The Hindu-Arabic Numerals

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.

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 Ph?nicians.[1]

The idea that our common numerals are Arabic in origin is not an old one. The medi?val and Renaissance writers generally recognized them as Indian, and many of them expressly stated that they were of Hindu origin.[2] 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."[3] Others, and among them such influential writers as Tartaglia[4] in Italy and K?bel[5] in Germany, asserted the Arabic origin of the numerals, while still others left the matter undecided[6] or simply dismissed them as "barbaric."[7] 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 Mo?ammed the Son of Moses, from Khowārezm, or, more after the manner of the Arab, Mo?ammed ibn Mūsā al-Khowārazmī,[8] a man of great learning and one to whom the world is much indebted for its present knowledge of algebra[9] and of arithmetic. Of him there will often be occasion to speak; and in the arithmetic which he wrote, and of which Adelhard of Bath[10] (c. 1130) may have made the translation or paraphrase,[11] he stated distinctly that the numerals were due to the Hindus.[12] This is as plainly asserted by later Arab writers, even to the present day.[13] Indeed the phrase 'ilm hindī, "Indian science," is used by them for arithmetic, as also the adjective hindī alone.[14]

Probably the most striking testimony from Arabic sources is that given by the Arabic traveler and scholar Mohammed ibn A?med, Abū 'l-Rī?ān al-Bīrūnī (973-1048), who spent many years in Hindustan. He wrote a large work on India,[15] one on ancient chronology,[16] 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 explicitly[17] 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 a?ka[18] 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 speaks[19] of "179, 876, 755, expressed in Indian ciphers," thus again attributing these forms to Hindu sources.

Preceding Al-Bīrūnī there was another Arabic writer of the tenth century, Mo?ahhar ibn ?āhir,[20] 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 Mo?ahhar'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.

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.[21] 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 states[22] that the wise men of India, assembled by the king, composed the Sindhind. Further on[23] he states, upon the authority of the historian Mo?ammed ibn 'Alī 'Abdī, that by order of Al-Man?ūr 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. Colebrooke[24] first pointed out the connection between Siddhānta and Sindhind. He ascribes to the word the meaning "the revolving ages."[25] Similar designations are collected by Sédillot,[26] who inclined to the Greek origin of the sciences commonly attributed to the Hindus.[27] Casiri,[28] citing the Tārīkh al-?okamā or Chronicles of the Learned,[29] refers to the work as the Sindum-Indum with the meaning "perpetuum ?ternumque." The reference[30] in this ancient Arabic work to Al-Khowārazmī is worthy of note.

This Sindhind is the book, says Mas'ūdī,[31] which gives all that the Hindus know of the spheres, the stars, arithmetic,[32] and the other branches of science. He mentions also Al-Khowārazmī and ?abash[33] as translators of the tables of the Sindhind. Al-Bīrūnī[34] 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-Man?ūr, in the year of the Hejira 154 (A.D. 771).

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.[35]

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 'Alī, 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 possibility[36] that some of the works ascribed to Sened ibn 'Alī are really works of Al-Khowārazmī, whose name immediately precedes his. However, it is to be noted in this connection that Casiri[37] also mentions the same writer as the author of a most celebrated work on arithmetic.

To Al-?ūfī, 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.

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[38] in 1202. In this work he refers frequently to the nine Indian figures,[39] thus showing again the general consensus of opinion in the Middle Ages that the numerals were of Hindu origin.

Some interest also attaches to the oldest documents on arithmetic in our own language. One of the earliest treatises on algorism is a commentary[40] on a set of verses called the Carmen de Algorismo, written by Alexander de Villa Dei (Alexandra de Ville-Dieu), a Minorite monk of about 1240 A.D. The text of the first few lines is as follows:

"Hec algorism' ars p'sens dicit' in qua

Talib; indor fruim bis quinq; figuris.[41]

"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."

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