work was published after Professor Fluegel'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 illustration 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 manuscript from which these are taken is the oldest (970 A.D.) Arabic document 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 ueber die Fortschritte der Mathematik_.

[274] This title was first applied to Roman emperors in posthumous coins of Julius Caesar. Subsequently the emperors assumed 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 Class. 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 institutions. While referred to by Baeda (672-735) and Hrabanus Maurus (c. 776-856), it was only after Gerbert's time that the _Boetii de institutione arithmetica libri duo_ was really a common work.

[285] Also spelled Cassiodorius.

[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 Cassiodorus in these statements in his _Decline and Fall of the Roman Empire_, chap. xxxix. 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 Cassiodore, de Symmaque, surpassent 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 Wuestenfeld, _Geschichte der arabischen Aerzte und Naturforscher_, Goettingen, 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_.