Chapter II

, p. 16.

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

[146] "Zur Geschichte des indischen Ziffernsystems," _Zeitschrift für 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:

[Illustration]

the same letter (_ka_) appearing in the successive consonant forms, _ka_, _kha_, _ga_, _gha_, etc. See C. I. Gerhardt, _Über die Entstehung und Ausbreitung des dekadischen Zahlensystems_, Programm, p. 17, Salzwedel, 1853, and _Études historiques sur l'arithmétique de position_, Programm, p. 24, Berlin, 1856; E. Jacquet, _Mode d'expression symbolique des nombres_, loc. cit., p. 97; L. Rodet, "Sur la véritable signification de la notation numérique inventée 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, _Pañcasiddh[=a]ntik[=a]_, translated by G. Thibaut and M. S. Dvived[=i], Benares, 1889; see Bühler, 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 Bühler 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 _Pañcasiddh[=a]ntik[=a]_ as early as 505 A.D. [Bühler, _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] Bühler, 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 Graecè & Hebraicè eximè 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.] Eneström 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; Bühler, 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 Bühler, 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 Bühler.

[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 Bühler. 7, 3, 5, from "Torkhede Copper-Plate Grant of the Time of Govindaraja of Gujerat," Fleet, _Epigraphia Indica_, Vol. III, pp. 53-58. See Bühler.

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

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

[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 Tübingen_, Bloomfield and Garbe, Baltimore, 1901. Somewhat similar forms are given under "Numération Cachemirienne," by Pihan, _Exposé_ 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, Münster 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 _Archæologia_, 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 Bháscara_, 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." [Bühler, _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 für Mathematik und Physik_, Vol. XLV, _Hist.-lit. Abt._, p. 47. The manuscript is No. C. 80, in the Dresden library.

[198] J. G. Prändel, _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, _Aperçu historique sur l'origine et le développement des méthodes en géométrie_, 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 numéraux_ etc., reprint p. 30, and J. Brandis, _Das Münz-, 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 über die Verbreitung der indisch-arabischen Rechenkunst und Zahlzeichen im christl. Abendlande," _Zeitschrift für 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, _Études historiques sur l'arithmétique de position_, Berlin, 1856, p. 12; J. Bowring, _The Decimal System in Numbers, Coins, & Accounts_, London, 1854, p. 33.

[208] Karabacek, _Wiener Zeitschrift für die Kunde des Morgenlandes_, Vol. XI, p. 13; _Führer 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 Hebræam originem sapit refértque: & ut docti arbitrantur, à verbo saphar, quod Ordine numerauit significat. Unde Sephar numerus est: hinc Siphra (vulgo corruptius). Etsi verò gens Iudaica his notis, quæ hodie Siphræ 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)?", _Études de philologie néo-grecque_, Paris, 1892.

[212] Bühler, 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 nommé par les uns, _sipos, rota, galgal_ ...; par les autres _tsiphra_ (de [Hebrew: TSPR], _couronne_ ou _diadème_) ou _ciphra_ (de [Hebrew: SPR], _numération_)." Ch. de Paravey, _Essai sur l'origine unique et hiéroglyphique des chiffres et des lettres de tous les peuples_, Paris, 1826, p. 165, a rather fanciful work, gives "vase, vase arrondi et fermé par un couvercle, qui est le symbole de la 10^e Heure, [symbol]," among the Chinese; also "Tsiphron Zéron, ou tout à fait vide en arabe, [Greek: tziphra] en grec ... d'où chiffre (qui dérive plutôt, suivant nous, de l'Hébreu _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 Florentiæ adservantur_. See Fazzari, loc. cit., p. 5.

[217] "Et doveto sapere chel zeuero per se solo non significa nulla ma è 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 François, "più 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 sè 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 für Kulturgeschichte_, Berlin, 1905, pp. 155-195, gives the following two schemes of derivation, (1) "zefiro, zeviro, zeiro, zero," (2) "zefiro, zefro, zevro, zero."

[221] Köbel (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 Eneström, _Öfversigt af Kongl. Vetenskaps-Akademiens Förhandlingar_, 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 "çefiro: 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 può significare: delle quali n'è 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 traicté 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 quæ 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." [Köbel'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] litteræ 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, cùm tamen hodie omnes hæ notæ vulgò ciphræ 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: thêkê]) 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'arithmétique_, 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 è 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; Günther, _Geschichte_, p. 316.)

[243] "La dixième 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 système de numération écrite_, 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, "Über die bei verschiedenen Völkern üblichen Systeme von Zahlzeichen und über den Ursprung des Stellenwerthes in den indischen Zahlen," Crelle's _Journal für 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 Néophytos," _Revue Archéologique_, 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_. Boissière, _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 quantité 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 ekôluse graphesthai Hellênisti tous dêmosious tôn logothesiôn kôdikas, all' Arabiois auta parasêmainesthai, chôris tôn psêphôn, epeidê adunaton têi ekeinôn glôssêi monada ê duada ê triada ê oktô hêmisu ê tria graphesthai; dio kai heôs sêmeron 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, _Études_, 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 (_Études_, p. 9) from a manuscript in the Bibliothèque 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 Bibliothèque Nationale has numbers like [pi alpha with 4 dots] for 8,100,000,000. See Gerhardt, _Études_, p. 19. Pihan, _Exposé_ 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 mathématiques chez les orientaux," _Journal Asiatique_, IV (5), 1859, p. 358, note.

[260] P. 56.

[261] Reinaud, _Mémoire 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, _Études, _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. Flügel, Leipzig, Vol. I, 1871, and Vol. II, 1872. This work was published after Professor Flügel'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 mathématiques 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 donnée 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 über die Fortschritte der Mathematik_.

[274] This title was first applied to Roman emperors in posthumous coins of Julius Cæsar. 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 Boëtius. 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 _Encyclopædia 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 Bæda (672-735) and Hrabanus Maurus (c. 776-856), it was only after Gerbert's time that the _Boëtii 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 numéraux_ etc., reprint, p. 4.]

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

[288] J. C. Scaliger in his _Poëtice_ 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 Théodoric, les lettres reprendre une nouvelle vie en Italie, les écoles florissantes et les savans honorés. Et certes les ouvrages de Boëce, de Cassiodore, de Symmaque, surpassent de beaucoup toutes les productions du siècle précédent." [_Histoire des mathématiques_, 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 Wüstenfeld, _Geschichte der arabischen Aerzte und Naturforscher_, Göttingen, 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, _Études_ 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 Irán_, London, 1902, p. 167. Sykes was the first European to follow the course of Alexander's army across eastern Persia.

[295] Bühler, _Indian Br[=a]hma Alphabet_, note, p. 27; _Palaeographie_, p. 2; _Herodoti Halicarnassei 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 Classical 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] Bühler, _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, Strassburg, 1898, pp. 3-7.

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

[304] J. Beloch, _Griechische Geschichte_, Vol. III, Strassburg, 1904, pp. 30-31.

[305] E.g., the denarius, the words for hour and minute ([Greek: hôra, 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: rhaphê]), 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, cum 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 Tanaïs." [_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 classe maris.

"The divine Cæsar 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 title 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_, Nürnberg, 1801. Even as late as 1827 Romagnosi (in his supplement to _Ricerche storiche sull' India_ etc., by Robertson, Vol. II, p. 580, 1827) asserted 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 passage 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 institutione arithmetica libri duo, de institutione musica libri quinque. Accedit geometria quae fertur Boetii_.... Leipzig.... MDCCCLXVII.

[336] See also P. Tannery, "Notes sur la pseudo-géometrie de Boèce," 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 manuscript 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) asserts.

[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 über die Fortschritte der Mathematik_ for 1907, asserts 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 römischen 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 Hände 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. klass. 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 Archéologique_ for 1856-1857, and in his treatise _Les signes numéraux_ etc. See also M. Chasles, "De la connaissance qu'ont eu les anciens d'une numération décimale écrite qui fait usage de neuf chiffres prenant les valeurs de position," _Comptes rendus_, Vol. VI, pp. 678-680; "Sur l'origine de notre système de numération," _Comptes rendus_, Vol. VIII, pp. 72-81; and note "Sur le passage du premier livre de la géométrie de Boèce, relatif à un nouveau système de numération," in his work _Aperçu historique sur l'origine et le devéloppement des méthodes en géométrie_, 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 römischen 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 Einführung der jetzigen Ziffern in Europa durch Gerbert_, Berlin, 1892, p. 11; Bayley, loc. cit., p. 59; Gerhardt, _Études_, 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 privée des Romains_, Vol. II (French trans.), p. 505, Paris, 1893.

[341] In a Plimpton manuscript 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 manuscript 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 passage is not in a manuscript 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'arithmétique indienne en Occident_.

[343] They are found in none of the very ancient manuscripts, 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 passage 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 illustration.

[348] "Habebant enim diverse formatos apices vel caracteres." See the reference to Gerbert on p. 117.

[349] C. Henry, "Sur l'origine de quelques notations mathématiques," _Revue Archéologique_, 1879, derives these from the initial letters used as abbreviations for the names of the numerals, a theory that finds few supporters.

[350] E.g., it appears in Schonerus, _Algorithmus Demonstratus_, Nürnberg, 1534, f. A4. In England it appeared in the earliest English arithmetical manuscript known, _The Crafte of Nombrynge_: "¶ fforthermore ye most vndirstonde that in this craft ben vsid teen figurys, as here bene writen for ensampul, [Numerals] ... in the quych we vse teen figurys of Inde. Questio. ¶ why ten fyguris of Inde? Solucio. for as I have sayd afore thei were fonde fyrst in Inde of a kynge of that Cuntre, that was called Algor." See Smith, _An Early English Algorism_, loc. cit.

[351] Friedlein ed., p. 397.

[352] Carlsruhe codex of Gerlando.

[353] Munich codex of Gerlando.

[354] Carlsruhe codex of Bernelinus.

[355] Munich codex of Bernelinus.

[356] Turchill, c. 1200.

[357] Anon. MS., thirteenth century, Alexandrian Library, Rome.

[358] Twelfth-century Boethius, Friedlein, p. 396.

[359] Vatican codex, tenth century, Boethius.

[360] a, h, i, are from the Friedlein ed.; the original in the manuscript from which a is taken contains a zero symbol, as do all of the six plates given by Friedlein. b-e from the Boncompagni _Bulletino_, Vol. X, p. 596; f ibid., Vol. XV, p. 186; g _Memorie della classe di sci., Reale Acc. dei Lincei_, An. CCLXXIV (1876-1877), April, 1877. A twelfth-century arithmetician, possibly John of Luna (Hispalensis, of Seville, c. 1150), speaks of the great diversity of these forms even in his day, saying: "Est autem in aliquibus figuram istarum apud multos diuersitas. Quidam enim septimam hanc figuram representant [Symbol] alii autem sic [Symbol], uel sic [Symbol]. Quidam vero quartam sic [Symbol]." [Boncompagni, _Trattati_, Vol. II, p. 28.]

[361] Loc. cit., p. 59.

[362] Ibid., p. 101.

[363] Loc. cit., p. 396.

[364] Khosr[=u] I, who began to reign in 531 A.D. See W. S. W Vaux, _Persia, _London, 1875, p. 169; Th. Nöldeke, _Aufsätze zur persichen Geschichte_, Leipzig, 1887, p. 113, and his article in the ninth edition of the _Encyclopædia Britannica_.

[365] Colebrooke, _Essays_, Vol. II, p. 504, on the authority of Ibn al-Adam[=i], astronomer, in a work published by his continuator Al-Q[=a]sim in 920 A.D.; Al-B[=i]r[=u]n[=i], _India, _Vol. II, p. 15.

[366] H. Suter, _Die Mathematiker_ etc., pp. 4-5, states that Al-Faz[=a]r[=i] died between 796 and 806.

[367] Suter, loc. cit., p. 63.

[368] Suter, loc. cit., p. 74.

[369] Suter, _Das Mathematiker-Verzeichniss im Fihrist_. The references to Suter, unless otherwise stated, are to his later work _Die Mathematiker und Astronomen der Araber_ etc.

[370] Suter, _Fihrist_, p. 37, no date.

[371] Suter, _Fihrist_, p. 38, no date.

[372] Possibly late tenth, since he refers to one arithmetical work which is entitled _Book of the Cyphers_ in his _Chronology_, English ed., p. 132. Suter, _Die Mathematiker_ etc., pp. 98-100, does not mention this work; see the _Nachträge und Berichtigungen_, pp. 170-172.

[373] Suter, pp. 96-97.

[374] Suter, p. 111.

[375] Suter, p. 124. As the name shows, he came from the West.

[376] Suter, p. 138.

[377] Hankel, _Zur Geschichte der Mathematik_, p. 256, refers to him as writing on the Hindu art of reckoning; Suter, p. 162.

[378] [Greek: Psêphophoria kat' Indous], Greek ed., C. I. Gerhardt, Halle, 1865; and German translation, _Das Rechenbuch des Maximus Planudes_, H. Wäschke, Halle, 1878.

[379] "Sur une donnée historique relative à l'emploi des chiffres indiens par les Arabes," Tortolini's _Annali di scienze mat. e fis._, 1855.

[380] Suter, p. 80.

[381] Suter, p. 68.

[382] Sprenger also calls attention to this fact, in the _Zeitschrift d. deutschen morgenländ. Gesellschaft_, Vol. XLV, p. 367.

[383] Libri, _Histoire des mathématiques_, Vol. I, p. 147.

[384] "Dictant la paix à l'empereur de Constantinople, l'Arabe victorieux demandait des manuscrits et des savans." [Libri, loc. cit., p. 108.]

[385] Persian _bagadata_, "God-given."

[386] One of the Abbassides, the (at least pretended) descendants of `Al-Abb[=a]s, uncle and adviser of Mo[h.]ammed.

[387] E. Reclus, _Asia_, American ed., N. Y., 1891, Vol. IV, p. 227.

[388] _Historical Sketches_, Vol. III, chap. iii.

[389] On its prominence at that period see Villicus, p. 70.

[390] See pp. 4-5.

[391] Smith, D. E., in the _Cantor Festschrift_, 1909, note pp. 10-11. See also F. Woepcke, _Propagation_.

[392] Eneström, in _Bibliotheca Mathematica_, Vol. I (3), p. 499; Cantor, _Geschichte_, Vol. I (3), p. 671.

[393] Cited in