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 Āryabhata," Journal Asiatique, Vol. XVI (7), pp. 440-485. On the two Āryabhaṭas see Kaye, Bibl. Math., Vol. X (3), p. 289.

[150] Using kha, a synonym of śūnya. [Bayley, loc. cit., p. 22, and L. Rodet, Journal Asiatique, Vol. XVI (7), p. 443.]

[151] Varāha-Mihira, Pañcasiddhāntikā, translated by G. Thibaut and M. S. Dvivedī, Benares, 1889; see Bühler, loc. cit., p. 78; Bayley, loc. cit., p. 23.

[152] Bṛhat Saṃhitā, 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 Bṛhat Saṃhitā. The system was also used in the Pañcasiddhāntikā 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 Rājputāna 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] Bakhṣālī 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 Bakshālī 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 Lāṭadeśa," 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] Śāradā 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 Bakshālī 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 śūnyabindu '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 ο for οὐδὲν, 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 ο, without the notion of place value, by the Greeks.

[201] Al-Battānī 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] τζίφρα, 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 צפר, couronne ou diadème) ou ciphra (de ספר, 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, τζίφρα 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, circulus 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 גלגל, galgal (the Hebrew for wheel), and ספרא, 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, θήκη) 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 Klos (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 τσίφρα 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 ἀριθμοὶ Ἰνδικοί of the monk Neophytos, supposed to be of the fourteenth century. In this work the forms τζύφρα and τζύμφρα 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 Walīd 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: καὶ ἐκώλυσε γράφεσθαι Ἑλληνιστὶ τοὺς δημοσίους τῶν λογοθεσίων κώδικας, ἀλλ' Ἀραβίοις αὐτὰ παρασημαίνεσθαι, χωρὶς τῶν ψήφων, ἐπειδὴ ἀδύνατον τῇ ἐκείνων γλώσσῃ μονάδα ἢ δυάδα ἢ τριάδα ἢ ὀκτὼ ἥμισυ ἢ τρία γράφεσθαι· διὸ καὶ ἕως σήμερόν εἰσιν σὺν αὐτοῖς νοτάριοι Χριστιανοί. 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. Ḥisāb-al-Ġobār, by an anonymous author, probably Abū 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 "Ghobār" 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 Sihāb al-Dīn 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 Abū Zakarījā el-Ḥaṣṣār," Bibliotheca Mathematica, Vol. II (3), p. 15.

[264] A. Devoulx, "Les chiffres arabes," Revue Africaine, Vol. XVI, pp. 455-458.

[265] Kitāb 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-Ḥaṣṣār's forms, Suter, Bibliotheca Mathematica, Vol. II (3), p. 15.

[269] Woepcke, Sur une donnée historique, etc., loc. cit. The name ġobār 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 Gobārī et Hawāī," Atti dell' accademia pontificia dei nuovi Lincei, Vol. XIX. The adjective applied to the forms in 5 is gobārī 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: