The Hindu-Arabic Numerals
The Hindu-Arabic Numerals by David Eugene Smith and Louis Charles Karpinski was published in 1911 to, as mentioned in the preface of the book, "bring together the fragmentary narrations and to set forth the general problem of the origin and development of these numerals."
- So familiar are we with the numerals that bear the misleading name of Arabic, and so extensive is their use... that it is difficult... to realize that their general acceptance... is a matter of only the last four centuries.
- It seems strange that such a labor-saving device should have struggled for nearly a thousand years after its system of place value was perfected before it replaced such crude notations as the one that the Roman conqueror made substantially universal in Europe.
- This story has often been told in part, but it is a long time since any effort has been made to bring together the fragmentary narrations and to set forth the general problem of the origin and development of these numerals.
- In this little work we have attempted to state the history of these forms in small compass, to place before the student materials for the investigation of the problems involved, and to express as clearly as possible the results, of the labors of scholars who have studied the subject...
- We have had no theory to exploit, for the history of mathematics has seen too much of this tendency... we have weighed the testimony and have set forth what seem to be the reasonable conclusions from the evidence...
- If this work shall show more clearly the value of our number system, and shall make the study of mathematics seem more real to the teacher and student, and shall offer material for interesting some pupil more fully in his work... the considerable labor involved in its preparation has not been in vain.
- We... acknowledge our especial indebtedness to Professor Alexander Ziwet for reading all the proof, [and] for the digest of a Russian work, to Professor Clarence L. Meader for Sanskrit transliterations, and to Mr. Steven T. Byington for Arabic transliterations... and... to other scholars in Oriental learning...
I. 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... had three systems of writing, with a numerical notation for each; the Greeks had two... sets of numerals, and the Roman symbols... changed... from century to century.
- It will be well... to think of the numerals... we... 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.
- 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 Phoenicians.
- The idea that our common numerals are Arabic in origin is not an old one. The mediaeval 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... [also made by] 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."
- Footnote: The Grouncd of Artes (1558) fol. C, 5.
- 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."
- [T]he Arabs... 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 Mūsā al-Khowārazmī, a man of great learning and one to whom the world is much indebted for its present knowledge of algebra and of arithmetic. ...[I]n 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 hindī, "Indian science," is used by them for arithmetic, as [is] also the adjective hindī alone.
- Probably the most striking testimony from Arabic sources is that given by the Arabic traveler and scholar Mohammed ibn Ahmed, Abū ’l-Rīahān al-Bīrūnī (973-1048), who spent many years in Hindustan. He wrote... the “Book of the Ciphers,” unfortunately lost, which treated... of the Hindu art of calculating... being versed in Arabic, Persian, Sanskrit, Hebrew, and Syriac, as well as in astronomy, chronology, and mathematics. In his work on India he... 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 aṅka 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... in three different systems...
- Preceding Al-Bīrūnī... another Arabic writer of the tenth century, Motahhar ibn Tāhir, author of the Book of the Creation and of History... gave... in Indian (Nāgarī symbols), a large number asserted by the people of India to represent the duration of the world.
- Al-Mas'ūdī (885?-956), whose journeys carried him from Bagdad to Persia, India, Ceylon, and... across the China sea, and at other times to Madagascar, Syria, and Palestine... neglected no accessible sources of information, examining also the history of the Persians, the Hindus, and the Romans. ...[H]is ...Meadows of Gold ...states that the wise men of India, assembled by the king, composed the Sindhind...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... Colebrooke ascribes... the meaning "the revolving ages." Similar designations are collected by Sedillot... Casiri... refers to the work as the Sindum-Indum... meaning "perpetuum aeternumque [eternal perpetuity]."
- This Sindhind is the book, says Mas'ūdī, which gives all that the Hindus know of the spheres, the stars, arithmetic, and the other branches of science. He mentions... Al-Khowārazmī and Habash as translators of the tables of the Sindhind.
- The oldest work... complete, on the history of Arabic literature and history is the Kitah al-Fihrist, written in the year 987 a.d., by Ibn Abī Ya'qūb al-Nadīm. ...Of the ten chief divisions of the work, the [second subdivision of the] seventh... treats of mathematicians and astronomers.
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... is also given as the author of a work on the Hindu method of reckoning. ...[T]here is a possibility that some of the works ascribed to Sened ibn 'Alī are really works of Al-Khowārazmī ...However, ...Casiri also mentions the same writer as the author of a most celebrated work on arithmetic.
To Al-Sūfī... is also credited a large work... and similar treatises by other writers...
- [T]herefore... the Arabs from the early ninth century on fully recognized the Hindu origin of the new numerals.
- Leonard of Pisa... wrote his Liber Abbaci in 1202. ...[H]e 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.
- One of the earliest treatises on algorism is... the Carmen de Algorismo [Poem about Arithmetic], 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:
"Hee 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 [arithmetic], 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."
II. 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... to... present day the Hindu has been wont to put his thought into rhythmic form. The first of this poetry... 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... from approximately 2000 B.C. to 1400 B.C. Following this work, or possibly contemporary... is the Brahmanic literature, which is partly ritualistic (the Brahmanas), and partly philosophical (the Upanishads). Our... interest is in the Sutras... which contain... geometric material used in connection with altar construction, and also numerous examples of... "Pythagorean numbers," although this was long before Pythagoras lived.
- Whitney places the whole of the Veda literature, including the Vedas, the Brahmanas, and the Sutras, between 1500 B.C. and 800 B.C., thus agreeing with [Albert] Bürk who holds that the knowledge of the Pythagorean theorem revealed in the Sütras goes back to the eighth century B.C.
- Footnotes: 1) W. D. Whitney, Sanskrit Grammar, 3d ed., Leipzig, 1806. 2) “ Das Āpastamba-Śulba-Sūtra,” Zeitschrift der deutschen Morgenländischen Gesellschaft, Vol. LV, p. 543, and Vol. LVI, p. 327. Additional Note: Bürk, Albert (1901). "Das Āpastamba-Śulba-Sūtra, herausgegeben, übersetzt und mit einer Einleitung versehen". Zeitschrift der Deutschen Morgenländischen Gesellschaft (in German). 55: 543–591.
- 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... 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.
- [W]e are not at all sure that the most ancient forms of the numerals commonly known as Arabic had their origin in India. ...[T]heir 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... 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... we consider the rise of the numerals in the land of the Sindhu... only the large movement... is meant, and that there must... be... numerous possible sources outside of India... and long anterior to the first prominent appearance of the number symbols.
- [I]n the history of ancient India... primary schools... existed in earliest times, and of the seventy-two recognized sciences writing and arithmetic were the most prized. In the Vedic period [~]2000 to 1400 B.C., there was the same attention to astronomy that was found in the earlier civilizations of Babylon, China, and Egypt... Such advance... presupposes a fair knowledge of calculation, but of the manner of calculating we are quite ignorant...
- [T]he Lalitavistara, relates that when the Bödhisattva 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... 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.
- Footnote: 100 kotis=100*107
- Sir Edwin Arnold, in The Light of Asia... 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 Kātha, used to count the stars of night,
The Kōti-Kātha, 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 thé which the gods compute their future and their past.'"
- Thereupon Visvamitra Ācārya expresses his approval... and asks to hear the "measure of the line" as far as yōjana, the longest measure bearing name. This given, Buddha adds:
..." 'And master! If it please,
I shall recite how many sun-motes
lie From end to end within a yōjana.’
Thereat, with instant skill, the little prince
Pronounced the total of the atoms true.
But Viswamitra heard it on his face
Prostrate before the boy; 'For thou,' he cried,
'Art Teacher of thy teachers—thou, not I,
Art Guru.' "
- [T]his is far from being history. And yet it puts in charming rhythm only what the ancient Lalitavistara relates of the number-series of the Buddha’s time. ...[I]t reveals a condition that would have been impossible unless arithmetic had attained a considerable degree of advancement.
V. The Question of the Introduction of the Numerals into Europe by Boethius
- [W]e are uncertain as to the time and place of [the] introduction [of these numeral forms] into Europe. There are two general theories... 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... The second, advanced by Woepcke, is 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.
- 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... 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... the Oriental led to its use... 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.
- 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...
- Footnote: Nicolaus Bubnov, The Origin and History of Our Numerals (in Russian), Kiev, 1908, & The Independence of European Arithmetic (in Russian), Kiev.
- The Spanish forms of the numerals were called the hurūf al-ģobār, the ģobār or dust numerals, as distinguished from the hurūf aljumal or alphabetic numerals. Probably the latter... were also used by the Arabs. ...[D]oubtless ...these numerals were written on the dust abacus, this plan being distinct from the counter method ...Al-Bīrūnī states that the Hindus often performed numerical computations in the sand. ...The system has nine characters, but no zero. A dot above a character indicates tens, two dots hundreds, and so on, meaning 50, and [ meaning 500].
- When we consider... that the dot is found for zero in the Bakhsālī manuscript, and that it was used in subscript form in the Kitāb al-Fihrist in the tenth century... we are forced to believe that this form may also have been of Hindu origin.
- The Indian use of subscript dots to indicate the tens, hundreds, thousands, etc., is established by a passage in the Kitāb al-Fihrist (987 A.D.)... The numeral forms given are those which have usually been called Indian, in opposition to ģobār. In this document the dots are placed below the characters, instead of being superposed... The significance was the same.
- Anicius Manlius Severinus Boethius was born at Rome c. 475. 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. He was accordingly looked upon as a saint, his bones were enshrined, and as a natural consequence his books were among the classics in the church schools for a thousand years. 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.
- 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.
- [I]t is one of the mistakes of scholars to believe that they are the sole transmitters of knowledge. ...[T]he 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... trade routes from the Orient to the Occident. [I]t was to the tradesman and the wandering scholar that the spread of such learning was due, rather than to the school man.
- Avicenna (980-1037 a.d.)... relates that when his people were living at Bokhara 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.
- 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. The products of the East were always finding their way to the West, the Greeks getting their ginger from Malabar, as the Phoenicians had long before brought gold from Malacca.
- 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, the game of finger-guessing, the water clock, the music system, the use of the myriad, the calendars, and in many other ways.
- 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. There is also a noteworthy resemblance between certain Greek and Chinese words, showing that in remote times there must have been more or less interchange of thought.
- 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." 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.
- Augustus speaks of envoys received by him from India... and it is not improbable that he also received an embassy from China. ...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, while by the time of Constantine Europe was in direct communication with the Far East.
- In the fifth century the Persian medical school at Jondi-Sapur admitted both the Hindu and the Greek doctrines...
- [N]ot 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.
VI. The Development of the Numerals Among the Arabs
- Mohammedanism was to the world from the eighth to the thirteenth century what Rome and Athens and the Italo-Hellenic influence generally had been to the ancient civilization. ...The Arab empire was an ellipse of learning with its foci at Bagdad and Cordova, and its rulers not infrequently took pride in demanding intellectual rather than commercial treasure as the result of conquest.
- [T]he Hindu numerals found their way to the North... in the eighth century they were taken to Bagdad. It was early in that century that the Mohammedans obtained their first foothold in northern India, thus foreshadowing an epoch of supremacy that endured with varied fortunes until after the golden age of Akbar the Great (1542-1605) and Shah Jehan. They also conquered Khorassan and Afghanistan, so that the learning and the commercial customs of India at once found easy access to the newly-established schools and the bazaars of Mesopotamia and western Asia.
- It was just after the Sindhind was brought to Bagdad that Muhammad ibn Mūsā al-Khwārizmī... was called to that city. ...Appreciating at once the value of the position system so recently brought from India, he wrote an arithmetic based upon these numerals, and this was translated into Latin...
- Contemporary with Al-Khowarazmi... Abū 'l-Teiyib, Sened ibn Allī... also wrote a work on Hindu arithmetic...[T]he struggle to have the Hindu numerals replace the Arabic did not cease for a long time thereafter.
- We thus have the numerals in Arabia, in two forms: one the form now used there, and the other the one used by Al-Khowarazmi. The question then remains, how did this second form find its way into Europe?
VII. The Definite Introduction of the Numerals into Europe
- [T]he probability [is] that it was the trader rather than the scholar... [who] carried these numerals from their original habitat to various commercial centers... we shall never know when they first made their inconspicuous entrance into Europe.
- The power of the Goths, who had held Spain for three centuries, was shattered at the battle of Jerez de la Frontera in 711, and almost immediately the Moors became masters of Spain and so remained for five hundred years, and masters of Granada for a much longer period. Until 850 the Christians were... free as to religion and... holding political office, so that priests and monks were not infrequently skilled... in Latin and Arabic, acting as official translators... [W]hile it lasted the learning and the customs of the East must have be come more or less the property of Christian Spain. At thie time the ġobār numerals were probably in that country, and these may well have made their way into Europe from the schools of Cordova, Granada, and Toledo.
Quotes about The Hindu-Arabic Numerals
- This book gives in compact form a readable and carefully prepared account of the numerous researches... made in the endeavor to trace the origin and development of the Hindu-Arabic numerals. Teachers of mathematics will welcome it, while students specializing in the history of mathematics will derive great help... Like the arithmetician Tonstall the authors read everything in every language and spent much time in licking what they found into shape ad ursi exemplum, as the bear does her cubs.
- A History of Mathematics by Florian Cajori
- A Short Account of the History of Mathematics by W. W. Rouse Ball.
- Indian mathematics