The Hindu-Arabic Numerals

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.

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

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

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]

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. ā, as in father.

c, (S) like ch in church (Italian c in cento).

ḍ, ṇ, ṣ, ṭ, (S) d, n, sh, t, made with the tip of the tongue turned up and back into the dome of the palate. ḍ, ṣ, ṭ, ẓ, (A) d, s, t, z, made with the tongue spread so that the sounds are produced largely against the side teeth. Europeans commonly pronounce ḍ, ṇ, ṣ, ṭ, ẓ, both (S) and (A), as simple d, n, sh (S) or s (A), t, z. ḏ (A), like th in this.

e, (S) as in they. (A) as in bed.

ġ, (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, ṭ, 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. ḥ, (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. ḥ (S) is final consonant h, like final h (A).

i, as in pin. ī, 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.

ṁ, ṅ, (S) like French final m or n, nasalizing the preceding vowel.

ṇ, see ḍ. 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. ṛ, (S) r used as vowel, as in apron when pronounced aprn and not apern; modern Hindus say ri, hence our amrita, Krishna, for a-mṛta, Kṛṣṇa.

s, as in same. ṣ, see ḍ. ś, (S) English sh (German sch).

ṭ, see ḍ.

u, as in put. ū, as in rule.

y, as in you.

ẓ, see ḍ.

‛, (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 ḥ. 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 ṁ and ṅ 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.

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,

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 Phœnicians.[^[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]]

Others argued that they were probably invented by the Chaldeans or the Jews because they increased in value from right to left, an argument that would apply quite as well to the Roman and Greek systems, or to any other. It was, indeed, to the general idea of notation that many of these writers referred, as is evident from the words of England's earliest arithmetical textbook-maker, Robert Recorde (c. 1542): "In that thinge all men do agree, that the Chaldays, whiche fyrste inuented thys arte, did set these figures as thei set all their letters. for they wryte backwarde as you tearme it, and so doo they reade. And that may appeare in all Hebrewe, Chaldaye and Arabike bookes ... where as the Greekes, Latines, and all nations of Europe, do wryte and reade from the lefte hand towarde the ryghte."[^[3]] Others, and

among them such influential writers as Tartaglia[^[4]] in Italy and 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 Moḥammed the Son of Moses, from Khowārezm, or, more after the manner of the Arab, Moḥammed ibn Mūsā al-Khowārazmī,[^[8]] a man of great

learning and one to whom the world is much indebted for its present knowledge of algebra[^[9]] and of arithmetic. Of him there will often be occasion to speak; and in the arithmetic which he wrote, and of which Adelhard of Bath[^[10]] (c. 1130) may have made the translation or paraphrase,[^[11]] he stated distinctly that the numerals were due to the Hindus.[^[12]] This is as plainly asserted by later Arab

writers, even to the present day.[^[13]] Indeed the phrase ‛ilm hindī, "Indian science," is used by them for arithmetic, as also the adjective hindī alone.[^[14]]

Probably the most striking testimony from Arabic sources is that given by the Arabic traveler and scholar Mohammed ibn Aḥmed, Abū 'l-Rīḥān al-Bīrūnī (973-1048), who spent many years in Hindustan. He wrote a large work on India,[^[15]] one on ancient chronology,[^[16]] the "Book of the Ciphers," unfortunately lost, which treated doubtless of the Hindu art of calculating, and was the author of numerous other works. Al-Bīrūnī was a man of unusual attainments, being versed in Arabic, Persian, Sanskrit, Hebrew, and Syriac, as well as in astronomy, chronology, and mathematics. In his work on India he gives detailed information concerning the language and

customs of the people of that country, and states explicitly[^[17]] that the Hindus of his time did not use the letters of their alphabet for numerical notation, as the Arabs did. He also states that the numeral signs called aṅka[^[18]] had different shapes in various parts of India, as was the case with the letters. In his Chronology of Ancient Nations he gives the sum of a geometric progression and shows how, in order to avoid any possibility of error, the number may be expressed in three different systems: with Indian symbols, in sexagesimal notation, and by an alphabet system which will be touched upon later. He also speaks[^[19]] of "179, 876, 755, expressed in Indian ciphers," thus again attributing these forms to Hindu sources.

Preceding Al-Bīrūnī there was another Arabic writer of the tenth century, Moṭahhar ibn Ṭāhir,[^[20]] author of the Book of the Creation and of History, who gave as a curiosity, in Indian (Nāgarī) symbols, a large number asserted by the people of India to represent the duration of the world. Huart feels positive that in Moṭahhar's time the present Arabic symbols had not yet come into use, and that the Indian symbols, although known to scholars, were not current. Unless this were the case, neither the author nor his readers would have found anything extraordinary in the appearance of the number which he cites.

Mention should also be made of a widely-traveled student, Al-Mas‛ūdī (885?-956), whose journeys carried him from Bagdad to Persia, India, Ceylon, and even

across the China sea, and at other times to Madagascar, Syria, and Palestine.[^[21]] He seems to have neglected no accessible sources of information, examining also the history of the Persians, the Hindus, and the Romans. Touching the period of the Caliphs his work entitled Meadows of Gold furnishes a most entertaining fund of information. He states[^[22]] that the wise men of India, assembled by the king, composed the Sindhind. Further on[^[23]] he states, upon the authority of the historian Moḥammed ibn ‛Alī ‛Abdī, that by order of Al-Manṣūr many works of science and astrology were translated into Arabic, notably the Sindhind (Siddhānta). Concerning the meaning and spelling of this name there is considerable diversity of opinion. Colebrooke[^[24]] first pointed out the connection between Siddhānta and Sindhind. He ascribes to the word the meaning "the revolving ages."[^[25]] Similar designations are collected by Sédillot,[^[26]] who inclined to the Greek origin of the sciences commonly attributed to the Hindus.[^[27]] Casiri,[^[28]] citing the Tārīkh al-ḥokamā or Chronicles of the Learned,[^[29]] refers to the work

as the Sindum-Indum with the meaning "perpetuum æternumque." The reference[^[30]] in this ancient Arabic work to Al-Khowārazmī is worthy of note.

This Sindhind is the book, says Mas‛ūdī,[^[31]] which gives all that the Hindus know of the spheres, the stars, arithmetic,[^[32]] and the other branches of science. He mentions also Al-Khowārazmī and Ḥabash[^[33]] as translators of the tables of the Sindhind. Al-Bīrūnī[^[34]] refers to two other translations from a work furnished by a Hindu who came to Bagdad as a member of the political mission which Sindh sent to the caliph Al-Manṣūr, in the year of the Hejira 154 (A.D. 771).

The oldest work, in any sense complete, on the history of Arabic literature and history is the Kitāb al-Fihrist, written in the year 987 A.D., by Ibn Abī Ya‛qūb al-Nadīm. It is of fundamental importance for the history of Arabic culture. Of the ten chief divisions of the work, the seventh demands attention in this discussion for the reason that its second subdivision treats of mathematicians and astronomers.[^[35]]

The first of the Arabic writers mentioned is Al-Kindī (800-870 A.D.), who wrote five books on arithmetic and four books on the use of the Indian method of reckoning. Sened ibn ‛Alī, the Jew, who was converted to Islam under the caliph Al-Māmūn, is also given as the author of a work on the Hindu method of reckoning. Nevertheless, there is a possibility[^[36]] that some of the works ascribed to Sened ibn ‛Alī are really works of Al-Khowārazmī, whose name immediately precedes his. However, it is to be noted in this connection that Casiri[^[37]] also mentions the same writer as the author of a most celebrated work on arithmetic.

To Al-Ṣūfī, who died in 986 A.D., is also credited a large work on the same subject, and similar treatises by other writers are mentioned. We are therefore forced to the conclusion that the Arabs from the early ninth century on fully recognized the Hindu origin of the new numerals.

Leonard of Pisa, of whom we shall speak at length in the chapter on the Introduction of the Numerals into Europe, wrote his Liber Abbaci[^[38]] in 1202. In this work he refers frequently to the nine Indian figures,[^[39]] thus showing again the general consensus of opinion in the Middle Ages that the numerals were of Hindu origin.

Some interest also attaches to the oldest documents on arithmetic in our own language. One of the earliest

treatises on algorism is a commentary[^[40]] on a set of verses called the Carmen de Algorismo, written by Alexander de Villa Dei (Alexandra de Ville-Dieu), a Minorite monk of about 1240 A.D. The text of the first few lines is as follows:

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

Talib; indor

"This boke is called the boke of algorim or augrym after lewder use. And this boke tretys of the Craft of Nombryng, the quych crafte is called also Algorym. Ther was a kyng of Inde the quich heyth Algor & he made this craft.... Algorisms, in the quych we use teen figurys of Inde."

CHAPTER 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 Brāhmaṇas), and partly philosophical (the Upanishads). Our especial interest is

in the Sūtras, 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 Brāhmaṇas, and the Sūtras, 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 Sūtras goes back to the eighth century B.C.

The importance of the Sūtras 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

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

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 Bōdhisattva[^[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 10⁵³, adding that he could proceed as far as 10⁴²¹,[^[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,

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 Viṣvamitra:

"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 kōti, nahut, ninnahut,

Khamba, viskhamba, abab, attata,

To kumuds, gundhikas, and utpalas,

By pundarīkas 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 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 the which

The gods compute their future and their past.'"

Thereupon Viṣvamitra Ācārya[^[61]] expresses his approval of the task, 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 Gūrū.'"

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 Vedāṅgas, or "limbs for supporting the Veda," part of that great branch of Hindu literature known as Smṛiti (recollection), that which was to be handed down by tradition. Of these the sixth is known as Jyotiṣa (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

also speak of eighteen ancient Siddhāntas 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 Mahābhārata[^[64]] and the Rāmāyana, 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]]

The early Hindu numerals[^[68]] may be classified into three great groups, (1) the Kharoṣṭhī, (2) the Brāhmī, and (3) the word and letter forms; and these will be considered in order.

The Kharoṣṭhī numerals are found in inscriptions formerly known as Bactrian, Indo-Bactrian, and Aryan, and appearing in ancient Gandhāra, 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 Aśoka, 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

various other parts of the world. These Aśoka[^[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 Kharoṣṭhī characters, and the rest in some form of Brāhmī. In the Kharoṣṭhī inscriptions only four numerals have been found, and these are merely vertical marks for one, two, four, and five, thus:

In the so-called Śaka 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:

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 Nānā Ghāt forms hereafter given, although the inscriptions themselves at Nānā Ghāt are later than those of the Aśoka period. The

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 Kharoṣṭhī 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 Brāhmī 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 Kharoṣṭhī 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 Kharoṣṭhī 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 Brāhmī characters. The alphabet attributed to Brahmā is the oldest of the several known in India, and was used from the earliest historic times. There are various theories of its origin,

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 Aśoka appear, some of these having been incised in Brāhmī as well as Kharoṣṭhī. 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 Aśoka edicts:[^[74]]

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 Nānā Ghāt 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 Kalyāna to Junar. It seems to have been cut out by a descendant

of King Śātavāhana,[^[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]]

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.

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:

From this time on, until the decimal system finally adopted the first nine characters and replaced the rest of the Brāhmī 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]]

Table showing the Progress of Number Forms in India

[Most of these numerals are given by Bühler, loc. cit., Tafel IX.]

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]]

[^[94]]

[^[95]]

[^[96]]

[^[97]]

[^[98]]

To these may be added the following numerals below one hundred, similar to those in the table:

We have thus far spoken of the Kharoṣṭhī and Brāhmī 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

Hieroglyphic

Hieratic

Demotic

The last of these is merely a cursive form as in the Arabic

There is, however, considerable ground for interesting speculation with respect to these first three numerals. The earliest Hindu forms were perpendicular. In the Nānā Ghāt inscriptions they are vertical. But long before either the Aśoka or the Nānā Ghāt 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

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 Aśoka's time in India. After Aśoka, and perhaps among the merchants of earlier centuries, the horizontal forms may have come down into India from China, thus giving those of the Nānā Ghāt 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.

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 Kṣatrapa coins, the nine to that of the Kuṣana 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 Kharoṣṭhī 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

have been criticized as being affected by his theory. The following is part of the hypothesis:[^[109]]

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 Brāhmī 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

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 akṣaras, 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 Phœnician, 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]]

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]]

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,

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

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 Lhāsa,[^[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 Phœnicia 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,

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

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.

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" (śūnya), or "heaven-space" (ambara ākāśa); one by "stick" (rupa), "moon" (indu śaśin), "earth" (bhū), "beginning" (ādi), "Brahma," or, in general, by anything markedly unique; two by "the twins" (yama), "hands" (kara), "eyes" (nayana), etc.; four by "oceans," five by "senses" (viṣaya) or "arrows" (the five arrows of Kāmadēva); 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.

As an example of this system, the date "Śaka Saṃvat, 867" (A.D. 945 or 946), is given by "giri-raṣa-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 Śrautasūtra of Kātyāyana and Lāṭyāyana.[^[132]] It was certainly known to Varāha-Mihira (d. 587),[^[133]] for he used it in the Bṛhat-Saṃhitā.[^[134]] It has also been asserted[^[135]] that Āryabhaṭa (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:

By this plan a numeral might be represented by any one of several letters, as shown in the preceding table, and thus it could the more easily be formed into a word for mnemonic purposes. For example, the word

has the value 1,565,132, reading from right to left.[^[138]] This, the oldest specimen (1184 A.D.) known of this notation, is given in a commentary on the Rigveda, representing the number of days that had elapsed from the beginning of the Kaliyuga. Burnell[^[139]] states that this system is even yet in use for remembering rules to calculate horoscopes, and for astronomical tables.

A second system of this kind is still used in the pagination of manuscripts in Ceylon, Siam, and Burma, having also had its rise in southern India. In this the thirty-four consonants when followed by a (as ka ... la) designate the numbers 1-34; by ā (as kā ... lā), those from 35 to 68; by i (ki ... li), those from 69 to 102, inclusive; and so on.[^[140]]

As already stated, however, the Hindu system as thus far described was no improvement upon many others of the ancients, such as those used by the Greeks and the Hebrews. Having no zero, it was impracticable to designate the tens, hundreds, and other units of higher order by the same symbols used for the units from one to nine. In other words, there was no possibility of place value without some further improvement. So the Nānā Ghāt

symbols required the writing of "thousand seven twenty-four" about like T 7, tw, 4 in modern symbols, instead of 7024, in which the seven of the thousands, the two of the tens (concealed in the word twenty, being originally "twain of tens," the -ty signifying ten), and the four of the units are given as spoken and the order of the unit (tens, hundreds, etc.) is given by the place. To complete the system only the zero was needed; but it was probably eight centuries after the Nānā Ghāt inscriptions were cut, before this important symbol appeared; and not until a considerably later period did it become well known. Who it was to whom the invention is due, or where he lived, or even in what century, will probably always remain a mystery.[^[141]] It is possible that one of the forms of ancient abacus suggested to some Hindu astronomer or mathematician the use of a symbol to stand for the vacant line when the counters were removed. It is well established that in different parts of India the names of the higher powers took different forms, even the order being interchanged.[^[142]] Nevertheless, as the significance of the name of the unit was given by the order in reading, these variations did not lead to error. Indeed the variation itself may have necessitated the introduction of a word to signify a vacant place or lacking unit, with the ultimate introduction of a zero symbol for this word.

To enable us to appreciate the force of this argument a large number, 8,443,682,155, may be considered as the Hindus wrote and read it, and then, by way of contrast, as the Greeks and Arabs would have read it.

Modern American reading, 8 billion, 443 million, 682 thousand, 155.

Hindu, 8 padmas, 4 vyarbudas, 4 kōṭis, 3 prayutas, 6 lakṣas, 8 ayutas, 2 sahasra, 1 śata, 5 daśan, 5.

Arabic and early German, eight thousand thousand thousand and four hundred thousand thousand and forty-three thousand thousand, and six hundred thousand and eighty-two thousand and one hundred fifty-five (or five and fifty).

Greek, eighty-four myriads of myriads and four thousand three hundred sixty-eight myriads and two thousand and one hundred fifty-five.

As Woepcke[^[143]] pointed out, the reading of numbers of this kind shows that the notation adopted by the Hindus tended to bring out the place idea. No other language than the Sanskrit has made such consistent application, in numeration, of the decimal system of numbers. The introduction of myriads as in the Greek, and thousands as in Arabic and in modern numeration, is really a step away from a decimal scheme. So in the numbers below one hundred, in English, eleven and twelve are out of harmony with the rest of the -teens, while the naming of all the numbers between ten and twenty is not analogous to the naming of the numbers above twenty. To conform to our written system we should have ten-one, ten-two, ten-three, and so on, as we have twenty-one, twenty-two, and the like. The Sanskrit is consistent, the units, however, preceding the tens and hundreds. Nor did any other ancient people carry the numeration as far as did the Hindus.[^[144]]

When the aṅkapalli,[^[145]] the decimal-place system of writing numbers, was perfected, the tenth symbol was called the śūnyabindu, generally shortened to śūnya (the void). Brockhaus[^[146]] has well said that if there was any invention for which the Hindus, by all their philosophy and religion, were well fitted, it was the invention of a symbol for zero. This making of nothingness the crux of a tremendous achievement was a step in complete harmony with the genius of the Hindu.

It is generally thought that this śūnya as a symbol was not used before about 500 A.D., although some writers have placed it earlier.[^[147]] Since Āryabhaṭa gives our common method of extracting roots, it would seem that he may have known a decimal notation,[^[148]] although he did not use the characters from which our numerals are derived.[^[149]] Moreover, he frequently speaks of the

void.[^[150]] If he refers to a symbol this would put the zero as far back as 500 A.D., but of course he may have referred merely to the concept of nothingness.

A little later, but also in the sixth century, Varāha-Mihira[^[151]] wrote a work entitled Bṛhat Saṃhitā[^[152]] in which he frequently uses śūnya in speaking of numerals, so that it has been thought that he was referring to a definite symbol. This, of course, would add to the probability that Āryabhaṭa was doing the same.

It should also be mentioned as a matter of interest, and somewhat related to the question at issue, that Varāha-Mihira used the word-system with place value[^[153]] as explained above.

The first kind of alphabetic numerals and also the word-system (in both of which the place value is used) are plays upon, or variations of, position arithmetic, which would be most likely to occur in the country of its origin.[^[154]]

At the opening of the next century (c. 620 A.D.) Bāṇa[^[155]] wrote of Subandhus's Vāsavadattā as a celebrated work,

and mentioned that the stars dotting the sky are here compared with zeros, these being points as in the modern Arabic system. On the other hand, a strong argument against any Hindu knowledge of the symbol zero at this time is the fact that about 700 A.D. the Arabs overran the province of Sind and thus had an opportunity of knowing the common methods used there for writing numbers. And yet, when they received the complete system in 776 they looked upon it as something new.[^[156]] Such evidence is not conclusive, but it tends to show that the complete system was probably not in common use in India at the beginning of the eighth century. On the other hand, we must bear in mind the fact that a traveler in Germany in the year 1700 would probably have heard or seen nothing of decimal fractions, although these were perfected a century before that date. The élite of the mathematicians may have known the zero even in Āryabhaṭa's time, while the merchants and the common people may not have grasped the significance of the novelty until a long time after. On the whole, the evidence seems to point to the west coast of India as the region where the complete system was first seen.[^[157]] As mentioned above, traces of the numeral words with place value, which do not, however, absolutely require a decimal place-system of symbols, are found very early in Cambodia, as well as in India.

Concerning the earliest epigraphical instances of the use of the nine symbols, plus the zero, with place value, there

is some question. Colebrooke[^[158]] in 1807 warned against the possibility of forgery in many of the ancient copper-plate land grants. On this account Fleet, in the Indian Antiquary,[^[159]] discusses at length this phase of the work of the epigraphists in India, holding that many of these forgeries were made about the end of the eleventh century. Colebrooke[^[160]] takes a more rational view of these forgeries than does Kaye, who seems to hold that they tend to invalidate the whole Indian hypothesis. "But even where that may be suspected, the historical uses of a monument fabricated so much nearer to the times to which it assumes to belong, will not be entirely superseded. The necessity of rendering the forged grant credible would compel a fabricator to adhere to history, and conform to established notions: and the tradition, which prevailed in his time, and by which he must be guided, would probably be so much nearer to the truth, as it was less remote from the period which it concerned."[^[161]] Bühler[^[162]] gives the copper-plate Gurjara inscription of Cedi-saṃvat 346 (595 A.D.) as the oldest epigraphical use of the numerals[^[163]] "in which the symbols correspond to the alphabet numerals of the period and the place." Vincent A. Smith[^[164]] quotes a stone inscription of 815 A.D., dated Saṃvat 872. So F. Kielhorn in the Epigraphia Indica[^[165]] gives a Pathari pillar inscription of Parabala, dated Vikrama-saṃvat 917, which corresponds to 861 A.D.,

and refers also to another copper-plate inscription dated Vikrama-saṃvat 813 (756 A.D.). The inscription quoted by V. A. Smith above is that given by D. R. Bhandarkar,[^[166]] and another is given by the same writer as of date Saka-saṃvat 715 (798 A.D.), being incised on a pilaster. Kielhorn[^[167]] also gives two copper-plate inscriptions of the time of Mahendrapala of Kanauj, Valhabī-saṃvat 574 (893 A.D.) and Vikrama-saṃvat 956 (899 A.D.). That there should be any inscriptions of date as early even as 750 A.D., would tend to show that the system was at least a century older. As will be shown in the further development, it was more than two centuries after the introduction of the numerals into Europe that they appeared there upon coins and inscriptions. While Thibaut[^[168]] does not consider it necessary to quote any specific instances of the use of the numerals, he states that traces are found from 590 A.D. on. "That the system now in use by all civilized nations is of Hindu origin cannot be doubted; no other nation has any claim upon its discovery, especially since the references to the origin of the system which are found in the nations of western Asia point unanimously towards India."[^[169]]

The testimony and opinions of men like Bühler, Kielhorn, V. A. Smith, Bhandarkar, and Thibaut are entitled to the most serious consideration. As authorities on ancient Indian epigraphy no others rank higher. Their work is accepted by Indian scholars the world over, and their united judgment as to the rise of the system with a place value—that it took place in India as early as the

sixth century A.D.—must stand unless new evidence of great weight can be submitted to the contrary.

Many early writers remarked upon the diversity of Indian numeral forms. Al-Bīrūnī was probably the first; noteworthy is also Johannes Hispalensis,[^[170]] who gives the variant forms for seven and four. We insert on p. 49 a table of numerals used with place value. While the chief authority for this is Bühler,[^[171]] several specimens are given which are not found in his work and which are of unusual interest.

The Śāradā forms given in the table use the circle as a symbol for 1 and the dot for zero. They are taken from the paging and text of The Kashmirian Atharva-Veda[^[172]], of which the manuscript used is certainly four hundred years old. Similar forms are found in a manuscript belonging to the University of Tübingen. Two other series presented are from Tibetan books in the library of one of the authors.

For purposes of comparison the modern Sanskrit and Arabic numeral forms are added.

Numerals used with Place Value

CHAPTER IV

THE SYMBOL ZERO

What has been said of the improved Hindu system with a place value does not touch directly the origin of a symbol for zero, although it assumes that such a symbol exists. The importance of such a sign, the fact that it is a prerequisite to a place-value system, and the further fact that without it the Hindu-Arabic numerals would never have dominated the computation system of the western world, make it proper to devote a chapter to its origin and history.

It was some centuries after the primitive Brāhmī and Kharoṣṭhī numerals had made their appearance in India that the zero first appeared there, although such a character was used by the Babylonians[^[185]] in the centuries immediately preceding the Christian era. The symbol is

"The earliest undoubted occurrence of a zero in India is an inscription at Gwalior, dated Samvat 933 (876 A.D.). Where 50 garlands are mentioned (line 20), 50 is written

Aside from its appearance in early inscriptions, there is still another indication of the Hindu origin of the symbol in the special treatment of the concept zero in the early works on arithmetic. Brahmagupta, who lived in Ujjain, the center of Indian astronomy,[^[190]] in the early part

of the seventh century, gives in his arithmetic[^[191]] a distinct treatment of the properties of zero. He does not discuss a symbol, but he shows by his treatment that in some way zero had acquired a special significance not found in the Greek or other ancient arithmetics. A still more scientific treatment is given by Bhāskara,[^[192]] although in one place he permits himself an unallowed liberty in dividing by zero. The most recently discovered work of ancient Indian mathematical lore, the Ganita-Sāra-Saṅgraha[^[193]] of Mahāvīrācārya (c. 830 A.D.), while it does not use the numerals with place value, has a similar discussion of the calculation with zero.

What suggested the form for the zero is, of course, purely a matter of conjecture. The dot, which the Hindus used to fill up lacunæ in their manuscripts, much as we indicate a break in a sentence,[^[194]] would have been a more natural symbol; and this is the one which the Hindus first used[^[195]] and which most Arabs use to-day. There was also used for this purpose a cross, like our X, and this is occasionally found as a zero symbol.[^[196]] In the Bakhṣālī manuscript above mentioned, the word śūnya, with the dot as its symbol, is used to denote the unknown quantity, as well as to denote zero. An analogous use of the

zero, for the unknown quantity in a proportion, appears in a Latin manuscript of some lectures by Gottfried Wolack in the University of Erfurt in 1467 and 1468.[^[197]] The usage was noted even as early as the eighteenth century.[^[198]]

The small circle was possibly suggested by the spurred circle which was used for ten.[^[199]] It has also been thought that the omicron used by Ptolemy in his Almagest, to mark accidental blanks in the sexagesimal system which he employed, may have influenced the Indian writers.[^[200]] This symbol was used quite generally in Europe and Asia, and the Arabic astronomer Al-Battānī[^[201]] (died 929 A.D.) used a similar symbol in connection with the alphabetic system of numerals. The occasional use by Al-Battānī of the Arabic negative, lā, to indicate the absence of minutes

(or seconds), is noted by Nallino.[^[202]] Noteworthy is also the use of the