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The Hinduâ?"Arabic Numerals
By DAVID EUGENE SMITH, Louis Charles Karpinski
Dover Publications, Inc.Copyright © 2004 Dover Publications, Inc.
All rights reserved.
EARLY IDEAS OF THEIR ORIGIN
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 Phnicians.
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. 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." Others, and among them such influential writers as Tartaglia in Italy and Köbel in Germany, asserted the Arabic origin of the numerals, while still others left the matter undecided or simply dismissed them as "barbaric." 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 Mohammed the Son of Moses, from Khowarezm, or, more after the manner of the Arab, Mohammed ibn Musa al-Khowarazmi, a man of great learning and one to whom the world is much indebted for its present knowledge of algebra 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 (c. 1130) may have made the translation or paraphrase, he stated distinctly that the numerals were due to the Hindus. This is as plainly asserted by later Arab writers, even to the present day. Indeed the phrase 'ilm hindi, "Indian science," is used by them for arithmetic, as also the adjective hindi alone.
Probably the most striking testimony from Arabic sources is that given by the Arabic traveler and scholar Mohammed ibn Ahmed, Abu '1-Rihan al-Biruni (973–1048), who spent many years in Hindustan. He wrote a large work on India, one on ancient chronology, 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-Biruni 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 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 anka 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 of "179, 876, 755, expressed in Indian ciphers," thus again attributing these forms to Hindu sources.
Preceding Al-Biruni there was another Arabic writer of the tenth century, Motahhar ibn Tahir, author of the Book of the Creation and of History, who gave as a curiosity, in Indian (Nagari) symbols, a large number asserted by the people of India to represent the duration of the world. Huart feels positive that in Motahhar'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'udi (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. 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 that the wise men of India, assembled by the king, composed the Sindhind. Further on he states, upon the authority of the historian Mohammed ibn 'Ali 'Abdi, that by order of Al-Mansur many works of science and astrology were translated into Arabic, notably the Sindhind (Siddhanta). Concerning the meaning and spelling of this name there is considerable diversity of opinion. Colebrooke first pointed out the connection between Siddhanta and Sindhind. He ascribes to the word the meaning "the revolving ages." Similar designations are collected by Sédillot, who inclined to the Greek origin of the sciences commonly attributed to the Hindus. Casiri, citing the Tarikh alhokama or Chronicles of the Learned, refers to the work as the Sindum-Indum with the meaning "perpetuum æternumque." The reference in this ancient Arabic work to Al-Khowarazmi is worthy of note.
This Sindhind is the book, says Mas'udi, which gives all that the Hindus know of the spheres, the stars, arithmetic, and the other branches of science. He mentions also Al-Khowarazmi and Habash as translators of the tables of the Sindhind. Al-Biruni 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-Mansur, 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 Kitab aL-Fihrist, written in the year 987 A.D., by Ibn Abi Ya'qub al-Nadim. 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.
The first of the Arabic writers mentioned is Al-Kindi (800–870 A.D.), who wrote five books on arithmetic and four books on the use of the Indian method of reckoning. Sened ibn 'Ali, the Jew, who was converted to Islam under the caliph Al-Mamun, is also given as the author of a work on the Hindu method of reckoning. Nevertheless, there is a possibility that some of the works ascribed to Sened ibn 'Ali are really works of Al-Khowarazmi, whose name immediately precedes his. However, it is to be noted in this connection that Casiri also mentions the same writer as the author of a most celebrated work on arithmetic.
To Al-Sufi, 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 Abbaei in 1202. In this work he refers frequently to the nine Indian figures, 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 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:
"Hec algorism' ars p'sens dicit' in qua
Talib; indor fruim bis quinq; figuris.
"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."CHAPTER 2
EARLY HINDU FORMS WITH NO PLACE VALUE
While it is generally conceded that the scientific development of astronomy among the Hindus towards the beginning of the Christian era rested upon Greek or Chinese sources, yet their ancient literature testifies to a high state of civilization, and to a considerable advance in sciences, in philosophy, and along literary lines, long before the golden age of Greece. From the earliest times even up to the present day the Hindu has been wont to put his thought into rhythmic form. The first of this poetry — it well deserves this name, being also worthy from a metaphysical point of view — consists of the Vedas, hymns of praise and poems of worship, collected during the Vedic period which dates from approximately 2000 B.C. to 1400 B.C. Following this work, or possibly contemporary with it, is the Brahmanic literature, which is partly ritualistic (the Brahmanas), and partly philosophical (the Upanishads). Our especial interest is in the Sutras, versified abridgments of the ritual and of ceremonial rules, which contain considerable geometric material used in connection with altar construction, and also numerous examples of rational numbers the sum of whose squares is also a square, i.e. "Pythagorean numbers," although this was long before Pythagoras lived. Whitney places the whole of the Veda literature, including the Vedas, the Brahamanas, and the Sutras, between 1500 B.C. and 800 B.C., thus agreeing with Bürk who holds that the knowledge of the Pythagorean theorem revealed in the Sutras goes back to the eighth century B.C.
The importance of the Sutras as showing an independent origin of Hindu geometry, contrary to the opinion long held by Cantor of a Greek origin, has been repeatedly emphasized in recent literature, especially since the appearance of the important work of Von Schroeder. Further fundamental mathematical notions such as the conception of irrationals and the use of gnomons, as well as the philosophical doctrine of the transmigration of souls, — all of these having long been attributed to the Greeks, — are shown in these works to be native to India. Although this discussion does not bear directly upon the origin of our numerals, yet it is highly pertinent as showing the aptitude of the Hindu for mathematical and mental work, a fact further attested by the independent development of the drama and of epic and lyric poetry.
It should be stated definitely at the outset, however, that we are not at all sure that the most ancient forms of the numerals commonly known as Arabic had their origin in India. As will presently be seen, their forms may have been suggested by those used in Egypt, or in Eastern Persia, or in China, or on the plains of Mesopotamia. We are quite in the dark as to these early steps; but as to their development in India, the approximate period of the rise of their essential feature of place value, their introduction into the Arab civilization, and their spread to the West, we have more or less definite information. When, therefore, we consider the rise of the numerals in the land of the Sindhu, it must be understood that it is only the large movement that is meant, and that there must further be considered the numerous possible sources outside of India itself and long anterior to the first prominent appearance of the number symbols.
No one attempts to examine any detail in the history of ancient India without being struck with the great dearth of reliable material. So little sympathy have the people with any save those of their own caste that a general literature is wholly lacking, and it is only in the observations of strangers that any all- round view of scientific progress is to be found. There is evidence that primary schools existed in earliest times, and of the seventy-two recognized sciences writing and arithmetic were the most prized. In the Vedic period, say from 2000 to 1400 B.C., there was the same attention to astronomy that was found in the earlier civilizations of Babylon, China, and Egypt, a fact attested by the Vedas themselves. Such advance in science presupposes a fair knowledge of calculation, but of the manner of calculating we are quite ignorant and probably always shall be. One of the Buddhist sacred books, the Lalitavistara, relates that when the Bdhisattva was of age to marry, the father of Gopa, his intended bride, demanded an examination of the five hundred suitors, the subjects including arithmetic, writing, the lute, and archery. Having vanquished his rivals in all else, he is matched against Arjuna the great arithmetician and is asked to express numbers greater than 100 kotis. In reply he gave a scheme of number names as high as 1053, adding that he could proceed as far as 10421, all of which suggests the system of Archimedes and the unsettled question of the indebtedness of the West to the East in the realm of ancient mathematics. Sir Edwin Arnold, in The Light of Asia, does not mention this part of the contest, but he speaks of Buddha's training at the hands of the learned Visvamitra:
"And Viswamitra said, 'It is enough,
Let us to numbers. After me repeat
Your numeration till we reach the lakh,
One, two, three, four, to ten, and then by tens
To hundreds, thousands.' After him the child
Named digits, decads, centuries, nor paused,
The round lakh reached, but softly murmured on,
Then comes the koti, nahut, ninnahut,
Khamba, viskhamba, abab, attata,
To kumuds, gundhikas, and utpalas,
By pundarikas into padumas,
Which last is how you count the utmost grains
Of Hastagiri ground to finest dust;
But beyond that a numeration is,
The Katha, used to count the stars of night, The Koti-Katha, for the ocean drops;
Ingga, the calculus of circulars;
Sarvanikchepa, by the which you deal
With all the sands of Gunga, till we come
To Antah-Kalpas, where the unit is
The sands of the ten crore Gungas. If one seeks
More comprehensive scale, th' arithmic mounts
By the Asankya, which is the tale
Of all the drops that in ten thousand years
Would fall on all the worlds by daily rain;
Thence unto Maha Kalpas, by the which
The gods compute their future and their past.'"
Excerpted from The Hinduâ?"Arabic Numerals by DAVID EUGENE SMITH, Louis Charles Karpinski. Copyright © 2004 Dover Publications, Inc.. Excerpted by permission of Dover Publications, Inc..
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Table of Contents
PRONUNCIATION OF ORIENTAL NAMES,
CHAPTER I - EARLY IDEAS OF THEIR ORIGIN,
CHAPTER II - EARLY HINDU FORMS WITH NO PLACE VALUE,
CHAPTER III - LATER HINDU FORMS, WITH A PLACE VALUE,
CHAPTER IV - THE SYMBOL ZERO,
CHAPTER V - THE QUESTION OF THE INTRODUCTION OF THE NUMERALS INTO EUROPE BY BOETHIUS,
CHAPTER VI - THE DEVELOPMENT OF THE NUMERALS AMONG THE ARABS,
CHAPTER VII - THE DEFINITE INTRODUCTION OF THE NUMERALS INTO EUROPE,
CHAPTER VIII - THE SPREAD OF THE NUMERALS IN EUROPE,
DOVER PHOENIX EDITIONS,