The HinduArabic Numerals, by David Eugene Smith and Louis Charles pot

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Hindu-David Eugene Smith and Louis Charles Karpinski

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Title: The Hindu-Arabic Numerals

Author: David Eugene Smith

Louis Charles Karpinski

Release Date: September 14, 2007 [EBook

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Language: English

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THE

HINDU-ARABIC NUMERALS

BYDAVID EUGENE SMITH

ANDLOUIS CHARLES KARPINSKI

BOSTON AND LONDON

GINN AND COMPANY, PUBLISHERS

The Athenæum Press

GINN AND COMPANY ·

PROPRIETORSBOSTON · U.S.A

such a labor-saving device should havestruggled for nearly a thousand years afterits system of place value was perfectedbefore it replaced such crude notations asthe one that the Roman conqueror madesubstantially universal in Europe Such,however, is the case, and there isprobably no one who has not at least someslight passing interest in the story of thisstruggle To the mathematician and thestudent of civilization the interest isgenerally a deep one; to the teacher of theelements of knowledge the interest may beless marked, but nevertheless it is real;and even the business man who makesdaily use of the curious symbols by which

we express the numbers of commerce,cannot fail to have some appreciation forthe 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 beenmade to bring together the fragmentarynarrations and to set forth the generalproblem of the origin and development ofthese numerals In this little work we haveattempted to state the history of theseforms in small compass, to place beforethe student materials for the investigation

of the problems involved, and to express

as clearly as possible the results of thelabors of scholars who have studied thesubject in different parts of the world Wehave had no theory to exploit, for thehistory of mathematics has seen too much

of this tendency already, but as far aspossible we have weighed the testimony

and have set forth what seem to be thereasonable conclusions from the evidence

at hand

To facilitate the work of students an indexhas been prepared which we hope may beserviceable In this the names of authorsappear only when some use has beenmade of their opinions or when theirworks are first mentioned in full in afootnote

If this work shall show more clearly thevalue of our number system, and shallmake the study of mathematics seem morereal to the teacher and student, and shalloffer material for interesting some pupilmore fully in his work with numbers, theauthors will feel that the considerablelabor involved in its preparation has not

been in vain.

We desire to acknowledge our especialindebtedness to Professor AlexanderZiwet for reading all the proof, as well asfor the digest of a Russian work, toProfessor Clarence L Meader for Sanskrittransliterations, and to Mr Steven T.Byington for Arabic transliterations andthe scheme of pronunciation of Orientalnames, and also our indebtedness to otherscholars in Oriental learning forinformation

DAVID EUGENE SMITHLOUIS CHARLES KARPINSKI

CONTENTS

PRONUNCIATION OF

ORIENTALNAMES 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 OFTHE

NUMERALS INTO

EUROPE BYBOETHIUS 63

VI THE DEVELOPMENT OF

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

ḍ, ṇ, ṣ, ṭ, (S) d, n, sh, t, made with the tip

of the tongue turned up and back into thedome 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 theside teeth Europeans commonlypronounce 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 belowthe 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) retainsits consonant sound at the end of a word

ḥ, (A) an unvoiced consonant formedbelow 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,

Ger man 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).

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 theplace of the accent ṁ and ṅ count as

consonants, but h after another consonant

does not (A), on the last syllable thatcontains a long vowel or a vowelfollowed by two consonants, except that afinal long vowel is not ordinarilyaccented; if there is no long vowel nortwo 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 thecommon numerals used in daily life are ofcomparatively recent origin The number

of systems of notation employed beforethe Christian era was about the same asthe number of written languages, and insome cases a single language had severalsystems The Egyptians, for example, hadthree systems of writing, with a numerical

notation for each; the Greeks had twowell-defined sets of numerals, and theRoman symbols for number changed more

or less from century to century Even day the number of methods of expressingnumerical concepts is much greater thanone would believe before making a study

to-of the subject, for the idea that ourcommon numerals are universal is farfrom being correct It will be well, then,

to think of the numerals that we stillcommonly call Arabic, as only one ofmany systems in use just before theChristian era As it then existed the systemwas no better than many others, it was oflate origin, it contained no zero, it wascumbersome and little used, and it had noparticular promise Not until centurieslater did the system have any standing in

the world of business and science; andhad the place value which nowcharacterizes it, and which requires azero, been worked out in Greece, wemight have been using Greek numerals to-day instead of the ones with which we arefamiliar.

Of the first number forms that the worldused this is not the place to speak Many

of them are interesting, but none had muchscientific value In Europe the invention ofnotation was generally assigned to theeastern shores of the Mediterranean untilthe critical period of about a century ago,

—sometimes to the Hebrews, sometimes

to the Egyptians, but more often to theearly trading Phœnicians.[1]

The idea that our common numerals areArabic in origin is not an old one Themediæval and Renaissance writersgenerally recognized them as Indian, andmany of them expressly stated that theywere of Hindu origin.[2] Others argued thatthey were probably invented by theChaldeans or the Jews because theyincreased in value from right to left, anargument that would apply quite as well tothe Roman and Greek systems, or to anyother It was, indeed, to the general idea

of notation that many of these writersreferred, as is evident from the words ofEngland's earliest arithmetical textbook-maker, Robert Recorde (c 1542): "In thatthinge all men do agree, that the Chaldays,whiche fyrste inuented thys arte, did setthese figures as thei set all their letters for

they wryte backwarde as you tearme it,and so doo they reade And that mayappeare in all Hebrewe, Chaldaye andArabike bookes where as the Greekes,Latines, and all nations of Europe, dowryte and reade from the lefte handtowarde the ryghte."[3] Others, and amongthem such influential writers asTartaglia[4] in Italy and Köbel[5] inGermany, asserted the Arabic origin of thenumerals, while still others left the matterundecided[6] or simply dismissed them as

"barbaric."[7] Of course the Arabsthemselves never laid claim to theinvention, always recognizing theirindebtedness to the Hindus both for thenumeral forms and for the distinguishingfeature of place value Foremost among

these writers was the great master of thegolden age of Bagdad, one of the first ofthe Arab writers to collect themathematical classics of both the East andthe West, preserving them and finallypassing them on to awakening Europe.This man was Moḥammed the Son ofMoses, from Khowārezm, or, more afterthe manner of the Arab, Moḥammed ibnMūsā al-Khowārazmī,[8] a man of greatlearning and one to whom the world ismuch indebted for its present knowledge

of algebra[9] and of arithmetic Of himthere will often be occasion to speak; and

in the arithmetic which he wrote, and ofwhich Adelhard of Bath[10] (c 1130) mayhave made the translation orparaphrase,[11] he stated distinctly that the

numerals were due to the Hindus.[12] This

is as plainly asserted by later Arabwriters, 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 fromArabic sources is that given by the Arabictraveler and scholar Mohammed ibn

Aḥmed, Abū 'l-Rīḥān al-Bīrūnī 1048), who spent many years inHindustan He wrote a large work onIndia,[15] one on ancient chronology,[16]the "Book of the Ciphers," unfortunatelylost, which treated doubtless of the Hinduart of calculating, and was the author ofnumerous other works Al-Bīrūnī was aman of unusual attainments, being versed

(973-in Arabic, Persian, Sanskrit, Hebrew, andSyriac, as well as in astronomy,chronology, and mathematics In his work

on India he gives detailed informationconcerning the language and customs ofthe people of that country, and statesexplicitly[17] that the Hindus of his timedid not use the letters of their alphabet fornumerical notation, as the Arabs did Healso states that the numeral signs called

aṅka[18] had different shapes in variousparts 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 toavoid any possibility of error, the numbermay be expressed in three differentsystems: with Indian symbols, insexagesimal 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 againattributing these forms to Hindu sources.Preceding Al-Bīrūnī there was anotherArabic 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 ofIndia to represent the duration of theworld Huart feels positive that in

Moṭahhar's time the present Arabicsymbols had not yet come into use, andthat the Indian symbols, although known toscholars, were not current Unless thiswere the case, neither the author nor hisreaders would have found anything

extraordinary in the appearance of thenumber which he cites.

Mention should also be made of a traveled student, Al-Mas'ūdī (885?-956),whose journeys carried him from Bagdad

widely-to Persia, India, Ceylon, and even acrossthe China sea, and at other times toMadagascar, Syria, and Palestine.[21] Heseems to have neglected no accessiblesources of information, examining also thehistory of the Persians, the Hindus, and theRomans Touching the period of the

Caliphs his work entitled Meadows of

Gold furnishes a most entertaining fund of

information He states[22] that the wisemen of India, assembled by the king,

composed the Sindhind Further on[23] hestates, upon the authority of the historian

Moḥammed ibn 'Alī 'Abdī, that by order

of Al-Manṣūr many works of science andastrology were translated into Arabic,notably the Sindhind (Siddhānta).

Concerning the meaning and spelling ofthis 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] Similardesignations are collected by Sédillot,[26]who inclined to the Greek origin of thesciences commonly attributed to theHindus.[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 "perpetuumæternumque." The reference[30] in this

ancient Arabic work to Al-Khowārazmī isworthy of note.

T h i s Sindhind is the book, saysMas'ūdī,[31] which gives all that theHindus know of the spheres, the stars,arithmetic,[32] and the other branches ofscience 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 awork furnished by a Hindu who came toBagdad as a member of the politicalmission 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 Nadīm It is of fundamental importance forthe history of Arabic culture Of the tenchief divisions of the work, the seventhdemands attention in this discussion forthe reason that its second subdivisiontreats of mathematicians andastronomers.[35]

al-The first of the Arabic writers mentioned

is Al-Kindī (800-870 A.D.), who wrotefive books on arithmetic and four books

on the use of the Indian method ofreckoning Sened ibn 'Alī, the Jew, whowas converted to Islam under the caliphAl-Māmūn, is also given as the author of awork 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 thisconnection that Casiri[37] also mentionsthe same writer as the author of a mostcelebrated work on arithmetic.

To Al-Ṣūfī, who died in 986 A.D., is alsocredited a large work on the same subject,and similar treatises by other writers arementioned We are therefore forced to theconclusion that the Arabs from the earlyninth century on fully recognized theHindu 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 herefers frequently to the nine Indianfigures,[39] thus showing again the generalconsensus of opinion in the Middle Agesthat the numerals were of Hindu origin.Some interest also attaches to the oldestdocuments on arithmetic in our ownlanguage One of the earliest treatises onalgorism 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 Minoritemonk of about 1240 A.D. The text of thefirst few lines is as follows:

"Hec algorism' ars p'sens dicit' in

qua

Talib; indor fruim bis quinq;

"This boke is called the boke of algorim

or augrym after lewder use And this boketretys of the Craft of Nombryng, the quychcrafte is called also Algorym Ther was akyng of Inde the quich heyth Algor & hemade this craft Algorisms, in the quych

we use teen figurys of Inde."

CHAPTER II

EARLY HINDU FORMS WITH NO

PLACE VALUEWhile it is generally conceded that thescientific development of astronomyamong the Hindus towards the beginning

of the Christian era rested upon Greek[42]

or Chinese[43] sources, yet their ancientliterature testifies to a high state ofcivilization, and to a considerableadvance in sciences, in philosophy, andalong literary lines, long before the goldenage of Greece From the earliest timeseven up to the present day the Hindu hasbeen wont to put his thought into rhythmicform The first of this poetry—it welldeserves this name, being also worthyfrom a metaphysical point of view[44]—consists of the Vedas, hymns of praise andpoems of worship, collected during theVedic period which dates fromapproximately 2000 B.C. to 1400 B.C.[45]

Following this work, or possiblycontemporary with it, is the Brahmanic

literature, which is partly ritualistic (theBrāhmaṇas), and partly philosophical (theUpanishads) Our especial interest is inthe Sūtras, versified abridgments of theritual and of ceremonial rules, whichcontain considerable geometric materialused in connection with altar construction,and also numerous examples of rationalnumbers the sum of whose squares is also

a square, i.e "Pythagorean numbers,"although this was long before Pythagoraslived Whitney[46] places the whole of theVeda literature, including the Vedas, theBrāhmaṇas, and the Sūtras, between 1500

B.C. and 800 B.C., thus agreeing withBürk[47] who holds that the knowledge ofthe Pythagorean theorem revealed in theSūtras 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 byCantor[48] of a Greek origin, has beenrepeatedly emphasized in recentliterature,[49] especially since theappearance of the important work of VonSchroeder.[50] Further fundamentalmathematical notions such as theconception of irrationals and the use ofgnomons, as well as the philosophicaldoctrine of the transmigration of souls,—all of these having long been attributed tothe Greeks,—are shown in these works to

be native to India Although thisdiscussion does not bear directly upon theorigin of our numerals, yet it is highlypertinent as showing the aptitude of theHindu for mathematical and mental work,

a fact further attested by the independentdevelopment of the drama and of epic andlyric poetry.

It should be stated definitely at the outset,however, that we are not at all sure thatthe most ancient forms of the numeralscommonly known as Arabic had theirorigin in India As will presently be seen,their forms may have been suggested bythose used in Egypt, or in Eastern Persia,

or in China, or on the plains ofMesopotamia We are quite in the dark as

to these early steps; but as to theirdevelopment in India, the approximateperiod of the rise of their essential feature

of place value, their introduction into theArab civilization, and their spread to theWest, we have more or less definite

information When, therefore, we considerthe rise of the numerals in the land of theSindhu,[51] it must be understood that it isonly the large movement that is meant, andthat there must further be considered thenumerous possible sources outside ofIndia itself and long anterior to the firstprominent appearance of the numbersymbols.

No one attempts to examine any detail inthe history of ancient India without beingstruck with the great dearth of reliablematerial.[52] So little sympathy have thepeople with any save those of their owncaste that a general literature is whollylacking, and it is only in the observations

of strangers that any all-round view ofscientific progress is to be found There is

evidence that primary schools existed inearliest times, and of the seventy-tworecognized sciences writing andarithmetic were the most prized.[53] In theVedic period, say from 2000 to 1400 B.C.,there was the same attention to astronomythat was found in the earlier civilizations

of Babylon, China, and Egypt, a factattested by the Vedas themselves.[54] Suchadvance in science presupposes a fairknowledge of calculation, but of themanner of calculating we are quiteignorant and probably always shall be.One of the Buddhist sacred books, the

Lalitavistara, relates that when the

Bōdhisattva[55] was of age to marry, thefather of Gopa, his intended bride,demanded an examination of the fivehundred suitors, the subjects including