Produced by David Newman, Chuck Greif, Keith Edkins and the Online Distributed Proofreading Team at https://www.pgdp.net (This file was produced from images from the Cornell University Library: Historical Mathematics Monographs collection.) Transcriber's Note: The following codes are used for characters that are not present in the character set used for this version of the book. [=a] a with macron (etc.) [.g] g with dot above (etc.) ['s] s with acute accent [d.] d with dot below (etc.) [d=] d with line below [H)] H with breve below THE HINDU-ARABIC NUMERALS BY DAVID EUGENE SMITH AND LOUIS CHARLES KARPINSKI BOSTON AND LONDON GINN AND COMPANY, PUBLISHERS 1911 COPYRIGHT, 1911, BY DAVID EUGENE SMITH AND LOUIS CHARLES KARPINSKI ALL RIGHTS RESERVED 811.7 THE ATHENÆUM PRESS GINN AND COMPANY · PROPRIETORS BOSTON · U.S.A. * * * * * {iii} 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 passing 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 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 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 pronunciation of Oriental names, and also our indebtedness to other scholars in Oriental learning for information. DAVID EUGENE SMITH LOUIS CHARLES KARPINSKI * * * * * {v} CONTENTS CHAPTER PRONUNCIATION OF ORIENTAL NAMES vi I. EARLY IDEAS OF THEIR ORIGIN 1 II. EARLY HINDU FORMS WITH NO PLACE VALUE 12 III. LATER HINDU FORMS, WITH A PLACE VALUE 38 IV. THE SYMBOL ZERO 51 V. THE QUESTION OF THE INTRODUCTION OF THE NUMERALS INTO EUROPE BY BOETHIUS 63 VI. THE DEVELOPMENT OF THE NUMERALS AMONG THE ARABS 91 VII. THE DEFINITE INTRODUCTION OF THE NUMERALS INTO EUROPE 99 VIII. THE SPREAD OF THE NUMERALS IN EUROPE 128 INDEX 153 * * * * * {vi} PRONUNCIATION OF ORIENTAL NAMES (S) = in Sanskrit names and words; (A) = in Arabic names and words. B, D, F, G, H, J, L, M, N, P, SH (A), T, TH (A), V, W, X, Z, as in English. A, (S) like _u_ in _but_: thus _pandit_, pronounced _pundit_. (A) like _a_ in _ask_ or in _man_. [=A], as in _father_. C, (S) like _ch_ in _church_ (Italian _c_ in _cento_). [D.], [N.], [S.], [T.], (S) _d_, _n_, _sh_, _t_, made with the tip of the tongue turned up and back into the dome of the palate. [D.], [S.], [T.], [Z.], (A) _d_, _s_, _t_, _z_, made with the tongue spread so that the sounds are produced largely against the side teeth. Europeans commonly pronounce [D.], [N.], [S.], [T.], [Z.], both (S) and (A), as simple _d_, _n_, _sh_ (S) or _s_ (A), _t_, _z_. [D=] (A), like _th_ in _this_. E, (S) as in _they_. (A) as in _bed_. [.G], (A) a voiced consonant formed below the vocal cords; its sound is compared by some to a _g_, by others to a guttural _r_; in Arabic words adopted into English it is represented by _gh_ (e.g. _ghoul_), less often _r_ (e.g. _razzia_). H preceded by _b_, _c_, _t_, _[t.]_, etc. does not form a single sound with these letters, but is a more or less distinct _h_ sound following them; cf. the sounds in _abhor, boathook_, etc., or, more accurately for (S), the "bhoys" etc. of Irish brogue. H (A) retains its consonant sound at the end of a word. [H.], (A) an unvoiced consonant formed below the vocal cords; its sound is sometimes compared to German hard _ch_, and may be represented by an _h_ as strong as possible. In Arabic words adopted into English it is represented by _h_, e.g. in _sahib_, _hakeem_. [H.] (S) is final consonant _h_, like final _h_ (A). I, as in _pin_. [=I], as in _pique_. K, as in _kick_. KH, (A) the hard _ch_ of Scotch _loch_, German _ach_, especially of German as pronounced by the Swiss. [.M], [.N], (S) like French final _m_ or _n_, nasalizing the preceding vowel. [N.], see [D.]. Ñ, like _ng_ in _singing_. O, (S) as in _so_. (A) as in _obey_. Q, (A) like _k_ (or _c_) in _cook_; further back in the mouth than in _kick_. R, (S) English _r_, smooth and untrilled. (A) stronger. [R.], (S) r used as vowel, as in _apron_ when pronounced _aprn_ and not _apern_; modern Hindus say _ri_, hence our _amrita_, _Krishna_, for _a-m[r.]ta, K[r.][s.][n.]a_. S, as in _same_. [S.], see [D.]. ['S], (S) English _sh_ (German _sch_). [T.], see [D.]. U, as in _put_. [=U], as in _rule_. Y, as in _you_. [Z.], see [D.]. `, (A) a sound kindred to the spiritus lenis (that is, to our ears, the mere distinct separation of a vowel from the preceding sound, as at the beginning of a word in German) and to _[h.]_. The ` is a very distinct sound in Arabic, but is more nearly represented by the spiritus lenis than by any sound that we can produce without much special training. That is, it should be treated as silent, but the sounds that precede and follow it should not run together. In Arabic words adopted into English it is treated as silent, e.g. in _Arab_, _amber_, _Caaba_ (_`Arab_, _`anbar_, _ka`abah_). (A) A final long vowel is shortened before _al_ (_'l_) or _ibn_ (whose _i_ is then silent). Accent: (S) as if Latin; in determining the place of the accent _[.m]_ and _[.n]_ count as consonants, but _h_ after another consonant does not. (A), on the last syllable that contains a long vowel or a vowel followed by two consonants, except that a final long vowel is not ordinarily accented; if there is no long vowel nor two consecutive consonants, the accent falls on the first syllable. The words _al_ and _ibn_ are never accented. * * * * * {1} 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 cumbersome 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 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.[1] 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.[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 Köbel[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[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 asserted 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 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-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 asserted 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 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[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 Sédillot,[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 æternumque." 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 documents 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 Bürk[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 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,[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 kumuds, 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] But beyond that a numeration is, The K[=a]tha, used to count the stars of night, The K[=o]ti-K[=a]tha, 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.'" {17} Thereupon Vi[s.]vamitra [=A]c[=a]rya[61] expresses his approval of the task, and asks to hear the "measure of the line" as far as y[=o]jana, 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 y[=o]jana.' 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 G[=u]r[=u].'" It is needless to say that this 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. While it extends beyond all reason, nevertheless it reveals a condition that would have been impossible unless arithmetic had attained a considerable degree of advancement. To this pre-Christian period belong also the _Ved[=a][.n]gas_, or "limbs for supporting the Veda," part of that great branch of Hindu literature known as _Sm[r.]iti_ (recollection), that which was to be handed down by tradition. Of these the sixth is known as _Jyoti[s.]a_ (astronomy), a short treatise of only thirty-six verses, written not earlier than 300 B.C., and affording us some knowledge of the extent of number work in that period.[62] The Hindus {18} also speak of eighteen ancient Siddh[=a]ntas or astronomical works, which, though mostly lost, confirm this evidence.[63] As to authentic histories, however, there exist in India none relating to the period before the Mohammedan era (622 A.D.). About all that we know of the earlier civilization is what we glean from the two great epics, the Mah[=a]bh[=a]rata[64] and the R[=a]m[=a]yana, from coins, and from a few inscriptions.[65] It is with this unsatisfactory material, then, that we have to deal in searching for the early history of the Hindu-Arabic numerals, and the fact that many unsolved problems exist and will continue to exist is no longer strange when we consider the conditions. It is rather surprising that so much has been discovered within a century, than that we are so uncertain as to origins and dates and the early spread of the system. The probability being that writing was not introduced into India before the close of the fourth century B.C., and literature existing only in spoken form prior to that period,[66] the number work was doubtless that of all primitive peoples, palpable, merely a matter of placing sticks or cowries or pebbles on the ground, of marking a sand-covered board, or of cutting notches or tying cords as is still done in parts of Southern India to-day.[67] {19} The early Hindu numerals[68] may be classified into three great groups, (1) the Kharo[s.][t.]h[=i], (2) the Br[=a]hm[=i], and (3) the word and letter forms; and these will be considered in order. The Kharo[s.][t.]h[=i] numerals are found in inscriptions formerly known as Bactrian, Indo-Bactrian, and Aryan, and appearing in ancient Gandh[=a]ra, now eastern Afghanistan and northern Punjab. The alphabet of the language is found in inscriptions dating from the fourth century B.C. to the third century A.D., and from the fact that the words are written from right to left it is assumed to be of Semitic origin. No numerals, however, have been found in the earliest of these inscriptions, number-names probably having been written out in words as was the custom with many ancient peoples. Not until the time of the powerful King A['s]oka, in the third century B.C., do numerals appear in any inscriptions thus far discovered; and then only in the primitive form of marks, quite as they would be found in Egypt, Greece, Rome, or in {20} various other parts of the world. These A['s]oka[69] inscriptions, some thirty in all, are found in widely separated parts of India, often on columns, and are in the various vernaculars that were familiar to the people. Two are in the Kharo[s.][t.]h[=i] characters, and the rest in some form of Br[=a]hm[=i]. In the Kharo[s.][t.]h[=i] inscriptions only four numerals have been found, and these are merely vertical marks for one, two, four, and five, thus: | || ||| |||| In the so-called ['S]aka inscriptions, possibly of the first century B.C., more numerals are found, and in more highly developed form, the right-to-left system appearing, together with evidences of three different scales of counting,--four, ten, and twenty. The numerals of this period are as follows: [Illustration] There are several noteworthy points to be observed in studying this system. In the first place, it is probably not as early as that shown in the N[=a]n[=a] Gh[=a]t forms hereafter given, although the inscriptions themselves at N[=a]n[=a] Gh[=a]t are later than those of the A['s]oka period. The {21} four is to this system what the X was to the Roman, probably a canceling of three marks as a workman does to-day for five, or a laying of one stick across three others. The ten has never been satisfactorily explained. It is similar to the A of the Kharo[s.][t.]h[=i] alphabet, but we have no knowledge as to why it was chosen. The twenty is evidently a ligature of two tens, and this in turn suggested a kind of radix, so that ninety was probably written in a way reminding one of the quatre-vingt-dix of the French. The hundred is unexplained, although it resembles the letter _ta_ or _tra_ of the Br[=a]hm[=i] alphabet with 1 before (to the right of) it. The two hundred is only a variant of the symbol for hundred, with two vertical marks.[70] This system has many points of similarity with the Nabatean numerals[71] in use in the first centuries of the Christian era. The cross is here used for four, and the Kharo[s.][t.]h[=i] form is employed for twenty. In addition to this there is a trace of an analogous use of a scale of twenty. While the symbol for 100 is quite different, the method of forming the other hundreds is the same. The correspondence seems to be too marked to be wholly accidental. It is not in the Kharo[s.][t.]h[=i] numerals, therefore, that we can hope to find the origin of those used by us, and we turn to the second of the Indian types, the Br[=a]hm[=i] characters. The alphabet attributed to Brahm[=a] is the oldest of the several known in India, and was used from the earliest historic times. There are various theories of its origin, {22} none of which has as yet any wide acceptance,[72] although the problem offers hope of solution in due time. The numerals are not as old as the alphabet, or at least they have not as yet been found in inscriptions earlier than those in which the edicts of A['s]oka appear, some of these having been incised in Br[=a]hm[=i] as well as Kharo[s.][t.]h[=i]. As already stated, the older writers probably wrote the numbers in words, as seems to have been the case in the earliest Pali writings of Ceylon.[73] The following numerals are, as far as known, the only ones to appear in the A['s]oka edicts:[74] [Illustration] These fragments from the third century B.C., crude and unsatisfactory as they are, are the undoubted early forms from which our present system developed. They next appear in the second century B.C. in some inscriptions in the cave on the top of the N[=a]n[=a] Gh[=a]t hill, about seventy-five miles from Poona in central India. These inscriptions may be memorials of the early Andhra dynasty of southern India, but their chief interest lies in the numerals which they contain. The cave was made as a resting-place for travelers ascending the hill, which lies on the road from Kaly[=a]na to Junar. It seems to have been cut out by a descendant {23} of King ['S][=a]tav[=a]hana,[75] for inside the wall opposite the entrance are representations of the members of his family, much defaced, but with the names still legible. It would seem that the excavation was made by order of a king named Vedisiri, and "the inscription contains a list of gifts made on the occasion of the performance of several _yagnas_ or religious sacrifices," and numerals are to be seen in no less than thirty places.[76] There is considerable dispute as to what numerals are really found in these inscriptions, owing to the difficulty of deciphering them; but the following, which have been copied from a rubbing, are probably number forms:[77] [Illustration] The inscription itself, so important as containing the earliest considerable Hindu numeral system connected with our own, is of sufficient interest to warrant reproducing part of it in facsimile, as is done on page 24. {24} [Illustration] The next very noteworthy evidence of the numerals, and this quite complete as will be seen, is found in certain other cave inscriptions dating back to the first or second century A.D. In these, the Nasik[78] cave inscriptions, the forms are as follows: [Illustration] From this time on, until the decimal system finally adopted the first nine characters and replaced the rest of the Br[=a]hm[=i] notation by adding the zero, the progress of these forms is well marked. It is therefore well to present synoptically the best-known specimens that have come down to us, and this is done in the table on page 25.[79] {25} TABLE SHOWING THE PROGRESS OF NUMBER FORMS IN INDIA NUMERALS 1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 1000 A['s]oka[80] [Illustration] ['S]aka[81] [Illustration] A['s]oka[82] [Illustration] N[=a]gar[=i][83] [Illustration] Nasik[84] [Illustration] K[s.]atrapa[85] [Illustration] Ku[s.]ana [86] [Illustration] Gupta[87] [Illustration] Valhab[=i][88] [Illustration] Nepal [89] [Illustration] Kali[.n]ga[90] [Illustration] V[=a]k[=a][t.]aka[91] [Illustration] [Most of these numerals are given by Bühler, loc. cit., Tafel IX.] {26} With respect to these numerals it should first be noted that no zero appears in the table, and as a matter of fact none existed in any of the cases cited. It was therefore impossible to have any place value, and the numbers like twenty, thirty, and other multiples of ten, one hundred, and so on, required separate symbols except where they were written out in words. The ancient Hindus had no less than twenty of these symbols,[92] a number that was afterward greatly increased. The following are examples of their method of indicating certain numbers between one hundred and one thousand: [93] [Numerals] for 174 [94] [Numerals] for 191 [95] [Numerals] for 269 [96] [Numerals] for 252 [97] [Numerals] for 400 [98] [Numerals] for 356 {27} To these may be added the following numerals below one hundred, similar to those in the table: [Numerals][99] for 90 [Numerals][100] for 70 We have thus far spoken of the Kharo[s.][t.]h[=i] and Br[=a]hm[=i] numerals, and it remains to mention the third type, the word and letter forms. These are, however, so closely connected with the perfecting of the system by the invention of the zero that they are more appropriately considered in the next chapter, particularly as they have little relation to the problem of the origin of the forms known as the Arabic. Having now examined types of the early forms it is appropriate to turn our attention to the question of their origin. As to the first three there is no question. The [1 vertical stroke] or [1 horizontal stroke] is simply one stroke, or one stick laid down by the computer. The [2 vertical strokes] or [2 horizontal strokes] represents two strokes or two sticks, and so for the [3 vertical strokes] and [3 horizontal strokes]. From some primitive [2 vertical strokes] came the two of Egypt, of Rome, of early Greece, and of various other civilizations. It appears in the three Egyptian numeral systems in the following forms: Hieroglyphic [2 vertical strokes] Hieratic [Hieratic 2] Demotic [Demotic 2] The last of these is merely a cursive form as in the Arabic [Arabic 2], which becomes our 2 if tipped through a right angle. From some primitive [2 horizontal strokes] came the Chinese {28} symbol, which is practically identical with the symbols found commonly in India from 150 B.C. to 700 A.D. In the cursive form it becomes [2 horizontal strokes joined], and this was frequently used for two in Germany until the 18th century. It finally went into the modern form 2, and the [3 horizontal strokes] in the same way became our 3. There is, however, considerable ground for interesting speculation with respect to these first three numerals. The earliest Hindu forms were perpendicular. In the N[=a]n[=a] Gh[=a]t inscriptions they are vertical. But long before either the A['s]oka or the N[=a]n[=a] Gh[=a]t inscriptions the Chinese were using the horizontal forms for the first three numerals, but a vertical arrangement for four.[101] Now where did China get these forms? Surely not from India, for she had them, as her monuments and literature[102] show, long before the Hindus knew them. The tradition is that China brought her civilization around the north of Tibet, from Mongolia, the primitive habitat being Mesopotamia, or possibly the oases of Turkestan. Now what numerals did Mesopotamia use? The Babylonian system, simple in its general principles but very complicated in many of its details, is now well known.[103] In particular, one, two, and three were represented by vertical arrow-heads. Why, then, did the Chinese write {29} theirs horizontally? The problem now takes a new interest when we find that these Babylonian forms were not the primitive ones of this region, but that the early Sumerian forms were horizontal.[104] What interpretation shall be given to these facts? Shall we say that it was mere accident that one people wrote "one" vertically and that another wrote it horizontally? This may be the case; but it may also be the case that the tribal migrations that ended in the Mongol invasion of China started from the Euphrates while yet the Sumerian civilization was prominent, or from some common source in Turkestan, and that they carried to the East the primitive numerals of their ancient home, the first three, these being all that the people as a whole knew or needed. It is equally possible that these three horizontal forms represent primitive stick-laying, the most natural position of a stick placed in front of a calculator being the horizontal one. When, however, the cuneiform writing developed more fully, the vertical form may have been proved the easier to make, so that by the time the migrations to the West began these were in use, and from them came the upright forms of Egypt, Greece, Rome, and other Mediterranean lands, and those of A['s]oka's time in India. After A['s]oka, and perhaps among the merchants of earlier centuries, the horizontal forms may have come down into India from China, thus giving those of the N[=a]n[=a] Gh[=a]t cave and of later inscriptions. This is in the realm of speculation, but it is not improbable that further epigraphical studies may confirm the hypothesis. {30} As to the numerals above three there have been very many conjectures. The figure one of the Demotic looks like the one of the Sanskrit, the two (reversed) like that of the Arabic, the four has some resemblance to that in the Nasik caves, the five (reversed) to that on the K[s.]atrapa coins, the nine to that of the Ku[s.]ana inscriptions, and other points of similarity have been imagined. Some have traced resemblance between the Hieratic five and seven and those of the Indian inscriptions. There have not, therefore, been wanting those who asserted an Egyptian origin for these numerals.[105] There has already been mentioned the fact that the Kharo[s.][t.]h[=i] numerals were formerly known as Bactrian, Indo-Bactrian, and Aryan. Cunningham[106] was the first to suggest that these numerals were derived from the alphabet of the Bactrian civilization of Eastern Persia, perhaps a thousand years before our era, and in this he was supported by the scholarly work of Sir E. Clive Bayley,[107] who in turn was followed by Canon Taylor.[108] The resemblance has not proved convincing, however, and Bayley's drawings {31} have been criticized as being affected by his theory. The following is part of the hypothesis:[109] _Numeral_ _Hindu_ _Bactrian_ _Sanskrit_ 4 [Symbol] [Symbol] = ch chatur, Lat. quattuor 5 [Symbol] [Symbol] = p pancha, Gk. [Greek:p/ente] 6 [Symbol] [Symbol] = s [s.]a[s.] 7 [Symbol] [Symbol] = [s.] sapta ( the s and [s.] are interchanged as occasionally in N. W. India) Bühler[110] rejects this hypothesis, stating that in four cases (four, six, seven, and ten) the facts are absolutely against it. While the relation to ancient Bactrian forms has been generally doubted, it is agreed that most of the numerals resemble Br[=a]hm[=i] letters, and we would naturally expect them to be initials.[111] But, knowing the ancient pronunciation of most of the number names,[112] we find this not to be the case. We next fall back upon the hypothesis {32} that they represent the order of letters[113] in the ancient alphabet. From what we know of this order, however, there seems also no basis for this assumption. We have, therefore, to confess that we are not certain that the numerals were alphabetic at all, and if they were alphabetic we have no evidence at present as to the basis of selection. The later forms may possibly have been alphabetical expressions of certain syllables called _ak[s.]aras_, which possessed in Sanskrit fixed numerical values,[114] but this is equally uncertain with the rest. Bayley also thought[115] that some of the forms were Phoenician, as notably the use of a circle for twenty, but the resemblance is in general too remote to be convincing. There is also some slight possibility that Chinese influence is to be seen in certain of the early forms of Hindu numerals.[116] {33} More absurd is the hypothesis of a Greek origin, supposedly supported by derivation of the current symbols from the first nine letters of the Greek alphabet.[117] This difficult feat is accomplished by twisting some of the letters, cutting off, adding on, and effecting other changes to make the letters fit the theory. This peculiar theory was first set up by Dasypodius[118] (Conrad Rauhfuss), and was later elaborated by Huet.[119] {34} A bizarre derivation based upon early Arabic (c. 1040 A.D.) sources is given by Kircher in his work[120] on number mysticism. He quotes from Abenragel,[121] giving the Arabic and a Latin translation[122] and stating that the ordinary Arabic forms are derived from sectors of a circle, [circle]. Out of all these conflicting theories, and from all the resemblances seen or imagined between the numerals of the West and those of the East, what conclusions are we prepared to draw as the evidence now stands? Probably none that is satisfactory. Indeed, upon the evidence at {35} hand we might properly feel that everything points to the numerals as being substantially indigenous to India. And why should this not be the case? If the king Srong-tsan-Gampo (639 A.D.), the founder of Lh[=a]sa,[123] could have set about to devise a new alphabet for Tibet, and if the Siamese, and the Singhalese, and the Burmese, and other peoples in the East, could have created alphabets of their own, why should not the numerals also have been fashioned by some temple school, or some king, or some merchant guild? By way of illustration, there are shown in the table on page 36 certain systems of the East, and while a few resemblances are evident, it is also evident that the creators of each system endeavored to find original forms that should not be found in other systems. This, then, would seem to be a fair interpretation of the evidence. A human mind cannot readily create simple forms that are absolutely new; what it fashions will naturally resemble what other minds have fashioned, or what it has known through hearsay or through sight. A circle is one of the world's common stock of figures, and that it should mean twenty in Phoenicia and in India is hardly more surprising than that it signified ten at one time in Babylon.[124] It is therefore quite probable that an extraneous origin cannot be found for the very sufficient reason that none exists. Of absolute nonsense about the origin of the symbols which we use much has been written. Conjectures, {36} however, without any historical evidence for support, have no place in a serious discussion of the gradual evolution of the present numeral forms.[125] TABLE OF CERTAIN EASTERN SYSTEMS Siam [Illustration: numerals] Burma[126] [Illustration: numerals] Malabar[127] [Illustration: numerals] Tibet[128] [Illustration: numerals] Ceylon[129] [Illustration: numerals] Malayalam[129] [Illustration: numerals] {37} We may summarize this chapter by saying that no one knows what suggested certain of the early numeral forms used in India. The origin of some is evident, but the origin of others will probably never be known. There is no reason why they should not have been invented by some priest or teacher or guild, by the order of some king, or as part of the mysticism of some temple. Whatever the origin, they were no better than scores of other ancient systems and no better than the present Chinese system when written without the zero, and there would never have been any chance of their triumphal progress westward had it not been for this relatively late symbol. There could hardly be demanded a stronger proof of the Hindu origin of the character for zero than this, and to it further reference will be made in Chapter IV. * * * * * {38} CHAPTER III LATER HINDU FORMS, WITH A PLACE VALUE Before speaking of the perfected Hindu numerals with the zero and the place value, it is necessary to consider the third system mentioned on page 19,--the word and letter forms. The use of words with place value began at least as early as the 6th century of the Christian era. In many of the manuals of astronomy and mathematics, and often in other works in mentioning dates, numbers are represented by the names of certain objects or ideas. For example, zero is represented by "the void" (_['s][=u]nya_), or "heaven-space" (_ambara [=a]k[=a]['s]a_); one by "stick" (_rupa_), "moon" (_indu ['s]a['s]in_), "earth" (_bh[=u]_), "beginning" (_[=a]di_), "Brahma," or, in general, by anything markedly unique; two by "the twins" (_yama_), "hands" (_kara_), "eyes" (_nayana_), etc.; four by "oceans," five by "senses" (_vi[s.]aya_) or "arrows" (the five arrows of K[=a]mad[=e]va); six by "seasons" or "flavors"; seven by "mountain" (_aga_), and so on.[130] These names, accommodating themselves to the verse in which scientific works were written, had the additional advantage of not admitting, as did the figures, easy alteration, since any change would tend to disturb the meter. {39} As an example of this system, the date "['S]aka Sa[m.]vat, 867" (A.D. 945 or 946), is given by "_giri-ra[s.]a-vasu_," meaning "the mountains" (seven), "the flavors" (six), and the gods "_Vasu_" of which there were eight. In reading the date these are read from right to left.[131] The period of invention of this system is uncertain. The first trace seems to be in the _['S]rautas[=u]tra_ of K[=a]ty[=a]yana and L[=a][t.]y[=a]yana.[132] It was certainly known to Var[=a]ha-Mihira (d. 587),[133] for he used it in the _B[r.]hat-Sa[m.]hit[=a]._[134] It has also been asserted[135] that [=A]ryabha[t.]a (c. 500 A.D.) was familiar with this system, but there is nothing to prove the statement.[136] The earliest epigraphical examples of the system are found in the Bayang (Cambodia) inscriptions of 604 and 624 A.D.[137] Mention should also be made, in this connection, of a curious system of alphabetic numerals that sprang up in southern India. In this we have the numerals represented by the letters as given in the following table: 1 2 3 4 5 6 7 8 9 0 k kh g gh [.n] c ch j jh ñ [t.] [t.]h [d.] [d.]h [n.] t th d th n p ph b bh m y r l v ['s] [s.] s h l {40} By this plan a numeral might be represented
{'title': 'The Hindu-Arabic Numerals by Louis Charles Karpinski and David Eugene Smith', 'source': 'Project Gutenberg', 'lang': 'en'}