LOUIS C RLE S RPI - Forgotten Books

HINDU-ARABIC NUMERALS

DAVID EUGENE SM ITH

AND

LOUI S CHARLE S KARPINSK I

BOSTON AND LONDONGINN AND COMPANY

,PUBLISHER S

1911

PREFACE

So familiar are we with the numerals that bear themisleading name of Arabic , and so extensive is their u sein Europe and the Americas, that it is difficult for us toreal ize that their general acceptance in the transactionsof commerce is a matter of only the last four centuries ,and that they are unknown to a very large part of thehuman race to-day. It seems strange that such a laborsaving device should have struggled for nearly a thousand years after its system of place value was perfectedbefore it replaced such crude notations as the one thatthe Roman conqueror made substantially universal inEurope. Such, however, is the case, and there is probably no one who has not at least some slight passinginterest in the story of this struggle. To the mathema

tician and the student of civiliz ation the i nterest is generally a deep one ; to the teacher of the elements ofknowledge the interest may be less marked, but nevertheles s it is real ; and even the business man who makesdaily u se of the curious symbols by which we expressthe numbers of commerce, cannot fail to have someappreciation for the story of the rise and progress ofthese

'

tools of his trade.

This story has often been told in part, but it is a longtim e since any efiort has been made to bring togetherthe fragmentary narrations and to set forth the gen

eral problem of the origin and development of theseiii

236299

iv THE HINDU—ARABIC NUMERALS

numerals. In this little work we have attempted to statethe history of these forms in small compass , to placebefore the student materials for the investigation of theproblems i nvolved, and to express as clearly as possiblethe results of the labors of scholars who have studiedthe subject in different parts of the world. We havehad no theory to exploit, for the history of mathematicshas seen too much of this tendency already, but as faras pos sible we have weighed the testimony and have set

forth what seem to be the reasonable conclusions fromthe evidence at hand.

To facilitate the work of students an index has beenprepared which we hope may be serviceable. In this thenames of authors appear only when some u se has beenmade of their Opinions or when their works are firstmentioned in full i n a footnote.

If this work shall show more clearly the value of ournumber system, and shall make the study of mathematicsseem more real to the teacher and student, and shall offermaterial for interes ting some pupil more fully in his workwith numbers , the authors will feel that the considerablelabor involved in its preparation has not been in vain.

We desire to acknowledge our especial indebtednessto Professor Alexander Z iwet for reading all the proof,as well as for the digest of a Russian work , to ProfessorClarence L . Meader for Sanskrit transliterations, and toMr. Steven T. Byington for Arabic transliterations andthe scheme of pronunciation of Oriental names, and alsoour indebtedness to other s cholars in Oriental learningfor information.

DAVID EUGENE SMITH

LOUIS CHARLES KARPINSKI

CONTENTS

CHAPTE R PAGEPRONUNCIATION OF ORIENTAL NAME S

I. EARLY IDEAS OF THE IR ORIGINII. EARLY HINDU FORMS WITH NO PLACE VALUEIII. LATER HINDU FORMS , WITH A PLACE VALUEIv. THE SYMBOL ZERO

V . THE QUE STION OF THE INTRODUCTION OI“ THE

NUMERAL S INTO EUROPE BY BOETHIUSVI . THE DEVELOPME NT OF THE NUMERALS AMONG THE

ARABS

VII . THE DEFINITE INTRODUCTION OF THE NUMERALS

INTO EUROPE

VIII. THE SPREAD OF THE NUMERALS IN EUROPE

INDEX

PRONUNCIATION OF ORIENTAL NAMES

(S) in Sanskrit names and words ; (A) in Arab ic names andwords .

b yd)f ) g , h , j) l ) m )

n, p )

Sh (A) , t ;th (A) , v ,

W , x ,Z, as in Engl ish .

a , (S) like u in bu t : thus p andi t,pronounced p undit. (A) l ike a in

ask or in man. a, as infa ther.

e, (S ) like ch in chu rch (Italian c in

cento) .d,n, s , t , (S) d, n, sh , it, made withthe tip of the tongue turned upand back into the dome of thepalate. d

,s, t, z

, (A) d , s , t, z ,

m ade with the tongue spread so

that the sounds are producedlargely aga inst the s ide teeth .

E uropeans commonly pronounced, n, s, t . z . both (S ) and (A) , as

s imple d, n, sh (S) or s (A) , t, z.

4 (A) , l ike ih in this .

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

g, (A) a voiced consonant formedbelow the vocal cords ; its soundis compared by som e to a g , byothers to a guttural r ; in Arab icwords adopted into Engl ish it isrepresented by gh (e .g . ghou l) ,less often r (e .g . razz ia) .

h preceded by b, c, t, i , etc . doesnot form a s ingle sound with theseletters , but is a m ore or les s distinct h sound following them ; cf.

the sounds in abhor, boat/wok,

etc ., or , more accurately for (S ) ,

the bhoys etc . of Irish brogue .

h (A) reta ins its consonant soundat the end of a word . 11

, (A) anunvoiced consonant form ed belowthe voca l cords ; its sound is som e

times compared to German hardch , and may be represented by an

h as strong as poss ible . InArab icwords adopted into English it i srepresented by h , e .g . in sah ib,

hakeem . h (S) is final consonanth ,like final h (A) .i , as inp in. i , as inp i que.

k, as in ki ck.

kh, (A) the hard ch of Scotch loch ,

German ach , espec ially of German

as pronounced by the Swiss .

m, ii , (8) l ike French final m or n,

nasaliz ing the preceding vowel .n, see (1 . ii

,l ike ng in s inging.

o, (S) as in so. (A) as in obey .

q , (A) l ike k (or c) in cook ; furtherback in the m outh than in kick.

r , (S) English r, sm ooth and un

trilled . (A) stronger.as vowel , as in ap ron when pronounced ap rnandnot ap ern; m od

ernHindus say r i , hence our um

rita , K rishna ,for a-mz'ta , K rsna .

s, as in same. s

, see "i . S, (S) Eng

l ish sh (German s ch) .15, see 11 .

u , as in p u t. (1, as in ru le.

y , as in you .

z , see (1.

(A) a sound k indred to the sp irituslenis (that is , to our ears , the meredistinct separationof avowel fromthe preceding sound , as at the beginning of a word in German) andto h. The is a very distinct soundin Arab ic , but i s more nearlyrepresented by the spiritus lenisthan by any sound that we can

produce without m uch spec ialtraining. That is , it should betreated as s ilent , but the soundsthat precede and follow it shouldnot run together . InArab icwordsadopted into English it i s treatedas s ilent , e .g . in Arab, amber,Caaba ( Arab, ‘

anbar, ka‘abah ) .

(A) A final long vowel is shortened before a l (’l) or ibn (whose i is then

s ilent) .Accent : (S ) as if Latin ; in determ ining the place of the accentmand

i t count as consonants , bu t h after another consonant does not . (A) , onthe last syllable that conta ins a long vowel or a vowel followed by twoconsonants , except that a final long vowel is not ordinarily accented ; i fthere is no long vowel nor two consecutive consonants , the accent falls onthe first syllable . The words a l and ibn are never accented .

VI

THE

HINDU—ARABIC NUMERALS

CHAPTER I

EARLY IDEAS OF THE IR ORIGIN

It has long been recognized that the common numeralsused in daily life are of comparatively recent origin .

The number of systems of notation employed beforethe Chris tian era was about the same as the number ofwritten languages , and i n some cases a s ingle languagehad several systems . The Egyptians , for example, hadthree systems of writing, with a numerical notation foreach ; the Greeks had two well-defined sets of numerals,and the Roman symbols for number changed more or lessfrom century to century. Even to-day the number ofmethods of express ing numerical concepts is muchgreater than one would believe before making a studyof the subject, for the idea that our comm on numeralsare universal is far from being correct. It will be well,then, to think of the numerals that we still commonlycall Arabic, as only one of many systems i n u se justbefore the Christian era. As it then existed the systemwas no better than many others, it was of late origin,

itcontained no zero, it was cumbersome and little used,

l ( ( G f

THE HINDU-ARABIC NUMERALS

and it had no particular promise. Not until centuries laterdid the sys tem have any standing i n the world of bus iness and science ; and had the place value which nowcharacteriz es it, and which requires a zero, been workedout i nGreece, we might have been us ing Greek numeralsto—day instead of the ones with which we are familiar.Of the firs t number forms that the world used thi s is

not the place to speak . Many of them are interesting,but none had much scientific value. In Europe the ihvention of notation was generally assigned to the easternshores of the Mediterranean until the critical period ofabout a century ago , sometimes to the Hebrews, som e

times to the Egyp tians , but more often to the earlytrading Phoenicians .

1

The idea that our comm on numerals are Arabic inorigin is not an Old one. The mediaeval and Renaissancewriters generally recognized them as Indian , and manyof them express ly stated that they were of Hindu origin.

2

1 “ D iscipu lus . Qu is primus invenit numeram apud Hebraeos et

E gyptios M agis ter . Abraham primu s invenit num erum apudHebrmos

,deinde M oses et Abraham tradidit istam scientiam numeri

ad E gyptios , et docu it eos : deinde Josephus .

”[Bede, De comp ute

dialogas (doubtful ly assigned to him) , Op era omnia , Paris , 1862, Vol . I,

p .

Al ii rcferunt ad Phoenices inventores arithm eticae, propter eandem

eommerciorum cau ssam Al i i ad Indos Ioannes de Sacrobosco,cu jus

sepu lchrum est Lu tetize in com itio Maturinens i,refert ad A rabes .

”

[Ramu s,A rithmeticce libri duo

,Basel

,1569

, p .

S im ilar notes are given by Peletariu s in h is commentary on the

arithmetic of Gemma Fris ius (1563 ed .,fol . and in h is own work

(1570 Lyons cd ., p . 14 ) La valeu r des F igu res commence au coste

dextre tirant vers le coste senestre : au rebou rs de notre maniered ’escrire par ce que la prem iere prattique est venue des Chaldees

ou des Pheniciens, qu i ont é té les prem iers traffiquers de marchan

dis c .

”

2 Maxiniu s Planudes (c . 1330) states that the nine symbols comefrom the Indians .

”[W éi schke

’s German translation

,Halle

,1878

,

EARLY IDEAS OF THE IR ORIGIN 3

Others argued that they were probably invented by theChaldeans or the Jews because they increased in valuefrom right to left, an argument that would apply quiteas well to the Roman and Greek systems, or to any

other. It was , indeed, to the general idea Of notationthat many of these writers referred, as is evident fromthe words of England ’s earliest arithmetical textbookmaker, Robert Recorde ( c. In that thinge all

men do agree, that the Chaldays , whiche fyrste inu entedthys arte, did set these figures as thei set all their letters .for they wryte backwarde as you tearme it, and so doothey 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 readefrom the lefte hand towarde the ryghte.

” 1 Others, and

p . Willichius speaks of the Zyphrae Indicae, in his Arithmeticoe

libri tres (Strasburg, 1540, p . and Cataneo of “ le noue figure degli Indi,” in hi s Lep ratiche delle dve p rime mathematiche (Venice, 1546,fol . Woepcke is not correct, therefore, in saying (“M émoire sur la

propagation des ch iffres indiens ,” hereafter referred to as Prop agation[Journal A s iatique, Vol . I 1863

, p . that Wallis (A Treatise onA lgebra , both h istorical and p ractical, London, 1685, p . 13

,and De

algebra tractatus,Latin edition in his Op era omnia , 1693, Vol . II ,

p . 10) was one of the first to give the Hindu origin.

1 From the 1558 edition of The Ground of A rtes , fol . C , 5. S im ilarlyBishop Tonstall writes Qu i a Chaldeis primum in finitimos

,deinde

in omnes pene gentes fluxit . Numerandi artem a Chaldeis esse

profectam : qu i dum s cribunt,a dextra incip iunt, et in leuam pro

grediuntur. [De arte supputandi , London, 1522, fol . B ,Gemma

Frisius,the great continental rival of Recorde, had the same idea

Primum autem appellamus dexterum locum ,eo quod haec ars vel a

Chaldaeis,vel ab Hebraeis ortum habere credatur, qu i etiam eo ordi ne

s cribunt ” but this refers more evidently to the A rabic numerals .

[Arithmeticce practicae methodvs faci lis , Antwerp , 1540 , fol . 4 of the1563 ed .] Sacrobosco (c . 1225) mentions the same thing. Even the

modern Jewish writers claim that one of their scholars,Mashallah

(c . introduced them to the M ohammedan world . [C . Levias,

The Jewish Encyclop edia , New York, 1905, Vol. IX , p .

4 THE HINDU—ARABIC NUMERALS

among them such influential writers as Tartaglia 1 inItaly and K '

obel 2 i nGermany, asserted the Arabic origi nOf the numerals , while still others left the matter undecided 3 or simply dismissed them as

“ barbaric.” 4 Of

course the Arabs themselves never laid claim to the i nvention,

always recogniz ing their indebtedness to theHindus both for the numeral forms and for the disti ngu ishing feature of place value. Foremost among thesewriters was the great master of the golden age of Bagdad, one of the first of the Arab writers to collect themathematical classics of both the E ast and theWest, preserving them and finally pass ing them on to awakeningEurope. This man was M ohammed the Son of Moses,from Khowarezm , or, more after the manner of the Arab,Mohammed ibn Musa al-Khowarazm i ,

5 a man of great

1que esto fu tronato di fare da gli A rab i con diece figure .

[La p rima p arte del general trattato di numeri , etmisere, Venice, 1556,fol . 9 of the 1592 edition.]

2 “ Vom welchen A rabischen au ch disz Kunst entsprungen ist.

[A in nerv geordnet Rechenbiechlin, Augsbu rg, 1514 , fol . 13 of the 1531edition. The printer used the letters rv for w in “ new in the firstedition

,as he had no w of the proper font ]

8 Among them Glareanus Characteres simplices sunt nouem s ignificatiu i

,ab Indis usque, s iue Chaldae is asc iti E st

item unus .0 ci rcu lus, qu i nih il s ignificat.

”[De VI . Arithmeticae

practicae sp eciebvs , Paris , 1539, fol . 9 of the 1543 edition.]4 Barbarische Oder gemeine Z i ffern.

”[Anonymous , Das E inmahl

E ins cum notis variorum,Dresden

,1703

, p . So Vossiu s (De universae

matheseos natura et constitutione liber, Amsterdam ,1650

, p . 34 ) callsthem Barbaras numeri notas .

” The word at that time was possiblysynonymous with A rabic .

6 His fu ll name was ‘

Abfi‘

Abdallah Mohammed ibn Masa al

Khowarazmi . He was born in Khowarezm ,

“ the lowlands ,” the

country about the present Kh iva and bordering on the Oxus,and

l ived at Bagdad under the caliph al-Mann'

m . He died probably between 220 and 230 of the Mohammedan era

,that is

,between 835and

845 A .D .,although some put the date as early as 812. The best ac

count o i th is great scholar may be found in an article by C . Nallino,

“A l-Huwarizm i,

” in theA tti della R.Accad.dei Lincei , Rome, 1896. See


learning and one to whom the world is much i ndebtedfor its present knowledge of algebra 1 and of arithm etic.Of him there will Often be occasion to speak ; and in thearithmetic whi ch he wrote, and Of which Adelhard ofBath 2 ( c. 1130) may have made the translation or paraphrase,3 he stated distinctly that the numerals were dueto the Hindus .4 This is as plainly asserted by later Arabalso Verhandlungen des 5. Congresses der Orientalisten, Berlin, 1882,Vol . II

, p . 19 ; W . Spittaf Bey in the Zeitschrift der deutschenMorgenla

'

nd . Gesellschaft, Vol . XXXIII , p . 224 ; Steinschneider in the Zeitschriftder’ deutschenMorgenla'nd. Gesellschaft, Vol . L , p . 214 ; Treutlein

in the Abhand lungen zur GeschichtederMathematik,Vol . I

, p . 5; Su ter,Die Mathematiker und Astronomen der A raber und ih re Werke

,

”

AbhandlungenzurGeschichtederMathematik,Vol .X

,Leipzig, 1900, p . 10

,

and Nachtrag e,” in Vol . X IV , p . 158 ; Cantor, Geschichte der Mathe

matik,Vol . 1

,3d ed.

, pp . 712—733 etc . ; F .Woepcke inProp agation, p . 489.

SO recently has he become known that Heilbronner,writing in 1742

merely ment ions him as Ben—M usa,inter A rabes celebris Geometra

,

scripsit de figuris planis sphericis .

”[H istoria matheseos universoe

Leipzig, 1742, p .

III this work m ost of the Arabic names will be transliterated substantially as laid down by Suter in his work Die Mathematiker etc.

,

except where th is Violates English pronunciation. The scheme of pronunciation of oriental names is set forth in the preface .

1 Ourword a lgebra is f rom the title of one of h is works,A l—jabr wa ’ l

muqabalah, Completion and Comparison. The work was translated intoEngl ish by F . Rosen

,London

,1831

,and treated in L ’A lgebre d

’al

K harizmi et les méthodes indienne et grecque, Leon Rodet, Paris , 1878,extract f rom the Journal A siatique. For the derivation of the worda lgebra , see Cossali , Scritti Inediti , pp . 381—383, Rome

,1857 ; Leo

nardo ’ s L iber A bbaci p . 4 10,Rome

,1857 ; both published by B .

Boncompagni .“Almu chabala ” also was used as a name for algebra .

2 Th is learned scholar,teacher of O ’

Creat who wrote the Helceph

(“Prologus N . Ocreati in Helcep h ad Adelardum Batensem magistrum

studied in Toledo, learned A rabic, traveled as far east as

Egypt, and brought from the Levant num erous manuscripts for studyand translation. See Henry in the Abhandlungen zur Geschichte derMathematik

,Vol . III

, p . 131 Woepcke in Prop agation, p . 518.

3 The title is A lgoritm i de numero Indorum . That he did not maketh is translation is asserted by EnestrOm in the Bibliotheca Mathematica

,

Vol . I p . 520 .

4 Thus he speaks de numero indorum per .IX . literas,

and pro

ceeds : D ixit algoritm i Cum u idissem yndos constitu isse .IX . literas


writers, even to the present day.

1 Indeed the phrase'i lrn hindi ,

“ Indian science,” is used by them for arithmetic, as also the adjective hindi alone.

2

Probably the most striking testimony from Arabicsources is that given by the Arabic traveler and scholarM ohammed ibn Ahmed, Abu ’

1-Rihan al-Biri'

m i ( 973

who spent many years in Hindustan. He wrotea large work on India,3 one on ancient chrono logy,4 theBook of the Ciphers ,” unfortunately lost, which treateddoubtless of the Hindu art of calculating, a nd was theauthor of numerous other works. Al-B iruni was a manof unusual attainments , being versed in Arabic, Persian,Sanskrit, Hebrew, and Syriac , as well as in astronomy,chronology, and mathematics . In his work on India hegives detailed information concerning the language andin uniuerso numero suo

, propter dispositionem suam quam posuerunt,

uolu i patefacere de Opera quod fit per eas aliqu id quod esset leniusdiscentibus

,si deus uoluerit .” [Boncompagni , Trattati d’A ritmetica

,

Rome,

Discussed by F .Woepcke, Sur l’introduction de l

’arith

mé tique indienne en Occident, Rome, 1859.

1 Thus in a commentary by ‘

Ali ibnAbi Bekr ibn al-Jamal al-Ansarial-M ekki on a treatise on gobar arithmetic (explained later) called A lmurshidah

,found by Woepcke in Paris (Prop agation, p . there is

mentioned the fact that there are“ nine Indian figu res and a sec

ond kind of Indian figu res although these are the figu res of thegobar writing.

” So in a commentary by Hosein ibn Mohammed al

Mahall i (died in 1756) on the Mokhtasarj i‘

i lm cl-hisab (Extract fromArithmetic) by ‘

Abdalqadir ibn‘

Ali al-Sakhawi (died c . 1000) it is related that “ the preface treats of the forms of the figures of Hindus igns , such as were established by the Hindu nation.

”[Woepcke,

Prop agation, p .

2 See alsoWoepcke, Prop agation, p . 505. The origin is discussed atmuch length by G . R . Kaye,

“Notes on IndianMathematics .—

‘

A rithmetical Notation

,

” Journ. and Proc. of theA s iatic Soc. of Bengal, Vol .

III,1907

, p . 489 .

3 A lberuni ’ s India,A rabic version, London, 1887 ; English transla

tion,ib id .

,1888 .

4 Chronology of AncientNations,London, 1879 . Arabic and English

versions,by C . E . Sachau .

EARLY IDEAS OF THEIR ORIGIN 7

customs of the people of that country, and states ex

plicitly1 that the Hi ndus of his time did not u se the

letters of their alphabet for numerical notation , as the

Arabs did. He also states that the numeral signs calledanka 2 had different shapes i n various parts of India, aswas the case with the letters. In his Chronology of An

cientNations he gives the sum of a geometric progress ionand shows how

,in order to avoid any possibility of error,

the number may be expressed in three different systemswith Indian symbols , in sexagesimal notation , and by analphabet system which will be touched upon later. He

also speaks 3 of “ 1 79, 876, 755, expressed in Indianciphers ,” thus again attributing these forms to Hi ndusources.Preceding A l-Biri'm i there was another Arabic ' writer

of the tenth century, Motahh ar ibn Tahir,4 author ofthe Book of the Creation and of E story, who gave as acuriosity, in Indian (Nagari) symbols , a large numberasserted by the people of India to represent the durationof the world. Huart feels positive that in Motahhar

’

s

tim e the present Arabic symbols had not yet come intouse, and that the Indian symbols , although known toscholars , were not current. Unless this were the case,neither the author nor his readers would have foundanything extraordinary in the appearance of the numberwhich he cites.Mention should also be made of a widely-traveled

student, A l-Mas‘

iidi ( 885— 9 whose journeys carried

him from Bagdad to Persia, India, Ceylon , and even1 India

,Vol . I

,chap . xvi .

2 The Hindu name for the symbols of the decimal place system .

3 Sachau ’s Engl ish ed ition Of the Chronology , p . 64 .

4 L ittérature arabe,C l . Huart

,Par is

,1902 .


across the China sea, and at other tim es to Madagascar,

Syria, and Palestine.

1 He seems to have neglected noaccess ible sources of information, examini ng also the

history Of the Persians , the Hindus, and the Romans .Touching the period of the Caliphs his work entitledMeadows of Gold furnishes a most entertai ning fund ofinformation. He states 2 that the wise men Of India,assembled by the king, composed the S indhind. Further on 3 he states, upon the authority of the historianM ohammed ibn °

A lit

Abdi , that by order Of A l-Mansfi r

many works of science and astrology were translated intoArabic, notably the S indhinol ( S iddhdnta) . Concerningthe meaning and spelling of this name there is cons iderablediversity of opinion. Colebrooke 4 first pointed outthe connection between S iddhanta and S indhind. He

ascribes to the word the meaning the revolving ages .” 5S imilar designations are

°collected by S é dillot,6 who inclined to the Greek origin of the sciences commonlyattributed to the Hi ndus .

7 Cas iri,8 citing the Tdrikh a l

hokarnd or Chronicles of the Learned,9 refers to the work

1 Huart,H istory of A rabic L iterature, Engl ish ed.

,New York

,1903

,

p . 182 seq .

2 A i-Mas'udi ’ s Meadows of Gold, translated in part by Aloys Spreu

ger, London,184 1 Les p ra ir ies d ’ or, trad . par C . Barb ier de Meynard

et Pavet de Courteille,Vols . I to IX

,Paris

,1861— 1877 .

3 Les p ra ir ies d’or, Vol . VIII, p . 289 seq .

4 E ssays , Vol . II , p . 428.

5Loc . cit., p . 504 .

6 Mater iano; p our servir a l’histoire comparée des sciences mathematiques chez les Grecs ci les Orientauas, 2 vols .

,Paris

,1845— 184 9

, pp . 4 38

7 Hemade an exception, however, in favor of the num erals,loc . cit .

,

Vol . II, p . 503.

8 B ibliotheca Arabico—Hi spana E scuria lmi s is,Madrid

,1 760— 1770

pp . 426—427 .

9 The author,Ibn al-Qifti , flou rished A .D . 1198 [Colebrooke, loc,

. cit.

note V ol . II, p


as the S indum-Indum with the meaning perpetu inn

aeternumqu e. The reference 1 i n this ancient Arabicwork to Al-Khowarazm i is worthy of note.

This S indhind is the book , says Mas'fidi

,

2 which givesall that the Hindus know Of the spheres , the stars, arithmetic ,3 and the other branches of science. He mentionsalso A l-Khowarazm i and Habash 4 as translators Of thetables of the S indhind. A l-B ir i'nu 5 refers to two othertranslations from a work furnished by a Hindu whocame to Bagdad as a member of the political missionwhich S indh sent to the caliph A l-Mans fi r

,in the year of

the Hej ira 154 (A .D .

The oldes t work , i nany sense complete, on the historyof Arabic l iterature and his tory is the Ifitdb a l-Fihrist,

written in the year 987 A .D., by Ibn Abi Ya

'

q i'

ib al-Nadim.

It is of fundamental importance for the history of Arabicculture. Of the ten chief divis ions of the work, the sev

enth demands attention i n thi s discussion for the reasonthat its second subdivision treats of mathematicians andastronomers .6

H

1 “ Liber A rtis Logisticae a Mohamado Ben Musa A lkhuarezmita

exornatu s, qu i ceteros omnes brqvitate methodi ac fac i litate praestat,

Indorum que in praeclarissim is inventis ingenium acumen ostendit. [Casi i i , loc . cit . p .

2 Macoudi , Le livre de l’ avertissement ci de la revision. Translationby B . Carra de Vaux, Paris , 1896 .

3 Verifying the hypothes is of Woepcke, Prop agation, that the S indh ind included a treatment of arithm etic .

4 Ahmed ibn ‘

Abdallah,Suter

,Die M athematiker

,etc .

, p . 12.

5 India,Vol . II

, p . 15.

6 See H. Suter, Das Mathematiker -V ei zeichniss im Fihrist,

A bhandlungen zur Geschichte der M athematik,Vol . VI

,Leipzig, 1892.

For fu rther references to early A 1 ab1c writers the reader is I eferred

to H . Suter, Die M athematiker und A stronomen der A raber und ihreWerke. Also “Nachtrage und Berichtigungen” to the same (Abhandlungen, Vol . X IV ,

1902, pp . 155


The first of the Arabic writers mentioned is A l-Kindi( 800

— 870 who wrote five books on arithm etic andfour books on the use Of the Indian method of reckoning.

Sened ibn ‘

Ali , the Jew , whowas converted to Islam underthe caliphA l-Mam i

'

m , is also given as theauthor of a workon the Hindu method of reckoning. Nevertheless, thereis a possibility I that some of the works ascribed to Senedibn '

A li are really works of A l-Khowarazm i , whose nameimmediately precedes hi s . However, it is to be noted inthis connection that Casiri 2 also mentions the samewriteras the author of a most celebrated work 0 11 arithm etic.ToAl-S i' i fi , who died in 986A.D ., is also credi ted a large

work on the same subject, and s im ilar treatises by otherwriters are mentioned. We are therefore forced to theconclusion that the Arabs from the early ninth centuryon fully recogniz ed the Hindu origin of the new numerals .

Leonard of Pisa, Of whom we shall speak at length inthe chapter on the Introduction Of the Numerals intoEurope, wrote hi s I / iber Abbaci 3 i n 1202. In thi s workhe refers frequently to the nine Indian figures ,4 thusshowing again the general consensus of Opinion i n theM iddle Ages that the numerals were of Hindu origin .

Some interest also attaches ‘to the oldes t documents onarithmfi ic in our own language. One of the earliest

1 Suter,loc. cit.

,note 165

, pp . 62—63.

2 Send Ben A li,

tum arithmetica scripta maxime celebrata,

quae publici ju ris fec it. [Loc . cit., p .

3 Scri tti di Leonardo P isano,Vol . I

,L iber Abbaci Vol . II

,

Scritti published by Baldassarre Boncompagni , Rome . A lsoTre Scritti Inediti

,and Intorno ad Op ere di Leonardo P isano, Rome

,

1854 .

4 Ubi ex mirabil i magisterio in arte per novem figuras indorum

introductus ” etc . In another place, as a head ing to a separate d ivis i on

,he writes

,

“ De cognitione novem figurarum yndorum”

etc.

Novem figu re indorum he sunt 9 8 7 6 54 3 2

https://www.forgottenbooks.com/join

CHAPTER II

EARLY HINDU FORMS WITH NO PLACE VALUE

While it is generally conceded that the scientific development of astronomy among the Hi ndus towardsthe beginning of the Christian era rested upon Greek 1or Chinese 2 sources, yet their ancient literature testifiesto a high state of civilization, and to a considerable ad

vance in s ciences, in philosophy, and along literary lines ,long before the golden age of Greece. From the earliesttimes even up to the present day the Hindu has beenwont to put his thought into rhythm ic form. The firstof this poetry — it well deserves this name, being alsoworthy from a metaphysical point of View 3

consists ofthe Vedas, hymns of praise and poems of worship , collected during the Vedie period which dates from approxi

mately 2000 B .C . to 14 00 Following this work , orpossibly contemporary with it, is the Brahmanic literature,whi ch is partly ritual is tic ( the Brahm anas) , and partlyphilosophical ( the Upanishads) . Our especial interest is

1 Th ibaut,A stronomie

,A strologie und Mathematik, Strassbu rg,

1899 .

2 Gu stave Schlegel, Uranographie ch inoise ou p reuves directes quel’ astronom ie p rimitive est origina ire de la Ch ine, et qu ’ elle a é té empruntee p ar les anciens p eup les occidentaua; a la sp here ch ino ise ; out rage accomp agne d’ un atlas céleste chinois et grec, The Hague and Leyden,1875.

3 E . W . Hopkins , The Religions of India , Boston, 1898, p . 7 .

4 R . C . Dutt, H istory of India , London, 1906.

12

EARLY HINDU FORMS WITH NO PLACE VALUE 13

in the Sutras, vers ified abridgments of the ritual and ofceremonial rules, which contain considerable geometricmaterial used in connection with altar construction , andalso numerous examples of rational numbers the sum ofwhose squares is also a square, i.e.

“ Pythagorean numbers ,” although this was long before Pythagoras lived.

Whi tney 1 places the whole of the Veda literature, including the Vedas , the Brahmanas, and the Sutras, between1500 B C . and 800 B .C . , thus agree ing with Burk 2 whoholds that the knowledge of the Pythagorean theorem re

vealed in the Sutras goes back to the eighth century BCThe im portance of the Sutras as showing an i ndepend

ent origin of Hindu geometry, contrary to the Opinionlong held by Cantor 3 of a Greek origi n, has been repeatedly emphasiz ed in recent literature,4 especially sincethe appearance of the important work of Von Schroeder.5Further fundamental mathematical notions such as the

conception of irrationals and the u se of gnomons , as well asthe phi losophical 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. A l

though this discussion does not bear directly upon the1 W . D .

_Wh itney , Sanskrit Grammar, 3d ed .

,Le ipzig, 1896.

2 Das Apastambaf Sulba-Sii tra,

” Zeitschrift der deutschen Morgenla

'

ndischen Gesellschaft,Vol . LV

, p . 543,and Vol . LVI , p . 327 .

3 Gesch ichte der Math,Vol . 1

,2d cd .

, p . 595.

4 L . von Schroeder,Pythagoras und die Inder

,Leipzig, 1884 ; H.

Vogt, Haben die alten Inder den Pythagoreischen Leh rsatz und dasIrrat ionale gekannt? Bibliotheca Mathematica

,V ol . VII pp . 6—20 ;

A . Burk,l oc . cit . ; Max Simon

,Geschichte derMathematik im A ltertum

,

Berlin,1909

, pp . 137—165; th ree S i'

Itras are translated in part byThibaut

,Journa l of the A siatic Society of Bengal, 1875, and one ap

peared in The Pandit,1875; Beppo Levi, Osservazioni e congettu re

Sopra la geometria degli indiani ,” Bibliotheca Mathematica,Vol . IX

1908, pp . 97— 105.

5Loc . cit . ; also Indiens Literatur and Gu itar, Leipzig, 1887 .

14 THE HINDU—ARABIC -NUMERALS

origin of our numerals , yet it is highly pertinent as showing the aptitude of the Hindu for mathematical and men

tal work , a fact further attested by the independentdevelopment 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 formsOf the numerals commonly known as Arabic had theirorigin in India. As will presently be seen, their formsmay have been suggested by those used in Egypt, or inEastern Persia, or in China, or on the plains of Mesopotami a. We are quite in the dark as to these early steps ;but as to their development in India, the approximateperiod of the rise Of their essential feature of place value,their introduction into the Arab civiliz ation, and theirSpread to the West, we have more or less definite information. When , therefore, we consider the rise of thenumerals in the land of the S indhu , 1 it must be understood that it is only the large movement that is meant,and that there must further be considered the numerouspossible sources outside of India itself and long

'

anterior

to the first prominent appearance of the number sym bols.No one attempts to examine any detail in the history of

ancient India without bei ng struck with the great dearthof reliable material ? SO l ittle sympathy have the peoplewith any save those of their Own caste that a general literature is wholly lacking, and it is only in the observationsof strangers that any all-round View of scientific progressis to be found. There is evidence that primary schools

1 It is generally agreed that the name of the river S indhu , corruptedby western peoples to Hindhu , Indos , Indus , is the root of Hindustanand of Ind ia . Reclus

,A sia

,English ed.

,Vol . III , p . 14 .

2 See the comments of Oppert, On the Original Inhabitants of Bharatavarsa or India, London, 1893, p. l ,


existed in earliest times, and of the seventy-two recogniz edsciences writing and arithmetic were the most prized.

1 In

the Vedic period, say from 2000 to 14 00 B .C ., there wasthe same attention to astronomy that was found in theearlier civilizations of Babylon, China, and Egypt, a fact attested by theVedas themselves ? Such advance in sciencepresupposes a fair knowledge of calculation , but of themanner of calculating we are quite ignorant and probably always shall be. One of the Buddhist sacred books ,the Lalitavistara , relates that when the Bodh isattva 3 wasof 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, andarchery. Having vanquished his rivals in all else

,he is

matched against Arjuna the great arithmetician and isasked to express numbers greater than 100 kotis .4 In

reply he gave a scheme of number names as high as 1053,adding that he could proceed as far as all of whichsuggests the system Of Archimedes and the unsettledquestion of the indebtedness of the West to the E ast inthe realm Of ancient mathematics .6 Sir Edwin Arnold,

1 A . Hillebrandt,A lt-Indien

,Breslau

,1899

, p . 111 . Fragmentaryrecords relate that Kharavela

,king of Kali iiga, learned as a boy lekha

(writing) , ganana (reckoning) , and rupa (arithmetic applied to monetary affairs and mensu ration) , probably in the 5th century B .C .

[Bii hler, Indische Pa laeographic, Strassbu rg, 1896, p .

2 R . C . Dutt,A H istory of Civilization in Ancient India , London,

1893,Vol . I

, p . 174 .

3 The Buddha . The date of h is birth is uncertain. S ir E dwin Ar

nold put it c . 6204

5 There is some uncertainty about this limit.6 This problem deserves more study than has yet been given it. A

beginning may be made with Comte Goblet d’A lviella, Ce que l

’ Inde

do it a la Grece, Paris , 1897, and H . G . Keene’s review

,The Greeks in

India,” in the Calcutta Review,Vol , CXIV , 1902, p . 1, See also F ,


in The Light of Asia , does not mention this part of thecontest, but he speaks of Buddha’ s training at the handsof the learned V isvam itra :

And Vi swam itra said, It i s enough ,

Let u s to numbers . A fter m e repeatYou r num eration till we reach the lakh ,1One, two , three, fou r, to ten, and then by tens

To hundreds , th ou sands .

’

A fter h im the childNam ed digits , decads , centu ries , nor pau sed ,The round lakh reached, but s oftly mu rm u red on,Then com es the koti, nahut, ninnahu t,Khamba, Vi skhamba , abab, attata,

To kumuds , gundh ikas , and u tpalas ,

By pundarikas into padumas ,

Wh ich last i s how you count the u tm o st grainsOf Hastagiri ground to finest du st ; 2But beyond that a num eration i s ,The Katha, u sed to count the stars o f night,The Koti-Katha, for the ocean drops ;Ingga, the calcu lu s of circu lars ;Sarvanikchepa, by the wh ich you deallVith all the sands o f Gunga, till we com e

To Antah-Kalpas , where the unit i sThe sands o f the ten crore Gungas . If one seeks

M ore comprehens ive scale, th’

arithm ic m ountsBy the Asankya , wh ich i s the taleOf all the drops that in ten thou sand yearsWou ld fall on all the worlds by daily rain ;Thence unto Maha Kalpas , by the whichThe gods compu te their futu re and their past.

Woepcke, Prop agation, p . 253 ; G . R . Kaye, loc. cit., p . 4 75seq .

,and

“The Sou rce of Hindu Mathematics,

” Jou rna l of the Royal A s iaticSociety , July , 1910, pp . 749— 760 ; G . Th ibaut

,A stronom ie

,A strologie

und Mathematik, pp . 43—50 and 76— 79 . It will be discussed more fully

in Chapter VI .

1 L e . to The lakh is still the common large unit in India,like the myriad in ancient Greece and the million in the West.

2 Th is again suggests the Psamm ites , or De harenae numero as it iscalled in the 154 4 edition of the Op era of A rchimedes , a work inwhichthe great Syracusan proposes to Show to the king by geometric proofswhich you can follow

,that the numbers wh ich have been named by

EARLY HINDU FORMS WITH NO PLACE VALUE 1 7

Thereupon V isvam itra A carya1expres ses his approval

of the task , and asks to hear the measure of the l ineas far as i ana, the longest measure bearing name. Thisgiven , Buddha adds

And m aster if it please,I shall recite how m any sun-m otes lieFrom end to end w ith in a i ana .

’

Thereat, with instant skill, the little princePronounced the total of the atom s tru e .

But Vi swam itra heard i t on h i s facePro strate before the boy ; For th ou ,

’

he cried ,Art Teacher o f thy teachers thou , not I ,

Art Gfi rfi.

’

It is needless to say that this is far from being history.

And yet it puts in charming rhythm onlyWhat the ancientLalitavistara relates of the number-series of the Buddha’ stime. While it extends beyond all reason, nevertheles sit reveals a condition that would have been i mpossibleunless arithmetic had attained a considerable degree ofadvancement.To this pre-Christian period belong also the Veddngas ,

or “ limbs for supporting the Veda,” part of that greatbranch of Hindu literature known as Smriti (recollec

tion) , that which was to be handed down by tradition.

Of these the sixth is known as Jyotisa ( astronomy) , a

short treatise of only thirty-six verses , written not earlierthan 300 B .C .

, and affording u s some knowledge of theextent of number work in that period ? The Hindus

us are sufficient to exceed not only the number of a sand-heap as

large as the whole earth,but one as large as the universe .

” For a

l ist of early editions of this work see D . E . Sm ith , Rara A rithmetica,

Boston,1909

, p . 227 .

1 L e . the Wise.

2 S ir M onier Monier—Williams,Indian Wisdom

,4 th ed .

,London

,

1893, pp . 14 4

,177 . See also J : C . Marshman

,Abridgment of theH istory

of India , London, 1893, p . 2.


also speak of eighteen ancient S iddhantas or astronomicalworks, which, though mostly lost, confirm this evidence.

1‘

As to authentic histories , however, there exist i nIndianone relating to the period before the Mohammedan era

( 622 About all that we know Of the earlier c ivilization is what we glean from the two great epics, theMahabharata 2

and the Ramayana, from coins, and froma few inscriptions ?It is with this unsatisfactory material, then , that we

have to deal in searching for the early history of theHindu-Arabic numerals , and the fact that many unsolvedproblems exist and will continue to exist is no longerstrange when we cons ider the conditions. It is rathersurprising that so much has been discovered withi n a

century, than that we are so uncertain as to origins a nddates and the early spread of the system. The probabil

ity being that writi ng was not i ntroduced into Indiabefore the close of the fourth century B .C . , and literatureexisting only in spoken form prior to that period,4 thenumber work was doubtless that of all primitive peoples ,palpable, merely a matter of placing sticks or cowries orpebbles on the ground, of marking a sand-covered board,or of cutting notches or tying cords as is s till done inparts of Southern India to-day.

5

1 For a l ist and for some description of these works see R . C . Dutt,

A H istory of Civi lization in Ancient India , Vol . I I , p . 121 .

2 Professor Ramkrishna Gopal Bhandarkar fixes the date as the

fifth century Cons ideration of the Date of the M ahabharata,

”

in the Journa l of the Bombay Branch of theR . A . Soc .,Bombay , 1873

Vol. X, p .

3 Marshm an,loc . cit .

, p . 2 . z

4 A . C . Burnell,South Indian Palaeography , 2d ed .

,London

,1878

,

p . 1,seq .

5Th is extens ive subject of palpable arithmetic,essentially the

hi story of the abacus,deserves to be treated in a work by itself .


various other parts of the world. These Asoka 1 inscriptions, some thirty in all, are found in widely separatedparts of India, often on columns , and are in the variousvernaculars that were familiar to the people. Two are i nthe Kharosthi characters, and the rest in some form ofBrahmi . In the Kharosthi inscriptions only four num erals have been found , and these are merely vertical marksfor one, two, four, and five, thus

/In the so-called Saka inscriptions, possibly of the firstcentury B .C ., more numerals are found, and in morehighly developed form, the right-to-left sys tem appearing,together with evidences of three different scales of counting, four, ten, and twenty. The numerals of thisperiod are as follows

1 2 3 4 5 6 8

I II m x l>< H>< X X 7

3 7 33 333 7 333 Al i n20 50 so 70 100 200

There are several noteworthy points to be Observed i nstudying this system. In the first place, it is probably notas early as that shown i n the Nana Ghat forms hereaftergiven , although the inscriptions themselves at NanaGhat are later than those Of the As’ oka period. The

1 The earliest work on the subject was by James Prinsep,“ On the

Inscriptions of Piyadas i or Asoka,” etc .,Journal of the A s iatic Society

of Benga l, 1838, following a prel im inary suggestion in the same j ou rnalin 1837 . See also “Asoka Notes

,

” by V . A . Smith,The Indian A n

tiquary ,V ol . XXXVII

,1908

, p . 24 seq .,Vol . XXXVIII , pp . 151—159

,

June,1909 ; The E arly H istory of India, 2d cd .

,Oxford

,1908

, p . 154 ;

J. F . Fleet,“The LastWords of Asoka,” Journal of the Royal A s iatic

S ociety , October, 1909, pp . 981—1016 ; E . Senart, Les inscrip tions dePiyadasi , 2 vols ,

Paris,1887 .


four is to this system what the X was to the Roman,probably a canceling of three marks as a workman doesto-day for five, or a laying of one stick across three others.The ten has never been satisfactorily explained ] It iss imi lar to the A of the Kharosthi alphabet, but we haveno knowledge as to Why it was chosen . The twenty isevidently 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 ofthe French. The hundred is unexplained, although itresembles the letter ta or tra of the Brahmi alphabet with1 before ( to the right of) it. The two hundred is onlya variant of the symbol for hundred, with two verticalmarks.1This system has many points of s imilarity with the

Nabatean numerals 2 in u se in the first centuries of theChristian era. The cross is here used for four, and theKharosthi form is employed for twenty. In addition tothis there is a trace of an analogous u se of a scale Of

twenty. While the symbol for 100 is quite different, themethod of forming the other hundreds is the same. The

correspondence seems to be too marked to be whollyaccidental.It is not in the Kharosthi numerals , therefore, that we

can hope to find the origin Of those used by us, and weturn to the second of the Indian typ es , the Brahmi characters . The alphabet attributed to Brahmais the oldest ofthe several known in India, and was used from the earliesthistoric t imes. There are various theories of its origin,

1 For a d iscuss ion of the minor details of this system ,see Bii hler

,

loc. cit ., p . 73 .

2 Jul ius Euting, Nabataische Inschriften aus Arabien, Berlin, 1885,pp . 96—97, with a table of numerals .


none of which has as yet any wide acceptance, 1 althoughthe problem offers hope of solution in due time. Thenumerals are not as Old as the alphabet, or at least theyhave not as yet been found in inscriptions earlier thanthose in which the edicts of ASoka appear, some of thesehaving been incised in Brahm i as well as Kharosthi . Asalready stated, the older writers probably wrote the numbers in words, as seems to have been the case i n theearliest Pali writings Of Ceylon ?The following numerals are, as far as known , the

only ones to appear in the As’ oka edicts : 3

i —r’E

50

These fragments from the third century B .C ., crude and

unsatisfactory as they are, are the undoubted early formsfrom which our present system developed. They nextappear in the second century B .C . in some inscriptions inthe cave on the tOp of the NanaGhat hill, about seventyfive miles from Poona in central India. These inscrip~

tions may be memorials of the early Andhra dynasty ofsouthern India, but their chief interest lies in the numerals which they contain .

The cave was made as a resting-place for travelers ascending the hi ll, which lies on the road from Kalyana toJunar. It seems to have been cut out by a descendant

1 For the five principal theories see Buh ler, loc . cit ., p . 10 .

2 Bayley , loc . cit .,reprint p . 3.

3 B iihler,loc . cit . ; Ep igraphia Indica ,

Vol . III , p . 134 ,Indian An

tiquary , Vol . VI , p . 155seq .,and Vol . X , p . 107 .


of King Satavahana, 1 for inside the wall Oppos ite the em

trance are representations of the members Of his family,much defaced, but With the names still legible. It wouldseem that the excavation was made by order of a kingnamed Vedis iri, and “ the inscription contains a list ofgifts made on the occasion of the performance of severalgagnas or religious sacrifices , and numerals are to beseen in no less than thirty places ?There is considerable dispute as to what numerals are

really found in these inscriptions , owing to the difficultyof deciphering them but the following, which have beencopied from a rubbing, are probably number forms 3

44

1— 1

50 0 1-111

T I P?”

The inscription itself, so important as contain ing theearliest Considerable Hindu numeral system

,connected

with our“ own , is of sufficient interest to warrant reprodu cing part of it in facsimile, as is done on page 24 .

1 Pandit Bhagavanlal Indraji , “On Ancient Nagari Numeration ;from an Inscription at Naneghat Jou rnal of the Bombay Branch of theRoyal A s i ati c Society , 1876, Vol . XII , p . 404 .

2 Ib . p . 405. He gives also a plate and an interpretation of eachnum eral .

3 These may be compared with Bii hler’ s drawings , loc . cit. ; wi thBayley

,loc . cit . , p . 337 and plates ; and with Bayley ’

s article in the

Encyclbpoedia Britannica ,9th ed .

,art .

“Numerals .

”

24 THE HINDU-ARABIC NUMERALS

The next very noteworthy evidence of the numerals ,and this quite complete as will be seen , is found in cer

tain other cave inscriptions dating back ‘ to the first orsecond century A .D . In these, the Nasik 1 cave Inscrip

tions , the forms are as follows

oc o < e x _a 7 7 771

7 7 7:

71 77 77

From thi s time on, until the decimal system finallyadopted the first nine characters and replaced the rest ofthe Brahmi notation by adding the zero, the progres s Ofthese forms is well marked. It is therefore well to present

1 E . Senart,

“The Inscriptions in the Caves at Nasik,” Ep igraphiaIndica

,Vol . VIII

, pp . 59—96 “ The Inscriptions in the Cave at Karle,”Ep igraphia Indica, Vol . VII , pp . 4 7— 74 ; B

’

uhler,Palaeographic, Tafel

IX .

EARLY HINDU FORM S IVITH NO PLACE VALUE 25

TABLE SHOWING THE PROGR E SS OF NUMB ER FORMSIN IND IA

NUM E RA LS I 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 1000

a ASoka l NHi ll”h a s I ll xx 7 3 733373 Al i /1

l ” A, C Ad Nagari 2r (9 7 q ‘ i d ) HK T

x"7 7 7

.K u sm — :1F é o SPoc 6 u x@ x coe

“Gupta ~ ~ 5fifb§n3 3 0r 82 09 64 9771?i Valh abi KFUI

‘Q

QQ-Q V HJ JKQOQ T

~ 1 e n N ews a ak K alinga

03 8 ° 94 (I)

‘

U1Vakataka r. 1

’

d 6a Kharosth i numerals

,Asoka inscripti ons , c . 250 Senart, Notes

d ’e’

p igrap hie indienne. Given by Buhler, loc . cit .,Tafel I .

b Same, Saka inscript ions , probably of the first centu ry B .C .

Senart, loc . cit . ; Biihler, loc . cit .

0 Brahm i numerals,Asoka inscriptions , c . 250 B .C . Indian Anti

guary , Vol . VI , p . 155 seq .

4 Same,Nana Ghat inscript ions , c . 150 B .c . Bhagavanlal Indraji ,

On Ancient Nagari Numeration,loc . cit . Copied from a squ eeze of

the original .9 Same, Nasik inscription,

c .1 003 .c . Bu rgess , A rcheological SurveyRep ort, Western India Senart, Ep igrap hia Indica , Vol . VII , pp . 4 7

79,and Vol . VIII

, pp . 59—96 .

f Ksatrapa coins,c . 200 A .D . Journal of the Roya l A s iatic Society

1890, p . 639 .

g Kusana inscriptions , c . 150 A .D . Ep igrap hia Indica , Vol . I , p . 381,

and Vol . II, p . 201 .

11 Gupta Inscriptions , c. 300 A .D . to 450 A .D . Fleet,loc . cit .

,Vol . III .

1 Valhabi,c . 600 A .D . Corpus , Vol . I II .

J'

Bendall’ s Table of Numerals,in Cat. Sansk. Budd . MSS .

,British

Museum .

k Indian Antiquary , Vol . XIII , 120 ; Ep igraphia Indica , Vol . III ,127 ff .

1 Fleet,loc. cit .

[Most of these numerals are g iven by Buhler , loc . cit ., Tafel IX .]


synoptically the best—known specim ens that have c ome

down to us, and thi s is done in the table on pageWith respect to these numerals it Should firs t be noted

that no zero appears i nthe table, and as a matter"of factnone existed in any of the cases cited. It was thereforeimpossible to have any place value, and the numbers liketwenty, thirty, and other multiples of ten, one hundred,and so on , requ ired separate symbols except Where theywere written out in words . The anc ient Hindus had noless than twenty of thes e symbols?a number that was

afterward greatly increased. The following are examplesof their method of i ndicating certain numbers betweenone hundred and one thousand

3for 174 4M (x) for 191

5

T t51 for 269 6

7 v for 252

7

CE;for 400 8

‘

J’

A/ 5for 356

1 See Fleet, loc. cit . See also T . Benfey , Sanskrit Grammar, London

,1863

, p . 217 ; M . R . Kale,H igher Sanskrit Grammar, 2d ed. ,

Bom

bay , 1898, p . 110,and other authorities as cited .

2 Bayley, loc . cit., p . 335.

3 From a copper plate of 4 93 A .D .,found at Karitalai

,Central

India . [Fleet, loc . cit.,Plate XVI .] It should be stated

,however

,that

m any of these copper plates , being deeds of property, have forgeddates so as to give the appearance of antiqu ity of title. On the otherhand

,as Colebrooke long ago pointed out

,a su ccessful forgery has

to im itate the writing of the period in question, so that it becomesevidence well worth cons idering, as shown in Chapter III .

4 From a copper plate of 510 A .D .,found at Majhgawain, Central

India . [Fleet, loc . cit.,Plate XIV .]

5 From an inscription of 588 A .D .,found at BOdh-Gaya

,Bengal

Pres idency . [Fleet, loc . cit .,Plate XXIV .]

6 From a copper plate of 571 A .D .,found at Maliya, Bombay Presi

deney . [Fleet, loc . cit. , Plate XXIV .]7 From a Bijayagadh pillar inscription of 372 A .D . [Fleet, loc . cit. ;

Plate XXXVI, C . ]

8 From a copper plate Of 434 A .D . [IndianAntiquary , Vol . I , p .


symbol, whi ch is practically identical with the symbolsfound commonly in India from 150 B .C . to 700 A .D . In

the cursive form it becomes 2 , and this was frequentlyused for two in Germany until the 18th century. It

finally went

'

into the modern form 2, and the i n the

same way became our 3.

There is , however, considerable ground for interestingspeculation with respect to these firs t three numerals .

The earliest Hindu forms were perpendicular. In the

NanaGhat inscriptions they are vertical. But long beforeeither the As’ oka or the Nana Ghat inscriptions the Chinese were using the horiz ontal forms for the firs t threenumerals , but a vertical arrangement for four.1 Nowwhere did China get these form s ? Surely not fromIndia, for she had them,

as her monuments and literature 2 show, long before the Hi ndus knew them. The

tradition is that China brought her civilization aroundthe north of Tibet, from M ongolia, the prim itive habitatbeing Mesopotamia, or possibly the oases of Turkestan .

NOW what num erals did Mesopotamia u se The Babylonian system, s imple in its general pri nciples but verycomplicated in many of its details , is now well known.

3

In particular, one, two, and three were represented byvertical arrow-heads . Why, then, did the Chinese write

1 It seems evident that the Ch inese fou r,cu riously enough called

eight in the m outh ,” is only a cu rs ive ll“.

2 Chalfont,F . H .

,M emo irs of the Carnegie M use um

,Vol . IV

,110 . 1

J . Hager, An E xp lanation of the E lementary Characters of the Ch inese,London

,1801 .

3 H. V . Hilprecht, Mathematica l,Metrological and Chronologica l

Tablets from the Temp le L ibrary at N ippur, Vol . XX, part I , of Series

A,Cune i form Texts Published by the Babylonian E xped i tion of the

Univers ity of Pennsylvania, 1906 ; A . E isenloh r, E in a ltbabylonischerFelderp lan, Leipzig, 1906 ; Maspero , Dawn of C ivi lization, p . 773.


theirs horiz ontally The problem now takes a new interest when we find that these Babylonian forms were notthe primitive ones of this region , but that the earlySumerian forms were horiz ontal.1What interpretation shall be given to these facts ?

Shall we s ay that it was mere accident that one peoplewrote one vertically and that another wrote it horiz ontally ‘? This may be the case ; but it may also be thecase that the tribal migrations that ended in the Mongoli nvasion of China started from the Euphrates While yetthe Sumerian c ivilization was prominent, or from somecommon source in Turkes tan, and that they carried tothe E ast the primitive numerals of their anc ient home,the firs t three, these being all that the people as a Wholeknew or needed. It is equally poss ible that these threehorizontal forms represent primitive s tick-laying, the mostnatural pos ition of a stick placed in front of a calculatorbeing the horiz ontal one. When , however, the cuneiformwriting developed more fully, the vertical form may havebeen proved the eas ier to make, so that by the time themigrations to the Wes t began these were i n u se, andfrom them came the upright forms of Egypt, Greece,Rome, and other M editerranean lands , and those Of

As’ oka’

s time in India. After Asoka, and perhaps amongthe merchants Of earlier centuries , the horiz ontal formsmay have come down into India from China, thus givingthose of the NanaGhat cave and of later inscriptions . Thisis in the realm of speculation , but it is not improbable thatfurther epigraphical s tudies may confirm the hypothesis.

1 SirH . H. Howard,On the E arl iest Inscriptions from Chaldea,

Proceedings of the Society of B iblica lA rchaeology , XXI , p . 301,L ondon

1899 .


As to the numerals above three there have been verymany conjectures. The figure one of the Demotic lookslike the one of the Sanskrit, the two ( reversed) l ike that ofthe Arabic, the four has some resemblance to t hat in theNas ik caves , the five ( reversed) to that on the Ksatrapa

coins , the nine to that of the Ku s ana inscriptions , andother points of S imilarity have been imagined. Somehave traced resemblance between the Hieratic five and

seven and those of the Indian inscriptions. There havenot, therefore, been wanting those who asserted an Egyptian origin for these numerals 1 There has already beenmentioned the fact that the Kharosthi numerals wereformerly known as Bactrian, IndO-Bactrian , and Aryan.

Cunningham 2 was the first to sugges t that these numerals were derived from the alphabet of the Bactrianciviliz ation of E astern Pers ia, perhaps a thousand yearsbefore our era, and in this he was supported by the

scholarly work of S ir E . Clive ' Bayley,3 who in turnwas followed by Canon Taylor.4 The resemblance hasnot proved convincing, however, and Bayley’ s drawings

1 For a bibliography of the principal hypotheses of th is natu re see

Buh ler,loc . cit.

, p . 77 . B ijhler (p . 78) feels that of all these hyp othesesthat wh ich connects the Brahm i with the Egy ptian num erals is themost plausible, although he does not addu ce any convincing proof .Th . Henri Martin

,

“L es s ignes numéraux et l’ arithm é tique chez lespeuples de l

’antiqui té et du moyen age ” (being an examination of

Cantor ’ s Mathematische Beitrage zum Cu lturleben der Annali dimatematica pura ed app licata , V Ol .V

,Rom e

,1864 , pp . 8

,70 . Also

,same

author, Recherches nouvelles sur l’origine de notre systeme de nu

m é ration écrite,

” Revue A rchéologique, 1857, pp . 36,55. See also the

tables given later in th is work.

2 Journal of the Roya lA s iatic Society , Bombay Branch, Vol .XXIII .3 Loc . cit.

,reprint, Part I , pp . 12, 17 . Bayley

’s deductions are

generally regarded as unwarranted .

4 The A lphabet, London, 1883, Vol . II , pp . 265,266

,and The A cad

emy of Jan. 28,1882.


have been criticized as being affected by hi s theory. Thefollowing is part of the hypothesis 1

Numera l H indu Bactrian Sanskri t

ch atur, Lat. q uattuor

panch a , Gk. 1réw e

Th e s and s are interch anged as

occas ionally in N .W. Indi a

Buhler 2 rejects this hypothesis, stating that in fourcases ( four, six , seven , and ten) the facts are abs olutelyagainst it.While the relation to ancient Bactrian forms has been

generally doubted, it is agreed that most of the numeralsresemble Brahmi letters , and we would naturally expectthem to be initials.3 But, knowing the ancient pronunciation of most of the number names ,4 we find this notto be the case. We next fall back upon the hypothesis

1 Taylor,The A lp habet, loc . cit .

,table on p . 266.

2 Buh ler,On the Origin of the Indian Brahma A lp habet, Strassbu rg,

1898,footnote

, pp . 52,53.

3 Albrecht Weber,H istory of Indian L iterature, English ed .

,Bos

ton,1878

, p . 256 : “ The Indian figures from 1—9 are abbreviated formsof the initial letters of the numerals them selves the zero

,too

,

has arisen out of the first letter of the word sunya (empty) (it occurseven in Pingala) . It is the decimal place value of these figures whichgives them s ignificance.

” C . Henry,Sur l

’origine de quelques nota

tions mathématiques ,” Revue A rchéologique, June and July, 1879, attempts to derive the Boeth ian form s from the initials of Latinwords .

See also J . Prinsep,“ E xam ination of the Inscriptions from Girnar in

Gu jerat,and Dhau li in Cuttach

,

” Journal of theA s iatic S ociety of Bengal, 1838, especially Plate XX , p . 348 this was the first work on the

subject.4 Buhler

,Palaeographic, p . 75

, gives the list, with the list of letters(p. 76) corresponding to the number symbols ,


that they represent the order of letters 1 in the ancientalphabet. From what we know of thi s order, however,there seems also no basis for this assumption. We have,therefore, to confess that we are not certain that thenumerals were alphabetic at all, and if they were alphabetie we have no evidence at present as to the basis ofselection. The later forms may possibly have been alphabetical expressions of certain syllables called aksaras ,

which possessed in Sanskrit fixed numerical values?butthis is equally uncertain with the rest. Bayley alsothought 3 that some of the forms were Phoenician, as

notably the u se Of a c ircle for twenty, but the resemblance is in general too remote to be convincing.

There is also some Slight possibility that Chinese infiuence is to be seen i n certai n Of the early forms of Hindunumerals .4

1 For a general discuss ion of the connection between the numeralsand the different kinds Of alphabets , see the articles by U . Ceretti

,

“ Sulla origine delle cifre numeral i moderne,

” R ivista di fis ica , mate

matica e scienze natura li,Pisa and Pavia

,1909

,anno X ,

numbers 114,

118,119

,and 120

,and continuation in 1910 .

2 Th is is one of Buhler ’ s hypotheses . See Bayley , loc . cit.,reprint

p . 4 ; a good bibliography Of original sou rces is given in th is work, p . 38.

3 Loc . cit .,reprint, part I , pp . 12

,17 . See also Burnell

,loc . cit .

p . 64,and tables in plate XXIII .

4 This was asserted by G . Hager (M emoria su lle cifre arabicheM ilan

,1813

,also published in Fundgruben des Orients , V ienna, 1811,

and in B ibliotheque Britannique, Geneva, See also the recentarticle by M aj or Charles E . Woodru ff

,

“ The Evolution of ModernNumerals from TallyMarks

,

”AmericanMathematica lMonthly , August

September, 1909 . Biernatzki,

“Die A rithmetik der Ch inesen,

” Cretle’ s

Journa l fiir die reine und angewandte Mathematik, Vol. LII , 1857,pp . 59—96

,also asserts the priority of the Chinese claim for a place

system and the zero,but upon the fiims iest au thority . Ch . de Para

vey , E ssa i sur l’origine unique et h iéroglyp hique des chifires et des lettres

dc tous les p eup les , Paris , 1826 ; G . Kleinwachter,

“The Origin of theA rab ic Numerals

,

” Ch ina Review,Vol . X I

,1882—1883, pp . 379- 381,

Vol . XII, pp . 28—30 ; B iot, “ Note sur la conna issance que les Ch inois

Ont euc de la valeur de pos ition des ch i ffres ,” JournalA siatique, 1839,


More absurd is the hypothesis of a Greek origin , supposedly supported by derivation of the current symbOlsfrom the first nine letters of the Greek alphabet ? Thisdifficult feat is accomplished by twisting some of theletters , cutting Off, adding on , and effecting other changesto make the letters fit the theory. This peculiar theorywas first set up by Dasypodiu s

2(Conrad Rauhfu ss) , and

was later elaborated by Huet.3

pp . 497—502. A . Terrien de Lacouperie,“The Old Numerals

,the

Counting-R ods and the Swan-Pan in China,

” Numismatic Chronicle,Vol . III pp . 297—340

,and Crowder B . Moseley

,

“Numeral Characters : Theory of Origin and Development

,

” American Antiquarian,Vol . XXII

, pp . 279—284,both propose to derive our numerals from

Ch inese characters,in much the same way as is done by Major Wood

ru ff,in the article above c ited .

1 The Greeks, probably following the Semitic cu stom

,used nine

letters of the alphabet for the numerals from 1 to 9,then nine others

for 10 to 90,and fu rther letters to represent 100 to 900 . A s the ord i

nary Greek alphabet was insufficient,containing only twenty-fou r

letters,an alphabet of twenty-seven letters was used .

2 Institutiones mathematicae, 2 vols ., Strassbu rg, 1593— 1596, a some

what rare work from wh ich the following quotation is takenQu is est harum Cyphrarum autorA qu ibus hae us itatae syphrarum notae s int inventae : hactenus

incertum fuit : meo tameh iudicio, qu od exiguum esse fateor : a grae

cis librarijs (quorum Olim magna fu it copia) literae Graecorum qu ibusveteres Graeci tamquam numerorum notis sunt us i : fuerunt corruptae .

vt ex his licet Videre .

Graecorum Literae corruptae .

“ Sed qua ratione graecorumo< l

" J'

s , 5“

2 ” A 9 literae ita fuerunt corrup tae?“ Finxerunt has corruptas

I 61 CU

ff 5.

9 < V 7 Graecorum literarum notas : vel

abiectione vt innota binarijnu2 3 4 S 6 7 5

/ C

7 meri,vel additione vt in terna

rij,vel inuersione vt in septe

marij,numeri nota

,nostrae notae

, qu ibus hod ie utimur : ab his soladifferunt elegantia, vt apparet .

”

See also Bayer,H istoria regni Graecorum Bactriani , St. Petersbu rg,

1738, pp . 129—130

, qu oted by M artin,Recherches nouvelles

,etc .

,loc . cit .

3 P . D . Huet,Demonstratlo evangelica , Paris , 1769, note to p . 139 on

p. 64 7 : “Ab Arabibus vel ab Indis inventas esse, nonvulgus eruditorum


A bizarre derivation based upon early Arabic ( c. 104 0A .D .) sources is given by Kircher in his work 1 0 11 numbermysticism. He quotes from Abenragel,2 giving the Ara

bic and a Latin translation 3 and stating that the ordinaryArabic forms are derived from sectors of a circle, 69 .

Ou t of all these confl icting theories , and from all the

resemblances seen or imagined between the numerals ofthe West and those of the E ast, what conclus ions are we

prepared to draw as the evidence now stands Probablynone that is satisfactory. Indeed, upon the evidence at

m odo,sed doctiss im i qu ique ad hanc diem arbitrati sunt. Ego vero

falsum id esse,merosque esse Graecorum characteres aio ; a librari is

G raecae linguae ignaris interpolatos , et diuturna scribendi consuetu

d ine corruptos . Nam primum I apex fuit, seu Virgula, nota t at t oos . 2,

est ipsum B extremis su is truncatum . y, si in sinistram partem ineli

naveris cauda mutilaveris s inistrum cornu s inistrorsum fiexeris,

fiet 3 . Res ipsa loqu itur 4 ips iss imum esse A,cu jus crus sinistrum

erigitur Ka'r ci. xdeerov

,infra basim descendit ; bas is vero ipsa ultra

crus producta em inet. V ides quam 5 s im ile s it 7 45a; infimo tantum

s em icircu lo, qu i s inistrorsum patebat, dextrorsum converso . én lanuov

Baaquod ita notabatur g,rotundato ventre

, pede detracto, peperit 7 0 6.

Ex Z basi sua mutilato,ortum est 7 0 7 . Si H infiexis introrsum ap i

c ibus in rotundiorem commodiorem formam mutaveris,exurget TO8.

At 9 ips iss imum est s .

”

I . We idler,Sp ici legium observationum ad h istoriam notaram nu

meralium,Wittenberg, 1755, derives them f rom the Hebrew letters ;

Dom Augustin Calmet,Recherches sur l ’Origine des ch i ffres d’

arith

m é tiqu e,” M émo ires p our l’histoire des sciences ci des beaua: arts , Tre

voux,1707 (pp . 1620—1635

,with two plates) , derives the cu rrent symbols

from the Romans,stating that they are rel ics of the ancient “ Notae

Tironianae.

” These notes ” were part of a system of shorthand invented

,or at least perfected, by Tiro, a slave who was f reed by C icero .

L . A . Sedillot,

“ Sur l’origine de nos ch iffres

,

” A tti dell’A ccademia

p ontificia dei nuovi L incei , Vol . XVII I , 1864—1865, pp . 316—322, derivesthe A rab ic forms from the Roman numerals .

1 Athanasius Kircher,Arithmoloyia s ive De abditis Numerorum

mysterijs qua origo, antiqu itas (2fabrica Numerorum exp onitur, Rome,1665.

2 See Suter, Die M athematiker und A stronomen der A raber, p . 100 .

3 E t h i numeri sunt numeri Indiani,a Brachmanis Indiae Sapi

entibus ex figura circuli secti inuenti ,”


however, without any historical evidence for support,have no place i n a serious dis cuss ion of the gradual evolu tion of the present numeral forms ?

TA BLE O F CERTA IN E AS TERN SYS TE M S

S iam O fl m m é é b ‘y d’ w q o

o 9 J 19 9 9 63 9 0 6 903M alabar

.90'

(C;PL /KJ é 7Tfifl d /2 "AU

4 T ibet 7 2 U'G

'

O) (A,

70 '

5Ceylon 67 W w vO GB/

QC’VN5M alayalam P (L 0 2. 6

0

3 9 1-1 {7 8 w

1 For a purely fanci fu l derivation from the corresponding numberof strokes

,see W .W . R . Ball

,A ShortA ccount of theH istory of M athe

matics,Ist ed .

,London

,1888

, p . 14 7 s im ilarly J . B . Reveillaud,E ssa i

sur les chifres arabes,Paris

,1883 ; P . Voizot,

“ Les ch i ffres arabes et

leu r origine,” La Nature,1899

, p . 222 G . Dumesnil,

“De la forme des

ch i ffres usuels,

” Annales de l’université de Grenoble

,1907

,Vol . XIX

,

pp . 657—674 , also a note in Revue A rchéologique, 1890, Vol . XVIpp . 342—34 8 ; one Of the earl iest references to a poss ible derivationfrom points is in a work by Bettino ent itled Ap iaria universae p h ilosop hiae mathematicae in qu ibus p aradoxa ci noua machinamenta ad usu s

exim ios traducta , ci faci llimis demonstrationibus confirmata , Bologna,1545

,Vol . II , Apiarium XI , p . 5.

2 A lphabetum Barmanum,Romae

,MDC CLxxvr

, p . 50 . The 1 is evi

dently Sanskrit, and the 4 , 7 , and poss ibly 9 are from India .

3 A lphabetum Grandonico-Malabaricum,Romae

,M DCC LX X I I

, p . 90 .

The zero is not u sed,but the symbols for 10, 100, and so on

,are jo ined

to the units to make the h igher numbers .

4 A lphabetum Tangutanum ,Romae

,M DCCLX X I I I

, p . 107 . In a Ti

betan MS . in the l ibrary of Professor Sm ith , probably Of the e ighteenth centu ry , substantially these forms are given.

5 Bayley , loc . cit ., plate 11 . S im ilar forms to these here shown

,and

numerou s other forms found in India,as well as those of other oriental

countries , are given by A . P. Pihan,Expose des s ignes de numeration

us ités chez les peup les orientaua: anc iens etmodernes , Paris , 1860 .

EARLY HINDU FORMS WITH No PLACE VALUE 37

We m ay summarize this chapter by saying that no oneknows What suggested certain of the early numeral formsused i n India. The origin of some is evident, but theorigin of others will probably never be known . There isno reason why they should not have been invented bysome priest or teacher or guild, by the order of someking, or as part of the mys ticism of some temple. Whatever the origin, they were no better than scores of otherancient systems and no better than the present Chi nesesystem when written without the zero, and there wouldnever have been any chance of their triumphal rogres s

westward had it not been for this relative symbol.There could hardly be demanded a s trong f of theHindu origin of the character for zero than this , and toit further reference will be made in Chapter IV .

CHAPTER III

LATER HINDU FORMS , WITH A PLACE VALUE

Before speaking of the perfected Hindu numerals withthe zero and the place value, it is necessary to considerthe third system mentioned on page 19, the word andletter forms . The u se of words With place value beganat 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 , numbersare represented by the names of certain objects or ideas .For example, zero is represented by the void ( s

'unya)or “ heaven-space ( ambara dkdsa) one by stick( rupa) ,

“ moon ( inclu sasin) , earth “ beginning ( ddi) ,

“ Brahma,” or, i n general, by anythingmarkedly unique ; two by “ the twins (yama) , “ hands(kara) ,

“eyes (nayana) , etc. ; four by “ oceans ,” five

by “ senses ”

(vi saya) or “arrows ” ( the five arrows of

Kamadeva) ; S ix by “ seasons ” or “ flavors ” ; seven bymountain ( aga) , and so on ? These names , accommo

dating themselves to the verse in which scientific workswere written , had the additional advantage of not admitting, as did the figures , easy alteration , S ince any changewould tend to disturb the meter.

1 Buhler,loc . c it.

, p . 80 ; J . F . Fleet,Corpus inscrip tionum Indica

rum,Vol . III

,Calcutta

,1888. L ists of such words are given also by

Al-Biri'

ini in his work India ; by Burnell,loc . cit. ; by E . Jacquet,

Mode d’express ion symbol ique des nombres employé par les Ind iens ,

les Tibétains et les Javanais,

” Journal A s iatique,Vol .XVI , Paris , 1835.

38

LATER HINDU FORMS WITH A PLACE VALUE 39

As an example of this system, the date Saka Samvat,867

”

(A .D . 945 or is given by “giri

—rasa-vasu ,”

meaning the mountains ” ( seven) , the flavors ( s ix) ,and the gods Vasu of which there were eight. In readi ng the date these are read from right to left ? Theperiod of i nvention of this system is uncertain . The firsttrace seems to be in the Srau tasutra of Katyayana andLatyayana

? It was certainly known to Varaha-M ihira

( d. for he used it in the E rhat-S amhita.4 It has

also been asserted 5 that Aryabhata ( c. 500 AD .) wasfamiliar with this system, but there is nothing to provethe statement.6 The earliest epigraphical examples ofthe system are found i n the Bayang (Cambodia) inscriptions of 604 and 624 AD ?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 repre

sented by the letters as given in the following table1 2 3 4 5 7 8 9 0

k kh g gh 11 ch j jh ii

t th (1 dh n th (1 db np ph b bh my r 1 V S h l1 Th is date is given by Fleet, loc . cit .

,Vol . 111

, p . 73,as the earliest

ep igraph ical instance of th is u sage in India proper .2 Weber

,Indische S tudien

,Vol . VIII , p . 166 seq .

3 Journal of the Royal A s iatic S ociety , Vol . I p . 407 .

4 VIII,20

,21 .

5 Th . H . Mart in,Les s ignes numéraua: Rome

,1864 ; Lassen,

Indische A lterthumskunde, Vol . 11, 2d cd .,Leipz ig and London

,1874

,

p . 1153 .

6 But see Bu rnell, loc . cit.,and Thibaut, A stronomie

,A strologie und

Mathematik, p . 71 .

7 A . Barth , Inscriptions Sanscrites du Cambodge, in theNotices etextra its des M ss . de la B ibliotheque nationa le, Vol . XXVII

,Part I

, pp . 1

180,1885; see also numerous articles inJournalA siatique, byAymonier.


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 wordfor mnemonic purposes . For example, the word

2 3 1 5 6 5 1

kha yontyanme sa mdp a

has the value reading from right to left ? This ,the oldest specimen ( 1184 AD .) known Of this notationis given in a commentary on the Rigveda, representi ngthe number of days that had elapsed from the beginningof the Kaliyuga. Burnell 2 states that this system iseven yet in u se 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 , S iam, and Burma,having also had its rise i n southern India. In this thethirty-four consonants when followed by a ( as ka la)designate the numbers 1— 34 ; by a ( as kc la) , thosefrom 35 to 68 ; by i (ki li) , those from 69 to 102,inclusive ; and so on.

3

As already stated, however, the Hindu system as thusfar des cribed was no impr

\ovement upon many others of

the ancients , such as those Used by the Greeks and the

Hebrews . Having no zero, it was impracticable to des ignate the tens , hundreds, and other units of higher orderby the same symbols used for the units from one to ni ne.

In other words, there was no poss ibility of place valuewithout some further improvem ent. So the Nana Ghat

1 Buhler,loc . cit.

, p . 82.

2 Loc . cit., p . 79 .

3 B iih ler,loc . c it.

, p . 83 . The Hindu astrologers still u se an alpha

betical system of numerals . [Bu rnell, loc . cit., p .

LATER HINDU FORMS WITH A PLACE VALUE 4 1

symbols required the writing of thousand seven twentyfour about like T 7, tW,

4 in modern symbols , insteadof 7024 , in which the seven Of the thousands , the twoof the tens ( concealed in the word twenty, being originally “ twain of tens , the -ty signifying ten) , and thefour of the units are given as spoken and the order ofthe unit ( tens , hundreds , etc.) is given by the place. T0complete the system only the zero was needed ; but itwas probably eight centuries after the NanaGhat inscriptions were cut, before this important symbol appeared ;and not until a considerably later period did it becomewell known . Who it was to whom the i nvention is due ,or Where he lived, or even in what century, will probablyalways remain a mystery ? It is possible that one of theforms of ancient abacus suggested to some Hindu astronomer or mathematician the use of a symbol to stand forthe vacant line when the counters were removed . It iswell established that in different parts of India the namesof the higher powers took different forms, even the orderbeing interchanged ? Nevertheless , as the significance ofthe 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 theultimate introduction of a zero symbol for this word.

To enable us to appreciate the force of this argumenta large number, 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.

1 Well cou ld Ramu s say , Qu icunq ; au tem fuerit inventor decemnotarum laudem magh am meru it .

”

2 Al-B i runi gives lists .

4 2 THE HINDU—ARABIC NUMERALS

Modern American reading, 8 billion, 4 4 3 million, 682thousand

,155.

Hindu , 8 padmas, 4 vyarbudas , 4 kOtis , 3 prayu tas ,

6 laksas , 8 ayu tas , 2 sahasra, 1 Sata, 5 dasan, 5.

Arabic and early German, eight thousand thousandthousand and four hundred thousand thousand and fortythree thousand thousand, and six hundred thousand and

eighty-two thousand and one hundred fifty-five ( or fiveand fifty) .

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

As Woepcke1 pointed out, the reading of numbers of

this kind shows that the notation adopted by the Hindustended to bring out the place idea. N 0 other languagethan the Sanskrit has made such consistent application ,in numeration , of the decimal system of numbers. Theintroduction of myriads as in the Greek, and thousandsas in Arabic and in modern numeration , is really a stepaway from a decimal scheme. So in the numbers belowone hundred in English, eleven and twelve are out ofharmony with the rest of the -teens , while the naming ofall the numbers between ten and twenty is not analogousto the naming of the numbers above twenty. To conformto our written system we should have ten-one, ten

-two,ten-three, and so 0 11 , as we have twenty-one, twenty-two,and the like. The Sanskrit is cons istent, the units , however, preceding the tens and hundreds . Nor did any .

other ancient people carry the numeration as far as didthe Hi ndus ?

1 Propagation, loc . cit ., p . 4 43 .

2 See the qu otation from The L ight of A s ia in Chapter I I , p . 16.

4 4 THE HINDU— ARABIC NUMERALS

void ? If he refers to a symbol this would put the zeroas far back as 500 A .D ., but of course he m ay have re

ferred merely to the concept of nothingness .A little later, but also in the s ixth century, Varaha

M ihira2 wrote a work entitled E rhat S amhitd 3 in whichhe frequently uses sunya in speaking of numerals , sothat it has been thought that he was referring to a definite symbol. This , of course, would add to the probability that Aryabhata was doing the s ame.

It should also be mentioned as a matter of interes t, andsomewhat related to the ques tion at issue, that Varaha

M ihi ra used the word-system with place value 4 as ex

plained above.

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

5

At the Opening of the next century ( c. 620 A .D .) Bana 6wrote of Suband hu s ’ s Vdsavadattd as a celebrated work

loc . cit., p . 97 ; L . R odet, Sur la veritable s ignification de la notat ion

num érique inventée parAryabhata,” Journa lA s iatique, Vol . XVI

pp . 4 40—485. On the two Aryabhatas see Kaye, B ibl. M ath .,Vol . X

p . 289.

1 Using kha, a synonym of sanya . [Bayley ,loc . cit .

, p . 22,and L .

Rodet,Journal A s iatique, Vol . XVI p .

2 Varaha—M ih ira,Pa ii casiddhantika, translated by G . Thibaut and

M . S . Dvivedi,Benares

,1889 ; see Bali ler, loc . cit .

, p . 78 ; Bayleyloc . cit.

, p . 23.

3 Brhat Samhita,translated by Kern,

Journal of the Royal A s iaticSociety, 1870 4 1875.

4 It is stated by Bali ler in a personal letter to Bayley (loc . cit ., p . 65)

that there are hundreds of instances of th is u sage in the Bfl tat Sam

h ita. The system was also u sed in the Pancas iddhantika as early as

505A .D . [Bijhler, Pa laeograph ic, p . 80,and Fleet

,Journal of the Royal

A s iatic Society , 1910 , p .

5 Cantor,Gesch ichte der Mathematik

,Vol . I p . 608.

6 Buh ler,loc . cit.

, p . 78.

LATER IIINDU FORM S WITH A PLACE VALUE 45

and mentioned that the stars dotting the sky are herecompared with zeros , these being points as in the modern Arabic system. On the other hand, a strong argument agains t any Hindu knowledge of the symbol zeroat this time is the fact that about 700 A .D . the Arabsoverran the province of S ind and thus had an opportu

nity of knowing the common methods used there forwriting numbers . And yet, when they received the complete sys tem in 776 they looked upon it as somethingnew ? Such evidence is not conclus ive, but it tends toShow that the complete system was probably not in common u se in India at the beginning Of the eighth century.

On the other hand , we must bear in mind the fact thata traveler in Germany in the year 1 700 would probablyhave heard or seen nothing of decimal fractions, althoughthese were perfected a century before that date. The

é lite of the mathematicians may have known the zeroeven i n Aryabhata

’

s time, while the merchants and thecommon people may not have grasped the s ignificance ofthe novelty until a long time after. On the whole, theevidence seems to point to the west coast of India as theregion where the complete sys tem was first seen ? AS

mentioned above, traces of the numeral words with placevalue, which do not, however, abs olutely require a decimal place-system of symbols , are found very early inCambodia, as well as in India.

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

1 Bayley, p . 38.

2 Noviomagus , in his De numeris libri duo,Paris

,1539

,confesses h is

ignorance as to the origin of the zero , but says : “D . Henricus Grau iu s,

vir Graece Hebraice exime doctu s,Hebraicam originem ostendit

,

”

adding that Valla “ Indis Orientalibus gent ibus inventionem tribu it.”


is some question. Colebrooke 1 in 1807 warned agains tthe poss ibility of forgery i nmany of the ancient copperplate land grants . On this account Fleet, in the IndianAntiquary?dis cusses at length this phase of the work ofthe epigraphists in India, holding that many of theseforgeries were made about the end of the eleventh cen

tury. Colebrooke 3 takes a more rational View of theseforgeries than does Kaye, who seems to hold that theytend to inval idate the whole Indian hypothes is. “ Buteven Where that m ay be suspected, the historical uses ofa monum ent fabricated so much nearer to the times towhich it assumes to belong, will not be entirely supers eded. The necess ity of rendering the forged grant credible would compel a fabricator to adhere to history, andconform to established notions : and the tradition, whichprevailed in his time, and by which he must be guided,wou ld

'

probably be so much nearer to the truth, as itwas less rem ote from the period which it concerned.

” 4

B iihler 5 gives the copper-plate Gurjara inscription Of

Cedi-samvat 34 6 (595AD .) as the Oldest epigraphicalu se of the numerals 6 “ in which the symbols correspondto the alphabet numerals of the period and the place.

V incent A . Smith 7 quotes a stone inscription of 815A .D .,

dated Samvat 872. So F . Kielhorn in the Ep igraphia

Indica 8 glves a Pathari pillar i nscription of Parabala,datedVikrama-samvat 91 7 ,which corresponds to 861A .D .,

1 See E ssays , Vol . II, pp . 287 and 288.

2 Vol . XXX , p . 205seqq .

3 Loc . cit., p . 284 seqq .

4 Colebrooke,loc . cit.

, p . 288.5Loc . cit. , p . 78.

6 Hereafter,unless expressly stated to the contrary ,

we shall us e

the word numerals ” to mean numerals with place value .

7 “The Gu rjaras of Rajputana and Kanauj ,” inJournal of the RoyalA s iatic Soc iety , January and April, 1909.

3 Vol. IX,1908

, p . 248.

LATER HINDU FORMS WIT II A PLACE VALUE 4 7

and refers also to another copper-plate inscription datedVikrama-samvat 813 ( 756 The inscription quotedby V. A . Smith above is that given by D . R. Bhandarkar,1 and another is given by the same writer as ofdate Saka-samvat 715 ( 798 being incised 0 11 apilaster. Kielhorn 2 also gives two Copper-plate inscriptions of the time of Mahendrapala of Kanauj , V alhabi

samvat 574 ( 893 A .D .) and V ikrama-samvat 956 ( 899That there should be any ins criptions of date as

early even as 750 A .D . , would tend to Show that the system was at least a century older. As w ill be shown inthe further development, it was more than two centuries after the introduction of the numerals into Europethat they appeared there upon coins and inscriptions .While Thibaut 3 does not consider it necessary to quoteany specific instances of the u se of the numerals, hestates that traces are found from 590 AD . 0 11 .

“ Thatthe system now in u se by all c ivilized nations is of Hinduorigin cannot be doubted ; no other nation has any claimupon its dis covery, especially since the references to theorigin of the system which are found i n the nations ofwestern Asia point unanimously towards India.

” 4

The testimony and Opinions Of men l ike Biihler, Kielhorn , V . A . Smith, Bhandarkar, and Thibaut are entitledto the mos t serious consideration. As authorities onancient Indian epigraphy no others rank higher. Theirwork is accepted by Indian scholars the world over, andtheir united judgment as to the rise of the system witha place value — that it took place in India as early as the

1 Ep igraphia Indica ,V ol . IX

, pp . 193 and 198.

2 Ep igraphia Indica , Vol . IX , p . 1 .

3 Loc . cit., p . 71 .

4 Th ibaut, p . 71.


s ixth century A .D . mus t s tand unless new evidence ofgreat weight can be submitted to the contrary.

Many early writers remarked upon the diversity ofIndian numeral forms . A l-B iruni was probably the firstnoteworthy is also Johannes Hispalens is , 1 who gives thevariant forms for seven and four. We insert on p . 4 9 a

table of numerals used with place value. While the chiefauthority for this is Biihler,2 several specimens are givenwhich are not found in his work and which are of unusualinterest.The Sarada forms given in the table u se the circle as a

symbol for 1 and the dot for zero. They are taken fromthe paging and text of The K ashmirian Atkarva Veda ,3

of which the manuscript used is certainly four hundredyears old. S imilar forms are found in a manus cript belonging to the Univers ity of T iibingen. Two other seriespresented are from Tibetan books in the library of oneof the authors.For purposes of comparison the modern Sanskrit and

Arabic numeral forms are added.

Sanskrit,

A rab ic ,

1 “ E st autem in aliqu ibus figurarum i starum apud mul tos diuers i

tas . Qu idam enim septimam hanc figuram representant,” etc . [Bon

compagni , Trattati , p . Enestrom has shown that very l ikely th iswork is incorrectly attributed to Johannes Hispalensis . [BibliothecaMathematica

,Vol . IX p .

2 Indische Pa laeograp h ic, Tafel IX .

3 Edited by Bloomfield and Garbe,Baltim ore, 1901, conta ining

photographic reproductions of the manuscript.

LATER HINDU FORMS WITH A PLACE VALUE 4 9

NUMERALS USED WITH PLACE VALUE

8

Bakhsali MS . See page 43 ; Hoernle, R .,The Indian Antiquary ,

Vol . XVII, pp . 1 plate Hoernle

,Verhandlungen des VI I . Inter

nationa len Orienta listen—Congresses , A rische S ection, Vienna , 1888, “Onthe Bakshali M anuscript,” pp . 127—14 7 , 3 plates ; Buhler, loc. cit.

b 3,4, 6, f rom H . H . Dhruva

,Three Land-Grants from Sankheda,

Ep igraphia Indica , Vol . II, pp . 19—24 with plates ; date 595A .D . 7

,1,5,


from Bhandarkar,Dau latabad Plates ,” Ep igrap hia Indica , Vol . IX ,

part V ; date 0 . 798 A .D .

c 8,7,2,from “Buckhala Inscription of Nao abhatta Bhandarkar

,

Ep igraphia Indica , Vol . IX, partV ; date 815A .D . 5from “The Morb i

Copper-Plate,” Bhandarkar, The Ind ian Antiq uary , Vol . II , pp . 257

258,with plate ; date 804 A .D . See Buhler, loc . cit .

d 8 from the above M orbi Copper-Plate ._4,5,7,9,and 0

,from “Asni

Inscription of M ahipala,” The Indian Antiquary , Vol . XVI , pp . 174 ~

175; inscription is on red sandstone,date 917 A .D . See Buh ler .

e 8,9,4,from Rashtrakuta Grant of Amoghavarsha,

” J. F . Fleet,

The Indian Antiquary ,Vol . XII , pp . 263- 272 ; copper-plate grant of

date c . 972 A .D . See Buhler . 7,3,5,from “ Torkhede Copper-Plate

G rant of the Time of Govindaraja of Gu jerat,” Fleet, Ep igraphia Indica , Vol . III , pp . 53—58. See Buhler .

f From “A Copper-Plate Grant of King Tritochanapala Chaii luky aof Latadesa ,” H. H. Dhruva

,Indian Antiquary , Vol . XII , pp . 196

205; date 1050 A .D . See Buhler .g Burnell

,A . C .

,South Indian Paloeography , plate XXIII , Telugu

Canarese numerals of the eleventh centu ry . See Buhler .h and i From a manu scrip t of the second half of the thirteenth

centu ry , reproduced in Della vita e delle Opere d i Leonardo Pisano ,Baldassare Boncompagni , Rome

,1852

,inA tti dell’A ccademia Pontificia

dei nuovi Lincei,anno V .

J'

and k From a fourteenth-centu ry manuscript, as reproduced inDella vita etc . ,

Boncompagni , loc . cit.

1 From a Tibetan MS . in the l ibrary of D . E . Smith .

m From a Tibetan block-book in the l ibrary of D . E . Smith .

11 Saradanum erals from The K ashm ir ianA tharva —Veda ,rep roduced

by chromophotography from the manuscrip t in .the Univers ity L ibraryat Tii bingen, Bloomfield and Garbe , Baltimore, 1901 . S omewhat s imilar forms are given under “Numération Cachemirienne,” by Pihan,Exp ose etc.

, p . 84 .


u se of this Babylonian symbol seems to have been in thefractions , 60ths , 3600ths , etc.

,and somewhat similar to

the Greek u se of o, for obBe’

v,with the meaning vacant.

“The earliest undoubted occurrence of a zero in India isan inscription at Gwal ior, dated Samvat 933 ( 876Where 50 garlands are mentioned ( line 50 is written

aO . 270 ( line 4 ) is written 2?o .

” 1 TheBakhsali Manuscript 2 probably antedates this , using the point or dot asa zero symbol. Bayley mentions a grant of Jaika Rashtraki

’

i ta of Bharuj, found at Okamandel, of date 738 A .D .,

which contains a zero, and also a coin with indistinctGupta date 707 ( 897 but the reliability of Bayley’ s work is ques tioned. As has been noted, the appearance of the numerals in i nscriptions and on coins wouldbe of much later occurrence than the origin and writtenexposition of the system. From the period mentionedthe spread was rapid over all of India, save the southernpart, where the Tam il and M alayalam people retain theold system even to the present day.

3

Aside from its appearance in early inscriptions , thereis still another i ndication of the Hindu origin of the symbol i n the special treatment of the concept zero in the

early works on arithmetic. Brahmagupta, who lived i nU jjain,

the center of Indian as tronomy,4 in the early part1 From a letter to D . E . Smith , from G . F . Hill of the British

Museum . See also his monograph “On the E arly Use of A rabic Num erals in Eu rope,” in Archaeologia , Vol . LXII p . 137 .

2 R . Hoernle,

“The Bakshali Manuscript,” Indian Antiquary, Vol .XVII

, pp . 33—4 8 and 275—279,1888 ; Th ibaut, Astronomie

,A strologie

und M athematik, p . 75; Hoernle, Verhandlungen, loc . cit., p . 132.

3 Bayley , loc . cit . ,Vol . XV

, p . 29. Also Bendall, “On a System ofNumerals used in South India ,” Journa l of the Royal A s iatic Society ,1896

, pp . 789— 792 .

4 V . A . Smith,The E arly History of India, 2d ed .

,Oxford

,1908

,

p . 14 .

THE SYMBOL ZERO 53

of the seventh century, gives in his arithmetic 1 a distincttreatment of the properties of zero. He does not discussa symbol, but he shows by his treatment that in someway zero had acquired a special s ignificance not found inthe Greek or other ancient arithmetics . A still morescientific treatment is given by Bhaskara,2 although i none place he permits him self an unallowed liberty individing by zero . The most recently discovered workof ancient Indian mathematical lore , the Ganita-SaraSangraba 3 of Mahaviracarya ( c. 830 whi le it doesnot u se the numerals with place value, has a sim ilar discu ss ion of the calculation with zero .What suggested the form for the zero is, of course,

purely a matter of conjecture. The dot, whi ch the Hindu s us ed to fi ll up lacunae in their manuscripts , much aswe indicate a break in a sentence,4 would have been amore natural sym bol ; and this is the one which the Hin~

dus first used .

5 and whi ch most Arabs u se to-day. Therewas also used for thi s purpose a cross , like our X ,

and thisis occasionally found as a z ero symbol.6 In the Bakhs '

ali

manus cript above mentioned, the word sunya , with thedot as its symbol, is used to denote the unknown quantity, as well as to denote zero. An analogous u se of the

1 Colebrooke,A lgebra , with A rithmetic and M ensu ration

, from the

S anskrit of Brahmegup ta and Bhdscara,London

,1817

, pp . 339—340 .

2 Ib id ., p . 138.

3 D . E . Sm ith,in the B ibliotheca Mathematica

,Vol . IX pp . 106~

110 .

4 A s when we u se th ree dots5 “ The Hindus call the nought explicitly sunyabindu the dot

marking a blank,

’and about 500 A .D . they marked it by a simple dot ,

which latter is comm only u sed in inscriptions and MSS . in order tomark a blank

,and wh ich was later converted into a small circle .

”

[Buhler , On the Origin of the Indian A lp habet, p . 53,note .]

6 Fazzari,Dell’ origine delle p arole zero 6 cifra , Naples , 1903.


zero, for the unknown quantity in a proportion , appearsin a Latin manuscript of some lectures by GottfriedWolack in the Univers ity of E rfurt in 14 67 and

The usage was noted even as early as the eighteenthcentury.

2

The small c ircle was possibly suggested by the spurredcircle which was used for ten.

3 It has als o been thoughtthat the omicron used by Ptolemy in his A lmagest, tomark accidental blanks in the sexages imal systemwhich heemployed , may have influenced the Indian writers .4 Thissymbol was used quite generally in Europe and Asia, andthe Arabic astronomer A l-Battani 5 (died 929 A .D .) useda s imilar symbol in connection with the alphabetic systemof numerals . The occas ional u se by A l-Battani of theArabic negative, la, to indicate the absence of minutes

1 E . Wappler,“ Zur Geschichte der M athematik im 15. Jahrhun

dert,

” in the Zeitschriftfilr M athematik und Phys ik, Vol . XLV ,H ist.

lit. A bt. , p . 4 7 . The manu script is No . C . 80,in the Dresden l ibrary .

2 J . G . Prandel , A lgebra nebst ihrer literarischen Geschichte, p . 572,

Munich , 1795.

3 See the table, p . 23. Does the fact that the early Eu ropean arithmetics

,following the A rab cu stom

,always pu t the 0 after the 9

,sug

gest that the 0 was derived from the old Hindu symbol for 104 Bayley , loc . cit.

, p . 48. From this fact Delambre (H isto ire de l’ astronomie ancienne) inferred that Ptolemy knew the zero

,a theory

accepted by Chasles , Ap ergu h istorique sur l ’origine et le développ ement

des mé thodes engeometric, 1875ed., p . 4 76 ; Nesselmann

,however

,showed

(A lgebra der Griechen, 1842, p . that Ptolemy merely u sed 0 foroz

’

zbé v,with no notion of zero . See also G . Fazzari, “Dell ’ origine delle

parole zero e cifra,

” A teneo,Anno I

,No . 11

,reprinted at Naples in

1903,where the use of the point and the small cross for zero is also

mentioned . Th . H . Mart in,Les signes numéraua; etc . ,

reprint p . 30,and

J. Brandis,Das Munz Mass and Gewichtswesen in Vorderasien bis auf

A lexander den Grossen,Berlin

,1866

, p . 10,also discuss this usage of 0 ,

without the notion of place value, by the Greeks .

5A l—Battani s ive A lbateni i op us astronomicum .

’

Ad fidem codicisescurialens is arab ice editum

,latineversum ,

adnotationibus instru ctum

a Carolo Alphonso Nallino , 1899—1907 . Publicazioni del R . Osservatorio d i Brera in M ilano

,No . XL .

THE SYMBOL ZERO 55

( or seconds) , is noted by Nallino.1 Noteworthy is also

the u se of the O for unity in the Sarada characters of theKashmirian Atharva-Veda, the writing being at least 4 00years old. Bhaskara (c . 1150) used a small circle abovea number to indicate subtraction, and in the Tartar writing a redundant word is removed by drawing an ovalaround it. It would be interesting to know whether ourscore mark Q1) , read “ four in the hole,” could trace itspedigree to the same sources . O

’

Creat 2 ( c. in aletter to his teacher, Adelhard of Bath, uses 7 for zero,being an abbreviation for the word teea which we shallsee was one of the names used for zero, although it couldquite as well be from r é

'

t’

q a . M ore rarely O ’

Creat uses0 , applyi ng the name cyfra to both forms. Frater S igsboto 3 ( c. 1150) uses the same symbol. Other peculiarforms are noted by Heiberg 4 as being in u se among theByzantine Greeks in the fifteenth century. It is evidentfrom the text that some of these writers did not understand the import of the new system.

5

Although the dot was used at first in India, as notedabove, the small circle later replaced it and continues inu se to thi s day. The Arabs , however, did not adopt the

1 Loc . cit .,Vol . II

, p . 271 .

2 C . Henry,Prologus N. Ocreati in Helceph ad Adelardum Baten

sem magistrum suum,

” A bhandlungen zur Geschichte der Mathematik,

Vol . I II,1880 .

3 M ax . Cu rtze,

“Ueber eine A lgorismu s—Schrift des X II . Jahrhun

derts,

” Abhandlungen zur Geschichte der Mathematik,Vol . VIII

,1898

,

pp . Alfred Nagl, Ueber eine Algorismus-Sch rift des X II . Jahr

hunderts undfiberdieVerbreitungder indisch-arabischenRechenkunstund Zah lzeichen im ch ristl . Abendlande,” Zeitschriftfilr Mathematikund Phys ik, H ist.-lit. Abth.

,Vol . XXXIV , pp . 129—146 and 161- 170

,

With one plate .

4 “Byzantinische Analekten,

” Abhandlungen zur Geschichte derM athematik

,Vol . IX , pp . 161—189 .

5

q or q for 0 .

qalso u sed for 5. w_ for 13. [Heiberg, loc . cit.]


circle, since it bore some resemblance to the letter whi chexpressed the number five in the alphabet system.

1 The

earliest Arabic zero known. is the dot, used in a manuscript of 873A .D .

2 Sometimes both the dot and the circleare used in the same work, having the same meaning,which is the case in an Arabic MS ., an abridged arithmetic of Jam shjd,

3 982 A .H . ( 1575 As given inthis work the numerals are ‘l Av

iQ 19

. The formfor 5 varies, in some works becoming (P or a) ; c isfound in Egypt and 1) appears in some fonts of type.

To-day the Arabs u se the 0 only when , under Europ‘eaninfluence, they adopt the ordinary system. Among theChinese the firs t definite tracefié f zero is in the work ofTs in 4 of 124 7 A .D . The form is the circular one of theHindus, and undoubtedly was brought tO

/ghina by some

traveler.The name of this all-important symbol als o demands

some attention , especially as we are ever/

i} yet quite un

decided as to what to call it. We speak of it to-day as

zero, naught, and even cipher ; the telephone operatoroften calls it 0 ,

and the illiterate or careless person callsit aught. In View of all this uncertainty we may wellinquire what it has been called in the past.5

1 Gerhardt,E tudes historiques sur l’ arithme’tique dep os ition, Berl in,

1856, p . 12

;J . Bowring, "The Decima l System inNumbers , Coins , (f: A c

counts,London

,1854

, p . 33.

2 Karabgeek, Wiener Zeitschrift fur die K unde des MorgenlandesVol . XI

, p . 13 Fuhrer durch diePapyrus—Ausstellung E rzherzogRa iner ,Vienna

,1894

, p . 216.

3 In the library of G . A . Plimpton,E sq .

4 Cantor,Geschichte

,Vol . I p . 674 ; Y . M ikam i , “A Remark on

the Ch inese Mathematics in Cantor ’ s Gesch ichte der Mathematik,”A rch iv der Mathematik und Phys ik, Vol . XV pp . 68- 70 .

5 Of cou rse the earlier h istorians made innumerable guesses as tothe origin of the word c ipher . E .g. Matthew Hos tus , De numeratione

THE SYMBOL ZERO 57

As already stated, the Hi ndus called it sang/ a, void .

” 1

This passed over into the Arabic as as—sifr or sifr.

2 WhenLeonard of P isa ( 1202) wrote upon the Hindu numeralshe spoke of this character as . eephirum .

3 M aximus Planudes writi ng under both the Greek and theArabic influence, called it tziphra.

4 In a treatise on arithmeticwritten in the Itali an language by Jacob of Florence 5

emendata , Antwerp , 1582, p . 10,says : Siphra vox Hebraeam originem

sap it refé rtqu e : u t docti arbitrantur,averbo saphar, quod Ord ine

numerau it s ignificat . Unde Sephar numerus est : hinc S iphra (vulgocorruptius) . E tsi verb gens Iudaica his notis

, quae hodie S iphrae

vocantur, u sa non fu it : mans it tamen rei appellatio apud mu ltas

gentes .

” Dasypodiu s , Institutiones mathematicae,Vol . I

,1593

, gives alarge part of th is qu otation word for word , without any mention Of

the sou rce. Hermannu s Hugo , De p rima scribendi origine, Trajecti adRhenum

,1738

, pp . 304 - 305,and note

, p . 305; Karl Krumbacher,Woher stammt das Wort Ziffer (Ch i ffre) E tudes de phi lologie

ne'

o-grecque, Paris , 1892.

1 B ii hler,loc . cit .

, p . 78 and p . 86.

2 Fazzari,loc . cit .

, p . 4 . So E lia M israch i (1455—1526) in his posthum ous Book of Number

,Constantinople, 1534 , explains s ifra as being

A rabic . See also Steinschneider,Bibliotheca M athematica

,1893

, p . 69,

and G .Wertheim,DieA r ithmetikdes E lia M israchi

,Programm ,

Frankfu rt

,1893.

3 “ Cum his novem figuris , et cum hoc signo O, qu od arabice zeph irum appellatur, scribitur qu ilibet numerus .

”

4 rg‘

lppa , a form also u sed by NeOphytos (date unknown, probably

c . It is cu riou s that Finaeu s (1555 ed .,f . 2) u sed the form izi

phra th roughout . A . J . H . Vincent [“ Sur l ’origine de nos ch i ffres ,”Notices etExtra its des MSS .

,Paris

,184 7 , pp . 143— 150] says : “ Ce cercle

fut nommé par les uns,sip os , rota , ga lgal par les au tres tsip hra

(de 193 ,couronne ou diademe) ou cip hra (de WED,

Ch .

de Paravey , E ssai sur l’origine unique et hie

’

roglyphique des chifres etdeslettres dc tous les p eup les , Paris , 1826, p . 165

,a rather fancifu l work

,

gives vase,vase arrondi et ferm é par umcouvercle

, qu i est le symbolede la l oe Heu re

,among the Ch inese also Ts iphron Z é ron,

ou

tout a fa it vide en arabe,rg

'

lppa en grec d ’ o il chiffre (qu i deriveplutot, su ivant nou s , de l ’Hebreu Sep her,

5 Compilatus a Magistro Jacobo de Florentia apu d m ontem pesa

lanum,

”and described by G . Lam i in h is Cata loga s codicum manu

scrip torum qu i in bibliotheca Riccardiana Florentioe adservantur. See

Fazzari,loc . cit .

, p . 5.


( 1307) it is called zeuero,1 while in an arithmetic of Gio

vanni di Danti of Arezz o ( 1370) the word appears asceuero.

2 Another form is cep tro,3 which was also a stepfrom zephirum to zero .4Of course the English cipher, French chif re, is derived

from the same Arabic word, as-s ifr, but in several languages it has come to mean the numeral figures in general.A trace of this appears in our word ciphering, meaningfiguring or computing.

5 Johann Hu sw irt 6 uses the wordwith both meanings ; he gives for the tenth characterthe four names theca , circu lu s , cifra , and figura nihili .

In this statement Hu swirt probably follows , as did manywriters of that period , the A lgorismu s Of Johannes deSacrobosco ( c. 1250 who was also known as Johnof Halifax or John Of Holyw ood. The commentary of

1 “ E t doveto sapere chel zeuero per se solo non s ignifica nu lla ma

e potentia di fare s ignificare, E t decina O cent ina ia o m iglia ianon s i puote scrivere senza questo segno 0 . la quale s i ch iama zeuero .

”

[Fazzari, loc . cit., p .

2 Ibid ., p . 6.

3 Avicenna (980 translation by Gasbarri et Francois , “ p i u ilpunto (gli A rabi adoperavano il punto in vece dello zero il cu i segno0 in arabo s i ch iama zep iro donde il vocabolo zero) , che per se stessonon esprime nessun numero .” This quotation is taken f rom D . C .

Martines,Origine e p rogressi dell ’ aritmetica ,

Messina,1865.

4 Leo Jordan, “Materialien zur Gesch ichte der arab ischen Zah lzeichen in Frankreich

,

” A rch iv fur K u lturgeschichte, Berlin, 1905,pp . 155—195, gives the following two schemes of derivation

, (1)“zefiro

,

zeviro,zeiro

,zero

,

”

(2) zefiro,zefro

,zevro

,zero .”

5 KObel (1518 cd . ,f . A 4 ) speaks of the numerals in general as “ die

der gemain mann er nendt .

” Recorde (Grounde of A rtes , 1558 ed .,

f . B6) says that the zero is “called priuatly a Cyphar, though all the

other sometimes be l ikewise named .

”

5 “ Decimo X 0 theca, circu l‘iJ cifra sive figura nihill

[Ench irid ion A lgorismi , Cologne, Later,

“quoniam de integris

tam in cifris quam in proiectilibu s , the word proiecti libus referringto markers “ th rown and used on an abacus

,whence the French

jetons and the English expression to cast an account.

”


Certain , 1 written in 14 85. This word soon became fairlywell known in Spain 2 and France.

3 The medieval writersalso spoke of it as the sip os ,

4 and occasionally as thewheel,

5circu lu s 5 ( i n German das R inglein

7) , circu lar

1 Leo Jordan,loc . cit . In the Catalogue of MSS .

,B ibl. de l’A rsena l

,

Vol . III, pp . 154— 155

,th is work is No . 2904 (184 B ibl . Nat.

,

and is also called Petit tra icté dc algorisme.

2 Texada (1546) says that there are mu ene letros yvn zero 0 cifra ’

f . 33

)Savonne (1563, 1751 cd.

,f . “Vne ansi formee (o) qu i s ’appelle

nulle, entre marehans zero ,” showing the influence of Ital ian names

on French mercant ile cu stoms . Trenchant (Lyons , 1566, 1578 ed ., p .

12) also says : “ La derniere qu i s ’apele nulle, ou zero but Champenois

,h is contemporary , writing in Paris in 1577 (although the work

was not publ ished until uses“cipher ,” the Ital ian influence

showing itself less in th is center of university cu lture than in the com

mercial atmosphere of Lyons .

4 Thus Radulph of Laon (c . Inscribitur in u ltimo ordine et

figura Q s ipos nomine, quae, licet numerum nu llum s ignitet, tan

tum ad alia quaedam utilis,u t insequentibu s declarabitur.

”[“ Der

A rithmetische Tractat des Radu lph von Laon,

” A bhandlungen zur Ge

sch ichte dcr M athematik,Vol . V

, p . 97,from a manuscript of the th i r

teenth centu ry ] Chasles (Comp tes rendus , t . 16,1843

, pp . 1393,1408)

calls attention to the fact that Radu lph did not know how to u se the

zero,and he doubts if the sipos was really identical with it. Radulph

says figuram ,cu i sipos nomen est O inmotum rotulae for

matam null iu s numeri s ignificatione inscribi solere praediximu s,

”and

thereafter u ses rotula . He uses the sipos simply as a kind of markeron the abacus .

5 Rabbi ben E zra (1092—1168) u sed both 5353, galgal (the Hebrew forwheel) , and SHED, s ifra . See M . Steinschne ider , D ie Mathematik beiden Juden,

” in B ibliotheca M athematica,1893

, p . 69,and Silberberg,

Das Buch der Zahl des R . A braham ibn E sra , Frankfu rt a. M .,1895

, p .

96,note 23 in this work theHebrew letters are u sed for numerals with

place value, having the zero .5 E .g. ,

in the twelfth -century Liber a lgorismi (see Boncompagni ’ sTrattati

,II

, p . So Ramu s (L ibri II , 1569 cd., p . 1) says :

“ Circu lus quae nota est ultima : nil per se significat .

”(See also the Scho

nerus ed . of Ramu s,1586, p .

7 “Und wirt das ringle in 0 . die Z i ffer genant die nichts bedeut.”[Kobel

’s Rechenbuch

,1549 cd.

,f . 10

,and other editions ]

THE SYMBOL ZERO 61

note,1 theca ,

2 long suppos ed to be from its resemblance tothe Greek theta, but explained by Petrus deDacia as beingderived from the name of the iron 3 used to brand thievesand robbers with a circular mark placed on the forehead oron the cheek . It was also called omicron 4 ( the Greekbeing sometimes written 6 or 4) to dis tinguish it from theletter 0 . It also went by the name nu ll 5 ( in the Latin books

1 L e .

“circu lar figu re, our word notation having come from the

medieval nota . Thus Tzwivel (1507 , f . 2) says : “Nota au tem c ircularis .o . per se sumpta nih il vsus habet. alijs tamen adiuncta earum

s ignificantiam et auget et ordinem permutat quantum quo ponit ordinem . vt adiuncta note binarij hoc m odo 20 facit eam s ignificare bis

decem etc .

” A lso (ibid .,f . “figura circu laris

,

”circul aris nota .

”

Clichtoveus (1503 ed.,f . xxxvn ) calls it

“nota aut circu laris O,

”

circularis nota,” and “figura c i rcularis .

” Tonstall (1522, f . B3) saysof it : Dec imo uero nota ad formam .O . litterae circulari figura est

quam alij circu lum,uulgus cyphram uocat

,and later (f . C 4 ) speaks

of the c i rculos .

” Grammateus,in his A lgorismus de integris (E rfu rt,

1523,f . A2) , speaking of the nine significant figu res , remarks : “ His au

tem superadditur decima figura ci rcu laris ut 0 existens que ratione suanihil significat.” Noviomagu s (DeNumeris libri I I

,Paris

,1539

,chap .

xvi,De notis numerorum

, quas zyphras vocant calls it c i rcularisnota

, quam ex h is solam,alij s ipheram ,

Georgiu s Valla zyphram .

”

2 Huswirt,as above . Ramu s (Scholaemathematicae, 1569 ed .

, p . 112)discusses the name interestingly, saying : Circu lum appellamus cum

multis, quam alii thecam

,alii figuram nih ili

,alii figuram privationis ,

seu figuram nu llam vocant,al i i ciphram ,

cum tamen h odie omnes haemotae vu lgo ciphrae nom inentur, h is notis numerare idem s it qu odciphrare .

” Tartaglia (1592 ed.,f . 9) says “

s i chiama da alcuni tecca,

da alcuni c i rcolo,da altri c if ra

,da altri zero

,da alcuni altri nu lla .

”

3 “Quare au tem ali is nominibus vocetur

,non d icit auctor, quia

omnia alia nom ina habent rationem suae lineationis sive figurationis .

Qu ia rotunda est,dicitur haec figura teca ad s im ilitudinem tecae.

Teca enim est ferrum figurae rotundae, qu od ignitum solet in qu ibus

dam regionibus imprim i fronti vel maxillae fu ris seu latronum .

”[Loc .

cit., p . But in Greek theca (GHKH,

arm) is a place to put some

th ing, a receptacle . If a vacant column,e .g. in the abacu s

,was so

called,the initial m ight have given the early forms (9 and a for the zero .

4 Buteo,Logistica , Lyons , 1559 . See also Wertheim in the B iblio

theca M athematica,1901

, p . 214 .

5 O est appellee ch iffre ou nu lle on figu re de nulle valeur .Roche, L ’

arithmé tique, Lyons ,


nihil 1 or nu lla ,2 and in the French rien and very com

mouly by the name cipher.4 Wallis 5gives one of the earli

est extended discus sions of the various forms of theword,giving certain other variations worthy of note, as ziphra, zifera, Siphra, ciphra, tsiphra, tziphra, and the Greek r é

’

v’

cbpafi

1 Dec ima autem figura nihil uocata,” figura nih ili (quam etiamc i fram [Stife1, A ri thmetica integra , 1544 , f .

2 Z ifra,Nu lla u el figura Nih ili .

”[Scheubel, 1545, p . 1 of ch .

Nu lla is also u sed by Ital ian writers . Thus Sfortunat i (1545ed .,f . 4 )

says et la decima nu lla e ch iamata questa decima zero Cataldi(1602, p .

“ La prima, che e o , si ch iama nu lla

,ouero zero

,ouero

niente .

”It also found its way into the Dutch arithmetics

,e .g. Raets

(1576, 1580 ed .,f . A3) : Nu llo dat ist niet Van der Schuere (1600 ,

1624 ed.,

VVi lkens (1669 cd., p . In Germany Johann A lbert

(Wittenberg, 1534 ) and Rudolff (1526) both adopted the Italian nu lla

and popularized it . (See also Kuckuck,D ie Rechenkunst im sechzehn

ten Jahrhundert, Berl in,1874

, p . 7 Gunther,Gesch ichte

, p .

3 “ La dixieme s’

appelle ch i fre vu lga i rement : les vu s l’

appellantzero : nou s la pou rrons appeller vn R ien.

”[Peletien 1607 ed.

, p .

4 It appears in the Pol ish arithm etic of Klos (1538) as cyfra .

“The

C iphra 0 augm enteth places , but of h im selfe s ignifieth not,” Digges ,

1579, p . 1 . Hodder (l 0th ed .

,1672

, p . 2) uses only this word (cypheror c ipher) , and the same is true of the first native American arithme

tic,written by Isaac Greenwood (1729, p . Petrus de Dacia derives

cyfra from c i rcumference .

“ Vocatur et iam cyfra , quas i circumfacta

vel c ircum ferenda, quod idem est, quod c i rcu lus non habito respectu

ad centrum .

”

[Loc . cit., p .

5 Op era mathematica , 1695, Oxford ,Vol . I , chap . ix,Mathes is univer

sa lis , “De figuris numeralibus ,”pp . 4 6—4 9 ; Vol . II , A lgebra , p . 10.

5 Martin,Origine de notre Systeme dc numeration écrite, note 149 , p . 36

of reprint, spells ro l¢pa from Maximus Planudes,c it ing Wallis as an

authority . This i s an error,for Wall is gives the correct form as above .

A lexander von Humboldt,

“Uber die bei versch iedenen VO’

lkern

ublichen Systeme von Zah lze ichen und uber denUrsprung des Stellenwerthes in den ind is chen Zahlen

,

” Grelle’ s Journal fur reine und

angewandte M athematik,Vol . IV

,1829

,called attention to the work

dptfluol’

I I/dtxoi of the monk NeOphy tos , supposed to be of the fou rteenth centu ry. In th is work the form s rg

‘

ziqbpa and Thimppa appear .See also Boeckh

,De abaco Graecorum ,

Berl in,184 1

,and Tannery , “ Le

Scholie du m o ine NeOphytos ,” Revue A rche’

ologique, 1885, pp . 99—102.

Jordan,10 0 . cit.

, gives f rom twelfth and th i rteenth centu ry manuscriptsthe forms cifra ,

cifre, ch ifras , and cifrus . Du Cange, Glossarium mediaeet infimae Latinitatis

,Paris

,1842

, gives also chi lerae. Dasypodius ,Institutiones Mathematicae

,S trassburg, 1593—1596, adds the forms

zyphra and syphra . Bo is s ierc,L

’art d’

arythmetiquc contenant toutedimentica

,tres -s ingu lier ct commode, tant p our l ’ art m i lita ire que autres

ca lcu lations , Paris , 1554 : “ I’u is y en a vn au tre dict zero lequel ne

des igne nu lle q uantité par soy, ains seu lement les loges vu ides ,”

CHAPTER V

THE QUE STION OF THE INTRODUCTION OF THE

NUMERALS INTO EUROPE BY BOETHIUS

Jus t as we were quite uncertain as to the origin ofthe numeral forms , so too are we uncertain as to thetime and place of their introduction into Europe. Thereare two general theories as to this introduction . The firstis that they were carried by the M oors to Spain in theeighth or ninth century, and thence were transmittedto Christian Europe, a theory which will be consideredlater. The second, advanced by Woepcke,

1 is that theywere not brought to Spain by the M oors , but that theywere already i nSpain when theArabs arrived there, havingreached theWest through the Neo-Pythagoreans. 'Thereare two facts to support this second theory : ( 1 ) the formsof these numerals are characteristic, differing materiallyfrom those which were brought by Leonardo of Pisafrom Northern Africa early in the thirteenth century( before 1202 A .D.) ( 2) they are essentially those which

1 Propagation, pp . 27,234

,4 42 . Treu tlein

,

“Das Rechnen im 16.

Jahrhundert ,” A bhandlungen zur Gesch ichte der Mathematik

,Vol . I

,

p . 5,favors the same view . It is combated by many writers , e .g. A .C .

Burnell,loc . cit.

, p . 59. L ong before Woepcke, I . F . and G . I .Weidler

,De characteribus numerorum vu lgaribus et corum aetatibus

,Witten

berg, 1727 , asserted the poss ib ility of their introduction into G reeceby Pythagoras or one of h is followers : Potuer unt autem ex oriente

,

uel ex phoenicia , ad graecos traduci , uel Pythagorae, u el e ius discipulorum auxilio

,cum aliqu is eo

, proficiendi in literis causa, iter faceret,et hoc quoque inuentum addisceret.

”


tradition has so persistently ass igned to Boethius ( c. 500and which he would naturally have received, if

at all,from these same Neo-Pythagoreans or from the

sources from which they derived them. Furthermore,Woepcke points out that the Arabs on entering Spain( 711 A .D .) would naturally have followed their customOf adopting for the computation of taxes the numericalsystems of the countries they conquered,1 so that thenumerals brought from Spain to Italy, not having undergone the sam e modifications as those of the E astern Arabempire, would have differed, as they certainly did , fromthose that came through Bagdad. The theory is that theHindu system,

without the zero, early reached Alexandria ( say 450 and that the Neo-Pythagorean lovefor the mysterious and especially for the Oriental ledto its u se as something bizarre and cabalistic ; that itwas then pass ed along the M editerranean , reaching Boethiu s i nAthens or i nRom e, and to the schools of Spai n,being dis covered i nAfrica and Spain by the Arabs evenbefore they themselves knew the improved system withthe place value.

1 E .g.,they adopted the Greek numerals in use in Damascu s and

Syria, and the Coptic in Egypt. Theophanes (758—818 Chronograp hia , Scriptores Historiae Byzantinae, Vol . XXXIX ,

Bonnae,1839

,

p . 575,relates that in 699 A .D . the cal iph Wal id forbade the u se of the

Greek language in the bookkeeping of the treasury of the caliphate,but perm itted the u se of the Greek alphabetic numerals , s ince the

A 1abs had no convenient number notation : rat e’

xwkw e 7 pd¢ea€a t‘

Ex

7\77Vt0’

7'l ‘

rovs bnuoo'lovs rdwhoyodea lwu c bcxa s

,a 7\7\ Apafliots a ura. napaa

'

n

ua lveoda t, xwpis"

‘

rwv ¢n¢wv, e’

7rec5ndbvua'r ov T y éKelvwv 7 hw0

'

a'

y yovdda 17

77 rptada 7) 61cm : r'

iuwv n rpla 7 pd¢eofia t da) Ka i 8111s onuepbv claw

0 01i abroi‘

s‘

vordpwt Xpw‘

r ta vol. The importance of th is contemporaneousdocument was po inted out by M artin

,loc . c it . Karabacek,

“Die In

volutio i in arab ischen Schriftwesen,

” Vol . CXXXV of S itzungsberichtcd . p hil.-hist. C lassc d . k. Akad . d . Wiss .

,Vienna , 1896, p . 25

, gives anA rabic date of 868 in Greek letters .

THE BOETHIUS QUESTION 65

A recent theory s et forth by Bubnov 1 also deservesmention, chiefly because of the seriousness of purposeshown by this well-known writer. Bubnov holds thatthe forms first found in Europe are derived from anc ientsymbols used On the abacus , but that the zero is of Hinduorigin . This theory does not seem tenable, however, inthe light of the evidence already set forth.

Two questions are presented by Woepcke’

s theory( 1) What was the nature of these Spanish numerals , andhow were they made known to Italy ( 2) Did Boethiusknow themThe Spanish forms of the numerals were called the

harr'

if al-gobdr, the gobar or dust numerals, as distingu ished from the huruf a l—juma l or alphabetic numerals. Probably the latter, under the influence of theSyrians or Jews ,2 were also used by the Arabs . Thesignificance of the term gobs-ir is doubtless that thesenumerals were written on the dust abacus, this planbeing distinct from the counter method of representingnumbers. It is also worthy of note that A l-B iruni statesthat the Hindus often performed numerical computationsin the sand. The term is found as early as c. 950,in the verses Of an anonymous writer of Kairw'

an, inTunis

,in which the author speaks of one of his works

on gobfir calculation 3and, much later, the Arab writer

Abfi Bekr M ohammed ibn '

Abdalle'

th, surnamed al-Hassar

1 The Origin and H istory of Our Numerals (inRussian) , Kiev, 1908 ;The Indep endence of E uropean A rithmetic (in Russian) , Kiev.

2 Woepcke, loc . cit ., pp . 4 62, 262.

3 Woepcke, loc. cit ., p . 24 0 . H isab-al-Gobar

,by an anonymou s

author, probably Abu Sahl Dunash ibn Tam im

,is given by Stein

schneider, Die Mathematikbei den Juden,” BibliothecaMathematica,

1895, p . 26.


( the arithmetician) , wrote a work of whi ch the secondchapter was On the dust figures .” 1

The gobar numerals themselves were first made knownto modern scholars by S ilvestrede Sacy, who discoveredthem in an Arabic manuscript from the library of theancient abbey of St.-Germain-des-Pré s .2 The system hasnine characters, but no zero. A dot above a characterindicates tens , two dots hundreds , and so on , 5meaning50, and 5 m eaning 5000. It has been suggested thatpossibly these dots, sprinkled like dust above the numerals , gave rise to the word g

’

obdr,3 but this is not at all

probable. This system of dots is found in Persia at amuch later date with numerals quite like the modernArabic 4 but that it was used at all is s ignificant, for itis hardly l ikely that the western system would go back toPers ia, when the perfected Hindu one was near at hand.

At first sight there would seem to be some reason forbeli eving that this feature of the gobé r system was of

1 Steinschneider in the Abhandlungen, Vol. I II , p . 110 .

2 See his Gramgnaire arabe,Vol . 1

,Paris

,1810

, plate VIII Ger

hardt,Etudes

, pp . 9—11,and E ntstehung etc.

, p . 1 . F . Weidler,

Sp ici legium observationum ad h istoriam notarum numera lium p ertinentium ,

Wittenberg, 1755, speaks of the figura cifrarum Saracenica

rum ”as being d i fferent from that of the characterum Boeth ianorum ,

”

wh ich are similar to the vulgar ” or commonnumerals ; see also Hum

boldt,loc . cit.

5 Gerhardt mentions it in his E ntstehung etc ., p . 8 ; Woepcke, Pro

p agation, states that these numerals were u sed not for calcu lation,but

very mu ch as we u se Roman numerals . These superposed dots are

found with both forms of numerals (Prop agation, pp . 24 44 Gerhardt (E tudes , p 9) from a manu sm ipt in the Bibliotheque

Nationale . The numeral forms are8 V L,o S P; Pl ,

20 beingindicated by Pand 200 by p. This scheme of zero dots was alsoadopted by the Byzantine Greeks , for a manu script of Planudes in theB ibliotheque Nati onale has numbers l ike irii for 8, See

Ge1ha1dt E tudes, p . 19 . Pihan

,E xpose etc .

,.p 208

, gives two forms ,\ s iatic and Maghreb ian, of “ Ghobar ” numei als .


likely have known the gobfir forms before the numeralsreached them again in The term “ gobar numerals was also used without any reference to the peculiaru se of dots.2 In this connection it is worthy of mentionthat the Algerians employed two different forms ofnumerals in manuscripts even of the fourteenth cen

tury,3 and that the Moroccans of to-day employ the

European forms ins tead of the present Arabic.

The Indian u se of subscript dots to indicate the tens ,hundreds , thousands , etc. , is established by a passage inthe Ifitab a l-Fihrist 4 ( 987 A .D .) in which the writer discusses the written language of the people of India. Notwithstandi ng the importance oi this reference for theearly history of the numerals , it has not been mentionedby previous writers on this subject. The numeral formsgiven are those which have usually been called Indian ,5in Opposition to gobs-i r. In this document the dots are

placed below the characters, instead of being superposedas described above. The s ignificance was the same.

In form these goba'

tr numerals resemble ou r own muchmore closely than the Arab numerals do. They variedmore or less , but were substantially as follows

manu script remarks : The science is called A lgobar becau se the

inventor had the habit of writing the figu res on a tablet covered ,wit11

sand .

”

[Gerhardt, E tudes , p . 11,note .]

1 Gerhardt,E ntstehung etc .

, p . 20 .

2 H . Su ter,

“Das Rechenbuch des Abu Zakarija el-Hassar,” Bibliotheca Mathematica

,Vol . II p . 15.

5A . Devou lx,

“Les chiffres arabes,

” RevueAfricaine, Vol .XVI , pp .

455—458.

4 K itdb al-Fihrist, G . Fliigel, Leipzig, Vol . I , 1871, and Vol . II

1872 . Th is work was published after Professor Fliigel’ s death by J .

Roediger and A . Mueller . The first volume contains the A rabic textand the second volume conta ins critical notes upon it.

5Like those of line5in the illustration on page 69,

THE BOETHIUS QUE STION

The question of the possible influence of the Egyptiandemotic and hieratic ordinal forms has been so oftensuggested that it seems well to introduce them at thispo int, for comparison with the gobe

'

tr forms. They wouldas appropriately be used in connection with the Hinduforms

,and the evidence of a relation Of the first three

with all these systems is apparent. The only furtherresemblance is in the Demotic 4 and in the 9, so that thestatement that the Hindu forms in general came from

1 Woepcke, Recherches sur l’histoire des sciences mathématiques chezles orientauzt , loc . cit. Prop agation, p . 57 .

2 A l-Hassar’ s forms , Suter, B ibliotheca Mathematica,Vol . II

p . 15.

3 Woepcke, Sur une donnée historique, etc .,loc. cit. The name gobar

is not used in the text . The manuscript f rom wh ich these are taken

is the oldest (970 A .D .) A rabic document known to conta in all of thenumerals .

4 Silvestre de Sacy , loc . cit . He gives the ordinary modern A rabicforms

,calling them Indien.

5 and 5 Woepcke,“ Introduction au calcul Gobari et Hawai

,

” A tti

dell’ accadem ia pontificia dei nuovi L incei , Vol . X IX . The adjective applied to the forms in 5 is gobari and to those in 5 indienne. This is thedirect opposite of Woepcke

’s u se of these adjectives in the Recherches

sur l’histoire cited above,in wh ich the ordinary A rabic forms (like

those in row 5) are called indiens .

These forms are usually written from right to left.


this source has no foundation . The firs t four Egyptiancardinal numerals 1 resemble more tl1e~m odern Arabic.

This theory of the very earlyintroduction of the numerals

2 into Europe fails in severalpoints . In the first place the

early Western forms are notknown ; in the s econd placesome early E astern forms are

like the gobar, as is s een in thethird line on p. 69, where theforms are from a manuscriptwritten at Shiraz about 970A .D.,

and in which some western Ara

bic forms, e.g. p for 2, are alsoused. Probablymos t significant

DEMOT IC AND 11 114 3 4 1 10 of all is the fact that the gobarORD INA LS numerals as g iven by Sacy are

all,with the exception of the symbol for eight, either s in

gleArabic letters or combinations of letters . SO much fortheWoepcke theory and the meaning of the g

'

obfir numerals . We now have to cons ider the question as to whetherBoethius knew these gobar forms , or forms akin to them.

This large question 2 suggests several minor ones( 1) Who was Boethius ( 2) Could he have knownthese numerals ( 3) Is there any pos itive or strong circumstantial evidence that he did know them ( 4 ) Whatare the probabilities in the case

1 J . G .Wilkinson,The Manners and Customs of the Ancient Egyp

tians,revised by S . Birch , London, 1878, Vol . II , p . 4 93

, plate XVI .

2 There is an extens ive literatu re 0 11 th is Boeth iu s—Frage .

” The

reader who cares to go fu lly into it shou ld consu lt the various volumes

of the Jahrbuch ubcr die Fortschritte dcr Mathematik.


Firs t, who“

was Boethius, Divus 1 Boethius as he wascalled in the M iddle Ages Aniciu s Manlius SeverinusBoethius 2 was born at Rome c . 4 75. He was a mem

ber of the distinguished family of the Anicii,3 which hadfor some time before his birth been Christian . E arlyleft an orphan , the tradition is that he was taken toAthens at about the age of ten, and that he remainedthere eighteen years .4 He married Ru sticiana, daughterof the senator Symmachus , and this union of two suchpowerful families allowed him to move in the highestcircles .5 Standing strictly for the right, and agai ns t allin iquity at court, he became the object of hatred on thepart of all the unscrupulous element near the throne,and his bold defense of the ex-consul Albinus , unjus tlyaccused of treason , led to his imprisonment at Pavia 5and his execution in Not many generations afterhis death, the period bei ng one in which historical criticism was at its lowest ebb, the church found it profitableto look upon his execution as a martyrdom.

8 He was1 Th is title was first applied to Roman emperors in posthumou s

co ins of Juliu s Caesar . Subsequently the emperors assumed it du ringthei r own l i fetimes

,thus deifying themselves . See F . Gnecch i , Monete

romane,2d ed .

,M ilan

,1900

, p . 299 .

2 This is the common spelling of the name,although the more cor

rect Latin form is Boé tiu s . See Harper’ s D ict. of Class . L it. and

Antiq .,New York

,1897

,Vol . I

, p . 213 . There is mu ch uncert a inty as.to his life. A good summary of the evidence is given in the last twoeditions of the E ncyclop oedia Britannica .

3 His father,Flavius Manl ius Boeth iu s

,was consul in 4 87 .

4 There is,however

,no good historic evidence Of th is soj ou rn in

Athens .

5 His arithmetic is dedicated to Symmachu s : Dom ino suo patricio Symmacho Boetiu s .

”[Friedlein cd .

, p .

5 It was while here that he wrote De consolationep hi losOphiae.

7 It is sometimes given as 525.

3 There was a medieval tradition that he was executed because of awork on the Trini ty .


accordingly looked upon as a saint,1 his bones were em

shrined ,2 and as a natural consequence his books wereamong the classics in the church schools for a thousandyears .3 It is pathetic, however, to think of the medievalstudent trying to extract mental nourishment from awork so abstract, SO meaningless, so unnecessarily complicated, as the arithmetic of Boethius .He was looked upon by his contemporaries and imme

diate successors as a master, for Cassiodorus 4 ( c. 4 90c. 585A .D .) says to him :

“ Through your translationsthe music of Pythagoras and the astronomy of Ptolemyare read by those of Italy, and the arithmetic of Nicomachus and the geometry of Euclid are known to those Of

the West.” 5 Founder of the medieval scholasticism ,

1 Hence the Divus in his name.

2 Thus Dante,speaking of his bu rial place in the monastery of St.

Pietro in Ciel d ’ Oro,at Pavia

,says :

The sa intly soul , that showsThe world ’ s deceitfulness , to all who hear him ,

Is , with the s ight of all the good that i s ,Blest there . The l imb s , whence it was driven,

lie

Down in C ieldauro ; and from martyrdomAnd exile came it here.

—Paradiso, Canto X.

3 -Not,however

,in the mercantile schools . The arithmetic of Boe

thins would have been about the last book to be thought of in such

institutions . While referred to by Baeda (672— 735) and Hrabanu s

Mau rus (c . 776 it was only after Gerbert’ s time that the Boetii

de institutione arithmetica libri duo was really a common work.

4 Also spelled Cass iodorius .

5AS a matter of fact,Boeth iu s cou ld not have translated any work

by Pythagoras on mu sic,because there was no su ch work

,but he did

make the theories of the Pythagoreans known. Neither d id he translate Nicomachu s

,alth ough he embodied many of the ideas of the Greek

writer in h is own arithmetic . G ibbo'n follows Cassiodorus in these

statements in his Decline and Fa ll of the Roman Emp ire, chap . xxxix .

Martin pointed ou t with positiveness the similarity of the first bookof Boeth ius to the first five books of Nicomachus . [Les s ignes numeraua: etc.

,reprint, p .


di stinguishing the trivium and quadrivium ,

1 writing theonly classics of his time , G ibbon well called him the lastof the Romans whom Cato or Tully could have acknowl

edged for their countryman.

” 2

The second question relating to Boethius is this : Couldhe possibly have known the H indu numerals In Vi ewof the relations that will be shown to have existed between the E ast and the West, there can only be anaffirmative answer to this question . The numerals hadexisted , without the z ero, for several centuries ; theyhad been well known in India ; there had been a continu ed interchange of thought between the E ast andWest ;andwarriors

,ambassadors , scholars , and the restless trader,

all had gone back and forth, by land or more frequentlyby sea, between the Mediterranean lands and the centersof Indian commerce and culture. Boethius could verywell have learned one or more forms of Hindu numeralsfrom some traveler or merchant.To justify this statement it is necessary to speak more

fully of these relations between the Far E ast and Europe.

It is true that we have no records of the interchange oflearning, in any large way, between eastern As ia and

central Europe in the century preceding the time ofBoethius . But it is one of the mis takes of scholars tobelieve that they are the sole transmitters of knowledge.

1 The general idea goes back to Pythagoras , however .2 J . C . Scaliger in h is Poe'tice also sa id of h im : Boeth ii Severini

ingenium ,eruditio

,ars

,sap ientia facile provocat omnes au ctores , sive

illi Graeci s int,s ive Latini [Heilbronner, H ist. math. univ.

, p .

L ibri,speaking of the t ime of Boeth iu s , remarks : “Nous voyons du

temps de Theodoric, les lettres reprendre une nouvelle vie en Ital ie, lesécoles florissantes et les savans honorés . E t certes les ouvrages de Boece,de Cass iodore

,de Symmaqu e, surpassent de beau coup tou tes les produc

t ions du siecle precedent.” [H istoire des mathématiques , Vol . I , p .

74 THE HINDU- ARABIC NUMERALS

As a matter of fact there is abundant reason for believing that Hindu numerals would naturally have beenknown to the Arabs, and even along every trade routeto the remote west, long before the zero entered to maketheir place-value possible, and that the characters , themethods of calculating, the improvements that took placefrom time to time, the zero when it appeared, and the

customs as to s olvi ng business problems, would all havebeen made known from generation to generation alongthese same trade routes from the Orient to the Occident.It must always be kept in mind that it was to the tradesman and the wandering scholar that the spread of suchlearning was due, rather than to the school m an. Indeed,Avicenna 1 ( 980— 1037 A .D . ) in a short biography of himself relates that when his people were livi ng at Bokhz

'

i ra

his father sent him to the house of a grocer to learn theHindu art of reckoning, in which this grocer ( oil dealer,possibly) was expert. Leonardo of P isa, too, had a similartraining.

Thewhole question of this spread of mercantile knowledge along the trade routes is so connected with the gobe‘rr numerals , the Boethius ques tion , Gerbert, Leonardoof Pisa, and other names and events , that a digress ionfor its consideration now becomes necessary.

2

1 Carra de Vaux,Avicenne

,Paris

,1900 ; Woepcke, Sur l

’

introduc

tion,etc . ; Gerhardt, E ntstehung etc .

, p . 20 . Avicenna is a corruptionfrom Ibn S ina, as pointed out by Wustenfeld

,Gesch ichte der arabischen

A erzte und Naturforscher, Gottingen,1840 . His full name is Abr

'

i‘

A li

al-Hosein ibn 811151 . For notes O11 Avicenna’ s arithmetic, seeWoepcke,

Prop agation, p . 502.

2 0 11 the early travel between the E ast and the “fest the following works may be consulted : A . Hillebrandt, A lt—Indien, containing“ Ch ines ische Reisende in Ind ien

,

” Bres lau,1899

, p . 179 ; C . A . Skeel,

Travel in the F irst Century after Christ, Cambridge, 1901, p . 14 2 ; M .

Reinaud,

Relations politiques ct commerc iales de l’ emp i re roma in


northwest India andwrote upon his ventures .1 He inducedthe nations along the Indus to acknowledge the Persiansupremacy, and such number systems as there were inthese lands would naturally have been known to a manof his attainments.A century after Scylax , Herodotus showed consider

able knowledge of India, speaking of its cotton and itsgold ,2 telling how Sesostris 3 fitted out ships to sail tothat country, and mentioning the routes to the east.These routes were generally by the Red Sea, and hadbeen followed by the Phoenicians and the Sabaeans , andlater were taken by the Greeks and Romans .4In the fourth century theWest and E ast came into

very close relations. As early as 330, Pytheas of Mas

s ilia (Marseilles) had explored as far north as the northern end of the British Isles and the coasts of the GermanSea, while Macedon , in close touch with s outhern France,was also sending her armies under Alexander 5 throughAfghanis tan as far east as the Punjab.

5 Pliny tells usthat Alexander the Great employed surveyors to measure

1 Buh ler,IndianBrahma A lphabet, note, p . 27 Palaeographic, p . 2

Herodoti Ha licarnassei h istoria , Amsterdam,1763

,Bk. IV

, p . 300 ;

Isaac Vossius , Perip lus Scylacis Caryandensis , 1639. It is doubtfulwhether the work attributed to Scylax was written by h im ,

but in

any case the work dates back to the fou rth centu ry See Sm ith ’s

D ictionary of Greek and Roman B iography .

2 Herodotus , Bk. III .

3 Rameses the Sesoosis of D iodoru s S iculu s .

4 Indian Antiquary , V ol . I, p . 229 ; F . B . Jevons

,Manual of Greek

A ntiqu ities , London,1895

, p . 386. On the relations , pol itical and com

m ercial, between India and Egypt 0 . 72 B .C .,under Ptolemy Auletes ,

see the Journa l A s iatique, 1863, p . 297 .

5 S ikandar, as the nam e still remains in northern India .

5Harp er’ s C lass ical D ict. , New York, 1897 , Vol . I, p . 724 ; F . B .

Jevons , lo c . cit ., p . 389 ; J . C . Marshman, Abridgment of the H istory

of India ,chaps . i and i i .

THE BOETHIUS QUE STION 77

the roads of India ; and one of the great highways isdescribed by Megasthenes , who in 295B .C .

, as the ambassador Of Seleucus , res ided at Patalipu tra, the presentPatna.1The Hindus also learned the art of coining from the

Greeks, or possibly from the Chinese, and the stores ofGreco-Hindu coins still found in northern India are a

constant source of historical information .

2 The Ramayana speaks of merchants traveling i n great caravansand embarking by s ea for foreign lands .3 Ceylon tradedwith Malacca and S iam, and Java was colonized byHindutraders, so that mercantile knowledge was being spreadabout the Indies during all the formative period of thenumerals .Moreover the results of the early Greek invas ion were

embodied by Dicaearchus of Messana ( about 320 ina map that long remained a standard. Furthermore ,Alexander did not allow his influence on the East tocease. He divided India into three satrapies , 4 placingGreek governors over two

‘

of them and leaving a Hinduruler in charge of the third , and in Bactriana, a part ofAriana or ancient Pers ia, he left governors and in thesethe wes tern civiliz ation was long i n evidence. Some ofthe Greek and Roman metrical and astronomical terms

1 Oppert, loc . cit ., p . 11 . It was at or near th is place that the first

great Indian mathematician,Aryabhata , was born in 4 76 A .D .

2 Buhler,Palaeograp h ic, p . 2, speaks of Greek coins of a period

anterior to Alexander,found in northern India . M ore complete infor

mation may be found in Indian Coins , by E . J . Rapson, Strassbu rg,1898

, pp . 3— 7 .

3 Oppert, loc . cit. , p . 14 ; and to h im is due other s im ilar inforvmation.

4 J . Beloch,Griechische Geschichte, Vol . III , Strassbu rg, 1904 , pp .

30—31.

78 THE HINDUr ARABIC NUMERALS

found their way, doubtles s at this time, into the Sanskritlanguage.

1 Even as late as from the second to the fifthcenturies A .D . , Indian coins showed the Hellenic influence. The Hindu astronomical terminology reveals thesame relationship to western thought, for Varfiha-M ihira

( 6th century a contemporary of Aryabhata, entitled a work of his the E rhat-S amhitd, a literal translationof

,u erydxnGui/rafts of Ptolemy ; 2 and in various ways is

thi s interchange of ideas apparent.3 It could not havebeen at all unusual for the ancient Greeks to go to India

,for Strabo lays down the route, saying that all who

make the journey start from Ephesus and traverse Phrygiaand Cappadocia before taking the direct road.

4 The produ cts of the E ast were always finding their way to theWest, the Greeks getting their ginger 5 from Malabar,as the Phdanicians had long before brought gold fromMalacca.Greece mus t also have had early relations with China,

for there is a notable similarity between the Greek andChinese l ife, as is Shown in their houses, their domesticcustoms , their marriage ceremonies , the public storytellers , the puppet shows which Herodotus says wereintroduced from Egypt, the street jugglers , the games ofdice,6 the game of finger-guessing ,7 the water clock , the

1 E .g.,the denarius , the words for hou r and minute (cb'pa , Nea rby) ,

and poss ibly the S igns of the zod iac . [R . Caldwell,Comp arative Gram

mar of the Dravidian Languages , London, 1856, p . 4 38 ] On the probable Ch inese origin Of the zodiac see Sch legel, loc . cit.

2 Marie,Vol . I I

, p . 73 ; R . Caldwell,loc . cit.

3 A . Cunningham ,loc . cit .

, p . 50 .

4 C . A . J . Skeel , Travel, loc . cit., p . 14 .

5 Inchiver,from inaki

,

“ the green root.

”[IndianAntiquary , Vol . I ,

p .

5 In Ch ina dating only from the second century A .D .,however.

7 The Italian morra .

THE BOETHIUS QUE STION 79

mus ic system , the u se of the myriad, 1 the calendars, andin many other ways .

2 In passing through the suburbs ofPeking tod ay, on the way to the Great Bell temple, oneis constantly reminded of the semi-Greek architecture ofPompeii, so closely does modern China touch the oldclass ical civilization of the Mediterranean . The Chinesehistorians tell us that about 200 their arms were successfu l in the far west, and that in 180 B .C . an ambassadorwent to Bactria, then a Greek city, and reported that Chinese products were 0 11 sale in the markets there.

3 Thereis also a noteworthy resemblance between certain Greekand Chinese words , 4 showing that in remote tim es theremust have been more or less interchange of thought.The Romans also exchanged products with the East.

Horace says , A busy trader, you hasten to the farthestIndies , flying from poverty over sea, over crags, overfires .

” 5 The products of the Orient, Spices and jewelsfrom India, frankincense from Persia, and silks fromChina, being more in demand than the exports from theMediterranean lands , the balance of trade was againstthe Wes t, and thus Roman coin found its way eastward . In 1898, for example, a number of Roman coinsdating from 114 B C . to Hadrian ’ s time were found at

Pakli , a part Of the Hazfira district, s ixteen miles northof Abbottz'tbfid,6 and numerous similar dis coveries havebeen made from time to time.

1 J . Bowring, TheDecimal System ,London

,1854

, p . 2.

2 H. A . G i les,lectu re at Columb ia University , M arch 12

,1902

,on

China and Ancient Greece .

” 3 Giles,loc . cit.

4 E .g.,the names for grape, rad ish (la-

p o, fidcpn) , water-lily (s i—kua“ west gou rds m ta

,

“gourds are much alike . [Giles , loc . cit.]

5 Ep istles , I , 1 , 4 5—46 . 0 11 the Roman trade routes,see Beazley

loc . c it.,V ol . I

, p . 179 .

5 Am . Journ. of A rcheo l., Vol . IV , p . 366.

80 THE HINDU-"ARABIC NUMERALS

Augustus speaks of envoys received by him from India,a thing never before known , 1 and it is not improbable thathe also received an embassy from China.

2 Suetonius (firstcentury A .D .) speaks in his history of these relations,3 asdo several of his contemporaries ,4 and Vergil 5 tells ofAugustus doing battle i n Persia. In Pliny

’ s time thetrade of the Roman Empire with Asia amounted to amillion and a quarter dollars a year, a sum far greaterrelatively then than now,

6 while by the tirne of Constantine Europe was in direct communication with the FarE ast.7In view of these relations it is not beyond the range of

possibility that proof may s ometime come to light to Showthat the Greeks and Rom ans knew something of the

1 M . Perrot gives th is conjectu ral restoration of h is words : “Ad

me ex India regum legationes saepe m issi sunt numquam antea visae

apu d quemquam principem Romanorum .

”[M . Reinaud

,

“ Relationspolitiques et commerciales de l’ empire romain avec l’As ie orientale

,

”

Journ. A siat.,Vol . I p .

2 Reinaud,loc . cit.

, p . 189 . Florus,II

,34 (IV ,

refers to itSeres etiam habitantesque sub ipso sole Indi , cum gemm is et margari

tis elephantes quoque inter munera trahentes nihil magis quam longinqu itatem viae imputabant.” Horace shows his geograph ical knowledgeby saying : Not those who drink of the deep Danube shall now breakthe Julian edicts ; not the Getae, not the Seres , nor the perfidiou sPers ians , nor those born on the river Tana‘

is .

”[Odes , Bk. IV

, Ode

15,213 Quavirtutis moderationisque fama Indos etiam ac Scythas auditu

modo cognitos pellexit ad am icitiam suam pOpulique Romani nltro perlegatos petendam .

”[Reih ana, loc . cit.

, p .


, p . 180 .

5 Georgics , 11, 170—172. So Propertius (E legies , II I , 4 )Arma deu s Caesar dites m editatur ad IndosE t freta gemm iferi findere classe maris .

The divine Caesar m editated carry ing arm s against opulent India , and

with h is sh ips to cut the gem-bearing seas .

”

5 Heyd , loc . c it.,Vol . I

, p . 4 .


, p . 393.


number system of India, as several writers have maintained.

1

Returning to the E ast, there are many evidences of theSpread of knowledge in and about India itself. In thethird century Buddhism began to be a connectingmedium of thought. It had already permeated the Himalaya territory, had reached eastern Turkestan , and hadprobably gone thence to China. Some centuries later ( i n62 A .D .) the Chinese emperor sent an ambassador toIndia, and in 67 A .D . a Buddhist monk was invited toChina.2 Then , too, in India itself Asoka, whose namehas already been mentioned in this work , extended theboundaries of his domains even into Afghanistan , so thatit was entirely poss ible for the numerals of the Punjabto have worked their way north even at that early date.

3

Furthermore, the influence of Persia must not be forgotten in considering this transmission of knowledge. In

the fifth century the Persian medical school at JondiSapur admitted both the Hindu and the Greek doctrines,and Firdusi tells us that during the brilliant reign of

1 The title page of Calandri for example, represents Pythagoras with these numerals before him . [Sm ith , RaraA rithmetica

, p .

Isaacus Vossiu s , Observationes ad Pomp onium Melam de situ orbis , 1658,maintained that the A rabs derived these numerals from the west . A

learned dissertation to this effect, bu t deriving them from the Romans

instead of th e Greeks,was written by Ginanni in 1753 (Dissertatio

mathematica critica de numeralium notarum minuscularum origine,Venice

,See also Mannert

,De numerorum quos arabicos vocant vera

origine Pythagorica , Nurnberg, 1801 . Even as late as 1827 Romagnos i

(in his supplement to R icerche storiche sull’ India etc.,by Robertson,

Vol . II, p . 580

,1827) asserted that Pythagoras originated them . [R .

Bombelli,L ’

antica numerazione italica , Rome,1876

, p . Gow (H ist.of GreekMath

, p . 98) th inks that Iamblichu s must have known a sim ilar system in order to have worked out certain of h is theorems

,but

this is an unwarranted deduction from the passage given.

2 A . Hillebrandt,A lt-Indien

, p . 179.

3 J. C . Marshman, loc . cit. , chaps . i and ii .


Khosru I,

1 the golden age of Pahlavi literature, theHindu game of chess was introduced into Pers ia, at a

time when wars with the Greeks were bringing prestigeto the Sassanid dynas ty.

Again, not far from the time of Boethius , in the sixthcentury, the Egyptian monk Cosmas , in hi s earlier yearsas a trader, made j ourneys to Abyssinia and even toIndia and Ceylon , receiving the name Indicop leus tes ( theIndian traveler) . His m ap shows some knowledge Of the earth from the Atlantic to India. Such a

man would, with hardly a doubt, have observed everynumeral system used by the people with whom he so

journed,2 and whether or not he recorded his s tudies in

permanent form he would have transmitted such scrapsof knowledge by word of mouth .

As to the Arabs , it is a mistake to feel that their activi

ties began with M ohammed. Commerce had always beenheld in honor by them , and the Qoreish 3 had annuallyfor many generations sent caravans bearing the spices andtextiles of Yemen to the shores of the Mediterranean. In

the fifth century they traded by sea with India and evenwith China, andHira was an emporium for the wares ofthe East,4 so that any numeral system of any part of thetrading world could hardly have remained isolated.

Long before the warlike activity of the Arabs, Alexandria had become the great market-place of theworld.

From this center caravans traversed Arabia to Hadramaut, where they met ships from India. O thers wentnorth to Damascus , wh ile still others made their way

1 He reigned 531—579 A .D . ; called NuSirwan, the holy one.

2 J . Keane,The E volution of Geography , London, 1899, p . 38.

3 The A rabs who lived in and abou t Mecca .

4 S . Guyard , in E ncyc. Brit.,9th ed .

,Vol. XVI , p . 597 .


nor is it much nearer being answered than it was overtwo centuries ago when Wallis ( 1693) expressed hisdoubts about it 1 soon after Voss ius ( 1658) had calledattention to the matter.2 Stated briefly, there are threeworks on mathematics attributed to Boethius 3

( 1) the

arithmetic, ( 2) a work 0 11 music, and ( 3) the geometry.

4

The genuineness of the arithmfi ic and the treatise onmusic is generally recognized, but the geometry, whichcontains the Hindu numerals with the zero, is undersuspicion.

5 There are plenty of supporters of the ideathat Boethius knew the numerals and i ncluded them in

this book ,5and on the other hand there are as many who

1 “At non credendum est id in Autographis contigisse, aut vetus tioribu s Codd . M SS .

”[Wallis

,Op era omnia , Vol . II , p .

2 In Observationes ad Pomp onium M elam de s itu orbis . The ques

t ionwas next taken up in a large way by Weidler, loc. cit.,De charac

teribus etc .,1727

,and in Sp icilegium etc . ,

1755.

3 The best edition of these works is that of G . Friedlein,Anicu

M anli i Torquati Sever ini Boeti i de institutione arithmetica libri duo,de

institutionemus ica libri quinque. A ccedit geometria quac fertur Boeti i .Leipzig. M DCCC LX V I I .

4 See also P. Tannery , Notes sur la pseudo—géom étrie de Boece,in B ibliotheca Mathematica

,Vol . I p . 39 . Th is is not the geometiy

in two books in wh ich are mentioned the numerals . There is a manu

script oi this pseudo-geometry of the ninth centu ry,but the earl iest

one of the other work is of the eleventh century (Tannery ) , unlessthe Vatican codex is of the tenth centu ry as Friedlein (p . 372) asserts .

5 Friedlein feels that it is partly spu rious , bu t he says :“ E orum

librorum, quos Boetius de geometria scrips isse dicitur

,investigare

veram inscriptionem nih il al iud esset nisi operam et tempus perdere.

”

[Preface, p . v.] N. Bubnov in the Russian Journal of the M inistry ofPublic Instruction

,1907

,in an article of wh ich a synopsis is given in

the Jahrbuch tiber die Fortschritte derMathematik for 1907 asserts thatthe geometry was written in the eleventh centu ry .

5 The most noteworthy of these was for a long tim e Cantor (Gesch ichte

,Vol . I .

,3d ed .

, pp . 587 who in h is earlier days even

believed that Pythagoras had k nown them . Cantor says (DierO’mischenAgrimensorcn,

Leipzig, 1875, p .

“ Uns also,wir wiederholen es

,

ist die Geometrie des Boetius echt, d ieselbe Schri ft, welche er nach

Euklid bearbe itete,von welcher ein Codex bereits in Jahre 821 im

THE BOETHIUS QUE ST ION 85

feel that the geometry, or at least the part mentioningthe numerals, is Spurious .1 The argument of those whodeny the authenticity of the particular passage in question may briefly be stated thus :1 . The falsification of texts has always been the sub

ject of complaint. It was so with the Romans ,2 it was common in the M iddle Ages ,3 and it is much more prevalent

Kloster Reichenau vorhandenwar,von welcher ein anderes E xemplar

im Jabre 982 zu Mantua in die Hande Gerbert’ s gelangte, von welchermannigfache Handsch riften noch heu te vorhanden s inc But againstthis Opinion of the antiquity of M SS . containing these numerals isthe important statement of P . Tannery

, perhaps the m ost critical ofmodern historians of mathematics , that none exists earlier than the

eleventh centu ry . See also J . L . Heiberg in Phi lologus , Zeitschrift f .

d . klass . A ltertum,Vol . XLIII

, p . 508.

Of Cantor’ s predecessors , Th . H . Martinwas one of the most prominent

,h is argum ent for au thentic ity appearing in the Revue Archeolo

gigue for 1856— 1857 , and in h is treatise Les s ignes numerau it etc .

See also M . Chasles,

“De la connaissance qu ’ont eu les anciens d ’une

numeration décimale écrite qu i fait u sage de neu f chiffres prenantles valeu rs de position,” Comp tes rendus , Vol . VI , pp . 678—680 ; Sur

l’origine de notre systeme de numeration,

” Comptes rendus , Vol .VIII

, pp . 72—81 and note Sur le passage du prem ier livre de la geomé trie de Boece

,relatif aun nouveau systeme de numeration,

” in his

workAp ercu h istorique sur l ’origine et le devé loppement des méthodes engé ome

’

trie,of which the first edition appeared in 1837 .

1 J . L . Heiberg places the book in the eleventh centu ry on philological grounds , Phi lologus , loc . cit . ; Woepcke, in Prop agation, p . 4 4 ;

Blume,Lachmann

,and Rudorff

,Die Schriftenderrb'mischenFeldmesser ,

Berlin,184 8 ; Boeckh , De abaco graecorum,

Berlin,184 1 ; Friedlein,

in h is Leipzig ed it ion of 1867 ; Weissenborn,Abhandlungen, Vol . II ,

p . 185,his Gerbert

, pp . 1 , 24 7 , and his Gesch ichte der E infu'hrung derjetz igen Z ifiern in E uropa durch Gerbert, Berlin, 1892, p . 11 ; Bayley ,

loc . cit ., p . 59 ; Gerhardt, E tudes , p . 17

,E ntstehung und Ausbreitung,

p . 14 ; Nagl, Gerbert, p . 57 ; Bubnov, loc . cit. See also the discu ss ionby Chasles , Halliwell

,and L ibri

,in the Comp tes rendus , 1839 , Vol . IX

p . 4 4 7,and in Vols . VIII , XVI , XVII of the sam e jou rnal .

2 J . Marquardt, La vie p rivée des Roma ins , Vol . 11 (French trans ) ,p . 505

,Paris

,1893.

3 In a Plimptonmanu script of the arithmetic of Boeth iu s of the th irteenth centu ry

,for example, the Roman numerals are all replaced by

theA rabic,and the same is true in the first printed edition of the book.


tod ay than we commonly think . We have but to s ee

how every hymn-book compiler feels him self authorized to change at will the claSS1cs of our language, andhow unknown editors have mutilated Shakespeare, to seehow much more easy it was for medieval scribes to insertor el iminate paragraphs without any protest from critics.12. IfBoethius had known these numerals hewould have

mentioned them in his arithm etic, but he does not do so.2

3. If he had known them, and had mentioned them inany of his works , his contemporaries , disciples, and succes sors would have known and mentioned them. Butneither Capella ( c. nor any of the numerous medieval writers who knew the works Of Boethius makes anyreference to the system.

4

(See Sm ith ’ s Rara Arithmetica, pp . 434

,25 D . E . Sm ith also Cop

ied from a manuscript of the arithmetic in the Lau rentian l ibrary at

Florence,of 1370

,the followingforms

, l 7 7 d q 1 X 7 °

wh ich ,of cou rse

,are interpolations . A 11 interesting example of a for

gery in ecclesiastical matters is in the charter sa id to have beengivenby St. Patrick, granting indu lgences to the benefactors of Glastonbu ry ,dated “ In nomine dom ini nostri Jhesu Christi Ego Patricius humil iss ervuncu lus Dei anno incarnationis ejusdem ccccxxx .

” Now if theBenedictines are right in saying that D ionysius Exiguus , a Scyth ianmonk

,first arranged the Christian ch ronology c . 532 A .D .

,this can

hardly be other than Spu rious . See A rbuthnot, loc. cit. , p . 38.

1 Halliwell,in his Rara Mathematica

, p . 107,states that the disputed

passage is not in a manuscript belonging to Mr. Ames,nor in one at

T rinity College . See also Woepcke, in Prop agation, pp . 37 and 42.

It was the evident corruption of the texts in such ed itions of Boethiusas those of Venice

,1499

,Basel

,1546 and 1570

,that led Woepcke

to publish his work Sur l’

introduction de l’arithmé tique indienne en

Occident.2 They are found in none of the very ancient manuscripts , as , for

example, in the ninth—centu ry codex in the Lau rentian l ibrarywh ich one of the auth ors has examined . It shou ld be said

,however,

that the disputed passage was written after the arithmetic, for it conta ins a reference to that work. See the Friedlein ed .

, p . 397 .

3 Sm ith,Rara A rithmetica

, p . 66.

4 J . L . Heiberg, Ph i lologus , Vol . XLIII, p . 507 .


4 . The passage in question has all the appearance ofan interpolation by some scribe. Boethius is speaking ofangles , in his work on geometry, when the text suddenlychanges to a discus sion of classes of numbers .1 Thi s isfollowed by a chapter in explanation of the abacus,2 inwhich are described those numeral forms which are calledap ices or caracteres .

3 The forms 4 of these characters varyin different manuscripts , but in general are about asshown 0 11 page 88. They are commonly written withthe 9 at the left, decreas ing to the unit at the right, numerou s writers stating that this was because they werederived from Semitic sources in which the direction ofwriting is the opposite of our own. Thi s practice continu ed until the S ixteenth century.

5 The writer thenleaves the subject entirely, using the Roman numerals

1 “Nosse autem hu iu s artis dispicientem , qu id s int digiti , qu id articu l i , qu id compositi, qu id incompositi numeri .” [Friedlein ed.

, p .395.]2 De ratione abaci . In this he describes quandam formu lam

, quam

ob h onorem su i praeceptoris m ensam Pythagoream nom inabant

a posterioribu s appellabatur abacu s .

” This,as pictu red in the text, is

the common Gerbert abacu s . In the ed ition in M igne’s Patrologia

Latina,Vol . LXIII

,an ordinary mu ltiplication table (sometimes called

Pythagorean abacu s) is given in the illu stration.

3 Habebant enim diverse formatos apices vel caracteres .

” See the

reference to Gerbert on p . 117 .

4 C . Henry ,Sur l

’origine de quelques notations math ématiques ,

RevueArche’

ologique, 1879, derives these f rom the initial letters u sed asabbreviations for the names of the numerals

,a theory that finds few

supporters .

5 E .g.,it appears in Schonerus , A lgorithmus Demonstratvs

,Nu rn

berg, 1534 , f . A 4 . In England it appeared in the earl iest Engl isharithmetical manuscript known, The Crafte ofNombrynge 1]fforther

more ye m ost vndirstonde that in this craft ben vsid teen figurys , as

here bene writen for ensampu l , e 9 8 A 6 11 ,Q 3 2 1 111 the quych we

vse teen figurys of Inde . Questio .

‘Hwhy ten fyguris of Inde Solucio . for as I have sayd afore thei were fonde fyrst in Inde of a kyngeof that Cuntre

,that was called A lgor .” See Sm ith

,An E arly E nglish

A lgorism ,loc . cit.


FORM S OF THE NUM ERAL S , LARGELY FR OM WORKS ON

THE ABACU S 1

a Friedlein ed., p . 397 5 Carlsruhe codex of Gerlando .

Munich codex of Gerlando . ‘1 Carlsruhe codex of Bernelinus .

e Munich codex of Bernelinu s .f Turchill

,c . 1200 .

g Anon. MS .,th irteenth centu ry , A lexandrian L ibrary , Rome .

11 Twelfth-century Boeth iu s,Friedlein

, p . 396.

1 Vatican codex,tenth centu ry

,Boeth ius .

1 5,11,1,are from the Friedlein ed. ; the original in the manu script

from wh ich a is taken contains a zero symbol , as do all of the s ix

plates given by Friedlein.5—5 from the Boncompagni

O

Bu lletino,Vol .

X, p . 596 ;

f ibid .,Vol . XV

, p . 136 ; gM emorie della classe di sci .,Reale

A cc . dei L incei,An. CCLXXIV (1876 April, 1877 . A twelfth

centu ry arithmetician, possibly John of Luna (Hispalens is , of Seville,

c . speaks of the great diversity of these forms even in h is day ,saying :

“ E st autem in aliqu ibu s figuram istarum apud multos diners itas . Quidam enim septimam hanc figuram representant .q ali iautem s ic f ly ,

uel s ic fl . Quidam vero quartam s ic Q [Boncompagni , Trattati , Vol . II , p .


for. the rest of his discussion, a proceeding so foreign tothe method of Boethius as to be inexplicable on the

hyp othesis of authenticity. Why should such a scholarlywriter have given them with no mention of their originor u se E ither he would have mentioned some historical interest attaching to them

,or he would have used

them in some discussion ; he certainly would not haveleft the passage as it is.Sir E . Clive Bayley has added 1 a further reason for

bel ieving them spurious , namely that the 4 is not Of theNani-1 Ghat type, but of the Kabul form which the Arabsdid not receive until 7 76 ; 2 so that it is not likely, evenif the characters were known in Europe i n the time ofB oethius , that this particular form was recognized. It

is worthy of mention , also, that in the Six abacus formsfrom the chief manuscripts as given by Friedlein,

3each

contains some form Of zero, which symbol probably originated in India about this time or later. It could hardlyhave reached Europe so soon.

As to the fourth question , Did Boethius probably knowthe numerals It seems to be a fair conclusion , according to our present evidence, that ( 1 ) Boethius mightvery easily have known these numerals without the zero,but, ( 2) there is no reliable evidence that he did knowthem. And just as Boethius might have come in contactwith them, so any other inquiring mind might have doneso either in his

'

tim e or at any time before they definitelyappeared in the tenth century. These centuries, five innumber, represented the darkest of the Dark Ages , andeven if these numerals were occasionally m et and studied,no trace of them would be ‘ l ikely to show itself in the

1 Loc . cit ., p . 59.

2 Ibid ., p . 101 .

3 Loc . cit ., p .

_396.


literature of the period, unless by chance it should getinto the writings of some man like Alcuin. A s a matterof fact, it was not until the ninth or tenth century thatthere is any tangible evidence of their presence in Christendom . They were probably known to merchants hereand there, but in their incomplete state they were not ofsufficient importance to attract any considerable attention.

As a result of this brief survey of the evidence severalconclusions seem reasonable : ( 1) commerce, and travelfor travel ’ s sake, never died out between the E ast and theWest ; ( 2) merchants had every opportunity of knowing,and would have been unreasonably s tupid if they hadnot known , the elementary number systems Of the peoples with whom they were trading, but they would nothave put thi s knowledge in permanent written form ;( 3) wandering s cholars would have known many andstrange things about the peoples they m et, but they toowere not, as a class , writers ; ( 4 ) there is every reasona priori for believing that the gobe

'

tr numerals wouldhave been known to merchants, and probably to some ofthe wandering scholars , long before the Arabs conquerednorthern Africa ; (5) the wonder is no

x

t‘

that the HinduArabic numerals were known about 1000 A .D., and

'

that

they were the subject of an elaborate work in 1202 byFibonacci, but rather that more extended manuscript evidence of their appearance before that time has not beenfound. That they were more or less known early in theM iddle Ages , certainly to many merchants of ChristianEurope, and probably to several scholars, but withoutthe zero, is hardly to be doubted. The lack of documentary evidence is not at all strange, in view of allof the circumstances.


74 The first definite trace that we have of the introdu ction of the Hindu system into Arabia dates from 7 73A .D .

,

1

when an Indian astronomer visited the court of the ca

liph , bringing with him astronomical tables which at thecaliph’ s command were trans lated into Arabic by A lFazz

’

rri .2 A l-Khowfirazm i and Habash (Ahmed ibn°

Ab

dalle‘

th, died c. 870) based their well-known tables uponthe work of A l-Fazeri . It may be asserted as highlyprobable that the numerals came at the same time as the

tables . They were certainly known a few decades later,and before 825A .D., about which time the original of theAlgoritmi dc numero Indorum was written , as that workmakes no pretense of being the first work to treat of theHindu numerals.The three writers mentioned cover the period from the

end of the eighth to the end of the ninth century. Whi lethe historians A l-M as'udi and A l-B iruni follow quiteclosely upon the men mentioned, it is well to note againthe Arab writers on Hindu arithmetic, contemporary withA l-Khowfirazm i , who were mentioned in chapter I , viz .Al-Kindi , Sened ibn

'

Ali , and A l-S iifi .For over five hundred years Arabic writers and others

continued to apply to works 0 11 arithmetic the name“ Indian. In the tenth century such writers are ‘Abdallt'

th ibn al-Hasan , Ab'u ’

l-Q5.s im 3( died 987 A .D .) of An

tioch, and M ohammed ibn‘Abdallah, AbuNasr 4 ( c .

of Kalwfidfi. near Bagdad. Others of the same period or1 Colebrooke

,E ssays , Vol . II

, p . 504,on the authority of Ibn al

Adam i,astronomer, in a work published by h is continuator Al-Qasim

in 920 A .D . ; Al-B i runi

,India

,Vol . II

, p . 15.

2 H. Suter,Die Mathematiker etc .

, pp . 4—5, states that Al-Fazari

died between 796 and 806.

3 Su ter,loc. cit., p . 63.

4 Suter,loc . cit.

, p . 74 .

DEVELOPMENT OF THE NUMERALS 93

earlier ( since they are mentioned in thewho explicitly u se the word “Hindu ” or “ Indian

,

”are

Sinan ibn al-Fath 2 of Harran , and Ahmed ibnc

Omar,

al-Karabisi .3 In the eleventh century come ‘

A l-B iruni 4

( 973— 104 8) and

'Ali ibn Ahmed,Abu ’

l-Hasan, Al

Nasawi 5 ( c . The following century brings similar works by Ishé q ibn Yusuf al-Sardafi 6 and Samu ’ i libn Yahyfi ibn

°

Abb€ts al-Magrebi al-Andalu si 7 ( c.and i n the thirteenth century are

°

Abdallatif ibn Yusufibn M ohammed , Muwaffaq al—Din Abi

'

i Mohammed al

Bagdfidi3( c. and Ibn al-Banne't.3

The'

Greek monk Maximus Planudes , writing in thefirst half of the fourteenth century, followed the Arabicusage in calling his work IndianArithmetic.

10 There werenumerous other Arabic writers upon arithmetic

,as that

subject occupied one of the high places among the sciences,but most of them did not feel it necessary to refer to theorigin of the symbols, the knowledge of which might wellhave been taken for granted.

1 Suter, Das M athematiker—Verzeichniss im Fihrist, The referencesto Suter, unless otherwise stated

,are to his later workDieMathemati

ker und A stronomen der A raber etc .

2 Suter, Fihrist, p . 37,no date .

3 Suter, Fihrist, p . 38,no date .

4 Possibly late tenth , since he refers to one arithmetical work wh ichis entitled Book of the Cyp hers in h is Chronology , English ed .

, p . 132.

Suter,Die M athematiker etc .

, pp . 98— 100,does not mention this work ;

see the Nachtra’

ge und Berichtigungen, pp . 170—172.

5 Suter, pp . 96—97 .

5 Suter , p . 111 .

7 Suter, p . 124 . AS the name shows

,he came from the West.

3 Suter, p . 138.

9 Hankel,Zur Geschichte der Mathematik

, p . 256,refers to h im as

writing on the Hindu art Of reckoning ; Suter, p . 162.

1° ‘I’nfpocpopla Kerr. ‘

Ivdous , Greek ed .,C . I . Gerhardt

,Halle

,1865;

and German translation,Das Rechenbuch des Maximus Planudes, H.

waschke, Halle, 1878.


One document, cited by Woepcke,l is of special inter

est since it shows at an early period, 970 A .D., the use

of the ordinary Arabic forms alongside the gobar. Thetitle of the work is Interesting and Beautifu l P roblems on

Numbers copied byAhmed ibn M ohammed ibn c

Abdaljalil,Abu Sa

'

id, al-S ijzi ,2 ( 951— 1024 ) from a work by a priestand physician , Naz if ibn Yumu,3 al-Qass ( died 0 .

Suter does not mention this work of Naz if.The second reason for not ascribing too much credit

to the purely Arab influence is that the Arab by h imselfnever showed any intellectual strength. What took placeafter Mohammed had lighted the fire i nthe hearts of hispeople was just what always takes place when differenttypes of strong races blend, —

.a great renaissance in

divers lines ° It was seen in the blending of such types atM iletus in the time of Thales, at Rome in the days ofthe early invaders, at Alexandria when the Greek setfirm foot O11 Egyptian soil, and we see it now when all

the nations mingle their vitality in the New World. So

when the Arab culture joined with the Persian , a newc ivilization rose and flourished.

4 The Arab influencecame not from its purity, but from its intermingling withan influence more cultured if less virile.

As a resu lt of this interactivity among peoples of diversei nterests and powers , Mohammedanism was to the world7 from the eighth to the thirteenth century what Rome andAthens and the Italo-Hellenic influence generally had

1 Sur une donnée h istorique relative a l ’ emplo i des ch iffres indiens par les A rabes ,” Tortolini ’ s Anna li d i scienze mat. efis .

,1855.

2 Suter, p . 80 .

3 Suter, p . 68.

4 Sprenger also calls attention to th is fact,in the Zei tschrift d.

deutschenmorgenland . Gesellschaft, Vol . XLV , p . 367 .


acces s to the newly-established schools and the bazaars ofMesopotamia and western Asia. The particular paths ofconquest and of commerce were either by way of theKhyber Pass and through Kabul, Herat and Khorassan ,or by sea through the strait of Ormuz to Basra (Busra)at the head of the Persian Gulf, and thence to Bagdad.

As a matter of fact, one form of Arabic numerals, the onenow in u se by the Arabs, is attributed to the influence ofKabul, while the other, which eventually became our numerals , may very likely have reached Arabia by the otherroute. It is in Bagdad,1 Dar al—Salfim “ the Abode ofPeace, that our special interest in the introduction of thenumerals centers. Built upon the ruins of an ancienttown by A l-Mansur 2 in the second half of the eighthcentury, it lies in one of those regions where the converging routes of trade give rise to large cities.3 Quite aswell of Bagdad as of Athens might Cardinal Newmanhave said : 4

“What it lost in conveniences of approach, it gainedin its neighborhood to the traditions of the mysteriousEast, and in the loveliness of the region in which it lay.

Hither, then , as to a sort of ideal land, where all archetypes of the great and the fair were found in substantialbeing, and all departments of truth explored, and all

diversities of intellectual power exhibited, where tasteand phi losophy were majestically enthroned as in a royalcourt, where there was no sovereignty but that of mind,and no nobility but that of genius, where professors were

1 Pers ian bagadata, God-given.

”

2 One of the Abbassides,the (at least pretended) descendants of

'

A l-Abbas,uncle and adviser of M ohammed .

3 E . Reclus,A s ia ,

Am erican ed .,N.Y .

,1891

,Vol . IV , p. 227 .

4 H i storical Sketches , Vol . I II , chap . i i i .

DEVELOPMENT OF THE NUMERALS 97

rulers, and princes did homage, thither flocked continuallyfrom the very corners of the orbis terrarum the manytongued generation , just rising, or just risen into manhood

,in order to gain wisdom.

” For here it was thatAl-Mansur and A l-Mfimun and Hart—in al-Rashi d (Aaronthe Just) made for a time the world

’ s center of intellectual ‘activity in general and in the domain of mathematicsin particular.1 It was just after the S indhind was broughtto Bag dad that M ohammed ibn Musa al-Khowarazmi ,

whose name has already been mentioned,2 was called tothat city. He was the most celebrated mathematician ofhis time, either in the E ast or West, writing treatises onarithmetic , the sundial, the astrolabe, chronology, geometry, and algebra, and giving through the Latin transliteration of his name, algoritmi , the name of algorism to theearly arithmetics using the newHindu numerals.3 Appreciating at once the value of the position system so recentlybrought from India, he wrote an arithmetic based uponthese numerals , and this was translated into Latin in thetime ofAdelhard of Bath ( c . although possibly byhis contemporary countryman Robert Cestrens is .

4 Thistranslation was found in Cambridge and was publishedby Boncompagni inContemporary with A l-Khowi razm i , and working also

underA l-M ‘Ztmun, was aJewish astronomer,Abu ’

l-Teiyib,

1 On its prominence at that period see Vill icu s , p . 70 .

2 See pp . 4—5.

3 Sm ith , D . E .,in the Cantor Festschrift, 1909, note pp . 10—11 . See

also F .Woepcke, P rop agation.

4 EnestrOm,in Bibliotheca Mathematica

,Vol . I p . 499 Cantor,

Geschichte,Vol . I p . 671 .

5 Cited in Chapter I . It begins : Dixit algoritmi laudes deo rec .

tori nostro atque defensori dicamu s dignas .

” It is devoted entirelyto the fundamental operations and contains. no appli cations.


Sened ibn 'Ali , who is said to have adopted the M oham

medan religion at the cal iph ’ s request. He also wrote a

work on Hindu arithm etic,1 so that the subject must havebeen attracting considerable attention at that time. In

deed, the struggle to have the Hindu numerals replacethe Arabic did not cease for a long time thereafter. c

Ali

ibn Ahm ed al-Nasawi , in his arithm etic of c . 1025, ’ tellsus that the symbolism of number was still unsettled inhi s day, although most people preferred the strictlyArabic forms .2We thus have the numerals in Arabia, in two forms

one the form now used there, and the other the one usedby A l-Khowfirazm i . The question then remains , how didthi s second form find its way into Europe and this question will be considered in the next chapter.

1 M . Steinschne ider, “Die M athematik bei den Juden,

” Bibliotheca

Mathematica,Vol . VIII p . 99. See also the reference to th is writer

in Chapter I .

2 Part of th is work has been trans lated from a Leyden M S . by F .

Woepcke, Prop agation, and more recently by H. Su ter, BibliothecaMathematica

,Vol . VII pp . 113— 119 .


remained as conquerors. The power of the Goths, whoh ad held Spain for three centuries , was Shattered at thebattle of Jerez de la Frontera in 711, and almost imm ediately the Moors became masters of Spain and so re

7Lm ained for five hundred years, and masters of Granadafor a much longer period. Until 850 the Christians wereabsolutely free as to religion and as to holding politicaloffice, so that priests and monks were not infrequentlySkilled both in Latin and Arabic, acting as official translators, and naturally reporting directly or indirectly toRome. There was indeed at this time a complaint thatChristian youths cultivated too assiduously a love forthe literature of the Saracen , and married too frequentlythe daughters of the infidel.1 It is true that this happystate of affairs was not permanent, but while it lastedthe learning and the customs of the E ast must have become more or less the property of Christian Spain . Atthis time the gobfir numerals were probably in that country, and these may well have made their way into Europe

7’ from the schools of Cordova, Granada, and Toledo.Furthermore, there was abundant opportunity for the

numerals of the East to reach Europe through the journeys of travelers and ambassadors. It was from the records of Suleiman the M erchant, a well-known Arab traderof the ninth century, that part of the story of S indbadthe Sailor was taken .

2 Such a merchant would have beenparticularly likely to know the numerals of the peoplewhom he m et, and he is a type of man that may well havetaken such symbols to European markets. A little later,

1 A . Neander,General H istory of the Christian Religion and Church,

5th American ed .,Boston

,1855

,Vol . 111

, p . 335,2 Beazley , loc. cit.

,Vol . I

, p . 49.

DEFINITE INTRODUCTION smith 131115131; T3151Abu ’

l-Hasan c

A li al-Mas‘udi ( d. 956) of Bagdad traveledto the China Sea on the east, at least as far south asZanz ibar, and to the Atlantic on the west, 1 and he speaksof the nine figures with which the Hindus reckoned.

2

There was also a Bagdad merchant, one Abu ’

l-Qasim°

Obeidall§th ibn Ahmed, better known by his Persianname Ibn Khordti dbeh ,

3 who wrote about 850 A .D . awork entitled B ook of Roads and P rovinces 4 in which thefollowing graphi c account appears 5 The Jewish mer

sian, Roman (Greek and Latin) , Arabic,and S lavic. They travel from the West

to the East, and from the E ast to the West, sometim esby land, sometimes by sea. They take ship from Franceon the Western Sea, and they voyage to Farama (nearthe ruins of the ancient Pelu s ium ) there they transfertheir goods to caravans and go by land to Colz

'

om ( on theRed Sea) . They there reémbark on the Oriental—(Red)Sea and go to Hejaz and to Jiddah, and thence to theS ind, India, and China. Returning, they bring back theproducts of the oriental lands . These journeys are

also made by land. The merchants, leaving France andSpain , cross to Tangier and thence pass through theAfrican provinces and Egyp t. They then go to Ramleh , visit Damascus, Kufa, Bagdad, and Basra, penetrateinto Ahwaz , Fars , Kerman , S ind, and thus reach Indi aand China.

”Such travelers, about 900 A .D.

, must necessarily have spread abroad a knowledge of all number

1 Beazley,loc. cit .

,Vol. I , pp . 50, 4 60 .

2 See pp . 7—8.

3 The name also appears as Mohamm ed Abu’ l-Qasim ,and IbnHau

qal . Beazley , loc . cit. ,Vol . I

, p . 45.

4 K itdb al—masalikwa’ l—mamalik.

5 Reinaud,M ém . sur l ’ Inde in Gerhardt

,E tudes

, p . 18.

162 j5,

—ARABIC NUMERALS

systems used in recording prices or in the computationsof the market. There is an interesting witness to thismovement, a cruciform brooch now in the British Mu

seum. It is English, certainly as early as the eleventhcentury, but it is inlaid with a piece of paste on whichis the M ohammedan inscription, i n Kufic characters,There is no God but God.

”How did such an inscrip

tion find its way, perhaps in the time of Alcuin of York,to England 7 And if these Kufic characters reachedthere, then why not the numeral forms as well 7Even in literature of the better class there appears

now and then some stray proof of the important factthat the great trade routes to the far E ast were neverclosed for long, and that the customs and marks of tradeendured from generation to generation. The Gu listan ofthe Persian poet Sa'di 1 contains such a passage :

I met a merchant who owned one hundred and fortycamels, and fifty slaves and porters. He answered tome : I want to carry sulphur of Persia to China, whichin that country, as I hear, bears a high price ; and thenceto take Chinese ware to Roum ; and from Roum to loadup with brocades for Hind ; and so to trade Indian steel(pulab) to Halib. From Halib I will convey its glass toYeman , and carry the painted cloths of Yeman back toPersia. ’ 2 On the other hand, these men were not of thelearned class , nor would they preserve in treatises anyknowledge that they might have, although this knowledge would occas ionally reach

‘

the ears of the learned asbits of curious information.

1 Born at Sh i raz in 1193. He himself had traveled from Indiato E urope.

2 Gu listan (Rose Garden) , Gateway the th i rd,XXII . Sir Edwin

A rnold ’ s translation, N.Y .,1899

, p . 177 .


from England as far as India,1 and generally in theM iddle Ages groceries came to Europe from Asia as nowthey come from the colonies and from America. Syria,Asia M inor

,and Cyprus furnished sugar and wool, and

India yielded her perfumes and Spices, while rich tapestries for the courts and the wealthy burghers came fromPersia and from China.2 Even in the time of Justinian( c. 550) there seems to have been a silk trade with China,which country in turn carried on commerce with Ceylon ,3and reached out to Turkestan where other merchantstransmitted the E astern products westward. In the sev

enth century there was a well-defined comm erce betweenPersia and India, as well as between Persia and Constantinople.

4 The Byzantine commerciarii were stationedat the outposts not merely as customs Officers but as

government purchasing agents.5Occasionally there went along these routes of trade

men of real learning, and such would surely have carriedthe knowledge of many customs back and forth. Thusat a period when the numerals are known to have beenpartly understood in Italy, at theopening of the eleventhcentury, one Constantine, anAfrican , traveled from Italythrough a great part of Africa and Asia, even on toIndia, for the purpose Of ' learning the sciences of theOrient. He Spent thirty-ni ne years in travel, havingbeen hospitably received in Babylon , and upon his returnhe was welcomed with great honor at Salerno.6A very interesting illustration of thi s i ntercourse also

appears in the tenth century, when the son of Otto I

1 Cunningham ,loc . cit .

, p . 81 .4 Ibid .

, p . 21 .

2 Heyd , loc . cit .,Vol . I , p . 4 .

5 Ibid ., p . 23.

3 Ib id ., p . 5.

5 L ibri,H istoire Vol . I , p . 167 .

DEFINITE INTRODUCTION INTO EUROPE 105

( 936— 973) married a princess from Constantinople. This

monarch was in touch with the Moors of Spain andinvited to his court numerous scholars from abroad,1and his intercourse with the E ast as well as the Westmust have brought together much of the learning ofeach.

Another powerful means for the circulation of mysticism and philosophy, and more or less of culture, took itsstart just before the conversion of Constantine ( c.

in the form of Christian pilgrim travel. This was a

feature peculiar to the zealots of early Christianity,found in only a slight degree among their Jewish predecessors in the annual pilgrimage to . Jerusalem, andalmost wholly wanting i n other pre-Christian peoples.Chief among these early pilgrims were the two Placentians, John and Antonine the E lder ( c . who

,in

their wanderings to Jerusalem, seem to have started amovement which culminated centuries later in the crusades.2 In 333 a Bordeaux pilgrim compiled the firstChristian guide-book , the Itinerary from Bordeaux to

Jeru salem ,

3 and from this time on the holy pilgrimagenever entirely ceased.Still another certain route for the entrance of the nu

merals into Christian Europe was through the pillagingand trading carried on by the Arabs on the northernshores of the Mediterranean. As early as 652 A .D ., in

the thirtieth year of the Hejira, the Mohamm edans descended upon the Shores of S icily and took much spoil.Hardly had the wretched Constans given place to the

1 Picavet,Gerbert

,un p ap e phi losophe, d ’

ap res l ’histoire et d ’ ap resla légende, Paris , 1897 , p . 19 .

2 Beazley,loc . cit .

, Vol . I , chap . 1,and p . 54 seq .

3 Ibid ., p . 57 .


young Constantine IV when they agai n attacked the

island and plundered ancient Syracuse. Again in 827,

under Asad, they ravaged the coasts. Although at thi stime they failed to conquer Syracuse, they soon held a

good part of the island, and a little later they succes sfully bes ieged the city. Before Syracuse fell, however,they had plundered the shores of Italy, even to the wallsof Rome itself ; and had not Leo IV ,

in 84 9, repaired theneglected fortifications, the effects of the Mos lem raid ofthat year might have been very far-reaching. Ibn Khordé dbeh, who left Bagdad in the latter part of the ninthcentury, gives a picture of the great commercial activityat that time i nthe Saracen city of Palermo. In this samecentury they had establ ished themselves in Piedmont,and i n 906 they pillaged Turin.

1 On the Sorrento peninsula the traveler who climbs the hill to the beautifulRavello sees still several traces of the Arab architecture,reminding him of the fact that about 900 A .D. Amalfiwasa commercial center of the M oors .2 Not only at this time,but even a century earlier, the artists of northern Indiasold their wares at such centers, and in the courts both ofHe

'

tri‘

m al-Rashid and of Charlemagne.

3 Thus the Arabsdominated the Mediterranean Sea long before Venice

held the gorgeou s East in feeAnd was the safegu ard of the West,

”

and long before Genoa had become her powerful rival.4

1 Libri,H istoire

,Vol . I

, p . 110,n.

, citing authorities,and p . 152.

2 Possibly the Old tradition,

“ Prima dedit nantis u sum magnetis

Amalphis , is true so far as it means the m odern form of compasscard . Sec Beazley , loc . cit .

,Vol . II

, p . 398.

3 R . C . Dutt,loc . cit .

,Vol . II

, p . 312 .

4 E . J . Payne, in The Cambridge Modern H istory , London,1902

,

Vol . I,chap . 1.


It is therefore reasonable to conclude that in the M iddle Ages, as in the time of Boethius, it was a simplematter for any inquiring scholar to become acquaintedwith such numerals of the Orient as merchants may haveused for warehouse or price marks . And the fact thatGerbert seems to have known only the forms of the simplest of these, not comprehending their full significance,seems to prove that he picked them up i n just this way.

Even if Gerbert did not bring his knowledge of theOriental numerals from Spain, he may easily have oh

tained them from the marks on merchant’ s goods, had hebeen so inclined. Such knowledge was probably oh

tainable in various parts of Italy, though as parts of meremercantile knowledge the forms might soon have beenlost, it needing the pen of the scholar to preserve them.

Trade at this time was not stagnant. During the eleventhand twelfth centuries the Slavs, for example, had verygreat commercial interests, their trade reaching to Kievand Novgorod, and thence to the East. Constantinoplewas a great clearing-house of comm erce with the Orient,1and the Byzantine merchants must have been entirelyfamiliar with the various numerals of the E astern peoples.In the eleventh century the Ital ian town of Amalfiestablished a factory 2 in Constantinople, and had trade re

lations with Antioch and Egypt. Venice, as early as theninth century

,had a valuable trade with Syria and Cairo.3

Fifty years after Gerbert died, i n the time of Cnut, theDane and the Norwegian pushed their commerce far beyond the northern seas, both by caravans through Russ iato the Orient, and by their venturesome barks which


, p . 194 .

2 L e . a comm iss ion house .


, p . 186 .


sailed through the Strait of G ibraltar into the Mediterranean.

1 Only a little later, probably before 1200 A .D .,

a clerk in the service of Thomas aBecket, present at thelatter ’ s death, wrote a life of the martyr, to whi ch (fortunately for our purposes) he prefixed a brief eulogy of

the city of London .

2 This clerk, William Fitz Stephenby name , thus speaks Of the British capital

Au rum m ittit Arabs : species et thura Sabeeu sArm a Sythes Oleum palm arum divite sylvaPingue solum Babylon : Nilu s lapides pretiososNorwegi , Ru s si , varium gri sum , s abdinaS

'

Seres , purpureas vestes : Galli , su a Vina.

Although, as a matter of fact, the Arabs had no gold tosend, and the Scythians no arms , and Egypt no preciousstones save only the turquoise, the Chinese ( S eres) mayhave sent their purple vestments, and the north her sablesand other furs , and France her wines . At any rate theverses show very clearly an extensive foreign trade.Then there were the Crusades, which in these times

brought the E ast in touch with the West. The Spirit ofthe Orient showed itself in the songs of the troubadours,and the baudekin,

3 the canopy of Bagdad, 4 became common i n the churches of Italy. In S ic ily and in Venicethe textile industries of the E ast found place, and madetheir way even to the Scandinavian peninsula.

5

We therefore have this state of affairs : There wasabundant i ntercourse between the E ast and West for

1 J . R . Green,ShortH istory of the E nglish Peop le, New York, 1890

p . 66.

2 W . Besant,London

,New York, 1892, p . 43:

3 Ba ldakin,baldekin

,ba ldachino .

4 Ital ian Ba ldacco .

5 J . K .Mumford, Oriental Rugs , New York, 1901, p . 18.


some centuries before the Hindu numerals appear inany manuscripts i n Christian Europe. The numeralsmust of necessity have been known to many traders ina country like Italy at least as early as the ninth century,and probably even earlier, but there was no reason forpreserving them in treatises. Therefore when a man likeGerbert made them known to the scholarly circles , hewas merely describing what had been familiar in a smallway to many people in a different walk of life.

Since Gerbert 1 was for a long time thought to havebeen the one to introduce the numerals into Italy,2 abrief sketch of this unique character is proper. Born ofhumble parents,3 this remarkable man became the counselor and companion of kings , and finally wore the papaltiara as Sylvester II, from 999 until his death inHe was early brought under the i nfluence of the monksat Aurillac, and particularly of Raimund, who had beena pupil of Odo of C luny, and there i ndue time he himself took holy orders. He vis ited Spain in about 967 incompany with Count Borel ,5 remain ing there three years,

1 Or Girbert, the Latin forms Gerbertus and Girbertus appearingindifferently in the documents of his time .

2 See, for example, J . C . He ilbronner,H istoria matheseos universoe

p . 74 0 .

3 Obscu ro loco natum ,

”as an old chroni cle of Au rillac has it.

4 N. Bubnov,Gerberti p ostea S ilvestri I I p ap ac op era mathematica

Berlin,1899

,is the most complete and reliable sou rce of information

Picavet,loc . cit . ,

Gerbert etc . ; Olleris , (E uvres de C erberi,Paris

,1867

Havet,Lettres de Gerbert

,Paris

,1889 ; H . Weissenborn,

Gerbert ; Bei

tra'ge zur K enntnis der M athematik (les M ittelalters,Berl in

,1888

,and

Zur Geschichte der E infuhrung dcr jetzigen Zifi'

ern in E urop a durchGerbert

,Berl in

,1892 ; BUdinger, Ueber C erberis wissenschaftliche und

p olitische S tellung, Cassel, 1851 ; R icher, Historiarum l iber III,

” in

Bubnov,loc . cit .

, pp . 376—381 Nagl, Gerbert und die Rechenkunst des

1 0 . Jahrhunderts,Vienna

,1888.

Richer tells of the vis it to Au rillac by Borel, a Spanish nobleman

,just as Gerbert was entering into young manhood . He relates


wfili of Saragossa,1 SO that there was more or less offriendly relation between Christian and M oor.After his three years in Spain , Gerbert went to Italy,

about 970, where he m et Pope John X III, being by himpresented to the emperor Otto 1. Two years laterat the emperor’ s request, he went to Rheim s , where hestudied philosophy, ass isting to make of that place an ed

u cational center ; and in 983 he became abbot at B obbio.The next year he returned to Rheim s , and became archbishop Of that diocese in 991. For political reasons hereturned to Italy in 996, became archbishop of Ravennai n 998, and the following year was elected to the papalchair. Far ahead of his age in wisdom, he suffered asmany such scholars have even in times not so remoteby being accused of heresy and witchcraft. AS late as1522, in a biography published at Venice, it is relatedthat by black art he attained the papacy, after havinggiven his soul to the devil.2 Gerbert was , however,interested in astrology,3 although thi s was merely theastronomy of that time and was such a science as any

learned man would wish to know, even as to-day we wishto be reasonably familiar with physics and chemistry.

That Gerbert and hi s pupils knew the gobar numerals is a fact no longer Open to controversy.4 Bernelinus and R icher 5 call them by the well-known name of

1 Picavet, loc . cit., p . 37 .

2 Con S inistre arti conseguri la dignita del Pontificato. La

sciato poi l ’ abito, e ’ l monasterio,e datos i tutto in potere del diavolo .”

[Quoted inBombelli ,L ’antica numeraz ione I talica , Rome

,1876

, p . 4 1n, ]

3 He writes from Rheims in 984 to one Lupitus , in Barcelona, saying Itaque librum de astrologia translatum a te mich i petenti dirige,” presumably referring to some A rab ic treatise. [Ep ist. no . 24

of the Havet collection, p .

4 See Bubnov, loc . cit. , p . x .

Olleris,loc . cit .

, p . 361, l . 15, for Bernelinus ; and Bubnov, loc . cit. ,

p . 381,l . 4

,for Richer .


“ caracteres , a word used by Radu lph of Laon in thesame sense a century later.1 It is probable that Gerbertwas the first to describe these gobar numerals in any.

scientific way in Clni stian Europe, but without the zero]If he knew the latter he certainly did not understandits u se 2The question still to be settled is as to where he

found these numerals . That he did not bring them fromSpain is the opinion of a number of careful investigators.3 Thi s is thought to be the more probable becausemost of the men who made Spain famous for learninglived after Gerbert was there. Such were Ibn Siné.

(Avicenna) who lived at the beginning, and Gerber ofSeville who flourished in the middle, of the eleventhcentury, and Abu Roshd (Averro

'

es) who lived at theend of the twelfth.

4 Others hold that hi s proximity to

1 Woepcke found thi s in a Paris MS . of Radulph of Laon, c . 1100 .

[Prop agation, p . E t prima quidem trium spaciorum superductio

uni tatis caractere inscribitur, qu i chaldeo nom ine dicitur igin.

” See

also A lfred Nagl, “ Der arithmetische Tractat des Radu lph von

Laon (Abhandlungen zur Geschichte der Mathematik,VOl .V

, pp . 85

133) , p . 97 .

2 Weissenborn,loc . cit. , p . 239 . When Olleris (Gl uvres de Gerbert,

Paris,1867

, p . cci) says , C ’ est a lui et non point aux A rabes, que

l ’Europe doit son systeme et ses signes de numeration,

” he exaggerates ,since the evidence is all against his knowing the place valu e. Friedlein

emphas i zes th is in the Zeitschriftfur Mathematikund Phys ik, Vol . X II

L iteraturzeitung, p . 70 : Fur das System unserer Numerationist dieNu ll das wesentlichsteMerkmal

,und diese kannte Gerbert ni cht.

Er selbst sch rieb alle Zahlen mit den rOmischen Zahlzeichen und man

kann ihm also nicht verdanken,was er selbst ni cht kannte.

”

3 E .g.,Chasles

,Budinger, Gerhardt, and R icher . So Martin (Re

cherches nouvelles etc .) believes that Gerbert rece ived them from Boe

thins or his followers . See Woepcke, Prop agation, p . 4 1 .

4 Budinger, loc . cit ., p . 10 . Nevertheless , in Gerbert’ s time one Al

Mansur, governi ng Spain under the name of Hisham (976 called

from the Orient A l-Begani to teach his son,so that scholars were

recognized . [Picavet, p .


the Arabs for three years makes it probable that he as

s im ilated some of their learning, in Spite of the factthat the lines between Christian and Moor at that timewere Sharply drawn .

1 Writers fail, however, to recognize that a comm ercial numeral system would havebeen more l ikely to be made known by merchants thanby scholars. The itinerant peddler knew no forbiddenpale in Spain , any more than he has known one in otherlands. If the gobar numerals were used for markingwares or keeping simple accounts, it was he who wouldhave known them ,

and who would have been the one

rather than any Arab scholar to bring them to the ii iqu iring

‘

m ind of the young French monk . The factsthat Gerbert knew them only imperfectly, that he usedthem s olely for calculations, and that the forms are evi

dently like the Spanish gobzi r, make it all the moreprobable that it was through the small tradesman of theMoors that this versatile scholar derived his knowledge.

Moreover the part of the geometry bearing his name, andthat seems unquestionably his, shows the Arab influ encefi

"

proving that'

he at least came i nto contact with the

transplanted Oriental learning, even though imperfectly.

2

There was also the persistent Jewish m erchant tradingwith both peoples then as now, always alive to the ac

quiring of useful knowledge, and it would be very natural for a man like Gerbert to welcome learning fromsuch a source.

On the other hand, the two leading sources of information as to the l ife of Gerbert reveal practically nothing

_to show that he came within the Moorish sphere of

i nfluence during his sojourn in Spai n. These sources1 We issenborn

,loc . cit.

, p . 235.

2 Ibid ., p . 234 .


it was he who contributed the m orsel of knowledge SO

imperfectly assim ilated by the young French monk .1

Within a few years after Gerbert’ s vis it two young Spanish monks of lesser fame, and doubtless with not thatkeen interest in mathematical matters whi ch Gerbert had,regarded the apparently Slight knowledge whi ch they hadof the Hindu numeral forms as worthy of somewhat permanent record 2 i nmanuscripts which they were transcribing. The fact that such knowledge had penetrated to theirmodest cloisters i n northern Spain the one Albelda orAlbaida indicates that it was rather widely diffused.

Gerbert’

s treatise L ibella s de numerorum divisione 3 ischaracterized by Chasles as one of the most obscuredocuments in the history of science.

” 4 The most complete information in regard to thi s and the o ther mathem atical works Of Gerbert is given by Bubnov,5 whoconsiders this work to be genuine.

6

1 H. Suter , “ Zur Frage iiber den Josephu s Sap iens ,” BibliothecaMathematica , Vol . VIII (2) , p . 84 ; Weissenborn

,E infuhrung, p . 14 ;

also his Gerbert ; M . Steinschneider,in B ibliotheca Mathematica

,1893

,

p . 68. Wallis (A lgebra , 1685, chap . 14 ) went over the list of Spani shJosephs very carefu lly , but could find nothing save that “ JosephusHispanu s seu Josephu s sapiens videtur aut M au rus fuisse au t al iusqu is in Hispania .

”

2 P . Ewald , M itthei lungen, Neues A rchiv d . Gesellschafi filr alteredeutsche Gesch ichtskunde, Vol . VIII , 1883, pp . 354—364 . One of themanu scripts is of 976 A .D . and the other of 992 A .D . See also FranzSteffens

,Lateinische Palaograp hie, Freibu rg (Schweiz) , 1903

, pp .

xxxix- xl . The forms are reproduced in the plate on page 140 .

3 It is entitled Constantino suo Gerbertus scolasticus,because it was

addressed to Constantine,a m onk of the Abbey of Fleu ry . The text

of the letter to Constantine, preceding the treatise on the Abacus

,is

given in the Comp tes rendus , V ol . XVI p . 295. Thi s book seems

to have been written 0 . 980 A .D . [Bubnov, loc . cit., p .

4 Histoire de l ’Arithm é tique,” Comp tes rendus , Vol. XVI (1843)

pp . 156,281 .

5 Loc. cit.,Gerberti Op era etc .

5 Friedlein thought it spu rious . See Zeitschriftfur M athematikundPhys ik, Vol . X II Hist.-lit. suppl . , p . 74 . It was discovered in


So little did Gerbert appreciate these numerals thatin hi s works known as the R egu la de abaco computi and

the L ibellu s he makes no u se of them at all, employingonly the Roman forms.1 Nevertheless Bernelinu s 2 refersto the nine gobs-tr characters.3 These Gerbert had markedon a thousand jetons or counters,4 using the latter on an

abacus which he had a sign-maker prepare for him.

5

Instead of putting eight counters in say the tens’ column,

Gerbert would put a single counter marked 8, and sofor the other places, leaving the column empty wherewe would place a zero , but where he, lacking the zero,had no counter to place. These counters he possiblycalled caracteres, a name which adhered also to the figures themselves . It is an interesting speculation to consider whether these apices, as they are called in theBoethius interpolations , were in any way suggested bythose Roman jetons generally known in numismaticsas tesserae, and bearing the figures I—XVI, the sixteenreferri ng to the number of assi in a sestertiu s.6 The

the l ibrary of the Benedictine monastry of St. Peter, at Salzbu rg, andwas published by Peter Bernhard Pez in 1721 . Doubt was first castupon it in the Olleris edition (GEuvres dc C erberi) . See Weissenborn

,

Gerbert, pp . 2

,6,168

,and Picavet

, p . 81 . Hock,Cantor

,and Th . M artin

place the compos ition of the work at c . 996 when Gerbert was in Germany

,wh ile Olleris and Picavet refer it to the period when he was at

Rheim s .

1 Picavet,loc. cit .

, p . 182.

2 Who wrote after Gerbert became pope, for he u ses , in his preface,the words

,

“a dom ino pape Gerberto .

” He was qu ite certa inly notlater than the eleventh centu ry ; we do not have exact informationabou t the time in wh ich he lived .

3 Picavet,loc . cit.

, p . 182. Weissenborn,Gerbert

, p . 227 . InOlleris

Liber Abaci (of Bernelinus) , p . 361 .

4 Richer,in Bubnov

,loc . cit.

, p . 381 .

5Weissenborn,Gerbert

, p . 24 1 .

5Writers on num ism atics are qu ite uncerta in as to thei r u se . See

F. Gnecchi,Monete Romane

,2d cd . ,

M ilan,1900

, cap . XXXVII . For


name ap ices adhered to the Hindu-Arabic numerals untilthe s ixteenth century.

1

To the figures 0 11 the ap ices were given the namesIgin, andras, ormis , arbas, u as , calctis or caltis , zenis ,temenias , celentis , sipos,2 the origin and meaning ofwhich still remai n a mystery. The Semitic origin ofseveral of the words seems probable. Wahud, thaneine,

p ictu res of old Greek tesserae of Sarmatia,see S . Ambrosoli

,Monete

Greche,M ilan

,1899

, p . 202.

1 Thu s Tzwivel ’s arithmetic of 1507,fol . 2

,v.

,Speaks of the tenfig

u res as characteres sive numerorum ap ices a dino Seuerino Boetio .

”

2 Weissenborn uses s ip os for 0 . It is not given by Bernelinu s , andappears in Radu lph of Laon,

in the twelfth centu ry . See Gunther ’ sGeschichte

, p . 98,n. Weissenborn

, p . 11 Pihan,E xpose etc .

, pp .

xvi- xxi i.InFriedlein’

s Boetiu s, p . 396, the plate shows that all of the six im

portant manuscripts from wh ich the illustrations are taken contain thesymbol , whi le fou r out of five wh ich give the words u se the word s ip osfor O. The nam es appear in a twelfth—centu ry anonymous manus criptin the Vatican

,in a passage beginning

Ordine prim igeno s ib i nomen possidet igin .

Andras ecce locum m ox u endicat ipse secundumOrm is post num eros incompos itus s ib i primu s .

[Boncompagni Bu lletino, XV , p . Turch ill (twelfth centu ry) givesthe names Igin, andras , horm is , arbas , qu imas

,caletis

,zenis

,temenias

,

celentis,saying : Has autem figuras , ut domnu s [dom inus] Gvillelmus

Rx testatur,a pytagoricis habemu s

,nom ina u ero ab arabibus .

”(Who

theWilliam R . was is not known. Boncompagni Bu lletino XV, p .

Radulph of Laon (d . 1131) asserted that they were Chaldean (Prop agation, p . 4 8 A discuss i on of the whole question is also given inE . C . Bayley , loc . cit. Huet

,writing in 1679, asserted that they were

of Sem itic origin,as did Nesselmann in spite of h is despair over ormis ,

calctis,and celentis ; see Woepcke, Prop agation, p . 48. The names

were u sed as late as the fifteenth centu ry , without the zero , but withthe superscript dot for 10 ’ s , two dots for 100 ’ s , etc .

,as among the

early A rabs . Gerhardt mentions having seen a fou rteenth or fifteenthcentu ry manuscript in the B ibliotheca Amploniana with the names

Ingnin, andras , arm is , arbas , quinas , calctis , zencis , zemenias,zcelen

tis,

”and the statement “ S i unum punctum super ingnin ponitur, X

s ignificat. Si duo puncta super figuras superponunter, fiet

decuplim illins quod cum uno puncto s ignificabatur,” in Monats

berichte der K . P . Akad . d . Wiss .,Berl in

,1867

, p . 40.


of Bath, used the zero with the Roman characters , incontrast to Gerbert’s u se of the gobe

‘

tr forms withoutthe z ero.1 O ’

Creat uses three forms for zero, 0 , o, andr , as in Maximus Planudes . With this u se of the zerogoes, naturally, a place value, for he writes III III for33, ICCOO and I. II. 7 . r for 1200, I . 0 .VIII. IX for 1089,and I . IIII. IIII. rrrr for the square of 1200.

The period from the time of Gerbert until after theappearance of Leonardo ’ s monumental work may becalled the period of the abacists . Even for many yearsafter the appearance early in the twelfth century of thebooks explaining the Hindu art of reckoning, there wasstrife between the abacists , the advocates of the abacus,and the algorists , those who favored the new numerals.The words cifra and algorismus cifra were used witha somewhat derisive S ignificance, indicative of absoluteuselessness, as indeed the zero is useless on an abacusin which the value of any unit is given by the columnwhich it occupies.2 So Gautier de Coincy ( 11 77— 1236)in a work o on the miracles of Mary says

A horned beast, a sheep,

An algori smu s—c ipher,Is a priest, who on such a feast dayDoes not celebrate the h oly M other.3

So the abacus held the field for a long time, evenagainst the new algorism employing the new numerals.

1 The title of h is work is Prologus N . Ocreati in Helcep h (Arabical—qeif, investigation or memoir) ad Adelardum Batensem magistrum

suum . The work was made known by C . Henry,in the Zeitschriftfur

Mathematik und Phys ik, Vol . XXV , p . 129,and in the A bhandlungen

zur Gesch ichte derMathematik,Vol. III ; Weissenborn

,Gerbert, p . 188.

2 The zero is indicated by a vacant column.

3 Lee Jordan,loc . cit .

, p . 170 .

“ Ch i f re en augorisme” is the ex

press i on u sed,wh ile a century later gi ffre en argorisme and cyffres

d’augorisme

”are similarly used .


Geoffrey Chaucer 1 describes in The Zlfi ller’ s Tale the clerkW ith “ Hi s Alm ageste and bokes grete and sm ale,

Hi s astrelabie, longinge for h i s art,

Hi s augrim -stones layen faire apartOn shelves cou ched at h i s beddes heed.

So , too, in Chaucer’ s explanation of the astrolabe,2written for his son Lewis , the number of degrees is expressed ou the instrument i n H indu-Arabic numeralsOver the whiche degrees ther ben noumbres of augrim,

that devyden thilke same degrees fro fyve to fyve,”

and the nombres ben writen in augrim,

meaning in the way of the algorism . Thomas Usk

about 1387 writes : 3 “ a sypher in augrim have 11 0 mightin signification of it-selve, yet he yeveth power in signification to other. So slow and so painful is the assimilation of new ideas .Bernelinu s 4 states that the abacus is a well-polished

board ( or table) , which is Covered with blue sand and

used by geometers i n drawing geometrical figures. We

have previously mentioned the fact that the Hindus alsoperformed mathematical computations in the sand, al

though there is no evidence to Show that they had anycolumn abacus .5 For the purposes of computation ,Bernelinu s continues, the board is divided into thirtyvertical columns , three of which are reserved for fractions . Beginning with the units columns, each set of

1 TheWorks of Geofi'

rey Chaucer, edited by W .W . Skeat,Vol . IV

,

Oxford,1894

, p . 92 .

2 Loc . cit .,V ol . 111

, pp . 179 and 180 .

3 In Book II,chap . vi i

,of The Testament of Love, printed with

Chau cer ’ s Works,loc . cit.

,Vol . VII , London,

1897 .

4 L iber Abacci, published in Olleris , (Euvres de Gerbert, pp . 357—400 .

5 G . R . Kaye ,“ The Use of the Abacus in Ancient India

,

” Journaland Proceedings of the A s iatic Society of Benga l, 1908, pp . 293—297.


three columns ( lineae is the word which Bernelinus uses)is grouped together by a semic ircular are placed abovethem

,while a smaller arc is placed over the units col

umn and another joins the tens and hundreds columns.Thus arose the designation arcu s p ictagore

1 or sometim essim ply arcu s.

2 The operations of addition , subtraction ,and multiplication upon this form of the abacus requiredlittle explanation, although they were rather extensivelytreated, especially the multiplication of different ordersof numbers . But the operation of divis ion was effectedwith some difficulty. For the explanation of the methodof division by the u se of the complementary difference,3long the s tumbling-block in the way of the m edievalarithm etician , the reader is referred to works on the hi story ofmathematics 4 and to works relating particularlyto the abacus .

5

Among the writers on the subject may be m entionedAbbo 6 of Fleury ( c . Heriger

7 of Lobbes or Laubach

1 Liber Abbaci , by Leonardo Pisano , loc . cit ., p . 1 .

2 Friedle in,

“D ie Entwickelung des Rechnens m i t Columnen,

” Zeitschriftfu’

r Mathematik und Phys ik, Vol . X , p . 24 7 .

3 The divisor 6 or 16 being increased by the d i fference 4 , to 10 or20 respectively .

4 E .g. Cantor,Vol . I

, p . 882.

5 Friedlein,loc . cit . ; Friedlein,

Gerbert’s Regeln der D ivis i on”

and “Das Rechnen m it Columnen vor dem 10 . Jahrhundert,

” ZeitschriftfiirM athematikund Phys ik, Vol. IX Bu bnov

,loc . cit .

, pp . 197~

245; M . Chasles,

“ Histo ire de l ’ arithm é t ique . Recherches des tracesdu systeme de l ’abacus

,apres que cette m éthode a pris le 110 111 d’

Algo

risme.— Preuves qu ’

atou tes les époques , jusqu ’au xv 1e s iecle

,0 11 a su

que l’arithm é tiquevu lgai re ava it pou r originecettem éthode anc ienne,”

Comptes rendus , Vol . XVII , pp . 143— 154,also “ Regles de l

’abacus ,

Comptes rendus , Vol . XVI, pp . 218- 24 6, and “Analyse et explication

du tra ité de Gerbert,

” Comptes rendus , Vo l . XVI, pp . 281—299.

5 Bubnov,loc . cit .

, pp . 203—204,Abbonis abacu s .

7 “ Regulae de numerorum abaci rationibu s ,” in Bubnov, loc . cit.,

pp . 205—225.


interested listeners. Another Raoul, or Radu lph, to whomwe have referred as Radu lph of Laon , 1 a teacher i n thecloister school of his c ity, and the brother of Anselm ofLaon 2 the celebrated theologian, wrote a treatise on music,extant but unpublished, and an arithm etic which Nagl

first published i n The latter work, preserved to usin a parchment manuscript of seventy-seven leaves, contai ns a curious mixture of Roman and gobs-tr numerals, theformer for express ing large results , the latter for practicalcalculation. These gobei r caracteres include the sipos( zero) , Q , of which, however, Radu lph did not knowthe full significance ; showing that at the Opening of thetwelfth century the system was still uncertain in its statusin the church schools of central France.

At the same time the words algorismu s and cifra werecoming into general u se even in non-mathematical literatu re. Jordan 4 cites numerous instances of such u se fromthe works of Alanu s ab Insu lis 5 (Alain de L ille) , Gautier de Coincy ( 11 7 7 and others.Another contributor to arithmfi ic during this interest

ing period was a prominent Spanish Jew called various lyJohn of Luna, John of Seville, Johannes Hispalensis ,Johannes Toletanu s , and Johannes Hispanensis de Luna.6

1 Cantor,Gesch ichte

,Vol . I

, p . 762. A . Nagl in the Abhandlungenzur Geschichte der Mathematik, Vol . V , p . 85.

2 1030—1117 .

3 Abhandlungen zur Geschichte der Mathematik,Vol . V , pp . 85—133.

The work begins “ Incip it L iber Radulfi laudunens is dc abaco .”4 Materia lien zur Geschichtederarabi schen Zahlzeichen inFrankrei ch

loc . cit. 5Who died in 1202.

5 Cantor,Gesch ichte

,Vol . I pp . 800—803 Boncompagni , Trattati ,

Part II . M . Ste inschneider (“D ie Mathematik bei den Juden,

”

B ibliotheca Mathemat ica,Vol . X p . 79) ingeniou sly derives another

name bywhich he is called (Abendeu th) from IbnDaud (SonOf David) .See also A bhandlungen, Vol . 111, p . 110.


His date is rather closely fixed by the fact that he dedicated a work to Raimund who was archbishop of Toledobetween 1130 and His interests were chiefly inthe translation of Arabic works, especially such as boreupon the Aristotelian philosophy. From the standpointof arithmetic, however, the chief interest centers about amanuscript entitled Joannis IIisp alensis liber Algorismi dePractica Arismetrice which Boncompagni found in whatis now the B ibliothe‘que nationale at Paris. Although thisdistinctly lays claim to being A l-Khowe‘ trazm i ’s work ,2

the evidence is altogether against the statement,3 bu t thebook is quite as valuable, since it represents the knowledge Of the time in which it was written. It relates to theoperations with integers and sexagesimal fractions, including roots

, and contains 11 0 applications .

4

Contemporary with John of Luna, and also living inToledo, was Gherard of Cremona,5 who has som etimesbeen identified, but erroneous ly, with Gernardu s ,6 the

1 John is sa id to have died in 1157 .

2 For it says ,“ Incipit prologus in libro alghoarism i de practica

arism etrice . Qu i editus est a magistro Johanne yspalensi .” It is pub

lished in fu ll in the second part of Boncompagni ’ s Trattati d ’aritmetica .

3 Possibly,indeed

,the meaning of “ libro alghoarism i is not “ to

Al-Khowarazm i ’ s book,

” but “ to a book of algorism .

” John of Lunasays of it : “Hoc idem est i llud etiam qu od alcorismu s dicerevidetu r .” [ Trattati , p .

4 For a résum é,see Cantor

,Vol . I pp . 800—803. A s to the au

thor,see Enestr

‘

Om in the B ibliotheca Mathematica , Vol . VI p . 114,

and Vol . IX p . 2.

5Born at Crem ona (although some have asserted at Carmona,in

Andalusia) in 1114 ; di ed at Toledo in 1187 . Cantor,loc . cit . ; Bon

compagni , A tti d . R . A ccad . d . n. L incei , 1851 .

5 See A bhandlungen zur Gesch ichtederMathematik,Vol .X IV

, p . 149 ;

B ibliotheca Mathematica,Vol . IV p . 206. Boncompagni had a

fou rteenth—centu ry manu script of h is work, Gerardi Cremonensis artismetrice p ractice. See also T . L . Heath , The Th irteen Books of E uclid ’ sE lements , 3 vols .

,Cambridge , 1908, Vol . I , pp. 92—94 A . A . Bj

'

ornbo,


author of a work on algorism. He was a physician, anas tronomer, and a mathematician , translating from the

Arabic both in Italy and in Spain. In arithm etic he wasinfluential in spreading the ideas of algorism.

Four Englishmen Adelhard of Bath ( 0 . Rob

ert of Chester (Robertu s Cestrens is , c. WilliamShelley, and Daniel M orley ( 1180) — are known 1 tohave journeyed to Spain in the twelfth century for thepurpose of studying m athematics and Arabic. Adelhardof Bath made translations from Arabic into Latin of A iKhowe

'

trazm i’

s astronomical tables 2 and of Euclid ’ s Elements,3while Robert Of Chester is known as the translator

l/OfA l-Khowfirazm i

’

s algebra.

4 There is 110 reason to doubtthat all of these m en, and others , were familiar with thenumerals which the Arabs were us ing.

The earliest trace we have of computation with Hindunumerals i nGermany is in an A lgorismu s of 114 3, nowin the Hofbibliothek in V ienna.

5 It is bound in with a

“ Gerhard von Cremonas Ubersetzung von Alkwarizm is Algebra und

von Euklids E lementen,

” B ibliotheca Mathematica,Vol . VI pp .

239- 248 1 Wallis,A lgebra , 1685, p . 12 seq .

2 Cantor,Geschichte

,Vol. I p . 906 ; A . A . Bjornbo

,

“Al-Chwa

rizm i ’ s trigonometriske Tavler,

” Festskrift ti l H . G . Zeuthen,Copen

hagen,1909

, pp . 1—17 .3 Heath

,loc . cit .

, pp . 93- 96.

4 M . Steinschneider,Zeitschrift der deutschen morgenla‘ndischen Cc

sellschaft, Vol . XXV ,1871

, p . 104,and Zeitschrift fur Mathematik und

Phys ik, Vol . XVI , 1871, pp . 392- 393 ; M . Curtze, Centralblatt fur

B ibliothekswesen,1899

, p . 289 ; E .Wappler, Zur Geschichte der deutschen A lgebra im 15. Jahrhundert

,Programm ,

Zwickau,1887 L . C .

Karpinski , “ Robert of Chester ’ s T ranslation of the A lgebfa of AlKhowarazm i

,

” B ibliotheca Mathematica ,Vol . XI p . 125. He is also

known as Robertus Retinens is,or Robert of Reading.

5Nagl, A ., Ueber e ine Algorismu s -Schrift des XII . Jah rhunderts

und uber die Verbreitung der indisch-arabischen Rechenkuns t und

Zahlze ichen im ch ristl . Abendlande,

” in the Zeitschriftfu'

rMathemat ikand Phys ik, II i st.-lit. Abik.

,Vol . XXXIV

, p . 129 . Cu rtze,Abhand

lungen zur Gesch ichte dcr Mathematik,Vol . VIII , pp . 1

—27 .

CHAPTER VIII

THE SPREAD OF THE NUMERALS IN EUROPE

Of all the medieval writers , probably the one most ihflu ential in introducing the new numerals to the scholarsof Europewas Leonardo Fibonacci, of Pisa.

1 This remarkable m an, the mos t noteworthy mathematical genius ofthe M iddle Ages , was born at Pis‘a aboutThe traveler of to-day may cross the V ia Fibonacci

on his way to the Campo Santo, and there he may see

at the end of the long corridor, across the quadrangle,the statue of Leonardo in Scholar’ s garb. Few townshave honored a mathematician more, and few mathem a

ticians have so distinctly honored their birthplace. Leo

nardo was born in the golden age of thi s city, the periodof its commercial, rel igious , and intellectual prosperity.

3

1 F . Bonaini,

“M emoria unica sincrona di Leonardo Fibonacc i,

”

Pisa,1858

,republished

°

in 1867,and appearing in the G iorna le A rca

dico,Vol . CXCVII (N. S . LII) ; Gaetano M ilanesi, Documento inedito e

sconosciuto a L ionardo Fibonacci , Roma,1867 ; Gugl ielmini, E logio

di L ionardo Pisano, Bologna, 1812, p . 35; L ibri , H isto ire des sci

ences mathématiques , Vol . II, p . 25; D . Mart ines

,Origine e p rogress i

dell’ aritmetica,Messina

,1865

, p . 4 7 ; Lucas , in Boncompagni Bu lle

tino, Vol . X , pp . 129,239 ; Besagne, ibid .

,Vol . IX

, p . 583 ; Boncompagni ,

three works as cited in Chap . I ; G . Enestr‘

Om,

“Ueber zwei angebl iche mathematische Schu len im christlichen M ittelalter,” BibliothecaM athematica

,V ol . VII I pp . 252- 262 ; Boncompagni ,

“Della vitae delle opere d i Leonardo Pisano ,” loc . cit.

2 The date is pu rely conjectu ral . See the B ibliotheca Mathematica,

Vol. IV p . 215.

3 An Old chronicle relates that in 1063 Pisa fough t a great battlewith the Saracens at Palermo , capturing s ix sh ips , one being “ full ofwondrous treasu re,” and th is was devoted to building the cathedral .

128

SPREAD OF THE NUMERALS IN EUROPE 129

S ituated practically at the mouth of the Arno , Pisaformed with Genoa and Venice the trio of the greatestcommercial centers of Italy at the opening of the thirteenthcentury. Even before Venice had captured the Levantine trade, Pisa had close relations with the E ast. Anold Latin chronicle relates that in 1005 “ Pisa was captured by the Saracens ,” that in the following year the

Pisans overthrew the Saracens at Reggie ,” and that in

1012 the Saracens came to P isa and destroyed it. The

city soon recovered, however, sending no fewer than ahundred and twenty ships to Syria in foundinga merchant colony in Constantinople a few years later,2and meanwhi le carrying on an interurban warfare in Italythat seemed to stimulate it to great activity.

3 A writerOf 1114 tells us that at that time there were many heathen people — Turks, L ibyans , Parthians, and Chaldeans — to be found in P isa. It was in the midst of suchwars, in a cosmopolitan and commercial town , in a center where literary work was not appreciated,4 that thegenius of Leonardo appears as one of the surprises ofhistory, warning us again that “we should draw 11 0

horoscope ; that we should expect little, for what weexpect will not come to pass .” 5Leonardo ’ s father was one William,

5 and he had abrother nam ed Bonaccingu s ,

7 but nothing further is

1 Heyd,loc . cit .

,Vol . I

, p . 149.2 Ibid .

, p . 211 .

3 J. A . Symonds , Rena issance in Italy . The Age of Desp ots . New

York,1883

, p . 62.4 Symonds

,loc . cit.

, p . 79.

5 J . A . Froude,The Science of H istory , London, 1864 .

“Um brevetd’apothicaire n

’empecha pas Dante d ’etre le plu s grand poete de

l ’Italie,et cc fut un petit marchand de Pise qu i donna l ’algebre aux

Chrétiens .

”[Libri , Histoire, Vol . I , p . xvi .]

5A document of 1226,found and published in 1858, reads : “Leo

nardo bigollo quondam Gu ilielmi .” 7 “Bonaccingo germane suo.

”


known of hi s family. As to Fibonacci, most writers 1 haveassumed that his fathei ’ s name was Bonaccio,2 whencefiliu s Bonaccii, or Fibonacci. Others 3 bel ieve that thename, even in the Latin form offilius B onacci i as usedin Leonardo’ s work , was simply a general one, like ourJohnson or Bronson (Brown

’ s son) ; and the only contemporary evidence that we have bears out this view.

As to the name Bigollo, us ed by Leonardo, some havethought it a self-as sumed one meaning blockhead, a termthat had been applied to him by the comm ercial worldor possibly by the university c ircle, and taken by himthat he might prove what a blockhead could do. Milanesi,4 however, has Shown that the word Bigollo ( orPigollo) was used in Tuscany to mean a traveler, andwas naturally assumed by one who had studied, as Leonardo had, in foreign lands.Leonardo ’ s father was a commercial agent at Bugia,

the modern Bougie,5 the ancient Saldae on the coast ofBarbary,6 a royal capital under the Vandals and again,

a century before Leonardo, under the Beni Hamm ad.

It had one of the best harbors on the coast, sheltered asit is by Mt. Lalla Guraia,7 and at the close of the twelfthcentury it was a center of African comm erce. It was herethat Leonardo was taken as a child, and here he went toschool to a M oorish master. When he reached the yearsof young manhood he started on a tour of the Mediterranean Sea, and vis ited Egypt, Syria, Greece, S ic ily,and Provence, meeting with s cholars as well as with

1 E .g. L ibri,Gugl ielm ini , T iraboschi . 2 Latin

,Bonaccius .

3 Boncompagni and M ilanes i . 4 Reprint, p . 5.

5Whence the French name for candle .

5Now part of Algiers .

7 E . Reclus , Africa , New York, 1893, Vol . I I , p . 253.


account such a book would attract even less attentionthan usual.1It would now be thought that the western world

would at once adopt the new numerals which Leonardohad made known , and which were so much superior toanything that had been in u se in Christian Europe. The

antagonism of the universities would avail but little, itwould seem, against such an im provement. It must beremembered, however, that there was great difficulty inspreading knowledge at this time, some two hunche d andfifty years before printing was invented.

“ Popes and

princes and even great religious institutions posses sed farfewer books than many farmers of the present age. Thelibrary belonging to the Cathedral Church of San Mar

tino at Lucca in the ninth century contained only nineteenvolumes of abridgments from ecclesiastical commentaries .” 2 Indeed, it was not until the early part of the fifteenth century that Palla degli Strozz i took steps to carryout the project that had been in the mind of Petrarch,the founding of a public library. It was largely byword of mouth, therefore, that this early knowledge hadto be transmitted. Fortunately the presence of-foreignstudents in Italy at this time made thi s transmissionfeasible. ( If human nature was the same then as now, itis not imposs ible that the very Opposition of the facultiesto the works of Leonardo led the students to inves tigate

1 On the commercial activity of the period , it is known that billsof exchange passed between M essina and Constantinople in 1161

,

and that a bank was founded at Venice in 1170,the Bank of San

Marco being established in the following year . The activity Of Pisawas very mani fest at this time . Heyd , loc . cit .

,Vol . II

, p . 5; V . Casagrandi , S toria e cronologia ,

3d ed .,M ilan,

1901, p . 56.

2 J. A . Symonds , loc. cit.,Vol. II , p . 127.


them the more zealously.) At V icenza in 1209, forexample, there were Bohemi ans, Poles, Frenchm en ,Burgundians , Germans, and Spaniards , not to speak ofrepresentatives of divers towns of Italy ; and what was

true there was also true of other intellectual centers .The knowledge could not fail to Spread, therefore, andas a matter Of fact we find numerous bits of evidencethat thi s was the case. Although the bankers of Florence were forbidden to u se these numerals in 1299, andthe statutes Of the university of Padua required stationers to keep the price lists of books “ non per cifras, sedper literas claros ,

” 1 the numerals really made muchheadway from about 1275 on .

It was, however, rather exceptional for the commonpeople of Germany to u se the Arabic numerals before thesixteenth century, a good witness to this fact being thepopular almanacs. Calendars of 1457— 14 96 2 have generally the Roman numerals, wh ile KObel ’s calendar of 1518gives the Arabic forms as subordinate to the Roman.

In the register of the Kreuzschule at Dresden the Romanforms were used even until 1539.

While not minimiz ing the importance of the scientificwork of Leonardo of Pisa, we may note that the more popular treatises by Alexander de V illa Dei ( c. 124 0 A .D .)and Joh n of Halifax ( Sacrobosco, c. 1250 A .D.) weremuch more widely used, and doubtless contributed moreto the spread of the numerals among the common people.

1 I . Taylor, The A lp habet, London,1883

,Vol . II

, p . 263.

2 Cited by Unger ’ s H istory , p . 15. The A rabic num erals appear ina-Regensbu rg ch ronicle of 1167 and in S ilesia in 1340 . See Schm idt’ sE ncyclopa

'

dieder E rziehung ,Vol .VI

, p . 726 ; A . Ku ckuk,Die Rechen

kunst im sechzehntenJahrhundert,

” Festschriftzurdritten Sacu larfeierdes Berlinischen Gymnasiums zum grauen K loster, Berlin, 1874 , p . 4 .


The Carmen de Algorismo1 of Alexander de V illa Dei

was written in verse, as indeed were many other textbooks of that time. That it waswidely used is evidencedby the large number of manuscripts 2 extant i nEuropeanlibraries. Sacrobosco ’ s Algorismu s ,3 in which some linesfrom the Carmen are quoted, enjoyed a wide popularityas a textbook for university instruction .

4 The work wasevidently written with this end in view, as numerouscommentaries by university lecturers are found. Probably the most widely used of these was that of Petrus deDacia 5written in 1291 . These works throw an i nteresting light upon the method of instruction m mathematicsin u se in the universities from the thirteenth even to thesixteenth century. Evidently the text was first read andcopied by students .6 Following this came line by lineexposition of the text, such as is given in Petrus deDacia

’

s commentary.

Sacrobosco ’s work is of interest also because it wasprobably due to the extended u se of this work that the

1 The text is given in Halliwell, Rara M athematica,London

,1839 .

2 Seven are given in A shmole ’ s Catalogue of Manuscripts in the

Oxford L ibrary , 1845.

3 Maxim il ian Cu rtze,Petri Phi lomeni dc Dacia in A lgorismum Vul

garem Johannis de Sacrobosco commentarius,una cum A lgorismo ip so,

Copenhagen, 1897 ; L . C . Karp inski , “Jordanus Nemorarius and Johnof Halifax

,

” AmericanMathematicalMonthly , Vol . XVII, pp . 108— 113.

4 J . Aschbach , Gesch ichte der Wiener Universita't im erstenJahrhunderte ihres Bestehens ,Wien, 1865, p . 93.

5 Curtze,loc . cit.

, gives the text.

5 Cu rtze,loc . cit.

,found some forty-five copies of the A lgoris

mus in three libraries of Munich,Venice, and E rfu rt (Amploniana ) .

Examination of two manuscripts from the Plimpton collection and

the Columbia l ibrary shows such marked d ivergence from each otherand from the text published by Cu rtze that the conclu s i on seem s legiti

mate that these were students ’ lectu re notes . The shorthand character of the writing fu rther confirms th is view ,

as it shows that theywere written largely for the personal use of the writers .


two pages ( one folio) relating to arithmetic. In these theforms of the numerals are given , and a very brief statementas to the Operations, it being evident that the writer himself had only the slightest understanding of the subject.Once the new system was known in France

, eventhus superficially, it would be passed across the Channel to England. Higden,1 writing soon after the openingof the fourteenth century, speaks of the French influenceat that time and for some generations preceding : 2 Fortwo hundred years chi ldren in scole, agenst the usageand manir of all other nations beeth compelled for toleave hire own language, and for to construe hir lessonsand hire thynges i n Frensche. Gentilmen chi ldrenbeeth taught to speke Frensche from the tyme that theybith rokked in hir cradell ; and uplondi ssche men willlikne himself to

'

gentylmen, and fondeth with greet besynesse for to speke Frensche.

”

The question is Often asked, why did not these newnumerals attract more immediate attention 7 Why didthey have to wait until the sixteenth century to be generally used in business and i n the schools 7 In reply itmay be said that in their elementary work the schoolsalways wait upon the demands of trade. That work whi chpretends to touch the life of the people must come reasonably near doing so. Now the computations of busines suntil about 1500 did not demand the new figures, fortwo reasons : First, cheap paper was not known. Papermaki ng of any kind was not introduced into Europe until

1 Ranulf Higden,a native of the west of England , entered St.

Werburgh’s monas tery at Chester in 1299 . He was a Bened ict ine

monk and ch ronicler,and died in 1364 . His Polychronicon, a history

in seven books,was printed by Caxton in 1480 .

2 Trevisa ’s translation,Higden having written in Latin.


the twelfth century, and cheap paper is a product ofthe nineteenth. Pencils, too, of the modern type, dateonly from the s ixteenth century. In the second place

,

modern methods of operating, particularly of multiplyingand dividing ( operations of relatively greater importancewhen all measures were in compound numbers requiringreductions at every step) , were not yet invented. The

Old plan required the eras ing of figures after they hadserved their purpose, an Operation very S imple with counters , since they could be removed. The new plan didnot as easily permit this. Hence we find the new numerals very tardily admitted to the counting-house , and notwelcomed with any enthusiasm by teachers.1Aside from their u se in the early treatises on the new

art of reckoning, the numerals appeared from time totime in the dating of manuscripts and upon monuments .The oldest definitely dated European document known

1 An illustration of this feeling is seen in the writings of Prosdocimode ’ Beldomandi (b . c . 1370— 1380

,(1.

“ Inveni in quam plu ribuslibris algorismi nuncupatis mores circa numeros Operandi satis variosatque diversos , qu i licet boni existerent atque veri erant, tamem fastidiosi

,tum propter ipsarum regu larum mu ltitudinem

,tum propter

earum deleationes,tum etiam propter ipsarum operationum proba

tiones , utrum Si bone fuerint vel ne . E rant et etiam isti modi interimfastidiosi

, quod s i in ali quo calcu lo astroloico error contigisset, calculatorem operationem suam a capite incipere oportebat, dato quoderror suu s adhu c satis prOpinquus ex isteret ; et hoc propter figuras insua Operatione deletas . Indigebat etiam calcu lator semper aliquo

lapide vel sibi conformi,super quo scribere atque faciliter delere

posset figuras cum qu ibus operabatur in calcu lo suo . E t quia haecomnia satis fastidiosa atque laboriosa m ihi visa sunt, disposu i libellumedere in quo omnia ista abicerentur : qu i etiam algorismu s sive liberde numeris denom inari poterit. Scias tamen quod in hoe libello ponere non intendo nisi ea quae ad calcu lum necessaria sunt

,alia quae

in aliis libris practice arismetrice tanguntur, ad calcu lum non neces

saria, propter brevitatem dim itendo .

”[Quoted by A . Nagl , Zeitschr ift

furM athematikund Physik,H ist.—lit.Abth .,Vol . XXXIV

, p . 143 ; Smith,Rara A rithmetica

, p . 14,in facs im ile ]


to contain the numerals is a Latin manuscript, 1 theCodex

,

V igilanu s , written in the Albelda Cloister notfar from Logrono in Spain,

in 976 A .D. The nine characters ( of gobzi r type) , without the zero, are given as anaddition to the first chapters of the third book of theOrigines by Isidorus of Seville, in which the Roman numerals are under discussion. Another Spanish copy ofthe same work, of 992 A .D ., contains the numerals in thecorresponding section . The writer ascribes an Indianorigin to them in the following words : “ Item de figurisarithmetice. Scire debemu s i n Indos subtiliss imum ingenium habere et ceteras gentes eis in arithmetica et geo

metria et ceteris liberalibu s disciplinis concedere. E t hocmanifestum est in nobem figuris , quibus des ignant unumqu emqu e gradum cu iu slibet gradus. Qu arum hec suntforma.” The nine gobei r characters follow. Some of theabacus forms 2 previously given are doubtless also ofthe tenth century. The earlies t Arabic documents con

taining the numerals are two manuscripts of 874 and888 A .D .

3 They appear about a century later in a work 4written at Shiraz in 970 A .D . There is also an earlytrace of their u se on a pillar recently discovered in achurch apparently destroyed as early as the tenth cen

tury,not far from the Jerem ias Monastery, in Egyp t.

1 P . Ewald , loc . cit . Franz Steffens , Lateinische Pa laographie, pp .

xxxix—xl . We are indebted to Professor J. M . Bu rnam for a photograph of this rare manuscript .

2 See the plate of form s on p . 88.

3 Karabacek, loc . cit ., p . 56 ; Karp inski , “ Hindu Numerals in the

Fihrist,

” Bibliotheca Mathematica ,V ol . XI p . 121 .

4 Woepcke,“ Sur une donnée h istorique ,” etc .

,loc . c it .

,and “ E ssa i

d ’une restitution de travaux perdus d ’Apollonius sur les quantitésirrat ionnelles

,d ’ apres des indicati ons ti rées d ’

un manuscrit arabe,”Tome X IV des Mémo ires p resentes p ar divers savants a l ’A cadémie des

sciences,Paris

,1856

,note

, pp . 6- 14 .


( eleventh) century, and the sixth and seventh are alsofrom an eleventh-century MS . of Boethius at Chartres.

EARL IES T MANUSCR IPT FORM S

These and other early forms are given by Mr. Hill in thistable, which is reproduced with his kind permission.

This is one of m ore than fifty tables given in Mr.

Hill’ s valuable paper, and to this monograph students

SPREAD OF THE NUMERALS IN EUROPE 14 1

are referred for details as to the development of numberforms in Europe from the tenth to the sixteenth cen

tury. It is of interest to add that he has found thatamong the earliest dates of European coins or medalsin these numerals , after the S icilian one already m en

tioned, are the following : Austria, 14 84 Germany, 14 89( Cologne) ; Switzerland, 14 24 (St. Gall) ; Netherlands ,14 74 ; France, 14 85; Italy,The earliest English coin dated in these numerals was

struck i n although there is a Scotch piece ofIn numbering pages Of a printed book these numeralswere first used in a work of Petrarch’ s published at C 0

logne in The date is given i n the following formin the Biblia P aup erum,

5 a block-book of 14 70, while in

another block-book which possibly goes back to c . 14 30 5the numerals appear in several illus trations, with formsas follows :

Many printed works anterior to 14 71 have pages or chapters numbered by hand, but many of these numerals are

1 See A rbuthnot, TheMysteries of Chronology , London,1900

, pp . 75,

78,98 ; F . Pich ler, Rep ertorium der steierischenMunzkunde, Gratz , 1875,

where the claim i s made of an Austrian coin of 1458 ; B ibliothecaM athematica

,Vol . X p . 120

,and Vol . X II p . 120 . There is a

B rabant p iece of 14 78 in the collect ion of D . E . Sm ith .

2 A specimen is in the British Mu seum . [A rbuthnot, p . 73 Ibid .

, p . 79 .

4 L iber de Remedi i s utriusque fortunae Coloniae.

5 Fr.Walthern et Hans B urning, Nordl ingen.

5Ars Memorandi,one of the Oldest Eu ropean block—books .

14 2 THE HINDU- ARABIC NUMERALS

of date much later than the printing of the work. Otherworks were probably numbered directly after printing.

Thus the chapters 2, 3, 4 , 5, 6 in a book of 14 70 1 arenumbered as follows : Capitulem zm . 4m .

v,

vi, and followed by Roman numerals.This appears in the body of the text, in Spaces left bythe printer to be filled in by hand. Another book 2 of14 70 has pages numbered by hand with a mixture ofRom an and Hindu numerals, thus,

As to monumental i nscriptions,3 there was oncethought to be a gravestone at Katharein , near Troppau ,with the date 1007 , and one at Biebrich of 1299. Thereis no doubt, however, of one at Pforzheim of 1371and one at U lm of Certain numerals on WellsCathedral have been assigned to the thirteenth century,but they are undoubtedly cons iderably later.5The table 0 11 page 1 4 3 will serve to supplement that

from Mr. Hill ’ s work .

6

1 E usebius Caesariensis ,Dep raep arationcevangelica ,Venice , Jenson,14 70 . The above statement holds for cop ies in the Astor L ibrary andin the Harvard Univers ity L ibrary .

2 Francisco de Retza,Comestorium vitiorum

,Nurnberg, 14 70 . The

copy referred to is in the Astor L ibrary .

3 See Mauch ,“ Ueber den Gebrauch arabischer Ziflern und die

Veranderungen derselben,

” A nzeiger fur K unde der deutschenVorzeit,1861

,columns 46

,81

,116

,151

,189

,229

,and 268 ; Calmet

,Recherches

sur l’originc des ch imes d’arithmé tique, plate, loc . cit.

4 G im ther, Gesch ichte, p . 175

,n. ; Mauch

,loc . cit .

5 These are given byW . R . Lethaby ,from drawings by J .T . Irvine

,

in the Proceedings of the Society of Antiquaries , 1906, p . 200 .

5 There are some ill-tabulated forms to be found in J . Bowring,The Decimal System ,

London,1854

, pp . 23,25

,and in L . A . Chassant,

Dictionna ire des abréviations latines et frangaises da moyen age,

14 4 THE HINDU —ARABIC NUMERALS

A"

—Qu. .

J11 3 .

\m th in"

?[I

For the sake of further comparison , three illustrations fromworks i nMr. Plimpton

’

s l ibrary,reproduced from the Ram Arith

metica, may be considered. The

first is from a Latin manuscripton arithmetic, 1 of which the original was written at Paris in 14 24by Bollandus, a Portuguese phys ician, who prepared the work atthe command of John of Lancaster, Duke of Bedford, at one

time Protector of England and

Regent of France, to whom the

work is dedicated. The figuresshow the successive powers of 2.

The second illustration is fromLuca da Firenze’s Inprencz

’

p io

darte dabaeho,2 c . 14 75, and the

third is from ananonymousmanuscript 3 of about 1500.

As to the forms of the numerals , fashion played a leading

part until printing was invented. This tended to fix theseforms

,although in writing there is still a great variation ,

as witness the French 5 and the German 7 and 9.

Even in printing there is not complete uniformity

1 See Ram Arithmetica, pp . 4 46—4 4 7 .

2 Ibid ., pp . 4 69—4 70 .

3 Ibid ., pp . 4 77—4 78.

SPREAD OF THE NUMERALS IN EUROPE 1 4 5

and it is ofteirdifficu lt for a foreigner to distinguishbetween the 3 and 5 of the French types.As to the particular numerals, the following are some

of the forms to be found in the later manuscripts andin the early printed books.1 . In the early printed books “one was often i, perhaps

to save types , just as some modern typewriters u se thesame character for l and In the manuscripts the one

”

appears i n such forms as 2

f,Ti c k et .

2.“Two often appears as z in the early printed books,

12 appearing as iz .

3 In the medieval manuscripts thefollowi ng forms are comm on : 4

9, 1 3 ,

—

LL ,

oz . p u ff, fl

1 The i is used for “ one in the Treviso arithmetic Clichto

veus (c . 1507 ed.,where both i and j are so u sed) , Ch iarini

Sacrobosco (14 88 and Tzwivel (1507 cd .,where jj and jz are used

for 11 and This was not universal,however

,for the A lgorithmus

linealis of c . 14 88 has a special typ e for 1 . In a student ’ s notebook oflectu res taken at the Univers ity of Wurzbu rg in 1660, inMr.Plimpton

’s

library,the ones are all in the form of i .

9 Thu s the date Ioffi for 1580,appears in a MS . in the Lau

rentian l ibrary at Florence . The second and the following five characters are taken from Cappelli

’s Dizionarz

’

o, p . 380, and are from

manu scripts of the twelfth , th i rteenth , fou rteenth , s ixteenth , seventeenth

, and e igh teenth centu ries,respectively .

3 E . g. Ch iarini’s work of 1481 Clichtoveu s (c .

4 The first is from an algorismus of the thirteenth century,in the

Hannover Library . [See Gerhardt, Ueber die Entstehung und

Ausbreitung des dekadischen Zahlensystems,

” loc. cit., p . The

second character is from a French algorismus , [Boncompagni Bu lletino, Vol . XV

, p . The third and the following sixteencharacters are given by Cappell i , loc . cit . , and are from manu scriptsof the twelfth th irteenth fourteenth fifteenth s ix

teenth seventeenth and eighteenth (1) centuries , respectively.


It is evident, from the early traces, that it is merelya curs ive form for the primitive just as 3 comes from

as in the Nana Ghat inscriptions.3. Three usuallyhad a special type in the first printed

books,although occasionally it appears as In the

medieval manuscripts it varied rather less than most ofthe others . The following are common forms : 2

4 .

“ Four has changed greatly ; and one of the firsttests as to the age of a manuscript on arithm etic, andthe place where it was written, is the examinationof this numeral . Until the time of printing the mostcommon form was R , although the Florentine manuscript of Leonard of P isa’

s work has the form 4 ;but the m anuscripts show that the Florentine arithme

ticians and astronomers rather early began to straightenthe first of these forms up to forms like 9 4

and 9 4

or 9 ,

5 more closely resembling our own. The firstprinted books generally used our present form 6 with theclosed top 4 the Open top used in writing ( 4 ) being

1 Thus Ch iarini (14 81) has 23 for 23.

2 The first of these is from a French algorismus , c . 1275. The

second and the following eight characters are given by Cappelli ,loc . cit .

,and are f rom manu scripts of the twelfth thi rteenth

,

fou rteenth,fifteenth seventeenth

,and eighteenth centuries ,

respectively .

3 See Nagl, loc . cit.

4 Hannover algorismus,th irteenth century .

5 See the Dagomari manuscript, in Ram Arithmetica, pp . 435,

437—4 40 .

6 But in the woodcuts of the Margarita Phi losophica (1503) the old

forms are used,although the new ones appear in the text. InCaxton’

s

Myrrour of the World (14 80) the old form is used .


8.“ E ight, l ike “

s ix, has changed but little. In

medieval times there are a few variants of interest as

In the sixteenth century, however, there was manifested a tendency to write it C4 1 2

9.“ Nine ”

has not varied as much as most of theothers. Among the medieval forms are the following 3

3" 7 . 2 p. 7 . a,

O. The shape of the zero also had a varied history.

The followi ng are common medieval forms 4

.aro

The explanation of the place value was a serious mat

ter to most of the early writers. If they had been usingan abacus constructed like the Rus s ian chotii , and had

placed this before all learners of the positional system,

there would have been little trouble. But the medieval

follows 1

thirteenth , fou rteenth fifteenth (2) , seventeenth , and e ighteenthcentu ries

,respectively .

1 All of these are given by Cappelli, thirteenth , fou rteenth , fifteenthand sixteenth centu ries

,respectively .

2 Smi th , Ram A rithmetica, p . 489 . This is also seen in several of the

Plimpton manu scripts , as in one written at Ancona in 1684 . See alsoCappelli, loc. cit.

3 French algorismu s,c . 1275

,for the first of these forms . Cap

pelli, th irteenth , fou rteenth , fifteenth and seventeenth centu ries,

respectively . The last th ree are taken from Byzantinische A na lekten,J. L . Heiberg, being form s of the fifteenth century , bu t not at all

common. 9 was the old Greek symbol for 90 .

4 For the first of these the reader is referred to the forms ascribedto Boeth ius

,in the illustration on p . 88 for the s econd

,to Radulph

of Laon,see p . 60 . The third is used occasionally in the B ollandus

(1424 ) manuscript, in Mr. Plimpton’s l ibrary . The remaining three

are f rom Cappelli, fou rteenth (2) and seventeenth centuries .

SPREAD OF THE NUMERALS IN EUROPE 14 9

line-reckoning, where the lines stood for powers of 10and the spaces for half of such powers , did not lenditself to thi s comparison. Accordingly we find suchlabored explanations as the following, from The C’mfte

Euery of these figuris bitokens bym selfe no more,yf he stonde in the first place of the rewele.

If it stonde in the secunde place of the rewle, he betokens ten tymes hym selfe, as thi s figure 2 here 20tokens ten tyme bym selfe, that is twenty, for he bymselfe betokens tweyne, ten tymes tw ene is twenty.

And for he stondis on the lyft s ide in the secundeplace , he betokens ten tyme hym selfe. And so goforth.

Nil cifra s ignificat s ed dat s ignare sequenti. Exponethi s verse. A cifre tokens nogt, bot he makes the figureto betoken that comes after bym more than he shu ld

he were away, as thus 10. here the figure of one tokensten, yf the cifre were away no figure byfore bym heschuld token bot one, for than he schuld stonde in thefirst place.

It would seem that a system that was thus used fordating documents , coins, and monuments, would havebeen generally adopted much earlier than it was, particu larly in those countries north of Italy where it didnot come into general u se until the sixteenth century.

This, however, has been the fate of many inventions, aswitness our neglect of logarithms and of contracted processes to-day.

.As to Germany, the fifteenth century saw the rise ofthe new sym bolism ; the sixteenth century saw it s lowly

1 Sm ith,An E arly English A lgorism .


gain the mastery ; the seventeenth century saw it finallyconquer the system that for two thousand years haddominated the arithm etic of business. Not a little ofthe success of the new plan was due to Luther’ s demandthat all learning should go into the vernacular.1During the transition period from the Roman to the

Arabic numerals, various anomalous forms found place.For example, we have in the fourteenth century ca for

1000. 300. 80 et 4 for and in a manuscriptof the fifteenth century 12901 for In the samecentury m. cccc. 8II appears for 14 82,5

'

wh'

ileM °CCCC °5O

( 1450) and MCCCCX L6 ( 14 4 6) are used by Theodorions Ruffi about the same time.

6 To the next centurybelongs the form 1vojj for 1502. Even in Sfortunati ’ S '

Nuovo lume7 the u se of ordinals is quite confused, the

propositions on a single page bei ng numbered “ tertia,”

4 , and “ V .

”

Although not connected with the Arabic numerals inany direct way, the medieval astrological numerals mayhere be mentioned. These are given by several earlywriters, but notably byNoviomagu s as follows 9 °

1 2 3 4 5 6 7 8 9 10

1 Kucku ck, p . 5.

2 A . Cappelli, loc . cit ., p . 372 .

3 Sm ith , Ram A rithmetica , p . 4 43.

4 Cu rtze,Petri Phi lomem'

de Dacia etc ., p . 1x .

5 Cappelli , loc.cit. , p . 376.6 Cu rtze

,loc . cit .

, pp . v 11 1— 1x,note.

7 Edition of 154 4— 1545, f . 52.

8 De numeris libri I I,154 4 cd .

,cap . xv. He ilbronner, loc . cit.

, p .

736,also gives them ,

and compares th is with other systems .

9 Noviomagu s says of them :“ De qu ibusdam Astrologicis , sive

Chaldaicis numerorum notis . Sunt aliae quaedam notae, qu ibu s

Chaldaei Astrologii quemlibet numerum artificiose argute describunt, sci tu periucundae, quas nobis communicau it Rodolphu s Paludanus Nou iomagus .

”

INDEX

Abbo of Fleu ry, “

122‘

Abdallah ibn al-Hasan,92

'Abdallatif ibn Yusuf , 93‘

Abdalqadir ibn‘

Ali al-Sakhawi 6Abenragel, 34

Abraham ibn Mei r ibn E zra,see

Rabbi ben E zraAbu

‘

Al i ai-Hosein ibn S ina, 74Abu ’l-Hasan

,93

,100

Abu ’ l-Qas im,92

Abu ’ l-Teiy ib, 97

Abu Nasr, 92Abu Roshd

,113

Abu Sahl Dunash ibn Tamim ,65

,

67

Adelhard of Bath,5,55

,97

,119

,

Adhemar of Chabanois , 111Ahmed al-Nasawi

,98

Ahmed ibn ‘

Abdallah,9,92

Ahmed ibn M ohamm ed,94

Ahmed ibn ‘

Omar,93

Aksaras , 32

Alanu s ab Insulis,124

A l-Bagdadi , 93Al-Battani

,54

Albelda (Alba ida) MS . ,116

Albert,J 62

Albert of York,103

AI-B iruni,6,4 1

,4 9

,65

,92

,93

Alcu in,103

Alexander the Great,76

Alexander de Villa Dei,11

,133

Alexandria,64

,82

92,97

,

Al-Fazari , 92

Alfred,103

A lgebra , etymology , 5A lgerian numerals

,68

Algorism ,97

Algorismus,124

,126

,135

Algorismus cifra,120

A l-Hassar,65

‘

A li ibn Abi Bekr,6

‘

A li ibn Ahmed,93

,98

Al-Karabi si,93

Al-Khowarazm i,4,9,10

Al-Kindi,10

,92

Almagest, 54

A l-Magrebi , 93

A l-Mahall i,6

A l-Mam i‘

m,10

,97

Al-Mansur,96

,97

A l-Mas'udi

,7,92

A l-Nadim,9

A l—Nasaw i,93

,98

Alphabetic numerals , 39,Al—Qasim ,

92

Al-Qass , 94

Al-Sakhawi , 6Al—Sardafi

,93

A l-S ijzi , 94

A l-Suf i,10

,92

Ambrosoli,118

Ankapalli , 43

Apices , 87 , 117, 118A rabs

,91- 98

A rbuthnot,14 1

154 THE HINDU—ARABIC NUMERALSA rchimedes

,15

,16

A rcu s Pictagore, 122A rjuna

,15

A rnold,E .

,15

,102

Ars memorandi,14 1

Aryabhata, 39, 43, 4 4

A ryan numerals,19

A schbach,134

A shmole,134

Asoka, 19, 20, 22, 81As-sifr

,57

,58

A strological numeralsA tharva-Veda, 4 8, 50,Augustus , 80

Averroes,113

Avicenna, 58, 74 , 113

Babylonian numerals , 28Babylonian zero , 51Bacon

,R .,

131

Bactrian numerals,19 30

Baeda,2,72

Bagdad , 4 , 96Bakhsali manu script, 43, 49 52Ball

,C . J .

,35

Ball,W . W . R .

,36

,131

Bana,4 4

Barth,A .

,39

Bayang inscriptions , 39Bayer

,33

Bayley,E . C .

,19

,23

,30

,32 52

Beazley,75

Bede,see Baeda

Beldomandi,137

Beloch,J.

,77

Bendall,25

,52

Benfey , T .,26

Bernelinus,88

,112

,117, 121 Calandri , 59, 81

Besagne, 128 Caldwell, R .,19

Besant,W .

,109 Calendars , 133

Bettino,36 Calmet

,34

Bhandarkar,18

,4 7

,50 Cantor

,M .

,5,13

,30

,43

,84

Bhaskara,53

,55

Biernatzki,32

B iot, 32

BjOrnbo, A . A .,125

,126

Blass iere,119

B loomfield,48

Blume,85

Boeckh,62

Boehmer,143

Boeschenstein,119

Boethius,63

,70—73 83—90

Boiss iere,63

Bombelli,81

Bonaini,128

Boncompagni , 5, 6, 10 , 48, 50 , 123,

125

Borghi , 59Borgo , 119Bougie, 130Bowring, J .

, 56

Brahm agupta, 52

Brahmanas,12

,13

Brahm i,19

,20

,31

,83

Brandis,J 54

Brhat-Samh ita,39

,4 4

,78

Brockhaus,43

Bubnov,65

,84

,110

,116

Buddha,education of

,15

,16

B iidinger, 110

Bugia , 130Buhler

,G .

,15

,19

,22

,31

,4 4

,50

Bu rgess , 25Burk

,13

Bu rmese numerals 36Bu rnell

,A . C .

,18

,4 0

Buteo,61


Florus,80

FlijgeI, G .,68

Franc isco de Retza,142

Francois , 58Friedlein

,G .

,84

,113

,116 122

Froude,J . A .

,129

Gandhara,19

Garbe,48

Gasbarri , 58

Gautier de Coincy , 120, 124Gemma Fris ius

,2,3,119

Gerber,113

Gerbert, 108, 110—120

,122

Gerhardt,C . I .

,4 3

,56

,93

,118

Gerland,88

,123

Gherard of Cremona, 125G ibbon

,72

G iles,H . A .

,79

G inanni,81

G iovanni d i Danti,58

Glareanu s,4,119

Gnecchi,71

,117

Gobar numerals,65

,100 112

Gow,J 81

Grammateus,61

Greek origin,33

Green,J . R .

,109

Greenwood,I .

,62

,119

Guglielmini , 128Gu listan

,102

Gunther, S .

,131

Guyard , S .,82

Habash,9,92

Hager, J . 28 32

Halliwell,59

,85

Hankel,93

Harun al-Rash id,97

,106

Havet, 110

Heath,T . L .

,125

Hebrew numerals,127

Hecataeus,75

He iberg, J . L ., 55, 85, 148

Heilbronner,5

Henry, C .

,5,31

,55

,87 120

135

Heriger, 122

Hermannu s Contractus,123

Herodotus,76

,78

Heyd , 75Higden, 136Hill

,G . F .

,52

,139

,142

Hillebrandt,A 15

,74

Hilprecht, H . V .,28

Hindu forms,early , 12

Hindu number names,42

Hodder,62

Hoernl e,43

,4 9

Holywood,see Sacrobosco

Hopkins , E . W .,12

Horace,79

,80

Hosein ibn Mohammed al-Ma

hall i,6

Hostus , M .,56

Howard,H . H .

,29

Hrabanus Mau rus,72

Huart,7

Huet,33

Hugo , H.,57

Humboldt,A . von

,62

Huswirt, 58

Iamblichu s , 81

Ibn Abi Ya‘

q fib, 9

Ibn al-Adam i,92

Ibn al-Banna, 93

Ibn Khordadbeh , 101 106

Ibn Wahab,103

India,history of , 14

writing in,18

Indicopleu stes , 83

Indo—Bactrian numerals 19

INDEX

Indraji , 23

I shaq ibn Yusuf al- Sardafi , 93

Jacob of Florence,57

Jacquet, E .,38

Jamshid, 56

Jehan Certa in,59

Jetons,58

,117

Jevons,F . B .

,76

Johannes Hispalensis , 48, 88, 124John of Hali fax

,see Sacrobosco

John of Luna,see Johannes His

palens is

Jordan,L .

,58

,124

Joseph Ispanus (Joseph Sapiens) ,115

Ju stinian,104

Kale,M . B .

,26

Karabacek,56

Karpinski , L . C . 126,134 138

Katyayana , 39

Kaye, c . R .,6,16

,43 4 6

,121

Keane,J .

,75

,82

Keene,H . G .

,15

Kern,44

Kharosthi,19

,20

Khosru,82

,91

Kielhorn,F .

,4 6

,4 7

Kircher,A .

,34

K itab ai-Fihrist,see Fihrist

K leinw'

achter,32

Klos,62

KObel,4,58

,60

,119

,123

Krumbacher,K .

,57

Kuckuck, 62, 133

Kugler, F . X .,51

Lachmann,85

Lacouperie, 33, 35

Lalitavistara,15

,17

Lami,G .

,57

157

La Roche,61

Lassen,39

Latyayana, 39

Leboeuf,135

Leonardo of Pisa,5,10

,57

,64

74,120

,128—133

Lethaby , W . R .,142

Levi,B .

,13

Levias,3

L ibri,73

,85

,95

L ight of A s ia,16

Lu ca da Firenze,14 4

Lucas,128

Mahabharata,18

Mahaviracarya , 53

Malabar numerals,36

M alayalam numerals,36

Mah nert,81

M argarita Philosophica , 146Marie

,78

Marquardt, J 85

M arshman,J . C .

,17

M artin,T . H .

,30

,62

,85

,113

Martines,D . C .

,58

Mashallah , 3

Maspero , 28Mauch

,142

Maximu sPlanudes 2 93,120

M egasthenes , 77

M erchants,114

Meynard , 8M igne, 87M ikam i

,Y .

,56

M ilanes i,128

M ohammed ibn ‘

Abdallah 92

Mohammed ibn Ahmed,6

Mohammed ibn ‘

A lit

Abdi,8

Mohammed ibn Musa,

see A l

Khowarazm i

Molinier,123

Moni er-Williams,17

158

Morley,D .

,126

Moroccan numerals,68

,

Mortet,V .

,11

Moseley,C . B .

,33

Motahhar ibn Tahir, 7Mueller

,A .

,68

Mumford,J . K .

,109

Muwaffaq al-D in,93

Nabatean forms,21

Nallino,4,54

,55

Nagl , A .,55

,110

,113

,126

Nana Ghat inscriptions , 20, 22,23

,40

Nardu cc i,123

Nasik cave inscriptions , 24Nazif ibn Yumu

,94

Neander,A .

,75

NeOphytos , 57 , 62

Neo-Pythagoreans , 64Nesselmann

,53

Newman,Cardinal

,96

Newman,F . W .

,131

NOldeke,Th .

,91

Notation,61

Note,61

,119

Noviomagu s , 45, 61, 119 150

Nul l,61

Numerals,

Algerian,68

astrological , 150Brahm i

,19- 22

,83

early ideas of origin,1

Hindu,26

Hindu,classified

,19

,38

Kharosth i , 19—22

Moroccan,68

Nabatean,21

origin, 27, 30, 31, 37supposed A rabic origin, 2supposed Babylonian origin,28

THE HINDU—ARABIC NUMERALS

Numerals,

supposed Chaldean and Jew

ish origin, 3supposed Chinese origin,

28,

32

supposed Egyptian origin, 27

supposed Greek origin, 33supposedPhoenicianorigin,

32

tables of,22- 27

,36

,48

,50

,

69,88

,14 0

,143

,145- 148

O’Creat, 5, 55, 119, 120

Olleris , 110 , 113

Oppert, G .,14

,75

Pal i , 22Paficasiddhantika, 4 4

Paravey , 32, 57

Patalipu tra, 77

Patna, 77

Patrick, R .,119

Payne , E . J .,106

Pegolotti , 107

Peletier,2,62

Perrot,80

Pers ia,66

,91

,107

Pertz,115

Petru s de Dac ia,59 61

,62

Pez,P. B .

,117

Ph ilalethes,

” 75

Ph illips , 107

Picavet, 105

Pichler, F .,14 1

P ihan,A . P . ,

36

Pisa, 128Place value , 26, 42, 4 6, 4 8

Planudes , see Maximus Planudes

Plimpton, G . A .,56

,59

,85

,143

,

Pl iny , 76PO10 , N. and M .

,107


Subandhu s,4 4

Suetoniu s , 80Suleiman

,100

Sunya , 43, 53, 57

Suter,5,9,68

,69

,93 116

Sutras,13

Sykes , P . M .,75

Sylvester I I , see Gerbert

Symonds,J . A .

,129

Tannery,P .

,62

,84

,85

Tartagl ia , 4 , 61Taylor, I .

,19

,30

Teca,55

,61

Tennent,J . E .

,75

Texada,60

Theca,58

,61

Theophanes , 64Thibau t

, G .,12

,13

,16

,4 4

,4 7

Tibetan numerals,36

Timotheu s,103

Tonstall,C .

,3,61

Trenchant,60

Treutlein,5,63, 123

Trevisa,136

Treviso arithmetic,145

T rivium and quadrivium ,73

Ts in,56

Tunis,65

Turchill,88

,118

,123

Turnour,G .

,75

Tz iphra, 57 , 62

T j‘

hppa , 55, 57 , 62

Tzwivel,61

,118

,145

Yul e,H.

,107

Uj jain,32

Unger , 133Upanishads , 12Usk, 121

Valla, G .

,61

Van der Schuere 62

Varaba-M ihira,39

,4 4

Vasavadatta, 4 4

Vaux,Carra de

,9,74

Vaux , W . S . W .,91

Vedangas , 17Vedas

,12

,15

,17

Vergil, 80Vincent

,A . J . H .

,57

Vogt , 13Voizot

,P.

,36

Vossiu s,4,76

,81

,84

Wall is, 3, 62, 84 , 116

Wappler, E .,54

,126

Waschke,H .

,2,93

Wattenbach,143

Weber,A .

,31

We idler,I . F .

,34

, 66

1Veidler, I . F . and G . I . 63

,66

Weissenborn,85

,110

Wertheim,G .

,57

,61

Whitney , W . D . . 13

Wilford,F .

,75

VVilkens,62

Wilkinson,J . G .

, 70

Willich ius,3

Woepcke, 3, 6, 42, 63, 64 , 65, 67,

69,70

,94

,113

,138

Wolack,G .

,54

Woodru ff,C . E .

,32

Word and letter numerals,38

,

4 4

Wustenfeld,74

Zeph irum ,57

,58

Z ephy r, 59Zepiro , 58

Zero,26

,38

,40

,43

,45 50

,51-62,

67

Zeuero,58

ANNOUNC EM ENTS

ALGEBRA FOR BEG INNERS

By DAVID EUGENE SMITH,

Professor of M athematics in Teachers College, Columbia Univers ity

I zmo,cloth

,154 pages , 50 cents

HIS work is intended to s erve as an introduction to

the study of algebra, and is adapted to the needs of

the seventh or eighth s chool year. It is arranged in

harmony with the leading courses of study that includealgebra in the curriculum of the grades .

The relation of algebra to arithmetic is emphas ized, thesubject

.

is treated topically, and each important po int istouched at least twice. The book begins by showing the

u ses of algebra, employing such practical applications as are

within the pupil’ s range of knowledge. When an interesthas thus been awakened in the subject, the fundamentaloperations are presented with the s imple explanations mecessary to make the s tudent independent of dogmatic rules .

Throughout the book abundant oral and written drill exercises are provided. The work includes linear equations withtwo unknown quantities , and easy quadratics .

The leading features may be summarized as follows ( I )an arrangement in harmony with existing courses of study ;(2 ) a presentation des igned to awaken the interes t of thepupils (3) a topical arrangement for each half year

, everyimportant topic being repeated ; (4 ) s implicity of explanations (5) development of the relation of algebra to arithmetic both in theory and in applications (6) emphas is laidon the importance of oral as well as written algebra.

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