See also:

• NUMERAL (from Lat. numerus, a number)

NUMERAL (from See also:

• LAT

Lat. numerus, a number)  , a figure used to represent a number . The use of visible signs to represent See also:

• NUMBERS, BOOK OF
• NUMBERS, PARTITION OF

numbers and aid reckoning is not only older than See also:

• WRITING
• WRITING (the verbal noun of " to write," O. Eng. writan, to inscribe)

writing, but older than the development of numerical See also:

• LANGUAGE (adapted from the Fr. langage, from langue, tongue, Lat. lingua)

language on the denary See also:

• SYSTEM

system; we See also:

• COUNT (Lat. comes, gen. comitis, Fr. comte, Ital. conte, Span. conde)

count by tens because our ancestors counted on their fingers and named numbers accordingly . So used, the fingers are really numerals, that is, visible numerical signs; and in antiquity the practice of counting by these natural signs prevailed in all classes of society . In the later times of antiquity the See also:

• FINGER

finger symbols were See also:

• DEVELOPED

developed into a system capable of expressing all numbers below 10,000 . The See also:

• LEFT

left See also:

• HAND
• HAND (a word common to Teutonic languages; cf. Ger. Hand, Goth. handus)
• HAND, FERDINAND GOTTHELF (1786-185r)

hand was held up See also:

• FLAT (a modification of O. Eng. flet, an obsolete word of Teutonic origin, meaning the ground beneath the feet)

flat with the fingers together . The See also:

• UNITS, DIMENSIONS OF
• UNITS, PHYSICAL

units from r to 9 were expressed by various positions of the third, See also:

• FOURTH

fourth, and fifth fingers alone, one or more of these being either closed on the See also:

• PALM (Lat. palma, Gr. iraX6.jn )
• PALM, JOHANN PHILIPP (1768-1806)

palm or simply See also:

• BENT
• BENT (E1. BENI)
• BENT, JAMES THEODORE (1852–1897)

bent at the See also:

• MIDDLE

middle See also:

• JOINT (through Fr. from Lat. junctum, jungere, to join)

joint, according to the number meant . The thumb and See also:

• INDEX

index were thus left See also:

• FREE

free to See also:

• EXPRESS (through the French from the past participle of the Lat. exprimere, to press out, by transference used of representing objects in painting or sculpture, or of thoughts, &c. in words)

express the tens by a variety of relative positions, e.g. for 30 their points were brought together and stretched forward; for 50 the thumb was bent like the See also:

• GREEK
• GREEK, ETRUSCAN AND

Greek P and brought against the See also:

• BALL (in Mid. Eng. bal; the word is probably cognate with " bale," Teutonic in origin, cf. also Lat. falls, and Gr. 7raXXa)
• BALL, JOHN (1585-1640)
• BALL, JOHN (1818-1889)
• BALL, JOHN (d. 1381)
• BALL, SIR ALEXANDER JOHN, BART
• BALL, THOMAS (1819- )

ball of the index . The same set of signs if executed with the thumb and index of the right hand meant hundreds instead of tens, and the unit signs if performed on the right hand meant thousands.' The fingers serve to express numbers, but to make a permanent See also:

• NOTE
• NOTE (Lat. nota, mark, sign, from noscere, to know)

note of numbers some See also:

• KIND (O. E. ge-cynde, from the same root as is seen in " kin," supra)

kind of See also:

• MARK
• MARK, CARL (1858– )
• MARK, GOSPEL OF ST
• MARK, ST

mark or See also:

• TALLY

tally is needed . A single stroke is the obvious See also:

• REPRESENTATION

representation of unity; higher numbers are indicated by See also:

• GROUPS
• GROUPS, THEORY

groups of strokes . But when the strokes become many they are confusing, and so a new sign i The system is described by Nicolaus Rhabda of See also:

• SMYRNA (Ismir)

Smyrna (8th See also:

• CENTURY (from Lat. centuria, a division of a hundred men)

century A.D.), ap . N . Caussinus, De eloquentia sacra et humana (See also:

• PARIS
• PARIS (also called ALEXANDROS)
• PARIS, ALEXIS PAULIN (1800-x881)
• PARIS, BRUNO PAULIN GASTON (1839-1903)
• PARIS, FRANCOIS DE (169o-1727)
• PARIS, LOUIS PHILIPPE ALBERT
• PARIS, TREATIES OF (1814-1815)

Paris, 1636) .

The See also:

• VENERABLE (Lat. venerabilis, worthy of reverence, venerari, to reverence, to worship, allied to Venus, love; the Indo-Germ. root is wen-, to desire, whence Eng. " win, properly to struggle for, hence to gain)

Venerable See also:

• BEDE, BEDA, or B
• BEDE, CUTHBERT

Bede gives essentially the same system, and it See also:

• LONG, GEORGE (1800-1879)
• LONG, JOHN DAVIS (1838– )

long survived in the See also:

• EAST
• EAST, ALFRED (1849- )

East; see especially Rodiger, " Uber See also:

• DIE
• DIE (Fr. de, from Lat. datum, given)

die See also:

• IM(K2+a2+h2—2 ah cos 0)

im Orient gebrauchliche Fingersprache, &c.," D.M.G . (1845), and See also:

• PALMER
• PALMER, EDWARD HENRY (1840-1882)
• PALMER, ERASTUS DOW (1817-1904)
• PALMER, GEORGE (1818-1897)
• PALMER, JOHN
• PALMER, RAY (1808-1887)
• PALMER, SAMUEL (1805-1881)
• PALMER, SIR CHARLES MARK, BART

Palmer in Journ. of See also:

• PHILOLOGY

Philology, ii . 247 sqq . See also:

• NUMERAL (from Lat. numerus, a number)

NUMERAL must be introduced, perhaps for 5, at any See also:

• RATE

rate for to, roo, torso, and so forth . Intermediate numbers are expressed by the addition of symbols, as in the See also:

• ROMAN

Roman system ccxxxvi= 236 . This simplest way of writing numbers is well seen in the Babylonian See also:

• INSCRIPTIONS (from Lat. inscribere, to write upon)

inscriptions, where all numbers from t to 99 are got by repetition of the See also:

• VERTICAL (from Lat. vertex, highest point)

vertical arrowhead T =I, and a barbed sign = to . But the most interesting See also:

• CASE
• CASE, JOHN (d. 1600)

case is the See also:

• EGYPTIAN

Egyptian, because from its See also:

• HIERATIC

hieratic See also:

• FORM (Lat. forma)

form sprang the Phoenician numerals, and from them in turn those of See also:

• PALMYRA

Palmyra and the Syrians, as illustrated in table 1 . Two things are to be noted in this table—first, the way in which groups of units come to be joined by a See also:

• CROSS

cross See also:

• LINE

line, and then run together into a single See also:

• SYMBOL (Gr. o-uµ(3oXov, a sign)

symbol, and, further, the substitution in the hundreds of a principle of multiplication for the See also:

• MERE
• MERE, ADALBERT (1838-1909)

mere addition of symbols . The same thing appears in Babylonia, where a smaller number put to the right of the sign for too (T-) is to be added to it, but put to the left gives the number of hundreds . Thus (I-=loco, but T-<=tro . The Egyptians had See also:

• HIEROGLYPHICS (Gr. iepos, sacred, and'yXv ni, carving)

hieroglyphics for a thousand, a myriad, roo,000 (a See also:

• FROG

frog), a million (a See also:

• MAN
• MAN, ISLE OF (anc. Mona)

man with arms stretched out in admiration), and even for ten millions . Alphabetic writing did not do away with the use of numerical symbols, which were more perspicuous, and compendious than words written at length .

But the letters of the See also:

• ALPHABET (see also WRITING)

alphabet them-selves came to be used as numerals . One way of doing this was to use the initial See also:

• LETTER (through Fr. lettre from Lat. littera or litera, letter of the alphabet; the origin of the Latin word is obscure; it has probably no connexion with the root of linere, to smear, i.e. with wax, for an inscription with a stilus)

letter of the name of a number as its sign . This was the old Greek notation, said to go back to the See also:

• TIME (0. Eng. Lima, cf. Icel. timi, Swed. timme, hour, Dan. time; from the root also seen in " tide," properly the time of between the flow and ebb of the sea, cf. O. Eng. getidan, to happen, " even-tide," &c.; it is not directly related to Lat. tempus)
• TIME, MEASUREMENT OF
• TIME, STANDARD

time of See also:

• SOLON (7th and 6th century n.c.)

Solon. and usually named after the grammarian Herodian, who described it about A.D . 200 . I stood for t, II for 5, A for ro, H for too, X for moo, and M for 10,000; II with A in its bosom was 50, with H in its bosom it was 500 . Another way See also:

• COMMON

common to the Greeks, See also:

• HEBREWS, EPISTLE TO THE

Hebrews, and Syrians, and which in See also:

• GREECE

Greece gradually See also:

• SYRIAC

Syriac . Palmyrene . Phoenician . Hieratic . Hieroglyphic . ! I 1 I Il 1 ! rl /u III ti "1 In 3 rr /1/I 111 4'41 01 1111 4 II111 '.I II 111 5 l5 111111 t III III 6 I1lJ 'I!lIII 1111111 7 rt~- 1112 II III III Gi 4 1111 1111 8 rr- /111':3 111111111 Z 111 1111n 9 7 -7 n I0 7 1--p 1—7 IA In II rr-'-j 111111111—7 111!IIUIn I 19 0 nn 20 10 lZ \z,A See also:

• INN

Inn 21 70 --'8 nnn 30 00 33 yy nnnn 40 700 -253 -7&y nnnnn 50 000 333 4/yti nnnnnn 6o 7000 '7333 nnn nnnn 70 0000 3333 bh'H1/ nnnn nnnn 8o 70000 —PS 33`3 -'HHHH A Mn nnn nnn 90 —pr ~pl lol ~l 9 See also:

• I00

I00 'Tr ~1I al.) 1011 99 200 l'1 See also:

• ILI

Ili 999 300 displaced the Herodian numbers, was to make the first nine letters stand for the units and the See also:

• REST (O. Eng. rest, reste, bed, cognate with other Teutonic forms, e.g. Ger. Rast, Riiste, rest, and probably Gothic Rasta, league, i.e. resting or stopping place)

rest for the tens and hundreds .

With the old Semitic alphabet of 22 letters this system See also:

• BROKE, or BROOKE, ARTHUR (d. 1563)
• BROKE, SIR PHILIP BOWES VERE, BART

broke down at n=400, and the higher hundreds had to be got by juxtaposition; but when the See also:

• HEBREW

Hebrew square See also:

• CHARACTER (Gr. xapareri7p, from xap&crew, to scratch)

character got the distinct final forms .1,: ^, 7,'i, r these served for the hundreds from 500 to 900 . The Greeks with their larger alphabet. required but three supplemental signs, which they got by keeping for this purpose two old Phoenician letters which were not used in writing (F or ss= =6, and -me= 9o), and by adding sampian for goo i Among the Greeks the first certain use of this system seems to be on coins of See also:

• PTOLEMY (CLAUDIUS PTOLEMAEUS)

Ptolemy II . The first trace of it on Semitic ground is an Jewish coins of the I asmoneans . It is,the See also:

• FOUNDATION (Lat. fundatio, from fundare, to found)

foundation of geniatria as we find it in Jewish See also:

• BOOK

book and in the apocalyptic number of the beast (10P rhl= 666) . But we do not know how old gematriais; the name is borrowed from the Greek . The most See also:

• FAMILIAR (through the Fr. familier, from Lat. familiaris, of or belonging to the familia, family)

familiar case of the use of letters as numerals is the Roman system . Here C is the initial of centum and Mof mille; but instead of these signs we find older forms, consisting of a circle divided vertically for :r000 and horizontally, e, or in the cognate See also:

• ETRUSCAN

Etruscan system divided intos quadrants, ED, for roo . From the sign for r000, still sometimes roughly shown in See also:

• PRINT

print as cIo; comes D; the See also:

• HALF

half of the symbol for half the number; and the older forms of L, viz . .1 or .L, suggest that this also was once half of the See also:

• HUNDRED

hundred symbol . So V (Etruscan n) is half of X, which itself is not a true Roman letter . The system„ therefore, is hardly alphabetic in origin, though the See also:

• IDEA (Gr. Ibia, connected with i&eiv, to see; cf. Lat. species from specere, to look at)

idea has been thrown out that the signs for ro, 50, and too were originally the Greek X, SY, 43, which were not used in writing Latin ? When. high numbers had to be expressed systems such as we have described became very cumbrous, and in alphabetic systems it became inevitable to introduce a principle of periodicity by which, for example, the signs for 1, 2, 31 &c., might be used with a difference to express the same pumber of thousands .

Language itself suggested this principle, and so we find in Hebrew a or in Greek ,a = Iwo . So further f3Mu, OM., or simply 16 . = 20,000 . (2 myriads) . If now the larger were always written to the left of the smaller elements of a number the diacritic mark could be dispensed with in such a case as,6coha (instead of „6mha) = 2831, for here it was See also:

• PLAIN (O. Fr. plain, from Lat. plenum)

plain that (3 =2000, not 2, since otherwise it would not have preceded to= Soo . We have here the germ of the very important notion that the value of a symbol may be periodic and defined by its position . The same idea ' had appeared much earlier among the Babylonians, who reckoned by See also:

• POWERS, HIRAM (1805-1873)

powers of 6o, calling 6o a loss and 6o sixties a See also:

• SARI

sari On the tablets of Senkerah a See also:

• LIST (O.E. lisle, a Teutonic word, cf. Dut. lust, Ger. Leiste, adapted in Ital. lista and Fr. lisle)
• LIST, FRIEDRICH (1789-1846)

list of squares and cubes is given on this principle, and here the square of 59 is written 58•I—that is, 58X6o+l; and the. See also:

• CUBE (Gr. K46os, a cube)

cube of 3o is 73o--that is, 7 sar+3o See also:

• BOSS

Boss =7 X 662+30 X'6o . Here again we have value by position; but, as there is no zero, it is`left to' the See also:

• JUDGMENT

judgment of the reader to know which See also:

• POWER [WILLIAM GRATTAN] TYRONE (1797-1841)

power of 6o is meant in each case . The sexegesimal system, long specially associated with See also:

• ASTRONOMY (from Gr. &arpov, a star, and v ,usiv, to classify or arrange)

astronomy, has left a trace in our See also:

• DIVISION (from Lat. dividere, to break up into parts, separate)

division of the See also:

• HOUR

hour and of the circle, but as language goes by powers of 10 it is practically very inconvenient for most purposes of reckoning . The Greek mathematicians used a sort of decimal system; thus See also:

• ARCHIMEDES (c. 287–212 B.C.)
• ARCHIMEDES, SCREW OF

Archimedes was able to solve his problem of stating a. number greater than that of the grains of See also:

• SAND
• SAND, GEORGE (1804-1876)

sand which would fill the See also:

• SPHERE (Gr. vclsa?pa, a ball or globe)

sphere of the fixed stars by dividing numbers into octades, the unit of the second octade being rob and of the third Io16 . So too See also:

• APOLLONIUS

Apollonius of See also:

• PERGA (mod. Murtana)

Perga teaches multiplication by regarding 7 as the iruBµ$v or 70, 700 and so forth . One must then find successively the product of the several pythmens of the multiplier and the multiplicand, noticing in each case what are tens, what hundreds, and so on, and adding the results .

The want of a sign for zero made it impossible mechanically to distinguish the tens, hundreds, &c., as we 'now do . I The See also:

• ARABS

Arabs, who quite changed the See also:

• ORDER
• ORDER (through Fr. ordre, for earlier ordene, from Lat. ordo, ordinis, rank, service, arrangement; the ultimate source is generally taken to be the root seen in Lat. oriri, rise, arise, begin; cf. " origin ")
• ORDER, HOLY

order of the alphabet and extended it to twenty-eight letters, kept the See also:

• ORIGINAL

original values of the old letters (putting ee for o and cy for ,m,), while the hundreds from 500 to loon were expressed by the new letters in order from v tot .. In the time of See also:

• CALIPH, CALIF

Caliph Walid (A.n . 705-715) the Arabs had as yet no signs of numeration . 2 See further See also:

• FABRETTI, RAPHAEL (1618–1700)

Fabretti, Paldoga•aphisahe Stiadien . Very See also:

• EARLY
• EARLY, JUBAL ANDERSON (1816-1894)

early, however, a See also:

• MECHANICAL

mechanical contrivance, the See also:

• ABACUS (Gr. 15t(34, a slab; Fr. abaque, tailloir)

abacus; had beep: introduced for keeping numbers oftdifferent dendminatlons apart . This was a table with compartments or colum s' for counters, each See also:

• COLUMN (Lat. columna)

column representing a different value to be given to a See also:

• COUNTER

counter placed on it . This might be used either for See also:

• CONCRETE
• CONCRETE (Lat. concretes, participle of concrescere, to grow together)

concrete See also:

• ARITHMETIC (Gr. apeOµ7run7, sc. TEXVn, the art of counting, from (ipLBµos, number)

arithmetic—say with columns for pence, shlfliii,Fs and pounds; or for abstract reckoning—say with the Ba yIc4ran, secagesimal system . An old Greek abacus found at Salamise has, columns which, taken from right to' left, give a counter the value:.of r, 10, 100, See also:

• I000

I000 drachms, and finally of 1 See also:

• TALENT (Lat. talentum, adaptation of Gr. TaXavrov, balance, ! Recollections of a First Visit to the Alps (1841); Vacation Rambles weight, from root raX-, to lift, as in rXi vac, to bear, 1-aXas, and Thoughts, comprising recollections of three Continental

talent (6000drachms) respectively . An abacus on the decimal system might be ruled on.. See also:

• PAPER
• PAPER (Fr. papier, from Lat. papyrus)

paper or on a. See also:

• BOARD (O. Eng. bord)

board strewed with See also:

• FINE

fine sand, and was 'then a first step to the decimal; system . Two important steps, however, were still lacking: the' first was to use instead of counters distinctive marks (ciphers) for the digits from one to nine; the second and more important was to get a sign for zero, so that the colurnns might be dispensed with, and the See also:

• DENOMINATION (Lat. denominare, to give a specific name to)

denomination of each See also:

• CIPHER, or CYPHER (from Arab. sifr, void)

cipher seen at once by counting the number of digits following These two steps taken, we have at once the See also:

• MODERN

modern so-called Arabic numerals and the possibility of modern arithmetic; but the invention of the ciphers and. zero came but slowly, and their See also:

• HISTORY

history is a most obscure problem . What is quite certain is that our See also:

• PRESENT

present decimal system, in its See also:

• COMPLETE

complete form, with the zero. which enables us to do without the ruled columns of the abacus, is of See also:

• INDIAN

Indian origin .

From the See also:

• INDIANS, NORTH AMERICAN

Indians it passed to 'the Arabians, probably along with the astronomical tables brought to See also:

• BAGDAD, or BAGHDAD

Bagdad by an Indian See also:

• AMBASSADOR (also EMBASSADOR, the form sometimes still used in America; from the Fr. ambassadeur, with which compare Ital. ambasciatore and Span. embajador, all variants of the Med. Lat. ambassiator, ambassiator, ambasator, &c., derived from Med. Lat. amba

ambassador in 773 A.D . At all events the system was explained in Arabic in the early See also:

• PART

part of the 9th century by the famous See also:

• ABU

Abu Ja'far Mohammed b . Musa al-Khwarizmi (Hovarezmi), and from that time continued to spread, though at first slowly, through the Arabian See also:

• WORLD

world . In See also:

• EUROPE

Europe the complete system with the zero was derived from the Arabs in the 12th century, and the arithmetic based an this system was known by the name of algoritmus, algorithm, algorism . This barbarous word is nothing more than a transcriptionof Al-Khwarizmi, as was conjectured by See also:

• REINAUD, JOSEPH TOUSSAINT (1795-1867)

Reinaud, and has become plain since the publication of a unique See also:

• CAMBRIDGE
• CAMBRIDGE, EARLS AND DUKES OF
• CAMBRIDGE, RICHARD OWEN (1717-1802)

Cambridge MS. containing a Latin See also:

• TRANSLATION

translation—perhaps by Adelhard of See also:

• BATH
• BATH, THOMAS THYNNE
• BATH, WILLIAM

Bath-of the lost arithmetical See also:

• TREATISE

treatise of the Arabian mathematician.' The arithmetical methods of Khwarizmi were simplified by later Eastern writers, and these simpler methods were introduced to Europe by Leonardo of See also:

• PISA
• PISA, COUNCIL OF (1409)

Pisa in the See also:

• WEST
• WEST, BENJAMIN (1738-1820)
• WEST, NICHOLAS (1461-1533)

West and See also:

• MAXIMUS
• MAXIMUS, ST (c. 58o-662)

Maximus See also:

• PLANUDES, MAXIMUS (c. 1260-1330)

Planudes in the East . The.See also:

• TERM

term zero appears to come from the Arabic sifr through the form •zephyro used by Leonardo . Thus farmodern inquirersare agreed . The disputed points are—(I) the origin and See also:

• AGE (Fr. age, through late Lat. aetaticum, from aetas)

age of the Indian system, and (2) whether or not a less developed Indian system, without the zero' but with the nine other ciphers used on an abacus, entered Europe' before the rise of See also:

• ISLAM

Islam, and prepared the. way for a complete decimal flotation . — * h Ghobar8 Boetius5 .' T V ? L _ 8 r . The use of nuiiietals in See also:

• INDIA
• INDIA, FRENCH

India can be followed See also:

• BECK (or BEEK), DAVID (1621—1656)
• BECK, CHRISTIAN DANIEL (1757-1832)
• BECK, JAKOB SIGISMUND (1761-1840)

beck to the Nana See also:

• GHAT, or RHAT

Ghat inscription's, supposed to date from the early part of the 3rd century B.C . These are signs for units, tens and hundreds, as 3 Published by Boncompagni in Trattati d' aritmetica (See also:

• ROME
• ROME (Roma)

Rome, 1857) .

4 From See also:

• SIR

Sir E . C . Bayley's paper in J.RL4 Si (18'$2): 6 From See also:

• BURNELL, ARTHUR COKE (1840–1882)
• BURNELL, ROBERT (d. 1292)

Burnell's .. ozyth Indian •Palaeogrophy(1874) . 6 Of the loth century . (From Burnell, op. See also:

• CAT
• CAT, CIVET
• CAT, HOUSE

cat) ' Of "the loth century; from 'a MS. written at See also:

• SHIRAZ

Shiraz . (From Woepcke Memoire See also:

• SUE, EUGENE [JosEPH MARIE] (1804—1857)

sue la See also:

• PROPAGATION

propagation See also:

• DES

des ehiff,e§ indieni.) "From a MS. at Paris . . (From Woepcke, 4. tit.) 9 See also:

• ERLANGEN

Erlangen (See also:

• ALTDORF

Altdorf)MS . (From Woepcke, op. cit.) Nana Ghat (Indian See also:

• CAVE (Lat. cavea, from caves, hollow)
• CAVE, EDWARD (1691–1754)
• CAVE, WILLIAM (1637–1713)

Cave Inscriptions (Indian) 5 See also:

• DEVA (mod. Chester)
• DEVA (Sanskrit " heavenly ")

Deva a'garis Eastern Arabic' . 1 ` in the other old systems we have dealt with . Like the Indian alphabet, they are probably derived from abroad, but, as in the case of the alphabet, their origin is obscure . The forms of the later Indian numerals for the nine digits appear to be clearly derived from the earlier system . In table II. the first two lines give forms earlier than the introduction of the system of position, while the Devanagari in the third line was used with a zero and position value .

The " cave " numerals were employed during the first centuries of the See also:

• CHRISTIAN
• CHRISTIAN, WILLIAM (1608-1663)

Christian era . The earliest known example of a date written on the modern system is of A.D . 738, while the old system is found in use as See also:

• LATE

late as the early part of the 7th century (Bayley) . On the other hand, there is some See also:

• EVIDENCE (Lat. evidentia, evideri, to appear clearly)

evidence that a system of value by position was known to See also:

• SANSKRIT

Sanskrit writers on arithmetic in the 6th Christian century . These writers, however, do not use ciphers, but symbolical words and letters, so that it is not quite clear whether they refer to a system which had a zero, or to a system worked on an abacus, where the zero is represented by a See also:

• BLANK (from the Fr. blanc, white)

blank column . There is no See also:

• PROOF (in M. Eng. preove, proeve, preve, &°c., from O. Fr . prueve, proeve, &c., mod. preuve, Late. Lat. proba, probate, to prove, to test the goodness of anything, probus, good)

proof as yet for the use of any system of position in India before tip 6th century, and nothing beyond conjecture can be offered as to its origin . 2 . In Europe, before the introduction of the algorithm or full Indo-Arabic system with the zero, we find a transition system in which calculations were made on the decimal system with an abacus, but instead of unit counters there were placed in the columns ciphers, with values from one to nine, and of forms that are at bottom the Indian forms and agree most nearly with the numerals used by the Arabs of See also:

• AFRICA
• AFRICA, ROMAN

Africa and See also:

• SPAIN
• SPAIN (Espana)

Spain . For among the Arabs themselves there were varieties in the forms of the Indian numeral, and in particular an eastern and a western type . The latter is called ghobar (dust), a name which seems to connect it with the use of a sand-spread tablet for calculation . The abacus with ciphers instead of counters was used at Rheims about 970–980 by See also:

• GERBERT, MARTIN (172o-1793)

Gerbert, who after-wards was See also:

• POPE (Gr. irarraas, post-classical Lat. papa, father)
• POPE, ALEXANDER (1688–1744)
• POPE, ALEXANDER (1763-1835)
• POPE, JANE (1742-1818)
• POPE, JOHN (1822-1892)
• POPE, SIR THOMAS (c. 1507-1559)

pope under the See also:

• TITLE (0. Fr. title, mod. titre, from Lat. titulus)

title of See also:

• SYLVESTER, JAMES
• SYLVESTER, JOSHUA (1563–1618)

Sylvester II., and it became well known in the i 1th century . Where did Gerbert learn the use of the abacus with ciphers ?

There is no See also:

• DIRECT

direct evidence as to this, for the See also:

• STORY, JOHN (c. 1510-1571)
• STORY, JOSEPH (1779-1845)
• STORY, ROBERT HERBERT (1835-1907)
• STORY, WILLIAM WETMORE (1819—1895)

story in See also:

• WILLIAM
• WILLIAM (1143-1214)
• WILLIAM (1227-1256)
• WILLIAM (1J33-1584)
• WILLIAM (A.S. Wilhelm, O. Norse Vilhidlmr; O. H. Ger. Willahelm, Willahalm, M. H. Ger. Willehelm, Willehalm, Mod.Ger. Wilhelm; Du. Willem; O. Fr. Villalme, Mod. Fr. Guillaume; from " will," Goth. vilja, and " helm," Goth. hilms, Old Norse hidlmr, meaning
• WILLIAM (c. 1130-C. 1190)
• WILLIAM, 13TH

William of See also:

• MALMESBURY
• MALMESBURY, JAMES HARRIS, IST EARL OF (1746-1820)
• MALMESBURY, JAMES HOWARD HARRIS, 3RD EARL OF (1807-1889)

Malmesbury, that he See also:

• STOLE (Lat. stela and orarium, Fr. etole, It. stola, Sp. estola, Ger. Stela)

stole it from an Arab in Spain, is generally given up as fabulous . On the other hand, no evidence is offered for an earlier use of the abacus with ciphers, except a passage describing the system in the Geometria ascribed to See also:

• BOETIUS (or BoETxlus), ANICIUS MANLIUS SEVERINUS (c. A.D. 480–524)

Boetius . If this book is genuine the Indian numerals were known in Europe and applied to the abacus in the 5th century, and Gerbert only revived the long-forgotten system . On this view we have to explain how Boetius got the ciphers . The Geometria ascribes the system to the " Pythagorici "—i.e. the Neo-Pythagoreans—and it has been thought possible that the Indian forms for the numerals reached See also:

• ALEXANDRIA
• ALEXANDRIA (Arab. Iskenderia)

Alexandria, along with the cruder form of value by position involved in the use of the abacus without a zero, before direct communication ,etween Europe and India ceased, which it did about the 4th century It is then further conjectured by Woepcke that the ghobar numerals of the western Arabs were by them borrowed from the system of Boetius before the full Indian method with the zero reached them; and thus the resemblance between these forms and those in See also:

• MSS

MSS. of Boetius, which are essentially the same as in other MSS. of the 11th century, would be explained . This view, however, presents See also:

• GREAT

great difficulties, of which the See also:

• TOTAL

total disappearance of all trace of the system between Boetius and Gerbert is only one . We have no proof that the Indians ever used such an abacus, or that they had value by position at so early a date as is required, and the ghobar numerals are too similar to those of the eastern Arabs to make it very credible that the two systems had been separated for centuries . The genuineness of the Geometria is maintained by See also:

• MORITZ, KARL PHILIPP (1757-1793)

Moritz Cantor, but it has been attacked on other grounds than that of the passage about the abacus; and on the whole it is still an open question whether the abacus with ciphers is not the outcome of an early imperfect knowledge of the Arabic system, Gerbert or some other having got the signs and a See also:

• GENERAL
• GENERAL (Lat. generalis, of or relating to a genus, kind or class)

general idea of value by position without having an explanation of the zero . See NI . Cantor, Geschichte der Mathematik, vol. i . (See also:

• LEIPZIG

Leipzig, 188o) ; also NI . See also:

• CHASLES, VICTOR EUPHEMIEN PHILARETE (1798-1873)

Chasles, papers in the Comptes rendus (1843) ; G .

Friedlein, Die Zahlzeichen and das elementare Rechnen der Griechen and Romer, &c . (1869); F . Woepcke, Sur l'introduction de l'arithmetique indien en occident (Rome, 1859), and Memoire sur la propagation des chiffres indiens (Paris, 1863) . For the See also:

• PALAEOGRAPHY (Gr. aaXalbs, ancient, and ypal6ew, to write)

palaeography of the Indian numerals see Burnell, Elements of S . Indian Palaeography (1874) ; and Sir E . C . Bayley in J.R.A.S . (1882, 1883) . For Boetius compare Friedlein's edition of his arithmetic and See also:

• GEOMETRY

geometry (Leipzig, 1857), and Weissenhorn in Zeitsch . Math . Plays. See also:

• XXIV

xxiv . Other references to the copious literature will be found in Cantor and Friedlein, who also discuss the subject of the notation for fractions, which cannot be entered on here .

For systems passed over here, see Pihan, Expose des signes de numeration usites chez See also:

• LES

les peuples orienlaux (Paris, I86o) . (W . R .