The Hindu-Arabic Numerals c9: Chapter V. The Question of the Introduction of the Numerals into Europe by Boethius
The Hindu-Arabic Numerals c9
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Chapter V. The Question of the Introduction of the Numerals into Europe by Boethius
c9.2
Just as we were quite uncertain as to the origin of the numeral forms, so too are we uncertain as to the time and place of their introduction into Europe. There are two general theories as to this introduction. The first is that they were carried by the Moors to Spain in the eighth or ninth century, and thence were transmitted to Christian Europe, a theory which will be considered later. The second, advanced by Woepcke,¹ is that they were not brought to Spain by the Moors, but that they were already in Spain when the Arabs arrived there, having reached the West through the Neo-Pythagoreans. There are two facts to support this second theory: (1) the forms of these numerals are characteristic, differing materially from those which were brought by Leonardo of Pisa from Northern Africa early in the thirteenth century (before 1202 A.D.); (2) they are essentially those which tradition has so persistently assigned to Boethius (*c.* 500 A.D.), and which he would naturally have received, if at all, from these same Neo-Pythagoreans or from the sources from which they derived them. Furthermore, Woepcke points out that the Arabs on entering Spain (711 A.D.) would naturally have followed their custom of adopting for the computation of taxes the numerical systems of the countries they conquered,¹ so that the numerals brought from Spain to Italy, not having undergone the same modifications as those of the Eastern Arab empire, would have differed, as they certainly did, from those that came through Bagdad. The theory is that the Hindu system, without the zero, early reached Alexandria (say 450 A.D.), and that the Neo-Pythagorean love for the mysterious and especially for the Oriental led to its use as something bizarre and cabalistic; that it was then passed along the Mediterranean, reaching Boethius in Athens or in Rome, and to the schools of Spain, being discovered in Africa and Spain by the Arabs even before they themselves knew the improved system with the place value.
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¹ *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."
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¹ E.g., they adopted the Greek numerals in use in Damascus and Syria, and the Coptic in Egypt. Theophanes (758–818 A.D.), *Chronographia*, Scriptores Historiae Byzantinae, Vol. XXXIX, Bonnae, 1839, p. 575, relates that in 699 A.D. the caliph Walīd forbade the use of the Greek language in the bookkeeping of the treasury of the caliphate, but permitted the use of the Greek alphabetic numerals, since the Arabs had no convenient number notation: καὶ ἐκώλυσε γράφεσθαι Ἑλληνιστὶ τοὺς δημοσίους τῶν λογοθεσίων κώδικας, ἀλλʼ Ἀραβίοις αὐτὰ παρασημαίνεσθαι, χωρὶς τῶν ψήφων, ἐπειδὴ ἀδύνατον τῇ ἐκείνων γλώσσῃ μονάδα ἢ δυάδα ἢ τριάδα ἢ ὀκτὼ ἥμισυ ἢ τρία γράφεσθαι· διὸ καὶ ἕως σήμερόν εἰσιν σὺν αὐτοῖς νοτάριοι Χριστιανοί. The importance of this contemporaneous document was pointed out by Martin, loc. cit. Karabacek, “Die Involutio im arabischen Schriftwesen,” Vol. CXXXV of *Sitzungsberichte d. phil.-hist. Classe d. k. Akad. d. Wiss.*, Vienna, 1896, p. 25, gives an Arabic date of 868 A.D. in Greek letters.
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A recent theory set forth by Bubnov¹ also deserves mention, chiefly because of the seriousness of purpose shown by this well-known writer. Bubnov holds that the forms first found in Europe are derived from ancient symbols used on the abacus, but that the zero is of Hindu origin. This theory does not seem tenable, however, in the light of the evidence already set forth.
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¹ *The Origin and History of Our Numerals* (in Russian), Kiev, 1908; *The Independence of European Arithmetic* (in Russian), Kiev.
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Two questions are presented by Woepcke's theory: (1) What was the nature of these Spanish numerals, and how were they made known to Italy? (2) Did Boethius know them?
c9.8
The Spanish forms of the numerals were called the *ḥurūf al-ġobār*, the *ġobār* or dust numerals, as distinguished from the *ḥurūf al-jumal* or alphabetic numerals. Probably the latter, under the influence of the Syrians or Jews,² were also used by the Arabs. The significance of the term *ġobār* is doubtless that these numerals were written on the dust abacus, this plan being distinct from the counter method of representing numbers. It is also worthy of note that Al-Bīrūnī states that the Hindus often performed numerical computations in the sand. The term is found as early as *c.* 950, in the verses of an anonymous writer of Kairwān, in Tunis, in which the author speaks of one of his works on *ġobār* calculation;³ and, much later, the Arab writer Abū Bekr Moḥammed ibn ʿAbdallāh, surnamed al-Ḥaṣṣār (the arithmetician), wrote a work of which the second chapter was "On the dust figures."¹
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² Woepcke, loc. cit., pp. 462, 262.
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³ 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*, 1895, p. 26.
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¹ Steinschneider in the *Abhandlungen*, Vol. III, p. 110.
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The ġobār numerals themselves were first made known to modern scholars by Silvestre de Sacy, who discovered them in an Arabic manuscript from the library of the ancient abbey of St.-Germain-des-Prés.² The system has nine characters, but no zero. A dot above a character indicates tens, two dots hundreds, and so on, 5 meaning 50, and 5 meaning 5000. It has been suggested that possibly these dots, sprinkled like dust above the numerals, gave rise to the word *ġobār*,³ but this is not at all probable. This system of dots is found in Persia at a much later date with numerals quite like the modern Arabic;⁴ but that it was used at all is significant, for it is hardly likely that the western system would go back to Persia, when the perfected Hindu one was near at hand.
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² See his *Grammaire arabe*, Vol. I, Paris, 1810, plate VIII; Gerhardt, *Études*, pp. 9–11, and *Entstehung* etc., p. 8; I. F. Weidler, *Spicilegium observationum ad historiam notarum numeralium pertinentium*, Wittenberg, 1755, speaks of the "figura cifrarum Saracenicarum" as being different from that of the "characterum Boethianorum," which are similar to the "vulgar" or common numerals; see also Humboldt, loc. cit.
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³ 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).
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⁴ Gerhardt (*Études*, p. 9) from a manuscript in the Bibliothèque Nationale. The numeral forms are 9 8 7 6 5 4 3 2 1, 20 being indicated by 2 and 200 by 2. This scheme of zero dots was also adopted by the Byzantine Greeks, for a manuscript of Planudes in the Bibliothèque Nationale has numbers like ·· (··)α for 8,100,000,000. See Gerhardt, *Études*, p. 19. Pihan, *Exposé* etc., p. 208, gives two forms, Asiatic and Maghrebian, of "Ghobār" numerals.
c9.16
At first sight there would seem to be some reason for believing that this feature of the ġobār system was of Arabic origin, and that the present zero of these people,¹ the dot, was derived from it. It was entirely natural that the Semitic people generally should have adopted such a scheme, since their diacritical marks would suggest it, not to speak of the possible influence of the Greek accents in the Hellenic number system. When we consider, however, that the dot is found for zero in the Bakhṣālī manuscript,² and that it was used in subscript form in the *Kitāb al-Fihrist*³ in the tenth century, and as late as the sixteenth century,⁴ although in this case probably under Arabic influence, we are forced to believe that this form may also have been of Hindu origin.
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¹ See Chap. IV.
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² Possibly as early as the third century A.D., but probably of the eighth or ninth. See Cantor, I (3), p. 598.
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³ Ascribed by the Arabic writer to India.
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⁴ See Woepcke's description of a manuscript in the Chasles library, "Recherches sur l'histoire des sciences mathématiques chez les orientaux," *Journal Asiatique*, IV (5), 1859, p. 358, note.
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The fact seems to be that, as already stated,⁵ the Arabs did not immediately adopt the Hindu zero, because it resembled their 5; they used the superscript dot as serving their purposes fairly well; they may, indeed, have carried this to the west and have added it to the *ġobār* forms already there, just as they transmitted it to the Persians. Furthermore, the Arab and Hebrew scholars of Northern Africa in the tenth century knew these numerals as Indian forms, for a commentary on the *Sēfer Yeṣīrāh* by Abū Sahl ibn Tamim (probably composed at Kairwān, c. 950) speaks of "the Indian arithmetic known under the name of *ġobār* or dust calculation."⁶ All this suggests that the Arabs may very likely have known the ġobār forms before the numerals reached them again in 773.¹ The term "ġobār numerals" was also used without any reference to the peculiar use of dots.² In this connection it is worthy of mention that the Algerians employed two different forms of numerals in manuscripts even of the fourteenth century,³ and that the Moroccans of to-day employ the European forms instead of the present Arabic.
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⁵ P. 56.
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⁶ Reinaud, *Mémoire sur l'Inde*, p. 399. In the fourteenth century one Sihāb al-Dīn wrote a work on which a scholiast to the Bodleian manuscript remarks: "The science is called Algobar because the inventor had the habit of writing the figures on a tablet covered with sand." [Gerhardt, *Études*, p. 11, note.]
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¹ Gerhardt, *Entstehung* etc., p. 20.
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² H. Suter, "Das Rechenbuch des Abū Zakarījā el-Ḥaṣṣār," *Bibliotheca Mathematica*, Vol. II (3), p. 15.
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³ A. Devoulx, "Les chiffres arabes," *Revue Africaine*, Vol. XVI, pp. 455-458.
c9.27
The Indian use of subscript dots to indicate the tens, hundreds, thousands, etc., is established by a passage in the *Kitāb al-Fihrist*⁴ (987 a.d.) in which the writer discusses the written language of the people of India. Notwithstanding the importance of this reference for the early history of the numerals, it has not been mentioned by previous writers on this subject. The numeral forms given are those which have usually been called Indian,⁵ in opposition to ġobār. In this document the dots are placed below the characters, instead of being superposed as described above. The significance was the same.
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⁴ *Kitāb al-Fihrist*, G. Flügel, Leipzig, Vol. I, 1871, and Vol. II, 1872. This work was published after Professor Flügel's death by J. Roediger and A. Mueller. The first volume contains the Arabic text and the second volume contains critical notes upon it.
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⁵ Like those of line 5 in the illustration on page 69.
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In form these ġobār numerals resemble our own much more closely than the Arab numerals do. They varied more or less, but were substantially as follows:
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The question of the possible influence of the Egyptian demotic and hieratic ordinal forms has been so often suggested that it seems well to introduce them at this point, for comparison with the ġobār forms. They would as appropriately be used in connection with the Hindu forms, and the evidence of a relation of the first three with all these systems is apparent. The only further resemblance is in the Demotic 4 and in the 9, so that the statement that the Hindu forms in general came from this source has no foundation. The first four Egyptian cardinal numerals¹ resemble more the modern Arabic.
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¹ Woepcke, *Recherches sur l'histoire des sciences mathématiques chez les orientaux*, loc. cit.; *Propagation*, p. 57.
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² Al-Ḥaṣṣār's forms, Suter, *Bibliotheca Mathematica*, Vol. II (3), p. 15.
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³ Woepcke, *Sur une donnée historique*, etc., loc. cit. The name *ġobār* is not used in the text. The manuscript from which these are taken is the oldest (970 A.D.) Arabic document known to contain all of the numerals.
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⁴ Silvestre de Sacy, loc. cit. He gives the ordinary modern Arabic forms, calling them *Indien*.
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⁵ and 6 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 *ġobārī* 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.
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¹ J. G. Wilkinson, *The Manners and Customs of the Ancient Egyptians*, revised by S. Birch, London, 1878, Vol. II, p. 493, plate XVI.
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DEMOTIC AND HIERATIC ORDINALS
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This theory of the very early introduction of the numerals into Europe fails in several points. In the first place the early Western forms are not known; in the second place some early Eastern forms are like the ġobār, as is seen in the third line on p. 69, where the forms are from a manuscript written at Shiraz about 970 A.D., and in which some western Arabic forms, e.g. p for 2, are also used. Probably most significant of all is the fact that the ġobār numerals as given by Sacy are all, with the exception of the symbol for eight, either single Arabic letters or combinations of letters. So much for the Woepcke theory and the meaning of the ġobār numerals. We now have to consider the question as to whether Boethius knew these ġobār forms, or forms akin to them.
c9.41
This large question² suggests several minor ones: (1) Who was Boethius? (2) Could he have known these numerals? (3) Is there any positive or strong circumstantial evidence that he did know them? (4) What are the probabilities in the case?
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² There is an extensive literature on this "Boethius-Frage." The reader who cares to go fully into it should consult the various volumes of the *Jahrbuch über die Fortschritte der Mathematik*.
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First, who was Boethius, — Divus¹ Boethius as he was called in the Middle Ages? Anicius Manlius Severinus Boethius² was born at Rome *c.* 475. He was a member of the distinguished family of the Anicii,³ which had for some time before his birth been Christian. Early left an orphan, the tradition is that he was taken to Athens at about the age of ten, and that he remained there eighteen years.⁴ He married Rusticiana, daughter of the senator Symmachus, and this union of two such powerful families allowed him to move in the highest circles.⁵ Standing strictly for the right, and against all iniquity at court, he became the object of hatred on the part of all the unscrupulous element near the throne, and his bold defense of the ex-consul Albinus, unjustly accused of treason, led to his imprisonment at Pavia⁶ and his execution in 524.⁷ Not many generations after his death, the period being one in which historical criticism was at its lowest ebb, the church found it profitable to look upon his execution as a martyrdom.⁸ He was accordingly looked upon as a saint,¹ his bones were enshrined,² and as a natural consequence his books were among the classics in the church schools for a thousand years.³ It is pathetic, however, to think of the medieval student trying to extract mental nourishment from a work so abstract, so meaningless, so unnecessarily complicated, as the arithmetic of Boethius.
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¹ This title was first applied to Roman emperors in posthumous coins of Julius Cæsar. Subsequently the emperors assumed it during their own lifetimes, thus deifying themselves. See F. Gnecchi, *Monete romane*, 2d ed., Milan, 1900, p. 299.
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² This is the common spelling of the name, although the more correct Latin form is Boëtius. See Harper's *Dict. of Class. Lit. and Antiq.*, New York, 1897, Vol. I, p. 213. There is much uncertainty as to his life. A good summary of the evidence is given in the last two editions of the *Encyclopædia Britannica*.
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³ His father, Flavius Manlius Boethius, was consul in 487.
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⁴ There is, however, no good historic evidence of this sojourn in Athens.
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⁵ His arithmetic is dedicated to Symmachus: "Domino suo patricio Symmacho Boetius." [Friedlein ed., p. 3.]
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⁶ It was while here that he wrote *De consolatione philosophiae*.
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⁷ It is sometimes given as 525.
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⁸ There was a medieval tradition that he was executed because of a work on the Trinity.
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¹ Hence the *Divus* in his name.
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² Thus Dante, speaking of his burial place in the monastery of St. Pietro in Ciel d' Oro, at Pavia, says:
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³ Not, however, in the mercantile schools. The arithmetic of Boethius would have been about the last book to be thought of in such institutions. While referred to by Bæda (672–735) and Hrabanus Maurus (*c.* 776–856), it was only after Gerbert's time that the *Boëtii de institutione arithmetica libri duo* was really a common work.
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He was looked upon by his contemporaries and immediate successors as a master, for Cassiodorus⁴ (*c.* 490–*c.* 585 a.d.) says to him: "Through your translations the music of Pythagoras and the astronomy of Ptolemy are read by those of Italy, and the arithmetic of Nicomachus and the geometry of Euclid are known to those of the West."⁵ Founder of the medieval scholasticism,
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⁴ Also spelled Cassiodorius.
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⁵ 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 numériques* etc., reprint, p. 4.]
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"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.
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distinguishing the trivium and quadrivium,¹ writing the only classics of his time, Gibbon well called him "the last of the Romans whom Cato or Tully could have acknowledged for their countryman."²
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¹ The general idea goes back to Pythagoras, however.
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² J. C. Scaliger in his *Poëtice* also said of him: "Boethii Severini ingenium, eruditio, ars, sapientia facile provocat omnes auctores, sive illi Graeci sint, sive Latini" [Heilbronner, *Hist. math. univ.*, p. 387]. Libri, speaking of the time of Boethius, remarks: "Nous voyons du temps de Théodoric, les lettres reprendre une nouvelle vie en Italie, les écoles florissantes et les savans honorés. Et certes les ouvrages de Boëce, de Cassiodore, de Symmaque, surpassent de beaucoup toutes les productions du siècle précédent." [*Histoire des mathématiques*, Vol. I, p. 78.]
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The second question relating to Boethius is this: Could he possibly have known the Hindu numerals? In view of the relations that will be shown to have existed between the East and the West, there can only be an affirmative answer to this question. The numerals had existed, without the zero, for several centuries; they had been well known in India; there had been a continued interchange of thought between the East and West; and warriors, ambassadors, scholars, and the restless trader, all had gone back and forth, by land or more frequently by sea, between the Mediterranean lands and the centers of Indian commerce and culture. Boethius could very well have learned one or more forms of Hindu numerals from some traveler or merchant.
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To justify this statement it is necessary to speak more fully of these relations between the Far East and Europe. It is true that we have no records of the interchange of learning, in any large way, between eastern Asia and central Europe in the century preceding the time of Boethius. But it is one of the mistakes of scholars to believe that they are the sole transmitters of knowledge.
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As a matter of fact there is abundant reason for believing that Hindu numerals would naturally have been known to the Arabs, and even along every trade route to the remote west, long before the zero entered to make their place-value possible, and that the characters, the methods of calculating, the improvements that took place from time to time, the zero when it appeared, and the customs as to solving business problems, would all have been made known from generation to generation along these same trade routes from the Orient to the Occident. It must always be kept in mind that it was to the tradesman and the wandering scholar that the spread of such learning was due, rather than to the school man. Indeed, Avicenna¹ (980–1037 A.D.) in a short biography of himself relates that when his people were living at Bokhāra his father sent him to the house of a grocer to learn the Hindu art of reckoning, in which this grocer (oil dealer, possibly) was expert. Leonardo of Pisa, too, had a similar training.
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¹ 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 Wüstenfeld, *Geschichte der arabischen Aerzte und Naturforscher*, Göttingen, 1840. His full name is Abū ʿAlī al-Ḥosein ibn Sīnā. For notes on Avicenna's arithmetic, see Woepcke, *Propagation*, p. 502.
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The whole question of this spread of mercantile knowledge along the trade routes is so connected with the gobār numerals, the Boethius question, Gerbert, Leonardo of Pisa, and other names and events, that a digression for its consideration now becomes necessary.²
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² 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.
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Even in very remote times, before the Hindu numerals were sculptured in the cave of Nānā Ghāt, there were trade relations between Arabia and India. Indeed, long before the Aryans went to India the great Turanian race had spread its civilization from the Mediterranean to the Indus.¹ At a much later period the Arabs were the intermediaries between Egypt and Syria on the west, and the farther Orient.² In the sixth century B.C., Hecatæus,³ the father of geography, was acquainted not only with the Mediterranean lands but with the countries as far as the Indus,⁴ and in Biblical times there were regular triennial voyages to India. Indeed, the story of Joseph bears witness to the caravan trade from India, across Arabia, and on to the banks of the Nile. About the same time as Hecatæus, Scylax, a Persian admiral under Darius, from Caryanda on the coast of Asia Minor, traveled to northwest India and wrote upon his ventures.¹ He induced the nations along the Indus to acknowledge the Persian supremacy, and such number systems as there were in these lands would naturally have been known to a man of his attainments.
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¹ G. Oppert, *On the Ancient Commerce of India*, Madras, 1879, p. 8.
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² Gerhardt, *Études* etc., pp. 8, 11.
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³ See Smith's *Dictionary of Greek and Roman Biography and Mythology*.
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⁴ P. M. Sykes, *Ten Thousand Miles in Persia, or Eight Years in Irán*, London, 1902, p. 167. Sykes was the first European to follow the course of Alexander's army across eastern Persia.
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¹ Bühler, *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*.
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A century after Scylax, Herodotus showed considerable knowledge of India, speaking of its cotton and its gold,² telling how Sesostris³ fitted out ships to sail to that country, and mentioning the routes to the east. These routes were generally by the Red Sea, and had been followed by the Phœnicians and the Sabæans, and later were taken by the Greeks and Romans.⁴
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² Herodotus, Bk. III.
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³ Rameses II(?), the *Sesoosis* of Diodorus Siculus.
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⁴ *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.
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In the fourth century B.C. the West and East came into very close relations. As early as 330, Pytheas of Massilia (Marseilles) had explored as far north as the northern end of the British Isles and the coasts of the German Sea, while Macedon, in close touch with southern France, was also sending her armies under Alexander⁵ through Afghanistan as far east as the Punjab.⁶ Pliny tells us that Alexander the Great employed surveyors to measure the roads of India; and one of the great highways is described by Megasthenes, who in 295 b.c., as the ambassador of Seleucus, resided at Pātalīpuṭra, the present Patna.¹
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⁵ Sikandar, as the name still remains in northern India.
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⁶ *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.
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¹ 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.
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The Hindus also learned the art of coining from the Greeks, or possibly from the Chinese, and the stores of Greco-Hindu coins still found in northern India are a constant source of historical information.² The Rāmāyana speaks of merchants traveling in great caravans and embarking by sea for foreign lands.³ Ceylon traded with Malacca and Siam, and Java was colonized by Hindu traders, so that mercantile knowledge was being spread about the Indies during all the formative period of the numerals.
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² Bühler, *Palaeographie*, p. 2, speaks of Greek coins of a period anterior to Alexander, found in northern India. More complete information may be found in *Indian Coins*, by E. J. Rapson, Strassburg, 1898, pp. 3–7.
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³ Oppert, loc. cit., p. 14; and to him is due other similar information.
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Moreover the results of the early Greek invasion were embodied by Dicæarchus of Messana (about 320 b.c.) in a map that long remained a standard. Furthermore, Alexander did not allow his influence on the East to cease. He divided India into three satrapies,⁴ placing Greek governors over two of them and leaving a Hindu ruler in charge of the third, and in Bactriana, a part of Ariana or ancient Persia, he left governors; and in these the western civilization was long in evidence. Some of the Greek and Roman metrical and astronomical terms found their way, doubtless at this time, into the Sanskrit language.¹ Even as late as from the second to the fifth centuries a.d., Indian coins showed the Hellenic influence. The Hindu astronomical terminology reveals the same relationship to western thought, for Varāha-Mihira (6th century a.d.), a contemporary of Āryabhaṭa, entitled a work of his the *Bṛhat-Saṁhitā*, a literal translation of μ ε γ α′ λ η σ υ ν τ α ξ ι s of Ptolemy;² and in various ways is this interchange of ideas apparent.³ It could not have been at all unusual for the ancient Greeks to go to India, for Strabo lays down the route, saying that all who make the journey start from Ephesus and traverse Phrygia and Cappadocia before taking the direct road.⁴ The products of the East were always finding their way to the West, the Greeks getting their ginger⁵ from Malabar, as the Phœnicians had long before brought gold from Malacca.
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⁴ J. Beloch, *Griechische Geschichte*, Vol. III, Strassburg, 1904, pp. 30–31.
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¹ E.g., the denarius, the words for hour and minute (ωρα, λεπν), 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.
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² Marie, Vol. II, p. 73; R. Caldwell, loc. cit.
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³ A. Cunningham, loc. cit., p. 50.
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⁴ C. A. J. Skeel, *Travel*, loc. cit., p. 14.
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⁵ *Inchiver*, from *inchi*, "the green root." [*Indian Antiquary*, Vol. I, p. 352.]
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Greece must also have had early relations with China, for there is a notable similarity between the Greek and Chinese life, as is shown in their houses, their domestic customs, their marriage ceremonies, the public story-tellers, the puppet shows which Herodotus says were introduced from Egypt, the street jugglers, the games of dice,⁶ the game of finger-guessing,⁷ the water clock, the music system, the use of the myriad,¹ the calendars, and in many other ways.² In passing through the suburbs of Peking to-day, on the way to the Great Bell temple, one is constantly reminded of the semi-Greek architecture of Pompeii, so closely does modern China touch the old classical civilization of the Mediterranean. The Chinese historians tell us that about 200 B.C. their arms were successful in the far west, and that in 180 B.C. an ambassador went to Bactria, then a Greek city, and reported that Chinese products were on sale in the markets there.³ There is also a noteworthy resemblance between certain Greek and Chinese words,⁴ showing that in remote times there must have been more or less interchange of thought.
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⁶ In China dating only from the second century a.d., however.
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⁷ The Italian *morra*.
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¹ J. Bowring, *The Decimal System*, London, 1854, p. 2.
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² H. A. Giles, lecture at Columbia University, March 12, 1902, on "China and Ancient Greece."
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³ Giles, loc. cit.
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⁴ E.g., the names for grape, radish (*la-po*, *ῥάφη*), water-lily (*si-kua*, "west gourds"; *σικύα*, "gourds"), are much alike. [Giles, loc. cit.]
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The Romans also exchanged products with the East. Horace says, "A busy trader, you hasten to the farthest Indies, flying from poverty over sea, over crags, over fires."⁵ The products of the Orient, spices and jewels from India, frankincense from Persia, and silks from China, being more in demand than the exports from the Mediterranean lands, the balance of trade was against the West, and thus Roman coin found its way eastward. In 1898, for example, a number of Roman coins dating from 114 B.C. to Hadrian's time were found at Paklī, a part of the Hazāra district, sixteen miles north of Abbottābād,⁶ and numerous similar discoveries have been made from time to time.
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⁵ *Epistles*, I, 1, 45-46. On the Roman trade routes, see Beazley, loc. cit., Vol. I, p. 179.
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⁶ *Am. Journ. of Archeol.*, Vol. IV, p. 366.
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Augustus speaks of envoys received by him from India, a thing never before known,¹ and it is not improbable that he also received an embassy from China.² Suetonius (first century A.D.) speaks in his history of these relations,³ as do several of his contemporaries,⁴ and Vergil⁵ tells of Augustus doing battle in Persia. In Pliny’s time the trade of the Roman Empire with Asia amounted to a million and a quarter dollars a year, a sum far greater relatively then than now,⁶ while by the time of Constantine Europe was in direct communication with the Far East.⁷
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¹ 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.]
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² Reinaud, loc. cit., p. 189. Florus, II, 34 (IV, 12), refers to it: “Seres etiam habitantesque sub ipso sole Indi, cum gemmis et margaritis elephantes quoque inter munera trahentes nihil magis quam longinquitatem viae imputabant.” Horace shows his geographical knowledge by saying: “Not those who drink of the deep Danube shall now break the Julian edicts; not the Getae, not the Seres, nor the perfidious Persians, nor those born on the river Tanaïs.” [*Odes*, Bk. IV, Ode 15, 21–24.]
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³ “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.]
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⁴ Reinaud, loc. cit., p. 180.
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⁵ *Georgics*, II, 170–172. So Propertius (*Elegies*, III, 4):
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⁶ Heyd, loc. cit., Vol. I, p. 4.
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⁷ Reinaud, loc. cit., p. 393.
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In view of these relations it is not beyond the range of possibility that proof may sometime come to light to show that the Greeks and Romans knew something of the
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Arma deus Caesar dites meditatur ad Indos
Et freta gemmiferi findere classe maris.
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“The divine Cæsar meditated carrying arms against opulent India, and with his ships to cut the gem-bearing seas.”
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number system of India, as several writers have maintained.¹
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¹ The title page of Calandri (1491), for example, represents Pythagoras with these numerals before him. [Smith, *Rara Arithmetica*, p. 46.] Isaacus Vossius, *Observationes ad Pomponium Melam de situ orbis*, 1658, maintained that the Arabs derived these numerals from the west. A learned dissertation to this effect, but deriving them from the Romans instead of the Greeks, was written by Ginanni in 1753 (*Dissertatio mathematica critica de numeralium notarum minuscularum origine*, Venice, 1753). See also Mannert, *De numerorum quos arabicos vocant vera origine Pythagorica*, Nürnberg, 1801. Even as late as 1827 Romagnosi (in his supplement to *Ricerche storiche sull' India* etc., by Robertson, Vol. II, p. 580, 1827) asserted that Pythagoras originated them. [R. Bombelli, *L'antica numerazione italica*, Rome, 1876, p. 59.] Gow (*Hist. of Greek Math.*, p. 98) thinks that Iamblichus must have known a similar system in order to have worked out certain of his theorems, but this is an unwarranted deduction from the passage given.
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Returning to the East, there are many evidences of the spread of knowledge in and about India itself. In the third century B.C. Buddhism began to be a connecting medium of thought. It had already permeated the Himalaya territory, had reached eastern Turkestan, and had probably gone thence to China. Some centuries later (in 62 A.D.) the Chinese emperor sent an ambassador to India, and in 67 A.D. a Buddhist monk was invited to China.² Then, too, in India itself Aśoka, whose name has already been mentioned in this work, extended the boundaries of his domains even into Afghanistan, so that it was entirely possible for the numerals of the Punjab to have worked their way north even at that early date.³
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² A. Hillebrandt, *Alt-Indien*, p. 179.
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³ J. C. Marshman, loc. cit., chaps. i and ii.
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Furthermore, the influence of Persia must not be forgotten in considering this transmission of knowledge. In the fifth century the Persian medical school at Jondi-Sapur admitted both the Hindu and the Greek doctrines, and Firdusī tells us that during the brilliant reign of Khosrū I,¹ the golden age of Pahlavī literature, the Hindu game of chess was introduced into Persia, at a time when wars with the Greeks were bringing prestige to the Sassanid dynasty.
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¹ He reigned 531–579 A.D.; called Nuśīrwān, *the holy one*.
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Again, not far from the time of Boethius, in the sixth century, the Egyptian monk Cosmas, in his earlier years as a trader, made journeys to Abyssinia and even to India and Ceylon, receiving the name *Indicopleustes* (the Indian traveler). His map (547 A.D.) shows some knowledge of the earth from the Atlantic to India. Such a man would, with hardly a doubt, have observed every numeral system used by the people with whom he sojourned,² and whether or not he recorded his studies in permanent form he would have transmitted such scraps of knowledge by word of mouth.
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² J. Keane, *The Evolution of Geography*, London, 1899, p. 38.
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As to the Arabs, it is a mistake to feel that their activities began with Mohammed. Commerce had always been held in honor by them, and the Qoreish³ had annually for many generations sent caravans bearing the spices and textiles of Yemen to the shores of the Mediterranean. In the fifth century they traded by sea with India and even with China, and Hira was an emporium for the wares of the East,⁴ so that any numeral system of any part of the trading world could hardly have remained isolated.
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³ The Arabs who lived in and about Mecca.
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⁴ S. Guyard, in *Encyc. Brit.*, 9th ed., Vol. XVI, p. 597.
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Long before the warlike activity of the Arabs, Alexandria had become the great market-place of the world. From this center caravans traversed Arabia to Hadramaut, where they met ships from India. Others went north to Damascus, while still others made their way along the southern shores of the Mediterranean. Ships sailed from the isthmus of Suez to all the commercial ports of Southern Europe and up into the Black Sea. Hindus were found among the merchants¹ who frequented the bazaars of Alexandria, and Brahmins were reported even in Byzantium.
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¹ Oppert, loc. cit., p. 29.
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Such is a very brief résumé of the evidence showing that the numerals of the Punjab and of other parts of India as well, and indeed those of China and farther Persia, of Ceylon and the Malay peninsula, might well have been known to the merchants of Alexandria, and even to those of any other seaport of the Mediterranean, in the time of Boethius. The Brāhmī numerals would not have attracted the attention of scholars, for they had no zero so far as we know, and therefore they were no better and no worse than those of dozens of other systems. If Boethius was attracted to them it was probably exactly as any one is naturally attracted to the bizarre or the mystic, and he would have mentioned them in his works only incidentally, as indeed they are mentioned in the manuscripts in which they occur.
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In answer therefore to the second question, Could Boethius have known the Hindu numerals? the reply must be, without the slightest doubt, that he could easily have known them, and that it would have been strange if a man of his inquiring mind did not pick up many curious bits of information of this kind even though he never thought of making use of them.
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Let us now consider the third question, Is there any positive or strong circumstantial evidence that Boethius did know these numerals? The question is not new, nor is it much nearer being answered than it was over two centuries ago when Wallis (1693) expressed his doubts about it¹ soon after Vossius (1658) had called attention to the matter.² Stated briefly, there are three works on mathematics attributed to Boethius:³ (1) the arithmetic, (2) a work on music, and (3) the geometry.⁴
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¹ "At non credendum est id in Autographis contigisse, aut vetustioribus Codd. MSS." [Wallis, *Opera omnia*, Vol. II, p. 11.]
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² 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.
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³ 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.
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⁴ See also P. Tannery, "Notes sur la pseudo-géométrie de Boèce," in *Bibliotheca Mathematica*, Vol. I (3), p. 39. This is not the geometry in two books in which are mentioned the numerals. There is a manuscript of this pseudo-geometry of the ninth century, but the earliest one of the other work is of the eleventh century (Tannery), unless the Vatican codex is of the tenth century as Friedlein (p. 372) asserts.
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The genuineness of the arithmetic and the treatise on music is generally recognized, but the geometry, which contains the Hindu numerals with the zero, is under suspicion.⁵ There are plenty of supporters of the idea that Boethius knew the numerals and included them in this book,⁶ and on the other hand there are as many who feel that the geometry, or at least the part mentioning the numerals, is spurious.¹ The argument of those who deny the authenticity of the particular passage in question may briefly be stated thus:
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⁵ Friedlein feels that it is partly spurious, but he says: "Eorum librorum, quos Boetius de geometria scripsisse dicitur, investigare veram inscriptionem nihil aliud esset nisi operam et tempus perdere." [*Preface*, p. v.] N. Bubnov in the Russian *Journal of the Ministry of Public Instruction*, 1907, in an article of which a synopsis is given in the *Jahrbuch über die Fortschritte der Mathematik for 1907*, asserts that the geometry was written in the eleventh century.
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⁶ The most noteworthy of these was for a long time Cantor (*Geschichte*, Vol. I., 3d ed., pp. 587–588), who in his earlier days even believed that Pythagoras had known them. Cantor says (*Die römischen Agrimensoren*, Leipzig, 1875, p. 130): "Uns also, wir wiederholen es, ist die Geometrie des Boetius echt, dieselbe Schrift, welche er nach Euklid bearbeitete, von welcher ein Codex bereits in Jahre 821 im Kloster Reichenau vorhanden war, von welcher ein anderes Exemplar im Jahre 982 zu Mantua in die Hände Gerbert's gelangte, von welcher mannigfache Handschriften noch heute vorhanden sind." But against this opinion of the antiquity of MSS. containing these numerals is the important statement of P. Tannery, perhaps the most critical of modern historians of mathematics, that none exists earlier than the eleventh century. See also J. L. Heiberg in *Philologus, Zeitschrift f. d. klass. Altertum*, Vol. XLIII, p. 508. Of Cantor's predecessors, Th. H. Martin was one of the most prominent, his argument for authenticity appearing in the *Revue Archéologique* for 1856–1857, and in his treatise *Les signes numéraux* etc. See also M. Chasles, "De la connaissance qu'ont eu les anciens d'une numération décimale écrite qui fait usage de neuf chiffres prenant les valeurs de position," *Comptes rendus*, Vol. VI, pp. 678–680; "Sur l'origine de notre système de numération," *Comptes rendus*, Vol. VIII, pp. 72–81; and note "Sur le passage du premier livre de la géométrie de Boèce, relatif à un nouveau système de numération," in his work *Aperçu historique sur l'origine et le devéloppement des méthodes en géométrie*, of which the first edition appeared in 1837.
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¹ J. L. Heiberg places the book in the eleventh century on philological grounds, *Philologus*, loc. cit.; Woepcke, in *Propagation*, p. 44; Blume, Lachmann, and Rudorff, *Die Schriften der römischen Feldmesser*, Berlin, 1848; Boeckh, *De abaco graecorum*, Berlin, 1841; Friedlein, in his Leipzig edition of 1867; Weissenborn, *Abhandlungen*, Vol. II, p. 185, his *Gerbert*, pp. 1, 247, and his *Geschichte der Einführung der jetzigen Ziffern in Europa durch Gerbert*, Berlin, 1892, p. 11; Bayley, loc. cit., p. 59; Gerhardt, *Études*, p. 17, *Entstehung und Ausbreitung*, p. 14; Nagl, *Gerbert*, p. 57; Bubnov, loc. cit. See also the discussion by Chasles, Halliwell, and Libri, in the *Comptes rendus*, 1839, Vol. IX, p. 447, and in Vols. VIII, XVI, XVII of the same journal.
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1. The falsification of texts has always been the subject of complaint. It was so with the Romans,² it was common in the Middle Ages,³ and it is much more prevalent to-day than we commonly think. We have but to see how every hymn-book compiler feels himself authorized to change at will the classics of our language, and how unknown editors have mutilated Shakespeare, to see how much more easy it was for medieval scribes to insert or eliminate paragraphs without any protest from critics.¹
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² J. Marquardt, *La vie privée des Romains*, Vol. II (French trans.), p. 505, Paris, 1893.
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³ 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, [Figure:] 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.
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¹ Halliwell, in his *Rara Mathematica*, p. 107, states that the disputed passage is not in a manuscript belonging to Mr. Ames, nor in one at Trinity College. See also Woepcke, in *Propagation*, pp. 37 and 42. It was the evident corruption of the texts in such editions of Boethius as those of Venice, 1499, Basel, 1546 and 1570, that led Woepcke to publish his work *Sur l'introduction de l'arithmétique indienne en Occident*.
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2. If Boethius had known these numerals he would have mentioned them in his arithmetic, but he does not do so.²
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² 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.
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3. If he had known them, and had mentioned them in any of his works, his contemporaries, disciples, and successors would have known and mentioned them. But neither Capella (c. 475)³ nor any of the numerous medieval writers who knew the works of Boethius makes any reference to the system.⁴ 4. The passage in question has all the appearance of an interpolation by some scribe. Boethius is speaking of angles, in his work on geometry, when the text suddenly changes to a discussion of classes of numbers.¹ This is followed by a chapter in explanation of the abacus,² in which are described those numeral forms which are called *apices* or *caracteres*.³ The forms⁴ of these characters vary in different manuscripts, but in general are about as shown on page 88. They are commonly written with the 9 at the left, decreasing to the unit at the right, numerous writers stating that this was because they were derived from Semitic sources in which the direction of writing is the opposite of our own. This practice continued until the sixteenth century.⁵ The writer then leaves the subject entirely, using the Roman numerals
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³ Smith, *Rara Arithmetica*, p. 66.
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⁴ J. L. Heiberg, *Philologus*, Vol. XLIII, p. 507.
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¹ "Nosse autem huius artis dispicientem, quid sint digiti, quid articuli, quid compositi, quid incompositi numeri." [Friedlein ed., p. 395.]
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² *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.
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³ "Habebant enim diverse formatos apices vel caracteres." See the reference to Gerbert on p. 117.
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⁴ C. Henry, "Sur l'origine de quelques notations mathématiques," *Revue Archéologique*, 1879, derives these from the initial letters used as abbreviations for the names of the numerals, a theory that finds few supporters.
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⁵ E.g., it appears in Schonerus, *Algorithmus Demonstratvs*, Nürnberg, 1534, f. A 4. 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, ø 9 8 7 6 ५ 8 3 2 1 . . . 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.
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Forms of the Numerals, Largely from Works on the Abacus¹
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¹ ^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. 136; ^g *Memorie della classe di sci., Reale Acc. dei Lincei*, An. CCLXXIV (1876–1877), April, 1877
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^a Friedlein ed., p. 397. ^b Carlsruhe codex of Gerlando.
^c Munich codex of Gerlando. ^d Carlsruhe codex of Bernelinus.
^e Munich codex of Bernelinus. ^f Turchill, c. 1200.
^g Anon. MS., thirteenth century, Alexandrian Library, Rome.
^h Twelfth-century Boethius, Friedlein, p. 396.
^i Vatican codex, tenth century, Boethius.
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for the rest of his discussion, a proceeding so foreign to the method of Boethius as to be inexplicable on the hypothesis of authenticity. Why should such a scholarly writer have given them with no mention of their origin or use? Either he would have mentioned some historical interest attaching to them, or he would have used them in some discussion; he certainly would not have left the passage as it is.
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Linked texts (5)
- Bṛhatsaṃhitā — Varāhamihira, Sanskrit
- Vālmīkirāmāyaṇa — Vālmīki, Sanskrit
- Pāṭaliputra
- Nānāghāṭ
- Bakhshālī Manuscript — Sanskrit