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

"Alii referunt ad Phœnices inventores arithmeticæ, propter eandem commerciorum caussam: Alii ad Indos: Ioannes de Sacrobosco, cujus sepulchrum est Lutetiæ in comitio Maturinensi, refert ad Arabes." [Ramus, Arithmeticæ 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 été les premiers traffiquers de marchandise."

[2] Maximus Planudes (c. 1330) states that "the nine symbols come from the Indians." [Wäschke's German translation, Halle, 1878, p. 3.] Willichius speaks of the "Zyphræ Indicæ," in his Arithmeticæ 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 ("Mémoire 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: "Primùm autem appellamus dexterum locum, eo quòd haec ars vel à Chaldæis, vel ab Hebræis ortum habere credatur, qui etiam eo ordine scribunt"; but this refers more evidently to the Arabic numerals. [Arithmeticæ practicæ 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 Chaldæis 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-Ḫuwārizmī" 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 Morgenländ. Gesellschaft, Vol. XXXIII, p. 224; Steinschneider in the Zeitschrift der deutschen Morgenländ. 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 "Nachträge," 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 universæ, 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'Algèbre d'al-Khārizmi et les méthodes indienne et grecque, Léon 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 Eneström 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'arithmétique indienne en Occident, Rome, 1859.

[13] Thus in a commentary by ‛Alī ibn Abī Bekr ibn al-Jamāl al-Anṣārī 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.]

[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] Littérature 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] Matériaux pour servir à l'histoire comparée des sciences mathématiques 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 à 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] Maçoudi, Le livre de l'avertissement et de la révision. 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 "Nachträge 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,... tùm arithmetica scripta maximè 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 traité 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:

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 été empruntée par les anciens peuples occidentaux à la sphère chinoise; ouvrage accompagné d'un atlas céleste 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 Morgenländischen 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. Bürk, 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. [Bühler, 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. 100·10⁷.

[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 à la Grèce, 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 Palæography, 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. Bühler, Indische Palaeographie, particularly chap. vi; A. C. Burnell, South Indian Palæography, 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 Bühler'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 Bühler, loc. cit., p. 73.

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

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

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

[74] Bühler, 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 Bühler's drawings, loc. cit.; with Bayley, loc. cit., p. 337 and plates; and with Bayley's article in the Encyclopædia 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; Bühler, Palaeographie, Tafel IX.

[79] See Fleet, loc. cit. See also T. Benfey, Sanskrit Grammar, London, 1863, p. 217; M. R. Kále, 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'épigraphie indienne. Given by Bühler, loc. cit., Tafel I.

[81] Same, Śaka inscriptions, probably of the first century B.C. Senart, loc. cit.; Bühler, 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 Archæology, XXI, p. 301, London, 1899.

[105] For a bibliography of the principal hypotheses of this nature see Bühler, loc. cit., p. 77. Bühler (p. 78) feels that of all these hypotheses that which connects the Brāhmī with the Egyptian numerals is the most plausible, although he does not adduce any convincing proof. Th. Henri Martin, "Les signes numéraux et l'arithmétique chez les peuples de l'antiquité et du moyen âge" (being an examination of Cantor's Mathematische Beiträge zum Culturleben der Völker), Annali di matematica pura ed applicata, Vol. V, Rome, 1864, pp. 8, 70. Also, same author, "Recherches nouvelles sur l'origine de notre système de numération écrite," Revue Archéologique, 1857, pp. 36, 55. See also the tables given later in this work.

[106] Journal of the Royal Asiatic Society, Bombay Branch, Vol. XXIII.

[107] Loc. cit., reprint, Part I, pp. 12, 17. Bayley's deductions are generally regarded as unwarranted.

[108] The Alphabet; London, 1883, Vol. II, pp. 265, 266, and The Academy of Jan. 28, 1882.

[109] Taylor, The Alphabet, loc. cit., table on p. 266.

[110] Bühler, On the Origin of the Indian 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 Piñgala). It is the decimal place value of these figures which gives them significance." C. Henry, "Sur l'origine de quelques notations mathématiques," Revue Archéologique, June and July, 1879, attempts to derive the Boethian forms from the initials of Latin words. See also J. Prinsep, "Examination of the Inscriptions from Girnar in Gujerat, and Dhauli in Cuttach," Journal of the Asiatic Society of Bengal, 1838, especially Plate XX, p. 348; this was the first work on the subject.

[112] Bühler, Palaeographie, p. 75, gives the list, with the list of letters (p. 76) corresponding to the number symbols.

[113] For a general discussion of the connection between the numerals and the different kinds of alphabets, see the articles by U. Ceretti, "Sulla origine delle cifre numerali moderne," Rivista di fisica, matematica e scienze naturali, Pisa and Pavia, 1909, anno X, numbers 114, 118, 119, and 120, and continuation in 1910.

[114] This is one of Bühler's hypotheses. See Bayley, loc. cit., reprint p. 4; a good bibliography of original sources is given in this work, p. 38.

[115] Loc. cit., reprint, part I, pp. 12, 17. See also Burnell, loc. cit., p. 64, and tables in plate XXIII.

[116] This was asserted by G. Hager (Memoria sulle cifre arabiche, Milan, 1813, also published in Fundgruben des Orients, Vienna, 1811, and in Bibliothèque Britannique, Geneva, 1812). See also the recent article by Major Charles E. Woodruff, "The Evolution of Modern Numerals from Tally Marks," American Mathematical Monthly, August-September, 1909. Biernatzki, "Die Arithmetik der Chinesen," Crelle's Journal für die reine und angewandte Mathematik, Vol. LII, 1857, pp. 59-96, also asserts the priority of the Chinese claim for a place system and the zero, but upon the flimsiest authority. Ch. de Paravey, Essai sur l'origine unique et hiéroglyphique des chiffres et des lettres de tous les peuples, Paris, 1826; G. Kleinwächter, "The Origin of the Arabic Numerals," China Review, Vol. XI, 1882-1883, pp. 379-381, Vol. XII, pp. 28-30; Biot, "Note sur la connaissance que les Chinois ont eue de la valeur de position des chiffres," Journal Asiatique, 1839, pp. 497-502. A. Terrien de Lacouperie, "The Old Numerals, the Counting-Rods and the Swan-Pan in China," Numismatic Chronicle, Vol. III (3), pp. 297-340, and Crowder B. Moseley, "Numeral Characters: Theory of Origin and Development," American Antiquarian, Vol. XXII, pp. 279-284, both propose to derive our numerals from Chinese characters, in much the same way as is done by Major Woodruff, in the article above cited.

[117] The Greeks, probably following the Semitic custom, used nine letters of the alphabet for the numerals from 1 to 9, then nine others for 10 to 90, and further letters to represent 100 to 900. As the ordinary Greek alphabet was insufficient, containing only twenty-four letters, an alphabet of twenty-seven letters was used.

[118] Institutiones mathematicae, 2 vols., Strassburg, 1593-1596, a somewhat rare work from which the following quotation is taken:

"Quis est harum Cyphrarum autor?

"A quibus hae usitatae syphrarum notae sint inventae: hactenus incertum fuit: meo tamen iudicio, quod exiguum esse fateor: a graecis librarijs (quorum olim magna fuit copia) literae Graecorum quibus veteres Graeci tamquam numerorum notis sunt usi: fuerunt corruptae. vt ex his licet videre.

"Graecorum Literae corruptae.

"Sed qua ratione graecorum literae ita fuerunt corruptae?

"Finxerunt has corruptas Graecorum literarum notas: vel abiectione vt in nota binarij numeri, vel additione vt in ternarij, vel inuersione vt in septenarij, numeri nota, nostrae notae, quibus hodie utimur: ab his sola differunt elegantia, vt apparet."

See also Bayer, Historia regni Graecorum Bactriani, St. Petersburg, 1788, pp. 129-130, quoted by Martin, Recherches nouvelles, etc., loc. cit.

[119] P. D. Huet, Demonstratio evangelica, Paris, 1769, note to p. 139 on p. 647: "Ab Arabibus vel ab Indis inventas esse, non vulgus eruditorum modo, sed doctissimi quique ad hanc diem arbitrati sunt. Ego vero falsum id esse, merosque esse Graecorum characteres aio; à librariis Graecae linguae ignaris interpolatos, et diuturna scribendi consuetudine corruptos. Nam primum 1 apex fuit, seu virgula, nota μονάδος. 2, est ipsum β extremis suis truncatum. γ, si in sinistram partem inclinaveris & cauda mutilaveris & sinistrum cornu sinistrorsum flexeris, fiet 3. Res ipsa loquitur 4 ipsissimum esse Δ, cujus crus sinistrum erigitur κατὰ κάθετον, & infra basim descendit; basis vero ipsa ultra crus producta eminet. Vides quam 5 simile sit τῷ epsilon; infimo tantum semicirculo, qui sinistrorsum patebat, dextrorsum converso. ἐπίσημον βαῦ quod ita notabatur digamma, rotundato ventre, pede detracto, peperit τὸ 6. Ex Ζ basi sua mutilato, ortum est τὸ 7. Si Η inflexis introrsum apicibus in rotundiorem & commodiorem formam mutaveris, exurget τὸ 8. At 9 ipsissimum est alt theta."

I. Weidler, Spicilegium observationum ad historiam notarum numeralium, Wittenberg, 1755, derives them from the Hebrew letters; Dom Augustin Calmet, "Recherches sur l'origine des chiffres d'arithmétique," Mémoires pour l'histoire des sciences et des beaux arts, Trévoux, 1707 (pp. 1620-1635, with two plates), derives the current symbols from the Romans, stating that they are relics of the ancient "Notae Tironianae." These "notes" were part of a system of shorthand invented, or at least perfected, by Tiro, a slave who was freed by Cicero. L. A. Sedillot, "Sur l'origine de nos chiffres," Atti dell' Accademia pontificia dei nuovi Lincei, Vol. XVIII, 1864-1865, pp. 316-322, derives the Arabic forms from the Roman numerals.

[120] Athanasius Kircher, Arithmologia sive De abditis Numerorum, mysterijs qua origo, antiquitas & fabrica Numerorum exponitur, Rome, 1665.

[121] See Suter, Die Mathematiker und Astronomen der Araber, p. 100.

[122] "Et hi numeri sunt numeri Indiani, a Brachmanis Indiae Sapientibus ex figura circuli secti inuenti."

[123] V. A. Smith, The Early History of India, Oxford, 2d ed., 1908, p. 333.

[124] C. J. Ball, "An Inscribed Limestone Tablet from Sippara," Proceedings of the Society of Biblical Archæology, Vol. XX, p. 25 (London, 1898). Terrien de Lacouperie states that the Chinese used the circle for 10 before the beginning of the Christian era. [Catalogue of Chinese Coins, London, 1892, p. xl.]

[125] For a purely fanciful derivation from the corresponding number of strokes, see W. W. R. Ball, A Short Account of the History of Mathematics, 1st ed., London, 1888, p. 147; similarly J. B. Reveillaud, Essai sur les chiffres arabes, Paris, 1883; P. Voizot, "Les chiffres arabes et leur origine," La Nature, 1899, p. 222; G. Dumesnil, "De la forme des chiffres usuels," Annales de l'université de Grenoble, 1907, Vol. XIX, pp. 657-674, also a note in Revue Archéologique, 1890, Vol. XVI (3), pp. 342-348; one of the earliest references to a possible derivation from points is in a work by Bettino entitled Apiaria universae philosophiae mathematicae in quibus paradoxa et noua machinamenta ad usus eximios traducta, et facillimis demonstrationibus confirmata, Bologna, 1545, Vol. II, Apiarium XI, p. 5.

[126] Alphabetum Barmanum, Romae, MDCCLXXVI, p. 50. The 1 is evidently Sanskrit, and the 4, 7, and possibly 9 are from India.

[127] Alphabetum Grandonico-Malabaricum, Romae, MDCCLXXII, p. 90. The zero is not used, but the symbols for 10, 100, and so on, are joined to the units to make the higher numbers.

[128] Alphabetum Tangutanum, Romae, MDCCLXXIII, p. 107. In a Tibetan MS. in the library of Professor Smith, probably of the eighteenth century, substantially these forms are given.

[129] Bayley, loc. cit., plate II. Similar forms to these here shown, and numerous other forms found in India, as well as those of other oriental countries, are given by A. P. Pihan, Exposé des signes de numération usités chez les peuples orientaux anciens et modernes, Paris, 1860.

[130] Bühler, loc. cit., p. 80; J. F. Fleet, Corpus inscriptionum Indicarum, Vol. III, Calcutta, 1888. Lists of such words are given also by Al-Bīrūnī in his work India; by Burnell, loc. cit.; by E. Jacquet, "Mode d'expression symbolique des nombres employé par les Indiens, les Tibétains et les Javanais," Journal Asiatique, Vol. XVI, Paris, 1835.

[131] This date is given by Fleet, loc. cit., Vol. III, p. 73, as the earliest epigraphical instance of this usage in India proper.

[132] Weber, Indische Studien, Vol. VIII, p. 166 seq.

[133] Journal of the Royal Asiatic Society, Vol. I (N.S.), p. 407.

[134] VIII, 20, 21.

[135] Th. H. Martin, Les signes numéraux ..., Rome, 1864; Lassen, Indische Alterthumskunde, Vol. II, 2d ed., Leipzig and London, 1874, p. 1153.

[136] But see Burnell, loc. cit., and Thibaut, Astronomie, Astrologie und Mathematik, p. 71.

[137] A. Barth, "Inscriptions Sanscrites du Cambodge," in the Notices et extraits des Mss. de la Bibliothèque nationale, Vol. XXVII, Part I, pp. 1-180, 1885; see also numerous articles in Journal Asiatique, by Aymonier.

[138] Bühler, loc. cit., p. 82.

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

[140] Bühler, loc. cit., p. 83. The Hindu astrologers still use an alphabetical system of numerals. [Burnell, loc. cit., p. 79.]

[141] Well could Ramus say, "Quicunq; autem fuerit inventor decem notarum laudem magnam meruit."

[142] Al-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 für 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 10¹⁸ and was not difficult, beginning as follows: