History of Hindu Mathematics: A Source Book v1.c7.s17: 17. Miscellaneous References to the Hindu Numerals

History of Hindu Mathematics: A Source Book v1.c7.s17

Language: English

v1.c7.s17.t1 Djàhiz (died 869) — 2 parts
v1.c7.s17.t2 Abul Hasan Al-Masûdî (943) — 2 parts
v1.c7.s17.t3 Abû Sahl Ibn Tamim (950) — 1 part
v1.c7.s17.t4 Al-Nadîm (987) — 1 part
v1.c7.s17.t5 Al-Bîrûnî (1030) — 45 parts

17. Miscellaneous References to the Hindu Numerals

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**Syrian Reference.** The following reference to a passage³ in a work of Severus Sebokht (662) shows that the fame of the Hindu numerals had reached the banks of the Euphrates in the beginning of the seventh century. Sebokht was a learned Syrian scholar who lived in the convent of Kenneshre on the Euphrates, in the time of the patriarch Athanasius Gam̄enale (died 631) and his successor John.¹ Sebokht seems to have been hurt by the arrogance of certain Greek scholars who looked down on the Syrians, and in defending the latter, he claims for them the invention of astronomy. He asserts that the Greeks were merely the pupils of the Chaldeans of Babylon and claims that these same Chaldeans were the very Syrians whom his opponents condemned. It is in this connection that he mentions the Hindus by way of illustration, using the following words:

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³ Attention was first drawn to this passage by F. Nau. *JA*, II, 1910, pp. 225-227; also see J. Ginsburg, *Bull. American Math. Soc.*, XXIII, 1917, p. 368.

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¹ W. Wright, *Short History of Syriac Literature*, pp. 137-139.

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"I will omit all discussion of the science of the Hindus, a people not the same as the Syrians, their subtle discoveries in the science of astronomy, discoveries that are more ingenious than those of the Greeks and the Babylonians; their computing that surpasses description. I wish only to say that this computation is done by means of nine signs. If those who believe because they speak Greek, that they have reached the limits of science should know these things, they would be convinced that there are also others who know something."

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This fragment clearly shows that not only did Sebokht know something of the numerals but that he understood their full significance, and may even have known the zero as Rabbi ben Ezra did (*vide infra* p. 103), in spite of the fact that he too speaks of nine numerals. In fact even today, we speak of nine numerals only, zero not being recognised a numeral.

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**Arabic References.** *Ibn Wahshiya (855).* In his work, *Ancient alphabets and hieroglyphic characters explained*, etc., Ibn Wahshiya gives three forms of Hindu numerals as three species of Hindu alphabets.¹ It seems that the forms were well known in his time in Arabia. In the first set of these forms, dots are used as zeros in the now customary way among the Arabs; in the second set the dots are superposed,² while in the third loops are attached to the ordinary symbols for the units.³ Karpinski thinks that the use of loops was probably the invention of Wahshiya.

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¹ Karpinski, *Bibl. Math.*, XIII, 1912-13, pp. 97-8.

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² Similar use of dots is found in certain manuscripts in India. Vide R. L. Mitra, *Notices of Sanskrit MSS.*, V, p. 299, MS. No. 1976, Plate No. I; *India Office Catalogue*, MS. Nos. 1946, 1947 and 1871.

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³ This form is similar to the use of ligatures in India when the numerals were written without place-value, (*vide supra*).

Djàhiz (died 869)

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The Arab philosopher Djâhiz calls the numerals "figures of Hind" and observes that with these numerals large numbers can be represented with great facility. He asks, "Who is the inventor of the figures of Hind, who are the authors of the *Sindhind* and of the *Arkhand* and the methods of calculation with the ciphers?"⁴

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⁴ Djâhiz's work is available in French translation (Caire, *Du rond et du carré*, 1324). Cf. Carra de Vaux, *Scientia*, XXI, 1921, pp. 273-282.

Abul Hasan Al-Masûdî (943)

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The Arab historian Al-Masûdî writes, "A congress of sages at the command of the Creator Brahmâ invented the nine figures and also their (the Hindus') astronomy and other sciences."⁵ Al-Masûdî's evidence is important as he had first-hand information of India. He was in Multan in 912 A.D. and in Cambay in 916 A.D.

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⁵ Reinaud, *Mémoire sur l'Inde*, Paris, 1894, p. 300. This shows that the Hindus in the tenth century had forgotten the inventor or inventors of the numerals, as a very long time had elapsed.

Abû Sahl Ibn Tamim (950)

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Ibn Tamim, a native of Kairwân, a village in Tunis in the north of Africa, wrote in his commentary on the *Sefer Yesîrâh*: "The Indians have invented the nine signs for marking the units. I have spoken sufficiently of them in a book that I have composed on the Hindu calculation, known under the name of *Hisâb al-ghobâr*."

Al-Nadîm (987)

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In the *Fihrist*, the author Al-Nadîm includes the Hindu numerals in a list of some two hundred alphabets of India (*Hind*.) These numerals are called *hindisah*.

Al-Bîrûnî (1030)

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Al-Bîrûnî resided in India for nearly thirteen years (1017-1030) and devoted himself to the study of the arts and sciences of the Hindus. He had also a remarkable knowledge of the Greek sciences and literature, so he was more qualified than any contemporary or even anterior Arab writer to speak with authority about the origin of the numerals. He wrote two books, viz., *Kitâb al-arqam* ("Book of Ciphers") and *Tazkira fî al-hisâb w'al-madd bi al-arqam al-Sind w'al-Hind* ("A treatise on arithmetic and the system of counting with the ciphers of Sindh and India"). In his *Tarikh al-Hind* ("Chronicles of India"), he says: "As in different parts of India the letters have different shapes, the numerical signs too, which are called *aṅka*, differ. The numerical signs which we use are derived from the finest forms of the Hindu signs." At another place he remarks: "The Hindus use the numeral signs in arithmetic in the same way as we do. I have composed a treatise showing how far possibly, the Hindus are ahead of us in this subject.”¹ In his *Athâr-ul-Bâkiya*² (“Vestiges of the Past,” written in 1000 A.D.) Al-Bîrûnî calls the modern numerals as *al-arqam al-hind*, *i.e.*, “the Indian Ciphers” and he has incidentally referred to their distinction from two other systems of expressing numbers, *viz.*, the sexagesimal system and the alphabetic system (*Harûf al-jumal*).

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¹ Reinaud, *l.c.*, p. 399.

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² *Kitâb al-Fihrist*, ed. G. Flügel, II, pp. 18-19.

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³ *Alberuni's India*, English translation by E. C. Sachau, London 2nd ed., 1910, Vol. I, p. 74.

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¹ Ibid, I, p. 177.

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² *The Chronology of Ancient Nations*, ed. by Sachau, London, 1879, pp. 62 and 132.

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*Abenragel* (1048)

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It has been stated by Ali bin Abil-Regal Abul-Hasan, called Abenragel, in the preface to his treatise on astronomy that the invention of reckoning with nine ciphers is due to the Hindu philosophers.³

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³ J. F. Montucla, *Historie des Mathématiques*, vol. I, p. 376.

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*Saraf-Eddin* (1172)

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Mahmûd bin Qajid al-Amûnî Saraf-Eddin of Mecca wrote a treatise, entitled *Fi al-handasa w'al arqam al-hindi* (“On geometry and the Indian ciphers”).⁴

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⁴ H. Suter, *Die Mathematiker und Astronomer der Ararbe und ihre Werke*, Leipzig, 1900, p. 126.

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*Alkalasâdî* (died 1486)

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In his commentary of the *Talkhis* of Ibn Albanna, Abul Hasan Ali Alkalasâdî states: “These nine signs, called the signs of the *ghobâr* (dust), are those that are employed very frequently in our Spanish provinces and in the countries of Maghrib and of Africa. Their origin is said to have been attributed by tradition to a man of the Indian nation. This man is said to have taken some fine dust, spread it upon a table and taught the people multiplication, division and other operations.”¹

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¹ *JA*, I, 1863, pp. 59f.

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*Behâ Eddin (c. 1600)*

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Referring to the numerals Behâ Eddin observes: “The Hindu savants have, in fact, invented the nine known characters.”²

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² *Kholasât al-hisâb*, translated into French by A. Marre, *Nouv. Ann. Math.*, V, 1864, p. 266.

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In the quotations from Arab scholars given above, the term *Hind* has been used for India, and *Hindi* for Indian. *Hind* is the term generally used in Arabic and Persian literature for India. In early writings distinction was sometimes made between Sind and Hind. Thus Al-Masûdî and Al-Bîrûnî used Sind to denote the countries to the west of the river Indus. This distinction is clearly in evidence in Ibn Hawkal’s map, reproduced in Elliot and Dawson’s *History of India*. There were others who did not make this distinction. Thus Istakri (912) uses Hind to denote the whole of India.³ Again in the *Shâhnâmâ* of Firdausi,⁴ Sind has been used for a river as well as for a country, and Hind for the whole of India. In later times this distinction disappeared completely. According to the lexicographers Ibn Seedeh (died 1066) and Firouzâbâdi (1328-1413), Hind is “the name of a well known nation” and according to El-Jowharee (1008) it denotes “the name of a country.” Instances of the use of *Hind* to denote India in the literature of the Arabs can be multiplied at pleasure.

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³ Elliot and Dawson’s *History of India*, II, p. 412.

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⁴ English translation by A.G. Warner and E. Warner, London, 1906.

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Carra de Vaux⁵ has suggested that the word *Hind* does not probably mean India but is really derived from *ènd* (or *hènd*) signifying "measure," "arithmetic" or "geometry." He concludes that the expression "the signs of *hind*" means "the arithmetical signs" and not "the signs of India." As regards the use of the adjective *hindi* by certain scholars in connection with the numerals, he conjectures that it has probably been employed through confusion for *hindasi*.

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⁵ Carra de Vaux, *Scientia*, XXI, 1917, p. 273.

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Carra de Vaux's derivation of the word *hind* from *ènd* or *hènd* cannot be accepted. It has no support from Arabic lexicography. Moreover, the word *hind* is a very ancient one. It occurs in the *Avesta*¹ both in the earlier *Yasna* and in the later (Sassanian) *Vendidad*. The word also occurs in the cuneiform inscriptions of Darius Hystaspes. The Pehlavi writings before the Arab conquest of Iran also show the word *hind*. In all those cases it means India.

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¹ *Yasna*, x. 141; *Yt.*, x. 104 (*Mihir Yast*).

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The word *hindi* is an adjective formed from *hind* and means "Indian". The fact that in a few isolated cases, it has been confused with the word *hindasi*, cannot make us conclude that this has happened in all cases.

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**The terms Hindasa, etc.** The words *hindasa*, *hindisa*, *handasa*, *hindasi*, *handasi*, etc., have been stated by competent authorities to be adjectives formed from *hind*, meaning "Indian". Kaye² and Carra de Vaux³ oppose this interpretation. Relying on the lexicon of Fîrouzâbâdi they assert that these terms are derived from the Persian *andâzah*, meaning "measure." There is no doubt that the word *hindasi* denotes "geometrical" in the Arabic language. But when this term is used in connection with an explanation of the rule of "double false position" or the method of "proof by nines" or in connection with the "numeral notation," we have to admit that it had some other significance also. As the arithmetical rules designated by the term *hindasi* are found in Hindu arithmetic prior to their occurrence in Arabia, it follows that *hindasi* also means Indian. The term *hindasi*, *hindasa* or *handasa* has, therefore, two meanings, one "geometrical" and the other "Indian". The controversy regarding the meaning of this term which was set at rest by Woepcke,¹ has arisen again because Kaye and Carra de Vaux have refused to recognise both meanings of this term.² It may be pointed out here that as one of the meanings of *hindasi* is synonymous with *hindi*, there is no wonder that the two words were sometimes confused with each other, especially by scribes who did not understand the text.

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² Kaye, *JASB*, III, 1907, p. 489, also *JASB*, VII, 1911, pp. 810f.

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³ Carra de Vaux, *l.c.*

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¹ Woepcke, *JA*, I, 1863, pp. 27f. See also Suter's article on *handasa* in the *Encyclopaedia of Islam* and Rosen's *Algebra of Mohammad Ben Musa*, London, 1831, pp. 196f.

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² It will not be difficult to point out in any literature words having more than one meaning. Occasionally these meanings have no connection. Whenever such a word is used, the appropriate meaning has to be deduced from the context.

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**European References.** *Isidorus of Seville.* The nine characters (of the *ghobār* type), without zero, are given as an addition to the first chapter of the third book of the *Origines* by Isidorus of Seville in which the Roman numerals are under discussion. Another Spanish copy of the same work (of 992 A.D.) contains the numerals in the corresponding section. The writer ascribes an Indian origin to them in the following words: "Item de figuris arithmetice. Scire debemus in Indos subtilissimum ingenium habere et ceteris gentes eis in arithmetica et geometria et ceteris liberalibus disciplinis concedere. Et hoc manifestum est in nobem figuris, quibus designant unum-quemque gradum cuinslibet gradus. Quarum hec sunt forma.”

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*Rabbi ben Ezra* (1092-1167)

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Rabbi Abraham ibn Meir ibn Ezra in his work, *Sefer ha-mispar* (“the Book of Number”), gives the Hindu forms of the numerals. He knew of the Hindu origin of the numerals for he states: “that is why the wise people of India have designated all their numbers through nine and have built forms for the nine ciphers.”

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*Leonardo of Pisa*

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Leonardo of Pisa in his work, *Liber Abaci* (1202), frequently refers to the nine Indian figures. At one place he says: “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.”

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*Alexander de Villa Dei*

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Alexander de Villa Dei (*c.* 1240) wrote a commentary on a set of verses called *Carmen de Algorismo*. In this commentary he writes: “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.”

Linked texts (2)

Vīdēvdād — Avestan
Yašt — Avestan