The Hindu-Arabic Numerals.

by David Eugene Smith and Louis Charles Karpinski.

PREFACE

So familiar are we with the numerals that bear the misleading name of Arabic, and so extensive is their use in Europe and the Americas, that it is difficult for us to realize that their general acceptance in the transactions of commerce is a matter of only the last four centuries, and that they are unknown to a very large part of the human race to-day. 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. Such, however, is the case, and there is probably no one who has not at least some slight pa.s.sing interest in the story of this struggle. To the mathematician and the student of civilization the interest is generally a deep one; to the teacher of the elements of knowledge the interest may be less marked, but nevertheless it is real; and even the business man who makes daily use of the curious symbols by which we express the numbers of commerce, cannot fail to have some appreciation for the story of the rise and progress of these tools of his trade.

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 {iv} numerals. In this little work we have attempted to state the history of these forms in small compa.s.s, 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 in different parts of the world. We have had no theory to exploit, for the history of mathematics has seen too much of this tendency already, but as far as possible we have weighed the testimony and have set forth what seem to be the reasonable conclusions from the evidence at hand.

To facilitate the work of students an index has been prepared which we hope may be serviceable. In this the names of authors appear only when some use has been made of their opinions or when their works are first mentioned in full in a footnote.

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 with numbers, the authors will feel that the considerable labor involved in its preparation has not been in vain.

We desire to acknowledge our especial indebtedness to Professor Alexander Ziwet for reading all the proof, as well as 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 the scheme of p.r.o.nunciation of Oriental names, and also our indebtedness to other scholars in Oriental learning for information.

DAVID EUGENE SMITH

LOUIS CHARLES KARPINSKI

THE HINDU-ARABIC NUMERALS

CHAPTER 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, 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 c.u.mbersome and little used, {2} 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 a.s.signed to the eastern sh.o.r.es 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.[1]

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.[2] {3} 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 {4} among them such influential writers as Tartaglia[4] in Italy and Kobel[5] in Germany, a.s.serted 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 cla.s.sics of both the East and the West, preserving them and finally pa.s.sing them on to awakening Europe. This man was Mo[h.]ammed the Son of Moses, from Khow[=a]rezm, or, more after the manner of the Arab, Mo[h.]ammed ibn M[=u]s[=a] al-Khow[=a]razm[=i],[8] a man of great {5} 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 a.s.serted by later Arab {6} writers, even to the present day.[13] Indeed the phrase _'ilm hind[=i]_, "Indian science," is used by them for arithmetic, as also the adjective _hind[=i]_ alone.[14]

Probably the most striking testimony from Arabic sources is that given by the Arabic traveler and scholar Mohammed ibn A[h.]med, Ab[=u]

'l-R[=i][h.][=a]n al-B[=i]r[=u]n[=i] (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-B[=i]r[=u]n[=i] 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 {7} 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[.n]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 s.e.xagesimal 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-B[=i]r[=u]n[=i] there was another Arabic writer of the tenth century, Mo[t.]ahhar ibn [T.][=a]hir,[20] author of the _Book of the Creation and of History_, who gave as a curiosity, in Indian (N[=a]gar[=i]) symbols, a large number a.s.serted by the people of India to represent the duration of the world. Huart feels positive that in Mo[t.]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'[=u]d[=i]

(885?-956), whose journeys carried him from Bagdad to Persia, India, Ceylon, and even {8} 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 ent.i.tled _Meadows of Gold_ furnishes a most entertaining fund of information. He states[22] that the wise men of India, a.s.sembled by the king, composed the _Sindhind_. Further on[23] he states, upon the authority of the historian Mo[h.]ammed ibn 'Al[=i] 'Abd[=i], that by order of Al-Man[s.][=u]r many works of science and astrology were translated into Arabic, notably the _Sindhind_ (_Siddh[=a]nta_). Concerning the meaning and spelling of this name there is considerable diversity of opinion.

Colebrooke[24] first pointed out the connection between _Siddh[=a]nta_ and _Sindhind_. He ascribes to the word the meaning "the revolving ages."[25]

Similar designations are collected by Sedillot,[26] who inclined to the Greek origin of the sciences commonly attributed to the Hindus.[27]

Casiri,[28] citing the _T[=a]r[=i]kh al-[h.]okam[=a]_ or _Chronicles of the Learned_,[29] refers to the work {9} as the _Sindum-Indum_ with the meaning "perpetuum aeternumque." The reference[30] in this ancient Arabic work to Al-Khow[=a]razm[=i] is worthy of note.

This _Sindhind_ is the book, says Mas'[=u]d[=i],[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-Khow[=a]razm[=i] and [H.]abash[33]

as translators of the tables of the _Sindhind_. Al-B[=i]r[=u]n[=i][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[s.][=u]r, 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 _Kit[=a]b al-Fihrist_, written in the year 987 A.D., by Ibn Ab[=i] Ya'q[=u]b al-Nad[=i]m. 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]

{10}

The first of the Arabic writers mentioned is Al-Kind[=i] (800-870 A.D.), who wrote five books on arithmetic and four books on the use of the Indian method of reckoning. Sened ibn 'Al[=i], the Jew, who was converted to Islam under the caliph Al-M[=a]m[=u]n, 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 'Al[=i] are really works of Al-Khow[=a]razm[=i], 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-[S.][=u]f[=i], 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 doc.u.ments on arithmetic in our own language. One of the earliest {11} 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[um] 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."

{12}

CHAPTER 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[42] or Chinese[43] 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[44]--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.[45] Following this work, or possibly contemporary with it, is the Brahmanic literature, which is partly ritualistic (the Br[=a]hma[n.]as), and partly philosophical (the Upanishads). Our especial interest is {13} in the S[=u]tras, 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[46]

places the whole of the Veda literature, including the Vedas, the Br[=a]hma[n.]as, and the S[=u]tras, between 1500 B.C. and 800 B.C., thus agreeing with Burk[47] who holds that the knowledge of the Pythagorean theorem revealed in the S[=u]tras goes back to the eighth century B.C.

The importance of the S[=u]tras as showing an independent origin of Hindu geometry, contrary to the opinion long held by Cantor[48] of a Greek origin, has been repeatedly emphasized in recent literature,[49] especially since the appearance of the important work of Von Schroeder.[50] 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 {14} origin of our numerals, yet it is highly pertinent as showing the apt.i.tude 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,[51] 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.[52] 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 {15} existed in earliest times, and of the seventy-two recognized sciences writing and arithmetic were the most prized.[53] 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.[54]

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 B[=o]dhisattva[55] 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.[56]

In reply he gave a scheme of number names as high as 10^{53}, adding that he could proceed as far as 10^{421},[57] 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.[58] Sir Edwin Arnold, {16} 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 Vi[s.]vamitra:

"And Viswamitra said, 'It is enough, Let us to numbers. After me repeat Your numeration till we reach the lakh,[59]

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 k[=o]ti, nahut, ninnahut, Khamba, viskhamba, abab, attata, To k.u.muds, gundhikas, and utpalas, By pundar[=i]kas into padumas, Which last is how you count the utmost grains Of Hastagiri ground to finest dust;[60]