Tannery, P., 62, 84, 85 Tartaglia, 4, 61 Taylor, I., 19, 30 Teca, 55, 61 Tennent, J. E., 75 Texada, 60 Theca, 58, 61 Theophanes, 64 Thibaut, G., 12, 13, 16, 44, 47 Tibetan numerals, 36 Timotheus, 103 Tonstall, C., 3, 61 Trenchant, 60 Treutlein, 5, 63, 123 Trevisa, 136 Treviso arithmetic, 145 Trivium and quadrivium, 73 Tsin, 56 Tunis, 65 Turchill, 88, 118, 123 Turnour, G., 75 Tziphra, 57, 62 [Greek: tziphra], 55, 57, 62 Tzwivel, 61, 118, 145

Ujjain, 32 Unger, 133 Upanishads, 12 Usk, 121

Valla, G., 61 Van der Schuere, 62 Varāha-Mihira, 39, 44, 78 Vāsavadattā, 44 Vaux, Carra de, 9, 74 Vaux, W. S. W., 91 Vedāṅgas, 17 Vedas, 12, 15, 17 Vergil, 80 Vincent, A. J. H., 57 Vogt, 13 Voizot, P., 36 Vossius, 4, 76, 81, 84

Wallis, 3, 62, 84, 116 Wappler, E., 54, 126 Waeschke, H., 2, 93 Wattenbach, 143 Weber, A., 31 Weidler, I. F., 34, 66 Weidler, I. F. and G. I., 63, 66 Weissenborn, 85, 110 Wertheim, G., 57, 61 Whitney, W. D., 13 Wilford, F., 75 Wilkens, 62 Wilkinson, J. G., 70 Willichius, 3 Woepcke, 3, 6, 42, 63, 64, 65, 67, 69, 70, 94, 113, 138 Wolack, G., 54 Woodruff, C. E., 32 Word and letter numerals, 38, 44 Wuestenfeld, 74

Yule, H., 107

Zephirum, 57, 58 Zephyr, 59 Zepiro, 58 Zero, 26, 38, 40, 43, 45, 49, 51-62, 67 Zeuero, 58

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Notes

al-Mekkī on a treatise on ġobār arithmetic (explained later) called Al-murshidah, found by Woepcke in Paris (Propagation, p. 66), there is mentioned the fact that there are "nine Indian figures" and "a second kind of Indian figures ... although these are the figures of the ġobār writing." So in a commentary by Ḥosein ibn Moḥammed al-Maḥallī (died in 1756) on the Mokhtaṣar fī'ilm el-ḥisāb (Extract from Arithmetic) by 'Abdalqādir ibn 'Alī al-Sakhāwī (died c. 1000) it is related that "the preface treats of the forms of the figures of Hindu signs, such as were established by the Hindu nation." [Woepcke, Propagation, p. 63.]]

which, of course, are interpolations. An interesting example of a forgery in ecclesiastical matters is in the charter said to have been given by St. Patrick, granting indulgences to the benefactors of Glastonbury, dated "In nomine domini nostri Jhesu Christi Ego Patricius humilis servunculus Dei anno incarnationis ejusdem ccccxxx." Now if the Benedictines are right in saying that Dionysius Exiguus, a Scythian monk, first arranged the Christian chronology c. 532 A.D., this can hardly be other than spurious. See Arbuthnot, loc. cit., p. 38.

[1] "Discipulus. Quis primus invenit numerum apud Hebraeos et AEgyptios? Magister. Abraham primus invenit numerum apud Hebraeos, deinde Moses; et Abraham tradidit istam scientiam numeri ad AEgyptios, et docuit eos: deinde Josephus." [Bede, De computo dialogus (doubtfully assigned to him), Opera omnia, Paris, 1862, Vol. I, p. 650.]

"Alii referunt ad Phoenices inventores arithmeticae, propter eandem commerciorum caussam: Alii ad Indos: Ioannes de Sacrobosco, cujus sepulchrum est Lutetiae in comitio Maturinensi, refert ad Arabes." [Ramus, Arithmeticae libri dvo, Basel, 1569, p. 112.]

Similar notes are given by Peletarius in his commentary on the arithmetic of Gemma Frisius (1563 ed., fol. 77), and in his own work (1570 Lyons ed., p. 14): "La valeur des Figures commence au coste dextre tirant vers le coste senestre: au rebours de notre maniere d'escrire par ce que la premiere prattique est venue des Chaldees: ou des Pheniciens, qui ont ete les premiers traffiquers de marchandise."

[2] Maximus Planudes (c. 1330) states that "the nine symbols come from the Indians." [Waeschke's German translation, Halle, 1878, p. 3.] Willichius speaks of the "Zyphrae Indicae," in his Arithmeticae libri tres (Strasburg, 1540, p. 93), and Cataneo of "le noue figure de gli Indi," in his Le pratiche delle dve prime mathematiche (Venice, 1546, fol. 1). Woepcke is not correct, therefore, in saying ("Memoire sur la propagation des chiffres indiens," hereafter referred to as Propagation [Journal Asiatique, Vol. I (6), 1863, p. 34]) that Wallis (A Treatise on Algebra, both historical and practical, London, 1685, p. 13, and De algebra tractatus, Latin edition in his Opera omnia, 1693, Vol. II, p. 10) was one of the first to give the Hindu origin.

[3] From the 1558 edition of The Grovnd of Artes, fol. C, 5. Similarly Bishop Tonstall writes: "Qui a Chaldeis primum in finitimos, deinde in omnes pene gentes fluxit.... Numerandi artem a Chaldeis esse profectam: qui dum scribunt, a dextra incipiunt, et in leuam progrediuntur." [De arte supputandi, London, 1522, fol. B, 3.] Gemma Frisius, the great continental rival of Recorde, had the same idea: "Primum autem appellamus dexterum locum, eo quod haec ars vel a Chaldaeis, vel ab Hebraeis ortum habere credatur, qui etiam eo ordine scribunt"; but this refers more evidently to the Arabic numerals. [Arithmeticae practicae methodvs facilis, Antwerp, 1540, fol. 4 of the 1563 ed.] Sacrobosco (c. 1225) mentions the same thing. Even the modern Jewish writers claim that one of their scholars, Māshāllāh (c. 800), introduced them to the Mohammedan world. [C. Levias, The Jewish Encyclopedia, New York, 1905, Vol. IX, p. 348.]

[4] "... & que esto fu trouato di fare da gli Arabi con diece figure." [La prima parte del general trattato di nvmeri, et misvre, Venice, 1556, fol. 9 of the 1592 edition.]

[5] "Vom welchen Arabischen auch disz Kunst entsprungen ist." [Ain nerv geordnet Rechenbiechlin, Augsburg, 1514, fol. 13 of the 1531 edition. The printer used the letters rv for w in "new" in the first edition, as he had no w of the proper font.]

[6] Among them Glareanus: "Characteres simplices sunt nouem significatiui, ab Indis usque, siue Chaldaeis asciti .1.2.3.4.5.6.7.8.9. Est item unus .0 circulus, qui nihil significat." [De VI. Arithmeticae practicae speciebvs, Paris, 1539, fol. 9 of the 1543 edition.]

[7] "Barbarische oder gemeine Ziffern." [Anonymous, Das Einmahl Eins cum notis variorum, Dresden, 1703, p. 3.] So Vossius (De universae matheseos natura et constitutione liber, Amsterdam, 1650, p. 34) calls them "Barbaras numeri notas." The word at that time was possibly synonymous with Arabic.

[8] His full name was 'Abū 'Abdallāh Moḥammed ibn Mūsā al-Khowārazmī. He was born in Khowārezm, "the lowlands," the country about the present Khiva and bordering on the Oxus, and lived at Bagdad under the caliph al-Māmūn. He died probably between 220 and 230 of the Mohammedan era, that is, between 835 and 845 A.D., although some put the date as early as 812. The best account of this great scholar may be found in an article by C. Nallino, "Al-Huwārizmī" in the Atti della R. Accad. dei Lincei, Rome, 1896. See also Verhandlungen des 5. Congresses der Orientalisten, Berlin, 1882, Vol. II, p. 19; W. Spitta-Bey in the Zeitschrift der deutschen Morgenlaend. Gesellschaft, Vol. XXXIII, p. 224; Steinschneider in the Zeitschrift der deutschen Morgenlaend. Gesellschaft, Vol. L, p. 214; Treutlein in the Abhandlungen zur Geschichte der Mathematik, Vol. I, p. 5; Suter, "Die Mathematiker und Astronomen der Araber und ihre Werke," Abhandlungen zur Geschichte der Mathematik, Vol. X, Leipzig, 1900, p. 10, and "Nachtraege," in Vol. XIV, p. 158; Cantor, Geschichte der Mathematik, Vol. I, 3d ed., pp. 712-733 etc.; F. Woepcke in Propagation, p. 489. So recently has he become known that Heilbronner, writing in 1742, merely mentions him as "Ben-Musa, inter Arabes celebris Geometra, scripsit de figuris planis & sphericis." [Historia matheseos universae, Leipzig, 1742, p. 438.]

In this work most of the Arabic names will be transliterated substantially as laid down by Suter in his work Die Mathematiker etc., except where this violates English pronunciation. The scheme of pronunciation of oriental names is set forth in the preface.

[9] Our word algebra is from the title of one of his works, Al-jabr wa'l-muqābalah, Completion and Comparison. The work was translated into English by F. Rosen, London, 1831, and treated in L'Algebre d'al-Khārizmi et les methodes indienne et grecque, Leon Rodet, Paris, 1878, extract from the Journal Asiatique. For the derivation of the word algebra, see Cossali, Scritti Inediti, pp. 381-383, Rome, 1857; Leonardo's Liber Abbaci (1202), p. 410, Rome, 1857; both published by B. Boncompagni. "Almuchabala" also was used as a name for algebra.

[10] This learned scholar, teacher of O'Creat who wrote the Helceph ("Prologus N. Ocreati in Helceph ad Adelardum Batensem magistrum suum"), studied in Toledo, learned Arabic, traveled as far east as Egypt, and brought from the Levant numerous manuscripts for study and translation. See Henry in the Abhandlungen zur Geschichte der Mathematik, Vol. III, p. 131; Woepcke in Propagation, p. 518.

[11] The title is Algoritmi de numero Indorum. That he did not make this translation is asserted by Enestroem in the Bibliotheca Mathematica, Vol. I (3), p. 520.

[12] Thus he speaks "de numero indorum per .IX. literas," and proceeds: "Dixit algoritmi: Cum uidissem yndos constituisse .IX. literas in uniuerso numero suo, propter dispositionem suam quam posuerunt, uolui patefacere de opera quod fit per eas aliquid quod esset leuius discentibus, si deus uoluerit." [Boncompagni, Trattati d'Aritmetica, Rome, 1857.] Discussed by F. Woepcke, Sur l'introduction de l'arithmetique indienne en Occident, Rome, 1859.

[13] Thus in a commentary by 'Alī ibn Abī Bekr ibn al-Jamāl al-Anṣār[=i

[14] See also Woepcke, Propagation, p. 505. The origin is discussed at much length by G. R. Kaye, "Notes on Indian Mathematics.—Arithmetical Notation," Journ. and Proc. of the Asiatic Soc. of Bengal, Vol. III, 1907, p. 489.

[15] Alberuni's India, Arabic version, London, 1887; English translation, ibid., 1888.

[16] Chronology of Ancient Nations, London, 1879. Arabic and English versions, by C. E. Sachau.

[17] India, Vol. I, chap. xvi.

[18] The Hindu name for the symbols of the decimal place system.

[19] Sachau's English edition of the Chronology, p. 64.

[20] Litterature arabe, Cl. Huart, Paris, 1902.

[21] Huart, History of Arabic Literature, English ed., New York, 1903, p. 182 seq.

[22] Al-Mas'ūdī's Meadows of Gold, translated in part by Aloys Sprenger, London, 1841; Les prairies d'or, trad. par C. Barbier de Meynard et Pavet de Courteille, Vols. I to IX, Paris, 1861-1877.

[23] Les prairies d'or, Vol. VIII, p. 289 seq.

[24] Essays, Vol. II, p. 428.

[25] Loc. cit., p. 504.

[26] Materiaux pour servir a l'histoire comparee des sciences mathematiques chez les Grecs et les Orientaux, 2 vols., Paris, 1845-1849, pp. 438-439.

[27] He made an exception, however, in favor of the numerals, loc. cit., Vol. II, p. 503.

[28] Bibliotheca Arabico-Hispana Escurialensis, Madrid, 1760-1770, pp. 426-427.

[29] The author, Ibn al-Qifṭī, flourished A.D. 1198 [Colebrooke, loc. cit., note Vol. II, p. 510].

[30] "Liber Artis Logisticae a Mohamado Ben Musa Alkhuarezmita exornatus, qui ceteros omnes brevitate methodi ac facilitate praestat, Indorum que in praeclarissimis inventis ingenium & acumen ostendit." [Casiri, loc. cit., p. 427.]

[31] Macoudi, Le livre de l'avertissement et de la revision. Translation by B. Carra de Vaux, Paris, 1896.

[32] Verifying the hypothesis of Woepcke, Propagation, that the Sindhind included a treatment of arithmetic.

[33] Aḥmed ibn 'Abdallāh, Suter, Die Mathematiker, etc., p. 12.

[34] India, Vol. II, p. 15.

[35] See H. Suter, "Das Mathematiker-Verzeichniss im Fihrist," Abhandlungen zur Geschichte der Mathematik, Vol. VI, Leipzig, 1892. For further references to early Arabic writers the reader is referred to H. Suter, Die Mathematiker und Astronomen der Araber und ihre Werke. Also "Nachtraege und Berichtigungen" to the same (Abhandlungen, Vol. XIV, 1902, pp. 155-186).

[36] Suter, loc. cit., note 165, pp. 62-63.

[37] "Send Ben Ali,... tum arithmetica scripta maxime celebrata, quae publici juris fecit." [Loc. cit., p. 440.]

[38] Scritti di Leonardo Pisano, Vol. I, Liber Abbaci (1857); Vol. II, Scritti (1862); published by Baldassarre Boncompagni, Rome. Also Tre Scritti Inediti, and Intorno ad Opere di Leonardo Pisano, Rome, 1854.

[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 Āpastamba-Śulba-Sūtra," 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 Sūtras 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 Bharatavarṣa or India, London, 1893, p. 1.

[53] A. Hillebrandt, Alt-Indien, Breslau, 1899, p. 111. Fragmentary records relate that Khāravela, king of Kaliṅga, learned as a boy lekhā (writing), gaṇanā (reckoning), and rūpa (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 Mahābhārata," 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 Brāhma Alphabet, is also of value.

[69] The earliest work on the subject was by James Prinsep, "On the Inscriptions of Piyadasi or Aśoka," etc., Journal of the Asiatic Society of Bengal, 1838, following a preliminary suggestion in the same journal in 1837. See also "Aśoka 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 Aśoka," 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 Bhagavānlāl Indrājī, "On Ancient Nāgāri Numeration; from an Inscription at Nāneghāt," 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] Kharoṣṭhī numerals, Aśoka inscriptions, c. 250 B.C. Senart, Notes d'epigraphie indienne. Given by Buehler, loc. cit., Tafel I.

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

[82] Brāhmī numerals, Aśoka inscriptions, c. 250 B.C. Indian Antiquary, Vol. VI, p. 155 seq.

[83] Same, Nānā Ghāt inscriptions, c. 150 B.C. Bhagavānlāl Indrājī, On Ancient Nāgarī 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] Kṣatrapa coins, c. 200 A.D. Journal of the Royal Asiatic Society, 1890, p. 639.

[86] Kuṣana 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] Valhabī, 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 Kārītalāī, 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 Majhgawāin, Central India. [Fleet, loc. cit., Plate XIV.]

[95] From an inscription of 588 A.D., found at Bōdh-Gayā, Bengal Presidency. [Fleet, loc. cit., Plate XXIV.]

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

[97] From a Bijayagaḍh 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] Kārītalāī 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 Brāhmī 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 Brāhma 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 ṣunya (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-Bīrūnī 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-Bīrūnī 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 aṅka.

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

[147] It is found in the Bakhṣālī 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.—Āryabhaṭa," 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 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, Pancasiddhāntikā, translated by G. Thibaut and M. S. Dvivedī, Benares, 1889; see Buehler, 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 Buehler 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 Pancasiddhāntikā 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 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.] 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] 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; 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 Lāṭadeśa," 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] Śāradā 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 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 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 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." [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-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 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. A4) speaks of the numerals in general as "die der gemain man Zyfer nendt." Recorde (Grounde of Artes, 1558 ed., f. B6) 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 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 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 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: [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. 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, 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 "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 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 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, Etudes, 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. 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-Ḥaṣṣār's forms, Suter, Bibliotheca Mathematica, Vol. II (3), p. 15.

[269] Woepcke, Sur une donnee 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 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 Sīnā, as pointed out by Wuestenfeld, Geschichte der arabischen Aerzte und Naturforscher, Goettingen, 1840. His full name is Abū 'Alī al-Ḥosein ibn Sīnā. 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.

[294] P. M. Sykes, Ten Thousand Miles in Persia, or Eight Years in Iran, London, 1902, p. 167. Sykes was the first European to follow the course of Alexander's army across eastern Persia.

[295] Buehler, Indian Brāhma 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, Āryabhaṭa, was born in 476 A.D.

[302] Buehler, 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: hora, 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: rhaphe]), 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 Tanais." [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 Caesar 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, Nuernberg, 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 Nuśīrwān, 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-geometrie de Boece," 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 ueber 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 roemischen 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 Haende 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 Archeologique for 1856-1857, and in his treatise Les signes numeraux etc. See also M. Chasles, "De la connaissance qu'ont eu les anciens d'une numeration decimale ecrite qui fait usage de neuf chiffres prenant les valeurs de position," Comptes rendus, Vol. VI, pp. 678-680; "Sur l'origine de notre systeme de numeration," Comptes rendus, Vol. VIII, pp. 72-81; and note "Sur le passage du premier livre de la geometrie de Boece, relatif a un nouveau systeme de numeration," in his work Apercu historique sur l'origine et le developpement des methodes en geometrie, 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 roemischen 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 Einfuehrung der jetzigen Ziffern in Europa durch Gerbert, Berlin, 1892, p. 11; Bayley, loc. cit., p. 59; Gerhardt, Etudes, 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 privee 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'arithmetique 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 mathematiques," Revue Archeologique, 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, Nuernberg, 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] Khosrū I, who began to reign in 531 A.D. See W. S. W Vaux, Persia, London, 1875, p. 169; Th. Noeldeke, Aufsaetze zur persichen Geschichte, Leipzig, 1887, p. 113, and his article in the ninth edition of the Encyclopaedia Britannica.

[365] Colebrooke, Essays, Vol. II, p. 504, on the authority of Ibn al-Adamī, astronomer, in a work published by his continuator Al-Qāsim in 920 A.D.; Al-Bīrūnī, India, Vol. II, p. 15.

[366] H. Suter, Die Mathematiker etc., pp. 4-5, states that Al-Fazārī 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 Nachtraege 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: Psephophoria kat' Indous], Greek ed., C. I. Gerhardt, Halle, 1865; and German translation, Das Rechenbuch des Maximus Planudes, H. Waeschke, Halle, 1878.

[379] "Sur une donnee historique relative a 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 morgenlaend. Gesellschaft, Vol. XLV, p. 367.

[383] Libri, Histoire des mathematiques, Vol. I, p. 147.

[384] "Dictant la paix a 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-Abbās, uncle and adviser of Moḥammed.

[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] Enestroem, in Bibliotheca Mathematica, Vol. I (3), p. 499; Cantor, Geschichte, Vol. I (3), p. 671.

[393] Cited in Chapter I. It begins: "Dixit algoritmi: laudes deo rectori nostro atque defensori dicamus dignas." It is devoted entirely to the fundamental operations and contains no applications.

[394] M. Steinschneider, "Die Mathematik bei den Juden," Bibliotheca Mathematica, Vol. VIII (2), p. 99. See also the reference to this writer in Chapter I.

[395] Part of this work has been translated from a Leyden MS. by F. Woepcke, Propagation, and more recently by H. Suter, Bibliotheca Mathematica, Vol. VII (3), pp. 113-119.

[396] A. Neander, General History of the Christian Religion and Church, 5th American ed., Boston, 1855, Vol. III, p. 335.

[397] Beazley, loc. cit., Vol. I, p. 49.

[398] Beazley, loc. cit., Vol. I, pp. 50, 460.

[399] See pp. 7-8.

[400] The name also appears as Moḥammed Abū'l-Qāsim, and Ibn Hauqal. Beazley, loc. cit., Vol. I, p. 45.

[401] Kitāb al-masālik wa'l-mamālik.

[402] Reinaud, Mem. sur l'Inde; in Gerhardt, Etudes, p. 18.

[403] Born at Shiraz in 1193. He himself had traveled from India to Europe.

[404] Gulistan (Rose Garden), Gateway the third, XXII. Sir Edwin Arnold's translation, N. Y., 1899, p. 177.