The Hindu-Arabic Numerals by Louis Charles Karpinski

Chapter 8 THE SPREAD OF THE NUMERALS IN EUROPE

Word Count: 23777 | Released on: 01/12/2017

als to the scholars of Europe was Leonardo Fibonacci, of Pisa.[515] This remarkable man, th

ers of Italy at the opening of the thirteenth century. Even before Venice had captured the Levantine trade, Pisa had close relations with the East. An old Latin chronicle relates that in 1005 "Pisa was captured by the Saracens," that in the following year "the Pisans overthrew the Saracens at Reggio," and that in 1012 "the Saracens came to Pisa and destroyed it." The city soon recovered, however, sending no fewer than a hundred and twenty ships to Syria in 1099,[518] founding a merchant colony in Constantinople a few years later,[519] and meanwhile carr

filius Bonaccii as used in Leonardo's work, was simply a general one, like our Johnson or Bronson (Brown's son); and the only contemporary evidence that we have bears out this view. As to the name Bigollo, used by Leonardo, some have thought it a self-assumed one meaning blockhead, a term that had been applied to him

started on a tour of the Mediterranean Sea, and visited Egypt, Syria, Greece, Sicily, and Provence, meeting with scholars as well as with merchants, and imbibing a knowledge of the various systems of numbers in use in the centers of trade. All these systems, however, he says he counted almost as errors compared with that of the Hindus.[532] Returning to Pisa, he wrote his Liber Abaci[533] in 1202, rewriting it in 1228.[534] In this work the numerals are explained and are used in the usual computations of business. Such a

ninth century contained only nineteen volumes of abridgments from ecclesiastical commentaries."[537] Indeed, it was not until the early part of the fifteenth century that Palla degli Strozzi took steps to carry out the project that had been in the mind of Petrarch, the founding of a public library. It was largely by word of mouth, therefore, that this early knowledge had to be transmitted. Fortunately the presence of foreign students in Italy at this time made this transmission feasible. (If human nature was the same then as now, it is not impossible that the very opposition of the faculties to the works of Leonardo led the students to investigate them the m

ss to this fact being the popular almanacs. Calendars of 1457-1496[539] have generally the Roman numerals, while K?bel's calendar of 1518

pular treatises by Alexander de Villa Dei (c. 1240 A.D.) and John of Halifax (Sacrobosco, c. 1250 A.D.) were

ted, enjoyed a wide popularity as a textbook for university instruction.[543] The work was evidently written with this end in view, as numerous commentaries by university lecturers are found. Probably the most widely used of these was that of Petrus de Dacia[544] written in 1291. These works throw an inter

troduction it is stated that this science of reckoning was due to a philosopher named Algus, whence the name algorismus,[546] and in the section on numeration reference is made to the Arabs as the inventors of this sc

then seems to have been lost in the multitude of Paris manuscripts; for although Chasles[552] relates his vain search for it, it was not rediscovered until 1882. In that year M. Ch. Henry found it, and to his care we owe our knowledge of the interesting manuscript. The work is anonymo

e and for some generations preceding:[554] "For two hundred years children in scole, agenst the usage and manir of all other nations beeth compelled for to leave hire own language, and for to construe hir lessons and hire thynges

ntil about 1500 did not demand the new figures, for two reasons: First, cheap paper was not known. Paper-making of any kind was not introduced into Europe until the twelfth century, and cheap paper is a product of the nineteenth. Pencils, too, of the modern type, date only from the sixteenth century. In the second place, modern methods of operating, particularly of multiplying and dividing (operations of relatively greater import

other Spanish copy of the same work, of 992 A.D., contains the numerals in the corresponding section. The writer ascribes an Indian origin to them in the following words: "Item de figuris arithmetic?. Scire debemus in Indos subtilissimum ingenium habere et ceteras gentes eis in arithmetica et geometria et ceteris liberalibus disciplinis concedere. Et hoc manifestum est in nobem figuris, quibus designant unumquemque gradum cuiuslibet gradus. Quarum hec sunt forma." The nine ?obār characters follow. Some of the abacus forms[557] previously given are doubtless also of th

il recently it was thought that the earliest such date was 1217 A.D. for an Arabic piece and 1388 for a Turkish one.[564] Most of the seals and medals containing dates that were at one time thought to be very early have been shown by Mr. Hill to be of relatively late workmanship. There are, however, in Eu

Erlangen MS. of Boethius, of the same (eleventh) century, and the sixth and seventh are also from an eleventh-century MS. of Boe

Manuscr

rope from the tenth to the sixteenth century. It is of interest to add that he has found that among the earliest dates of European coins or medals in these numerals, after t

39.[567] In numbering pages of a printed book these numerals were first used in a work of Petrarch's published at

es back to c. 1430[570] the numerals appear i

red directly after printing. Thus the chapters 2, 3, 4, 5, 6 in a book of 1470[571] are numbered as follows: Capitulem m.,... m.,... 4m.,... v,... vi, and followed by Roman numerals. This a

25 fo

47 fo

7, and one at Biebrich of 1299. There is no doubt, however, of one at Pforzheim of 1371 and one at Ulm of 1388.[574] Certai

serve to supplement that

anuscri

welfth ce

8] 11

9] 12

] c. 1

original was written at Paris in 1424 by Rollandus, a Portuguese physician, who prepared the work at the command of John of Lancaster, Duke of Bedford, at one time Protector of England and Regent of France, to whom the

although in writing there is still a great variation, as witness the French 5 and the German 7 and 9. Even in printing there

are some of the forms to be found in the late

pes, just as some modern typewriters use the same character for l a

books, 12 appearing as iz.[589] In the medieval

merely a cursive form for the primitive , just as

h occasionally it appears as .[591] In the medieval manuscripts it varied

lorentine manuscript of Leonard of Pisa's work has the form ;[593] but the manuscripts show that the Florentine arithmeticians and astronomers rather early began to straighten the first of these forms up to forms like [594] and [594] or ,[595]

ore the time of printing. The fol

others. The chief variation has been in the slope

nt erect form only since the fifteenth century

little. In medieval times there are a

wever, there was manifested

s most of the others. Among the me

d a varied history. The followin

d this before all learners of the positional system, there would have been little trouble. But the medieval line-reckoning, where the lines stood for powers of 10 and t

ym selfe & no more, yf he stonde i

okens ten tyme hym selfe, that is twenty, for he hym selfe betokens tweyne, & ten tymes twene is twenty. And

token that comes after hym more than he shuld & he were away, as thus 10. here the figure of one tokens ten, & yf the

earlier than it was, particularly in those countries north of Italy where it did not come into general use until the sixteenth cen

the seventeenth century saw it finally conquer the system that for two thousand years had dominated the arithmetic of business

n a manuscript of the fifteenth century 12901 for 1291.[609] In the same century m. cccc. 8II appears for 1482,[610] while MoCCCCo50 (1450) and MCCCCXL6 (1446) are used by Theodoricus Ruffi about the same tim

ieval astrological numerals may here be mentioned. These are given by seve

and China, in Siam and generally about the Malay Peninsula, in Tibet, and among the East India islands, the natives still adhere to their own numeral forms. Only as Western civilization is making its way into the commercial life of the East do the numerals as used by us find place, save as the Sanskrit for

entries refer to footnotes lin

f Fleu

h ibn al

tīf ibn

ibn 'Alī a

rage

r ibn Ezra, see

l-?osein i

?asan,

l-Qās

l-?ei

Na?r

Rosh

nash ibn Ta

th, 5, 55, 97,

of Chaba

al-Nas

n 'Abdal

bn Mo?a

ibn 'O

ara

ab Insu

a?dā

attā

(Albaida

rt,

of Yo

6, 41, 49

uin

er the

de Villa D

dria,

azār

red

, etymo

n numer

rism

us, 124,

mus cif

a??ā

bn Abī

n A?med

rābīs

4, 9, 10, 92, 9

ndī,

gest

a?re

a?al

mūn,

n?ūr,

s'ūdī

Nad

sawī,

numerals,

āsim

Qas

akhā

arda

ijzī

fī, 1

osol

pall

, 87,

s, 9

thno

edes,

Pictag

una

, E.,

morand

?a, 39,

numera

bach

ole,

19, 20,

fr, 5

ical num

Veda, 48

stus

ro?s

a, 58,

ian num

nian z

n, R

numeral

ad,

nuscript, 4

C. J

. W. R.

h, A

inscrip

er,

., 19, 23, 3

zle

, se

mandi

ch,

ll, 2

ey,

s, 88, 11

gne,

t, W.

tin

kar, 18

ara,

natz

A. A.,

sièr

mfie

me,

ckh

mer,

enstei

, 63, 70

sièr

elli

ini,

5, 6, 10, 48,

ghi

go,

gie

ng, J

agupt

a?as,

19, 20

is, J

?hita, 3

khau

65, 84,

ducation

nger

ia,

15, 19, 22

ges

e nume

, A. C.

eo,

dri,

ell,

ndar

met

., 5, 13,

ell

elli

s, 87, 11

dan

Algorism

randi

ri,

odoru

ald

ane

n, 14

ett

numer

nt, F.

peno

rs, see

emagn

, 60, 85, 1

nt, L.

cer,

ni, 14

ffr

numerals

se ze

, 120

her

lus,

eus, 61,

Vigila

gton,

date

ke, 8, 2

tine, 1

mas

sal

ters

teil

Nombrynge,

ades

ham, A.

55, 59,

ra,

mari

viel

te,

ius, 33

nou

mbre

nāga

lx, A

uva

us of Me

its

us Sic

ange

snil

C., 12, 1

ved

, relations,

n numer

nloh

Misra

ion Algo

, 48, 59, 9

rals in, 63,

Caesari

ing

d, P

ri, 5

see Leona

a nih

119. See

t, 67,

aeu

dus

ephen,

. C., 19

rus

el,

co de R

?ois

G., 84, 1

, J. A

hāra

be,

arri

e Coincy,

isius, 2

ber

108, 11

C. I., 43,

nd, 8

of Cre

bon

, H.

ann

i di Da

anus,

hi, 7

ls, 65, 100,

, J

mate

orig

, J.

od, I.,

elmin

stān

er, S

rd,

sh,

J. (G.)

well,

kel

-Rashīd,

et,

, T.

numer

t?us

J. L., 55

bron

5, 31, 55,

ger,

s Contra

otus,

den

F., 52,

andt, A

cht, H

orms, e

umber n

der

le, 4

, see Sa

s, E.

ce,

Mo?ammed al

us,

d, H.

us Mau

, H.

dt, A.

wir

lich

bī Ya

l-Ada

l-Ban

dā?beh,

Waha

history

ing

pleust

trian nu

rāj

Yūsuf al-?

f Flore

et, E

shi

Cert

s, 58

s, F.

spalensis,

lifax, see

, see Johann

, L.,

us (Joseph S

inia

M. R

bace

L. C., 12

āyan

., 6, 16,

, J.,

, H.

??hī,

rū,

rn, F.

er, A

Fihrist,

w?cht

, 58, 60

cher,

ck, 6

r, F.

mann

erie,

istara,

, G.

oche

sen

āyan

?uf

, 5, 10, 57, 64,

y, W.

, B.

ias

, 73,

of A

a Fire

as,

hārat

īrācā

r nume

am nume

ner

Philosop

ie,

ardt,

an, J.

H., 30, 6

es, D.

āllā

per

ch,

udes, 2, 57,

thene

hant

nar

ne,

mi,

nesi

ibn 'Ab

d ibn A

ibn 'Al

Mūsā, see A

nier

-Willi

y, D.

numeral

et,

y, C.

r ibn ?

er, A

d, J.

aq al-

an for

o, 4,

55, 110,

scriptions,

ucci

e inscrip

ibn Yu

er, A

ytos,

thagor

elma

, Card

, F. W

ke, T

tion

, 61

s, 45, 61

era

rian

logic

ī, 19

eas of o

du,

lassifie

??hī,

ccan

tean

27, 30

Arabic

abylonian

ldean and Je

hinese ori

tian origin,

Greek o

Ph?nician

36, 48, 49, 69, 88

5, 55,

is, 1

, G.,

iddhān

ey, 3

īpu?r

na,

ck, R

, E.

lott

ier,

rot

, 66,

tz,

Dacia, 5

P. B.

aleth

ips,

vet,

er, F

, A.

ue, 26, 4

see Maximu

, 56, 59, 85, 1

ny,

N. and

l, J.

p, J.,

erti

de' Beldo

my, 5

nam

agor

rean nu

of Mas

en Ezra

Laon, 60, 1

ts,

see Gemm

yana

2, 41

Glab

son

, see Da

, K. v

E., 14

rde,

d, 67,

illa

110, 1

e, A

rtso

Cestrensi

t, 5

ger,

andu

gnos

n, F

ula

olf

ph, 6

fi,

hau

sco, 3,

S. de,

dī,

scripti

ibn Ya

charact

onn

er, J.

ubel

egel

idt,

rus, 8

er, L.

lax

lot,

, 20,

bn 'Alī

nati,

ey, W

e nume

ānta,

boto

al-D

erbe

on,

bn al-F

bad,

hind

os,

H. C

, C.

, 11, 17, 53

A., 20,

h, W

ti,

64, 6

ta-B

nger

asūtr

ens,

er, 5, 57, 6

fel

ndhu

oniu

imān

, 43,

, 68, 69, 9

ras

, P.

r II, se

s, J.

, P., 6

glia,

, I.,

, 55

t, J.

ada

a, 5

phan

., 12, 13,

n nume

theu

ll, C.

chan

in, 5,

isa,

arithm

and quad

is,

l, 88,

ur, G

ra, 5

, 55,

, 61, 1

ain

er,

isha

a, G

r Schu

ihira, 3

adatt

arra de

W. S.

?gas

, 12,

gil

, A. J.

ot,

, 4, 76

3, 62,

r, E.,

e, H.,

nbach

r, A

, I. F.

. F. and G

nborn,

im, G.

y, W.

rd, F

ken

son, J

ichi

63, 64, 65, 67, 6

ck,

ff, C.

etter nume

enfe

, H.

rum,

hyr

iro

40, 43, 45,

ero

UNCE

TWO

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um apud Hebr?os, deinde Moses; et Abraham tradidit istam scientiam numeri ad ?gyptios, et docuit eos: deinde

sam: Alii ad Indos: Ioannes de Sacrobosco, cujus sepulchrum est Luteti? in comitio

570 Lyons ed., p. 14): "La valeur des Figures commence au coste dextre tirant vers le coste senestre: au rebours de notre maniere d'es

"le noue figure de gli Indi," in his Le pratiche delle dve prime mathematiche (Venice, 1546, fol. 1). Woepcke is not correct, therefore, in saying ("Mémoire sur la propagation des chiffres indiens," hereafter referred to as Propagation [Journal Asia

n, 1522, fol. B, 3.] Gemma Frisius, the great continental rival of Recorde, had the same idea: "Primùm autem appellamus dexterum locum, eo quòd haec ars vel à Chald?is, vel ab Hebr?is ortum habere credatur, qui etiam eo ordine scribunt"; but this refers more evidently to the Arabic numerals. [Arithmetic?

iece figure." [La prima parte del general trattato di nvm

echenbiechlin, Augsburg, 1514, fol. 13 of the 1531 edition. The printer used the l

e, siue Chald?is asciti .1.2.3.4.5.6.7.8.9. Est item unus .0 circulus, qui nihil signifi

1703, p. 3.] So Vossius (De universae matheseos natura et constitutione liber, Amsterdam, 1650, p.

in the Atti della R. Accad. dei Lincei, Rome, 1896. See also Verhandlungen des 5. Congresses der Orientalisten, Berlin, 1882, Vol. II, p. 19; W. Spitta-Bey in the Zeitschrift der deutschen Morgenl?nd. Gesellschaft, Vol. XXXIII, p. 224; Steinschneider in the Zeitschrift der deutschen Morgenl?nd. Gesellschaft, Vol. L, p. 214; Treutlein in the Abhandlungen zur Geschichte der Mathematik, Vol. I, p. 5; Suter, "Die Mathematiker und Astronomen der Araber und

wn by Suter in his work Die Mathematiker etc., except where this violates English pro

eated in L'Algèbre d'al-Khārizmi et les méthodes indienne et grecque, Léon Rodet, Paris, 1878, extract from the Journal Asiatique. For the derivation of the word algebra, see Cossal

m suum"), studied in Toledo, learned Arabic, traveled as far east as Egypt, and brought from the Levant numerous manuscripts for st

e did not make this translation is asserted by Enestr

mero suo, propter dispositionem suam quam posuerunt, uolui patefacere de opera quod fit per eas aliquid quod esset leuius discentibus, si deus uoluer

there are "nine Indian figures" and "a second kind of Indian figures ... although these are the figures of the ?obār writing." So in a commentary by ?osein ibn Mo?ammed al-Ma?allī (died in 1756) on the Mokhta?ar fī'ilm el-?isāb (Ex

h length by G. R. Kaye, "Notes on Indian Mathematics.-Arithmetical Notatio

c version, London, 1887; Eng

ons, London, 1879. Arabic and E

, Vol. I,

or the symbols of the

ish edition of the

e arabe, Cl. Hu

ic Literature, English ed.,

prenger, London, 1841; Les prairies d'or, trad. par C. Barbier d

es d'or, Vol. V

ys, Vol.

c. cit.

des sciences mathématiques chez les Grecs et les

wever, in favor of the numera

ispana Escurialensis, Madr

lourished A.D. 1198 [Colebrooke,

ceteros omnes brevitate methodi ac facilitate praestat, Indorum que in praec

sement et de la révision. Translati

pcke, Propagation, that the Sindhind

lāh, Suter, Die Mathe

a, Vol. I

ig, 1892. For further references to early Arabic writers the reader is referred to H. Suter, Die Mathematiker und Astrono

c. cit., note

a scripta maximè celebrata, quae publ

, Scritti (1862); published by Baldassarre Boncompagni, Rome. Also Tr

tc. In another place, as a heading to a separate division, he writes, "De cogniti

t Moritz Cantor, Leipzig, 1909. See also Victor Mortet, "Le plus ancien tr

Carmen de Algorismo that the anonymous autho

s ars praesens

m fruimur bis

ws is the

e, Astrologie und Mathe

originaire de la Chine, et qu'elle a été empruntée par les anciens peuples occidentaux à la sp

he Religions of India

History of Indi

Sanskrit Grammar, 3d

ft der deutschen Morgenl?ndischen Gesellsch

der Math., Vol.

(3), pp. 6-20; A. Bürk, loc. cit.; Max Simon, Geschichte der Mathematik im Altertum, Berlin, 1909, pp. 137-165; three Sūtras are translated in part by Thibaut, Journal of the Asiatic Soci

Indiens Literatur und

rupted by western peoples to Hindhu, Indos, Indus, is the root of

the Original Inhabitants of Bharat

li?ga, learned as a boy lekhā (writing), ga?anā (reckoning), and rūpa (arithmetic applied to monetary affairs

Civilization in Ancient India,

his birth is uncertain. Sir E

.e. 10

me uncertainty a

ew, "The Greeks in India," in the Calcutta Review, Vol. CXIV, 1902, p. 1. See also F. Woepeke, Propagation, p. 253; G. R. Kaye, loc. cit., p. 475 seq., and "The Source of Hindu Mathemati

common large unit in India, like the myriad

s to show to the king "by geometric proofs which you can follow, that the numbers which have been named by us ... are sufficient to exceed not only the number of a sand

.e. th

, London, 1893, pp. 144, 177. See also J. C. Marshman,

these works see R. C. Dutt, A History of Civ

entury B.C. ["Consideration of the Date of the Mahābhārata," in the Jour

man, loc.

Indian Pal?ography, 2d ed.

hmetic, essentially the history of the abacu

and reprint, London, 1882; I. Taylor, in The Academy, January 28, 1882, with a repetition of his argument in his work The Alphabet, London, 1883, Vol. II, p. 265, based on Bayley; G. R. Kaye, loc. cit., in some respects one of the most critical articles thus far published; J. C. Fleet, Corpus inscriptionum Indicarum, London, 1888, Vol. III, with facsimiles of many Indian inscriptions, and Indian

in the same journal in 1837. See also "A?oka Notes," by V. A. Smith, The Indian Antiquary, Vol. XXXVII, 1908, p. 24 seq., Vol. XXXVIII, pp. 151-159, June, 1909; The Early History of India, 2d e

minor details of this system,

chriften aus Arabien, Berlin, 1885,

cipal theories see Büh

loc. cit.,

a, Vol. III, p. 134; Indian Antiquary, V

on; from an Inscription at Nāneghāt," Journal of the Bombay

also a plate and an inter

with Bayley, loc. cit., p. 337 and plates; and with Bayley's a

ndica, Vol. VIII, pp. 59-96; "The Inscriptions in the Cave at Karle,"

r, London, 1863, p. 217; M. R. Kále, Higher Sanskrit Grammar,

, c. 250 B.C. Senart, Notes d'épigraphie in

bably of the first century B.C. S

scriptions, c. 250 B.C. Indian

hagavānlāl Indrājī, On Ancient Nāgarī Numeration

ological Survey Report, Western India; Senart, Epigraphi

A.D. Journal of the Royal As

0 A.D. Epigraphia Indica, Vol.

c. 300 A.D. to 450 A.D. F

c. 600 A.D. Co

umerals, in Cat. Sansk. B

l. XIII, 120; Epigraphia

eet, lo

y, loc. ci

per plates, being deeds of property, have forged dates so as to give the appearance of antiquity of title. On the other hand, as Colebrooke long ago pointed

D., found at Majhgawāin, Central I

, found at Bōdh-Gayā, Bengal Preside

., found at Māliyā, Bombay Presiden

inscription of 372 A.D. [Fle

e of 434 A.D. [Indian A

, c. 417 A.D. [Fleet, l

late of 493 A.D.,

se four, curiously enough called "eig

um, Vol. IV, no. 1; J. Hager, An Explanation of the

ol. XX, part I, of Series A, Cuneiform Texts Published by the Babylonian Expedition of the University of Pennsyl

tions from Chaldea," Proceedings of the Society

does not adduce any convincing proof. Th. Henri Martin, "Les signes numéraux et l'arithmétique chez les peuples de l'antiquité et du moyen age" (being an examination of Cantor's Mathematische Beitr?ge zum Culturleben der V?lker), Annali di

yal Asiatic Society, Bo

pp. 12, 17. Bayley's deductions ar

83, Vol. II, pp. 265, 266, and

Alphabet, loc. cit

he Indian Brāhma Alphabet, Stras

ya (empty) (it occurs even in Pi?gala). It is the decimal place value of these figures which gives them significance." C. Henry, "Sur l'origine de quelques notations mathématiques," Revue Archéologique, June and July, 1879, attempts to derive the Boethian f

s the list, with the list of letters (p.

see the articles by U. Ceretti, "Sulla origine delle cifre numerali moderne," Rivista di fisica, matematic

ley, loc. cit., reprint p. 4; a good bibliography

. 12, 17. See also Burnell, loc. cit

I, 1857, pp. 59-96, also asserts the priority of the Chinese claim for a place system and the zero, but upon the flimsiest authority. Ch. de Paravey, Essai sur l'origine unique et hiéroglyphique des chiffres et des lettres de tous les peuples, Paris, 1826; G. Kleinw?chter, "The Origin of the Arabic Numerals," China Review, Vol. XI, 1882-1883, pp. 379-381, Vol. XII, pp. 28-30; Biot, "Note sur la connaissance que les Chinois ont eue de la valeur de positio

rom 1 to 9, then nine others for 10 to 90, and further letters to represent 100 to 900. As the ordinary Greek

Strassburg, 1593-1596, a somewhat rare work

harum Cyph

quod exiguum esse fateor: a graecis librarijs (quorum olim magna fuit copia) literae Graecorum

m Literae

aecorum literae ita

ij numeri, vel additione vt in ternarij, vel inuersione vt in septenarij, numeri no

ani, St. Petersburg, 1788, pp. 129-130, quoted

e corruptos. Nam primum 1 apex fuit, seu virgula, nota μον?δο?. 2, est ipsum β extremis suis truncatum. γ, si in sinistram partem inclinaveris & cauda mutilaveris & sinistrum cornu sinistrorsum flexeris, fiet 3. Res ipsa loquitur 4 ipsissimum esse Δ, cujus crus sinistrum erigitur κατ? κ?θετον, & infra basim descendit; basis vero ipsa ultra crus prod

et des beaux arts, Trévoux, 1707 (pp. 1620-1635, with two plates), derives the current symbols from the Romans, stating that they are relics of the ancient "Notae Tironianae." These "notes" were part of a system of shorthand invented, or at l

abditis Numerorum, mysterijs qua origo, antiq

athematiker und Astron

iani, a Brachmanis Indiae Sapientib

rly History of India, Oxf

Arch?ology, Vol. XX, p. 25 (London, 1898). Terrien de Lacouperie states that the Chinese used the cir

iffres arabes et leur origine," La Nature, 1899, p. 222; G. Dumesnil, "De la forme des chiffres usuels," Annales de l'université de Grenoble, 1907, Vol. XIX, pp. 657-674, also a note in Revue Archéologique, 1890, Vol. XVI (3), pp. 342-348; one of the earliest referenc

VI, p. 50. The 1 is evidently Sanskrit, a

p. 90. The zero is not used, but the symbols for 10, 100, a

Tibetan MS. in the library of Professor Smith, probably of

found in India, as well as those of other oriental countries, are given by A. P. Pihan, Expos

words are given also by Al-Bīrūnī in his work India; by Burnell, loc. cit.; by E. Jacquet, "Mode d'expression symbo

., Vol. III, p. 73, as the earliest epigrap

sche Studien, Vol

oyal Asiatic Society,

VIII,

ome, 1864; Lassen, Indische Alterthumskunde, Vo

t., and Thibaut, Astronomie, A

extraits des Mss. de la Bibliothèque nationale, Vol. XXVII, Part I, pp. 1

er, loc. c

oc. cit

astrologers still use an alphabetical syst

cunq; autem fuerit inventor dece

Bīrūnī gi

ation, loc.

n from The Light of As

e ciphers wer

ernsystems," Zeitschrift für die Kunde d

ing of our era, but the date is much in question. G. Thibaut, loc. cit., places it between 700 and 900 A.D.; Canto

es on Indian Mathematics, No. 2.-āryabha?a," Journ. and Proc

ms explained above. This ran up to 1018 a

7, Salzwedel, 1853, and études historiques sur l'arithmétique de position, Programm, p. 24, Berlin, 1856; E. Jacquet, Mode d'expression symbolique des nombres, loc. cit., p. 97; L. Rodet, "S

ayley, loc. cit., p. 22, and L. Rodet,

y G. Thibaut and M. S. Dvivedī, Benares, 1889; see

ed by Kern, Journal of the Ro

instances of this usage in the B?hat Sa?hitā. The system was also used in the Pa?casiddhāntikā as early

chte der Mathematik,

er, loc. c

Bayley

origin of the zero, but says: "D. Henricus Grauius, vir Graecè & Hebraicè eximè doctus, Hebr

ys, Vol. II, p

. XXX, p.

. cit., p

ooke, loc. c

oc. cit

to the contrary, we shall use the word "n

Kanauj," in Journal of the Royal Asia

. IX, 190

Indica, Vol. IX

hia Indica, V

oc. cit

hibaut,

guram representant," etc. [Boncompagni, Trattati, p. 28.] Enestr?m has shown that very likely this w

e Palaeograph

Baltimore, 1901, containing photogra

plate; Hoernle, Verhandlungen des VII. Internationalen Orientalisten-Congresses, Arische Se

ia Indica, Vol. II, pp. 19-24 with plates; date 595 A.D. 7, 1, 5, from Bhanda

ca, Vol. IX, part V; date 815 A.D. 5 from "The Morbi Copper-Plate," Bhandarkar, The I

"Asni Inscription of Mahipala," The Indian Antiquary, Vol. XVI, pp.

pp. 263-272; copper-plate grant of date c. 972 A.D. See Bühler. 7, 3, 5, from "Torkhede Copper-Plate G

la Chanlukya of Lā?ade?a," H.H. Dhruva, Indian Antiqu

graphy, plate XXIII, Telugu-Canarese num

duced in "Della vita e delle opere di Leonardo Pisano," Baldassare Boncompa

anuscript, as reproduced in Della

an MS. in the libr

block-book in the l

he manuscript in the University Library at Tübingen, Bloomfield and Garbe, Baltimore, 1901. So

in the numerous plates at the end of the book; practically all of these con

h, from the beginnings to the time of Christ, Münster i. Westfa

tish Museum. See also his monograph "On the Early Use of Arabi

. XVII, pp. 33-48 and 275-279, 1888; Thibaut, Astronomie, Astrolog

l, "On a System of Numerals used in South India," Jo

rly History of India, 2d

nd Mensuration, from the Sanskrit of Brahmeg

Ibid.,

Bibliotheca Mathematica,

n we use thr

rked it by a simple dot, which latter is commonly used in inscriptions and MSS. in order to mark a blank, and

igine delle parole zero

," in the Zeitschrift für Mathematik und Physik, Vol. XLV, Hist.-li

nebst ihrer literarischen Ges

arithmetics, following the Arab custom, always put the 0 after the

showed (Algebra der Griechen, 1842, p. 138), that Ptolemy merely used ο for ο?δ?ν, with no notion of zero. See also G. Fazzari, "Dell' origine delle parole zero e cifra," Ateneo, Anno I, No. 11, reprinted at Naples in 1903, where the use of the point and the small cross for

is arabice editum, latine versum, adnotationibus instructum a Carolo Alphonso N

cit., Vol.

ad Adelardum Batensem magistrum suum," Abhandlun

-27; Alfred Nagl, "Ueber eine Algorismus-Schrift des XII. Jahrhunderts und über die Verbreitung der indisch-arabischen Rechenkunst und Zahlz

Abhandlungen zur Geschichte der

used for 5. for 13.

position, Berlin, 1856, p. 12; J. Bowring, The Decimal

Morgenlandes, Vol. XI, p. 13; Führer durch die Papyr

ibrary of G. A

ark on the Chinese Mathematics in Cantor's Geschichte der Mathe

umerauit significat. Unde Sephar numerus est: hinc Siphra (vulgo corruptius). Etsi verò gens Iudaica his notis, qu? hodie Siphr? vocantur, usa non fuit: mansit tamen rei appellatio apud multas gentes." Dasypodius, Institutiones mathematicae, Vol. I, 1593, gives a large part

loc. cit., p

onstantinople, 1534, explains sifra as being Arabic. See also Steinschneider, Bibliotheca Mathem

hoc signo 0, quod arabice zephirum ap

cle fut nommé par les uns, sipos, rota, galgal ...; par les autres tsiphra (de ???, couronne ou diadème) ou ciphra (de ???, numération)." Ch. de Paravey, Essai sur l'origine unique et hiéroglyphique des chiffres et des lettres de tous les peuples, Paris, 1826, p. 165, a rathe

and described by G. Lami in his Catalogus codicum manuscriptorum qui in b

ia di fare significare, ... Et decina o centinaia o migliaia non si puote scriv

Ibid.,

vece dello zero il cui segno 0 in arabo si chiama zepiro donde il vocabolo zero), che per sè stesso non esprime

," Archiv für Kulturgeschichte, Berlin, 1905, pp. 155-195, gives the following two sch

n man Zyfer nendt." Recorde (Grounde of Artes, 1558 ed., f. B_6) says that the zer

Later, "quoniam de integris tam in cifris quam in proiectilibus,"-the word proiectilibus referring to mark

mmentarius, una cum Algorismo ipso, Copenhagen, 1897, p. 2.] Curtze cites five manuscripts (fourteenth and fifteenth centuries) of Dacia's commentary in the libraries at

ze, loc. c

841, chap, i, "Joannis de Sacro-B

ara Arithmetic

n, f. 3, appears "Chiamata zero, ouero nulla." Woepcke asserted that it first appeared in Calandri (1491) in this sentence: "Sono dieci le figure con

Bulletino, Vol.

l'Arsenal, Vol. III, pp. 154-156, this work is No. 2904 (184 S.

hat there are "nueue letro

. Trenchant (Lyons, 1566, 1578 ed., p. 12) also says: "La derniere qui s'apele nulle, ou zero;" but Champenois, his contemporary, writing in Paris in 1577 (although the wo

ngen zur Geschichte der Mathematik, Vol. V, p. 97, from a manuscript of the thirteenth century.] Chasles (Comptes rendus, t. 16, 1843, pp. 1393, 1408) calls attention to the fact that Radulph did not know how to use the zero, and he doubts if the sipo

athematik bei den Juden," in Bibliotheca Mathematica, 1893, p. 69, and Silberberg, Das Buch der Zahl des R. Abraham ibn Esra,

II, p. 28). So Ramus (Libri II, 1569 ed., p. 1) says: "Circulus qu? nota est ul

genant die nichts bedeut." [K?bel's Reche

"figura circularis," "circularis nota." Clichtoveus (1503 ed., f. XXXVII) calls it "nota aut circularis o," "circularis nota," and "figura circularis." Tonstall (1522, f. B_3) says of it: "Decimo uero nota ad formam litter? circulari figura est: quam alij circulum, uulgus cyphram uocat," and later (f. C_4) speaks of the "circulos." Grammateus, in his Algorismus de

, alii figuram nihili, alii figuram privationis, seu figuram nullam vocant, alii ciphram, cùm tamen hodie omnes h? not? vulgò ciphr? nominentur, & his notis numera

d similitudinem tecae. Teca enim est ferrum figurae rotundae, quod ignitum solet in quibusdam regionibus imprimi fronti vel maxillae furis seu latronum." [Loc. cit., p. 26.] But in

59. See also Wertheim in the Bibl

lle ou figure de nulle valeur." [La

"figura nihili (quam etiam cifram uocant).

a prima, che è o, si chiama nulla, ouero zero, ouero niente." It also found its way into the Dutch arithmetics, e.g. Raets (1576, 1580 ed., f. A_3): "Nullo dat ist niet;" Van der Schuere (1600, 1624 ed., f. 7); Wilkens (1669 ed., p

t: les vns l'appellant zero: nous la pourrons

672, p. 2) uses only this word (cypher or cipher), and the same is true of the first native American arithmetic, written by Isaac Greenwood (1729, p. 1). Petrus de Dacia de

chap. ix, Mathesis universalis, "De figuris n

. 36 of reprint, spells τσ?φρα from Maximus Planudes, citing Wallis as a

τζ?φρα and τζ?μφρα appear. See also Boeckh, De abaco Graecorum, Berlin, 1841, and Tannery, "Le Scholie du moine Néophytos," Revue Archéologique, 1885, pp. 99-102. Jordan, loc. cit., gives from twelfth and thirteenth century manuscripts the forms cifra, ciffre, chifras, and cifrus. Du Cange, Glossarium mediae et infimae Latinitatis, Paris, 1842, gives also ch

it., p. 59. Long before Woepcke, I. F. and G. I. Weidler, De characteribus numerorum vulgaribus et eorum aetatibus, Wittenberg, 1727, asserted the possibility of their introduction into Greece by Pythagoras or one of his

hate, but permitted the use of the Greek alphabetic numerals, since the Arabs had no convenient number notation: κα? ?κ?λυσε γρ?φεσθαι ?λληνιστ? το?? δημοσ?ου? τ?ν λογοθεσ?ων κ?δικα?, ?λλ' ?ραβ?οι? α?τ? παρασημα?νεσθαι, χωρ?? τ?ν ψ?φων, ?πειδ? ?δ?νατον τ? ?κε?νων γλ?σσ? μον?δα ? δυ?δα ? τρι?δα ? ?κτ? ?μισυ ? τρ?α γρ?φεσθαι? δι? κα? ?

s (in Russian), Kiev, 1908; The Independenc

, loc. cit.,

hor, probably Abū Sahl Dunash ibn Tamim, is given by Steinschneider,

r in the Abhandlung

ilegium observationum ad historiam notarum numeralium pertinentium, Wittenberg, 1755, speaks of the "figura cifrarum Saracenicarum" as bei

at these numerals were used not for calculation, but very much as we use Roman numerals.

his scheme of zero dots was also adopted by the Byzantine Greeks, for a manuscript of Planudes in the Bibliothèque Nationale has numbers li

See Ch

entury A.D., but probably of the eight

by the Arabic w

ibrary, "Recherches sur l'histoire des sciences mathématiques ch

] P.

hich, a scholiast to the Bodleian manuscript remarks: "The science is called Algobar because the invent

t, Entstehung

Abū Zakarījā el-?a??ār," Bibliothe

ffres arabes," Revue Afric

was published after Professor Flügel's death by J. Roediger and A. Mueller. The first

line 5 in the illu

e des sciences mathématiques chez les o

Suter, Bibliotheca Mathem

is not used in the text. The manuscript from which these are taken is t

t. He gives the ordinary modern

ive applied to the forms in 5 is gobārī and to those in 6 indienne. This is the direct opposite of Woepcke's use of these adjec

sually written fr

s of the Ancient Egyptians, revised by S. Bir

The reader who cares to go fully into it should consult the var

us C?sar. Subsequently the emperors assumed it during their own lifetimes, thus

er's Dict. of Class. Lit. and Antiq., New York, 1897, Vol. I, p. 213. There is much uncertainty as to hi

avius Manlius Boethi

no good historic evidence

Symmachus: "Domino suo patricio Sym

e that he wrote De con

sometimes

dition that he was executed be

the Divus

rial place in the monastery of St. P

tly soul,

eitfulness, to

ight of all th

limbs, whence i

dauro; and f

it here."-Par

to be thought of in such institutions. While referred to by B?da (672-735) and Hrabanus Maurus (c. 776-856), i

spelled C

did he translate Nicomachus, although he embodied many of the ideas of the Greek writer in his own arithmetic. Gibbon follows Cassiodorus in these statements in his Decline and Fall of the

idea goes back to

st. math. univ., p. 387]. Libri, speaking of the time of Boethius, remarks: "Nous voyons du temps de Théodoric, les lettres reprendre une nouvelle vie en Italie, les écoles florissantes et le

is a corruption from Ibn Sīnā, as pointed out by Wüstenfeld, Geschichte der arabischen Aerzte und Naturforscher, G?ttingen

f Exploration and Geographical Science from the Conversion of the Roman Empire to A.D. 1420, London, 1897-1906, 3 vols.; Heyd, Geschichte des Levanthandels im Mittelalter, Stuttgart, 1897; J. Keane, The Evolution of Geography, London, 1899, p. 38; A. Cunningham, Corpus inscriptionum Indicarum, Calcutta, 1877, Vol. I; A. Neander, General History of the Christian Religion and Church, 5th American ed., Boston, 1855, Vol. III, p. 89; R. C. Dutt, A History of Civilization

Ancient Commerce of In

t, études etc

nary of Greek and Roman

s in Irán, London, 1902, p. 167. Sykes was the first European

IV, p. 300; Isaac Vossius, Periplus Scylacis Caryandensis, 1639. It is doubtful whether the work attributed to Scylax was writ

rodotus,

), the Sesoosis of

London, 1895, p. 386. On the relations, political and commercial, between India a

he name still remain

p. 724; F. B. Jevons, loc. cit., p. 389; J. C. Marshma

near this place that the first great Indian m

erior to Alexander, found in northern India. More complete information m

p. 14; and to him is due

he Geschichte, Vol. III, S

signs of the zodiac. [R. Caldwell, Comparative Grammar of the Dravidian Languages, Lon

II, p. 73; R. Ca

ningham, loc

Skeel, Travel,

"the green root." [Indian

only from the second

e Italia

he Decimal System,

lumbia University, March 12, 190

iles, l

??φη), water-lily (si-kua, "west gourds"; σικ?

the Roman trade routes, see Be

. of Archeol.,

aepe missi sunt numquam antea visae apud quemquam principem Romanorum." [M. Reinaud, "Relations pol

inter munera trahentes nihil magis quam longinquitatem viae imputabant." Horace shows his geographical knowledge by saying: "Not those who drink of the deep Danu

s auditu modo cognitos pellexit ad amicitiam suam populique Ro

ud, loc. ci

170-172. So Properti

sar dites med

iferi findere

arms against opulent India, and with

loc. cit., V

ud, loc. ci

e Romans instead of the Greeks, was written by Ginanni in 1753 (Dissertatio mathematica critica de numeralium notarum minuscularum origine, Venice, 1753). See also Mannert, De numerorum quos arabicos vocant vera origine Pythagorica, Nürnberg, 1801. Even as late as 1827 Romagnosi (in his supplement to Ricerche storiche sull' India etc., by

ebrandt, Alt-

hman, loc. cit.,

-579 A.D.; called Nu

volution of Geography

who lived in a

Encyc. Brit., 9th ed

rt, loc. c

his contigisse, aut vetustioribus Codd. MS

question was next taken up in a large way by Weidler, loc. ci

uati Severini Boetii de institutione arithmetica libri duo, de institutione music

ry in two books in which are mentioned the numerals. There is a manuscript of this pseudo-geometry of the ninth century, but the earliest

nihil aliud esset nisi operam et tempus perdere." [Preface, p. v.] N. Bubnov in the Russian Journal of the Ministry of Public Instruction, 1907, in an artic

, dieselbe Schrift, welche er nach Euklid bearbeitete, von welcher ein Codex bereits in Jahre 821 im Kloster Reichenau vorhanden war, von welcher ein anderes Exemplar im Jahre 982 zu Mantua in die H?nde Gerbert's gelangte, von welcher mannigfache Handschriften noch heute vorhanden sind." But against this

t eu les anciens d'une numération décimale écrite qui fait usage de neuf chiffres prenant les valeurs de position," Comptes rendus, Vol. VI, pp. 678-680; "Sur l'origine de notre système de numération," Comptes rendus, Vol. VIII, pp. 72-81; and

Berlin, 1841; Friedlein, in his Leipzig edition of 1867; Weissenborn, Abhandlungen, Vol. II, p. 185, his Gerbert, pp. 1, 247, and his Geschichte der Einführung der jetzigen Ziffern in Europa durch Gerbert, Berlin, 1892, p. 11; Bayley, loc. cit., p. 59;

vée des Romains, Vol. II (Fren

arithmetic in the Laurentian library at Florence, of 1370, the following forms, which, of course, are interpolations. An interesting example of a forgery in ecclesiastical matters is in the charter said to have been given by St. Patrick, granting indulgences to the benefactors of Glastonbury, dated "In nomine d

t Trinity College. See also Woepcke, in Propagation, pp. 37 and 42. It was the evident corruption of the texts in such editions of Boethius a

n the Laurentian library which one of the authors has examined. It should be said, however, that the disputed p

Rara Arithm

rg, Philologus, V

d sint digiti, quid articuli, quid compositi, qu

. a posterioribus appellabatur abacus." This, as pictured in the text, is the common Gerbert abacus. In the edition in Migne's Pa

atos apices vel caracteres." See

Archéologique, 1879, derives these from the initial letters used as abbre

brynge: "? fforthermore ye most vndirstonde that in this craft ben vsid teen figurys, as here bene writen for ensampul, ... in the quych we vse teen figurys of Inde. Questio. ? why

edlein ed

ruhe codex

ch codex o

uhe codex of

h codex of

rchill,

teenth century, Alexa

ntury Boethius,

odex, tenth cen

V, p. 186; g Memorie della classe di sci., Reale Acc. dei Lincei, An. CCLXXIV (1876-1877), April, 1877. A twelfth-century arithmetician, possibly John of Luna (Hispalensis, of Seville, c. 1150), speaks of the great diversity of these f

oc. cit

Ibid.,

c. cit.,

n, 1875, p. 169; Th. N?ldeke, Aufs?tze zur persichen Geschichte, Leipzig, 1887

f Ibn al-Adamī, astronomer, in a work published by his contin

r etc., pp. 4-5, states that Al

r, loc. ci

references to Suter, unless otherwise stated, are to his

Fihrist, p.

ok of the Cyphers in his Chronology, English ed., p. 132. Suter, Die Mathematiker etc.,

uter, p

Suter,

As the name shows,

Suter,

ik, p. 256, refers to him as writing on t

rdt, Halle, 1865; and German translation, Das Reche

emploi des chiffres indiens par les Arabes," T

Suter,

is fact, in the Zeitschrift d. deutschen

ire des mathématiq

inople, l'Arabe victorieux demandait des manusc

n bagadata,

least pretended) descendants of 'Al

American ed., N. Y.,

l Sketches, Vol

ence at that period

See pp

Festschrift, 1909, note pp. 10-11.

hematica, Vol. I (3), p. 499; Cant

rectori nostro atque defensori dicamus dignas." It is devoted en

den," Bibliotheca Mathematica, Vol. VIII (2), p. 99

MS. by F. Woepcke, Propagation, and more recently by H.

Christian Religion and Church, 5th Amer

, loc. cit.,

loc. cit., Vol.

See pp

ammed Abū'l-Qāsim, and Ibn Hauqal.

al-masālik w

sur l'Inde; in Ger

193. He himself had trave

ay the third, XXII. Sir Edwin Arnol

ngham, loc.

, Books, Vol

ernas peregrin

ovans, sophiae

novi libroru

et, terris repe

uleum venit de

his zeal for knowledge and seeking to discover in foreign lands novelties in books or in studie

ligion and Church, 5th American ed., Boston, 1855, Vo

ngham, loc.

loc. cit., V

Ibid.,

Histoire, V

losophe, d'après l'histoire et d'ap

cit., Vol. I, chap

Ibid.,

ol. I, p. 110, n., citin

magnetis Amalphis," is true so far as it means the modern f

t, loc. cit., V

ambridge Modern History, Lo

ao," Archives pour servir à l'étude de l'histoire de l'Asie orientale, Leyden, 1890,

in Venice. The best description of the Polo travels, and of other travels of the later Middl

rticle "China." The handbook cited is Pegolotti's Libro di divisamenti di paesi, chapters i-ii, where

gham, loc. c

a commiss

gham, loc. c

istory of the English Peop

, London, New Y

in, baldekin

talian

, Oriental Rugs, Ne

bertus and Girbertus appearing indiff

C. Heilbronner, Historia

tum," as an old chroni

t, Lettres de Gerbert, Paris, 1889 ; H. Weissenborn, Gerbert; Beitr?ge zur Kenntnis der Mathematik des Mittelalters, Berlin, 1888, and Zur Geschichte der Einführung der jetzigen Ziffern in Europa durch Gerbert, Berlin, 18

es how affectionately the abbot received him, asking if there were men in Spain well versed in the arts. Upon Borel's reply in the

Hatto also appears

ustworthy contemporary, of his going to Cordova, are unsupported. (See e.g. Picavet, p. 34.) Nevertheless this testimony is still accepted: K. von Raumer, for example (Geschicht

ia, praeclarissima quoque figurarum geometri?, aliaque non minus admiranda" (Epist. 8). Also in a letter to Rainard

vet, loc.

t, loc. ci

vet, loc.

to poi l' abito, e 'l monasterio, e datosi tutto in potere del diavolo."

ue librum de astrologia translatum a te michi petenti dirige," presumably refe

ubnov, loc.

. 15, for Bernelinus; and Bubnov, l

spaciorum superductio unitatis caractere inscribitur, qui chaldeo nomine dicitur igin." See also Alfred Nagl, "Der

tes, since the evidence is all against his knowing the place value. Friedlein emphasizes this in the Zeitschrift für Mathematik und Physik, Vol. XII (1867), Literaturzeitung, p. 70: "Für das System unserer N

(Recherches nouvelles etc.) believes that Gerbert received them

r, governing Spain under the name of Hishām (976-1002), called from the Orient

nborn, loc.

Ibid.,

oc. cit., and, less completely, by Olleris, loc. cit. Those touchin

," Vol. III, and at least three other editions have since appeared, viz

Richerus Monchus, Gallorum congressibus in volumine regerendis, im

rum libellum a Joseph Ispano editum abbas Warnerius" (a person otherwise unknown). In epistle 2

Gerbert; M. Steinschneider, in Bibliotheca Mathematica, 1893, p. 68. Wallis (Algebra, 1685, chap. 14) went over the list of Spanish Josephs v

83, pp. 354-364. One of the manuscripts is of 976 A.D. and the other of 992 A.D. See also Franz Steffens, Late

e Abbey of Fleury. The text of the letter to Constantine, preceding the treatise on the Abacus, is given in the C

étique," Comptes rendus, Vo

it., Gerbert

. Peter, at Salzburg, and was published by Peter Bernhard Pez in 1721. Doubt was first cast upon it in the Olleris edition (?uvres de Gerbert). See Weissenborn, Gerbert, pp. 2, 6, 168, and

et, loc. ci

rds, "a domino pape Gerberto." He was quite certainly not later than the eleve

ssenborn, Gerbert, p. 227. In Olleris

in Bubnov, loc

enborn, Ger

chi, Monete Romane, 2d ed., Milan, 1900, cap. XXXVII. For pictures of old Gre

, v., speaks of the ten figures as "characteres

appears in Radulph of Laon, in the twelfth century. See Günther's Gesch

illustrations are taken contain the symbol, while four out of five which give the words use the word sipo

no sibi nomen

cum mox uendic

eros incomposi

. A discussion of the whole question is also given in E. C. Bayley, loc. cit. Huet, writing in 1679, asserted that they were of Semitic origin, as did Nesselmann in spite of his despair over ormis, calctis, and celentis; see Woepcke, Propagation, p. 48. The names were used as late as the fifteenth century, without the zero, but with the superscript dot for 10's, two dots for 100's, etc., as among the early Arabs. Gerhardt men

als in 200 tongues, by Rev

"Aristoteles enim uoces rerum σ?μβολα uocat: id translatum, sonat notas." [Noviomagus, De Numeris Libri II, cap. vi.] "Alphabetum decem notar

arithmetic by an American (Greenwood, 1729) the author speaks of "a few Arabian Charecters or Numeral Figures, called Digits" (p. 1), and as late as 1790, in the third edition of J. J. Blassière's arithmetic (1st ed. 1769), the name characters is still in use, both for "de Latynsche en de Arabische" (p. 4), as is also the term "Cyfferletters" (p. 6, n.). Ziffer, the modern German form of cipher, was commonly used

rdum Batensem magistrum suum. The work was made known by C. Henry, in the Zeitschrift für Mathematik und Physik,

is indicated by

" is the expression used, while a century later "giffre e

haucer, edited by W. W. Skeat

., Vol. III, p

tament of Love, printed with Chaucer's

ished in Olleris, ?uvres

ncient India," Journal and Proceedings of the

, by Leonardo Pisa

Rechnens mit Columnen," Zeitschrift für

ng increased by the differenc

Cantor, Vol

oire de l'arithmétique. Recherches des traces du système de l'abacus, après que cette méthode a pris le nom d'Algorisme.-Preuves qu'à toutes les époques, jusq'au XVIe siècle, on a su que l'arithmétique vulgaire avait pour or

cit., pp. 203-204

abaci rationibus," in Bubn

vi al calcolo dell' abaco," Bulletino di bibliografia e di st

rdo di Bath intitolato 'Regulae Abaci,'" B. Bon

, "Intorno al Tractatus de Abaco di Ger

d'abaco contenuti in due codici Vaticani del secolo

es de l'histoire de France, V

62. A. Nagl in the Abhandlungen zur Ge

1030

matik, Vol. V, pp. 85-133. The work begins

chte der arabischen Zahlzeic

ho died

e Mathematik bei den Juden," Bibliotheca Mathematica, Vol. X (2), p. 79) ingeniously derives another name

said to have

arismetrice. Qui editus est a magistro Johanne yspalensi." It is publi

wārazmī's book," but "to a book of algorism." John of Luna says of it: "Hoc

3. As to the author, see Enestr?m in the Bibliotheca M

na, in Andalusia) in 1114; died at Toledo in 1187. Cantor,

ript of his work, Gerardi Cremonensis artis metrice practice. See also T. L. Heath, The Thirteen Books of Euclid's Elements, 3 vols., Cambridge, 1908, Vol. I, pp

Algebra, 168

j?rnbo, "Al-Chwārizmī's trigonometriske Tavler," Fest

, loc. cit.

93; M. Curtze, Centralblatt für Bibliothekswesen, 1899, p. 289; E. Wappler, Zur Geschichte der deutschen Algebra im 15. Jahrhundert, Programm, Zwickau, 1887; L. C. Karpinski,

abischen Rechenkunst und Zahlzeichen im christl. Abendlande," in the Zeitschrift für Mathematik und Physik,

e a in the pl

ebr?isch-arithmetisches Werk des R. Abraham ibn

ing's "Rabb

Zahlen durch neun bezeichnet und Formen für die 9 Zi

ogio di Lionardo Pisano, Bologna, 1812, p. 35; Libri, Histoire des sciences mathématiques, Vol. II, p. 25; D. Martines, Origine e progressi dell' aritmetica, Messina, 1865, p. 47; Lucas, in Boncompagni Bulletino, Vol. X, pp. 129, 239; Besagne, ibid., Vol. IX, p.

ctural. See the Bibliotheca Ma

e with the Saracens at Palermo, capturing six ships, one being "full

loc. cit., V

Ibid.,

ance in Italy. The Age of De

nds, loc.

n'empêcha pas Dante d'être le plus grand poète de l'Italie, et ce fut un petit marc

nd published in 1858, reads: "Leo

accingo ge

ri, Guglielmin

atin, B

ompagni an

Reprin

the French na

w part o

rica, New York, 189

i indorum." Woepcke, Propagation etc., regards this as referring to two different systems, bu

being used, and long afterwards in Italy, t

ab eodem anno 1228." Three MSS. of the thirteenth century are known, viz. at Milan, at

still more gloomy view of Oxford in his time in his Opus minus, in the Rerum Britannicarum medii aevi scriptores, London, 1859, Vol. I, p. 327. For a picture of Cambridge at this time consult F. W. Newman, The English Universities, translated from the German of

61, and that a bank was founded at Venice in 1170, the Bank of San Marco being established in the following year. The activity of

nds, loc. cit.,

Alphabet, London, 1

in 1340. See Schmidt's Encyclop?die der Erziehung, Vol. VI, p. 726; A. Kuckuk, "Die Rechenkunst im sechzehnten Ja

in Halliwell, Rara Mat

ole's Catalogue of Manuscript

robosco commentarius, una cum Algorismo ipso, Copenhagen, 1897; L. C. Karpinski, "Jordan

ner Universit?t im ersten Jahrhundert

loc. cit., g

Plimpton collection and the Columbia library shows such marked divergence from each other and from the text published by Curtze that the conclusion seems legitimate that these

mine Algus, unde et Algorismus nunc

sive iudaico, huius scientiae inventorum." [Curtze, loc. cit.,

dorum repraesentatur, sed tantum determinatus per deter

risme et de géométrie," Boncompagni Bulletino, Vol. XV, p. 49; Vic

rt du Roy Robert, arrivée en 1031, jusqu'à celle

des lettres en France au

torique, Paris,

astery at Chester in 1299. He was a Benedictine monk and chronicler, and died in

slation, Higden havi

i interim fastidiosi, quod si in aliquo calculo astroloico error contigisset, calculatorem operationem suam a capite incipere oportebat, dato quod error suus adhuc satis propinquus existeret; et hoc propter figuras in sua operatione deletas. Indigebat etiam calculator semper aliquo lapide vel sibi conformi, super quo scribere atque faciliter delere posset figuras cum quibus operabatur in calculo suo. Et quia haec omnia satis fastidiosa atque laboriosa mihi visa sunt,

Pal?ographie, pp. xxxix-xl. We are indebted to Professo

e plate of f

ski, "Hindu Numerals in the Fihrist," Bib

us d'Apollonius sur les quantités irrationnelles, d'après des indications tirées d'un manuscrit arabe," T

the Egypt Exploration Fund for

ni's library bearing this date: "Nota quod anno dni nri i

," read before the Society of Antiquaries April 14,

The date is part of

Manual of Musalman Nu

ierischen Münzkunde, Gr?tz, 1875, where the claim is made of an Austrian coin of 1458; Bibliotheca Mathematica,

in the British Museu

Ibid.,

mediis utriusque

ern et Hans Hur

one of the oldest E

Venice, Jenson, 1470. The above statement holds for copies

m vitiorum, Nürnberg, 1470. The copy

," Anzeiger für Kunde der deutschen Vorzeit, 1861, columns 46, 81, 116, 151, 189, 229,

chichte, p. 175, n

drawings by J. T. Irvine, in the Proceeding

nograph, are P. Treutlein, Geschichte unserer Zahlzeichen, Karlsruhe, 1875; Cantor's Geschichte, Vol. I, table; M. Prou, Manuel de paléographie latine et fran?aise, 2d ed., Paris, 1892, p. 164; A. Cappelli, Dizionario di abbreviature latine ed italiane, Milan, 1899. An interesting early sour

oc. cit., Abhandlungen, and Nagl, loc. cit. The forms are copied f

historica, "Scriptores" Vol. XVII, plate to p. 184; Wattenbach, Anleitung zur lateinischen Palaeog

published photograph of the original, in th

c. 1294, in Mr. Plimpton's library

ript in Sigmaringen, copied fro

1360 A.D. The work is a computus in which the date 1360 appears,

ismus in Mr. Plimpton's library. Date c.

a Arithmetica

id., pp.

ed.), and Tzwivel (1507 ed., where jj and jz are used for 11 and 12). This was not universal, however, for the Algorithmus linealis of c. 1488 has a specia

the following five characters are taken from Cappelli's Dizionario, p. 380, and are from manuscripts

's work of 1481; Cl

cit., p. 28.] The second character is from a French algorismus, c. 1275. [Boncompagni Bulletino, Vol. XV, p. 51.] The third and the following sixteen characters are given by Cappel

hiarini (148

ight characters are given by Cappelli, loc. cit., and are from manuscripts of the twelfth (2)

e Nagl,

algorismus, thi

anuscript, in Rara Arith

the old forms are used, although the new ones appear in the tex

teenth (3), fourteenth (7), fifteenth (6), and eighteenth centuries, respectively. Tho

s manuscript, 1424. The others in the first two lines are from Cappelli, twelfth (3), fourteenth (6), fift

irteenth century. The following are from Cappelli, fourteenth (3),

century. The following forms are from Cappelli, twelfth, thirteenth, fourte

i, thirteenth, fourteenth, fifteenth (2

en in several of the Plimpton manuscripts, as in one w