The Hindu-Arabic Numerals c5: Chapter I. Early Ideas of Their Origin
The Hindu-Arabic Numerals c5
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Chapter I. Early Ideas of Their Origin
c5.2
It has long been recognized that the common numerals used in daily life are of comparatively recent origin. The number of systems of notation employed before the Christian era was about the same as the number of written languages, and in some cases a single language had several systems. The Egyptians, for example, had three systems of writing, with a numerical notation for each; the Greeks had two well-defined sets of numerals, and the Roman symbols for number changed more or less from century to century. Even to-day the number of methods of expressing numerical concepts is much greater than one would believe before making a study of the subject, for the idea that our common numerals are universal is far from being correct. It will be well, then, to think of the numerals that we still commonly call Arabic, as only one of many systems in use just before the Christian era. As it then existed the system was no better than many others, it was of late origin, it contained no zero, it was cumbersome and little used, and it had no particular promise. Not until centuries later did the system have any standing in the world of business and science; and had the place value which now characterizes it, and which requires a zero, been worked out in Greece, we might have been using Greek numerals to-day instead of the ones with which we are familiar.
c5.3
Of the first number forms that the world used this is not the place to speak. Many of them are interesting, but none had much scientific value. In Europe the invention of notation was generally assigned to the eastern shores of the Mediterranean until the critical period of about a century ago, — sometimes to the Hebrews, sometimes to the Egyptians, but more often to the early trading Phœnicians.¹
c5.4
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.”
c5.5
The idea that our common numerals are Arabic in origin is not an old one. The mediæval and Renaissance writers generally recognized them as Indian, and many of them expressly stated that they were of Hindu origin.²
c5.6
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.
c5.7
Others argued that they were probably invented by the Chaldeans or the Jews because they increased in value from right to left, an argument that would apply quite as well to the Roman and Greek systems, or to any other. It was, indeed, to the general idea of notation that many of these writers referred, as is evident from the words of England’s earliest arithmetical textbook-maker, Robert Recorde (c. 1542): “In that thinge all men do agree, that the Chaldays, whiche fyrste inuented thys arte, did set these figures as thei set all their letters. for they wryte backwarde as you tearme it, and so doo they reade. And that may appeare in all Hebrewe, Chaldaye and Arabike bookes . . . where as the Greekes, Latines, and all nations of Europe, do wryte and reade from the lefte hand towarde the ryghte.” ¹ Others, and among them such influential writers as Tartaglia¹ in Italy and Köbel² in Germany, asserted the Arabic origin of the numerals, while still others left the matter undecided³ or simply dismissed them as "barbaric."⁴ Of course the Arabs themselves never laid claim to the invention, always recognizing their indebtedness to the Hindus both for the numeral forms and for the distinguishing feature of place value. Foremost among these writers was the great master of the golden age of Bagdad, one of the first of the Arab writers to collect the mathematical classics of both the East and the West, preserving them and finally passing them on to awakening Europe. This man was Moḥammed the Son of Moses, from Khowārezm, or, more after the manner of the Arab, Moḥammed ibn Mūsā al-Khowārazmī,⁵ a man of great learning and one to whom the world is much indebted for its present knowledge of algebra¹ and of arithmetic. Of him there will often be occasion to speak; and in the arithmetic which he wrote, and of which Adelhard of Bath² (*c.* 1130) may have made the translation or paraphrase,³ he stated distinctly that the numerals were due to the Hindus.⁴ This is as plainly asserted by later Arab writers, even to the present day.¹ Indeed the phrase *'ilm hindī*, "Indian science," is used by them for arithmetic, as also the adjective *hindī* alone.²
c5.8
¹ 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.]
c5.9
¹ " . . . & 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.]
c5.10
² "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.]
c5.11
³ 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.]
c5.12
⁴ "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.
c5.13
⁵ His full name was ʿAbū ʿAbdallāh 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 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.
c5.14
¹ 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.
c5.15
² 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.
c5.16
³ 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.
c5.17
⁴ 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.
c5.18
¹ 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 *Mokhlaṣar 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.]
c5.19
² 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.
c5.20
Probably the most striking testimony from Arabic sources is that given by the Arabic traveler and scholar Moḥammed ibn Aḥmed, Abū 'l-Rīḥān al-Bīrūnī (973–1048), who spent many years in Hindustan. He wrote a large work on India,³ one on ancient chronology,⁴ the "Book of the Ciphers," unfortunately lost, which treated doubtless of the Hindu art of calculating, and was the author of numerous other works. Al-Bīrūnī was a man of unusual attainments, being versed in Arabic, Persian, Sanskrit, Hebrew, and Syriac, as well as in astronomy, chronology, and mathematics. In his work on India he gives detailed information concerning the language and customs of the people of that country, and states explicitly¹ that the Hindus of his time did not use the letters of their alphabet for numerical notation, as the Arabs did. He also states that the numeral signs called *aṅka*² had different shapes in various parts of India, as was the case with the letters. In his *Chronology of Ancient Nations* he gives the sum of a geometric progression and shows how, in order to avoid any possibility of error, the number may be expressed in three different systems: with Indian symbols, in sexagesimal notation, and by an alphabet system which will be touched upon later. He also speaks³ of "179, 876, 755, expressed in Indian ciphers," thus again attributing these forms to Hindu sources.
c5.21
³ *Alberuni's India*, Arabic version, London, 1887; English translation, ibid., 1888.
c5.22
⁴ *Chronology of Ancient Nations*, London, 1879. Arabic and English versions, by C. E. Sachau.
c5.23
¹ *India*, Vol. I, chap. xvi.
c5.24
² The Hindu name for the symbols of the decimal place system.
c5.25
³ Sachau's English edition of the *Chronology*, p. 64.
c5.26
Preceding Al-Bīrūnī there was another Arabic writer of the tenth century, Moṭahhar ibn Ṭāhir,⁴ author of the *Book of the Creation and of History*, who gave as a curiosity, in Indian (Nāgarī) symbols, a large number asserted by the people of India to represent the duration of the world. Huart feels positive that in Moṭahhar's time the present Arabic symbols had not yet come into use, and that the Indian symbols, although known to scholars, were not current. Unless this were the case, neither the author nor his readers would have found anything extraordinary in the appearance of the number which he cites.
c5.27
⁴ *Littérature arabe*, Cl. Huart, Paris, 1902.
c5.28
Mention should also be made of a widely-traveled student, Al-Mas'ūdī (885?-956), whose journeys carried him from Bagdad to Persia, India, Ceylon, and even across the China sea, and at other times to Madagascar, Syria, and Palestine.¹ He seems to have neglected no accessible sources of information, examining also the history of the Persians, the Hindus, and the Romans. Touching the period of the Caliphs his work entitled *Meadows of Gold* furnishes a most entertaining fund of information. He states² that the wise men of India, assembled by the king, composed the *Sindhind*. Further on³ he states, upon the authority of the historian Moḥammed ibn ʿAlī ʿAbdī, that by order of Al-Manṣūr many works of science and astrology were translated into Arabic, notably the *Sindhind* (*Siddhānta*). Concerning the meaning and spelling of this name there is considerable diversity of opinion. Colebrooke⁴ first pointed out the connection between *Siddhānta* and *Sindhind*. He ascribes to the word the meaning "the revolving ages."⁵ Similar designations are collected by Sédillot,⁶ who inclined to the Greek origin of the sciences commonly attributed to the Hindus.⁷ Casiri,⁸ citing the *Tārīkh al-ḥokamā* or *Chronicles of the Learned*,⁹ refers to the work as the *Sindum-Indum* with the meaning "perpetuum æternumque." The reference¹ in this ancient Arabic work to Al-Khowārazmī is worthy of note.
c5.29
¹ Huart, *History of Arabic Literature*, English ed., New York, 1903, p. 182 seq.
c5.30
² Al-Masʿūdī'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.
c5.31
³ *Les prairies d'or*, Vol. VIII, p. 289 seq.
c5.32
⁴ *Essays*, Vol. II, p. 428.
c5.33
⁵ Loc. cit., p. 504.
c5.34
⁶ *Matériaux pour servir à l'histoire comparée des sciences mathématiques chez les Grecs et les Orientaux*, 2 vols., Paris, 1845–1849, pp. 438–439.
c5.35
⁷ He made an exception, however, in favor of the numerals, loc. cit., Vol. II, p. 503.
c5.36
⁸ *Bibliotheca Arabico-Hispana Escurialensis*, Madrid, 1760–1770, pp. 426–427.
c5.37
⁹ The author, Ibn al-Qifṭī, flourished A.D. 1198 [Colebrooke, loc. cit., note Vol. II, p 510].
c5.38
¹ "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.]
c5.39
This *Sindhind* is the book, says Mas'ūdī,² which gives all that the Hindus know of the spheres, the stars, arithmetic,³ and the other branches of science. He mentions also Al-Khowārazmī and Ḥabash⁴ as translators of the tables of the *Sindhind*. Al-Bīrūnī⁵ refers to two other translations from a work furnished by a Hindu who came to Bagdad as a member of the political mission which Sindh sent to the caliph Al-Manṣūr, in the year of the Hejira 154 (A.D. 771).
c5.40
² Maçoudi, *Le livre de l'avertissement et de la révision*. Translation by B. Carra de Vaux, Paris, 1896.
c5.41
³ Verifying the hypothesis of Woepcke, *Propagation*, that the *Sindhind* included a treatment of arithmetic.
c5.42
⁴ Aḥmed ibn 'Abdallāh, Suter, *Die Mathematiker*, etc., p. 12.
c5.43
⁵ *India*, Vol. II, p. 15.
c5.44
The oldest work, in any sense complete, on the history of Arabic literature and history is the *Kitāb al-Fihrist*, written in the year 987 A.D., by Ibn Abī Ya'qūb al-Nadīm. It is of fundamental importance for the history of Arabic culture. Of the ten chief divisions of the work, the seventh demands attention in this discussion for the reason that its second subdivision treats of mathematicians and astronomers.⁶
c5.45
⁶ 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).
c5.46
The first of the Arabic writers mentioned is Al-Kindī (800–870 A.D.), who wrote five books on arithmetic and four books on the use of the Indian method of reckoning. Sened ibn ʿAlī, the Jew, who was converted to Islam under the caliph Al-Māmūn, is also given as the author of a work on the Hindu method of reckoning. Nevertheless, there is a possibility¹ that some of the works ascribed to Sened ibn ʿAlī are really works of Al-Khowārazmī, whose name immediately precedes his. However, it is to be noted in this connection that Casiri² also mentions the same writer as the author of a most celebrated work on arithmetic.
c5.47
¹ Suter, loc. cit., note 165, pp. 62–63.
c5.48
² "Send Ben Ali, ... tùm arithmetica scripta maximè celebrata, quae publici juris fecit." [Loc. cit., p. 440.]
c5.49
To Al-Ṣūfī, who died in 986 A.D., is also credited a large work on the same subject, and similar treatises by other writers are mentioned. We are therefore forced to the conclusion that the Arabs from the early ninth century on fully recognized the Hindu origin of the new numerals.
c5.50
Leonard of Pisa, of whom we shall speak at length in the chapter on the Introduction of the Numerals into Europe, wrote his *Liber Abbaci*³ in 1202. In this work he refers frequently to the nine Indian figures,⁴ thus showing again the general consensus of opinion in the Middle Ages that the numerals were of Hindu origin.
c5.51
³ *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.
c5.52
⁴ "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."
c5.53
Some interest also attaches to the oldest documents on arithmetic in our own language. One of the earliest treatises on algorism is a commentary¹ on a set of verses called the *Carmen de Algorismo*, written by Alexander de Villa Dei (Alexandre de Ville-Dieu), a Minorite monk of about 1240 A.D. The text of the first few lines is as follows:
c5.54
¹ 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.
c5.55
"Hec algorism' ars p'sens dicit' in qua
Talib; indor\v{u} fruim bis quinq; figuris.²
c5.56
² These are the two opening lines of the *Carmen de Algorismo* that the anonymous author is explaining. They should read as follows:
c5.57
"This boke is called the boke of algorim or augrym after lewder use. And this boke tretys of the Craft of Nombryng, the quych crafte is called also Algorym. Ther was a kyng of Inde the quich heyth Algor & he made this craft. . . . Algorisms, in the quych we use teen figurys of Inde."
c5.58
Haec algorismus ars praesens dicitur, in qua
Talibus Indorum fruimur bis quinque figuris.
c5.59
What follows is the translation.
Linked texts (2)
- Kauṭilīya Arthaśāstra — Kauṭilya, Sanskrit
- Śvetāmbara Āgama — Ardhamāgadhī Prakrit