f ! ,cornell ,univ]sity ,libr>y3 ,hi/orical ,ma!matics ,monographs collection47 ,transcrib]'s ,note3 ,! foll[+ codes are us$ = *>act]s t are n pres5t 9 ! "* set us$ = ?is v]sion ( ! book4 a a ) macron 7etc47 4g g ) dot above 7etc47 's s ) acute acc5t d4 d ) dot bel[ 7etc47 d d ) l9e bel[ ,h7 ,h ) breve bel[ ,,! ,,h9du-,,arabic ,,num]als ,,by ,,david ,,eug5e ,,smi? ,,& ,,l\is ,,*>les ,,k>p9ski ,,bo/on ,,& ,,london ,,g9n ,,& ,,company1 ,,publi%]s #aiaa ,,copyright1 #aiaa1 ,,by ,,david ,,eug5e ,,smi? ,,& ,,l\is ,,*>les ,,k>p9ski ,,all ,,rights ,,res]v$ #haa4g ,,! ,,a!naeum ,,press ,,g9n ,,& ,,company 4 ,,proprietors ,,bo/on 4 ,u4,s4,a4 99 99 99 99 99 iii ,,preface ,s famili> are we ) ! num]als t be> ! mislead+ "n ( ,arabic1 & s ext5sive is _! use 9 ,europe & ! ,am]icas1 t x is difficult = u to realize t _! g5]al acceptance 9 ! transactions ( comm]ce is a matt] ( only ! la/ f\r c5turies1 & t !y are unkn[n to a v l>ge "p ( ! human race to-day4 ,x seems /range t su* a labor-sav+ device %\ld h /ruggl$ = ne>ly a ?\s& ye>s aft] its sy/em ( place value 0 p]fect$ be=e x replac$ su* crude notations z ! "o t ! ,roman conqu]or made sub/antially univ]sal 9 ,europe4 ,su*1 h[ev]1 is ! case1 & !re is probably no "o :o has n at lea/ "s slight pass+ int]e/ 9 ! /ory ( ?is /ruggle4 ,to ! ma!matician & ! /ud5t ( civilization ! int]e/ is g5]ally a deep "o2 to ! tea*] ( ! elem5ts ( k ! int]e/ may be less m>k$1 b nev]!less x is real2 & ev5 ! bus9ess man :o makes daily use ( ! curi\s symbols by :i* we express ! numb]s ( comm]ce1 _c fail to h "s appreciation = ! /ory ( ! rise & progress ( ^! tools ( 8 trade4 ,?is /ory has (t5 be5 told 9 "p1 b x is a long "t s9ce any ef=t has be5 made to br+ toge!r ! fragm5t>y n>rations & to set =? ! g5]al problem ( ! orig9 & developm5t ( ^! iv num]als4 ,9 ?is little "w we h attempt$ to /ate ! hi/ory ( ^! =ms 9 small compass1 to place be=e ! /ud5t mat]ials = ! inve/igation ( ! problems involv$1 & to express z cle>ly z possible ! results ( ! labors ( s*ol>s :o h /udi$ ! subject 9 diff]5t p>ts ( ! _w4 ,we h _h no !ory to exploit1 = ! hi/ory ( ma!matics has se5 too mu* ( ?is t5d5cy already1 b z f> z possible we h weigh$ ! te/imony & h set =? :at seem to be ! reasonable conclusions f ! evid5ce at h&4 ,to facilitate ! "w ( /ud5ts an index has be5 prep>$ :i* we hope may be s]viceable4 ,9 ?is ! names ( au?ors appe> only :5 "s use has be5 made ( _! op9ions or :5 _! works are fir/ m5tion$ 9 full 9 a footnote4 ,if ?is "w %all %[ m cle>ly ! value ( our numb] sy/em1 & %all make ! /udy ( ma!matics seem m real to ! tea*] & /ud5t1 & %all (f] mat]ial = int]e/+ "s pupil m fully 9 8 "w ) numb]s1 ! au?ors w feel t ! consid]able labor involv$ 9 its prep>ation has n be5 9 va94 ,we desire to ackn[l$ge our especial indebt$ness to ,pr(essor ,alex&] ,ziwet = read+ all ! pro(1 z well z = ! dige/ ( a ,russian "w1 to ,pr(essor ,cl>5ce ,l4 ,mead] = ,sanskrit translit]ations1 & to ,mr4 ,/ev5 ,t4 ,by+ton = ,arabic translit]ations & ! s*eme ( pronunciation ( ,ori5tal names1 & also our indebt$ness to o!r s*ol>s 9 ,ori5tal le>n+ = in=mation4 ,,david ,,eug5e ,,smi? ,,l\is ,,*>les ,,k>p9ski 99 99 99 99 99 v ,,cont5ts ,,*apt] ,,pronunciation ,,( ,,ori5tal ,,names vi ,i4 ,,e>ly ,,ideas ,,( ,,_! ,,orig9 #a ,,ii4 ,,e>ly ,,h9du ,,=ms ,,) ,,no ,,place ,,value #ab ,,iii4 ,,lat] ,,h9du ,,=ms1 ,,) ,a ,,place ,,value #ch ,,iv4 ,,! ,,symbol ,,z]o #ea ,v4 ,,! ,,"q ,,( ,,! ,,introduction ,,( ,,! ,,num]als ,,into ,,europe ,,by ,,boe?ius #fc ,,vi4 ,,! ,,developm5t ,,( ,,! ,,num]als ,,among ,,! ,,arabs #ia ,,vii4 ,,! ,,def9ite ,,introduction ,,( ,,! ,,num]als ,,into ,,europe #ii ,,viii4 ,,! ,,spread ,,( ,,! ,,num]als ,,9 ,,europe #abh ,,index #aec 99 99 99 99 99 vi ,,pronunciation ,,( ,,ori5tal ,,names 7,s7 9 ,sanskrit names & words2 7,a7 9 ,arabic names & words4 ,b1 ,d1 ,f1 ,g1 ,h1 ,j1 ,l1 ,m1 ,n1 ,p1 ,,% 7,a71 ,t1 ,,? 7,a71 ,v1 ,w1 ,x1 ,z1 z 9 ,engli%4 ,a1 7,s7 l u 9 b3 ?us p&it1 pron\nc$ pundit4 7,a7 l a 9 ask or 9 man4 ,a1 z 9 fa!r4 ,c1 7,s7 l * 9 *ur* 7,italian c 9 c5to74 ,d41 ,n41 ,s41 ,t41 7,s7 d1 n1 %1 t1 made ) ! tip ( ! tongue turn$ up & back into ! dome ( ! palate4 ,d41 ,s41 ,t41 ,z41 7,a7 d1 s1 t1 z1 made ) ! tongue spread s t ! s\nds are produc$ l>gely aga9/ ! side tee?4 ,europeans commonly pron\nce ,d41 ,n41 ,s41 ,t41 ,z41 bo? 7,s7 & 7,a71 z simple d1 n1 % 7,s7 or s 7,a71 t1 z4 ,d 7,a71 l ? 9 ?is4 ,e1 7,s7 z 9 !y4 7,a7 z 9 b$4 4,g1 7,a7 a voic$ consonant =m$ bel[ ! vocal cords2 its s\nd is comp>$ by "s to a g1 by o!rs to a guttural r2 9 ,arabic words adopt$ into ,engli% x is repres5t$ by gh 7e4g4 gh\l71 less (t5 r 7e4g4 razzia74 ,h prec$$ by b1 c1 t1 t41 etc4 does n =m a s+le s\nd ) ^! lett]s1 b is a m or less di/9ct h s\nd foll[+ !m2 cf4 ! s\nds 9 abhor1 boa?ook1 etc41 or1 m accurately = 7,s71 ! bhoys etc4 ( ,iri% brogue4 ,h 7,a7 reta9s its consonant s\nd at ! end ( a ^w4 ,h41 7,a7 an unvoic$ consonant =m$ bel[ ! vocal cords2 its s\nd is sometimes comp>$ to ,g]man h>d *1 & may be repres5t$ by an h z /rong z possible4 ,9 ,arabic words adopt$ into ,engli% x is repres5t$ by h1 e4g4 9 sahib1 hakeem4 ,h4 7,s7 is f9al consonant h1 l f9al h 7,a74 ,i1 z 9 p94 ,i1 z 9 pique4 ,k1 z 9 kick4 ,,kh1 7,a7 ! h>d * ( ,scot* lo*1 ,g]man a*1 especially ( ,g]man z pron\nc$ by ! ,swiss4 4,m1 4,n1 7,s7 l ,fr5* f9al m or n1 nasaliz+ ! prec$+ v[el4 ,n41 see ,d44 ,n1 l ng 9 s++4 ,o1 7,s7 z 9 s4 7,a7 z 9 obey4 ,q1 7,a7 l k 7or c7 9 cook2 fur!r back 9 ! m\? ?an 9 kick4 ,r1 7,s7 ,engli% r1 smoo? & untrill$4 7,a7 /rong]4 ,r41 7,s7 r us$ z v[el1 z 9 apron :5 pron\nc$ aprn & n ap]n2 mod]n ,h9dus say ri1 h5ce our amrita1 ,kri%na1 = a-mr4ta1 ,kr4s4n4a4 ,s1 z 9 same4 ,s41 see ,d44 ',s1 7,s7 ,engli% % 7,g]man s*74 ,t41 see ,d44 ,u1 z 9 put4 ,u1 z 9 rule4 ,y1 z 9 y4 ,z41 see ,d44 1 7,a7 a s\nd k9dr$ to ! spiritus l5is 7t is1 to our e>s1 ! m]e di/9ct sep>ation ( a v[el f ! prec$+ s\nd1 z at ! beg9n+ ( a ^w 9 ,g]man7 & to h44 ,! is a v di/9ct s\nd 9 ,arabic1 b is m ne>ly repres5t$ by ! spiritus l5is ?an by any s\nd t we c produce )\t mu* special tra9+4 ,t is1 x %\ld be treat$ z sil5t1 b ! s\nds t prec$e & foll[ x %\ld n run toge!r4 ,9 ,arabic words adopt$ into ,engli% x is treat$ z sil5t1 e4g4 9 ,arab1 amb]1 ,caaba 7,arab1 anb>1 kaabah74 7,a7 ,a f9al long v[el is %ort5$ be=e al 7'l7 or ibn 7^: i is !n sil5t74 ,acc5t3 7,s7 z if ,lat92 9 det]m9+ ! place ( ! acc5t 4m & 4n c\nt z consonants1 b h aft] ano!r consonant does n4 7,a71 on ! la/ syllable t conta9s a long v[el or a v[el foll[$ by two consonants1 except t a f9al long v[el is n ord9>ily acc5t$2 if !re is no long v[el nor two consecutive consonants1 ! acc5t falls on ! fir/ syllable4 ,! words al & ibn are nev] acc5t$4 99 99 99 99 99 #a ,,! ,,h9du-,,arabic ,,num]als ,,*apt] ,i ,,e>ly ,,ideas ,,( ,,_! ,,orig9 ,x has long be5 recogniz$ t ! common num]als us$ 9 daily life are ( comp>atively rec5t orig94 ,! numb] ( sy/ems ( notation employ$ be=e ! ,*ri/ian era 0 ab\t ! same z ! numb] ( writt5 languages1 & 9 "s cases a s+le language _h sev]al sy/ems4 ,! ,egyptians1 = example1 _h ?ree sy/ems ( writ+1 ) a num]ical notation = ea*2 ! ,greeks _h two well-def9$ sets ( num]als1 & ! ,roman symbols = numb] *ang$ m or less f c5tury to c5tury4 ,ev5 to-day ! numb] ( me?ods ( express+ num]ical concepts is mu* great] ?an "o w\ld believe be=e mak+ a /udy ( ! subject1 = ! idea t our common num]als are univ]sal is f> f be+ correct4 ,x w be well1 !n1 to ?9k ( ! num]als t we /ill commonly call ,arabic1 z only "o ( _m sy/ems 9 use j be=e ! ,*ri/ian era4 ,z x !n exi/$ ! sy/em 0 no bett] ?an _m o!rs1 x 0 ( late orig91 x conta9$ no z]o1 x 0 cumb]some & little us$1 #b & x _h no p>ticul> promise4 ,n until c5turies lat] did ! sy/em h any /&+ 9 ! _w ( bus9ess & sci5ce2 & _h ! place value :i* n[ *>act]izes x1 & :i* requires a z]o1 be5 work$ out 9 ,greece1 we might h be5 us+ ,greek num]als to-day in/ead ( ! ones ) :i* we are famili>4 ,( ! fir/ numb] =ms t ! _w us$ ?is is n ! place to speak4 ,_m ( !m are int]e/+1 b none _h mu* sci5tific value4 ,9 ,europe ! inv5tion ( notation 0 g5]ally assign$ to ! ea/]n %ores ( ! ,m$it]ranean until ! critical p]iod ( ab\t a c5tury ago1--sometimes to ! ,hebrews1 sometimes to ! ,egyptians1 b m (t5 to ! e>ly trad+ ,pho5icians4#a ,! idea t our common num]als are ,arabic 9 orig9 is n an old "o4 ,! m$iaeval & ,r5aissance writ]s g5]ally recogniz$ !m z ,indian1 & _m ( !m expressly /at$ t !y 7 ( ,h9du orig94#b #c ,o!rs argu$ t !y 7 probably inv5t$ by ! ,*aldeans or ! ,jews because !y increas$ 9 value f "r to left1 an argum5t t w\ld apply q z well to ! ,roman & ,greek sy/ems1 or to any o!r4 ,x 01 inde$1 to ! g5]al idea ( notation t _m ( ^! writ]s ref]r$1 z is evid5t f ! words ( ,engl&'s e>lie/ ari?metical textbook-mak]1 ,rob]t ,recorde 7c4 #aedb73 ,9 t ?+e all m5 d agree1 t ! ,*aldays1 :i*e fyr/e inu5t$ ?ys arte1 did set ^! figures z !i set all _! lett]s4 = !y wryte backw>de z y te>me x1 & s doo !y reade4 ,& t may appe>e 9 all ,hebrewe1 ,*aldaye & ,arabike bookes 444 :]e z ! ,greekes1 ,lat9es1 & all nations ( ,europe1 d wryte & reade f ! lefte h& t[>de ! ryghte4#c ,o!rs1 & #d among !m su* influ5tial writ]s z ,t>taglia#d 9 ,italy & ,koebel#e 9 ,g]many1 ass]t$ ! ,arabic orig9 ( ! num]als1 :ile /ill o!rs left ! matt] undecid$#f or simply dismiss$ !m z b>b>ic4#g ,( c\rse ! ,arabs !mselves nev] laid claim to ! inv5tion1 always recogniz+ _! indebt$ness to ! ,h9dus bo? = ! num]al =ms & = ! di/+ui%+ feature ( place value4 ,=emo/ among ^! writ]s 0 ! great ma/] ( ! gold5 age ( ,bagdad1 "o ( ! fir/ ( ! ,arab writ]s to collect ! ma!matical classics ( bo? ! ,ea/ & ! ,we/1 pres]v+ !m & f9ally pass+ !m on to awak5+ ,europe4 ,?is man 0 ,moh4amm$ ! ,son ( ,moses1 f ,kh[arezm1 or1 m aft] ! mann] ( ! ,arab1 ,moh4amm$ ibn ,musa al-,kh[arazmi1#h a man ( great #e le>n+ & "o to :om ! _w is mu* indebt$ = its pres5t k ( algebra#i & ( ari?metic4 ,( him !re w (t5 be occasion to speak2 & 9 ! ari?metic :i* he wrote1 & ( :i* ,adelh>d ( ,ba?#aj 7c4 #aacj7 may h made ! translation or p>aphrase1#aa he /at$ di/9ctly t ! num]als 7 due to ! ,h9dus4#ab ,?is is z pla9ly ass]t$ by lat] ,arab #f writ]s1 ev5 to ! pres5t day4#ac ,inde$ ! phrase ilm h9di1 ,indian sci5ce1 is us$ by !m = ari?metic1 z also ! adjective h9di alone4#ad ,probably ! mo/ /rik+ te/imony f ,arabic s\rces is t giv5 by ! ,arabic travel] & s*ol> ,mohamm$ ibn ,ah4m$1 ,abu 'l-,rih4an al-,biruni 7#igc-ajdh71 :o sp5t _m ye>s 9 ,h9du/an4 ,he wrote a l>ge "w on ,india1#ae "o on anci5t *ronology1#af ! ,book ( ! ,ciph]s1 un=tunately lo/1 :i* treat$ d\btless ( ! ,h9du art ( calculat+1 & 0 ! au?or ( num]\s o!r works4 ,al-,biruni 0 a man ( unusual atta9m5ts1 be+ v]s$ 9 ,arabic1 ,p]sian1 ,sanskrit1 ,hebrew1 & ,syriac1 z well z 9 a/ronomy1 *ronology1 & ma!matics4 ,9 8 "w on ,india he gives detail$ in=mation conc]n+ ! language & #g cu/oms ( ! p ( t c\ntry1 & /ates explicitly#ag t ! ,h9dus ( 8 "t did n use ! lett]s ( _! alphabet = num]ical notation1 z ! ,arabs did4 ,he also /ates t ! num]al signs call$ a4nka#ah _h diff]5t %apes 9 v>i\s p>ts ( ,india1 z 0 ! case ) ! lett]s4 ,9 8 ,*ronology ( ,anci5t ,nations he gives ! sum ( a geometric progression & %[s h[1 9 ord] to avoid any possibility ( error1 ! numb] may be express$ 9 ?ree diff]5t sy/ems3 ) ,indian symbols1 9 sexagesimal notation1 & by an alphabet sy/em :i* w be t\*$ ^u lat]4 ,he also speaks#ai ( #agi1 #hgf1 #gee1 express$ 9 ,indian ciph]s1 ?us aga9 attribut+ ^! =ms to ,h9du s\rces4 ,prec$+ ,al-,biruni !re 0 ano!r ,arabic writ] ( ! t5? c5tury1 ,mot4ahh> ibn ,t4ahir1#bj au?or ( ! ,book ( ! ,creation & ( ,hi/ory1 :o gave z a curiosity1 9 ,indian 7,nag>i7 symbols1 a l>ge numb] ass]t$ by ! p ( ,india to repres5t ! duration ( ! _w4 ,hu>t feels positive t 9 ,mot4ahh>'s "t ! pres5t ,arabic symbols _h n yet come into use1 & t ! ,indian symbols1 al?\gh kn[n to s*ol>s1 7 n curr5t4 ,unless ?is 7 ! case1 nei!r ! au?or nor 8 read]s w\ld h f\nd any?+ extraord9>y 9 ! appe>ance ( ! numb] :i* he cites4 ,m5tion %\ld also be made ( a widely-travel$ /ud5t1 ,al-,masudi 7#hhe8-ief71 ^: j\rneys c>ri$ him f ,bagdad to ,p]sia1 ,india1 ,ceylon1 & ev5 #h across ! ,*9a sea1 & at o!r times to ,madagasc>1 ,syria1 & ,pale/9e4#ba ,he seems to h neglect$ no accessible s\rces ( in=mation1 exam9+ also ! hi/ory ( ! ,p]sians1 ! ,h9dus1 & ! ,romans4 ,t\*+ ! p]iod ( ! ,caliphs 8 "w entitl$ ,mead[s ( ,gold furni%es a mo/ ent]ta9+ fund ( in=mation4 ,he /ates#bb t ! wise m5 ( ,india1 assembl$ by ! k+1 compos$ ! ,s9dh9d4 ,fur!r on#bc he /ates1 ^u ! au?ority ( ! hi/orian ,moh4amm$ ibn ,ali ,abdi1 t by ord] ( ,al-,mans4ur _m works ( sci5ce & a/rology 7 translat$ into ,arabic1 notably ! ,s9dh9d 7,siddhanta74 ,conc]n+ ! mean+ & spell+ ( ?is "n !re is consid]able div]sity ( op9ion4 ,colebrooke#bd fir/ po9t$ out ! connection betwe5 ,siddhanta & ,s9dh9d4 ,he ascribes to ! ^w ! mean+ ! revolv+ ages4#be ,simil> designations are collect$ by ,s$illot1#bf :o incl9$ to ! ,greek orig9 ( ! sci5ces commonly attribut$ to ! ,h9dus4#bg ,casiri1#bh cit+ ! ,tarikh al-h4okama or ,*ronicles ( ! ,le>n$1#bi ref]s to ! "w #i z ! ,s9dum-,indum ) ! mean+ p]petuum aet]numque4 ,! ref]5ce#cj 9 ?is anci5t ,arabic "w to ,al-,kh[arazmi is wor?y ( note4 ,?is ,s9dh9d is ! book1 says ,masudi1#ca :i* gives all t ! ,h9dus "k ( ! sph]es1 ! />s1 ari?metic1#cb & ! o!r bran*es ( sci5ce4 ,he m5tions also ,al-,kh[arazmi & ,h4aba%#cc z translators ( ! tables ( ! ,s9dh9d4 ,al-,biruni#cd ref]s to two o!r translations f a "w furni%$ by a ,h9du :o came to ,bagdad z a memb] ( ! political mission :i* ,s9dh s5t to ! caliph ,al-,mans4ur1 9 ! ye> ( ! ,hejira #aed 7,a4,d4 #gga74 ,! olde/ "w1 9 any s5se complete1 on ! hi/ory ( ,arabic lit]ature & hi/ory is ! ,kitab al-,fihri/1 writt5 9 ! ye> #ihg ,a4,d41 by ,ibn ,abi ,yaqub al-,nadim4 ,x is ( fundam5tal importance = ! hi/ory ( ,arabic culture4 ,( ! t5 *ief divisions ( ! "w1 ! sev5? dem&s att5tion 9 ?is discussion = ! reason t its second subdivision treats ( ma!maticians & a/ronom]s4#ce #aj ,! fir/ ( ! ,arabic writ]s m5tion$ is ,al-,k9di 7#hjj-hgj ,a4,d471 :o wrote five books on ari?metic & f\r books on ! use ( ! ,indian me?od ( reckon+4 ,s5$ ibn ,ali1 ! ,jew1 :o 0 conv]t$ to ,islam "u ! caliph ,al-,mamun1 is also giv5 z ! au?or ( a "w on ! ,h9du me?od ( reckon+4 ,nev]!less1 !re is a possibility#cf t "s ( ! works ascrib$ to ,s5$ ibn ,ali are really works ( ,al-,kh[arazmi1 ^: "n imm$iately prec$es 84 ,h[ev]1 x is to be not$ 9 ?is connection t ,casiri#cg also m5tions ! same writ] z ! au?or ( a mo/ celebrat$ "w on ari?metic4 ,to ,al-,s4ufi1 :o di$ 9 #ihf ,a4,d41 is also cr$it$ a l>ge "w on ! same subject1 & simil> treatises by o!r writ]s are m5tion$4 ,we are !re=e =c$ to ! conclusion t ! ,arabs f ! e>ly n9? c5tury on fully recogniz$ ! ,h9du orig9 ( ! new num]als4 ,leon>d ( ,pisa1 ( :om we %all speak at l5g? 9 ! *apt] on ! ,introduction ( ! ,num]als into ,europe1 wrote 8 ,lib] ,abbaci#ch 9 #abjb4 ,9 ?is "w he ref]s frequ5tly to ! n9e ,indian figures1#ci ?us %[+ aga9 ! g5]al cons5sus ( op9ion 9 ! ,middle ,ages t ! num]als 7 ( ,h9du orig94 ,"s int]e/ also atta*es to ! olde/ docum5ts on ari?metic 9 our own language4 ,"o ( ! e>lie/ #aa treatises on algorism is a comm5t>y#dj on a set ( v]ses call$ ! ,c>m5 de ,algorismo1 writt5 by ,alex&] de ,villa ,dei 7,alex&ra de ,ville-,dieu71 a ,m9orite monk ( ab\t #abdj ,a4,d4 ,! text ( ! fir/ few l9es is z foll[s3 ,hec algorism' ars p's5s dicit' 9 qua ,talib2 indorum fruim bis qu9q2 figuris4#da ,?is boke is call$ ! boke ( algorim or augrym aft] lewd] use4 ,& ?is boke tretys ( ! ,craft ( ,nombryng1 ! quy* crafte is call$ also ,algorym4 ,!r 0 a kyng ( ,inde ! qui* hey? ,algor he made ?is craft4444 ,algorisms1 9 ! quy* we use te5 figurys ( ,inde4 99 99 99 99 99 #ab ,,*apt] ,,ii ,,e>ly ,,h9du ,,=ms ,,) ,,no ,,place ,,value ,:ile x is g5]ally conc$$ t ! sci5tific developm5t ( a/ronomy among ! ,h9dus t[>ds ! beg9n+ ( ! ,*ri/ian era re/$ ^u ,greek#db or ,*9ese#dc s\rces1 yet _! anci5t lit]ature te/ifies to a high /ate ( civilization1 & to a consid]able advance 9 sci5ces1 9 philosophy1 & along lit]>y l9es1 long be=e ! gold5 age ( ,greece4 ,f ! e>lie/ times ev5 up to ! pres5t day ! ,h9du has be5 wont to put 8 ?\ght into rhy?mic =m4 ,! fir/ ( ?is poetry--x well des]ves ?is "n1 be+ also wor?y f a metaphysical po9t ( view#dd--;consi/s ( ! ,v$as1 hymns ( praise & poems ( wor%ip1 collect$ dur+ ! ,v$ic p]iod :i* dates f approximately #bjjj ,b4,c4 to #adjj ,b4,c4#de ,foll[+ ?is "w1 or possibly contempor>y ) x1 is ! ,brahmanic lit]ature1 :i* is p>tly rituali/ic 7! ,brahman4z71 & p>tly philosophical 7! ,upani%ads74 ,our especial int]e/ is #ac 9 ! ,sutras1 v]sifi$ abridgm5ts ( ! ritual & ( c]emonial rules1 :i* conta9 consid]able geometric mat]ial us$ 9 connection ) alt> con/ruction1 & also num]\s examples ( rational numb]s ! sum ( ^: squ>es is also a squ>e1 i4e4 ,py?agorean numb]s1 al?\gh ?is 0 long be=e ,py?agoras liv$4 ,:itney#df places ! :ole ( ! ,v$a lit]ature1 includ+ ! ,v$as1 ! ,brahman4z1 & ! ,sutras1 betwe5 #aejj ,b4,c4 & #hjj ,b4,c41 ?us agree+ ) ,bu]k#dg :o holds t ! k ( ! ,py?agorean !orem reveal$ 9 ! ,sutras goes back to ! eigh? c5tury ,b4,c4 ,! importance ( ! ,sutras z %[+ an indep5d5t orig9 ( ,h9du geometry1 contr>y to ! op9ion long held by ,cantor#dh ( a ,greek orig91 has be5 repeat$ly emphasiz$ 9 rec5t lit]ature1#di especially s9ce ! appe>ance ( ! important "w ( ,von ,s*ro$]4#ej ,fur!r fundam5tal ma!matical notions su* z ! conception ( irrationals & ! use ( gnomons1 z well z ! philosophical doctr9e ( ! transmigration ( s\ls1--all ( ^! hav+ long be5 attribut$ to ! ,greeks1--are %[n 9 ^! works to be native to ,india4 ,al?\gh ?is discussion does n be> directly ^u ! #ad orig9 ( our num]als1 yet x is highly p]t95t z %[+ ! aptitude ( ! ,h9du = ma!matical & m5tal "w1 a fact fur!r atte/$ by ! indep5d5t developm5t ( ! drama & ( epic & lyric poetry4 ,x %\ld be /at$ def9itely at ! outset1 h[ev]1 t we are n at all sure t ! mo/ anci5t =ms ( ! num]als commonly kn[n z ,arabic _h _! orig9 9 ,india4 ,z w pres5tly be se51 _! =ms may h be5 sugge/$ by ^? us$ 9 ,egypt1 or 9 ,ea/]n ,p]sia1 or 9 ,*9a1 or on ! pla9s ( ,mesopotamia4 ,we are q 9 ! d>k z to ^! e>ly /eps2 b z to _! developm5t 9 ,india1 ! approximate p]iod ( ! rise ( _! ess5tial feature ( place value1 _! introduction into ! ,arab civilization1 & _! spread to ! ,we/1 we h m or less def9ite in=mation4 ,:51 !re=e1 we consid] ! rise ( ! num]als 9 ! l& ( ! ,s9dhu1#ea x mu/ be und]/ood t x is only ! l>ge movem5t t is meant1 & t !re mu/ fur!r be consid]$ ! num]\s possible s\rces outside ( ,india itself & long ant]ior to ! fir/ prom95t appe>ance ( ! numb] symbols4 ,no "o attempts to exam9e any detail 9 ! hi/ory ( anci5t ,india )\t be+ /ruck ) ! great de>? ( reliable mat]ial4#eb ,s little sympa?y h ! p ) any save ^? ( _! own ca/e t a g5]al lit]ature is :olly lack+1 & x is only 9 ! obs]vations ( /rang]s t any all-r\nd view ( sci5tific progress is to be f\nd4 ,!re is evid5ce t prim>y s*ools #ae exi/$ 9 e>lie/ times1 & ( ! sev5ty-two recogniz$ sci5ces writ+ & ari?metic 7 ! mo/ priz$4#ec ,9 ! ,v$ic p]iod1 say f #bjjj to #adjj ,b4,c41 !re 0 ! same att5tion to a/ronomy t 0 f\nd 9 ! e>li] civilizations ( ,babylon1 ,*9a1 & ,egypt1 a fact atte/$ by ! ,v$as !mselves4#ed ,su* advance 9 sci5ce presupposes a fair k ( calculation1 b ( ! mann] ( calculat+ we are q ignorant & probably always %all be4 ,"o ( ! ,buddhi/ sacr$ books1 ! ,lalitavi/>a1 relates t :5 ! ,bodhisattva#ee 0 ( age to m>ry1 ! fa!r ( ,gopa1 8 int5d$ bride1 dem&$ an exam9ation ( ! five hundr$ suitors1 ! subjects includ+ ari?metic1 writ+1 ! lute1 & ar*]y4 ,hav+ vanqui%$ 8 rivals 9 all else1 he is mat*$ aga9/ ,arjuna ! great ari?metician & is ask$ to express numb]s great] ?an #ajj kotis4#ef ,9 reply he gave a s*eme ( numb] names z high z #ajec1 add+ t he c\ld proce$ z f> z #ajdba1eg all ( :i* sugge/s ! sy/em ( ,ar*im$es & ! unsettl$ "q ( ! indebt$ness ( ! ,we/ to ! ,ea/ 9 ! realm ( anci5t ma!matics4#eh ,sir ,edw9 ,arnold1 #af 9 ,! ,light ( ,asia1 does n m5tion ?is "p ( ! conte/1 b he speaks ( ,buddha's tra9+ at ! h&s ( ! le>n$ ,vis4vamitra3 ,& ,viswamitra said1 ',x is en\gh1 ,let u to numb]s4 ,aft] me repeat ,y\r num]ation till we rea* ! lakh1#ei ,"o1 two1 ?ree1 f\r1 to t51 & !n by t5s ,to hundr$s1 ?\s&s4' ,aft] him ! *ild ,nam$ digits1 decads1 c5turies1 nor paus$1 ,! r\nd lakh rea*$1 b s(tly murmur$ on1 ,!n comes ! koti1 nahut1 n9nahut1 ,khamba1 viskhamba1 abab1 attata1 ,to kumuds1 gundhikas1 & utpalas1 ,by pund>ikas into padumas1 ,:i* la/ is h[ y c\nt ! utmo/ gra9s ,( ,ha/agiri gr\nd to f9e/ du/2#fj ,b beyond t a num]ation is1 ,! ,ka?a1 us$ to c\nt ! />s ( night1 ,! ,koti-,ka?a1 = ! ocean drops2 ,+ga1 ! calculus ( circul>s2 ,s>vanik*epa1 by ! :i* y deal ,) all ! s&s ( ,gunga1 till we come ,to ,antah-,kalpas1 :]e ! unit is ,! s&s ( ! t5 crore ,gungas4 ,if "o seeks ,m compreh5sive scale1 ?' ari?mic m\nts ,by ! ,asankya1 :i* is ! tale ,( all ! drops t 9 t5 ?\s& ye>s ,w\ld fall on all ! worlds by daily ra92 ,!nce unto ,maha ,kalpas1 by ! :i* ,! gods compute _! future & _! pa/4' #ag ,!reupon ,vis4vamitra ,acarya#fa expresses 8 approval ( ! task1 & asks to he> ! measure ( ! l9e z f> z yojana1 ! longe/ measure be>+ "n4 ,?is giv51 ,buddha adds3 444 ',& ma/]6 if x please1 ,i %all recite h[ _m sun-motes lie ,f end to end )9 a yojana4' ,!reat1 ) in/ant skill1 ! little pr9ce ,pron\nc$ ! total ( ! atoms true4 ,b ,viswamitra he>d x on 8 face ,pro/rate be=e ! boy2 ',= ?\1' he cri$1 ',art ,tea*] ( ?y tea*]s--?\1 n ,i1 ,art ,guru4' ,x is ne$less to say t ?is is f> f be+ hi/ory4 ,& yet x puts 9 *>m+ rhy?m only :at ! anci5t ,lalitavi/>a relates ( ! numb]-s]ies ( ! ,buddha's "t4 ,:ile x ext5ds beyond all reason1 nev]!less x reveals a condition t w\ld h be5 impossible unless ari?metic _h atta9$ a consid]able degree ( advancem5t4 ,to ?is pre-,*ri/ian p]iod belong also ! ,v$a4ngas1 or limbs = support+ ! ,v$a1 "p ( t great bran* ( ,h9du lit]ature kn[n z ,smr4iti 7recollection71 t :i* 0 to be h&$ d[n by tradition4 ,( ^! ! six? is kn[n z ,jyotis4a 7a/ronomy71 a %ort treatise ( only ?irty-six v]ses1 writt5 n e>li] ?an #cjj ,b4,c41 & af=d+ u "s k ( ! ext5t ( numb] "w 9 t p]iod4#fb ,! ,h9dus #ah also speak ( eighte5 anci5t ,siddhantas or a/ronomical works1 :i*1 ?\gh mo/ly lo/1 confirm ?is evid5ce4#fc ,z to au!ntic hi/ories1 h[ev]1 !re exi/ 9 ,india none relat+ to ! p]iod be=e ! ,mohamm$an era 7#fbb ,a4,d474 ,ab\t all t we "k ( ! e>li] civilization is :at we glean f ! two great epics1 ! ,mahabharata#fd & ! ,ramayana1 f co9s1 & f a few inscriptions4#fe ,x is ) ?is unsatisfactory mat]ial1 !n1 t we h to deal 9 se>*+ = ! e>ly hi/ory ( ! ,h9du-,arabic num]als1 & ! fact t _m unsolv$ problems exi/ & w cont9ue to exi/ is no long] /range :5 we consid] ! conditions4 ,x is r surpris+ t s mu* has be5 discov]$ )9 a c5tury1 ?an t we are s unc]ta9 z to orig9s & dates & ! e>ly spread ( ! sy/em4 ,! probability be+ t writ+ 0 n introduc$ into ,india be=e ! close ( ! f\r? c5tury ,b4,c41 & lit]ature exi/+ only 9 spok5 =m prior to t p]iod1#ff ! numb] "w 0 d\btless t ( all primitive peoples1 palpable1 m]ely a matt] ( plac+ /icks or c[ries or pebbles on ! gr\nd1 ( m>k+ a s&-cov]$ bo>d1 or ( cutt+ not*es or ty+ cords z is /ill done 9 p>ts ( ,s\!rn ,india to-day4#fg #ai ,! e>ly ,h9du num]als#fh may be classifi$ into ?ree great gr\ps1 7#a7 ! ,kh>os4t4hi1 7#b7 ! ,brahmi1 & 7#c7 ! ^w & lett] =ms2 & ^! w be consid]$ 9 ord]4 ,! ,kh>os4t4hi num]als are f\nd 9 inscriptions =m]ly kn[n z ,bactrian1 ,indo-,bactrian1 & ,aryan1 & appe>+ 9 anci5t ,g&hara1 n[ ea/]n ,afghani/an & nor!rn ,punjab4 ,! alphabet ( ! language is f\nd 9 inscriptions dat+ f ! f\r? c5tury ,b4,c4 to ! ?ird c5tury ,a4,d41 & f ! fact t ! words are writt5 f "r to left x is assum$ to be ( ,semitic orig94 ,no num]als1 h[ev]1 h be5 f\nd 9 ! e>lie/ ( ^! inscriptions1 numb]-names probably hav+ be5 writt5 out 9 words z 0 ! cu/om ) _m anci5t peoples4 ,n until ! "t ( ! p[]ful ,k+ ,a'soka1 9 ! ?ird c5tury ,b4,c41 d num]als appe> 9 any inscriptions ?us f> discov]$2 & !n only 9 ! primitive =m ( m>ks1 q z !y w\ld be f\nd 9 ,egypt1 ,greece1 ,rome1 or 9 #bj v>i\s o!r p>ts ( ! _w4 ,^! ,a'soka#fi inscriptions1 "s ?irty 9 all1 are f\nd 9 widely sep>at$ p>ts ( ,india1 (t5 on columns1 & are 9 ! v>i\s v]nacul>s t 7 famili> to ! p4 ,two are 9 ! ,kh>os4t4hi *>act]s1 & ! re/ 9 "s =m ( ,brahmi4 ,9 ! ,kh>os4t4hi inscriptions only f\r num]als h be5 f\nd1 & ^! are m]ely v]tical m>ks = "o1 two1 f\r1 & five1 ?us3 ,9 ! s-call$ ',saka inscriptions1 possibly ( ! fir/ c5tury ,b4,c41 m num]als are f\nd1 & 9 m highly develop$ =m1 ! "r-to-left sy/em appe>+1 toge!r ) evid5ces ( ?ree diff]5t scales ( c\nt+1--f\r1 t51 & tw5ty4 ,! num]als ( ?is p]iod are z foll[s3 ,illu/ration ,!re are sev]al notewor?y po9ts to be obs]v$ 9 /udy+ ?is sy/em4 ,9 ! fir/ place1 x is probably n z e>ly z t %[n 9 ! ,nana ,ghat =ms h]eaft] giv51 al?\gh ! inscriptions !mselves at ,nana ,ghat are lat] ?an ^? ( ! ,a'soka p]iod4 ,! #ba f\r is to ?is sy/em :at ! ,x 0 to ! ,roman1 probably a cancel+ ( ?ree m>ks z a workman does to-day = five1 or a lay+ ( "o /ick across ?ree o!rs4 ,! t5 has nev] be5 satisfactorily expla9$4 ,x is simil> to ! ,a ( ! ,kh>os4t4hi alphabet1 b we h no k z to :y x 0 *os54 ,! tw5ty is evid5tly a ligature ( two t5s1 & ?is 9 turn sugge/$ a k9d ( radix1 s t n9ety 0 probably writt5 9 a way rem9d+ "o ( ! quatre-v+t-dix ( ! ,fr5*4 ,! hundr$ is unexpla9$1 al?\gh x resembles ! lett] ta or tra ( ! ,brahmi alphabet ) #a be=e 7to ! "r (7 x4 ,! two hundr$ is only a v>iant ( ! symbol = hundr$1 ) two v]tical m>ks4#gj ,?is sy/em has _m po9ts ( simil>ity ) ! ,nabatean num]als#ga 9 use 9 ! fir/ c5turies ( ! ,*ri/ian era4 ,! cross is "h us$ = f\r1 & ! ,kh>os4t4hi =m is employ$ = tw5ty4 ,9 addition to ?is !re is a trace ( an analog\s use ( a scale ( tw5ty4 ,:ile ! symbol = #ajj is q diff]5t1 ! me?od ( =m+ ! o!r hundr$s is ! same4 ,! correspond5ce seems to be too m>k$ to be :olly accid5tal4 ,x is n 9 ! ,kh>os4t4hi num]als1 !re=e1 t we c hope to f9d ! orig9 ( ^? us$ by u1 & we turn to ! second ( ! ,indian types1 ! ,brahmi *>act]s4 ,! alphabet attribut$ to ,brahma is ! olde/ ( ! sev]al kn[n 9 ,india1 & 0 us$ f ! e>lie/ hi/oric times4 ,!re are v>i\s !ories ( its orig91 #bb none ( :i* has z yet any wide acceptance1#gb al?\gh ! problem (f]s hope ( solution 9 due "t4 ,! num]als are n z old z ! alphabet1 or at lea/ !y h n z yet be5 f\nd 9 inscriptions e>li] ?an ^? 9 :i* ! edicts ( ,a'soka appe>1 "s ( ^! hav+ be5 incis$ 9 ,brahmi z well z ,kh>os4t4hi4 ,z already /at$1 ! old] writ]s probably wrote ! numb]s 9 words1 z seems to h be5 ! case 9 ! e>lie/ ,pali writ+s ( ,ceylon4#gc ,! foll[+ num]als are1 z f> z kn[n1 ! only ones to appe> 9 ! ,a'soka edicts3#gd ,illu/ration ,^! fragm5ts f ! ?ird c5tury ,b4,c41 crude & unsatisfactory z !y are1 are ! und\bt$ e>ly =ms f :i* our pres5t sy/em develop$4 ,!y next appe> 9 ! second c5tury ,b4,c4 9 "s inscriptions 9 ! cave on ! top ( ! ,nana ,ghat hill1 ab\t sev5ty-five miles f ,poona 9 c5tral ,india4 ,^! inscriptions may be memorials ( ! e>ly ,&hra dyna/y ( s\!rn ,india1 b _! *ief int]e/ lies 9 ! num]als :i* !y conta94 ,! cave 0 made z a re/+-place = travel]s asc5d+ ! hill1 :i* lies on ! road f ,kalyana to ,jun>4 ,x seems to h be5 cut out by a desc5dant #bc ( ,k+ ',satavahana1#ge = inside ! wall opposite ! entrance are repres5tations ( ! memb]s ( 8 family1 mu* defac$1 b ) ! names /ill legible4 ,x w\ld seem t ! excavation 0 made by ord] ( a k+ nam$ ,v$isiri1 & ! inscription conta9s a li/ ( gifts made on ! occasion ( ! p]=mance ( sev]al yagnas or religi\s sacrifices1 & num]als are to be se5 9 no less ?an ?irty places4#gf ,!re is consid]able dispute z to :at num]als are really f\nd 9 ^! inscriptions1 ow+ to ! difficulty ( deciph]+ !m2 b ! foll[+1 :i* h be5 copi$ f a rubb+1 are probably numb] =ms3#gg ,illu/ration ,! inscription itself1 s important z conta9+ ! e>lie/ consid]able ,h9du num]al sy/em connect$ ) our own1 is ( suffici5t int]e/ to w>rant reproduc+ "p ( x 9 facsimile1 z is done on page #bd4 #bd ,illu/ration ,! next v notewor?y evid5ce ( ! num]als1 & ?is q complete z w be se51 is f\nd 9 c]ta9 o!r cave inscriptions dat+ back to ! fir/ or second c5tury ,a4,d4 ,9 ^!1 ! ,nasik#gh cave inscriptions1 ! =ms are z foll[s3 ,illu/ration ,f ?is "t on1 until ! decimal sy/em f9ally adopt$ ! fir/ n9e *>act]s & replac$ ! re/ ( ! ,brahmi notation by add+ ! z]o1 ! progress ( ^! =ms is well m>k$4 ,x is !re=e well to pres5t synoptically ! be/-kn[n specim5s t h come d[n to u1 & ?is is done 9 ! table on page #be4gi #be ,,table ,,%[+ ,,! ,,progress ,,( ,,numb] ,,=ms ,,9 ,,india ,,num]als #a #b #c #d #e #f #g #h #i #aj #bj #cj #dj #ej #fj #gj #hj #ij #ajj #bjj #ajjj ,a'soka#hj ,illu/ration ',saka#ha ,illu/ration ,a'soka#hb ,illu/ration ,nag>i#hc ,illu/ration ,nasik#hd ,illu/ration ,ks4atrapa#he ,illu/ration ,kus4ana #hf ,illu/ration ,gupta#hg ,illu/ration ,valhabi#hh ,illu/ration ,nepal #hi ,illu/ration ,kali4nga#ij ,illu/ration ,vakat4aka#ia ,illu/ration ,mo/ ( ^! num]als are giv5 by ,buehl]1 loc4 cit41 ,tafel ,,ix4 #bf ,) respect to ^! num]als x %\ld fir/ be not$ t no z]o appe>s 9 ! table1 & z a matt] ( fact none exi/$ 9 any ( ! cases cit$4 ,x 0 !re=e impossible to h any place value1 & ! numb]s l tw5ty1 ?irty1 & o!r multiples ( t51 "o hundr$1 & s on1 requir$ sep>ate symbols except :]e !y 7 writt5 out 9 words4 ,! anci5t ,h9dus _h no less ?an tw5ty ( ^! symbols1#ib a numb] t 0 aft]w>d greatly increas$4 ,! foll[+ are examples ( _! me?od ( indicat+ c]ta9 numb]s betwe5 "o hundr$ & "o ?\s&3 #ic ,num]als = #agd #id ,num]als = #aia #ie ,num]als = #bfi #if ,num]als = #beb #ig ,num]als = #djj #ih ,num]als = #cef #bg ,to ^! may be add$ ! foll[+ num]als bel[ "o hundr$1 simil> to ^? 9 ! table3 ,num]als#ii = #ij ,num]als#ajj = #gj ,we h ?us f> spok5 ( ! ,kh>os4t4hi & ,brahmi num]als1 & x rema9s to m5tion ! ?ird type1 ! ^w & lett] =ms4 ,^! are1 h[ev]1 s closely connect$ ) ! p]fect+ ( ! sy/em by ! inv5tion ( ! z]o t !y are m appropriately consid]$ 9 ! next *apt]1 p>ticul>ly z !y h little relation to ! problem ( ! orig9 ( ! =ms kn[n z ! ,arabic4 ,hav+ n[ exam9$ types ( ! e>ly =ms x is appropriate to turn our att5tion to ! "q ( _! orig94 ,z to ! fir/ ?ree !re is no "q4 ,! #a v]tical /roke or #a horizontal /roke is simply "o /roke1 or "o /ick laid d[n by ! comput]4 ,! #b v]tical /rokes or #b horizontal /rokes repres5ts two /rokes or two /icks1 & s = ! #c v]tical /rokes & #c horizontal /rokes4 ,f "s primitive #b v]tical /rokes came ! two ( ,egypt1 ( ,rome1 ( e>ly ,greece1 & ( v>i\s o!r civilizations4 ,x appe>s 9 ! ?ree ,egyptian num]al sy/ems 9 ! foll[+ =ms3 ,hi]oglyphic #b v]tical /rokes ,hi]atic ,hi]atic #b ,demotic ,demotic #b ,! la/ ( ^! is m]ely a cursive =m z 9 ! ,arabic ,arabic #b1 :i* becomes our #b if tipp$ ?r\gh a "r angle4 ,f "s primitive #b horizontal /rokes came ! ,*9ese #bh symbol1 :i* is practically id5tical ) ! symbols f\nd commonly 9 ,india f #aej ,b4,c4 to #gjj ,a4,d4 ,9 ! cursive =m x becomes #b horizontal /rokes jo9$1 & ?is 0 frequ5tly us$ = two 9 ,g]many until ! #ah? c5tury4 ,x f9ally w5t into ! mod]n =m #b1 & ! #c horizontal /rokes 9 ! same way became our #c4 ,!re is1 h[ev]1 consid]able gr\nd = int]e/+ speculation ) respect to ^! fir/ ?ree num]als4 ,! e>lie/ ,h9du =ms 7 p]p5dicul>4 ,9 ! ,nana ,ghat inscriptions !y are v]tical4 ,b long be=e ei!r ! ,a'soka or ! ,nana ,ghat inscriptions ! ,*9ese 7 us+ ! horizontal =ms = ! fir/ ?ree num]als1 b a v]tical arrangem5t = f\r4#aja ,n[ :]e did ,*9a get ^! =ms8 ,surely n f ,india1 = %e _h !m1 z h] monum5ts & lit]ature#ajb %[1 long be=e ! ,h9dus knew !m4 ,! tradition is t ,*9a br\ght h] civilization ar\nd ! nor? ( ,tibet1 f ,mongolia1 ! primitive habitat be+ ,mesopotamia1 or possibly ! oases ( ,turke/an4 ,n[ :at num]als did ,mesopotamia use8 ,! ,babylonian sy/em1 simple 9 its g5]al pr9ciples b v complicat$ 9 _m ( its details1 is n[ well kn[n4#ajc ,9 p>ticul>1 "o1 two1 & ?ree 7 repres5t$ by v]tical arr[-heads4 ,:y1 !n1 did ! ,*9ese write #bi !irs horizontally8 ,! problem n[ takes a new int]e/ :5 we f9d t ^! ,babylonian =ms 7 n ! primitive ones ( ?is region1 b t ! e>ly ,sum]ian =ms 7 horizontal4#ajd ,:at int]pretation %all be giv5 to ^! facts8 ,%all we say t x 0 m]e accid5t t "o p wrote "o v]tically & t ano!r wrote x horizontally8 ,?is may be ! case2 b x may also be ! case t ! tribal migrations t end$ 9 ! ,mongol invasion ( ,*9a />t$ f ! ,euphrates :ile yet ! ,sum]ian civilization 0 prom95t1 or f "s common s\rce 9 ,turke/an1 & t !y c>ri$ to ! ,ea/ ! primitive num]als ( _! anci5t home1 ! fir/ ?ree1 ^! be+ all t ! p z a :ole knew or ne$$4 ,x is equally possible t ^! ?ree horizontal =ms repres5t primitive /ick-lay+1 ! mo/ natural position ( a /ick plac$ 9 front ( a calculator be+ ! horizontal "o4 ,:51 h[ev]1 ! cunei=m writ+ develop$ m fully1 ! v]tical =m may h be5 prov$ ! easi] to make1 s t by ! "t ! migrations to ! ,we/ began ^! 7 9 use1 & f !m came ! upright =ms ( ,egypt1 ,greece1 ,rome1 & o!r ,m$it]ranean l&s1 & ^? ( ,a'soka's "t 9 ,india4 ,aft] ,a'soka1 & p]haps among ! m]*ants ( e>li] c5turies1 ! horizontal =ms may h come d[n into ,india f ,*9a1 ?us giv+ ^? ( ! ,nana ,ghat cave & ( lat] inscriptions4 ,?is is 9 ! realm ( speculation1 b x is n improbable t fur!r epigraphical /udies may confirm ! hypo!sis4 #cj ,z to ! num]als above ?ree !re h be5 v _m conjectures4 ,! figure "o ( ! ,demotic looks l ! "o ( ! ,sanskrit1 ! two 7rev]s$7 l t ( ! ,arabic1 ! f\r has "s resemblance to t 9 ! ,nasik caves1 ! five 7rev]s$7 to t on ! ,ks4atrapa co9s1 ! n9e to t ( ! ,kus4ana inscriptions1 & o!r po9ts ( simil>ity h be5 imag9$4 ,"s h trac$ resemblance betwe5 ! ,hi]atic five & sev5 & ^? ( ! ,indian inscriptions4 ,!re h n1 !re=e1 be5 want+ ^? :o ass]t$ an ,egyptian orig9 = ^! num]als4#aje ,!re has already be5 m5tion$ ! fact t ! ,kh>os4t4hi num]als 7 =m]ly kn[n z ,bactrian1 ,indo-,bactrian1 & ,aryan4 ,cunn+ham#ajf 0 ! fir/ to sugge/ t ^! num]als 7 d]iv$ f ! alphabet ( ! ,bactrian civilization ( ,ea/]n ,p]sia1 p]haps a ?\s& ye>s be=e our era1 & 9 ?is he 0 support$ by ! s*ol>ly "w ( ,sir ,e4 ,clive ,bayley1#ajg :o 9 turn 0 foll[$ by ,canon ,taylor4#ajh ,! resemblance has n prov$ conv9c+1 h[ev]1 & ,bayley's draw+s #ca h be5 criticiz$ z be+ affect$ by 8 !ory4 ,! foll[+ is "p ( ! hypo!sis3#aji ,num]al ,h9du ,bactrian ,sanskrit #d ,symbol ,symbol * *atur1 ,lat4 quattuor #e ,symbol ,symbol p pan*a1 ,gk4 ,greek3p_/ente #f ,symbol ,symbol s s4as4 #g ,symbol ,symbol s4 sapta 7 ! s & s4 are int]*ang$ z occasionally 9 ,n4 ,w4 ,india7 ,buehl]#aaj rejects ?is hypo!sis1 /at+ t 9 f\r cases 7f\r1 six1 sev51 & t57 ! facts are absolutely aga9/ x4 ,:ile ! relation to anci5t ,bactrian =ms has be5 g5]ally d\bt$1 x is agre$ t mo/ ( ! num]als resemble ,brahmi lett]s1 & we w\ld naturally expect !m to be initials4#aaa ,b1 kn[+ ! anci5t pronunciation ( mo/ ( ! numb] names1#aab we f9d ?is n to be ! case4 ,we next fall back ^u ! hypo!sis #cb t !y repres5t ! ord] ( lett]s#aac 9 ! anci5t alphabet4 ,f :at we "k ( ?is ord]1 h[ev]1 !re seems also no basis = ?is assumption4 ,we h1 !re=e1 to confess t we are n c]ta9 t ! num]als 7 alphabetic at all1 & if !y 7 alphabetic we h no evid5ce at pres5t z to ! basis ( selection4 ,! lat] =ms may possibly h be5 alphabetical expressions ( c]ta9 syllables call$ aks4aras1 :i* possess$ 9 ,sanskrit fix$ num]ical values1#aad b ?is is equally unc]ta9 ) ! re/4 ,bayley also ?\ght#aae t "s ( ! =ms 7 ,pho5ician1 z notably ! use ( a circle = tw5ty1 b ! resemblance is 9 g5]al too remote to be conv9c+4 ,!re is also "s slight possibility t ,*9ese influ5ce is to be se5 9 c]ta9 ( ! e>ly =ms ( ,h9du num]als4#aaf #cc ,m absurd is ! hypo!sis ( a ,greek orig91 suppos$ly support$ by d]ivation ( ! curr5t symbols f ! fir/ n9e lett]s ( ! ,greek alphabet4#aag ,?is difficult feat is accompli%$ by twi/+ "s ( ! lett]s1 cutt+ (f1 add+ on1 & effect+ o!r *anges to make ! lett]s fit ! !ory4 ,?is peculi> !ory 0 fir/ set up by ,dasypodius#aah 7,conrad ,rauhfuss71 & 0 lat] elaborat$ by ,huet4#aai #cd ,a biz>re d]ivation bas$ ^u e>ly ,arabic 7c4 #ajdj ,a4,d47 s\rces is giv5 by ,kir*] 9 8 "w#abj on numb] my/icism4 ,he quotes f ,ab5ragel1#aba giv+ ! ,arabic & a ,lat9 translation#abb & /at+ t ! ord9>y ,arabic =ms are d]iv$ f sectors ( a circle1 circle4 ,out ( all ^! conflict+ !ories1 & f all ! resemblances se5 or imag9$ betwe5 ! num]als ( ! ,we/ & ^? ( ! ,ea/1 :at conclusions are we prep>$ to draw z ! evid5ce n[ /&s8 ,probably none t is satisfactory4 ,inde$1 ^u ! evid5ce at #ce h& we might prop]ly feel t ev]y?+ po9ts to ! num]als z be+ sub/antially indig5\s to ,india4 ,& :y %\ld ?is n be ! case8 ,if ! k+ ,srong-tsan-,gampo 7#fci ,a4,d471 ! f\nd] ( ,lhasa1#abc c\ld h set ab\t to devise a new alphabet = ,tibet1 & if ! ,siamese1 & ! ,s+halese1 & ! ,burmese1 & o!r peoples 9 ! ,ea/1 c\ld h creat$ alphabets ( _! own1 :y %\ld n ! num]als also h be5 fa%ion$ by "s temple s*ool1 or "s k+1 or "s m]*ant guild8 ,by way ( illu/ration1 !re are %[n 9 ! table on page #cf c]ta9 sy/ems ( ! ,ea/1 & :ile a few resemblances are evid5t1 x is also evid5t t ! creators ( ea* sy/em endeavor$ to f9d orig9al =ms t %\ld n be f\nd 9 o!r sy/ems4 ,?is1 !n1 w\ld seem to be a fair int]pretation ( ! evid5ce4 ,a human m9d _c readily create simple =ms t are absolutely new2 :at x fa%ions w naturally resemble :at o!r m9ds h fa%ion$1 or :at x has kn[n ?r\gh he>say or ?r\gh sight4 ,a circle is "o ( ! _w's common /ock ( figures1 & t x %\ld mean tw5ty 9 ,pho5icia & 9 ,india is h>dly m surpris+ ?an t x signifi$ t5 at "o "t 9 ,babylon4#abd ,x is !re=e q probable t an extrane\s orig9 _c be f\nd = ! v suffici5t reason t none exi/s4 ,( absolute nons5se ab\t ! orig9 ( ! symbols :i* we use mu* has be5 writt54 ,conjectures1 #cf h[ev]1 )\t any hi/orical evid5ce = support1 h no place 9 a s]i\s discussion ( ! gradual evolution ( ! pres5t num]al =ms4#abe ,,table ,,( ,,c]ta9 ,,ea/]n ,,sy/ems ,siam ,illu/ration3 num]als ,burma#abf ,illu/ration3 num]als ,malab>#abg ,illu/ration3 num]als ,tibet#abh ,illu/ration3 num]als ,ceylon#abi ,illu/ration3 num]als ,malayalam#abi ,illu/ration3 num]als #cg ,we may summ>ize ?is *apt] by say+ t no "o kn[s :at sugge/$ c]ta9 ( ! e>ly num]al =ms us$ 9 ,india4 ,! orig9 ( "s is evid5t1 b ! orig9 ( o!rs w probably nev] be kn[n4 ,!re is no reason :y !y %\ld n h be5 inv5t$ by "s prie/ or tea*] or guild1 by ! ord] ( "s k+1 or z "p ( ! my/icism ( "s temple4 ,:atev] ! orig91 !y 7 no bett] ?an scores ( o!r anci5t sy/ems & no bett] ?an ! pres5t ,*9ese sy/em :5 writt5 )\t ! z]o1 & !re w\ld nev] h be5 any *ance ( _! triumphal progress we/w>d _h x n be5 = ?is relatively late symbol4 ,!re c\ld h>dly be dem&$ a /rong] pro( ( ! ,h9du orig9 ( ! "* = z]o ?an ?is1 & to x fur!r ref]5ce w be made 9 ,*apt] ,,iv4 99 99 99 99 99 #ch ,,*apt] ,,iii ,,lat] ,,h9du ,,=ms1 ,,) ,a ,,place ,,value ,be=e speak+ ( ! p]fect$ ,h9du num]als ) ! z]o & ! place value1 x is necess>y to consid] ! ?ird sy/em m5tion$ on page #ai1--! ^w & lett] =ms4 ,! use ( words ) place value began at lea/ z e>ly z ! #f? c5tury ( ! ,*ri/ian era4 ,9 _m ( ! manuals ( a/ronomy & ma!matics1 & (t5 9 o!r works 9 m5tion+ dates1 numb]s are repres5t$ by ! names ( c]ta9 objects or ideas4 ,= example1 z]o is repres5t$ by ! void 7'sunya71 or heav5-space 7amb>a aka'sa72 "o by /ick 7rupa71 moon 7indu 'sa's971 e>? 7bhu71 beg9n+ 7adi71 ,brahma1 or1 9 g5]al1 by any?+ m>k$ly unique2 two by ! tw9s 7yama71 h&s 7k>a71 eyes 7nayana71 etc42 f\r by oceans1 five by s5ses 7vis4aya7 or arr[s 7! five arr[s ( ,kamadeva72 six by seasons or flavors2 sev5 by m\nta9 7aga71 & s on4#acj ,^! names1 accommodat+ !mselves to ! v]se 9 :i* sci5tific works 7 writt51 _h ! additional advantage ( n admitt+1 z did ! figures1 easy alt]ation1 s9ce any *ange w\ld t5d to di/urb ! met]4 #ci ,z an example ( ?is sy/em1 ! date ',saka ,sam4vat1 #hfg 7,a4,d4 #ide or #idf71 is giv5 by giri-ras4a-vasu1 mean+ ! m\nta9s 7sev571 ! flavors 7six71 & ! gods ,vasu ( :i* !re 7 eight4 ,9 read+ ! date ^! are read f "r to left4#aca ,! p]iod ( inv5tion ( ?is sy/em is unc]ta94 ,! fir/ trace seems to be 9 ! ',srautasutra ( ,katyayana & ,lat4yayana4#acb ,x 0 c]ta9ly kn[n to ,v>aha-,mihira 7d4 #ehg71acc = he us$ x 9 ! ,br4hat-,sam4hita4#acd ,x has also be5 ass]t$#ace t ,aryabhat4a 7c4 #ejj ,a4,d47 0 famili> ) ?is sy/em1 b !re is no?+ to prove ! /atem5t4#acf ,! e>lie/ epigraphical examples ( ! sy/em are f\nd 9 ! ,bayang 7,cambodia7 inscriptions ( #fjd & #fbd ,a4,d4#acg ,m5tion %\ld also be made1 9 ?is connection1 ( a curi\s sy/em ( alphabetic num]als t sprang up 9 s\!rn ,india4 ,9 ?is we h ! num]als repres5t$ by ! lett]s z giv5 9 ! foll[+ table3 #a #b #c #d #e #f #g #h #i #j k kh g gh 4n c * j jh n t4 t4h d4 d4h n4 t ? d ? n p ph b bh m y r l v 's s4 s h l #dj ,by ?is plan a num]al might be repres5t$ by any "o ( sev]al lett]s1 z %[n 9 ! prec$+ table1 & ?us x c\ld ! m easily be =m$ into a ^w = mnemonic purposes4 ,= example1 ! ^w #b #c #a #e #f #e #a kha gont yan me s4a ma pa has ! value #a1efe1acb1 read+ f "r to left4#ach ,?is1 ! olde/ specim5 7#aahd ,a4,d47 kn[n ( ?is notation1 is giv5 9 a comm5t>y on ! ,rigv$a1 repres5t+ ! numb] ( days t _h elaps$ f ! beg9n+ ( ! ,kaliyuga4 ,burnell#aci /ates t ?is sy/em is ev5 yet 9 use = rememb]+ rules to calculate horoscopes1 & = a/ronomical tables4 ,a second sy/em ( ?is k9d is /ill us$ 9 ! pag9ation ( manuscripts 9 ,ceylon1 ,siam1 & ,burma1 hav+ also _h its rise 9 s\!rn ,india4 ,9 ?is ! ?irty-f\r consonants :5 foll[$ by a 7z ka 444 la7 designate ! numb]s #a-cd2 by a 7z ka 444 la71 ^? f #ce to #fh2 by i 7ki 444 li71 ^? f #fi to #ajb1 inclusive2 & s on4#adj ,z already /at$1 h[ev]1 ! ,h9du sy/em z ?us f> describ$ 0 no improvem5t ^u _m o!rs ( ! anci5ts1 su* z ^? us$ by ! ,greeks & ! ,hebrews4 ,hav+ no z]o1 x 0 impracticable to designate ! t5s1 hundr$s1 & o!r units ( high] ord] by ! same symbols us$ = ! units f "o to n9e4 ,9 o!r words1 !re 0 no possibility ( place value )\t "s fur!r improvem5t4 ,s ! ,nana ,ghat #da symbols requir$ ! writ+ ( ?\s& sev5 tw5ty-f\r ab\t l ,t #g1 tw1 #d 9 mod]n symbols1 in/ead ( #gjbd1 9 :i* ! sev5 ( ! ?\s&s1 ! two ( ! t5s 7conceal$ 9 ! ^w tw5ty1 be+ orig9ally twa9 ( t5s1 ! -ty signify+ t571 & ! f\r ( ! units are giv5 z spok5 & ! ord] ( ! unit 7t5s1 hundr$s1 etc47 is giv5 by ! place4 ,to complete ! sy/em only ! z]o 0 ne$$2 b x 0 probably eight c5turies aft] ! ,nana ,ghat inscriptions 7 cut1 be=e ?is important symbol appe>$2 & n until a consid]ably lat] p]iod did x become well kn[n4 ,:o x 0 to :om ! inv5tion is due1 or :]e he liv$1 or ev5 9 :at c5tury1 w probably always rema9 a my/]y4#ada ,x is possible t "o ( ! =ms ( anci5t abacus sugge/$ to "s ,h9du a/ronom] or ma!matician ! use ( a symbol to /& = ! vacant l9e :5 ! c\nt]s 7 remov$4 ,x is well e/abli%$ t 9 diff]5t p>ts ( ,india ! names ( ! high] p[]s took diff]5t =ms1 ev5 ! ord] be+ int]*ang$4#adb ,nev]!less1 z ! significance ( ! "n ( ! unit 0 giv5 by ! ord] 9 read+1 ^! v>iations did n lead to error4 ,inde$ ! v>iation itself may h necessitat$ ! introduction ( a ^w to signify a vacant place or lack+ unit1 ) ! ultimate introduction ( a z]o symbol = ?is ^w4 ,to enable u to appreciate ! =ce ( ?is argum5t a l>ge numb]1 #h1ddc1fhb1aee1 may be consid]$ z ! ,h9dus wrote & read x1 & !n1 by way ( contra/1 z ! ,greeks & ,arabs w\ld h read x4 #db ,mod]n ,am]ican read+1 #h billion1 #ddc million1 #fhb ?\s&1 #aee4 ,h9du1 #h padmas1 #d vy>budas1 #d kot4is1 #c prayutas1 #f laks4z1 #h ayutas1 #b sahasra1 #a 'sata1 #e da'san1 #e4 ,arabic & e>ly ,g]man1 eight ?\s& ?\s& ?\s& & f\r hundr$ ?\s& ?\s& & =ty-?ree ?\s& ?\s&1 & six hundr$ ?\s& & eighty-two ?\s& & "o hundr$ fifty-five 7or five & fifty74 ,greek1 eighty-f\r myriads ( myriads & f\r ?\s& ?ree hundr$ sixty-eight myriads & two ?\s& & "o hundr$ fifty-five4 ,z ,woepcke#adc po9t$ out1 ! read+ ( numb]s ( ?is k9d %[s t ! notation adopt$ by ! ,h9dus t5d$ to br+ out ! place idea4 ,no o!r language ?an ! ,sanskrit has made su* consi/5t application1 9 num]ation1 ( ! decimal sy/em ( numb]s4 ,! introduction ( myriads z 9 ! ,greek1 & ?\s&s z 9 ,arabic & 9 mod]n num]ation1 is really a /ep away f a decimal s*eme4 ,s 9 ! numb]s bel[ "o hundr$1 9 ,engli%1 elev5 & twelve are out ( h>mony ) ! re/ ( ! -te5s1 :ile ! nam+ ( all ! numb]s betwe5 t5 & tw5ty is n analog\s to ! nam+ ( ! numb]s above tw5ty4 ,to con=m to our writt5 sy/em we %\ld h t5-"o1 t5-two1 t5-?ree1 & s on1 z we h tw5ty-"o1 tw5ty-two1 & ! l4 ,! ,sanskrit is consi/5t1 ! units1 h[ev]1 prec$+ ! t5s & hundr$s4 ,nor did any o!r anci5t p c>ry ! num]ation z f> z did ! ,h9dus4#add #dc ,:5 ! a4nkapalli1#ade ! decimal-place sy/em ( writ+ numb]s1 0 p]fect$1 ! t5? symbol 0 call$ ! 'sunyab9du1 g5]ally %ort5$ to 'sunya 7! void74 ,brockhaus#adf has well said t if !re 0 any inv5tion = :i* ! ,h9dus1 by all _! philosophy & religion1 7 well fitt$1 x 0 ! inv5tion ( a symbol = z]o4 ,?is mak+ ( no?+ness ! crux ( a trem5d\s a*ievem5t 0 a /ep 9 complete h>mony ) ! g5ius ( ! ,h9du4 ,x is g5]ally ?\ght t ?is 'sunya z a symbol 0 n us$ be=e ab\t #ejj ,a4,d41 al?\gh "s writ]s h plac$ x e>li]4#adg ,s9ce ,aryabhat4a gives our common me?od ( extract+ roots1 x w\ld seem t he may h kn[n a decimal notation1#adh al?\gh he did n use ! *>act]s f :i* our num]als are d]iv$4#adi ,moreov]1 he frequ5tly speaks ( ! #dd void4#aej ,if he ref]s to a symbol ?is w\ld put ! z]o z f> back z #ejj ,a4,d41 b ( c\rse he may h ref]r$ m]ely to ! concept ( no?+ness4 ,a little lat]1 b also 9 ! six? c5tury1 ,v>aha-,mihira#aea wrote a "w entitl$ ,br4hat ,sam4hita#aeb 9 :i* he frequ5tly uses 'sunya 9 speak+ ( num]als1 s t x has be5 ?\ght t he 0 ref]r+ to a def9ite symbol4 ,?is1 ( c\rse1 w\ld add to ! probability t ,aryabhat4a 0 do+ ! same4 ,x %\ld also be m5tion$ z a matt] ( int]e/1 & some:at relat$ to ! "q at issue1 t ,v>aha-,mihira us$ ! ^w-sy/em ) place value#aec z expla9$ above4 ,! fir/ k9d ( alphabetic num]als & also ! ^w-sy/em 79 bo? ( :i* ! place value is us$7 are plays ^u1 or v>iations (1 position ari?metic1 :i* w\ld be mo/ likely to occur 9 ! c\ntry ( its orig94#aed ,at ! op5+ ( ! next c5tury 7c4 #fbj ,a4,d47 ,ban4a#aee wrote ( ,sub&hus's ,vasavadatta z a celebrat$ "w1 #de & m5tion$ t ! />s dott+ ! sky are "h comp>$ ) z]os1 ^! be+ po9ts z 9 ! mod]n ,arabic sy/em4 ,on ! o!r h&1 a /rong argum5t aga9/ any ,h9du k ( ! symbol z]o at ?is "t is ! fact t ab\t #gjj ,a4,d4 ! ,arabs ov]ran ! prov9ce ( ,s9d & ?us _h an opportunity ( kn[+ ! common me?ods us$ !re = writ+ numb]s4 ,& yet1 :5 !y receiv$ ! complete sy/em 9 #ggf !y look$ ^u x z some?+ new4#aef ,su* evid5ce is n conclusive1 b x t5ds to %[ t ! complete sy/em 0 probably n 9 common use 9 ,india at ! beg9n+ ( ! eigh? c5tury4 ,on ! o!r h&1 we mu/ be> 9 m9d ! fact t a travel] 9 ,g]many 9 ! ye> #agjj w\ld probably h he>d or se5 no?+ ( decimal fractions1 al?\gh ^! 7 p]fect$ a c5tury be=e t date4 ,! elite ( ! ma!maticians may h kn[n ! z]o ev5 9 ,aryabhat4a's "t1 :ile ! m]*ants & ! common p may n h grasp$ ! significance ( ! novelty until a long "t aft]4 ,on ! :ole1 ! evid5ce seems to po9t to ! we/ coa/ ( ,india z ! region :]e ! complete sy/em 0 fir/ se54#aeg ,z m5tion$ above1 traces ( ! num]al words ) place value1 :i* d n1 h[ev]1 absolutely require a decimal place-sy/em ( symbols1 are f\nd v e>ly 9 ,cambodia1 z well z 9 ,india4 ,conc]n+ ! e>lie/ epigraphical in/ances ( ! use ( ! n9e symbols1 plus ! z]o1 ) place value1 !re #df is "s "q4 ,colebrooke#aeh 9 #ahjg w>n$ aga9/ ! possibility ( =g]y 9 _m ( ! anci5t copp]-plate l& grants4 ,on ?is acc\nt ,fleet1 9 ! ,indian ,antiqu>y1#aei discusses at l5g? ?is phase ( ! "w ( ! epigraphi/s 9 ,india1 hold+ t _m ( ^! =g]ies 7 made ab\t ! end ( ! elev5? c5tury4 ,colebrooke#afj takes a m rational view ( ^! =g]ies ?an does ,kaye1 :o seems to hold t !y t5d to invalidate ! :ole ,indian hypo!sis4 ,b ev5 :]e t may be suspect$1 ! hi/orical uses ( a monum5t fabricat$ s mu* ne>] to ! times to :i* x assumes to belong1 w n be entirely sup]s$$4 ,! necessity ( r5d]+ ! =g$ grant cr$ible w\ld compel a fabricator to adh]e to hi/ory1 & con=m to e/abli%$ notions3 & ! tradition1 :i* prevail$ 9 8 "t1 & by :i* he mu/ be guid$1 w\ld probably be s mu* ne>] to ! tru?1 z x 0 less remote f ! p]iod :i* x conc]n$4#afa ,buehl]#afb gives ! copp]-plate ,gurj>a inscription ( ,c$i-sam4vat #cdf 7#eie ,a4,d47 z ! olde/ epigraphical use ( ! num]als#afc 9 :i* ! symbols correspond to ! alphabet num]als ( ! p]iod & ! place4 ,v9c5t ,a4 ,smi?#afd quotes a /one inscription ( #hae ,a4,d41 dat$ ,sam4vat #hgb4 ,s ,f4 ,kielhorn 9 ! ,epigraphia ,indica#afe gives a ,pa?>i pill> inscription ( ,p>abala1 dat$ ,vikrama-sam4vat #iag1 :i* corresponds to #hfa ,a4,d41 #dg & ref]s also to ano!r copp]-plate inscription dat$ ,vikrama-sam4vat #hac 7#gef ,a4,d474 ,! inscription quot$ by ,v4 ,a4 ,smi? above is t giv5 by ,d4 ,r4 ,bh&>k>1#aff & ano!r is giv5 by ! same writ] z ( date ,saka-sam4vat #gae 7#gih ,a4,d471 be+ incis$ on a pila/]4 ,kielhorn#afg also gives two copp]-plate inscriptions ( ! "t ( ,mah5drapala ( ,kanauj1 ,valhabi-sam4vat #egd 7#hic ,a4,d47 & ,vikrama-sam4vat #ief 7#hii ,a4,d474 ,t !re %\ld be any inscriptions ( date z e>ly ev5 z #gej ,a4,d41 w\ld t5d to %[ t ! sy/em 0 at lea/ a c5tury old]4 ,z w be %[n 9 ! fur!r developm5t1 x 0 m ?an two c5turies aft] ! introduction ( ! num]als into ,europe t !y appe>$ !re ^u co9s & inscriptions4 ,:ile ,?ibaut#afh does n consid] x necess>y to quote any specific in/ances ( ! use ( ! num]als1 he /ates t traces are f\nd f #eij ,a4,d4 on4 ,t ! sy/em n[ 9 use by all civiliz$ nations is ( ,h9du orig9 _c be d\bt$2 no o!r nation has any claim ^u its discov]y1 especially s9ce ! ref]5ces to ! orig9 ( ! sy/em :i* are f\nd 9 ! nations ( we/]n ,asia po9t unanim\sly t[>ds ,india4#afi ,! te/imony & op9ions ( m5 l ,buehl]1 ,kielhorn1 ,v4 ,a4 ,smi?1 ,bh&>k>1 & ,?ibaut are entitl$ to ! mo/ s]i\s consid]ation4 ,z au?orities on anci5t ,indian epigraphy no o!rs rank high]4 ,_! "w is accept$ by ,indian s*ol>s ! _w ov]1 & _! unit$ judgm5t z to ! rise ( ! sy/em ) a place value--t x took place 9 ,india z e>ly z ! #dh six? c5tury ,a4,d4--mu/ /& unless new evid5ce ( great weight c be submitt$ to ! contr>y4 ,_m e>ly writ]s rem>k$ ^u ! div]sity ( ,indian num]al =ms4 ,al-,biruni 0 probably ! fir/2 notewor?y is also ,johannes ,hispal5sis1#agj :o gives ! v>iant =ms = sev5 & f\r4 ,we ins]t on p4 #di a table ( num]als us$ ) place value4 ,:ile ! *ief au?ority = ?is is ,buehl]1#aga sev]al specim5s are giv5 :i* are n f\nd 9 8 "w & :i* are ( unusual int]e/4 ,! ',sarada =ms giv5 9 ! table use ! circle z a symbol = #a & ! dot = z]o4 ,!y are tak5 f ! pag+ & text ( ,! ,ka%mirian ,a?>va-,v$a#agb1 ( :i* ! manuscript us$ is c]ta9ly f\r hundr$ ye>s old4 ,simil> =ms are f\nd 9 a manuscript belong+ to ! ,univ]sity ( ,tueb+54 ,two o!r s]ies pres5t$ are f ,tibetan books 9 ! libr>y ( "o ( ! au?ors4 ,= purposes ( comp>ison ! mod]n ,sanskrit & ,arabic num]al =ms are add$4 ,sanskrit1 ,illu/ration ,arabic1 ,illu/ration #di ,,num]als ,,us$ ,,) ,,place ,,value #a #b #c #d #e #f #g #h #i #j a#agc ,illu/ration b#agd ,illu/ration c#age ,illu/ration d#agf ,illu/ration e#agg ,illu/ration f#agh ,illu/ration g#agi ,illu/ration h#ahj ,illu/ration i#ahj ,illu/ration j#aha ,illu/ration k#aha ,illu/ration l#ahb ,illu/ration m#ahc ,illu/ration n#ahd ,illu/ration 99 99 99 99 99 #ea ,,*apt] ,,iv ,,! ,,symbol ,,z]o ,:at has be5 said ( ! improv$ ,h9du sy/em ) a place value does n t\* directly ! orig9 ( a symbol = z]o1 al?\gh x assumes t su* a symbol exi/s4 ,! importance ( su* a sign1 ! fact t x is a pr]equisite to a place-value sy/em1 & ! fur!r fact t )\t x ! ,h9du-,arabic num]als w\ld nev] h dom9at$ ! computation sy/em ( ! we/]n _w1 make x prop] to devote a *apt] to its orig9 & hi/ory4 ,x 0 "s c5turies aft] ! primitive ,brahmi & ,kh>os4t4hi num]als _h made _! appe>ance 9 ,india t ! z]o fir/ appe>$ !re1 al?\gh su* a "* 0 us$ by ! ,babylonians#ahe 9 ! c5turies imm$iately prec$+ ! ,*ri/ian era4 ,! symbol is ,babylonian z]o symbol or ,babylonian z]o symbol1 & app>5tly x 0 n us$ 9 calculation4 ,nor does x always occur :5 units ( any ord] are lack+2 ?us #ahj is writt5 ,babylonian num]als #ahj ) ! mean+ ?ree sixties & no units1 s9ce #aha imm$iately foll[+ is ,babylonian num]als #aha1 ?ree sixties & "o unit4#ahf ,! ma9 #eb use ( ?is ,babylonian symbol seems to h be5 9 ! fractions1 #fj?s1 #cfjj?s1 etc41 & some:at simil> to ! ,greek use ( ,greek3 o1 = ,greek3 oud51 ) ! mean+ vacant4 ,! e>lie/ und\bt$ occurr5ce ( a z]o 9 ,india is an inscription at ,gwalior1 dat$ ,samvat #icc 7#hgf ,a4,d474 ,:]e #ej g>l&s are m5tion$ 7l9e #bj71 #ej is writt5 ,gwalior num]als #ej4 #bgj 7l9e #d7 is writt5 ,gwalior num]als #bgj4ahg ,! ,bakhs4ali ,manuscript#ahh probably ant$ates ?is1 us+ ! po9t or dot z a z]o symbol4 ,bayley m5tions a grant ( ,jaika ,ra%trakuta ( ,bh>uj1 f\nd at ,okam&el1 ( date #gch ,a4,d41 :i* conta9s a z]o1 & also a co9 ) indi/9ct ,gupta date #gjg 7#hig ,a4,d471 b ! reliability ( ,bayley's "w is que/ion$4 ,z has be5 not$1 ! appe>ance ( ! num]als 9 inscriptions & on co9s w\ld be ( mu* lat] occurr5ce ?an ! orig9 & writt5 exposition ( ! sy/em4 ,f ! p]iod m5tion$ ! spread 0 rapid ov] all ( ,india1 save ! s\!rn "p1 :]e ! ,tamil & ,malayalam p reta9 ! old sy/em ev5 to ! pres5t day4#ahi ,aside f its appe>ance 9 e>ly inscriptions1 !re is /ill ano!r indication ( ! ,h9du orig9 ( ! symbol 9 ! special treatm5t ( ! concept z]o 9 ! e>ly works on ari?metic4 ,brahmagupta1 :o liv$ 9 ,ujja91 ! c5t] ( ,indian a/ronomy1#aij 9 ! e>ly "p #ec ( ! sev5? c5tury1 gives 9 8 ari?metic#aia a di/9ct treatm5t ( ! prop]ties ( z]o4 ,he does n discuss a symbol1 b he %[s by 8 treatm5t t 9 "s way z]o _h acquir$ a special significance n f\nd 9 ! ,greek or o!r anci5t ari?metics4 ,a /ill m sci5tific treatm5t is giv5 by ,bhask>a1#aib al?\gh 9 "o place he p]mits himself an unall[$ lib]ty 9 divid+ by z]o4 ,! mo/ rec5tly discov]$ "w ( anci5t ,indian ma!matical lore1 ! ,ganita-,sara-,sa4ngraha#aic ( ,mahaviracarya 7c4 #hcj ,a4,d471 :ile x does n use ! num]als ) place value1 has a simil> discussion ( ! calculation ) z]o4 ,:at sugge/$ ! =m = ! z]o is1 ( c\rse1 purely a matt] ( conjecture4 ,! dot1 :i* ! ,h9dus us$ to fill up lacunae 9 _! manuscripts1 mu* z we indicate a break 9 a s5t5ce1#aid w\ld h be5 a m natural symbol2 & ?is is ! "o :i* ! ,h9dus fir/ us$#aie & :i* mo/ ,arabs use to-day4 ,!re 0 also us$ = ?is purpose a cross1 l our ,x1 & ?is is occasionally f\nd z a z]o symbol4#aif ,9 ! ,bakhs4ali manuscript above m5tion$1 ! ^w 'sunya1 ) ! dot z its symbol1 is us$ to d5ote ! unkn[n quantity1 z well z to d5ote z]o4 ,an analog\s use ( ! #ed z]o1 = ! unkn[n quantity 9 a proportion1 appe>s 9 a ,lat9 manuscript ( "s lectures by ,gottfri$ ,wolack 9 ! ,univ]sity ( ,erfurt 9 #adfg & #adfh4aig ,! usage 0 not$ ev5 z e>ly z ! eighte5? c5tury4#aih ,! small circle 0 possibly sugge/$ by ! spurr$ circle :i* 0 us$ = t54#aii ,x has also be5 ?\ght t ! omicron us$ by ,ptolemy 9 8 ,almage/1 to m>k accid5tal blanks 9 ! sexagesimal sy/em :i* he employ$1 may h influ5c$ ! ,indian writ]s4#bjj ,?is symbol 0 us$ q g5]ally 9 ,europe & ,asia1 & ! ,arabic a/ronom] ,al-,battani#bja 7di$ #ibi ,a4,d47 us$ a simil> symbol 9 connection ) ! alphabetic sy/em ( num]als4 ,! occasional use by ,al-,battani ( ! ,arabic negative1 la1 to indicate ! abs5ce ( m9utes #ee 7or seconds71 is not$ by ,nall9o4#bjb ,notewor?y is also ! use ( ! ,circle = unity 9 ! ',sarada *>act]s ( ! ,ka%mirian ,a?>va-,v$a1 ! writ+ be+ at lea/ #djj ye>s old4 ,bhask>a 7c4 #aaej7 us$ a small circle above a numb] to indicate subtraction1 & 9 ! ,t>t> writ+ a r$undant ^w is remov$ by draw+ an oval ar\nd x4 ,x w\ld be int]e/+ to "k :e!r our score m>k score m>k1 read f\r 9 ! hole1 c\ld trace its p$igree to ! same s\rces4 ,o',creat#bjc 7c4 #aacj71 9 a lett] to 8 tea*]1 ,adelh>d ( ,ba?1 uses ,greek3 t = z]o1 be+ an abbreviation = ! ^w teca :i* we %all see 0 "o ( ! names us$ = z]o1 al?\gh x c\ld q z well be f ,greek3 tziphra4 ,m r>ely ,o',creat uses circle ) b>1 apply+ ! "n cyfra to bo? =ms4 ,frat] ,sigsboto#bjd 7c4 #aaej7 uses ! same symbol4 ,o!r peculi> =ms are not$ by ,heib]g#bje z be+ 9 use among ! ,byzant9e ,greeks 9 ! fifte5? c5tury4 ,x is evid5t f ! text t "s ( ^! writ]s did n und]/& ! import ( ! new sy/em4#bjf ,al?\gh ! dot 0 us$ at fir/ 9 ,india1 z not$ above1 ! small circle lat] replac$ x & cont9ues 9 use to ?is day4 ,! ,arabs1 h[ev]1 did n adopt ! #ef circle1 s9ce x bore "s resemblance to ! lett] :i* express$ ! numb] five 9 ! alphabet sy/em4#bjg ,! e>lie/ ,arabic z]o kn[n is ! dot1 us$ 9 a manuscript ( #hgc ,a4,d4#bjh ,sometimes bo? ! dot & ! circle are us$ 9 ! same "w1 hav+ ! same mean+1 :i* is ! case 9 an ,arabic ,,ms41 an abridg$ ari?metic ( ,jam%id1#bji #ihb ,a4,h4 7#aege ,a4,d474 ,z giv5 9 ?is "w ! num]als are symbols4 ,! =m = #e v>ies1 9 "s works becom+ symbol or symbol2 symbol is f\nd 9 ,egypt & symbol appe>s 9 "s fonts ( type4 ,to-day ! ,arabs use ! #j only :51 "u ,european influ5ce1 !y adopt ! ord9>y sy/em4 ,among ! ,*9ese ! fir/ def9ite trace ( z]o is 9 ! "w ( ,ts9#baj ( #abdg ,a4,d4 ,! =m is ! circul> "o ( ! ,h9dus1 & und\bt$ly 0 br\ght to ,*9a by "s travel]4 ,! "n ( ?is all-important symbol also dem&s "s att5tion1 especially z we are ev5 yet q undecid$ z to :at to call x4 ,we speak ( x to-day z z]o1 naught1 & ev5 ciph]2 ! telephone op]ator (t5 calls x ,o1 & ! illit]ate or c>eless p]son calls x aught4 ,9 view ( all ?is unc]ta9ty we may well inquire :at x has be5 call$ 9 ! pa/4#baa #eg ,z already /at$1 ! ,h9dus call$ x 'sunya1 void4#bab ,?is pass$ ov] into ! ,arabic z as4-s4ifr or s4ifr4#bac ,:5 ,leon>d ( ,pisa 7#abjb7 wrote ^u ! ,h9du num]als he spoke ( ?is "* z zephirum4#bad ,maximus ,planudes 7#accj71 writ+ "u bo? ! ,greek & ! ,arabic influ5ce1 call$ x tziphra4#bae ,9 a treatise on ari?metic writt5 9 ! ,italian language by ,jacob ( ,flor5ce#baf #eh 7#acjg7 x is call$ zeu]o1#bag :ile 9 an ari?metic ( ,giovanni di ,danti ( ,arezzo 7#acgj7 ! ^w appe>s z ceu]o4#bah ,ano!r =m is zepiro1#bai :i* 0 also a /ep f zephirum to z]o4#bbj ,( c\rse ! ,engli% ciph]1 ,fr5* *iffre1 is d]iv$ f ! same ,arabic ^w1 as4-s4ifr1 b 9 sev]al languages x has come to mean ! num]al figures 9 g5]al4 ,a trace ( ?is appe>s 9 our ^w ciph]+1 mean+ figur+ or comput+4#bba ,johann ,huswirt#bbb uses ! ^w ) bo? mean+s2 he gives = ! t5? "* ! f\r names !ca1 circulus1 cifra1 & figura nihili4 ,9 ?is /atem5t ,huswirt probably foll[s1 z did _m writ]s ( t p]iod1 ! ,algorismus ( ,johannes de ,sacrobosco 7c4 #abej ,a4,d471 :o 0 also kn[n z ,john ( ,halifax or ,john ( ,holywood4 ,! comm5t>y ( #ei ,petrus de ,dacia#bbc 7c4 #abia ,a4,d47 on ! ,algorismus vulg>is ( ,sacrobosco 0 also widely us$4 ,! widespread use ( ?is ,engli%man's "w on ari?metic 9 ! univ]sities ( t "t is atte/$ by ! l>ge numb]#bbd ( ,,mss4 f ! ?irte5? to ! sev5te5? c5tury /ill extant1 tw5ty 9 ,muni*1 twelve 9 ,vi5na1 ?irte5 9 ,erfurt1 sev]al 9 ,engl& giv5 by ,halliwell1#bbe t5 li/$ 9 ,coxe's ,catalogue ( ! ,ox=d ,college ,libr>y1 "o 9 ! ,plimpton collection1#bbf "o 9 ! ,columbia ,univ]sity ,libr>y1 &1 ( c\rse1 _m o!rs4 ,f as4-s4ifr has come zephyr1 ciph]1 & f9ally ! abridg$ =m z]o4 ,! e>lie/ pr9t$ "w 9 :i* is f\nd ?is f9al =m appe>s to be ,cal&ri's ari?metic ( #adia1bbg :ile 9 manuscript x appe>s at lea/ z e>ly z ! middle ( ! f\rte5? c5tury4#bbh ,x also appe>s 9 a "w1 ,le ,kadran des m>*ans1 by ,jehan #fj ,c]ta91#bbi writt5 9 #adhe4 ,?is ^w soon became fairly well kn[n 9 ,spa9#bcj & ,france4#bca ,! m$ieval writ]s also spoke ( x z ! sipos1#bcb & occasionally z ! :eel1#bcc circulus#bcd 79 ,g]man das ,r+le9#bce71 circul> #fa note1#bcf !ca1#bcg long suppos$ to be f its resemblance to ! ,greek !ta1 b expla9$ by ,petrus de ,dacia z be+ d]iv$ f ! "n ( ! iron#bch us$ to br& ?ieves & robb]s ) a circul> m>k plac$ on ! =ehead or on ! *eek4 ,x 0 also call$ omicron#bci 7! ,greek o71 be+ sometimes writt5 o or ,greek3 ph to di/+ui% x f ! lett] o4 ,x also w5t by ! "n null#bdj 79 ! ,lat9 books #fb nihil#bda or nulla1#bdb & 9 ! ,fr5* ri5#bdc71 & v commonly by ! "n ciph]4#bdd ,wallis#bde gives "o ( ! e>lie/ ext5d$ discussions ( ! v>i\s =ms ( ! ^w1 giv+ c]ta9 o!r v>iations wor?y ( note1 z ziphra1 zif]a1 siphra1 ciphra1 tsiphra1 tziphra1 & ! ,greek ,greek3 tziphra4#bdf 99 99 99 99 99 #fc ,,*apt] ,v ,,! ,,"q ,,( ,,! ,,introduction ,,( ,,! ,,num]als ,,into ,,europe ,,by ,,boe?ius ,j z we 7 q unc]ta9 z to ! orig9 ( ! num]al =ms1 s too are we unc]ta9 z to ! "t & place ( _! introduction into ,europe4 ,!re are two g5]al !ories z to ?is introduction4 ,! fir/ is t !y 7 c>ri$ by ! ,moors to ,spa9 9 ! eigh? or n9? c5tury1 & !nce 7 transmitt$ to ,*ri/ian ,europe1 a !ory :i* w be consid]$ lat]4 ,! second1 advanc$ by ,woepcke1#bdg is t !y 7 n br\ght to ,spa9 by ! ,moors1 b t !y 7 already 9 ,spa9 :5 ! ,arabs arriv$ !re1 hav+ rea*$ ! ,we/ ?r\gh ! ,neo-,py?agoreans4 ,!re are two facts to support ?is second !ory3 7#a7 ! =ms ( ^! num]als are *>act]i/ic1 diff]+ mat]ially f ^? :i* 7 br\ght by ,leon>do ( ,pisa f ,nor!rn ,africa e>ly 9 ! ?irte5? c5tury 7be=e #abjb ,a4,d472 7#b7 !y are ess5tially ^? :i* #fd tradition has s p]si/5tly assign$ to ,boe?ius 7c4 #ejj ,a4,d471 & :i* he w\ld naturally h receiv$1 if at all1 f ^! same ,neo-,py?agoreans or f ! s\rces f :i* !y d]iv$ !m4 ,fur!rmore1 ,woepcke po9ts out t ! ,arabs on ent]+ ,spa9 7#gaa ,a4,d47 w\ld naturally h foll[$ _! cu/om ( adopt+ = ! computation ( taxes ! num]ical sy/ems ( ! c\ntries !y conqu]$1#bdh s t ! num]als br\ght f ,spa9 to ,italy1 n hav+ und]gone ! same modifications z ^? ( ! ,ea/]n ,arab empire1 w\ld h diff]$1 z !y c]ta9ly did1 f ^? t came ?r\gh ,bagdad4 ,! !ory is t ! ,h9du sy/em1 )\t ! z]o1 e>ly rea*$ ,alex&ria 7say #dej ,a4,d471 & t ! ,neo-,py?agorean love = ! my/]i\s & especially = ! ,ori5tal l$ to its use z some?+ biz>re & cabali/ic2 t x 0 !n pass$ along ! ,m$it]ranean1 rea*+ ,boe?ius 9 ,a!ns or 9 ,rome1 & to ! s*ools ( ,spa91 be+ discov]$ 9 ,africa & ,spa9 by ! ,arabs ev5 be=e !y !mselves knew ! improv$ sy/em ) ! place value4 #fe ,a rec5t !ory set =? by ,bubnov#bdi also des]ves m5tion1 *iefly because ( ! s]i\sness ( purpose %[n by ?is well-kn[n writ]4 ,bubnov holds t ! =ms fir/ f\nd 9 ,europe are d]iv$ f anci5t symbols us$ on ! abacus1 b t ! z]o is ( ,h9du orig94 ,?is !ory does n seem t5able1 h[ev]1 9 ! light ( ! evid5ce already set =?4 ,two que/ions are pres5t$ by ,woepcke's !ory3 7#a7 ,:at 0 ! nature ( ^! ,spani% num]als1 & h[ 7 !y made kn[n to ,italy8 7#b7 ,did ,boe?ius "k !m8 ,! ,spani% =ms ( ! num]als 7 call$ ! h4uruf al-4gobar1 ! 4gobar or du/ num]als1 z di/+ui%$ f ! h4uruf al-jumal or alphabetic num]als4 ,probably ! latt]1 "u ! influ5ce ( ! ,syrians or ,jews1#bej 7 also us$ by ! ,arabs4 ,! significance ( ! t]m 4gobar is d\btless t ^! num]als 7 writt5 on ! du/ abacus1 ?is plan be+ di/9ct f ! c\nt] me?od ( repres5t+ numb]s4 ,x is also wor?y ( note t ,al-,biruni /ates t ! ,h9dus (t5 p]=m$ num]ical computations 9 ! s&4 ,! t]m is f\nd z e>ly z c4 #iej1 9 ! v]ses ( an anonym\s writ] ( ,kairwan1 9 ,tunis1 9 :i* ! au?or speaks ( "o ( 8 works on 4gobar calculation2#bea &1 mu* lat]1 ! ,arab writ] ,abu ,bekr ,moh4amm$ ibn ,abdallah1 surnam$ al-,h4as4s4ar #ff 7! ari?metician71 wrote a "w ( :i* ! second *apt] 0 ,on ! du/ figures4#beb ,! 4gobar num]als !mselves 7 fir/ made kn[n to mod]n s*ol>s by ,silve/re de ,sacy1 :o discov]$ !m 9 an ,arabic manuscript f ! libr>y ( ! anci5t abbey ( ,/4-,g]ma9-des-,pres4#bec ,! sy/em has n9e *>act]s1 b no z]o4 ,a dot above a "* indicates t5s1 two dots hundr$s1 & s on1 #e ) dot mean+ #ej1 & #e ) #c dots mean+ #ejjj4 ,x has be5 sugge/$ t possibly ^! dots1 spr9kl$ l du/ above ! num]als1 gave rise to ! ^w 4gobar1#bed b ?is is n at all probable4 ,?is sy/em ( dots is f\nd 9 ,p]sia at a mu* lat] date ) num]als q l ! mod]n ,arabic2#bee b t x 0 us$ at all is significant1 = x is h>dly likely t ! we/]n sy/em w\ld g back to ,p]sia1 :5 ! p]fect$ ,h9du "o 0 ne> at h&4 ,at fir/ sight !re w\ld seem to be "s reason = believ+ t ?is feature ( ! 4gobar sy/em 0 ( #fg ,arabic orig91 & t ! pres5t z]o ( ^! p1#bef ! dot1 0 d]iv$ f x4 ,x 0 entirely natural t ! ,semitic p g5]ally %\ld h adopt$ su* a s*eme1 s9ce _! diacritical m>ks w\ld sugge/ x1 n to speak ( ! possible influ5ce ( ! ,greek acc5ts 9 ! ,hell5ic numb] sy/em4 ,:5 we consid]1 h[ev]1 t ! dot is f\nd = z]o 9 ! ,bakhs4ali manuscript1#beg & t x 0 us$ 9 subscript =m 9 ! ,kitab al-,fihri/#beh 9 ! t5? c5tury1 & z late z ! sixte5? c5tury1#bei al?\gh 9 ?is case probably "u ,arabic influ5ce1 we are =c$ to believe t ?is =m may also h be5 ( ,h9du orig94 ,! fact seems to be t1 z already /at$1#bfj ! ,arabs did n imm$iately adopt ! ,h9du z]o1 because x resembl$ _! #e2 !y us$ ! sup]script dot z s]v+ _! purposes fairly well2 !y may1 inde$1 h c>ri$ ?is to ! we/ & h add$ x to ! 4gobar =ms already !re1 j z !y transmitt$ x to ! ,p]sians4 ,fur!rmore1 ! ,arab & ,hebrew s*ol>s ( ,nor!rn ,africa 9 ! t5? c5tury knew ^! num]als z ,indian =ms1 = a comm5t>y on ! ,sef] ,yes4irah by ,abu ,sahl ibn ,tamim 7probably compos$ at ,kairwan1 c4 #iej7 speaks ( ! ,indian ari?metic kn[n "u ! "n ( 4gobar or du/ calculation4#bfa ,all ?is sugge/s t ! ,arabs may v #fh likely h kn[n ! 4gobar =ms be=e ! num]als rea*$ !m aga9 9 #ggc4bfb ,! t]m 4gobar num]als 0 also us$ )\t any ref]5ce to ! peculi> use ( dots4#bfc ,9 ?is connection x is wor?y ( m5tion t ! ,alg]ians employ$ two diff]5t =ms ( num]als 9 manuscripts ev5 ( ! f\rte5? c5tury1#bfd & t ! ,moroccans ( to-day employ ! ,european =ms in/ead ( ! pres5t ,arabic4 ,! ,indian use ( subscript dots to indicate ! t5s1 hundr$s1 ?\s&s1 etc41 is e/abli%$ by a passage 9 ! ,kitab al-,fihri/#bfe 7#ihg ,a4,d47 9 :i* ! writ] discusses ! writt5 language ( ! p ( ,india4 ,not)/&+ ! importance ( ?is ref]5ce = ! e>ly hi/ory ( ! num]als1 x has n be5 m5tion$ by previ\s writ]s on ?is subject4 ,! num]al =ms giv5 are ^? :i* h usually be5 call$ ,indian1#bff 9 opposition to 4gobar4 ,9 ?is docum5t ! dots are plac$ bel[ ! *>act]s1 in/ead ( be+ sup]pos$ z describ$ above4 ,! significance 0 ! same4 ,9 =m ^! 4gobar num]als resemble our own mu* m closely ?an ! ,arab num]als d4 ,!y v>i$ m or less1 b 7 sub/antially z foll[s3 #fi #abfg;,illu/ration #bbfh;,illu/ration #cbfi;,illu/ration #dbgj;,illu/ration #ebga;,illu/ration #fbga;,illu/ration ,! "q ( ! possible influ5ce ( ! ,egyptian demotic & hi]atic ord9al =ms has be5 s (t5 sugge/$ t x seems well to introduce !m at ?is po9t1 = comp>ison ) ! 4gobar =ms4 ,!y w\ld z appropriately be us$ 9 connection ) ! ,h9du =ms1 & ! evid5ce ( a relation ( ! fir/ ?ree ) all ^! sy/ems is app>5t4 ,! only fur!r resemblance is 9 ! ,demotic #d & 9 ! #i1 s t ! /atem5t t ! ,h9du =ms 9 g5]al came f #gj ?is s\rce has no f\ndation4 ,! fir/ f\r ,egyptian c>d9al num]als#bgb resemble m ! mod]n ,arabic4 ,illu/ration3 ,,demotic ,,& ,,hi]atic ,,ord9als ,?is !ory ( ! v e>ly introduction ( ! num]als into ,europe fails 9 sev]al po9ts4 ,9 ! fir/ place ! e>ly ,we/]n =ms are n kn[n2 9 ! second place "s e>ly ,ea/]n =ms are l ! 4gobar1 z is se5 9 ! ?ird l9e on p4 #fi1 :]e ! =ms are f a manuscript writt5 at ,%iraz ab\t #igj ,a4,d41 & 9 :i* "s we/]n ,arabic =ms1 e4g4 symbol = #b1 are also us$4 ,probably mo/ significant ( all is ! fact t ! 4gobar num]als z giv5 by ,sacy are all1 ) ! exception ( ! symbol = eight1 ei!r s+le ,arabic lett]s or comb9ations ( lett]s4 ,s mu* = ! ,woepcke !ory & ! mean+ ( ! 4gobar num]als4 ,we n[ h to consid] ! "q z to :e!r ,boe?ius knew ^! 4gobar =ms1 or =ms ak9 to !m4 ,?is l>ge "q#bgc sugge/s sev]al m9or ones3 7#a7 ,:o 0 ,boe?ius8 7#b7 ,c\ld he h kn[n ^! num]als8 7#c7 ,is !re any positive or /rong circum/antial evid5ce t he did "k !m8 7#d7 ,:at are ! probabilities 9 ! case8 #ga ,fir/1 :o 0 ,boe?ius1--,divus#bgd ,boe?ius z he 0 call$ 9 ! ,middle ,ages8 ,anicius ,manlius ,sev]9us ,boe?ius#bge 0 born at ,rome c4 #dge4 ,he 0 a memb] ( ! di/+ui%$ family ( ! ,anicii1#bgf :i* _h = "s "t be=e 8 bir? be5 ,*ri/ian4 ,e>ly left an orphan1 ! tradition is t he 0 tak5 to ,a!ns at ab\t ! age ( t51 & t he rema9$ !re eighte5 ye>s4#bgg ,he m>ri$ ,ru/iciana1 daught] ( ! s5ator ,symma*us1 & ?is union ( two su* p[]ful families all[$ him to move 9 ! highe/ circles4#bgh ,/&+ /rictly = ! "r1 & aga9/ all iniquity at c\rt1 he became ! object ( hatr$ on ! "p ( all ! unscrupul\s elem5t ne> ! ?rone1 & 8 bold def5se ( ! ex-consul ,alb9us1 unju/ly accus$ ( treason1 l$ to 8 imprisonm5t at ,pavia#bgi & 8 execution 9 #ebd4bhj ,n _m g5]ations aft] 8 dea?1 ! p]iod be+ "o 9 :i* hi/orical criticism 0 at its l[e/ ebb1 ! *ur* f\nd x pr(itable to look ^u 8 execution z a m>tyrdom4#bha ,he 0 #gb accord+ly look$ ^u z a sa9t1#bhb 8 bones 7 en%r9$1#bhc & z a natural consequ5ce 8 books 7 among ! classics 9 ! *ur* s*ools = a ?\s& ye>s4#bhd ,x is pa!tic1 h[ev]1 to ?9k ( ! m$ieval /ud5t try+ to extract m5tal n\ri%m5t f a "w s ab/ract1 s mean+less1 s unnecess>ily complicat$1 z ! ari?metic ( ,boe?ius4 ,he 0 look$ ^u by 8 contempor>ies & imm$iate successors z a ma/]1 = ,cassiodorus#bhe 7c4 #dij-;c4 #ehe ,a4,d47 says to him3 ,?r\gh y\r translations ! music ( ,py?agoras & ! a/ronomy ( ,ptolemy are read by ^? ( ,italy1 & ! ari?metic ( ,nicoma*us & ! geometry ( ,euclid are kn[n to ^? ( ! ,we/4#bhf ,f\nd] ( ! m$ieval s*ola/icism1 #gc di/+ui%+ ! trivium & quadrivium1#bhg writ+ ! only classics ( 8 "t1 ,gibbon well call$ him ! la/ ( ! ,romans :om ,cato or ,tully c\ld h ackn[l$g$ = _! c\ntryman4#bhh ,! second "q relat+ to ,boe?ius is ?is3 ,c\ld he possibly h kn[n ! ,h9du num]als8 ,9 view ( ! relations t w be %[n to h exi/$ betwe5 ! ,ea/ & ! ,we/1 !re c only be an affirmative answ] to ?is "q4 ,! num]als _h exi/$1 )\t ! z]o1 = sev]al c5turies2 !y _h be5 well kn[n 9 ,india2 !re _h be5 a cont9u$ int]*ange ( ?\ght betwe5 ! ,ea/ & ,we/2 & w>riors1 ambassadors1 s*ol>s1 & ! re/less trad]1 all _h gone back & =?1 by l& or m frequ5tly by sea1 betwe5 ! ,m$it]ranean l&s & ! c5t]s ( ,indian comm]ce & culture4 ,boe?ius c\ld v well h le>n$ "o or m =ms ( ,h9du num]als f "s travel] or m]*ant4 ,to ju/ify ?is /atem5t x is necess>y to speak m fully ( ^! relations betwe5 ! ,f> ,ea/ & ,europe4 ,x is true t we h no records ( ! int]*ange ( le>n+1 9 any l>ge way1 betwe5 ea/]n ,asia & c5tral ,europe 9 ! c5tury prec$+ ! "t ( ,boe?ius4 ,b x is "o ( ! mi/akes ( s*ol>s to believe t !y are ! sole transmitt]s ( k4 #gd ,z a matt] ( fact !re is abundant reason = believ+ t ,h9du num]als w\ld naturally h be5 kn[n to ! ,arabs1 & ev5 along e trade r\te to ! remote we/1 long be=e ! z]o ent]$ to make _! place-value possible1 & t ! *>act]s1 ! me?ods ( calculat+1 ! improvem5ts t took place f "t to "t1 ! z]o :5 x appe>$1 & ! cu/oms z to solv+ bus9ess problems1 w\ld all h be5 made kn[n f g5]ation to g5]ation along ^! same trade r\tes f ! ,ori5t to ! ,occid5t4 ,x mu/ always be kept 9 m9d t x 0 to ! tradesman & ! w&]+ s*ol> t ! spread ( su* le>n+ 0 due1 r ?an to ! s*ool man4 ,inde$1 ,avic5na#bhi 7#ihj-ajcg ,a4,d47 9 a %ort biography ( himself relates t :5 8 p 7 liv+ at ,bokhara 8 fa!r s5t him to ! h\se ( a groc] to le>n ! ,h9du art ( reckon+1 9 :i* ?is groc] 7oil deal]1 possibly7 0 exp]t4 ,leon>do ( ,pisa1 too1 _h a simil> tra9+4 ,! :ole "q ( ?is spread ( m]cantile k along ! trade r\tes is s connect$ ) ! 4gobar num]als1 ! ,boe?ius "q1 ,g]b]t1 ,leon>do ( ,pisa1 & o!r names & ev5ts1 t a digression = its consid]ation n[ becomes necess>y4#bij #ge ,ev5 9 v remote times1 be=e ! ,h9du num]als 7 sculptur$ 9 ! cave ( ,nana ,ghat1 !re 7 trade relations betwe5 ,arabia & ,india4 ,inde$1 long be=e ! ,aryans w5t to ,india ! great ,turanian race _h spread its civilization f ! ,m$it]ranean to ! ,indus4#bia ,at a mu* lat] p]iod ! ,arabs 7 ! int]m$i>ies betwe5 ,egypt & ,syria on ! we/1 & ! f>!r ,ori5t4#bib ,9 ! six? c5tury ,b4,c41 ,hecataeus1#bic ! fa!r ( geography1 0 acqua9t$ n only ) ! ,m$it]ranean l&s b ) ! c\ntries z f> z ! ,indus1#bid & 9 ,biblical times !re 7 regul> tri5nial voyages to ,india4 ,inde$1 ! /ory ( ,joseph be>s witness to ! c>avan trade f ,india1 across ,arabia1 & on to ! banks ( ! ,nile4 ,ab\t ! same "t z ,hecataeus1 ,scylax1 a ,p]sian admiral "u ,d>ius1 f ,c>y&a on ! coa/ ( ,asia ,m9or1 travel$ to #gf nor?we/ ,india & wrote ^u 8 v5tures4#bie ,he induc$ ! nations along ! ,indus to ackn[l$ge ! ,p]sian supremacy1 & su* numb] sy/ems z !re 7 9 ^! l&s w\ld naturally h be5 kn[n to a man ( 8 atta9m5ts4 ,a c5tury aft] ,scylax1 ,h]odotus %[$ consid]able k ( ,india1 speak+ ( its cotton & its gold1#bif tell+ h[ ,seso/ris#big fitt$ out %ips to sail to t c\ntry1 & m5tion+ ! r\tes to ! ea/4 ,^! r\tes 7 g5]ally by ! ,r$ ,sea1 & _h be5 foll[$ by ! ,pho5icians & ! ,sabaeans1 & lat] 7 tak5 by ! ,greeks & ,romans4#bih ,9 ! f\r? c5tury ,b4,c4 ! ,we/ & ,ea/ came into v close relations4 ,z e>ly z #ccj1 ,py!as ( ,massilia 7,m>seilles7 _h explor$ z f> nor? z ! nor!rn end ( ! ,briti% ,isles & ! coa/s ( ! ,g]man ,sea1 :ile ,mac$on1 9 close t\* ) s\!rn ,france1 0 also s5d+ h] armies "u ,alex&]#bii ?r\gh ,afghani/an z f> ea/ z ! ,punjab4#cjj ,pl9y tells u t ,alex&] ! ,great employ$ surveyors to measure #gg ! roads ( ,india2 & "o ( ! great highways is describ$ by ,mega/h5es1 :o 9 #bie ,b4,c41 z ! ambassador ( ,seleucus1 resid$ at ,pataliput4ra1 ! pres5t ,patna4#cja ,! ,h9dus also le>n$ ! art ( co9+ f ! ,greeks1 or possibly f ! ,*9ese1 & ! /ores ( ,greco-,h9du co9s /ill f\nd 9 nor!rn ,india are a con/ant s\rce ( hi/orical in=mation4#cjb ,! ,ramayana speaks ( m]*ants travel+ 9 great c>avans & emb>k+ by sea = =eign l&s4#cjc ,ceylon trad$ ) ,malacca & ,siam1 & ,java 0 coloniz$ by ,h9du trad]s1 s t m]cantile k 0 be+ spread ab\t ! ,indies dur+ all ! =mative p]iod ( ! num]als4 ,moreov] ! results ( ! e>ly ,greek invasion 7 embodi$ by ,dicae>*us ( ,messana 7ab\t #cbj ,b4,c47 9 a map t long rema9$ a /&>d4 ,fur!rmore1 ,alex&] did n all[ 8 influ5ce on ! ,ea/ to cease4 ,he divid$ ,india into ?ree satrapies1#cjd plac+ ,greek gov]nors ov] two ( !m & leav+ a ,h9du rul] 9 *>ge ( ! ?ird1 & 9 ,bactriana1 a "p ( ,ariana or anci5t ,p]sia1 he left gov]nors2 & 9 ^! ! we/]n civilization 0 long 9 evid5ce4 ,"s ( ! ,greek & ,roman metrical & a/ronomical t]ms #gh f\nd _! way1 d\btless at ?is "t1 into ! ,sanskrit language4#cje ,ev5 z late z f ! second to ! fif? c5turies ,a4,d41 ,indian co9s %[$ ! ,hell5ic influ5ce4 ,! ,h9du a/ronomical t]m9ology reveals ! same relation%ip to we/]n ?\ght1 = ,v>aha-,mihira 7#f? c5tury ,a4,d471 a contempor>y ( ,aryabhat4a1 entitl$ a "w ( 8 ! ,br4hat-,sam4hita1 a lit]al translation ( ,greek3 megale suntaxis ( ,ptolemy2#cjf & 9 v>i\s ways is ?is int]*ange ( ideas app>5t4#cjg ,x c\ld n h be5 at all unusual = ! anci5t ,greeks to g to ,india1 = ,/rabo lays d[n ! r\te1 say+ t all :o make ! j\rney />t f ,ephesus & trav]se ,phrygia & ,cappadocia be=e tak+ ! direct road4#cjh ,! products ( ! ,ea/ 7 always f9d+ _! way to ! ,we/1 ! ,greeks gett+ _! g+]#cji f ,malab>1 z ! ,pho5icians _h long be=e br\ght gold f ,malacca4 ,greece mu/ also h _h e>ly relations ) ,*9a1 = !re is a notable simil>ity betwe5 ! ,greek & ,*9ese life1 z is %[n 9 _! h\ses1 _! dome/ic cu/oms1 _! m>riage c]emonies1 ! public /ory-tell]s1 ! puppet %[s :i* ,h]odotus says 7 introduc$ f ,egypt1 ! /reet juggl]s1 ! games ( dice1#caj ! game ( f+]-guess+1#caa ! wat] clock1 ! #gi music sy/em1 ! use ( ! myriad1#cab ! cal5d>s1 & 9 _m o!r ways4#cac ,9 pass+ ?r\gh ! suburbs ( ,pek+ to-day1 on ! way to ! ,great ,bell temple1 "o is con/antly rem9d$ ( ! semi-,greek ar*itecture ( ,pompeii1 s closely does mod]n ,*9a t\* ! old classical civilization ( ! ,m$it]ranean4 ,! ,*9ese hi/orians tell u t ab\t #bjj ,b4,c4 _! arms 7 successful 9 ! f> we/1 & t 9 #ahj ,b4,c4 an ambassador w5t to ,bactria1 !n a ,greek city1 & report$ t ,*9ese products 7 on sale 9 ! m>kets !re4#cad ,!re is also a notewor?y resemblance betwe5 c]ta9 ,greek & ,*9ese words1#cae %[+ t 9 remote times !re mu/ h be5 m or less int]*ange ( ?\ght4 ,! ,romans also ex*ang$ products ) ! ,ea/4 ,horace says1 ,a busy trad]1 y ha/5 to ! f>!/ ,indies1 fly+ f pov]ty ov] sea1 ov] crags1 ov] fires4#caf ,! products ( ! ,ori5t1 spices & jewels f ,india1 frank9c5se f ,p]sia1 & silks f ,*9a1 be+ m 9 dem& ?an ! exports f ! ,m$it]ranean l&s1 ! balance ( trade 0 aga9/ ! ,we/1 & ?us ,roman co9 f\nd its way ea/w>d4 ,9 #ahih1 = example1 a numb] ( ,roman co9s dat+ f #aad ,b4,c4 to ,hadrian's "t 7 f\nd at ,pakli1 a "p ( ! ,hazara di/rict1 sixte5 miles nor? ( ,abbottabad1#cag & num]\s simil> discov]ies h be5 made f "t to "t4 #hj ,augu/us speaks ( envoys receiv$ by him f ,india1 a ?+ nev] be=e kn[n1#cah & x is n improbable t he also receiv$ an embassy f ,*9a4#cai ,suetonius 7fir/ c5tury ,a4,d47 speaks 9 8 hi/ory ( ^! relations1#cbj z d sev]al ( 8 contempor>ies1#cba & ,v]gil#cbb tells ( ,augu/us do+ battle 9 ,p]sia4 ,9 ,pl9y's "t ! trade ( ! ,roman ,empire ) ,asia am\nt$ to a million & a qu>t] doll>s a ye>1 a sum f> great] relatively !n ?an n[1#cbc :ile by ! "t ( ,con/ant9e ,europe 0 9 direct communication ) ! ,f> ,ea/4#cbd ,9 view ( ^! relations x is n beyond ! range ( possibility t pro( may sometime come to light to %[ t ! ,greeks & ,romans knew some?+ ( ! #ha numb] sy/em ( ,india1 z sev]al writ]s h ma9ta9$4#cbe ,return+ to ! ,ea/1 !re are _m evid5ces ( ! spread ( k 9 & ab\t ,india itself4 ,9 ! ?ird c5tury ,b4,c4 ,buddhism began to be a connect+ m$ium ( ?\ght4 ,x _h already p]meat$ ! ,himalaya t]ritory1 _h rea*$ ea/]n ,turke/an1 & _h probably gone !nce to ,*9a4 ,"s c5turies lat] 79 #fb ,a4,d47 ! ,*9ese emp]or s5t an ambassador to ,india1 & 9 #fg ,a4,d4 a ,buddhi/ monk 0 invit$ to ,*9a4#cbf ,!n1 too1 9 ,india itself ,a'soka1 ^: "n has already be5 m5tion$ 9 ?is "w1 ext5d$ ! b\nd>ies ( 8 doma9s ev5 into ,afghani/an1 s t x 0 entirely possible = ! num]als ( ! ,punjab to h work$ _! way nor? ev5 at t e>ly date4#cbg ,fur!rmore1 ! influ5ce ( ,p]sia mu/ n be =gott5 9 consid]+ ?is transmission ( k4 ,9 ! fif? c5tury ! ,p]sian m$ical s*ool at ,jondi-,sapur admitt$ bo? ! ,h9du & ! ,greek doctr9es1 & ,firdusi tells u t dur+ ! brilliant reign ( #hb ,khosru ,i1#cbh ! gold5 age ( ,pahlavi lit]ature1 ! ,h9du game ( *ess 0 introduc$ into ,p]sia1 at a "t :5 w>s ) ! ,greeks 7 br++ pre/ige to ! ,sassanid dyna/y4 ,aga91 n f> f ! "t ( ,boe?ius1 9 ! six? c5tury1 ! ,egyptian monk ,cosmas1 9 8 e>li] ye>s z a trad]1 made j\rneys to ,abyss9ia & ev5 to ,india & ,ceylon1 receiv+ ! "n ,indicopleu/es 7! ,indian travel]74 ,8 map 7#edg ,a4,d47 %[s "s k ( ! e>? f ! ,atlantic to ,india4 ,su* a man w\ld1 ) h>dly a d\bt1 h obs]v$ e num]al sy/em us$ by ! p ) :om he soj\rn$1#cbi & :e!r or n he record$ 8 /udies 9 p]man5t =m he w\ld h transmitt$ su* scraps ( k by ^w ( m\?4 ,z to ! ,arabs1 x is a mi/ake to feel t _! activities began ) ,mohamm$4 ,comm]ce _h always be5 held 9 honor by !m1 & ! ,qorei%#ccj _h annually = _m g5]ations s5t c>avans be>+ ! spices & textiles ( ,yem5 to ! %ores ( ! ,m$it]ranean4 ,9 ! fif? c5tury !y trad$ by sea ) ,india & ev5 ) ,*9a1 & ,h4ira 0 an emporium = ! w>es ( ! ,ea/1#cca s t any num]al sy/em ( any "p ( ! trad+ _w c\ld h>dly h rema9$ isolat$4 ,long be=e ! w>like activity ( ! ,arabs1 ,alex&ria _h become ! great m>ket-place ( ! _w4 ,f ?is c5t] c>avans trav]s$ ,arabia to ,hadramaut1 :]e !y met %ips f ,india4 ,o!rs w5t nor? to ,damascus1 :ile /ill o!rs made _! way #hc along ! s\!rn %ores ( ! ,m$it]ranean4 ,%ips sail$ f ! i/hmus ( ,suez to all ! comm]cial ports ( ,s\!rn ,europe & up into ! ,black ,sea4 ,h9dus 7 f\nd among ! m]*ants#ccb :o frequ5t$ ! baza>s ( ,alex&ria1 & ,brahm9s 7 report$ ev5 9 ,byzantium4 ,su* is a v brief resume ( ! evid5ce %[+ t ! num]als ( ! ,punjab & ( o!r p>ts ( ,india z well1 & inde$ ^? ( ,*9a & f>!r ,p]sia1 ( ,ceylon & ! ,malay p59sula1 might well h be5 kn[n to ! m]*ants ( ,alex&ria1 & ev5 to ^? ( any o!r seaport ( ! ,m$it]ranean1 9 ! "t ( ,boe?ius4 ,! ,brahmi num]als w\ld n h attract$ ! att5tion ( s*ol>s1 = !y _h no z]o s f> z we "k1 & !re=e !y 7 no bett] & no worse ?an ^? ( doz5s ( o!r sy/ems4 ,if ,boe?ius 0 attract$ to !m x 0 probably exactly z any "o is naturally attract$ to ! biz>re or ! my/ic1 & he w\ld h m5tion$ !m 9 8 works only incid5tally1 z inde$ !y are m5tion$ 9 ! manuscripts 9 :i* !y occur4 ,9 answ] !re=e to ! second "q1 ,c\ld ,boe?ius h kn[n ! ,h9du num]als8 ! reply mu/ be1 )\t ! slighte/ d\bt1 t he c\ld easily h kn[n !m1 & t x w\ld h be5 /range if a man ( 8 inquir+ m9d did n pick up _m curi\s bits ( in=mation ( ?is k9d ev5 ?\gh he nev] ?\ght ( mak+ use ( !m4 ,let u n[ consid] ! ?ird "q1 ,is !re any positive or /rong circum/antial evid5ce t ,boe?ius did "k ^! num]als8 ,! "q is n new1 #hd nor is x mu* ne>] be+ answ]$ ?an x 0 ov] two c5turies ago :5 ,wallis 7#afic7 express$ 8 d\bts ab\t x#ccc soon aft] ,vossius 7#afeh7 _h call$ att5tion to ! matt]4#ccd ,/at$ briefly1 !re are ?ree works on ma!matics attribut$ to ,boe?ius3#cce 7#a7 ! ari?metic1 7#b7 a "w on music1 & 7#c7 ! geometry4#ccf ,! g5u95ess ( ! ari?metic & ! treatise on music is g5]ally recogniz$1 b ! geometry1 :i* conta9s ! ,h9du num]als ) ! z]o1 is "u suspicion4#ccg ,!re are pl5ty ( support]s ( ! idea t ,boe?ius knew ! num]als & includ$ !m 9 ?is book1#cch & on ! o!r h& !re are z _m :o #he feel t ! geometry1 or at lea/ ! "p m5tion+ ! num]als1 is spuri\s4#cci ,! argum5t ( ^? :o d5y ! au!nticity ( ! p>ticul> passage 9 "q may briefly be /at$ ?us3 #a4 ,! falsification ( texts has always be5 ! subject ( compla9t4 ,x 0 s ) ! ,romans1#cdj x 0 common 9 ! ,middle ,ages1#cda & x is mu* m preval5t #hf to-day ?an we commonly ?9k4 ,we h b to see h[ e hymn-book compil] feels himself au?oriz$ to *ange at w ! classics ( our language1 & h[ unkn[n editors h mutilat$ ,%akespe>e1 to see h[ mu* m easy x 0 = m$ieval scribes to ins]t or elim9ate p>agraphs )\t any prote/ f critics4#cdb #b4 ,if ,boe?ius _h kn[n ^! num]als he w\ld h m5tion$ !m 9 8 ari?metic1 b he does n d s4#cdc #c4 ,if he _h kn[n !m1 & _h m5tion$ !m 9 any ( 8 works1 8 contempor>ies1 disciples1 & successors w\ld h kn[n & m5tion$ !m4 ,b nei!r ,capella 7c4 #dge7cdd nor any ( ! num]\s m$ieval writ]s :o knew ! works ( ,boe?ius makes any ref]5ce to ! sy/em4#cde #hg #d4 ,! passage 9 "q has all ! appe>ance ( an int]polation by "s scribe4 ,boe?ius is speak+ ( angles1 9 8 "w on geometry1 :5 ! text sudd5ly *anges to a discussion ( classes ( numb]s4#cdf ,?is is foll[$ by a *apt] 9 explanation ( ! abacus1#cdg 9 :i* are describ$ ^? num]al =ms :i* are call$ apices or c>act]es4#cdh ,! =ms#cdi ( ^! *>act]s v>y 9 diff]5t manuscripts1 b 9 g5]al are ab\t z %[n on page #hh4 ,!y are commonly writt5 ) ! #i at ! left1 decreas+ to ! unit at ! "r1 num]\s writ]s /at+ t ?is 0 because !y 7 d]iv$ f ,semitic s\rces 9 :i* ! direction ( writ+ is ! opposite ( our own4 ,?is practice cont9u$ until ! sixte5? c5tury4#cej ,! writ] !n leaves ! subject entirely1 us+ ! ,roman num]als = ! re/ ( 8 discussion1 a proce$+ s =eign to ! me?od ( ,boe?ius z to be inexplicable on ! hypo!sis ( au!nticity4 ,:y %\ld su* a s*ol>ly writ] h giv5 !m ) no m5tion ( _! orig9 or use8 ,ei!r he w\ld h m5tion$ "s hi/orical int]e/ atta*+ to !m1 or he w\ld h us$ !m 9 "s discussion2 he c]ta9ly w\ld n h left ! passage z x is4 #hh ,,=ms ,,( ,,! ,,num]als1 ,,l>gely ,,f ,,works ,,on ,,! ,,abacus#cea a#ceb ,illu/ration b#cec ,illu/ration c#ced ,illu/ration d#cee ,illu/ration e#cef ,illu/ration f#ceg ,illu/ration g#ceh ,illu/ration h#cei ,illu/ration i#cfj ,illu/ration #hi ,sir ,e4 ,clive ,bayley has add$#cfa a fur!r reason = believ+ !m spuri\s1 namely t ! #d is n ( ! ,nana ,ghat type1 b ( ! ,kabul =m :i* ! ,arabs did n receive until #ggf2cfb s t x is n likely1 ev5 if ! *>act]s 7 kn[n 9 ,europe 9 ! "t ( ,boe?ius1 t ?is p>ticul> =m 0 recogniz$4 ,x is wor?y ( m5tion1 also1 t 9 ! six abacus =ms f ! *ief manuscripts z giv5 by ,fri$le91#cfc ea* conta9s "s =m ( z]o1 :i* symbol probably orig9at$ 9 ,india ab\t ?is "t or lat]4 ,x c\ld h>dly h rea*$ ,europe s soon4 ,z to ! f\r? "q1 ,did ,boe?ius probably "k ! num]als8 ,x seems to be a fair conclusion1 accord+ to our pres5t evid5ce1 t 7#a7 ,boe?ius might v easily h kn[n ^! num]als )\t ! z]o1 b1 7#b7 !re is no reliable evid5ce t he did "k !m4 ,& j z ,boe?ius might h come 9 contact ) !m1 s any o!r inquir+ m9d might h done s ei!r 9 8 "t or at any "t be=e !y def9itely appe>$ 9 ! t5? c5tury4 ,^! c5turies1 five 9 numb]1 repres5t$ ! d>ke/ ( ! ,d>k ,ages1 & ev5 if ^! num]als 7 occasionally met & /udi$1 no trace ( !m w\ld be likely to %[ itself 9 ! #ij lit]ature ( ! p]iod1 unless by *ance x %\ld get into ! writ+s ( "s man l ,alcu94 ,z a matt] ( fact1 x 0 n until ! n9? or t5? c5tury t !re is any tangible evid5ce ( _! pres5ce 9 ,*ri/5dom4 ,!y 7 probably kn[n to m]*ants "h & !re1 b 9 _! incomplete /ate !y 7 n ( suffici5t importance to attract any consid]able att5tion4 ,z a result ( ?is brief survey ( ! evid5ce sev]al conclusions seem reasonable3 7#a7 comm]ce1 & travel = travel's sake1 nev] di$ out betwe5 ! ,ea/ & ! ,we/2 7#b7 m]*ants _h e opportunity ( kn[+1 & w\ld h be5 unreasonably /upid if !y _h n kn[n1 ! elem5t>y numb] sy/ems ( ! peoples ) :om !y 7 trad+1 b !y w\ld n h put ?is k 9 p]man5t writt5 =m2 7#c7 w&]+ s*ol>s w\ld h kn[n _m & /range ?+s ab\t ! peoples !y met1 b !y too 7 n1 z a class1 writ]s2 7#d7 !re is e reason a priori = believ+ t ! 4gobar num]als w\ld h be5 kn[n to m]*ants1 & probably to "s ( ! w&]+ s*ol>s1 long be=e ! ,arabs conqu]$ nor!rn ,africa2 7#e7 ! wond] is n t ! ,h9du-,arabic num]als 7 kn[n ab\t #ajjj ,a4,d41 & t !y 7 ! subject ( an elaborate "w 9 #abjb by ,fibonacci1 b r t m ext5d$ manuscript evid5ce ( _! appe>ance be=e t "t has n be5 f\nd4 ,t !y 7 m or less kn[n e>ly 9 ! ,middle ,ages1 c]ta9ly to _m m]*ants ( ,*ri/ian ,europe1 & probably to sev]al s*ol>s1 b )\t ! z]o1 is h>dly to be d\bt$4 ,! lack ( docum5t>y evid5ce is n at all /range1 9 view ( all ( ! circum/ances4 99 99 99 99 99 #ia ,,*apt] ,,vi ,,! ,,developm5t ,,( ,,! ,,num]als ,,among ,,! ,,arabs ,if ! num]als _h _! orig9 9 ,india1 z seems mo/ probable1 :5 did ! ,arabs come to "k ( !m8 ,x is cu/om>y to say t x 0 due to ! influ5ce ( ,mohamm$anism t le>n+ spread ?r\gh ,p]sia & ,arabia2 & s x 01 9 "p4 ,b le>n+ 0 already respect$ 9 ^! c\ntries long be=e ,mohamm$ appe>$1 & comm]ce fl\ri%$ all ?r\gh ?is region4 ,9 ,p]sia1 = example1 ! reign ( ,khosru ,nu'sirwan1#cfd ! great contempor>y ( ,ju/9ian ! law-mak]1 0 *>act]iz$ n only by an improvem5t 9 social & economic conditions1 b by ! cultivation ( lett]s4 ,khosru fo/]$ le>n+1 invit+ to 8 c\rt s*ol>s f ,greece1 & enc\rag+ ! introduction ( culture f ! ,we/ z well z f ! ,ea/4 ,at ?is "t ,ari/otle & ,plato 7 translat$1 & portions ( ! ,hito-pade'sa1 or ,fables ( ,pilpay1 7 r5d]$ f ! ,sanskrit into ,p]sian4 ,all ?is means t "s ?ree c5turies be=e ! great intellectual asc5dancy ( ,bagdad a simil> fo/]+ ( le>n+ 0 tak+ place 9 ,p]sia1 & "u pre-,mohamm$an influ5ces4 #ib ,! fir/ def9ite trace t we h ( ! introduction ( ! ,h9du sy/em into ,arabia dates f #ggc ,a4,d41#cfe :5 an ,indian a/ronom] visit$ ! c\rt ( ! caliph1 br++ ) him a/ronomical tables :i* at ! caliph's comm& 7 translat$ into ,arabic by ,al-,fazari4#cff ,al-,kh[arazmi & ,h4aba% 7,ah4m$ ibn ,abdallah1 di$ c4 #hgj7 bas$ _! well-kn[n tables ^u ! "w ( ,al-,faz>i4 ,x may be ass]t$ z highly probable t ! num]als came at ! same "t z ! tables4 ,!y 7 c]ta9ly kn[n a few decades lat]1 & be=e #hbe ,a4,d41 ab\t :i* "t ! orig9al ( ! ,algoritmi de num]o ,indorum 0 writt51 z t "w makes no pret5se ( be+ ! fir/ "w to treat ( ! ,h9du num]als4 ,! ?ree writ]s m5tion$ cov] ! p]iod f ! end ( ! eigh? to ! end ( ! n9? c5tury4 ,:ile ! hi/orians ,al-,ma'sudi & ,al-,biruni foll[ q closely ^u ! m5 m5tion$1 x is well to note aga9 ! ,arab writ]s on ,h9du ari?metic1 contempor>y ) ,al-,kh[arazmi1 :o 7 m5tion$ 9 *apt] ,i1 viz4 ,al-,k9di1 ,s5$ ibn ,ali1 & ,al-,s4ufi4 ,= ov] five hundr$ ye>s ,arabic writ]s & o!rs cont9u$ to apply to works on ari?metic ! "n ,indian4 ,9 ! t5? c5tury su* writ]s are ,abdallah ibn al-,h4asan1 ,abu 'l-,qasim#cfg 7di$ #ihg ,a4,d47 ( ,antio*1 & ,moh4amm$ ibn ,abdallah1 ,abu ,nas4r#cfh 7c4 #ihb71 ( ,kalwada ne> ,bagdad4 ,o!rs ( ! same p]iod or #ic e>li] 7s9ce !y are m5tion$ 9 ! ,fihri/1#cfi #ihg ,a4,d471 :o explicitly use ! ^w ,h9du or ,indian1 are ,s9an ibn al-,fath4#cgj ( ,h4arran1 & ,ahm$ ibn ,om>1 al-,k>abisi4#cga ,9 ! elev5? c5tury come ,al-,biruni#cgb 7#igc-ajdh7 & ,ali ibn ,ah4m$1 ,abu 'l-,h4asan1 ,al-,nasawi#cgc 7c4 #ajcj74 ,! foll[+ c5tury br+s simil> works by ,i%aq ibn ,yusuf al-,s4ardafi#cgd & ,samu'il ibn ,yah4ya ibn ,abbas al-,ma4grebi al-,&alusi#cge 7c4 #aagd71 & 9 ! ?irte5? c5tury are ,abdallatif ibn ,yusuf ibn ,moh4amm$1 ,muwaffaq al-,din ,abu ,moh4amm$ al-,ba4gdadi#cgf 7c4 #abca71 & ,ibn al-,banna4#cgg ,! ,greek monk ,maximus ,planudes1 writ+ 9 ! fir/ half ( ! f\rte5? c5tury1 foll[$ ! ,arabic usage 9 call+ 8 "w ,indian ,ari?metic4#cgh ,!re 7 num]\s o!r ,arabic writ]s ^u ari?metic1 z t subject occupi$ "o ( ! high places among ! sci5ces1 b mo/ ( !m did n feel x necess>y to ref] to ! orig9 ( ! symbols1 ! k ( :i* might well h be5 tak5 = grant$4 #id ,"o docum5t1 cit$ by ,woepcke1#cgi is ( special int]e/ s9ce x %[s at an e>ly p]iod1 #igj ,a4,d41 ! use ( ! ord9>y ,arabic =ms alongside ! 4gobar4 ,! title ( ! "w is ,int]e/+ & ,beautiful ,problems on ,numb]s copi$ by ,ah4m$ ibn ,moh4amm$ ibn ,abdaljalil1 ,abu ,said1 al-,sijzi1#chj 7#iea-ajbd7 f a "w by a prie/ & physician1 ,naz4if ibn ,yumn1#cha al-,qass 7di$ c4 #iij74 ,sut] does n m5tion ?is "w ( ,naz4if4 ,! second reason = n ascrib+ too mu* cr$it to ! purely ,arab influ5ce is t ! ,arab by himself nev] %[$ any intellectual /r5g?4 ,:at took place aft] ,moh4amm$ _h light$ ! fire 9 ! he>ts ( 8 p 0 j :at always takes place :5 diff]5t types ( /rong races bl5d1--a great r5aissance 9 div]s l9es4 ,x 0 se5 9 ! bl5d+ ( su* types at ,miletus 9 ! "t ( ,?ales1 at ,rome 9 ! days ( ! e>ly invad]s1 at ,alex&ria :5 ! ,greek set firm foot on ,egyptian soil1 & we see x n[ :5 all ! nations m+le _! vitality 9 ! ,new ,_w4 ,s :5 ! ,arab culture jo9$ ) ! ,p]sian1 a new civilization rose & fl\ri%$4#chb ,! ,arab influ5ce came n f its purity1 b f its int]m+l+ ) an influ5ce m cultur$ if less virile4 ,z a result ( ?is int]activity among peoples ( div]se int]e/s & p[]s1 ,mohamm$anism 0 to ! _w f ! eigh? to ! ?irte5? c5tury :at ,rome & ,a!ns & ! ,italo-,hell5ic influ5ce g5]ally _h #ie be5 to ! anci5t civilization4 ,if !y did n possess ! _s ( inv5tion :i* di/+ui%$ ! ,greeks & ! ,h9dus1 if !y did n %[ ! p]sev]ance 9 _! obs]vations t *>act]iz$ ! ,*9ese a/ronom]s1 !y at lea/ possess$ ! virility ( a new & victori\s p1 ) a desire to und]/& :at o!rs _h accompli%$1 & a ta/e :i* l$ !m ) equal ardor to ! /udy ( algebra & ( poetry1 ( philosophy & ( language4#chc ,x 0 9 #fbb ,a4,d4 t ,moh4amm$ fl$ f ,mecca1 & )9 a c5tury f t "t ! cresc5t _h replac$ ! cross 9 ,*ri/ian ,asia1 9 ,nor!rn ,africa1 & 9 a goodly portion ( ,spa94 ,! ,arab empire 0 an ellipse ( le>n+ ) its foci at ,bagdad & ,cordova1 & its rul]s n infrequ5tly took pride 9 dem&+ intellectual r ?an comm]cial treasure z ! result ( conque/4#chd ,x 0 "u ^! influ5ces1 ei!r pre-,mohamm$an or lat]1 t ! ,h9du num]als f\nd _! way to ! ,nor?4 ,if !y 7 kn[n be=e ,moh4amm$'s "t1 ! pro( ( ?is fact is n[ lo/4 ,?is mu*1 h[ev]1 is kn[n1 t 9 ! eigh? c5tury !y 7 tak5 to ,bagdad4 ,x 0 e>ly 9 t c5tury t ! ,mohamm$ans obta9$ _! fir/ foo?old 9 nor!rn ,india1 ?us =e%ad[+ an epo* ( supremacy t endur$ ) v>i$ =tunes until aft] ! gold5 age ( ,akb> ! ,great 7#aedb-afje7 & ,%ah ,jehan4 ,!y also conqu]$ ,khorassan & ,afghani/an1 s t ! le>n+ & ! comm]cial cu/oms ( ,india at once f\nd easy #if access to ! newly-e/abli%$ s*ools & ! baza>s ( ,mesopotamia & we/]n ,asia4 ,! p>ticul> pa?s ( conque/ & ( comm]ce 7 ei!r by way ( ! ,khyb] ,pass & ?r\gh ,kabul1 ,h]at & ,khorassan1 or by sea ?r\gh ! /rait ( ,ormuz to ,basra 7,busra7 at ! head ( ! ,p]sian ,gulf1 & !nce to ,bagdad4 ,z a matt] ( fact1 "o =m ( ,arabic num]als1 ! "o n[ 9 use by ! ,arabs1 is attribut$ to ! influ5ce ( ,kabul1 :ile ! o!r1 :i* ev5tually became our num]als1 may v likely h rea*$ ,arabia by ! o!r r\te4 ,x is 9 ,bagdad1#che ,dar al-,salam--! ,abode ( ,peace1 t our special int]e/ 9 ! introduction ( ! num]als c5t]s4 ,built ^u ! ru9s ( an anci5t t[n by ,al-,mans4ur#chf 9 ! second half ( ! eigh? c5tury1 x lies 9 "o ( ^? regions :]e ! conv]g+ r\tes ( trade give rise to l>ge cities4#chg ,q z well ( ,bagdad z ( ,a!ns might ,c>d9al ,newman h said3#chh ,:at x lo/ 9 conv5i5ces ( approa*1 x ga9$ 9 its neighborhood to ! traditions ( ! my/]i\s ,ea/1 & 9 ! lovel9ess ( ! region 9 :i* x lay4 ,hi!r1 !n1 z to a sort ( ideal l&1 :]e all ar*etypes ( ! great & ! fair 7 f\nd 9 sub/antial be+1 & all dep>tm5ts ( tru? explor$1 & all div]sities ( intellectual p[] exhibit$1 :]e ta/e & philosophy 7 maje/ically en?ron$ z 9 a royal c\rt1 :]e !re 0 no sov]eignty b t ( m9d1 & no nobility b t ( g5ius1 :]e pr(essors 7 #ig rul]s1 & pr9ces did homage1 ?i!r flock$ cont9ually f ! v corn]s ( ! orbis t]r>um ! _m-tongu$ g5]ation1 j ris+1 or j ris5 into manhood1 9 ord] to ga9 wisdom4 ,= "h x 0 t ,al-,mans4ur & ,al-,mamun & ,harun al-,ra%id 7,a>on ! ,j7 made = a "t ! _w's c5t] ( intellectual activity 9 g5]al & 9 ! doma9 ( ma!matics 9 p>ticul>4#chi ,x 0 j aft] ! ,s9dh9d 0 br\ght to ,bagdad t ,moh4amm$ ibn ,musa al-,kh[arazmi1 ^: "n has already be5 m5tion$1#cij 0 call$ to t city4 ,he 0 ! mo/ celebrat$ ma!matician ( 8 "t1 ei!r 9 ! ,ea/ or ,we/1 writ+ treatises on ari?metic1 ! sundial1 ! a/rolabe1 *ronology1 geometry1 & algebra1 & giv+ ?r\gh ! ,lat9 translit]ation ( 8 "n1 algoritmi1 ! "n ( algorism to ! e>ly ari?metics us+ ! new ,h9du num]als4#cia ,appreciat+ at once ! value ( ! position sy/em s rec5tly br\ght f ,india1 he wrote an ari?metic bas$ ^u ^! num]als1 & ?is 0 translat$ into ,lat9 9 ! "t ( ,adelh>d ( ,ba? 7c4 #aahj71 al?\gh possibly by 8 contempor>y c\ntryman ,rob]t ,ce/r5sis4#cib ,?is translation 0 f\nd 9 ,cambridge & 0 publi%$ by ,boncompagni 9 #aheg4cic ,contempor>y ) ,al-,kh[arazmi1 & work+ also "u ,al-,mamun1 0 a ,jewi% a/ronom]1 ,abu 'l-,t4eiyib1 #ih ,s5$ ibn ,ali1 :o is said to h adopt$ ! ,mohamm$an religion at ! caliph's reque/4 ,he also wrote a "w on ,h9du ari?metic1#cid s t ! subject mu/ h be5 attract+ consid]able att5tion at t "t4 ,inde$1 ! /ruggle to h ! ,h9du num]als replace ! ,arabic did n cease = a long "t !reaft]4 ,ali ibn ,ah4m$ al-,nasawi1 9 8 ari?metic ( c4 #ajbe1 tells u t ! symbolism ( numb] 0 /ill unsettl$ 9 8 day1 al?\gh mo/ p pref]r$ ! /rictly ,arabic =ms4#cie ,we ?us h ! num]als 9 ,arabia1 9 two =ms3 "o ! =m n[ us$ !re1 & ! o!r ! "o us$ by ,al-,kh[arazmi4 ,! "q !n rema9s1 h[ did ?is second =m f9d its way into ,europe8 & ?is "q w be consid]$ 9 ! next *apt]4 99 99 99 99 99 #ii ,,*apt] ,,vii ,,! ,,def9ite ,,introduction ,,( ,,! ,,num]als ,,into ,,europe ,x be+ d\btful :e!r ,boe?ius ev] knew ! ,h9du num]al =ms1 c]ta9ly )\t ! z]o 9 any case1 x becomes necess>y n[ to consid] ! "q ( _! def9ite introduction into ,europe4 ,f :at has be5 said ( ! trade relations betwe5 ! ,ea/ & ! ,we/1 & ( ! probability t x 0 ! trad] r ?an ! s*ol> :o c>ri$ ^! num]als f _! orig9al habitat to v>i\s comm]cial c5t]s1 x is evid5t t we %all nev] "k :5 !y fir/ made _! inconspicu\s entrance into ,europe4 ,curi\s cu/oms f ! ,ea/ & f ! tropics1--conc]n+ games1 social peculi>ities1 oddities ( dress1 & ! l1--are cont9ually be+ relat$ by sailors & trad]s 9 _! resorts 9 ,new ,york1 ,london1 ,hamburg1 & ,rott]dam to-day1 cu/oms t no s*ol> has yet describ$ 9 pr9t & t may n become kn[n = _m ye>s1 if ev]4 ,& if ?is be s n[1 h[ mu* m w\ld x h be5 true a ?\s& ye>s be=e ! inv5tion ( pr9t+1 :5 le>n+ 0 at its l[e/ ebb4 ,x 0 at ?is p]iod ( l[ e/eem ( culture t ! ,h9du num]als und\bt$ly made _! fir/ appe>ance 9 ,europe4 ,!re 7 _m opportunities = su* k to rea* ,spa9 & ,italy4 ,9 ! fir/ place ! ,moors w5t into ,spa9 z help]s ( a claimant ( ! ?rone1 & #ajj rema9$ z conqu]ors4 ,! p[] ( ! ,go?s1 :o _h held ,spa9 = ?ree c5turies1 0 %att]$ at ! battle ( ,j]ez de la ,front]a 9 #gaa1 & almo/ imm$iately ! ,moors became ma/]s ( ,spa9 & s rema9$ = five hundr$ ye>s1 & ma/]s ( ,granada = a mu* long] p]iod4 ,until #hej ! ,*ri/ians 7 absolutely free z to religion & z to hold+ political (fice1 s t prie/s & monks 7 n infrequ5tly skill$ bo? 9 ,lat9 & ,arabic1 act+ z (ficial translators1 & naturally report+ directly or indirectly to ,rome4 ,!re 0 inde$ at ?is "t a compla9t t ,*ri/ian y\?s cultivat$ too assidu\sly a love = ! lit]ature ( ! ,s>ac51 & m>ri$ too frequ5tly ! daught]s ( ! infidel4#cif ,x is true t ?is happy /ate ( affairs 0 n p]man5t1 b :ile x la/$ ! le>n+ & ! cu/oms ( ! ,ea/ mu/ h become m or less ! prop]ty ( ,*ri/ian ,spa94 ,at ?is "t ! 4gobar num]als 7 probably 9 t c\ntry1 & ^! may well h made _! way into ,europe f ! s*ools ( ,cordova1 ,granada1 & ,tol$o4 ,fur!rmore1 !re 0 abundant opportunity = ! num]als ( ! ,ea/ to rea* ,europe ?r\gh ! j\rneys ( travel]s & ambassadors4 ,x 0 f ! records ( ,suleiman ! ,m]*ant1 a well-kn[n ,arab trad] ( ! n9? c5tury1 t "p ( ! /ory ( ,s9dbad ! ,sailor 0 tak54#cig ,su* a m]*ant w\ld h be5 p>ticul>ly likely to "k ! num]als ( ! p :om he met1 & he is a type ( man t may well h tak5 su* symbols to ,european m>kets4 ,a little lat]1 #aja ,abu 'l-,h4asan ,ali al-,masudi 7d4 #ief7 ( ,bagdad travel$ to ! ,*9a ,sea on ! ea/1 at lea/ z f> s\? z ,zanzib>1 & to ! ,atlantic on ! we/1#cih & he speaks ( ! n9e figures ) :i* ! ,h9dus reckon$4#cii ,!re 0 also a ,bagdad m]*ant1 "o ,abu 'l-,qasim ,obeidallah ibn ,ah4m$1 bett] kn[n by 8 ,p]sian "n ,ibn ,khordad4beh1#djj :o wrote ab\t #hej ,a4,d4 a "w entitl$ ,book ( ,roads & ,prov9ces#dja 9 :i* ! foll[+ graphic acc\nt appe>s3#djb ,! ,jewi% m]*ants speak ,p]sian1 ,roman 7,greek & ,lat971 ,arabic1 ,fr5*1 ,spani%1 & ,slavic4 ,!y travel f ! ,we/ to ! ,ea/1 & f ! ,ea/ to ! ,we/1 sometimes by l&1 sometimes by sea4 ,!y take %ip f ,france on ! ,we/]n ,sea1 & !y voyage to ,f>ama 7ne> ! ru9s ( ! anci5t ,pelusium72 !re !y transf] _! goods to c>avans & g by l& to ,colzom 7on ! ,r$ ,sea74 ,!y !re reemb>k on ! ,ori5tal 7,r$7 ,sea & g to ,hejaz & to ,jiddah1 & !nce to ! ,s9d1 ,india1 & ,*9a4 ,return+1 !y br+ back ! products ( ! ori5tal l&s4444 ,^! j\rneys are also made by l&4 ,! m]*ants1 leav+ ,france & ,spa91 cross to ,tangi] & !nce pass ?r\gh ! ,african prov9ces & ,egypt4 ,!y !n g to ,ramleh1 visit ,damascus1 ,kufa1 ,bagdad1 & ,basra1 p5etrate into ,ahwaz1 ,f>s1 ,k]man1 ,s9d1 & ?us rea* ,india & ,*9a4 ,su* travel]s1 ab\t #ijj ,a4,d41 mu/ necess>ily h spread abroad a k ( all numb] #ajb sy/ems us$ 9 record+ prices or 9 ! computations ( ! m>ket4 ,!re is an int]e/+ witness to ?is movem5t1 a cruci=m broo* n[ 9 ! ,briti% ,museum4 ,x is ,engli%1 c]ta9ly z e>ly z ! elev5? c5tury1 b x is inlaid ) a piece ( pa/e on :i* is ! ,mohamm$an inscription1 9 ,kufic *>act]s1 ,!re is no ,god b ,god4 ,h[ did su* an inscription f9d its way1 p]haps 9 ! "t ( ,alcu9 ( ,york1 to ,engl&8 ,& if ^! ,kufic *>act]s rea*$ !re1 !n :y n ! num]al =ms z well8 ,ev5 9 lit]ature ( ! bett] class !re appe>s n[ & !n "s /ray pro( ( ! important fact t ! great trade r\tes to ! f> ,ea/ 7 nev] clos$ = long1 & t ! cu/oms & m>ks ( trade endur$ f g5]ation to g5]ation4 ,! ,guli/an ( ! ,p]sian poet ,sadi#djc conta9s su* a passage3 ,i met a m]*ant :o own$ "o hundr$ & =ty camels1 & fifty slaves & port]s4444 ,he answ]$ to me3 ',i want to c>ry sulphur ( ,p]sia to ,*9a1 :i* 9 t c\ntry1 z ,i he>1 be>s a high price2 & !nce to take ,*9ese w>e to ,r\m2 & f ,r\m to load up ) brocades = ,h9d2 & s to trade ,indian /eel 7pulab7 to ,halib4 ,f ,halib ,i w convey its glass to ,yeman1 & c>ry ! pa9t$ clo?s ( ,yeman back to ,p]sia4'#djd ,on ! o!r h&1 ^! m5 7 n ( ! le>n$ class1 nor w\ld !y pres]ve 9 treatises any k t !y might h1 al?\gh ?is k w\ld occasionally rea* ! e>s ( ! le>n$ z bits ( curi\s in=mation4 #ajc ,!re 7 also ambassadors pass+ back & =? f "t to "t1 betwe5 ! ,ea/ & ! ,we/1 & 9 p>ticul> dur+ ! p]iod :5 ^! num]als probably began to ent] ,europe4 ,?us ,*>lemagne 7c4 #hjj7 s5t emiss>ies to ,bagdad j at ! "t ( ! op5+ ( ! ma!matical activity !re4#dje ,& ) su* ambassadors mu/ h gone ! adv5tur\s s*ol>1 inspir$1 z ,alcu9 says ( ,ar*bi%op ,alb]t ( ,york 7#gff-ghj71djf to seek ! le>n+ ( o!r l&s4 ,fur!rmore1 ! ,ne/orian communities1 e/abli%$ 9 ,ea/]n ,asia & 9 ,india at ?is "t1 7 favor$ bo? by ! ,p]sians & by _! ,mohamm$an conqu]ors4 ,! ,ne/orian ,patri>* ( ,syria1 ,timo!us 7#ggh-hbj71 s5t mission>ies bo? to ,india & to ,*9a1 & a bi%op 0 appo9t$ = ! latt] field4 ,ibn ,wahab1 :o travel$ to ,*9a 9 ! n9? c5tury1 f\nd images ( ,*ri/ & ! apo/les 9 ! ,emp]or's c\rt4#djg ,su* a le>n$ body ( m51 kn[+ intimately ! c\ntries 9 :i* !y labor$1 c\ld h>dly h fail$ to make /range cu/oms kn[n z !y return$ to _! home /ations4 ,!n1 too1 9 ,alfr$'s "t 7#hdi-ija7 emiss>ies w5t #ajd f ,engl& z f> z ,india1#djh & g5]ally 9 ! ,middle ,ages groc]ies came to ,europe f ,asia z n[ !y come f ! colonies & f ,am]ica4 ,syria1 ,asia ,m9or1 & ,cyprus furni%$ sug> & wool1 & ,india yield$ h] p]fumes & spices1 :ile ri* tape/ries = ! c\rts & ! weal?y burgh]s came f ,p]sia & f ,*9a4#dji ,ev5 9 ! "t ( ,ju/9ian 7c4 #eej7 !re seems to h be5 a silk trade ) ,*9a1 :i* c\ntry 9 turn c>ri$ on comm]ce ) ,ceylon1#daj & rea*$ out to ,turke/an :]e o!r m]*ants transmitt$ ! ,ea/]n products we/w>d4 ,9 ! sev5? c5tury !re 0 a well-def9$ comm]ce betwe5 ,p]sia & ,india1 z well z betwe5 ,p]sia & ,con/ant9ople4#daa ,! ,byzant9e comm]ci>ii 7 /ation$ at ! outpo/s n m]ely z cu/oms (fic]s b z gov]nm5t pur*as+ ag5ts4#dab ,occasionally !re w5t along ^! r\tes ( trade m5 ( real le>n+1 & su* w\ld surely h c>ri$ ! k ( _m cu/oms back & =?4 ,?us at a p]iod :5 ! num]als are kn[n to h be5 p>tly und]/ood 9 ,italy1 at ! op5+ ( ! elev5? c5tury1 "o ,con/ant9e1 an ,african1 travel$ f ,italy ?r\gh a great "p ( ,africa & ,asia1 ev5 on to ,india1 = ! purpose ( le>n+ ! sci5ces ( ! ,ori5t4 ,he sp5t ?irty-n9e ye>s 9 travel1 hav+ be5 hospitably receiv$ 9 ,babylon1 & ^u 8 return he 0 welcom$ ) great honor at ,sal]no4#dac ,a v int]e/+ illu/ration ( ?is int]c\rse also appe>s 9 ! t5? c5tury1 :5 ! son ( ,otto ,i #aje 7#icf-igc7 m>ri$ a pr9cess f ,con/ant9ople4 ,?is mon>* 0 9 t\* ) ! ,moors ( ,spa9 & invit$ to 8 c\rt num]\s s*ol>s f abroad1#dad & 8 int]c\rse ) ! ,ea/ z well z ! ,we/ mu/ h br\ght toge!r mu* ( ! le>n+ ( ea*4 ,ano!r p[]ful means = ! circulation ( my/icism & philosophy1 & m or less ( culture1 took its />t j be=e ! conv]sion ( ,con/ant9e 7c4 #cab71 9 ! =m ( ,*ri/ian pilgrim travel4 ,?is 0 a feature peculi> to ! zealots ( e>ly ,*ri/ianity1 f\nd 9 only a slight degree among _! ,jewi% pr$ecessors 9 ! annual pilgrimage to ,j]usalem1 & almo/ :olly want+ 9 o!r pre-,*ri/ian peoples4 ,*ief among ^! e>ly pilgrims 7 ! two ,plac5tians1 ,john & ,anton9e ! ,eld] 7c4 #cjc71 :o1 9 _! w&]+s to ,j]usalem1 seem to h />t$ a movem5t :i* culm9at$ c5turies lat] 9 ! crusades4#dae ,9 #ccc a ,bordeaux pilgrim compil$ ! fir/ ,*ri/ian guide-book1 ! ,it9]>y f ,bordeaux to ,j]usalem1#daf & f ?is "t on ! holy pilgrimage nev] entirely ceas$4 ,/ill ano!r c]ta9 r\te = ! entrance ( ! num]als into ,*ri/ian ,europe 0 ?r\gh ! pillag+ & trad+ c>ri$ on by ! ,arabs on ! nor!rn %ores ( ! ,m$it]ranean4 ,z e>ly z #feb ,a4,d41 9 ! ?irtie? ye> ( ! ,hejira1 ! ,mohamm$ans desc5d$ ^u ! %ores ( ,sicily & took mu* spoil4 ,h>dly _h ! wret*$ ,con/ans giv5 place to ! #ajf "y ,con/ant9e ,,iv :5 !y aga9 attack$ ! isl& & plund]$ anci5t ,syracuse4 ,aga9 9 #hbg1 "u ,asad1 !y ravag$ ! coa/s4 ,al?\gh at ?is "t !y fail$ to conqu] ,syracuse1 !y soon held a good "p ( ! isl&1 & a little lat] !y successfully besieg$ ! city4 ,be=e ,syracuse fell1 h[ev]1 !y _h plund]$ ! %ores ( ,italy1 ev5 to ! walls ( ,rome itself2 & _h n ,leo ,,iv1 9 #hdi1 repair$ ! neglect$ =tifications1 ! effects ( ! ,moslem raid ( t ye> might h be5 v f>-rea*+4 ,ibn ,khordad4beh1 :o left ,bagdad 9 ! latt] "p ( ! n9? c5tury1 gives a picture ( ! great comm]cial activity at t "t 9 ! ,s>ac5 city ( ,pal]mo4 ,9 ?is same c5tury !y _h e/abli%$ !mselves 9 ,pi$mont1 & 9 #ijf !y pillag$ ,tur94#dag ,on ! ,sorr5to p59sula ! travel] :o climbs ! hill to ! beautiful ,ravello sees /ill sev]al traces ( ! ,arab ar*itecture1 rem9d+ him ( ! fact t ab\t #ijj ,a4,d4 ,amalfi 0 a comm]cial c5t] ( ! ,moors4#dah ,n only at ?is "t1 b ev5 a c5tury e>li]1 ! arti/s ( nor!rn ,india sold _! w>es at su* c5t]s1 & 9 ! c\rts bo? ( ,harun al-,ra%id & ( ,*>lemagne4#dai ,?us ! ,arabs dom9at$ ! ,m$it]ranean ,sea long be=e ,v5ice held ! gorge\s ,ea/ 9 fee ,& 0 ! safegu>d ( ! ,we/1 & long be=e ,g5oa _h become h] p[]ful rival4#dbj #ajg ,only a little lat] ?an ?is ! bro!rs ,nicolo & ,maffeo ,polo ent]$ ^u _! fam\s w&]+s4#dba ,leav+ ,con/ant9ople 9 #abfj1 !y w5t by ! ,sea ( ,azov to ,bokh>a1 & !nce to ! c\rt ( ,kublai ,khan1 p5etrat+ ,*9a1 & return+ by way ( ,acre 9 #abfi ) a commission :i* requir$ !m to g back to ,*9a two ye>s lat]4 ,?is "t !y took ) !m ,nicolo's son ,m>co1 ! hi/orian ( ! j\rney1 & w5t across ! plateau ( ,pamir2 !y sp5t ab\t tw5ty ye>s 9 ,*9a1 & came back by sea f ,*9a to ,p]sia4 ,! v5tures ( ! ,poli 7 n long unique1 h[ev]3 ! ?irte5? c5tury _h n clos$ be=e ,roman mission>ies & ! m]*ant ,petrus de ,lucolongo _h p5etrat$ ,*9a4 ,be=e #acej ! company ( mission>ies 0 l>ge1 conv]ts 7 num]\s1 *ur*es & ,franciscan conv5ts _h be5 organiz$ 9 ! ,ea/1 travel]s 7 appeal+ = ! tru? ( _! acc\nts to ! _m p]sons 9 ,v5ice :o _h be5 9 ,*9a1 ,tsuan-*au-fu _h a ,european m]*ant community1 & ,italian trade & travel to ,*9a 0 a ?+ t occupi$ two *apt]s ( a comm]cial h&book4#dbb #ajh ,x is !re=e reasonable to conclude t 9 ! ,middle ,ages1 z 9 ! "t ( ,boe?ius1 x 0 a simple matt] = any inquir+ s*ol> to become acqua9t$ ) su* num]als ( ! ,ori5t z m]*ants may h us$ = w>eh\se or price m>ks4 ,& ! fact t ,g]b]t seems to h kn[n only ! =ms ( ! simple/ ( ^!1 n compreh5d+ _! full significance1 seems to prove t he pick$ !m up 9 j ?is way4 ,ev5 if ,g]b]t did n br+ 8 k ( ! ,ori5tal num]als f ,spa91 he may easily h obta9$ !m f ! m>ks on m]*ant's goods1 _h he be5 s incl9$4 ,su* k 0 probably obta9able 9 v>i\s p>ts ( ,italy1 ?\gh z p>ts ( m]e m]cantile k ! =ms might soon h be5 lo/1 x ne$+ ! p5 ( ! s*ol> to pres]ve !m4 ,trade at ?is "t 0 n /agnant4 ,dur+ ! elev5? & twelf? c5turies ! ,slavs1 = example1 _h v great comm]cial int]e/s1 _! trade rea*+ to ,kiev & ,novgorod1 & !nce to ! ,ea/4 ,con/ant9ople 0 a great cle>+-h\se ( comm]ce ) ! ,ori5t1#dbc & ! ,byzant9e m]*ants mu/ h be5 entirely famili> ) ! v>i\s num]als ( ! ,ea/]n peoples4 ,9 ! elev5? c5tury ! ,italian t[n ( ,amalfi e/abli%$ a factory#dbd 9 ,con/ant9ople1 & _h trade relations ) ,antio* & ,egypt4 ,v5ice1 z e>ly z ! n9? c5tury1 _h a valuable trade ) ,syria & ,cairo4#dbe ,fifty ye>s aft] ,g]b]t di$1 9 ! "t ( ,cnut1 ! ,dane & ! ,norwegian pu%$ _! comm]ce f> beyond ! nor!rn seas1 bo? by c>avans ?r\gh ,russia to ! ,ori5t1 & by _! v5turesome b>ks :i* #aji sail$ ?r\gh ! ,/rait ( ,gibralt> into ! ,m$it]ranean4#dbf ,only a little lat]1 probably be=e #abjj ,a4,d41 a cl]k 9 ! s]vice ( ,?omas a ,becket1 pres5t at ! latt]'s dea?1 wrote a life ( ! m>tyr1 to :i* 7=tunately = our purposes7 he prefix$ a brief eulogy ( ! city ( ,london4#dbg ,?is cl]k1 ,william ,fitz ,/eph5 by "n1 ?us speaks ( ! ,briti% capital3 ,aurum mittit ,arabs3 species et ?ura ,sabaeus3 ,arma ,sy!s3 oleum palm>um divite sylva ,p+ue solum ,babylon3 ,nilus lapides pretiosos3 ,norwegi1 ,russi1 v>ium grisum1 sabd9as3 ,s]es1 purpureas ve/es3 ,galli1 sua v9a4 ,al?\gh1 z a matt] ( fact1 ! ,arabs _h no gold to s5d1 & ! ,scy?ians no arms1 & ,egypt no preci\s /ones save only ! turquoise1 ! ,*9ese 7,s]es7 may h s5t _! purple ve/m5ts1 & ! nor? h] sables & o!r furs1 & ,france h] w9es4 ,at any rate ! v]ses %[ v cle>ly an ext5sive =eign trade4 ,!n !re 7 ! ,crusades1 :i* 9 ^! times br\ght ! ,ea/ 9 t\* ) ! ,we/4 ,! _s ( ! ,ori5t %[$ itself 9 ! songs ( ! tr\bad\rs1 & ! baudek91#dbh ! canopy ( ,bagdad1#dbi became common 9 ! *ur*es ( ,italy4 ,9 ,sicily & 9 ,v5ice ! textile indu/ries ( ! ,ea/ f\nd place1 & made _! way ev5 to ! ,sc&9avian p59sula4#dcj ,we !re=e h ?is /ate ( affairs3 ,!re 0 abundant int]c\rse betwe5 ! ,ea/ & ,we/ = #aaj "s c5turies be=e ! ,h9du num]als appe> 9 any manuscripts 9 ,*ri/ian ,europe4 ,! num]als mu/ ( necessity h be5 kn[n to _m trad]s 9 a c\ntry l ,italy at lea/ z e>ly z ! n9? c5tury1 & probably ev5 e>li]1 b !re 0 no reason = pres]v+ !m 9 treatises4 ,!re=e :5 a man l ,g]b]t made !m kn[n to ! s*ol>ly circles1 he 0 m]ely describ+ :at _h be5 famili> 9 a small way to _m p 9 a diff]5t walk ( life4 ,s9ce ,g]b]t#dca 0 = a long "t ?\ght to h be5 ! "o to introduce ! num]als into ,italy1#dcb a brief sket* ( ?is unique "* is prop]4 ,born ( humble p>5ts1#dcc ?is rem>kable man became ! c\nselor & companion ( k+s1 & f9ally wore ! papal ti>a z ,sylve/] ,,ii1 f #iii until 8 dea? 9 #ajjc4dcd ,he 0 e>ly br\ght "u ! influ5ce ( ! monks at ,aurillac1 & p>ticul>ly ( ,raimund1 :o _h be5 a pupil ( ,odo ( ,cluny1 & !re 9 due "t he himself took holy ord]s4 ,he visit$ ,spa9 9 ab\t #ifg 9 company ) ,c\nt ,borel1#dce rema9+ !re ?ree ye>s1 #aaa & /udy+ "u ,bi%op ,hatto ( ,vi*1#dcf a city 9 ! prov9ce ( ,b>celona1#dcg !n entirely "u ,*ri/ian rule4 ,inde$1 all ( ,g]b]t's te/imony is z to ! influ5ce ( ! ,*ri/ian civilization ^u 8 education4 ,?us he speaks (t5 ( 8 /udy ( ,boe?ius1#dch s t if ! latt] knew ! num]als ,g]b]t w\ld h le>n$ !m f him4#dci ,if ,g]b]t _h /udi$ 9 any ,moori% s*ools he w\ld1 "u ! decree ( ! emir ,hi%am 7#ghg-hbb71 h be5 oblig$ to "k ,arabic1 :i* w\ld h tak5 mo/ ( 8 ?ree ye>s 9 ,spa91 & ( :i* /udy we h n ! slighte/ h9t 9 any ( 8 lett]s4#ddj ,on ! o!r h&1 ,b>celona 0 ! only ,*ri/ian prov9ce 9 imm$iate t\* ) ! ,moori% civilization at t "t4#dda ,fur!rmore we "k t e>li] 9 ! same c5tury ,k+ ,alonzo ( ,a/urias 7d4 #iaj7 confid$ ! education ( 8 son ,ordono to ! ,arab s*ol>s ( ! c\rt ( ! #aab wali ( ,s>agossa1#ddb s t !re 0 m or less ( fri5dly relation betwe5 ,*ri/ian & ,moor4 ,aft] 8 ?ree ye>s 9 ,spa91 ,g]b]t w5t to ,italy1 ab\t #igj1 :]e he met ,pope ,john ,,xiii1 be+ by him pres5t$ to ! emp]or ,otto ,i4 ,two ye>s lat] 7#igb71 at ! emp]or's reque/1 he w5t to ,rheims1 :]e he /udi$ philosophy1 assi/+ to make ( t place an educational c5t]2 & 9 #ihc he became abbot at ,bobbio4 ,! next ye> he return$ to ,rheims1 & became ar*bi%op ( t diocese 9 #iia4 ,= political reasons he return$ to ,italy 9 #iif1 became ar*bi%op ( ,rav5na 9 #iih1 & ! foll[+ ye> 0 elect$ to ! papal *air4 ,f> ahead ( 8 age 9 wisdom1 he suff]$ z _m su* s*ol>s h ev5 9 times n s remote by be+ accus$ ( h]esy & wit*craft4 ,z late z #aebb1 9 a biography publi%$ at ,v5ice1 x is relat$ t by black art he atta9$ ! papacy1 aft] hav+ giv5 8 s\l to ! devil4#ddc ,g]b]t 01 h[ev]1 int]e/$ 9 a/rology1#ddd al?\gh ?is 0 m]ely ! a/ronomy ( t "t & 0 su* a sci5ce z any le>n$ man w\ld wi% to "k1 ev5 z to-day we wi% to be reasonably famili> ) physics & *emi/ry4 ,t ,g]b]t & 8 pupils knew ! 4gobar num]als is a fact no long] op5 to controv]sy4#dde ,b]nel9us & ,ri*]#ddf call !m by ! well-kn[n "n ( #aac c>act]es1 a ^w us$ by ,radulph ( ,laon 9 ! same s5se a c5tury lat]4#ddg ,x is probable t ,g]b]t 0 ! fir/ to describe ^! 4gobar num]als 9 any sci5tific way 9 ,*ri/ian ,europe1 b )\t ! z]o4 ,if he knew ! latt] he c]ta9ly did n und]/& its use4#ddh ,! "q /ill to be settl$ is z to :]e he f\nd ^! num]als4 ,t he did n br+ !m f ,spa9 is ! op9ion ( a numb] ( c>eful inve/igators4#ddi ,?is is ?\ght to be ! m probable because mo/ ( ! m5 :o made ,spa9 fam\s = le>n+ liv$ aft] ,g]b]t 0 !re4 ,su* 7 ,ibn ,sina 7,avic5na7 :o liv$ at ! beg9n+1 & ,g]b] ( ,seville :o fl\ri%$ 9 ! middle1 ( ! elev5? c5tury1 & ,abu ,ro%d 7,av]roes7 :o liv$ at ! end ( ! twelf?4#dej ,o!rs hold t 8 proximity to #aad ! ,arabs = ?ree ye>s makes x probable t he assimilat$ "s ( _! le>n+1 9 spite ( ! fact t ! l9es betwe5 ,*ri/ian & ,moor at t "t 7 %>ply drawn4#dea ,writ]s fail1 h[ev]1 to recognize t a comm]cial num]al sy/em w\ld h be5 m likely to be made kn[n by m]*ants ?an by s*ol>s4 ,! it9]ant p$dl] knew no =bidd5 pale 9 ,spa91 any m ?an he has kn[n "o 9 o!r l&s4 ,if ! 4gobar num]als 7 us$ = m>k+ w>es or keep+ simple acc\nts1 x 0 he :o w\ld h kn[n !m1 & :o w\ld h be5 ! "o r ?an any ,arab s*ol> to br+ !m to ! inquir+ m9d ( ! "y ,fr5* monk4 ,! facts t ,g]b]t knew !m only imp]fectly1 t he us$ !m solely = calculations1 & t ! =ms are evid5tly l ! ,spani% 4gobar1 make x all ! m probable t x 0 ?r\gh ! small tradesman ( ! ,moors t ?is v]satile s*ol> d]iv$ 8 k4 ,moreov] ! "p ( ! geometry be>+ 8 "n1 & t seems unque/ionably 81 %[s ! ,arab influ5ce1 prov+ t he at lea/ came into contact ) ! transplant$ ,ori5tal le>n+1 ev5 ?\gh imp]fectly4#deb ,!re 0 also ! p]si/5t ,jewi% m]*ant trad+ ) bo? peoples !n z n[1 always alive to ! acquir+ ( useful k1 & x w\ld be v natural = a man l ,g]b]t to welcome le>n+ f su* a s\rce4 ,on ! o!r h&1 ! two lead+ s\rces ( in=mation z to ! life ( ,g]b]t reveal practically no?+ to %[ t he came )9 ! ,moori% sph]e ( influ5ce dur+ 8 soj\rn 9 ,spa94 ,^! s\rces #aae are 8 lett]s & ! hi/ory writt5 by ,ri*]4 ,g]b]t 0 a ma/] ( ! epi/ol>y art1 & 8 exalt$ position l$ to ! pres]vation ( 8 lett]s to a degree t w\ld n h be5 v\*saf$ ev5 by _! classic excell5ce4#dec ,ri*] 0 a monk at ,/4 ,remi de ,rheims1 & 0 d\btless a pupil ( ,g]b]t4 ,! latt]1 :5 ar*bi%op ( ,rheims1 ask$ ,ri*] to write a hi/ory ( 8 times1 & ?is 0 done4 ,! "w lay 9 manuscript1 entirely =gott5 until ,p]tz discov]$ x at ,bamb]g 9 #ahcc4ded ,! "w is d$icat$ to ,g]b]t z ar*bi%op ( ,rheims1#dee & w\ld assur$ly h te/ifi$ to su* ef=ts z he may h made to secure ! le>n+ ( ! ,moors4 ,n[ x is a fact t nei!r ! lett]s nor ?is hi/ory makes any /atem5t z to ,g]b]t's contact ) ! ,s>ac5s4 ,! lett]s d n speak ( ! ,moors1 ( ! ,arab num]als1 nor ( ,cordova4 ,spa9 is n ref]r$ to by t "n1 & only "o ,spani% s*ol> is m5tion$4 ,9 "o ( 8 lett]s he speaks ( ,joseph ,ispanus1#def or ,joseph ,sapi5s1 b :o ?is ,joseph ! ,wise ( ,spa9 may h be5 we d n "k4 ,possibly #aaf x 0 he :o contribut$ ! morsel ( k s imp]fectly assimilat$ by ! "y ,fr5* monk4#deg ,)9 a few ye>s aft] ,g]b]t's visit two "y ,spani% monks ( less] fame1 & d\btless ) n t ke5 int]e/ 9 ma!matical matt]s :i* ,g]b]t _h1 reg>d$ ! app>5tly slight k :i* !y _h ( ! ,h9du num]al =ms z wor?y ( some:at p]man5t record#deh 9 manuscripts :i* !y 7 transcrib+4 ,! fact t su* k _h p5etrat$ to _! mode/ cloi/]s 9 nor!rn ,spa9--! "o ,albelda or ,albaida--indicates t x 0 r widely diffus$4 ,g]b]t's treatise ,libellus de num]orum divisione#dei is *>act]iz$ by ,*asles z "o ( ! mo/ obscure docum5ts 9 ! hi/ory ( sci5ce4#dfj ,! mo/ complete in=mation 9 reg>d to ?is & ! o!r ma!matical works ( ,g]b]t is giv5 by ,bubnov1#dfa :o consid]s ?is "w to be g5u9e4#dfb #aag ,s little did ,g]b]t appreciate ^! num]als t 9 8 works kn[n z ! ,regula de abaco computi & ! ,libellus he makes no use ( !m at all1 employ+ only ! ,roman =ms4#dfc ,nev]!less ,b]nel9us#dfd ref]s to ! n9e 4gobar *>act]s4#dfe ,^! ,g]b]t _h m>k$ on a ?\s& jetons or c\nt]s1#dff us+ ! latt] on an abacus :i* he _h a sign-mak] prep>e = him4#dfg ,in/ead ( putt+ eight c\nt]s 9 say ! t5s' column1 ,g]b]t w\ld put a s+le c\nt] m>k$ #h1 & s = ! o!r places1 leav+ ! column empty :]e we w\ld place a z]o1 b :]e he1 lack+ ! z]o1 _h no c\nt] to place4 ,^! c\nt]s he possibly call$ c>act]es1 a "n :i* adh]$ also to ! figures !mselves4 ,x is an int]e/+ speculation to consid] :e!r ^! apices1 z !y are call$ 9 ! ,boe?ius int]polations1 7 9 any way sugge/$ by ^? ,roman jetons g5]ally kn[n 9 numismatics z tess]ae1 & be>+ ! figures ,i-,,xvi1 ! sixte5 ref]r+ to ! numb] ( assi 9 a se/]tius4#dfh ,! #aah "n apices adh]$ to ! ,h9du-,arabic num]als until ! sixte5? c5tury4#dfi ,to ! figures on ! apices 7 giv5 ! names ,ig91 &ras1 ormis1 arbas1 quimas1 calctis or caltis1 z5is1 tem5ias1 cel5tis1 sipos1#dgj ! orig9 & mean+ ( :i* /ill rema9 a my/]y4 ,! ,semitic orig9 ( sev]al ( ! words seems probable4 ,wahud1 ?ane9e1 #aai ?alata1 arba1 kumsa1 setta1 sebba1 tim9ia1 taseud are giv5 by ! ,rev4 ,r4 ,patrick#dga z ! names1 9 an ,arabic dialect us$ 9 ,morocco1 = ! num]als f "o to n9e4 ,( ^! ! words = f\r1 five1 & eight are /rik+ly l ^? giv5 above4 ,! "n apices 0 n1 h[ev]1 a common "o 9 lat] times4 ,notae 0 m (t5 us$1 & x f9ally gave ! "n to notation4#dgb ,/ill m common 7 ! names figures1 ciph]s1 signs1 elem5ts1 & *>act]s4#dgc ,s little effect did ! tea*+s ( ,g]b]t h 9 mak+ kn[n ! new num]als1 t ,o',creat1 :o liv$ a c5tury lat]1 a fri5d & pupil ( ,adelh>d #abj ( ,ba?1 us$ ! z]o ) ! ,roman *>act]s1 9 contra/ to ,g]b]t's use ( ! 4gobar =ms )\t ! z]o4#dgd ,o',creat uses ?ree =ms = z]o1 o1 o1 & ,greek3 t1 z 9 ,maximus ,planudes4 ,) ?is use ( ! z]o goes1 naturally1 a place value1 = he writes ,,iii ,,iii = #cc1 ,,iccoo & ,i4 ,,ii4 tau4 tau = #abjj1 ,i4 ,o4 ,,viii4 ,,ix = #ajhi1 & ,i4 ,,iiii4 ,,iiii4 tautautautau = ! squ>e ( #abjj4 ,! p]iod f ! "t ( ,g]b]t until aft] ! appe>ance ( ,leon>do's monum5tal "w may be call$ ! p]iod ( ! abaci/s4 ,ev5 = _m ye>s aft] ! appe>ance e>ly 9 ! twelf? c5tury ( ! books expla9+ ! ,h9du art ( reckon+1 !re 0 /rife betwe5 ! abaci/s1 ! advocates ( ! abacus1 & ! algori/s1 ^? :o favor$ ! new num]als4 ,! words cifra & algorismus cifra 7 us$ ) a some:at d]isive significance1 indicative ( absolute uselessness1 z inde$ ! z]o is useless on an abacus 9 :i* ! value ( any unit is giv5 by ! column :i* x occupies4#dge ,s ,gauti] de ,co9cy 7#aagg-abcf7 9 a "w on ! miracles ( ,m>y says3 ,a horn$ bea/1 a %eep1 ,an algorismus-ciph]1 ,is a prie/1 :o on su* a fea/ day ,does n celebrate ! holy ,"m4#dgf ,s ! abacus held ! field = a long "t1 ev5 aga9/ ! new algorism employ+ ! new num]als4 #aba ,ge(frey ,*auc]#dgg describes 9 ,! ,mill]'s ,tale ! cl]k ) ,8 ,almage/e & bokes grete & smale1 ,8 a/relabie1 long+e = 8 art1 ,8 augrim-/ones lay5 faire ap>t ,on %elves c\*$ at 8 b$des he$4 ,s1 too1 9 ,*auc]'s explanation ( ! a/rolabe1#dgh writt5 = 8 son ,lewis1 ! numb] ( degrees is express$ on ! in/rum5t 9 ,h9du-,arabic num]als3 ,ov] ! :i*e degrees !r b5 n\mbres ( augrim1 t devyd5 ?ilke same degrees fro fyve to fyve1 & 444 ! nombres 444 b5 writ5 9 augrim1 mean+ 9 ! way ( ! algorism4 ,?omas ,usk ab\t #achg writes3#dgi a syph] 9 augrim h no might 9 signification ( x-selve1 yet he yeve? p[] 9 signification to o!r4 ,s sl[ & s pa9ful is ! assimilation ( new ideas4 ,b]nel9us#dhj /ates t ! abacus is a well-poli%$ bo>d 7or table71 :i* is cov]$ ) blue s& & us$ by geomet]s 9 draw+ geometrical figures4 ,we h previ\sly m5tion$ ! fact t ! ,h9dus also p]=m$ ma!matical computations 9 ! s&1 al?\gh !re is no evid5ce to %[ t !y _h any column abacus4#dha ,= ! purposes ( computation1 ,b]nel9us cont9ues1 ! bo>d is divid$ into ?irty v]tical columns1 ?ree ( :i* are res]v$ = fractions4 ,beg9n+ ) ! units columns1 ea* set ( #abb ?ree columns 7l9eae is ! ^w :i* ,b]nel9us uses7 is gr\p$ toge!r by a semicircul> arc plac$ above !m1 :ile a small] arc is plac$ ov] ! units column & ano!r jo9s ! t5s & hundr$s columns4 ,?us arose ! designation arcus pictagore#dhb or sometimes simply arcus4#dhc ,! op]ations ( addition1 subtraction1 & multiplication ^u ?is =m ( ! abacus requir$ little explanation1 al?\gh !y 7 r ext5sively treat$1 especially ! multiplication ( diff]5t ord]s ( numb]s4 ,b ! op]ation ( division 0 effect$ ) "s difficulty4 ,= ! explanation ( ! me?od ( division by ! use ( ! complem5t>y diff]5ce1#dhd long ! /umbl+-block 9 ! way ( ! m$ieval ari?metician1 ! read] is ref]r$ to works on ! hi/ory ( ma!matics#dhe & to works relat+ p>ticul>ly to ! abacus4#dhf ,among ! writ]s on ! subject may be m5tion$ ,abbo#dhg ( ,fleury 7c4 #igj71 ,h]ig]#dhh ( ,lobbes or ,lauba* #abc 7c4 #iej-ajjg71 & ,h]mannus ,contractus#dhi 7#ajac-ajed71 all ( :om employ$ only ! ,roman num]als4 ,simil>ly ,adelh>d ( ,ba? 7c4 #aacj71 9 8 "w ,regulae ,abaci1#dij gives no ref]5ce to ! new num]als1 al?\gh x is c]ta9 t he knew !m4 ,o!r writ]s on ! abacus :o us$ "s =m ( ,h9du num]als 7 ,g]l&#dia 7fir/ half ( twelf? c5tury7 & ,tur*ill#dib 7c4 #abjj74 ,= ! =ms us$ at ?is p]iod ! read] is ref]r$ to ! plate on page #hh4 ,aft] ,g]b]t's dea?1 little by little ! s*ol>s ( ,europe came to "k ! new figures1 *iefly ?r\gh ! introduction ( ,arab le>n+4 ,! ,d>k ,ages _h pass$1 al?\gh ari?metic did n f9d ano!r advocate z prom95t z ,g]b]t = two c5turies4 ,speak+ ( ?is great revival1 ,ra\l ,glab]#dic 7#ihe-;c4 #ajdf71 a monk ( ! great ,b5$ict9e abbey ( ,cluny1 ( ! elev5? c5tury1 says3 ,x 0 z ?\gh ! _w _h aris5 & toss$ aside ! worn-out g>m5ts ( anci5t "t1 & wi%$ to app>el itself 9 a :ite robe ( *ur*es4 ,& ) ?is activity 9 religion came a correspond+ int]e/ 9 o!r l9es4 ,algorisms began to appe>1 & k f ! outside _w f\nd #abd int]e/$ li/5]s4 ,ano!r ,ra\l1 or ,radulph1 to :om we h ref]r$ z ,radulph ( ,laon1#did a tea*] 9 ! cloi/] s*ool ( 8 city1 & ! bro!r ( ,anselm ( ,laon#die ! celebrat$ !ologian1 wrote a treatise on music1 extant b unpubli%$1 & an ari?metic :i* ,nagl fir/ publi%$ 9 #ahij4dif ,! latt] "w1 pres]v$ to u 9 a p>*m5t manuscript ( sev5ty-sev5 leaves1 conta9s a curi\s mixture ( ,roman & 4gobar num]als1 ! =m] = express+ l>ge results1 ! latt] = practical calculation4 ,^! 4gobar c>act]es include ! sipos 7z]o71 ,symbol1 ( :i*1 h[ev]1 ,radulph did n "k ! full significance2 %[+ t at ! op5+ ( ! twelf? c5tury ! sy/em 0 /ill unc]ta9 9 its /atus 9 ! *ur* s*ools ( c5tral ,france4 ,at ! same "t ! words algorismus & cifra 7 com+ into g5]al use ev5 9 non-ma!matical lit]ature4 ,jordan #dig cites num]\s in/ances ( su* use f ! works ( ,alanus ab ,insulis#dih 7,ala9 de ,lille71 ,gauti] de ,co9cy 7#aagg-abcf71 & o!rs4 ,ano!r contributor to ari?metic dur+ ?is int]e/+ p]iod 0 a prom95t ,spani% ,jew call$ v>i\sly ,john ( ,luna1 ,john ( ,seville1 ,johannes ,hispal5sis1 ,johannes ,toletanus1 & ,johannes ,hispan5sis de ,luna4#dii #abe ,8 date is r closely fix$ by ! fact t he d$icat$ a "w to ,raimund :o 0 ar*bi%op ( ,tol$o betwe5 #aacj & #aaej4ejj ,8 int]e/s 7 *iefly 9 ! translation ( ,arabic works1 especially su* z bore ^u ! ,ari/otelian philosophy4 ,f ! /&po9t ( ari?metic1 h[ev]1 ! *ief int]e/ c5t]s ab\t a manuscript entitl$ ,joannis ,hispal5sis lib] ,algorismi de ,practica ,arismetrice :i* ,boncompagni f\nd 9 :at is n[ ! ,biblio!que nationale at ,p>is4 ,al?\gh ?is di/9ctly lays claim to be+ ,al-,kh[arazmi's "w1#eja ! evid5ce is altoge!r aga9/ ! /atem5t1#ejb b ! book is q z valuable1 s9ce x repres5ts ! k ( ! "t 9 :i* x 0 writt54 ,x relates to ! op]ations ) integ]s & sexagesimal fractions1 includ+ roots1 & conta9s no applications4#ejc ,contempor>y ) ,john ( ,luna1 & also liv+ 9 ,tol$o1 0 ,gh]>d ( ,cremona1#ejd :o has sometimes be5 id5tifi$1 b errone\sly1 ) ,g]n>dus1#eje ! #abf au?or ( a "w on algorism4 ,he 0 a physician1 an a/ronom]1 & a ma!matician1 translat+ f ! ,arabic bo? 9 ,italy & 9 ,spa94 ,9 ari?metic he 0 influ5tial 9 spread+ ! ideas ( algorism4 ,f\r ,engli%m5--,adelh>d ( ,ba? 7c4 #aacj71 ,rob]t ( ,*e/] 7,rob]tus ,ce/r5sis1 c4 #aadc71 ,william ,%elley1 & ,daniel ,morley 7#aahj7--;are kn[n#ejf to h j\rney$ to ,spa9 9 ! twelf? c5tury = ! purpose ( /udy+ ma!matics & ,arabic4 ,adelh>d ( ,ba? made translations f ,arabic into ,lat9 ( ,al-,kh[arazmi's a/ronomical tables#ejg & ( ,euclid's ,elem5ts1#ejh :ile ,rob]t ( ,*e/] is kn[n z ! translator ( ,al-,kh[arazmi's algebra4#eji ,!re is no reason to d\bt t all ( ^! m51 & o!rs1 7 famili> ) ! num]als :i* ! ,arabs 7 us+4 ,! e>lie/ trace we h ( computation ) ,h9du num]als 9 ,g]many is 9 an ,algorismus ( #aadc1 n[ 9 ! ,h(biblio!k 9 ,vi5na4#eaj ,x is b\nd 9 ) a #abg ,computus by ! same au?or & be>+ ! date giv54 ,x conta9s *apt]s ,de additione1 ,de dim9utione1 ,de m$iatione1 ,de divisione1 & "p ( a *apt] on multiplication4 ,! num]als are 9 ! usual m$ieval =ms except ! #b :i*1 z w be se5 f ! illu/ration1#eaa is some:at diff]5t1 & ! #c1 :i* takes ! peculi> %ape ,symbol1 a =m *>act]i/ic ( ! twelf? c5tury4 ,x 0 ab\t ! same "t t ! ,sef] ha-,misp>1#eab ! ,book ( ,numb]1 appe>$ 9 ! ,hebrew language4 ,! au?or1 ,rabbi ,abraham ibn ,meir ibn ,ezra1#eac 0 born 9 ,tol$o 7c4 #ajib74 ,9 #aaci he w5t to ,egypt1 ,pale/9e1 & ! ,ori5t1 sp5d+ also "s ye>s 9 ,italy4 ,lat] he liv$ 9 s\!rn ,france & 9 ,engl&4 ,he di$ 9 #aafg4 ,! probability is t he acquir$ 8 k ( ! ,h9du ari?metic#ead 9 8 native t[n ( ,tol$o1 b x is also likely t ! k ( o!r sy/ems :i* he acquir$ on travels increas$ 8 appreciation ( ?is "o4 ,we h m5tion$ ! fact t he us$ ! fir/ lett]s ( ! ,hebrew alphabet1 ,hebrew3 ,a ,b ,g ,d ,h ,w ,z ,,* ,t1 = ! num]als #i #h #g #f #e #d #c #b #a1 & a circle = ! z]o4 ,! quotation 9 ! note giv5 bel[ %[s t he knew ( ! ,h9du orig92 b 9 8 manuscript1 al?\gh he set d[n ! ,h9du =ms1 he us$ ! above n9e ,hebrew lett]s ) place value = all computations4 99 99 99 99 99 #abh ,,*apt] ,,viii ,,! ,,spread ,,( ,,! ,,num]als ,,9 ,,europe ,( all ! m$ieval writ]s1 probably ! "o mo/ influ5tial 9 introduc+ ! new num]als to ! s*ol>s ( ,europe 0 ,leon>do ,fibonacci1 ( ,pisa4#eae ,?is rem>kable man1 ! mo/ notewor?y ma!matical g5ius ( ! ,middle ,ages1 0 born at ,pisa ab\t #aage4eaf ,! travel] ( to-day may cross ! ,via ,fibonacci on 8 way to ! ,campo ,santo1 & !re he may see at ! end ( ! long corridor1 across ! quadrangle1 ! /atue ( ,leon>do 9 s*ol>s g>b4 ,few t[ns h honor$ a ma!matician m1 & few ma!maticians h s di/9ctly honor$ _! bir?place4 ,leon>do 0 born 9 ! gold5 age ( ?is city1 ! p]iod ( its comm]cial1 religi\s1 & intellectual prosp]ity4#eag #abi ,situat$ practically at ! m\? ( ! ,arno1 ,pisa =m$ ) ,g5oa & ,v5ice ! trio ( ! greate/ comm]cial c5t]s ( ,italy at ! op5+ ( ! ?irte5? c5tury4 ,ev5 be=e ,v5ice _h captur$ ! ,levant9e trade1 ,pisa _h close relations ) ! ,ea/4 ,an old ,lat9 *ronicle relates t 9 #ajje ,pisa 0 captur$ by ! ,s>ac5s1 t 9 ! foll[+ ye> ! ,pisans ov]?rew ! ,s>ac5s at ,reggio1 & t 9 #ajab ! ,s>ac5s came to ,pisa & de/roy$ x4 ,! city soon recov]$1 h[ev]1 s5d+ no few] ?an a hundr$ & tw5ty %ips to ,syria 9 #ajii1eah f\nd+ a m]*ant colony 9 ,con/ant9ople a few ye>s lat]1#eai & mean:ile c>ry+ on an int]urban w>f>e 9 ,italy t seem$ to /imulate x to great activity4#ebj ,a writ] ( #aaad tells u t at t "t !re 7 _m hea!n p--,turks1 ,libyans1 ,p>?ians1 & ,*aldeans--to be f\nd 9 ,pisa4 ,x 0 9 ! mid/ ( su* w>s1 9 a cosmopolitan & comm]cial t[n1 9 a c5t] :]e lit]>y "w 0 n appreciat$1#eba t ! g5ius ( ,leon>do appe>s z "o ( ! surprises ( hi/ory1 w>n+ u aga9 t we %\ld draw no horoscope2 t we %\ld expect little1 = :at we expect w n come to pass4#ebb ,leon>do's fa!r 0 "o ,william1#ebc & he _h a bro!r nam$ ,bonacc+us1#ebd b no?+ fur!r is #acj kn[n ( 8 family4 ,z to ,fibonacci1 mo/ writ]s#ebe h assum$ t 8 fa!r's "n 0 ,bonaccio1#ebf :5ce filius ,bonaccii1 or ,fibonacci4 ,o!rs#ebg believe t ! "n1 ev5 9 ! ,lat9 =m ( filius ,bonaccii z us$ 9 ,leon>do's "w1 0 simply a g5]al "o1 l our ,johnson or ,bronson 7,br[n's son72 & ! only contempor>y evid5ce t we h be>s out ?is view4 ,z to ! "n ,bigollo1 us$ by ,leon>do1 "s h ?\ght x a self-assum$ "o mean+ blockhead1 a t]m t _h be5 appli$ to him by ! comm]cial _w or possibly by ! univ]sity circle1 & tak5 by him t he might prove :at a blockhead c\ld d4 ,milanesi1#ebh h[ev]1 has %[n t ! ^w ,bigollo 7or ,pigollo7 0 us$ 9 ,tuscany to mean a travel]1 & 0 naturally assum$ by "o :o _h /udi$1 z ,leon>do _h1 9 =eign l&s4 ,leon>do's fa!r 0 a comm]cial ag5t at ,bugia1 ! mod]n ,b\gie1#ebi ! anci5t ,saldae on ! coa/ ( ,b>b>y1#ecj a royal capital "u ! ,v&als & aga91 a c5tury be=e ,leon>do1 "u ! ,b5i ,hammad4 ,x _h "o ( ! be/ h>bors on ! coa/1 %elt]$ z x is by ,mt4 ,lalla ,guraia1#eca & at ! close ( ! twelf? c5tury x 0 a c5t] ( ,african comm]ce4 ,x 0 "h t ,leon>do 0 tak5 z a *ild1 & "h he w5t to s*ool to a ,moori% ma/]4 ,:5 he rea*$ ! ye>s ( "y manhood he />t$ on a t\r ( ! ,m$it]ranean ,sea1 & visit$ ,egypt1 ,syria1 ,greece1 ,sicily1 & ,prov5ce1 meet+ ) s*ol>s z well z ) #aca m]*ants1 & imbib+ a k ( ! v>i\s sy/ems ( numb]s 9 use 9 ! c5t]s ( trade4 ,all ^! sy/ems1 h[ev]1 he says he c\nt$ almo/ z errors comp>$ ) t ( ! ,h9dus4#ecb ,return+ to ,pisa1 he wrote 8 ,lib] ,abaci#ecc 9 #abjb1 rewrit+ x 9 #abbh4ecd ,9 ?is "w ! num]als are expla9$ & are us$ 9 ! usual computations ( bus9ess4 ,su* a treatise 0 n de/9$ to be popul>1 h[ev]1 because x 0 too advanc$ = ! m]cantile class1 & too novel = ! cons]vative univ]sity circles4 ,inde$1 at ?is "t ma!matics _h only slight place 9 ! newly e/abli%$ univ]sities1 z witness ! olde/ kn[n /atute ( ! ,sorbonne at ,p>is1 dat$ #abae1 :]e ! subject is ref]r$ to only 9 an incid5tal way4#ece ,! p]iod 0 "o ( great comm]cial activity1 & on ?is v #acb acc\nt su* a book w\ld attract ev5 less att5tion ?an usual4#ecf ,x w\ld n[ be ?\ght t ! we/]n _w w\ld at once adopt ! new num]als :i* ,leon>do _h made kn[n1 & :i* 7 s mu* sup]ior to any?+ t _h be5 9 use 9 ,*ri/ian ,europe4 ,! antagonism ( ! univ]sities w\ld avail b little1 x w\ld seem1 aga9/ su* an improvem5t4 ,x mu/ be rememb]$1 h[ev]1 t !re 0 great difficulty 9 spread+ k at ?is "t1 "s two hundr$ & fifty ye>s be=e pr9t+ 0 inv5t$4 ,popes & pr9ces & ev5 great religi\s in/itutions possess$ f> few] books ?an _m f>m]s ( ! pres5t age4 ,! libr>y belong+ to ! ,ca!dral ,*ur* ( ,san ,m>t9o at ,lucca 9 ! n9? c5tury conta9$ only n9ete5 volumes ( abridgm5ts f ecclesia/ical comm5t>ies4#ecg ,inde$1 x 0 n until ! e>ly "p ( ! fifte5? c5tury t ,palla degli ,/rozzi took /eps to c>ry out ! project t _h be5 9 ! m9d ( ,petr>*1 ! f\nd+ ( a public libr>y4 ,x 0 l>gely by ^w ( m\?1 !re=e1 t ?is e>ly k _h to be transmitt$4 ,=tunately ! pres5ce ( =eign /ud5ts 9 ,italy at ?is "t made ?is transmission feasible4 7,if human nature 0 ! same !n z n[1 x is n impossible t ! v opposition ( ! faculties to ! works ( ,leon>do l$ ! /ud5ts to inve/igate #acc !m ! m zeal\sly47 ,at ,vic5za 9 #abji1 = example1 !re 7 ,bohemians1 ,poles1 ,fr5*m51 ,burgundians1 ,g]mans1 & ,spani>ds1 n to speak ( repres5tatives ( div]s t[ns ( ,italy2 & :at 0 true !re 0 also true ( o!r intellectual c5t]s4 ,! k c\ld n fail to spread1 !re=e1 & z a matt] ( fact we f9d num]\s bits ( evid5ce t ?is 0 ! case4 ,al?\gh ! bank]s ( ,flor5ce 7 =bidd5 to use ^! num]als 9 #abii1 & ! /atutes ( ! univ]sity ( ,padua requir$ /ation]s to keep ! price li/s ( books non p] cifras1 s$ p] lit]as cl>os1#ech ! num]als really made mu* headway f ab\t #abge on4 ,x 01 h[ev]1 r exceptional = ! common p ( ,g]many to use ! ,arabic num]als be=e ! sixte5? c5tury1 a good witness to ?is fact be+ ! popul> almanacs4 ,cal5d>s ( #adeg-adifeci h g5]ally ! ,roman num]als1 :ile ,koebel's cal5d> ( #aeah gives ! ,arabic =ms z subord9ate to ! ,roman4 ,9 ! regi/] ( ! ,kreuzs*ule at ,dresd5 ! ,roman =ms 7 us$ ev5 until #aeci4 ,:ile n m9imiz+ ! importance ( ! sci5tific "w ( ,leon>do ( ,pisa1 we may note t ! m popul> treatises by ,alex&] de ,villa ,dei 7c4 #abdj ,a4,d47 & ,john ( ,halifax 7,sacrobosco1 c4 #abej ,a4,d47 7 mu* m widely us$1 & d\btless contribut$ m to ! spread ( ! num]als among ! common p4 #acd ,! ,c>m5 de ,algorismo#edj ( ,alex&] de ,villa ,dei 0 writt5 9 v]se1 z inde$ 7 _m o!r textbooks ( t "t4 ,t x 0 widely us$ is evid5c$ by ! l>ge numb] ( manuscripts#eda extant 9 ,european libr>ies4 ,sacrobosco's ,algorismus1#edb 9 :i* "s l9es f ! ,c>m5 are quot$1 enjoy$ a wide popul>ity z a textbook = univ]sity in/ruction4#edc ,! "w 0 evid5tly writt5 ) ?is end 9 view1 z num]\s comm5t>ies by univ]sity lectur]s are f\nd4 ,probably ! mo/ widely us$ ( ^! 0 t ( ,petrus de ,dacia#edd writt5 9 #abia4 ,^! works ?r[ an int]e/+ light ^u ! me?od ( in/ruction 9 ma!matics 9 use 9 ! univ]sities f ! ?irte5? ev5 to ! sixte5? c5tury4 ,evid5tly ! text 0 fir/ read & copi$ by /ud5ts4#ede ,foll[+ ?is came l9e by l9e an exposition ( ! text1 su* z is giv5 9 ,petrus de ,dacia's comm5t>y4 ,sacrobosco's "w is ( int]e/ also because x 0 probably due to ! ext5d$ use ( ?is "w t ! #ace t]m ,arabic num]als became common4 ,9 two places !re is m5tion ( ! inv5tors ( ?is sy/em4 ,9 ! introduction x is /at$ t ?is sci5ce ( reckon+ 0 due to a philosoph] nam$ ,algus1 :5ce ! "n algorismus1#edf & 9 ! section on num]ation ref]5ce is made to ! ,arabs z ! inv5tors ( ?is sci5ce4#edg ,:ile "s ( ! comm5tators1 ,petrus de ,dacia#edh among !m1 knew ( ! ,h9du orig91 mo/ ( !m und\bt$ly took ! text z x /ood2 & s ! ,arabs 7 cr$it$ ) ! inv5tion ( ! sy/em4 ,! fir/ def9ite trace t we h ( an algorism 9 ! ,fr5* language is f\nd 9 a manuscript writt5 ab\t #abge4edi ,?is int]e/+ leaf1 = ! "p on algorism consi/s ( a s+le folio1 0 notic$ by ! ,abbe ,leboeuf z e>ly z #agda1eej & by ,daun\ 9 #ahbd4eea ,x !n seems to h be5 lo/ 9 ! multitude ( ,p>is manuscripts2 = al?\gh ,*asles#eeb relates 8 va9 se>* = x1 x 0 n r$iscov]$ until #ahhb4 ,9 t ye> ,m4 ,*4 ,h5ry f\nd x1 & to 8 c>e we owe our k ( ! int]e/+ manuscript4 ,! "w is anonym\s & is devot$ almo/ entirely to geometry1 only #acf two pages 7"o folio7 relat+ to ari?metic4 ,9 ^! ! =ms ( ! num]als are giv51 & a v brief /atem5t z to ! op]ations1 x be+ evid5t t ! writ] himself _h only ! slighte/ und]/&+ ( ! subject4 ,once ! new sy/em 0 kn[n 9 ,france1 ev5 ?us sup]ficially1 x w\ld be pass$ across ! ,*annel to ,engl&4 ,higd51#eec writ+ soon aft] ! op5+ ( ! f\rte5? c5tury1 speaks ( ! ,fr5* influ5ce at t "t & = "s g5]ations prec$+3#eed ,= two hundr$ ye>s *ildr5 9 scole1 ag5/ ! usage & manir ( all o!r nations bee? compell$ = to leave hire own language1 & = to con/rue hir lessons & hire ?ynges 9 ,fr5s*e4444 ,g5tilm5 *ildr5 bee? taught to speke ,fr5s*e f ! tyme t !y bi? rokk$ 9 hir cradell2 & uplondiss*e m5 w likne himself to g5tylm51 & fonde? ) greet besynesse = to speke ,fr5s*e4 ,! "q is (t5 ask$1 :y did n ^! new num]als attract m imm$iate att5tion8 ,:y did !y h to wait until ! sixte5? c5tury to be g5]ally us$ 9 bus9ess & 9 ! s*ools8 ,9 reply x may be said t 9 _! elem5t>y "w ! s*ools always wait ^u ! dem&s ( trade4 ,t "w :i* pret5ds to t\* ! life ( ! p mu/ come reasonably ne> do+ s4 ,n[ ! computations ( bus9ess until ab\t #aejj did n dem& ! new figures1 = two reasons3 ,fir/1 *eap pap] 0 n kn[n4 ,pap]-mak+ ( any k9d 0 n introduc$ into ,europe until #acg ! twelf? c5tury1 & *eap pap] is a product ( ! n9ete5?4 ,p5cils1 too1 ( ! mod]n type1 date only f ! sixte5? c5tury4 ,9 ! second place1 mod]n me?ods ( op]at+1 p>ticul>ly ( multiply+ & divid+ 7op]ations ( relatively great] importance :5 all measures 7 9 comp\nd numb]s requir+ r$uctions at e /ep71 7 n yet inv5t$4 ,! old plan requir$ ! eras+ ( figures aft] !y _h s]v$ _! purpose1 an op]ation v simple ) c\nt]s1 s9ce !y c\ld be remov$4 ,! new plan did n z easily p]mit ?is4 ,h5ce we f9d ! new num]als v t>dily admitt$ to ! c\nt+-h\se1 & n welcom$ ) any en?usiasm by tea*]s4#eee ,aside f _! use 9 ! e>ly treatises on ! new art ( reckon+1 ! num]als appe>$ f "t to "t 9 ! dat+ ( manuscripts & ^u monum5ts4 ,! olde/ def9itely dat$ ,european docum5t kn[n #ach to conta9 ! num]als is a ,lat9 manuscript1#eef ! ,codex ,vigilanus1 writt5 9 ! ,albelda ,cloi/] n f> f ,logrono 9 ,spa91 9 #igf ,a4,d4 ,! n9e *>act]s 7( 4gobar type71 )\t ! z]o1 are giv5 z an addition to ! fir/ *apt]s ( ! ?ird book ( ! ,orig9es by ,isidorus ( ,seville1 9 :i* ! ,roman num]als are "u discussion4 ,ano!r ,spani% copy ( ! same "w1 ( #iib ,a4,d41 conta9s ! num]als 9 ! correspond+ section4 ,! writ] ascribes an ,indian orig9 to !m 9 ! foll[+ words3 ,item de figuris ari?metice14 ,scire debemus 9 ,indos subtilissimum +5ium hab]e et cet]as g5tes eis 9 ari?metica et geometria et cet]is lib]alibus discipl9is conc$]e4 ,et hoc manife/um e/ 9 nobem figuris1 quibus designant unumquemque gradum cuiuslibet gradus4 ,qu>um hec sunt =ma4 ,! n9e 4gobar *>act]s foll[4 ,"s ( ! abacus =ms#eeg previ\sly giv5 are d\btless also ( ! t5? c5tury4 ,! e>lie/ ,arabic docum5ts conta9+ ! num]als are two manuscripts ( #hgd & #hhh ,a4,d4#eeh ,!y appe> ab\t a c5tury lat] 9 a "w#eei writt5 at ,%iraz 9 #igj ,a4,d4 ,!re is also an e>ly trace ( _! use on a pill> rec5tly discov]$ 9 a *ur* app>5tly de/roy$ z e>ly z ! t5? c5tury1 n f> f ! ,j]emias ,mona/]y1 9 ,egypt4 #aci ,a graffito 9 ,arabic on ?is pill> has ! date #cdi ,a4,h41 :i* corresponds to #ifa ,a4,d4#efj ,= ! dat+ ( ,lat9 docum5ts ! ,arabic =ms 7 us$ z e>ly z ! ?irte5? c5tury4#efa ,on ! e>ly use ( ^! num]als 9 ,europe ! only sci5tific /udy wor?y ! "n is t made by ,mr4 ,g4 ,f4 ,hill ( ! ,briti% ,museum4#efb ,f 8 inve/igations x appe>s t ! e>lie/ occurr5ce ( a date 9 ^! num]als on a co9 is f\nd 9 ! reign ( ,rog] ( ,sicily 9 #aach4efc ,until rec5tly x 0 ?\ght t ! e>lie/ su* date 0 #abag ,a4,d4 = an ,arabic piece & #achh = a ,turki% "o4#efd ,mo/ ( ! seals & m$als conta9+ dates t 7 at "o "t ?\ght to be v e>ly h be5 %[n by ,mr4 ,hill to be ( relatively late workman%ip4 ,!re are1 h[ev]1 9 ,european manuscripts1 num]\s in/ances ( ! use ( ^! num]als be=e ! twelf? c5tury4 ,besides ! example 9 ! ,codex ,vigilanus1 ano!r ( ! t5? c5tury has be5 f\nd 9 ! ,/4 ,gall ,,ms4 n[ 9 ! ,univ]sity ,libr>y at ,zu]i*1 ! =ms diff]+ mat]ially f ^? 9 ! ,spani% codex4 ,! ?ird specim5 9 po9t ( "t 9 ,mr4 ,hill's li/ is f a ,vatican ,,ms4 ( #ajgg4 ,! f\r? & fif? specim5s are f ! ,erlang5 ,,ms4 ( ,boe?ius1 ( ! same #adj 7elev5?7 c5tury1 & ! six? & sev5? are also f an elev5?-c5tury ,,ms4 ( ,boe?ius at ,*>tres4 ,^! & o!r e>ly =ms are giv5 by ,mr4 ,hill 9 ?is table1 :i* is reproduc$ ) 8 k9d p]mission4 ,,e>lie/ ,,manuscript ,,=ms ,illu/ration ,?is is "o ( m ?an fifty tables giv5 9 ,mr4 ,hill's valuable pap]1 & to ?is monograph /ud5ts #ada are ref]r$ = details z to ! developm5t ( numb]-=ms 9 ,europe f ! t5? to ! sixte5? c5tury4 ,x is ( int]e/ to add t he has f\nd t among ! e>lie/ dates ( ,european co9s or m$als 9 ^! num]als1 aft] ! ,sicilian "o already m5tion$1 are ! foll[+3 ,au/ria1 #adhd2 ,g]many1 #adhi 7,cologne72 ,switz]l&1 #adbd 7,/4 ,gall72 ,ne!rl&s1 #adgd2 ,france1 #adhe2 ,italy1 #acij4efe ,! e>lie/ ,engli% co9 dat$ 9 ^! num]als 0 /ruck 9 #aeea1eff al?\gh !re is a ,scot* piece ( #aeci4efg ,9 numb]+ pages ( a pr9t$ book ^! num]als 7 fir/ us$ 9 a "w ( ,petr>*'s publi%$ at ,cologne 9 #adga4efh ,! date is giv5 9 ! foll[+ =m 9 ! ,biblia ,paup]um1#efi a block-book ( #adgj1 ,illu/ration :ile 9 ano!r block-book :i* possibly goes back to c4 #adcjegj ! num]als appe> 9 sev]al illu/rations1 ) =ms z foll[s3 ,illu/ration ,_m pr9t$ works ant]ior to #adga h pages or *apt]s numb]$ by h&1 b _m ( ^! num]als are #adb ( date mu* lat] ?an ! pr9t+ ( ! "w4 ,o!r works 7 probably numb]$ directly aft] pr9t+4 ,?us ! *apt]s #b1 #c1 #d1 #e1 #f 9 a book ( #adgjega are numb]$ z foll[s3 ,capitulem ,symbol #bm41444 ,symbol #cm41444 #dm41444 v1444 vi1 & foll[$ by ,roman num]als4 ,?is appe>s 9 ! body ( ! text1 9 spaces left by ! pr9t] to be fill$ 9 by h&4 ,ano!r book#egb ( #adgj has pages numb]$ by h& ) a mixture ( ,roman & ,h9du num]als1 ?us1 ,illu/ration = #abe ,illu/ration = #aej ,illu/ration = #adg ,illu/ration = #bjb ,z to monum5tal inscriptions1#egc !re 0 once ?\ght to be a grave/one at ,ka?>e91 ne> ,troppau1 ) ! date #ajjg1 & "o at ,biebri* ( #abii4 ,!re is no d\bt1 h[ev]1 ( "o at ,p=zheim ( #acga & "o at ,ulm ( #achh4egd ,c]ta9 num]als on ,wells ,ca!dral h be5 assign$ to ! ?irte5? c5tury1 b !y are und\bt$ly consid]ably lat]4#ege ,! table on page #adc w s]ve to supplem5t t f ,mr4 ,hill's "w4#egf #adc ,,e>ly ,,manuscript ,,=ms #egg ,illu/ration ,twelf? c5tury ,a4,d4 #egh ,illu/ration #aaig ,a4,d4 #egi ,illu/ration #abge ,a4,d4 #ehj ,illu/ration c4 #abid ,a4,d4 #eha ,illu/ration c4 #acjc ,a4,d4 #ehb ,illu/ration c4 #acfj ,a4,d4 #ehc ,illu/ration c4 #addb ,a4,d4 #add ,illu/ration ,= ! sake ( fur!r comp>ison1 ?ree illu/rations f works 9 ,mr4 ,plimpton's libr>y1 reproduc$ f ! ,r>a ,ari?metica1 may be consid]$4 ,! fir/ is f a ,lat9 manuscript on ari?metic1#ehd ( :i* ! orig9al 0 writt5 at ,p>is 9 #adbd by ,roll&us1 a ,portuguese physician1 :o prep>$ ! "w at ! comm& ( ,john ( ,lanca/]1 ,duke ( ,b$=d1 at "o "t ,protector ( ,engl& & ,reg5t ( ,france1 to :om ! "w is d$icat$4 ,! figures %[ ! successive p[]s ( #b4 ,! second illu/ration is f ,luca da ,fir5ze's ,inpr5cipio d>te daba*o1#ehe c4 #adge1 & ! ?ird is f an anonym\s manuscript#ehf ( ab\t #aejj4 ,illu/ration ,z to ! =ms ( ! num]als1 fa%ion play$ a lead+ "p until pr9t+ 0 inv5t$4 ,?is t5d$ to fix ^! =ms1 al?\gh 9 writ+ !re is /ill a great v>iation1 z witness ! ,fr5* #e & ! ,g]man #g & #i4 ,ev5 9 pr9t+ !re is n complete uni=mity1 #ade & x is (t5 difficult = a =eign] to di/+ui% betwe5 ! #c & #e ( ! ,fr5* types4 ,illu/ration ,z to ! p>ticul> num]als1 ! foll[+ are "s ( ! =ms to be f\nd 9 ! lat] manuscripts & 9 ! e>ly pr9t$ books4 #a4 ,9 ! e>ly pr9t$ books "o 0 (t5 i1 p]haps to save types1 j z "s mod]n typewrit]s use ! same "* = l & #a4ehg ,9 ! manuscripts ! "o appe>s 9 su* =ms z#ehh ,illu/ration #b4 ,two (t5 appe>s z z 9 ! e>ly pr9t$ books1 #ab appe>+ z iz4#ehi ,9 ! m$ieval manuscripts ! foll[+ =ms are common3#eij ,illu/ration #adf ,x is evid5t1 f ! e>ly traces1 t x is m]ely a cursive =m = ! primitive #b horizontal /rokes1 j z #c comes f #c horizontal /rokes1 z 9 ! ,nana ,ghat inscriptions4 #c4 ,?ree usually _h a special type 9 ! fir/ pr9t$ books1 al?\gh occasionally x appe>s z ,symbol4#eia ,9 ! m$ieval manuscripts x v>i$ r less ?an mo/ ( ! o!rs4 ,! foll[+ are common =ms3#eib ,illu/ration #d4 ,f\r has *ang$ greatly2 & "o ( ! fir/ te/s z to ! age ( a manuscript on ari?metic1 & ! place :]e x 0 writt51 is ! exam9ation ( ?is num]al4 ,until ! "t ( pr9t+ ! mo/ common =m 0 ,symbol1 al?\gh ! ,flor5t9e manuscript ( ,leon>d ( ,pisa's "w has ! =m ,symbol2#eic b ! manuscripts %[ t ! ,flor5t9e ari?meticians & a/ronom]s r e>ly began to /raight5 ! fir/ ( ^! =ms up to =ms l ,symbol#eid & ,symbol#eid or ,symbol1#eie m closely resembl+ our own4 ,! fir/ pr9t$ books g5]ally us$ our pres5t =m#eif ) ! clos$ top ,symbol1 ! op5 top us$ 9 writ+ 7 ,symbol7 be+ #adg purely mod]n4 ,! foll[+ are o!r =ms ( ! f\r1 f v>i\s manuscripts3#eig ,illu/ration #e4 ,five also v>i$ greatly be=e ! "t ( pr9t+4 ,! foll[+ are "s ( ! =ms3#eih ,illu/ration #f4 ,six has *ang$ r less ?an mo/ ( ! o!rs4 ,! *ief v>iation has be5 9 ! slope ( ! top1 z w be se5 9 ! foll[+3#eii ,illu/ration #g4 ,sev51 l f\r1 has assum$ its pres5t erect =m only s9ce ! fifte5? c5tury4 ,9 m$ieval times x appe>$ z foll[s3#fjj ,illu/ration #adh #h4 ,eight1 l six1 has *ang$ b little4 ,9 m$ieval times !re are a few v>iants ( int]e/ z foll[s3#fja ,illu/ration ,9 ! sixte5? c5tury1 h[ev]1 !re 0 manife/$ a t5d5cy to write x ,symbol4#fjb #i4 ,n9e has n v>i$ z mu* z mo/ ( ! o!rs4 ,among ! m$ieval =ms are ! foll[+3#fjc ,illu/ration #j4 ,! %ape ( ! z]o also _h a v>i$ hi/ory4 ,! foll[+ are common m$ieval =ms3#fjd ,illu/ration ,! explanation ( ! place value 0 a s]i\s matt] to mo/ ( ! e>ly writ]s4 ,if !y _h be5 us+ an abacus con/ruct$ l ! ,russian *otue1 & _h plac$ ?is be=e all le>n]s ( ! positional sy/em1 !re w\ld h be5 little tr\ble4 ,b ! m$ieval #adi l9e-reckon+1 :]e ! l9es /ood = p[]s ( #aj & ! spaces = half ( su* p[]s1 did n l5d itself to ?is comp>ison4 ,accord+ly we f9d su* labor$ explanations z ! foll[+1 f ,! ,crafte ( ,nombrynge3 ,eu]y ( ^! figuris bitok5s hym selfe no m1 yf he /onde 9 ! fir/ place ( ! rewele4444 ,if x /onde 9 ! secunde place ( ! rewle1 he betok5s t5 tymes hym selfe1 z ?is figure #b "h #bj tok5s t5 tyme hym selfe1 t is tw5ty1 = he hym selfe betok5s tweyne1 t5 tymes tw5e is tw5ty4 ,& = he /ondis on ! lyft side 9 ! secunde place1 he betok5s t5 tyme hym selfe4 ,& s g =?4444 ,nil cifra significat s$ dat sign>e sequ5ti4 ,expone ?is v]se4 ,a cifre tok5s noght1 bot he makes ! figure to betok5 t comes aft] hym m ?an he %uld he 7 away1 z ?us #aj4 "h ! figure ( "o tok5s t51 yf ! cifre 7 away no figure by=e hym he s*uld tok5 bot "o1 = ?an he s*uld /onde 9 ! fir/ place4444#fje ,x w\ld seem t a sy/em t 0 ?us us$ = dat+ docum5ts1 co9s1 & monum5ts1 w\ld h be5 g5]ally adopt$ mu* e>li] ?an x 01 p>ticul>ly 9 ^? c\ntries nor? ( ,italy :]e x did n come into g5]al use until ! sixte5? c5tury4 ,?is1 h[ev]1 has be5 ! fate ( _m inv5tions1 z witness our neglect ( log>i?ms & ( contract$ processes to-day4 ,z to ,g]many1 ! fifte5? c5tury saw ! rise ( ! new symbolism2 ! sixte5? c5tury saw x sl[ly #aej ga9 ! ma/]y2 ! sev5te5? c5tury saw x f9ally conqu] ! sy/em t = two ?\s& ye>s _h dom9at$ ! ari?metic ( bus9ess4 ,n a little ( ! success ( ! new plan 0 due to ,lu!r's dem& t all le>n+ %\ld g into ! v]nacul>4#fjf ,dur+ ! transition p]iod f ! ,roman to ! ,arabic num]als1 v>i\s anomal\s =ms f\nd place4 ,= example1 we h 9 ! f\rte5? c5tury calpha = #ajd2fjg #ajjj4 #cjj4 #hj et #d = #achd2fjh & 9 a manuscript ( ! fifte5? c5tury #abija = #abia4fji ,9 ! same c5tury m4 cccc4 #h;,,ii appe>s = #adhb1faj :ile ,mo,c,c,c,co#ej 7#adej7 & ,,mccccxl#f 7#addf7 are us$ by ,!odoricus ,ruffi ab\t ! same "t4#faa ,to ! next c5tury belongs ! =m #avojj = #aejb4 ,ev5 9 ,s=tunati's ,nuovo lume#fab ! use ( ord9als is q confus$1 ! propositions on a s+le page be+ numb]$ t]tia1 #d1 & ,v4 ,al?\gh n connect$ ) ! ,arabic num]als 9 any direct way1 ! m$ieval a/rological num]als may "h be m5tion$4 ,^! are giv5 by sev]al e>ly writ]s1 b notably by ,noviomagus 7#aeci71fac z foll[s#fad3 ,illu/ration #aea ,?us we f9d ! num]als gradually replac+ ! ,roman =ms all ov] ,europe1 f ! "t ( ,leon>do ( ,pisa until ! sev5te5? c5tury4 ,b 9 ! ,f> ,ea/ to-day !y are q unkn[n 9 _m c\ntries1 & !y /ill h _! way to make4 ,9 _m p>ts ( ,india1 among ! common p ( ,japan & ,*9a1 9 ,siam & g5]ally ab\t ! ,malay ,p59sula1 9 ,tibet1 & among ! ,ea/ ,india isl&s1 ! natives /ill adh]e to _! own num]al =ms4 ,only z ,we/]n civilization is mak+ its way into ! comm]cial life ( ! ,ea/ d ! num]als z us$ by u f9d place1 save z ! ,sanskrit =ms appe> 9 p>ts ( ,india4 ,x is !re=e ) surprise t ! /ud5t ( ma!matics comes to realize h[ mod]n are ^! =ms s common 9 ! ,we/1 h[ limit$ is _! use ev5 at ! pres5t "t1 & h[ sl[ ! _w has be5 & is 9 adopt+ su* a simple device z ! ,h9du-,arabic num]als4 99 99 99 99 99 #aec ,,index ,transcrib]'s note3 _m ( ! entries ref] to footnotes l9k$ f ! page numb]s giv54 ,abbo ( ,fleury1 #abb ,abdallah ibn al-,h4asan1 #ib ,abdallatif ibn ,yusuf1 #ic ,abdalqadir ibn ,ali al-,sakhawi1 #f ,ab5ragel1 #cd ,abraham ibn ,meir ibn ,ezra1 see ,rabbi b5 ,ezra ,abu ,ali al-,h4ose9 ibn ,sina1 #gd ,abu 'l-,h4asan1 #ic1 #ajj ,abu 'l-,qasim1 #ib ,abu 'l-,t4eiyib1 #ig ,abu ,nas4r1 #ib ,abu ,ro%d1 #aac ,abu ,sahl ,duna% ibn ,tamim1 #fe1 #fg ,adelh>d ( ,ba?1 #e1 #ee1 #ig1 #aai1 #abc1 #abf ,adhem> ( ,*abanois1 #aaa ,ah4m$ al-,nasawi1 #ih ,ah4m$ ibn ,abdallah1 #i1 #ib ,ah4m$ ibn ,moh4amm$1 #id ,ah4m$ ibn ,om>1 #ic ,aks4aras1 #cb ,alanus ab ,insulis1 #abd ,al-,ba4gdadi1 #ic ,al-,battani1 #ed ,albelda 7,albaida7 ,,ms41 #aaf ,alb]t1 ,j41 #fb ,alb]t ( ,york1 #ajc ,al-,biruni1 #f1 #da1 #di1 #fe1 #ib1 #ic ,alcu91 #ajc ,alex&] ! ,great1 #gf ,alex&] de ,villa ,dei1 #aa1 #acc ,alex&ria1 #fd1 #hb ,al-,fazari1 #ib ,alfr$1 #ajc ,algebra1 etymology1 #e ,alg]ian num]als1 #fh ,algorism1 #ig ,algorismus1 #abd1 #abf1 #ace ,algorismus cifra1 #abj ,al-,h4as4s4ar1 #fe ,ali ibn ,abi ,bekr1 #f ,ali ibn ,ah4m$1 #ic1 #ih ,al-,k>abisi1 #ic ,al-,kh[arazmi1 #d1 #i1 #aj1 #ib1 #ig1 #ih1 #abe1 #abf ,al-,k9di1 #aj1 #ib ,almage/1 #ed ,al-,ma4grebi1 #ic ,al-,mah4alli1 #f ,al-,mamun1 #aj1 #ig ,al-,mans4ur1 #if1 #ig ,al-,masudi1 #g1 #ib ,al-,nadim1 #i ,al-,nasawi1 #ic1 #ih ,alphabetic num]als1 #ci1 #dj1 #dc ,al-,qasim1 #ib ,al-,qass1 #id ,al-,sakhawi1 #f ,al-,s4ardafi1 #ic ,al-,sijzi1 #id ,al-,sufi1 #aj1 #ib ,ambrosoli1 #aah ,a4nkapalli1 #dc ,apices1 #hg1 #aag1 #aah ,arabs1 #ia-ih ,arbu?not1 #ada #aed ,ar*im$es1 #ae1 #af ,arcus ,pictagore1 #abb ,arjuna1 #ae ,arnold1 ,e41 #ae1 #ajb ,ars memor&i1 #ada ,aryabhat4a1 #ci1 #dc1 #dd ,aryan num]als1 #ai ,as*ba*1 #acd ,a%mole1 #acd ,a'soka1 #ai1 #bj1 #bb1 #ha ,as4-s4ifr1 #eg1 #eh ,a/rological num]als1 #aej ,a?>va-,v$a1 #dh1 #di1 #ee ,augu/us1 #hj ,av]roes1 #aac ,avic5na1 #eh1 #gd1 #aac ,babylonian num]als1 #bh ,babylonian z]o1 #ea ,bacon1 ,r41 #aca ,bactrian num]als1 #ai1 #cj ,ba$a1 #b1 #gb ,bagdad1 #d1 #if ,bakhs4ali manuscript1 #dc1 #di1 #eb1 #ec ,ball1 ,c4 ,j41 #ce ,ball1 ,w4 ,w4 ,r41 #cf1 #aca ,ban4a1 #dd ,b>?1 ,a41 #ci ,bayang inscriptions1 #ci ,bay]1 #cc ,bayley1 ,e4 ,c41 #ai1 #bc1 #cj1 #cb1 #eb1 #hi ,beazley1 #ge ,b$e1 see ,ba$a ,beldom&i1 #acg ,belo*1 ,j41 #gg ,b5dall1 #be1 #eb ,b5fey1 ,t41 #bf ,b]nel9us1 #hh1 #aab1 #aag1 #aba ,besagne1 #abh ,besant1 ,w41 #aji ,bett9o1 #cf ,bh&>k>1 #ah1 #dg1 #di ,bhask>a1 #ec1 #ee ,bi]natzki1 #cb ,biot1 #cb ,bjo]nbo1 ,a4 ,a41 #abe1 #abf ,blassi]e1 #aai ,bloomfield1 #dh ,blume1 #he ,boeckh1 #fb ,boehm]1 #adc ,boes*5/e91 #aai ,boe?ius1 #fc1 #gj-gc1 #hc-ij ,boissi]e1 #fc ,bombelli1 #ha ,bona9i1 #abh ,boncompagni1 #e1 #f1 #aj1 #dh1 #di1 #abc1 #abe ,borghi1 #ei ,borgo1 #aai ,b\gie1 #acj ,b[r+1 ,j41 #ef ,brahmagupta1 #eb ,brahman4z1 #ab1 #ac ,brahmi1 #ai1 #bj1 #ca1 #hc ,br&is1 ,j41 #ed ,br4hat-,sam4hita1 #ci1 #dd1 #gh ,brockhaus1 #dc ,bubnov1 #fe1 #hd1 #aaj1 #aaf ,buddha1 education (1 #ae1 #af ,bu$+]1 #aaj ,bugia1 #acj ,buehl]1 ,g41 #ae1 #ai1 #bb1 #ca1 #dd1 #di ,burgess1 #be ,bu]k1 #ac ,burmese num]als1 #cf ,burnell1 ,a4 ,c41 #ah1 #dj ,buteo1 #fa ,cal&ri1 #ei1 #ha ,caldwell1 ,r41 #ai ,cal5d>s1 #acc ,calmet1 #cd ,cantor1 ,m41 #e1 #ac1 #cj1 #dc1 #hd #aee ,capella1 #hf ,cappelli1 #adc ,c>act]es1 #hg1 #aac1 #aag1 #aai ,c>dan1 #aai ,c>m5 de ,algorismo1 #aa1 #acd ,casagr&i1 #acb ,casiri1 #h1 #aj ,cassiodorus1 #gb ,cataldi1 #fb ,cataneo1 #c ,caxton1 #adc1 #adf ,c]etti1 #cb ,ceylon num]als1 #cf ,*alfont1 ,f4 ,h41 #bh ,*amp5ois1 #fj ,*>act]s1 see ,c>act]es ,*>lemagne1 #ajc ,*asles1 #ed1 #fj1 #he1 #aaf1 #abb1 #ace ,*assant1 ,l4 ,a41 #adb ,*auc]1 #aba ,*i>9i1 #ade1 #adf ,*iffre1 #eh ,*9ese num]als1 #bh1 #ef ,*9ese z]o1 #ef ,cifra1 #abj1 #abd ,ciph]1 #eh ,circulus1 #eh1 #fj ,cli*toveus1 #fa1 #aai1 #ade ,codex ,vigilanus1 #ach ,codr+ton1 ,o41 #aci ,co9s dat$1 #ada ,colebrooke1 #h1 #bf1 #df1 #ec ,con/ant9e1 #ajd1 #aje ,cosmas1 #hb ,cossali1 #e ,c\nt]s1 #aag ,c\rteille1 #h ,coxe1 #ei ,crafte ( ,nombrynge1 #aa1 #hg1 #adi ,crusades1 #aji ,cunn+ham1 ,a41 #cj1 #ge ,curtze1 #ee1 #ei1 #abf1 #acd ,cyfra1 #ee ,dagom>i1 #adf ,d',alviella1 #ae ,dante1 #gb ,dasypodius1 #cc1 #fg1 #fc ,daun\1 #ace ,delambre1 #ed ,devanag>i1 #g ,dev\lx1 ,a41 #fh ,dhruva1 #di ,dicae>*us ( 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9 ! preface4 #i ,our ^w algebra is f ! title ( "o ( 8 works1 ,al-jabr wa'l-muqabalah1 ,completion & ,comp>ison4 ,! "w 0 translat$ into ,engli% by ,f4 ,ros51 ,london1 #ahca1 & treat$ 9 ,l',algebre d'al-,kharizmi et les me?odes indi5ne et grecque1 ,leon ,rodet1 ,p>is1 #ahgh1 extract f ! ,j\rnal ,asiatique4 ,= ! d]ivation ( ! ^w algebra1 see ,cossali1 ,scritti ,in$iti1 pp4 #cha-chc1 ,rome1 #aheg2 ,leon>do's ,lib] ,abbaci 7#abjb71 p4 #daj1 ,rome1 #aheg2 bo? publi%$ by ,b4 ,boncompagni4 ,almu*abala also 0 us$ z a "n = algebra4 #aj ,?is le>n$ s*ol>1 tea*] ( ,o',creat :o wrote ! ,helceph 7,prologus ,n4 ,ocreati 9 ,helceph ad ,adel>dum ,bat5sem magi/rum suum71 /udi$ 9 ,tol$o1 le>n$ ,arabic1 travel$ z f> ea/ z ,egypt1 & br\ght f ! ,levant num]\s manuscripts = /udy & translation4 ,see ,h5ry 9 ! ,abh&lung5 zur ,ges*i*te d] ,ma!matik1 ,vol4 ,,iii1 p4 #aca2 ,woepcke 9 ,propagation1 p4 #eah4 #aa ,! title is ,algoritmi de num]o ,indorum4 ,t he did n make ?is translation is ass]t$ by ,ene/roem 9 ! ,biblio!ca ,ma!matica1 ,vol4 ,i 7#c71 p4 #ebj4 #ab ,?us he speaks de num]o indorum p] 4,,ix4 lit]as1 & proce$s3 ,dixit algoritmi3 ,cum uidissem yndos con/ituisse 4,,ix4 lit]as 9 uniu]so num]o suo1 propt] dispositionem suam quam posu]unt1 uolui patefac]e de op]a quod fit p] eas aliquid quod esset leuius disc5tibus1 si deus uolu]it4 ,boncompagni1 ,trattati d',aritmetica1 ,rome1 #aheg4 ,discuss$ by ,f4 ,woepcke1 ,sur l'introduction de l'ari?metique indi5ne en ,occid5t1 ,rome1 #ahei4 #ac ,?us 9 a comm5t>y by ,ali ibn ,abi ,bekr ibn al-,jamal al-,ans4ari #ad ,see also ,woepcke1 ,propagation1 p4 #eje4 ,! orig9 is discuss$ at mu* l5g? by ,g4 ,r4 ,kaye1 ,notes on ,indian ,ma!matics4--,ari?metical ,notation1 ,j\rn4 & ,proc4 ( ! ,asiatic ,soc4 ( ,b5gal1 ,vol4 ,,iii1 #aijg1 p4 #dhi4 #ae ,alb]uni's ,india1 ,arabic v]sion1 ,london1 #ahhg2 ,engli% translation1 ibid41 #ahhh4 #af ,*ronology ( ,anci5t ,nations1 ,london1 #ahgi4 ,arabic & ,engli% v]sions1 by ,c4 ,e4 ,sa*au4 #ag ,india1 ,vol4 ,i1 *ap4 xvi4 #ah ,! ,h9du "n = ! symbols ( ! decimal place sy/em4 #ai ,sa*au's ,engli% edition ( ! ,*ronology1 p4 #fd4 #bj ,litt]ature arabe1 ,cl4 ,hu>t1 ,p>is1 #aijb4 #ba ,hu>t1 ,hi/ory ( ,arabic ,lit]ature1 ,engli% ed41 ,new ,york1 #aijc1 p4 #ahb seq4 #bb ,al-,masudi's ,mead[s ( ,gold1 translat$ 9 "p by ,aloys ,spr5g]1 ,london1 #ahda2 ,les prairies d'or1 trad4 p> ,c4 ,b>bi] de ,meyn>d et ,pavet de ,c\rteille1 ,vols4 ,i to ,,ix1 ,p>is1 #ahfa-ahgg4 #bc ,les prairies d'or1 ,vol4 ,,viii1 p4 #bhi seq4 #bd ,essays1 ,vol4 ,,ii1 p4 #dbh4 #be ,loc4 cit41 p4 #ejd4 #bf ,mat]iaux p\r s]vir a l'hi/oire comp>ee des sci5ces ma!matiques *ez les ,grecs et les ,ori5taux1 #b vols41 ,p>is1 #ahde-ahdi1 pp4 #dch-dci4 #bg ,he made an exception1 h[ev]1 9 favor ( ! num]als1 loc4 cit41 ,vol4 ,,ii1 p4 #ejc4 #bh ,biblio!ca ,arabico-,hispana ,escurial5sis1 ,madrid1 #agfj-aggj1 pp4 #dbf-dbg4 #bi ,! au?or1 ,ibn al-,qift4i1 fl\ri%$ ,a4,d4 #aaih ,colebrooke1 loc4 cit41 note ,vol4 ,,ii1 p4 #eaj4 #cj ,lib] ,artis ,logi/icae a ,mohamado ,b5 ,musa ,alkhu>ezmita exornatus1 qui cet]os omnes brevitate me?odi ac facilitate prae/at1 ,indorum que 9 praecl>issimis inv5tis +5ium acum5 o/5dit4 ,casiri1 loc4 cit41 p4 #dbg4 #ca ,mac\di1 ,le livre de l'av]tissem5t et de la revision4 ,translation by ,b4 ,c>ra de ,vaux1 ,p>is1 #ahif4 #cb ,v]ify+ ! hypo!sis ( ,woepcke1 ,propagation1 t ! ,s9dh9d includ$ a treatm5t ( ari?metic4 #cc ,ah4m$ ibn ,abdallah1 ,sut]1 ,die ,ma!matik]1 etc41 p4 #ab4 #cd ,india1 ,vol4 ,,ii1 p4 #ae4 #ce ,see ,h4 ,sut]1 ,das ,ma!matik]-,v]zei*niss im ,fihri/1 ,abh&lung5 zur ,ges*i*te d] ,ma!matik1 ,vol4 ,,vi1 ,leipzig1 #ahib4 ,= fur!r ref]5ces to e>ly ,arabic writ]s ! read] is ref]r$ to ,h4 ,sut]1 ,die ,ma!matik] und ,a/ronom5 d] ,arab] und ihre ,w]ke4 ,also ,na*traege und ,b]i*tigung5 to ! same 7,abh&lung51 ,vol4 ,,xiv1 #aijb1 pp4 #aee-ahf74 #cf ,sut]1 loc4 cit41 note #afe1 pp4 #fb-fc4 #cg ,s5d ,b5 ,ali1444 tum ari?metica scripta maxime celebrata1 quae publici juris fecit4 ,loc4 cit41 p4 #ddj4 #ch ,scritti di ,leon>do ,pisano1 ,vol4 ,i1 ,lib] ,abbaci 7#aheg72 ,vol4 ,,ii1 ,scritti 7#ahfb72 publi%$ by ,baldass>re ,boncompagni1 ,rome4 ,also ,tre ,scritti ,in$iti1 & ,intorno ad ,op]e di ,leon>do ,pisano1 ,rome1 #ahed4 #ci ,ubi ex mirabili magi/]io 9 arte p] novem figuras indorum introductus etc4 ,9 ano!r place1 z a head+ to a sep>ate division1 he writes1 ,de cognitione novem figur>um yndorum etc4 ,novem figure indorum he sunt #i #h #g #f #e #d #c #b #a4 #dj ,see ,an ,anci5t ,engli% ,algorism1 by ,david ,eug5e ,smi?1 9 ,fe/s*rift ,moritz ,cantor1 ,leipzig1 #aiji4 ,see also ,victor ,mortet1 ,le plus anci5 traite francais d'algorisme1 ,biblio!ca ,ma!matica1 ,vol4 ,,ix 7#c71 pp4 #ee-fd4 #da ,^! are ! two op5+ l9es ( ! ,c>m5 de ,algorismo t ! anonym\s au?or is expla9+4 ,!y %\ld read z foll[s3 ,haec algorismus ars praes5s dicitur1 9 qua ,talibus ,indorum fruimur bis qu9que figuris4 ,:at foll[s is ! translation4 #db ,?ibaut1 ,a/ronomie1 ,a/rologie und ,ma!matik1 ,/rassburg1 #ahii4 #dc ,gu/ave ,s*legel1 ,uranographie *9oise ou preuves directes que l'a/ronomie primitive e/ orig9aire de la ,*9e1 et qu'elle a ete empruntee p> les anci5s peuples occid5taux a la sph]e *9oise2 ouvrage accompagne d'un atlas cele/e *9ois et grec1 ,! ,hague & ,leyd51 #ahge4 #dd ,e4 ,w4 ,hopk9s1 ,! ,religions ( ,india1 ,bo/on1 #ahih1 p4 #g4 #de ,r4 ,c4 ,dutt1 ,hi/ory ( ,india1 ,london1 #aijf4 #df ,w4 ,d4 ,:itney1 ,sanskrit ,gramm>1 #c;d ed41 ,leipzig1 #ahif4 #dg ,das ,apa/amba-',sulba-,sutra1 ,zeits*rift d] deuts*5 ,morg5la5dis*5 ,gesells*aft1 ,vol4 ,,lv1 p4 #edc1 & ,vol4 ,,lvi1 p4 #cbg4 #dh ,ges*i*te d] ,ma?41 ,vol4 ,i1 #b;d ed41 p4 #eie4 #di ,l4 von ,s*ro$]1 ,py?agoras und die ,ind]1 ,leipzig1 #ahhd2 ,h4 ,vogt1 ,hab5 die alt5 ,ind] d5 ,py?agoreis*5 ,lehrsatz und das ,irrationale gekannt8 ,biblio!ca ,ma!matica1 ,vol4 ,,vii 7#c71 pp4 #f-bj2 ,a4 ,bu]k1 loc4 cit42 ,max ,simon1 ,ges*i*te d] ,ma!matik im ,alt]tum1 ,b]l91 #aiji1 pp4 #acg-afe2 ?ree ,sutras are translat$ 9 "p by ,?ibaut1 ,j\rnal ( ! ,asiatic ,society ( ,b5gal1 #ahge1 & "o appe>$ 9 ,! ,p&it1 #ahge2 ,beppo ,levi1 ,oss]vazioni e congetture sopra la geometria degli indiani1 ,biblio!ca ,ma!matica1 ,vol4 ,,ix 7#c71 #aijh1 pp4 #ig-aje4 #ej ,loc4 cit42 also ,indi5s ,lit]atur und ,cultur1 ,leipzig1 #ahhg4 #ea ,x is g5]ally agre$ t ! "n ( ! riv] ,s9dhu1 corrupt$ by we/]n peoples to ,h9dhu1 ,indos1 ,indus1 is ! root ( ,h9du/an & ( ,india4 ,reclus1 ,asia1 ,engli% ed41 ,vol4 ,,iii1 p4 #ad4 #eb ,see ! comm5ts ( ,opp]t1 ,on ! ,orig9al ,inhabitants ( ,bh>atav>s4a or ,india1 ,london1 #ahic1 p4 #a4 #ec ,a4 ,hillebr&t1 ,alt-,indi51 ,breslau1 #ahii1 p4 #aaa4 ,fragm5t>y records relate t ,kharavela1 k+ ( ,kali4nga1 le>n$ z a boy lekha 7writ+71 gan4ana 7reckon+71 & rupa 7ari?metic appli$ to monet>y affairs & m5suration71 probably 9 ! #e? c5tury ,b4,c4 ,buehl]1 ,indis*e ,palaeographie1 ,/rassburg1 #ahif1 p4 #e4 #ed ,r4 ,c4 ,dutt1 ,a ,hi/ory ( ,civilization 9 ,anci5t ,india1 ,london1 #ahic1 ,vol4 ,i1 p4 #agd4 #ee ,! ,buddha4 ,! date ( 8 bir? is unc]ta94 ,sir ,edw9 ,arnold put x c4 #fbj ,b4,c4 #ef ,i4e4 #ajj4ajg4 #eg ,!re is "s unc]ta9ty ab\t ?is limit4 #eh ,?is problem des]ves m /udy ?an has yet be5 giv5 x4 ,a beg9n+ may be made ) ,comte ,goblet d',alviella1 ,ce que l',inde doit a la ,grece1 ,p>is1 #ahig1 & ,h4 ,g4 ,ke5e's review1 ,! ,greeks 9 ,india1 9 ! ,calcutta ,review1 ,vol4 ,,cxiv1 #aijb1 p4 #a4 ,see also ,f4 ,woepeke1 ,propagation1 p4 #bec2 ,g4 ,r4 ,kaye1 loc4 cit41 p4 #dge seq41 & ,! ,s\rce ( ,h9du ,ma!matics1 ,j\rnal ( ! ,royal ,asiatic ,society1 ,july1 #aiaj1 pp4 #gdi-gfj2 ,g4 ,?ibaut1 ,a/ronomie1 ,a/rologie und ,ma!matik1 pp4 #dc-ej & #gf-gi4 ,x w be discuss$ m fully 9 ,*apt] ,,vi4 #ei ,i4e4 to #ajj1jjj4 ,! lakh is /ill ! common l>ge unit 9 ,india1 l ! myriad 9 anci5t ,greece & ! million 9 ! ,we/4 #fj ,?is aga9 sugge/s ! ,psammites1 or ,de h>5ae num]o z x is call$ 9 ! #aedd edition ( ! ,op]a ( ,ar*im$es1 a "w 9 :i* ! great ,syracusan proposes to %[ to ! k+ by geometric pro(s :i* y c foll[1 t ! numb]s :i* h be5 nam$ by u 444 are suffici5t to exce$ n only ! numb] ( a s&-heap z l>ge z ! :ole e>?1 b "o z l>ge z ! univ]se4 ,= a li/ ( e>ly editions ( ?is "w see ,d4 ,e4 ,smi?1 ,r>a ,ari?metica1 ,bo/on1 #aiji1 p4 #bbg4 #fa ,i4e4 ! ,wise4 #fb ,sir ,moni] ,moni]-,williams1 ,indian ,wisdom1 #d? ed41 ,london1 #ahic1 pp4 #add1 #agg4 ,see also ,j4 ,c4 ,m>%man1 ,abridgm5t ( ! ,hi/ory ( ,india1 ,london1 #ahic1 p4 #b4 #fc ,= a li/ & = "s description ( ^! works see ,r4 ,c4 ,dutt1 ,a ,hi/ory ( ,civilization 9 ,anci5t ,india1 ,vol4 ,,ii1 p4 #aba4 #fd ,pr(essor ,ramkri%na ,gopal ,bh&>k> fixes ! date z ! fif? c5tury ,b4,c4 ,consid]ation ( ! ,date ( ! ,mahabharata1 9 ! ,j\rnal ( ! ,bombay ,bran* ( ! ,r4 ,a4 ,soc41 ,bombay1 #ahgc1 ,vol4 ,x1 p4 #b44 #fe ,m>%man1 loc4 cit41 p4 #b4 #ff ,a4 ,c4 ,burnell1 ,s\? ,indian ,palaeography1 #b;d ed41 ,london1 #ahgh1 p4 #a1 seq4 #fg ,?is ext5sive subject ( palpable ari?metic1 ess5tially ! hi/ory ( ! abacus1 des]ves to be treat$ 9 a "w by itself4 #fh ,! foll[+ are ! lead+ s\rces ( in=mation ^u ?is subject3 ,g4 ,buehl]1 ,indis*e ,palaeographie1 p>ticul>ly *ap4 vi2 ,a4 ,c4 ,burnell1 ,s\? ,indian ,palaeography1 #b;d ed41 ,london1 #ahgh1 :]e tables ( ! v>i\s ,indian num]als are giv5 9 ,plate ,,xxiii2 ,e4 ,c4 ,bayley1 ,on ! ,g5ealogy ( ,mod]n ,num]als1 ,j\rnal ( ! ,royal ,asiatic ,society1 ,vol4 ,,xiv1 "p #c1 & ,vol4 ,,xv1 "p #a1 & repr9t1 ,london1 #ahhb2 ,i4 ,taylor1 9 ,! ,academy1 ,janu>y #bh1 #ahhb1 ) a repetition ( 8 argum5t 9 8 "w ,! ,alphabet1 ,london1 #ahhc1 ,vol4 ,,ii1 p4 #bfe1 bas$ on ,bayley2 ,g4 ,r4 ,kaye1 loc4 cit41 9 "s respects "o ( ! mo/ critical articles ?us f> publi%$2 ,j4 ,c4 ,fleet1 ,corpus inscriptionum ,indic>um1 ,london1 #ahhh1 ,vol4 ,,iii1 ) facsimiles ( _m ,indian inscriptions1 & ,indian ,epigraphy1 ,ox=d1 #aijg1 repr9t$ f ! ,imp]ial ,gazette] ( ,india1 ,vol4 ,,ii1 pp4 #a-hh1 #aijg2 ,g4 ,?ibaut1 loc4 cit41 ,a/ronomie etc42 ,r4 ,caldwell1 ,comp>ative ,gramm> ( ! ,dravidian ,languages1 ,london1 #ahef1 p4 #bfb seq42 & ,epigraphia ,indica 7(ficial publication ( ! gov]nm5t ( ,india71 ,vols4 ,i-,,ix4 ,ano!r "w ( ,buehl]'s1 ,on ! ,orig9 ( ! ,indian ,brahma ,alphabet1 is also ( value4 #fi ,! e>lie/ "w on ! subject 0 by ,james ,pr9sep1 ,on ! ,inscriptions ( ,piyadasi or ,a'soka1 etc41 ,j\rnal ( ! ,asiatic ,society ( ,b5gal1 #ahch1 foll[+ a prelim9>y sugge/ion 9 ! same j\rnal 9 #ahcg4 ,see also ,a'soka ,notes1 by ,v4 ,a4 ,smi?1 ,! ,indian ,antiqu>y1 ,vol4 ,,xxxvii1 #aijh1 p4 #bd seq41 ,vol4 ,,xxxviii1 pp4 #aea-aei1 ,june1 #aiji2 ,! ,e>ly ,hi/ory ( ,india1 #b;d ed41 ,ox=d1 #aijh1 p4 #aed2 ,j4 ,f4 ,fleet1 ,! ,la/ ,words ( ,a'soka1 ,j\rnal ( ! ,royal ,asiatic ,society1 ,octob]1 #aiji1 pp4 #iha-ajaf2 ,e4 ,s5>t1 ,les inscriptions de ,piyadasi1 #b vols41 ,p>is1 #ahhg4 #gj ,= a discussion ( ! m9or details ( ?is sy/em1 see ,buehl]1 loc4 cit41 p4 #gc4 #ga ,julius ,eut+1 ,nabataeis*e ,ins*rift5 aus ,arabi51 ,b]l91 #ahhe1 pp4 #if-ig1 ) a table ( num]als4 #gb ,= ! five pr9cipal !ories see ,buehl]1 loc4 cit41 p4 #aj4 #gc ,bayley1 loc4 cit41 repr9t p4 #c4 #gd ,buehl]1 loc4 cit42 ,epigraphia ,indica1 ,vol4 ,,iii1 p4 #acd2 ,indian ,antiqu>y1 ,vol4 ,,vi1 p4 #aee seq41 & ,vol4 ,x1 p4 #ajg4 #ge ,p&it ,bhagavanlal ,indraji1 ,on ,anci5t ,nagari ,num]ation2 f an ,inscription at ,naneghat1 ,j\rnal ( ! ,bombay ,bran* ( ! ,royal ,asiatic ,society1 #ahgf1 ,vol4 ,,xii1 p4 #djd4 #gf ,ib41 p4 #dje4 ,he gives also a plate & an int]pretation ( ea* num]al4 #gg ,^! may be comp>$ ) ,buehl]'s draw+s1 loc4 cit42 ) ,bayley1 loc4 cit41 p4 #ccg & plates2 & ) ,bayley's article 9 ! ,encyclopa$ia ,britannica1 #i? ed41 art4 ,num]als4 #gh ,e4 ,s5>t1 ,! ,inscriptions 9 ! ,caves at ,nasik1 ,epigraphia ,indica1 ,vol4 ,,viii1 pp4 #ei-if2 ,! ,inscriptions 9 ! ,cave at ,k>le1 ,epigraphia ,indica1 ,vol4 ,,vii1 pp4 #dg-gd2 ,buehl]1 ,palaeographie1 ,tafel ,,ix4 #gi ,see ,fleet1 loc4 cit4 ,see also ,t4 ,b5fey1 ,sanskrit ,gramm>1 ,london1 #ahfc1 p4 #bag2 ,m4 ,r4 ,kale1 ,high] ,sanskrit ,gramm>1 #b;d ed41 ,bombay1 #ahih1 p4 #aaj1 & o!r au?orities z cit$4 #hj ,kh>os4t4hi num]als1 ,a'soka inscriptions1 c4 #bej ,b4,c4 ,s5>t1 ,notes d'epigraphie indi5ne4 ,giv5 by ,buehl]1 loc4 cit41 ,tafel ,i4 #ha ,same1 ',saka inscriptions1 probably ( ! fir/ c5tury ,b4,c4 ,s5>t1 loc4 cit42 ,buehl]1 loc4 cit4 #hb ,brahmi num]als1 ,a'soka inscriptions1 c4 #bej ,b4,c4 ,indian ,antiqu>y1 ,vol4 ,,vi1 p4 #aee seq4 #hc ,same1 ,nana ,ghat inscriptions1 c4 #aej ,b4,c4 ,bhagavanlal ,indraji1 ,on ,anci5t ,nag>i ,num]ation1 loc4 cit4 ,copi$ f a squeeze ( ! orig9al4 #hd ,same1 ,nasik inscription1 c4 #ajj ,b4,c4 ,burgess1 ,ar*eological ,survey ,report1 ,we/]n ,india2 ,s5>t1 ,epigraphia ,indica1 ,vol4 ,,vii1 pp4 #dg-gi1 & ,vol4 ,,viii1 pp4 #ei-if4 #he ,ks4atrapa co9s1 c4 #bjj ,a4,d4 ,j\rnal ( ! ,royal ,asiatic ,society1 #ahij1 p4 #fci4 #hf ,kus4ana inscriptions1 c4 #aej ,a4,d4 ,epigraphia ,indica1 ,vol4 ,i1 p4 #cha1 & ,vol4 ,,ii1 p4 #bja4 #hg ,gupta ,inscriptions1 c4 #cjj ,a4,d4 to #dej ,a4,d4 ,fleet1 loc4 cit41 ,vol4 ,,iii4 #hh ,valhabi1 c4 #fjj ,a4,d4 ,corpus1 ,vol4 ,,iii4 #hi ,b5dall's ,table ( ,num]als1 9 ,cat4 ,sansk4 ,budd4 ,,mss41 ,briti% ,museum4 #ij ,indian ,antiqu>y1 ,vol4 ,,xiii1 #abj2 ,epigraphia ,indica1 ,vol4 ,,iii1 #abg ff4 #ia ,fleet1 loc4 cit4 #ib ,bayley1 loc4 cit41 p4 #cce4 #ic ,f a copp] plate ( #dic ,a4,d41 f\nd at ,karitalai1 ,c5tral ,india4 ,fleet1 loc4 cit41 ,plate ,,xvi4 ,x %\ld be /at$1 h[ev]1 t _m ( ^! copp] plates1 be+ de$s ( prop]ty1 h =g$ dates s z to give ! appe>ance ( antiquity ( title4 ,on ! o!r h&1 z ,colebrooke long ago po9t$ out1 a successful =g]y has to imitate ! writ+ ( ! p]iod 9 "q1 s t x becomes evid5ce well wor? consid]+1 z %[n 9 ,*apt] ,,iii4 #id ,f a copp] plate ( #eaj ,a4,d41 f\nd at ,majhgawa91 ,c5tral ,india4 ,fleet1 loc4 cit41 ,plate ,,xiv4 #ie ,f an inscription ( #ehh ,a4,d41 f\nd at ,bodh-,gaya1 ,b5gal ,presid5cy4 ,fleet1 loc4 cit41 ,plate ,,xxiv4 #if ,f a copp] plate ( #ega ,a4,d41 f\nd at ,maliya1 ,bombay ,presid5cy4 ,fleet1 loc4 cit41 ,plate ,,xxiv4 #ig ,f a ,bijayagad4h pill> inscription ( #cgb ,a4,d4 ,fleet1 loc4 cit41 ,plate ,,xxxvi1 ,c4 #ih ,f a copp] plate ( #dcd ,a4,d4 ,indian ,antiqu>y1 ,vol4 ,i1 p4 #fj4 #ii ,gadhwa inscription1 c4 #dag ,a4,d4 ,fleet1 loc4 cit41 ,plate ,,iv1 ,d4 #ajj ,karitalai plate ( #dic ,a4,d41 ref]r$ to above4 #aja ,x seems evid5t t ! ,*9ese f\r1 curi\sly en\gh call$ eight 9 ! m\?1 is only a cursive #d v]tical /rokes4 #ajb ,*alfont1 ,f4 ,h41 ,memoirs ( ! ,c>negie ,museum1 ,vol4 ,,iv1 no4 #a2 ,j4 ,hag]1 ,an ,explanation ( ! ,elem5t>y ,*>act]s ( ! ,*9ese1 ,london1 #ahja4 #ajc ,h4 ,v4 ,hilpre*t1 ,ma!matical1 ,metrological & ,*ronological ,tablets f ! ,temple ,libr>y at ,nippur1 ,vol4 ,,xx1 "p ,i1 ( ,s]ies ,a1 ,cunei=m ,texts ,publi%$ by ! ,babylonian ,exp$ition ( ! ,univ]sity ( ,p5nsylvania1 #aijf2 ,a4 ,eis5lohr1 ,e9 altbabylonis*] ,feld]plan1 ,leipzig1 #aijf2 ,masp]o1 ,dawn ( ,civilization1 p4 #ggc4 #ajd ,sir ,h4 ,h4 ,h[>d1 ,on ! ,e>lie/ ,inscriptions f ,*aldea1 ,proce$+s ( ! ,society ( ,biblical ,ar*aeology1 ,,xxi1 p4 #cja1 ,london1 #ahii4 #aje ,= a bibliography ( ! pr9cipal hypo!ses ( ?is nature see ,buehl]1 loc4 cit41 p4 #gg4 ,buehl] 7p4 #gh7 feels t ( all ^! hypo!ses t :i* connects ! ,brahmi ) ! ,egyptian num]als is ! mo/ plausible1 al?\gh he does n adduce any conv9c+ pro(4 ,?4 ,h5ri ,m>t91 ,les signes num]aux et l'ari?metique *ez les peuples de l'antiquite et du moy5 age 7be+ an exam9ation ( ,cantor's ,ma!matis*e ,beitraege zum ,culturleb5 d] ,voelk]71 ,annali di matematica pura ed applicata1 ,vol4 ,v1 ,rome1 #ahfd1 pp4 #h1 #gj4 ,also1 same au?or1 ,re*]*es n\velles sur l'orig9e de notre sy/eme de num]ation ecrite1 ,revue ,ar*eologique1 #aheg1 pp4 #cf1 #ee4 ,see also ! tables giv5 lat] 9 ?is "w4 #ajf ,j\rnal ( ! ,royal ,asiatic ,society1 ,bombay ,bran*1 ,vol4 ,,xxiii4 #ajg ,loc4 cit41 repr9t1 ,"p ,i1 pp4 #ab1 #ag4 ,bayley's d$uctions are g5]ally reg>d$ z unw>rant$4 #ajh ,! ,alphabet2 ,london1 #ahhc1 ,vol4 ,,ii1 pp4 #bfe1 #bff1 & ,! ,academy ( ,jan4 #bh1 #ahhb4 #aji ,taylor1 ,! ,alphabet1 loc4 cit41 table on p4 #bff4 #aaj ,buehl]1 ,on ! ,orig9 ( ! ,indian ,brahma ,alphabet1 ,/rassburg1 #ahih1 footnote1 pp4 #eb1 #ec4 #aaa ,albre*t ,web]1 ,hi/ory ( ,indian ,lit]ature1 ,engli% ed41 ,bo/on1 #ahgh1 p4 #bef3 ,! ,indian figures f #a-i are abbreviat$ =ms ( ! initial lett]s ( ! num]als !mselves4443 ! z]o1 too1 has aris5 out ( ! fir/ lett] ( ! ^w s4unya 7empty7 7x occurs ev5 9 ,p+ala74 ,x is ! decimal place value ( ^! figures :i* gives !m significance4 ,c4 ,h5ry1 ,sur l'orig9e de quelques notations ma!matiques1 ,revue ,ar*eologique1 ,june & ,july1 #ahgi1 attempts to d]ive ! ,boe?ian =ms f ! initials ( ,lat9 words4 ,see also ,j4 ,pr9sep1 ,exam9ation ( ! ,inscriptions f ,girn> 9 ,guj]at1 & ,dhauli 9 ,cutta*1 ,j\rnal ( ! ,asiatic ,society ( ,b5gal1 #ahch1 especially ,plate ,,xx1 p4 #cdh2 ?is 0 ! fir/ "w on ! subject4 #aab ,buehl]1 ,palaeographie1 p4 #ge1 gives ! li/1 ) ! li/ ( lett]s 7p4 #gf7 correspond+ to ! numb] symbols4 #aac ,= a g5]al discussion ( ! connection betwe5 ! num]als & ! diff]5t k9ds ( alphabets1 see ! articles by ,u4 ,c]etti1 ,sulla orig9e delle cifre num]ali mod]ne1 ,rivi/a di fisica1 matematica e sci5ze naturali1 ,pisa & ,pavia1 #aiji1 anno ,x1 numb]s #aad1 #aah1 #aai1 & #abj1 & cont9uation 9 #aiaj4 #aad ,?is is "o ( ,buehl]'s hypo!ses4 ,see ,bayley1 loc4 cit41 repr9t p4 #d2 a good bibliography ( orig9al s\rces is giv5 9 ?is "w1 p4 #ch4 #aae ,loc4 cit41 repr9t1 "p ,i1 pp4 #ab1 #ag4 ,see also ,burnell1 loc4 cit41 p4 #fd1 & tables 9 plate ,,xxiii4 #aaf ,?is 0 ass]t$ by ,g4 ,hag] 7,memoria sulle cifre arabi*e1 ,milan1 #ahac1 also publi%$ 9 ,fundgrub5 des ,ori5ts1 ,vi5na1 #ahaa1 & 9 ,biblio!que ,britannique1 ,g5eva1 #ahab74 ,see also ! rec5t article by ,major ,*>les ,e4 ,woodruff1 ,! ,evolution ( ,mod]n ,num]als f ,tally ,m>ks1 ,am]ican ,ma!matical ,mon?ly1 ,augu/-,septemb]1 #aiji4 ,bi]natzki1 ,die ,ari?metik d] ,*9es51 ,crelle's ,j\rnal fu] die re9e und angew&te ,ma!matik1 ,vol4 ,,lii1 #aheg1 pp4 #ei-if1 also ass]ts ! priority ( ! ,*9ese claim = a place sy/em & ! z]o1 b ^u ! flimsie/ au?ority4 ,*4 de ,p>avey1 ,essai sur l'orig9e unique et hi]oglyphique des *iffres et des lettres de t\s les peuples1 ,p>is1 #ahbf2 ,g4 ,kle9wae*t]1 ,! ,orig9 ( ! ,arabic ,num]als1 ,*9a ,review1 ,vol4 ,,xi1 #ahhb-ahhc1 pp4 #cgi-cha1 ,vol4 ,,xii1 pp4 #bh-cj2 ,biot1 ,note sur la connaissance que les ,*9ois ont eue de la valeur de position des *iffres1 ,j\rnal ,asiatique1 #ahci1 pp4 #dig-ejb4 ,a4 ,t]ri5 de ,lac\p]ie1 ,! ,old ,num]als1 ! ,c\nt+-,rods & ! ,swan-,pan 9 ,*9a1 ,numismatic ,*ronicle1 ,vol4 ,,iii 7#c71 pp4 #big-cdj1 & ,cr[d] ,b4 ,moseley1 ,num]al ,*>act]s3 ,!ory ( ,orig9 & ,developm5t1 ,am]ican ,antiqu>ian1 ,vol4 ,,xxii1 pp4 #bgi-bhd1 bo? propose to d]ive our num]als f ,*9ese *>act]s1 9 mu* ! same way z is done by ,major ,woodruff1 9 ! article above cit$4 #aag ,! ,greeks1 probably foll[+ ! ,semitic cu/om1 us$ n9e lett]s ( ! alphabet = ! num]als f #a to #i1 !n n9e o!rs = #aj to #ij1 & fur!r lett]s to repres5t #ajj to #ijj4 ,z ! ord9>y ,greek alphabet 0 insuffici5t1 conta9+ only tw5ty-f\r lett]s1 an alphabet ( tw5ty-sev5 lett]s 0 us$4 #aah ,in/itutiones ma!maticae1 #b vols41 ,/rassburg1 #aeic-aeif1 a some:at r>e "w f :i* ! foll[+ quotation is tak53 ,quis e/ h>um ,cyphr>um autor8 ,a quibus hae usitatae syphr>um notae s9t inv5tae3 hact5us inc]tum fuit3 meo tam5 iudicio1 quod exiguum esse fateor3 a graecis libr>ijs 7quorum olim magna fuit copia7 lit]ae ,graecorum quibus vet]es ,graeci tamquam num]orum notis sunt usi3 fu]unt corruptae4 vt ex 8 licet vid]e4 ,graecorum ,lit]ae corruptae4 ,illu/ration ,s$ qua ratione graecorum lit]ae ita fu]unt corruptae8 ,f9x]unt has corruptas ,graecorum lit]>um notas3 vel abiectione vt 9 nota b9>ij num]i1 vel additione vt 9 t]n>ij1 vel inu]sione vt 9 sept5>ij1 num]i nota1 no/rae notae1 quibus hodie utimur3 ab 8 sola diff]unt elegantia1 vt app>et4 ,see also ,bay]1 ,hi/oria regni ,graecorum ,bactriani1 ,/4 ,pet]sburg1 #aghh1 pp4 #abi-acj1 quot$ by ,m>t91 ,re*]*es n\velles1 etc41 loc4 cit4 #aai ,p4 ,d4 ,huet1 ,demon/ratio evangelica1 ,p>is1 #agfi1 note to p4 #aci on p4 #fdg3 ,ab ,arabibus vel ab ,indis inv5tas esse1 non vulgus eruditorum modo1 s$ doctissimi quique ad hanc diem arbitrati sunt4 ,ego v]o falsum id esse1 m]osque esse ,graecorum *>act]es aio2 a libr>iis ,graecae l+uae ign>is int]polatos1 et diuturna scrib5di consuetud9e corruptos4 ,nam primum #a apex fuit1 seu virgula1 nota ,greek3 monados4 #b1 e/ ipsum beta extremis suis truncatum4 gamma1 si 9 s9i/ram p>tem incl9av]is cauda mutilav]is s9i/rum cornu s9i/rorsum flex]is1 fiet #c4 ,res ipsa loquitur #d ipsissimum esse ,delta1 cujus crus s9i/rum erigitur ,greek3 kata ka!ton1 infra basim desc5dit2 basis v]o ipsa ultra crus producta em9et4 ,vides quam #e simile sit ,greek3 toi epsilon2 infimo tantum semicirculo1 qui s9i/rorsum patebat1 dextrorsum conv]so4 ,greek3 episemon bau quod ita notabatur digamma1 rotundato v5tre1 p$e detracto1 pep]it ,greek3 to #f4 ,ex ,zeta basi sua mutilato1 ortum e/ ,greek3 to #g4 ,si ,eta inflexis introrsum apicibus 9 rotundiorem commodiorem =mam mutav]is1 exurget ,greek3 to #h4 ,at #i ipsissimum e/ alt !ta4 ,i4 ,weidl]1 ,spicilegium obs]vationum ad hi/oriam not>um num]alium1 ,witt5b]g1 #agee1 d]ives !m f ! ,hebrew lett]s2 ,dom ,augu/9 ,calmet1 ,re*]*es sur l'orig9e des *iffres d'ari?metique1 ,memoires p\r l'hi/oire des sci5ces et des beaux arts1 ,trev\x1 #agjg 7pp4 #afbj-afce1 ) two plates71 d]ives ! curr5t symbols f ! ,romans1 /at+ t !y are relics ( ! anci5t ,notae ,tironianae4 ,^! notes 7 "p ( a sy/em ( %or?& inv5t$1 or at lea/ p]fect$1 by ,tiro1 a slave :o 0 fre$ by ,cic]o4 ,l4 ,a4 ,s$illot1 ,sur l'orig9e de nos *iffres1 ,atti dell' ,accademia pontificia dei nuovi ,l9cei1 ,vol4 ,,xviii1 #ahfd-ahfe1 pp4 #caf-cbb1 d]ives ! ,arabic =ms f ! ,roman num]als4 #abj ,a?anasius ,kir*]1 ,ari?mologia sive ,de abditis ,num]orum1 my/]ijs qua origo1 antiquitas fabrica ,num]orum exponitur1 ,rome1 #affe4 #aba ,see ,sut]1 ,die ,ma!matik] und ,a/ronom5 d] ,arab]1 p4 #ajj4 #abb ,et hi num]i sunt num]i ,indiani1 a ,bra*manis ,indiae ,sapi5tibus ex figura circuli secti inu5ti4 #abc ,v4 ,a4 ,smi?1 ,! ,e>ly ,hi/ory ( ,india1 ,ox=d1 #b;d ed41 #aijh1 p4 #ccc4 #abd ,c4 ,j4 ,ball1 ,an ,inscrib$ ,lime/one ,tablet f ,sipp>a1 ,proce$+s ( ! ,society ( ,biblical ,ar*aeology1 ,vol4 ,,xx1 p4 #be 7,london1 #ahih74 ,t]ri5 de ,lac\p]ie /ates t ! ,*9ese us$ ! circle = #aj be=e ! beg9n+ ( ! ,*ri/ian era4 ,catalogue ( ,*9ese ,co9s1 ,london1 #ahib1 p4 xl4 #abe ,= a purely fanciful d]ivation f ! correspond+ numb] ( /rokes1 see ,w4 ,w4 ,r4 ,ball1 ,a ,%ort ,acc\nt ( ! ,hi/ory ( ,ma!matics1 #a/ ed41 ,london1 #ahhh1 p4 #adg2 simil>ly ,j4 ,b4 ,reveillaud1 ,essai sur les *iffres arabes1 ,p>is1 #ahhc2 ,p4 ,voizot1 ,les *iffres arabes et leur orig9e1 ,la ,nature1 #ahii1 p4 #bbb2 ,g4 ,dumesnil1 ,de la =me des *iffres usuels1 ,annales de l'univ]site de ,gr5oble1 #aijg1 ,vol4 ,,xix1 pp4 #feg-fgd1 also a note 9 ,revue ,ar*eologique1 #ahij1 ,vol4 ,,xvi 7#c71 pp4 #cdb-cdh2 "o ( ! e>lie/ ref]5ces to a possible d]ivation f po9ts is 9 a "w by ,bett9o entitl$ ,api>ia univ]sae philosophiae ma!maticae 9 quibus p>adoxa et n\a ma*9am5ta ad usus eximios traducta1 et facillimis demon/rationibus confirmata1 ,bologna1 #aede1 ,vol4 ,,ii1 ,api>ium ,,xi1 p4 #e4 #abf ,alphabetum ,b>manum1 ,romae1 ,,mdcclxxvi1 p4 #ej4 ,! #a is evid5tly ,sanskrit1 & ! #d1 #g1 & possibly #i are f ,india4 #abg ,alphabetum ,gr&onico-,malab>icum1 ,romae1 ,,mdcclxxii1 p4 #ij4 ,! z]o is n us$1 b ! symbols = #aj1 #ajj1 & s on1 are jo9$ to ! units to make ! high] numb]s4 #abh ,alphabetum ,tangutanum1 ,romae1 ,,mdcclxxiii1 p4 #ajg4 ,9 a ,tibetan ,,ms4 9 ! libr>y ( ,pr(essor ,smi?1 probably ( ! eighte5? c5tury1 sub/antially ^! =ms are giv54 #abi ,bayley1 loc4 cit41 plate ,,ii4 ,simil> =ms to ^! "h %[n1 & num]\s o!r =ms f\nd 9 ,india1 z well z ^? ( o!r ori5tal c\ntries1 are giv5 by ,a4 ,p4 ,pihan1 ,expose des signes de num]ation usites *ez les peuples ori5taux anci5s et mod]nes1 ,p>is1 #ahfj4 #acj ,buehl]1 loc4 cit41 p4 #hj2 ,j4 ,f4 ,fleet1 ,corpus inscriptionum ,indic>um1 ,vol4 ,,iii1 ,calcutta1 #ahhh4 ,li/s ( su* words are giv5 also by ,al-,biruni 9 8 "w ,india2 by ,burnell1 loc4 cit42 by ,e4 ,jacquet1 ,mode d'expression symbolique des nombres employe p> les ,indi5s1 les ,tibeta9s et les ,javanais1 ,j\rnal ,asiatique1 ,vol4 ,,xvi1 ,p>is1 #ahce4 #aca ,?is date is giv5 by ,fleet1 loc4 cit41 ,vol4 ,,iii1 p4 #gc1 z ! e>lie/ epigraphical in/ance ( ?is usage 9 ,india prop]4 #acb ,web]1 ,indis*e ,/udi51 ,vol4 ,,viii1 p4 #aff seq4 #acc ,j\rnal ( ! ,royal ,asiatic ,society1 ,vol4 ,i 7,n4,s471 p4 #djg4 #acd ,,viii1 #bj1 #ba4 #ace ,?4 ,h4 ,m>t91 ,les signes num]aux 4441 ,rome1 #ahfd2 ,lass51 ,indis*e ,alt]?umskunde1 ,vol4 ,,ii1 #b;d ed41 ,leipzig & ,london1 #ahgd1 p4 #aaec4 #acf ,b see ,burnell1 loc4 cit41 & ,?ibaut1 ,a/ronomie1 ,a/rologie und ,ma!matik1 p4 #ga4 #acg ,a4 ,b>?1 ,inscriptions ,sanscrites du ,cambodge1 9 ! ,notices et extraits des ,mss4 de la ,biblio!que nationale1 ,vol4 ,,xxvii1 ,"p ,i1 pp4 #a-ahj1 #ahhe2 see also num]\s articles 9 ,j\rnal ,asiatique1 by ,aymoni]4 #ach ,buehl]1 loc4 cit41 p4 #hb4 #aci ,loc4 cit41 p4 #gi4 #adj ,buehl]1 loc4 cit41 p4 #hc4 ,! ,h9du a/rolog]s /ill use an alphabetical sy/em ( num]als4 ,burnell1 loc4 cit41 p4 #gi4 #ada ,well c\ld ,ramus say1 ,quicunq2 autem fu]it inv5tor decem not>um laudem magnam m]uit4 #adb ,al-,biruni gives li/s4 #adc ,propagation1 loc4 cit41 p4 #ddc4 #add ,see ! quotation f ,! ,light ( ,asia 9 ,*apt] ,,ii1 p4 #af4 #ade ,! n9e ciph]s 7 call$ a4nka4 #adf ,zur ,ges*i*te des indis*5 ,ziff]nsy/ems1 ,zeits*rift fu] die ,kunde des ,morg5l&es1 ,vol4 ,,iv1 #ahdb1 pp4 #gd-hc4 #adg ,x is f\nd 9 ! ,bakhs4ali ,,ms4 ( an elem5t>y ari?metic :i* ,ho]nle plac$1 at fir/1 ab\t ! beg9n+ ( our era1 b ! date is mu* 9 "q4 ,g4 ,?ibaut1 loc4 cit41 places x betwe5 #gjj & #ijj ,a4,d42 ,cantor places ! body ( ! "w ab\t ! ?ird or f\r? c5tury ,a4,d41 ,ges*i*te d] ,ma!matik1 ,vol4 ,i 7#c71 p4 #eih4 #adh ,= ! opposite side ( ! case see ,g4 ,r4 ,kaye1 ,notes on ,indian ,ma!matics1 ,no4 #b4--;,aryabhat4a1 ,j\rn4 & ,proc4 ( ! ,asiatic ,soc4 ( ,b5gal1 ,vol4 ,,iv1 #aijh1 pp4 #aaa-ada4 #adi ,he us$ "o ( ! alphabetic sy/ems expla9$ above4 ,?is ran up to #ajah & 0 n difficult1 beg9n+ z foll[s3 ,illu/ration ! same lett] 7ka7 appe>+ 9 ! successive consonant =ms1 ka1 kha1 ga1 gha1 etc4 ,see ,c4 ,i4 ,g]h>dt1 ,ueb] die ,ent/ehung und ,ausbreitung des dekadis*5 ,zahl5sy/ems1 ,programm1 p4 #ag1 ,salzw$el1 #ahec1 & ,etudes hi/oriques sur l'ari?metique de position1 ,programm1 p4 #bd1 ,b]l91 #ahef2 ,e4 ,jacquet1 ,mode d'expression symbolique des nombres1 loc4 cit41 p4 #ig2 ,l4 ,rodet1 ,sur la v]itable signification de la notation num]ique inv5tee p> ,aryabhata1 ,j\rnal ,asiatique1 ,vol4 ,,xvi 7#g71 pp4 #ddj-dhe4 ,on ! two ,aryabhat4z see ,kaye1 ,bibl4 ,ma?41 ,vol4 ,x 7#c71 p4 #bhi4 #aej ,us+ kha1 a synonym ( 'sunya4 ,bayley1 loc4 cit41 p4 #bb1 & ,l4 ,rodet1 ,j\rnal ,asiatique1 ,vol4 ,,xvi 7#g71 p4 #ddc4 #aea ,v>aha-,mihira1 ,pancasiddhantika1 translat$ by ,g4 ,?ibaut & ,m4 ,s4 ,dviv$i1 ,b5>es1 #ahhi2 see ,buehl]1 loc4 cit41 p4 #gh2 ,bayley1 loc4 cit41 p4 #bc4 #aeb ,br4hat ,sam4hita1 translat$ by ,k]n1 ,j\rnal ( ! ,royal ,asiatic ,society1 #ahgj-ahge4 #aec ,x is /at$ by ,buehl] 9 a p]sonal lett] to ,bayley 7loc4 cit41 p4 #fe7 t !re are hundr$s ( in/ances ( ?is usage 9 ! ,br4hat ,sam4hita4 ,! sy/em 0 also us$ 9 ! ,pancasiddhantika z e>ly z #eje ,a4,d4 ,buehl]1 ,palaeographie1 p4 #hj1 & ,fleet1 ,j\rnal ( ! ,royal ,asiatic ,society1 #aiaj1 p4 #hai4 #aed ,cantor1 ,ges*i*te d] ,ma!matik1 ,vol4 ,i 7#c71 p4 #fjh4 #aee ,buehl]1 loc4 cit41 p4 #gh4 #aef ,bayley1 p4 #ch4 #aeg ,noviomagus1 9 8 ,de num]is libri duo1 ,p>is1 #aeci1 confesses 8 ignorance z to ! orig9 ( ! z]o1 b says3 ,d4 ,h5ricus ,grauius1 vir ,graece ,hebraice exime doctus1 ,hebraicam orig9em o/5dit1 add+ t ,valla ,indis ,ori5talibus g5tibus inv5tionem tribuit4 #aeh ,see ,essays1 ,vol4 ,,ii1 pp4 #bhg & #bhh4 #aei ,vol4 ,,xxx1 p4 #bje seqq4 #afj ,loc4 cit41 p4 #bhd seqq4 #afa ,colebrooke1 loc4 cit41 p4 #bhh4 #afb ,loc4 cit41 p4 #gh4 #afc ,h]eaft]1 unless expressly /at$ to ! contr>y1 we %all use ! ^w num]als to mean num]als ) place value4 #afd ,! ,gurj>as ( ,rajputana & ,kanauj1 9 ,j\rnal ( ! ,royal ,asiatic ,society1 ,janu>y & ,april1 #aiji4 #afe ,vol4 ,,ix1 #aijh1 p4 #bdh4 #aff ,epigraphia ,indica1 ,vol4 ,,ix1 pp4 #aic & #aih4 #afg ,epigraphia ,indica1 ,vol4 ,,ix1 p4 #a4 #afh ,loc4 cit41 p4 #ga4 #afi ,?ibaut1 p4 #ga4 #agj ,e/ autem 9 aliquibus figur>um i/>am apud multos diu]sitas4 ,quidam enim septimam hanc figuram repres5tant1 etc4 ,boncompagni1 ,trattati1 p4 #bh4 ,ene/roem has %[n t v likely ?is "w is incorrectly attribut$ to ,johannes ,hispal5sis4 ,biblio!ca ,ma!matica1 ,vol4 ,,ix 7#c71 p4 #b4 #aga ,indis*e ,palaeographie1 ,tafel ,,ix4 #agb ,edit$ by ,bloomfield & ,g>be1 ,baltimore1 #aija1 conta9+ photographic reproductions ( ! manuscript4 #agc ,bakhs4ali ,,ms4 ,see page #dc2 ,ho]nle1 ,r41 ,! ,indian ,antiqu>y1 ,vol4 ,,xvii1 pp4 #cc-dh1 #a plate2 ,ho]nle1 ,v]h&lung5 des ,,vii4 ,int]national5 ,ori5tali/5-,congresses1 ,aris*e ,section1 ,vi5na1 #ahhh1 ,on ! ,bak%ali ,manuscript1 pp4 #abg-adg1 #c plates2 ,buehl]1 loc4 cit4 #agd #c1 #d1 #f1 f ,h4 ,h4 ,dhruva1 ,?ree ,l&-,grants f ,sankh$a1 ,epigraphia ,indica1 ,vol4 ,,ii1 pp4 #ai-bd ) plates2 date #eie ,a4,d4 #g1 #a1 #e1 f ,bh&>k>1 ,daulatabad ,plates1 ,epigraphia ,indica1 ,vol4 ,,ix1 "p ,v2 date c4 #gih ,a4,d4 #age #h1 #g1 #b1 f ,buckhala ,inscription ( ,nagabhatta1 ,bh&>k>1 ,epigraphia ,indica1 ,vol4 ,,ix1 "p ,v2 date #hae ,a4,d4 #e f ,! ,morbi ,copp]-,plate1 ,bh&>k>1 ,! ,indian ,antiqu>y1 ,vol4 ,,ii1 pp4 #beg-beh1 ) plate2 date #hjd ,a4,d4 ,see ,buehl]1 loc4 cit4 #agf #h f ! above ,morbi ,copp]-,plate4 #d1 #e1 #g1 #i1 & #j1 f ,asni ,inscription ( ,mahipala1 ,! ,indian ,antiqu>y1 ,vol4 ,,xvi1 pp4 #agd-age2 inscription is on r$ s&/one1 date #iag ,a4,d4 ,see ,buehl]4 #agg #h1 #i1 #d1 f ,ra%trakuta ,grant ( ,amoghav>%a1 ,j4 ,f4 ,fleet1 ,! ,indian ,antiqu>y1 ,vol4 ,,xii1 pp4 #bfc-bgb2 copp]-plate grant ( date c4 #igb ,a4,d4 ,see ,buehl]4 #g1 #c1 #e1 f ,torkh$e ,copp]-,plate ,grant ( ! ,"t ( ,gov9d>aja ( ,guj]at1 ,fleet1 ,epigraphia ,indica1 ,vol4 ,,iii1 pp4 #ec-eh4 ,see ,buehl]4 #agh ,f ,a ,copp]-,plate ,grant ( ,k+ ,trito*anapala ,*anlukya ( ,lat4ade'sa1 ,h4,h4 ,dhruva1 ,indian ,antiqu>y1 ,vol4 ,,xii1 pp4 #aif-bje2 date #ajej ,a4,d4 ,see ,buehl]4 #agi ,burnell1 ,a4 ,c41 ,s\? ,indian ,palaeography1 plate ,,xxiii1 ,telugu-,can>ese num]als ( ! elev5? c5tury4 ,see ,buehl]4 #ahj ,f a manuscript ( ! second half ( ! ?irte5? c5tury1 reproduc$ 9 ,della vita e delle op]e di ,leon>do ,pisano1 ,baldass>e ,boncompagni1 ,rome1 #aheb1 9 ,atti dell' ,accademia ,pontificia dei nuovi ,l9cei1 anno ,v4 #aha ,f a f\rte5?-c5tury manuscript1 z reproduc$ 9 ,della vita etc41 ,boncompagni1 loc4 cit4 #ahb ,f a ,tibetan ,,ms4 9 ! libr>y ( ,d4 ,e4 ,smi?4 #ahc ,f a ,tibetan block-book 9 ! libr>y ( ,d4 ,e4 ,smi?4 #ahd ',sarada num]als f ,! ,ka%mirian ,a?>va-,v$a1 reproduc$ by *romophotography f ! manuscript 9 ! ,univ]sity ,libr>y at ,tueb+51 ,bloomfield & ,g>be1 ,baltimore1 #aija4 ,some:at simil> =ms are giv5 "u ,num]ation ,ca*emiri5ne1 by ,pihan1 ,expose etc41 p4 #hd4 #ahe ,franz ,x4 ,kugl]1 ,die ,babylonis*e ,mondre*nung1 ,freiburg i4 ,br41 #aijj1 9 ! num]\s plates at ! end ( ! book2 practically all ( ^! conta9 ! symbol to :i* ref]5ce is made4 ,cantor1 ,ges*i*te1 ,vol4 ,i1 p4 #ca4 #ahf ,f4 ,x4 ,kugl]1 ,/]nkunde und ,/]ndi5/ 9 ,babel1 ,i4 ,bu*1 f ! beg9n+s to ! "t ( ,*ri/1 ,mu5/] i4 ,we/fal51 #aijg4 ,x also has num]\s tables conta9+ ! above z]o4 #ahg ,f a lett] to ,d4 ,e4 ,smi?1 f ,g4 ,f4 ,hill ( ! ,briti% ,museum4 ,see also 8 monograph ,on ! ,e>ly ,use ( ,arabic ,num]als 9 ,europe1 9 ,ar*aeologia1 ,vol4 ,,lxii 7#aiaj71 p4 #acg4 #ahh ,r4 ,ho]nle1 ,! ,bak%ali ,manuscript1 ,indian ,antiqu>y1 ,vol4 ,,xvii1 pp4 #cc-dh & #bge-bgi1 #ahhh2 ,?ibaut1 ,a/ronomie1 ,a/rologie und ,ma!matik1 p4 #ge2 ,ho]nle1 ,v]h&lung51 loc4 cit41 p4 #acb4 #ahi ,bayley1 loc4 cit41 ,vol4 ,,xv1 p4 #bi4 ,also ,b5dall1 ,on a ,sy/em ( ,num]als us$ 9 ,s\? ,india1 ,j\rnal ( ! ,royal ,asiatic ,society1 #ahif1 pp4 #ghi-gib4 #aij ,v4 ,a4 ,smi?1 ,! ,e>ly ,hi/ory ( ,india1 #b;d ed41 ,ox=d1 #aijh1 p4 #ad4 #aia ,colebrooke1 ,algebra1 ) ,ari?metic & ,m5suration1 f ! ,sanskrit ( ,brahmegupta & ,bhasc>a1 ,london1 #ahag1 pp4 #cci-cdj4 #aib ,ibid41 p4 #ach4 #aic ,d4 ,e4 ,smi?1 9 ! ,biblio!ca ,ma!matica1 ,vol4 ,,ix 7#c71 pp4 #ajf-aaj4 #aid ,z :5 we use ?ree dots 744474 #aie ,! ,h9dus call ! n\ght explicitly 'sunyab9du '! dot m>k+ a blank1' & ab\t #ejj ,a4,d4 !y m>k$ x by a simple dot1 :i* latt] is commonly us$ 9 inscriptions & ,,mss4 9 ord] to m>k a blank1 & :i* 0 lat] conv]t$ into a small circle4 ,buehl]1 ,on ! ,orig9 ( ! ,indian ,alphabet1 p4 #ec1 note4 #aif ,fazz>i1 ,dell' orig9e delle p>ole z]o e cifra1 ,naples1 #aijc4 #aig ,e4 ,wappl]1 ,zur ,ges*i*te d] ,ma!matik im #ae4 ,jahrhund]t1 9 ! ,zeits*rift fu] ,ma!matik und ,physik1 ,vol4 ,,xlv1 ,hi/4-lit4 ,abt41 p4 #dg4 ,! manuscript is ,no4 ,c4 #hj1 9 ! ,dresd5 libr>y4 #aih ,j4 ,g4 ,pra5del1 ,algebra neb/ ihr] lit]>is*5 ,ges*i*te1 p4 #egb1 ,muni*1 #agie4 #aii ,see ! table1 p4 #bc4 ,does ! fact t ! e>ly ,european ari?metics1 foll[+ ! ,arab cu/om1 always put ! #j aft] ! #i1 sugge/ t ! #j 0 d]iv$ f ! old ,h9du symbol = #aj8 #bjj ,bayley1 loc4 cit41 p4 #dh4 ,f ?is fact ,delambre 7,hi/oire de l'a/ronomie anci5ne7 inf]r$ t ,ptolemy knew ! z]o1 a !ory accept$ by ,*asles1 ,ap]cu hi/orique sur l'orig9e et le developpem5t des me?odes en geometrie1 #ahge ed41 p4 #dgf2 ,nesselmann1 h[ev]1 %[$ 7,algebra d] ,grie*51 #ahdb1 p4 #ach71 t ,ptolemy m]ely us$ ,greek3 o = ,greek3 oud51 ) no notion ( z]o4 ,see also ,g4 ,fazz>i1 ,dell' orig9e delle p>ole z]o e cifra1 ,at5eo1 ,anno ,i1 ,no4 #aa1 repr9t$ at ,naples 9 #aijc1 :]e ! use ( ! po9t & ! small cross = z]o is also m5tion$4 ,?4 ,h4 ,m>t91 ,les signes num]aux etc41 repr9t p4 #cj1 & ,j4 ,br&is1 ,das ,mu5z-1 ,mass- und ,gewi*tswes5 9 ,vord]asi5 bis auf ,alex&] d5 ,gross51 ,b]l91 #ahff1 p4 #aj1 also discuss ?is usage ( ,greek3 o1 )\t ! notion ( place value1 by ! ,greeks4 #bja ,al-,battani sive ,albat5ii opus a/ronomicum4 ,ad fidem codicis escurial5sis arabice editum1 lat9e v]sum1 adnotationibus in/ructum a ,c>olo ,alphonso ,nall9o1 #ahii-aijg4 ,publicazioni del ,r4 ,oss]vatorio di ,br]a 9 ,milano1 ,no4 ,,xl4 #bjb ,loc4 cit41 ,vol4 ,,ii1 p4 #bga4 #bjc ,c4 ,h5ry1 ,prologus ,n4 ,ocreati 9 ,helceph ad ,adel>dum ,bat5sem magi/rum suum1 ,abh&lung5 zur ,ges*i*te d] ,ma!matik1 ,vol4 ,,iii1 #ahhj4 #bjd ,max4 ,curtze1 ,ueb] e9e ,algorismus-,s*rift des ,,xii4 ,jahrhund]ts1 ,abh&lung5 zur ,ges*i*te d] ,ma!matik1 ,vol4 ,,viii1 #ahih1 pp4 #a-bg2 ,alfr$ ,nagl1 ,ueb] e9e ,algorismus-,s*rift des ,,xii4 ,jahrhund]ts und ueb] die ,v]breitung d] indis*-arabis*5 ,re*5kun/ und ,zahlzei*5 im *ri/l4 ,ab5dl&e1 ,zeits*rift fu] ,ma!matik und ,physik1 ,hi/4-lit4 ,ab?41 ,vol4 ,,xxxiv1 pp4 #abi-adf & #afa-agj1 ) "o plate4 #bje ,byzant9is*e ,analekt51 ,abh&lung5 zur ,ges*i*te d] ,ma!matik1 ,vol4 ,,ix1 pp4 #afa-ahi4 #bjf symbol or symbol = #j4 symbol also us$ = #e4 symbols = #ac4 ,heib]g1 loc4 cit4 #bjg ,g]h>dt1 ,etudes hi/oriques sur l'ari?metique de position1 ,b]l91 #ahef1 p4 #ab2 ,j4 ,b[r+1 ,! ,decimal ,sy/em 9 ,numb]s1 ,co9s1 ,acc\nts1 ,london1 #ahed1 p4 #cc4 #bjh ,k>abacek1 ,wi5] ,zeits*rift fu] die ,kunde des ,morg5l&es1 ,vol4 ,,xi1 p4 #ac2 ,fuehr] dur* die ,papyrus-,aus/ellung ,erzh]zog ,ra9]1 ,vi5na1 #ahid1 p4 #baf4 #bji ,9 ! libr>y ( ,g4 ,a4 ,plimpton1 ,esq4 #baj ,cantor1 ,ges*i*te1 ,vol4 ,i 7#c71 p4 #fgd2 ,y4 ,mikami1 ,a ,rem>k on ! ,*9ese ,ma!matics 9 ,cantor's ,ges*i*te d] ,ma!matik1 ,ar*iv d] ,ma!matik und ,physik1 ,vol4 ,,xv 7#c71 pp4 #fh-gj4 #baa ,( c\rse ! e>li] hi/orians made innum]able guesses z to ! orig9 ( ! ^w ciph]4 ,e4g4 ,mat!w ,ho/us1 ,de num]atione em5data1 ,antw]p1 #aehb1 p4 #aj1 says3 ,siphra vox ,hebraeam orig9em sapit ref]tque3 ut docti arbitrantur1 a v]bo saph>1 quod ,ord9e num]auit significat4 ,unde ,seph> num]us e/3 h9c ,siphra 7vulgo corruptius74 ,etsi v]o g5s ,iudaica 8 notis1 quae hodie ,siphrae vocantur1 usa non fuit3 mansit tam5 rei appellatio apud multas g5tes4 ,dasypodius1 ,in/itutiones ma!maticae1 ,vol4 ,i1 #aeic1 gives a l>ge "p ( ?is quotation ^w = ^w1 )\t any m5tion ( ! s\rce4 ,h]mannus ,hugo1 ,de prima scrib5di orig9e1 ,trajecti ad ,rh5um1 #agch1 pp4 #cjd-cje1 & note1 p4 #cje2 ,k>l ,krumba*]1 ,woh] /ammt das ,wort ,ziff] 7,*iffre781 ,etudes de philologie neo-grecque1 ,p>is1 #ahib4 #bab ,buehl]1 loc4 cit41 p4 #gh & p4 #hf4 #bac ,fazz>i1 loc4 cit41 p4 #d4 ,s ,elia ,misra*i 7#adee-aebf7 9 8 po/hum\s ,book ( ,numb]1 ,con/ant9ople1 #aecd1 expla9s sifra z be+ ,arabic4 ,see also ,/e9s*neid]1 ,biblio!ca ,ma!matica1 #ahic1 p4 #fi1 & ,g4 ,w]!im1 ,die ,ari?metik des ,elia ,misra*i1 ,programm1 ,frankfurt1 #ahic4 #bad ,cum 8 novem figuris1 et cum hoc signo #j1 quod arabice zephirum appellatur1 scribitur quilibet num]us4 #bae ,greek3 tziphra1 a =m also us$ by ,neophytos 7date unkn[n1 probably c4 #accj74 ,x is curi\s t ,f9aeus 7#aeee ed41 f4 #b7 us$ ! =m tziphra ?r\gh\t4 ,a4 ,j4 ,h4 ,v9c5t ,sur l'orig9e de nos *iffres1 ,notices et ,extraits des ,,mss41 ,p>is1 #ahdg1 pp4 #adc-aej says3 ,ce c]cle fut nomme p> les uns1 sipos1 rota1 galgal 4442 p> les autres tsiphra 7de ,hebrew3 ,,tspr1 c\ronne ou diademe7 ou ciphra 7de ,hebrew3 ,,spr1 num]ation74 ,*4 de ,p>avey1 ,essai sur l'orig9e unique et hi]oglyphique des *iffres et des lettres de t\s les peuples1 ,p>is1 #ahbf1 p4 #afe1 a r fanciful "w1 gives vase1 vase arrondi et f]me p> un c\v]cle1 qui e/ le symbole de la #aj;e ,heure1 symbol1 among ! ,*9ese2 also ,tsiphron ,z]on1 ou t\t a fait vide en arabe1 ,greek3 tziphra en grec 444 d'ou *iffre 7qui d]ive plutot1 suivant n\s1 de l',hebreu ,seph]1 compt]47 #baf ,compilatus a ,magi/ro ,jacobo de ,flor5tia apud montem pesalanum1 & describ$ by ,g4 ,lami 9 8 ,catalogus codicum manuscriptorum qui 9 biblio!ca ,ricc>diana ,flor5tiae ads]vantur4 ,see ,fazz>i1 loc4 cit41 p4 #e4 #bag ,et doveto sap]e *el zeu]o p] se solo non significa nulla ma e pot5tia di f>e signific>e1 444 ,et dec9a o c5t9aia o migliaia non si puote scriv]e s5za que/o segno #j4 la quale si *iama zeu]o4 ,fazz>i1 loc4 cit41 p4 #e4 #bah ,ibid41 p4 #f4 #bai ,avic5na 7#ihj-ajcf71 translation by ,gasb>ri et ,francois1 piu il punto 7gli ,arabi adop]avano il punto 9 vece dello z]o il cui segno #j 9 arabo si *iama zepiro donde il vocabolo z]o71 *e p] se /esso non esprime nessun num]o4 ,?is quotation is tak5 f ,d4 ,c4 ,m>t9es1 ,orig9e e progressi dell' aritmetica1 ,mess9a1 #ahfe4 #bbj ,leo ,jordan1 ,mat]iali5 zur ,ges*i*te d] arabis*5 ,zahlzei*5 9 ,frankrei*1 ,ar*iv fu] ,kulturges*i*te1 ,b]l91 #aije1 pp4 #aee-aie1 gives ! foll[+ two s*emes ( d]ivation1 7#a7 zefiro1 zeviro1 zeiro1 z]o1 7#b7 zefiro1 zefro1 zevro1 z]o4 #bba ,koebel 7#aeah ed41 f4 ,a#d7 speaks ( ! num]als 9 g5]al z die d] gema9 man ,zyf] n5dt4 ,recorde 7,gr\nde ( ,artes1 #aeeh ed41 f4 ,b#f7 says t ! z]o is call$ priuatly a ,cyph>1 ?\gh all ! o!r sometimes be likewise nam$4 #bbb ,decimo ,x #j !ca1 circulu cifra sive figura nihili appelat'4 ,en*iridion ,algorismi1 ,cologne1 #aeja4 ,lat]1 quoniam de integris tam 9 cifris quam 9 proiectilibus1--! ^w proiectilibus ref]r+ to m>k]s ?r[n & us$ on an abacus1 :5ce ! ,fr5* jetons & ! ,engli% expression to ca/ an acc\nt4 #bbc ,decima v]o o dicitur teca1 circulus1 vel cyfra vel figura ni*ili4 ,maximilian ,curtze1 ,petri ,philom5i de ,dacia 9 ,algorismum ,vulg>em ,johannis de ,sacrobosco comm5t>ius1 una cum ,algorismo ipso1 ,cop5hag51 #ahig1 p4 #b4 ,curtze cites five manuscripts 7f\rte5? & fifte5? c5turies7 ( ,dacia's comm5t>y 9 ! libr>ies at ,erfurt1 ,leipzig1 & ,salzburg1 9 addition to ^? giv5 by ,ene/roem1 ,oefv]sigt af ,kongl4 ,vet5skaps-,akademi5s ,fo]h&l+>1 #ahhe1 pp4 #ae-bg1 #fe-gj2 #ahhf1 pp4 #eg-fj4 #bbd ,curtze1 loc4 cit41 p4 ,,vi4 #bbe ,r>a ,ma!matica1 ,london1 #ahda1 *ap1 i1 ,joannis de ,sacro-,bosco ,tractatus de ,arte ,num]&i4 #bbf ,smi?1 ,r>a ,ari?metica1 ,bo/on1 #aiji4 #bbg ,9 ! #adhd edition1 ,borghi uses ! =m cefiro3 ou]o nulla3 :ile 9 ! #adhh edition he uses zefiro3 ou]o nulla1 & 9 ! #aedj edition1 f4 #c1 appe>s ,*iamata z]o1 ou]o nulla4 ,woepcke ass]t$ t x fir/ appe>$ 9 ,cal&ri 7#adia7 9 ?is s5t5ce3 ,sono dieci le figure con le quali ciascuno num]o si puo signific>e3 delle quali n'e una *e si *iama z]o3 et p] se sola nulla significa4 7f4 #d74 ,see ,propagation1 p4 #ebb4 #bbh ,boncompagni ,bullet9o1 ,vol4 ,,xvi1 pp4 #fgc-fhe4 #bbi ,leo ,jordan1 loc4 cit4 ,9 ! ,catalogue ( ,,mss41 ,bibl4 de l',ars5al1 ,vol4 ,,iii1 pp4 #aed-aef1 ?is "w is ,no4 #bijd 7#ahd ,s4,a4,f471 ,bibl4 ,nat41 & is also call$ ,petit traicte de algorisme4 #bcj ,texada 7#aedf7 says t !re are nueue letros yvn z]o o cifra 7f4 #c74 #bca ,savonne 7#aefc1 #agea ed41 f4 #a73 ,vne ansi =mee 7o7 qui s'appelle nulle1 entre m>*ans z]o1 %[+ ! influ5ce ( ,italian names on ,fr5* m]cantile cu/oms4 ,tr5*ant 7,lyons1 #aeff1 #aegh ed41 p4 #ab7 also says3 ,la d]ni]e qui s'apele nulle1 ou z]o2 b ,*amp5ois1 8 contempor>y1 writ+ 9 ,p>is 9 #aegg 7al?\gh ! "w 0 n publi%$ until #aegh71 uses ciph]1 ! ,italian influ5ce %[+ itself less 9 ?is c5t] ( univ]sity culture ?an 9 ! comm]cial atmosph]e ( ,lyons4 #bcb ,?us ,radulph ( ,laon 7c4 #aajj73 ,inscribitur 9 ultimo ord9e et figura symbol sipos nom9e1 quae1 licet num]um nullum signitet1 tantum ad alia qua$am utilis1 ut insequ5tibus decl>abitur4 ,d] ,ari?metis*e ,tractat des ,radulph von ,laon1 ,abh&lung5 zur ,ges*i*te d] ,ma!matik1 ,vol4 ,v1 p4 #ig1 f a manuscript ( ! ?irte5? c5tury4 ,*asles 7,comptes r5dus1 t4 #af1 #ahdc1 pp4 #acic1 #adjh7 calls att5tion to ! fact t ,radulph did n "k h[ to use ! z]o1 & he d\bts if ! sipos 0 really id5tical ) x4 ,radulph says3 444 figuram1 cui sipos nom5 e/ symbol 9 motum rotulae =matam nullius num]i significatione inscribi sol]e pra$iximus1 & !reaft] uses rotula4 ,he uses ! sipos simply z a k9d ( m>k] on ! abacus4 #bcc ,rabbi b5 ,ezra 7#ajib-aafh7 us$ bo? ,hebrew3 ,,glgl1 galgal 7! ,hebrew = :eel71 & ,hebrew3 ,,spr'1 sifra4 ,see ,m4 ,/e9s*neid]1 ,die ,ma!matik bei d5 ,jud51 9 ,biblio!ca ,ma!matica1 #ahic1 p4 #fi1 & ,silb]b]g1 ,das ,bu* d] ,zahl des ,r4 ,abraham ibn ,esra1 ,frankfurt a4 ,m41 #ahie1 p4 #if1 note #bc2 9 ?is "w ! ,hebrew lett]s are us$ = num]als ) place value1 hav+ ! z]o4 #bcd ,e4g41 9 ! twelf?-c5tury ,lib] aligorismi 7see ,boncompagni's ,trattati1 ,,ii1 p4 #bh74 ,s ,ramus 7,libri ,,ii1 #aefi ed41 p4 #a7 says3 ,circulus quae nota e/ ultima3 nil p] se significat4 7,see also ! ,s*on]us ed4 ( ,ramus1 #aehf1 p4 #a47 #bce ,und wirt das r+le9 o4 die ,ziff] g5ant die ni*ts b$eut4 ,koebel's ,re*5bu*1 #aedi ed41 f4 #aj1 & o!r editions4 #bcf ,i4e4 circul> figure1 our ^w notation hav+ come f ! m$ieval nota4 ,?us ,tzwivel 7#aejg1 f4 #b7 says3 ,nota autem circul>is 4o4 p] se sumpta nihil vsus habet4 alijs tam5 adiuncta e>um significantiam et auget et ord9em p]mutat quantum quo ponit ord9em4 vt adiuncta note b9>ij hoc modo #bj facit eam signific>e bis decem etc4 ,also 7ibid41 f4 #d71 figura circul>is1 circul>is nota4 ,cli*toveus 7#aejc ed41 f4 ,,xxxvii7 calls x nota aut circul>is o1 circul>is nota1 & figura circul>is4 ,ton/all 7#aebb1 f4 ,b#c7 says ( x3 ,decimo u]o nota ad =mam symbol litt]ae circul>i figura e/3 quam alij circulum1 uulgus cyphram uocat1 & lat] 7f4 ,c#d7 speaks ( ! circulos4 ,grammateus1 9 8 ,algorismus de integris 7,erfurt1 #aebc1 f4 ,a#b71 speak+ ( ! n9e significant figures1 rem>ks3 ,8 autem sup]additur decima figura circul>is ut #j exi/5s que ratione sua nihil significat4 ,noviomagus 7,de ,num]is libri ,,ii1 ,p>is1 #aeci1 *ap4 xvi1 ,de notis num]orum1 quas zyphras vocant7 calls x circul>is nota1 quam ex 8 solam1 alij siph]am1 ,georgius ,valla zyphram4 #bcg ,huswirt1 z above4 ,ramus 7,s*olae ma!maticae1 #aefi ed41 p4 #aab7 discusses ! "n int]e/+ly1 say+3 ,circulum appellamus cum multis1 quam alii !cam1 alii figuram nihili1 alii figuram privationis1 seu figuram nullam vocant1 alii ciphram1 cum tam5 hodie omnes hae notae vulgo ciphrae nom95tur1 8 notis num]>e idem sit quod ciphr>e4 ,t>taglia 7#aeib ed41 f4 #i7 says3 si *iama da alcuni tecca1 da alcuni circolo1 da altri cifra1 da altri z]o1 da alcuni altri nulla4 #bch ,qu>e autem aliis nom9ibus vocetur1 non dicit auctor1 quia omnia alia nom9a hab5t rationem suae l9eationis sive figurationis4 ,quia rotunda e/1 dicitur haec figura teca ad similitud9em tecae4 ,teca enim e/ f]rum figurae rotundae1 quod ignitum solet 9 quibusdam regionibus imprimi fronti vel maxillae furis seu latronum4 ,loc4 cit41 p4 #bf4 ,b 9 ,greek !ca 7,,!ke1 ,greek3 !ke7 is a place to put some?+1 a receptacle4 ,if a vacant column1 e4g4 9 ! abacus1 0 s call$1 ! initial might h giv5 ! e>ly =ms symbol & symbol = ! z]o4 #bci ,buteo1 ,logi/ica1 ,lyons1 #aeei4 ,see also ,w]!im 9 ! ,biblio!ca ,ma!matica1 #aija1 p4 #bad4 #bdj #j e/ appellee *iffre ou nulle ou figure de nulle valeur4 ,la ,ro*e1 ,l'ari?metique1 ,lyons1 #aebj4 #bda ,decima autem figura nihil uocata1 figura nihili 7quam etiam cifram uocant74 ,/ifel1 ,ari?metica integra1 #aedd1 f4 #a4 #bdb ,zifra1 ,nulla uel figura ,nihili4 ,s*eubel1 #aede1 p4 #a ( *4 #a4 ,nulla is also us$ by ,italian writ]s4 ,?us ,s=tunati 7#aede ed41 f4 #d7 says3 et la decima nulla e *iamata que/a decima z]o2 ,cataldi 7#afjb1 p4 #a73 ,la prima1 *e e o1 si *iama nulla1 ou]o z]o1 ou]o ni5te4 ,x also f\nd its way into ! ,dut* ari?metics1 e4g4 ,raets 7#aegf1 #aehj ed41 f4 ,a#c73 ,nullo dat i/ niet2 ,van d] ,s*u]e 7#afjj1 #afbd ed41 f4 #g72 ,wilk5s 7#affi ed41 p4 #a74 ,9 ,g]many ,johann ,alb]t 7,witt5b]g1 #aecd7 & ,rudolff 7#aebf7 bo? adopt$ ! ,italian nulla & popul>iz$ x4 7,see also ,kuckuck1 ,die ,re*5kun/ im se*zehnt5 ,jahrhund]t1 ,b]l91 #ahgd1 p4 #g2 ,gu5!r1 ,ges*i*te1 p4 #caf47 #bdc ,la dixieme s'appelle *ifre vulgairem5t3 les vns l'appellant z]o3 n\s la p\rrons appell] vn ,ri54 ,peleti]1 #afjg ed41 p4 #ad4 #bdd ,x appe>s 9 ! ,poli% ari?metic ( ,klos 7#aech7 z cyfra4 ,! ,ciphra #j augm5te? places1 b ( himselfe signifie? n1 ,digges1 #aegi1 p4 #a4 ,hodd] 7#aj? ed41 #afgb1 p4 #b7 uses only ?is ^w 7cyph] or ciph]71 & ! same is true ( ! fir/ native ,am]ican ari?metic1 writt5 by ,isaac ,gre5wood 7#agbi1 p4 #a74 ,petrus de ,dacia d]ives cyfra f circumf]5ce4 ,vocatur etiam cyfra1 quasi circumfacta vel circumf]5da1 quod idem e/1 quod circulus non habito respectu ad c5trum4 ,loc4 cit41 p4 #bf4 #bde ,op]a ma!matica1 #afie1 ,ox=d1 ,vol4 ,i1 *ap4 ix1 ,ma!sis univ]salis1 ,de figuris num]alibus1 pp4 #df-di2 ,vol4 ,,ii1 ,algebra1 p4 #aj4 #bdf ,m>t91 ,orig9e de notre sy/eme de num]ation ecrite1 note #adi1 p4 #cf ( repr9t1 spells ,greek3 tsiphra f ,maximus ,planudes1 cit+ ,wallis z an au?ority4 ,?is is an error1 = ,wallis gives ! correct =m z above4 ,alex&] von ,humboldt1 ,ueb] die bei v]s*i$55 ,voelk]n uebli*5 ,sy/eme von ,zahlzei*5 und ueb] d5 ,ursprung des ,/ell5w]!s 9 d5 indis*5 ,zahl51 ,crelle's ,j\rnal fu] re9e und angew&te ,ma!matik1 ,vol4 ,,iv1 #ahbi1 call$ att5tion to ! "w ,greek3 ari?moi ,indikoi ( ! monk ,neophytos1 suppos$ to be ( ! f\rte5? c5tury4 ,9 ?is "w ! =ms ,greek3 tzuphra & ,greek3 tzumphra appe>4 ,see also ,boeckh1 ,de abaco ,graecorum1 ,b]l91 #ahda1 & ,tann]y1 ,le ,s*olie du mo9e ,neophytos1 ,revue ,ar*eologique1 #ahhe1 pp4 #ii-ajb4 ,jordan1 loc4 cit41 gives f twelf? & ?irte5? c5tury manuscripts ! =ms cifra1 ciffre1 *ifras1 & cifrus4 ,du ,cange1 ,gloss>ium m$iae et infimae ,lat9itatis1 ,p>is1 #ahdb1 gives also *il]ae4 ,dasypodius1 ,in/itutiones ,ma!maticae1 ,/rassburg1 #aeic-aeif1 adds ! =ms zyphra & syphra4 ,boissi]e1 ,l'art d'ary?metique cont5ant t\te dim5tion1 tres-s+uli] et commode1 tant p\r l'art militaire que autres calculations1 ,p>is1 #aeed3 ,puis y en a vn autre dict z]o lequel ne designe nulle quantite p> soy1 a9s seulem5t les loges vuides4 #bdg ,propagation1 pp4 #bg1 #bcd1 #ddb4 ,treutle91 ,das ,re*n5 im #af4 ,jahrhund]t1 ,abh&lung5 zur ,ges*i*te d] ,ma!matik1 ,vol4 ,i1 p4 #e1 favors ! same view4 ,x is combat$ by _m writ]s1 e4g4 ,a4 ,c4 ,burnell1 loc4 cit41 p4 #ei4 ,long be=e ,woepcke1 ,i4 ,f4 & ,g4 ,i4 ,weidl]1 ,de *>act]ibus num]orum vulg>ibus et eorum aetatibus1 ,witt5b]g1 #agbg1 ass]t$ ! possibility ( _! introduction into ,greece by ,py?agoras or "o ( 8 foll[]s3 ,potu]unt autem ex ori5te1 uel ex pho5icia1 ad graecos traduci1 uel ,py?agorae1 uel eius discipulorum auxilio1 cum aliquis eo1 pr(ici5di 9 lit]is causa1 it] fac]et1 et hoc quoque inu5tum addisc]et4 #bdh ,e4g41 !y adopt$ ! ,greek num]als 9 use 9 ,damascus & ,syria1 & ! ,coptic 9 ,egypt4 ,!ophanes 7#geh-hah ,a4,d471 ,*ronographia1 ,scriptores ,hi/oriae ,byzant9ae1 ,vol4 ,,xxxix1 ,bonnae1 #ahci1 p4 #ege1 relates t 9 #fii ,a4,d4 ! caliph ,walid =bade ! use ( ! ,greek language 9 ! bookkeep+ ( ! treasury ( ! caliphate1 b p]mitt$ ! use ( ! ,greek alphabetic num]als1 s9ce ! ,arabs _h no conv5i5t numb] notation3 ,greek3 kai ekoluse graphe/hai ,hell5i/i t\s demosi\s ton logo!sion kodikas1 all' ,arabiois auta p>asema9e/hai1 *oris ton psephon1 epeide adunaton tei eke9on glossei monada e duada e triada e okto hemisu e tria graphe/hai2 dio kai heos sem]on eis9 sun autois not>ioi ,*ri/ianoi4 ,! importance ( ?is contemporane\s docum5t 0 po9t$ out by ,m>t91 loc4 cit4 ,k>abacek1 ,die ,involutio im arabis*5 ,s*riftwes51 ,vol4 ,,cxxxv ( ,sitzungsb]i*te d4 phil4-hi/4 ,classe d4 k4 ,akad4 d4 ,wiss41 ,vi5na1 #ahif1 p4 #be1 gives an ,arabic date ( #hfh ,a4,d4 9 ,greek lett]s4 #bdi ,! ,orig9 & ,hi/ory ( ,our ,num]als 79 ,russian71 ,kiev1 #aijh2 ,! ,indep5d5ce ( ,european ,ari?metic 79 ,russian71 ,kiev4 #bej ,woepcke1 loc4 cit41 pp4 #dfb1 #bfb4 #bea ,woepcke1 loc4 cit41 p4 #bdj4 ,h4isab-al-4,gobar1 by an anonym\s au?or1 probably ,abu ,sahl ,duna% ibn ,tamim1 is giv5 by ,/e9s*neid]1 ,die ,ma!matik bei d5 ,jud51 ,biblio!ca ,ma!matica1 #ahif1 p4 #bf4 #beb ,/e9s*neid] 9 ! ,abh&lung51 ,vol4 ,,iii1 p4 #aaj4 #bec ,see 8 ,grammaire arabe1 ,vol4 ,i1 ,p>is1 #ahaj1 plate ,,viii2 ,g]h>dt1 ,etudes1 pp4 #i-aa1 & ,ent/ehung etc41 p4 #h2 ,i4 ,f4 ,weidl]1 ,spicilegium obs]vationum ad hi/oriam not>um num]alium p]t95tium1 ,witt5b]g1 #agee1 speaks ( ! figura cifr>um ,s>ac5ic>um z be+ diff]5t f t ( ! *>act]um ,boe?ianorum1 :i* are simil> to ! vulg> or common num]als2 see also ,humboldt1 loc4 cit4 #bed ,g]h>dt m5tions x 9 8 ,ent/ehung etc41 p4 #h2 ,woepcke1 ,propagation1 /ates t ^! num]als 7 us$ n = calculation1 b v mu* z we use ,roman num]als4 ,^! sup]pos$ dots are f\nd ) bo? =ms ( num]als 7,propagation1 pp4 #bdd-bdf74 #bee ,g]h>dt 7,etudes1 p4 #i7 f a manuscript 9 ! ,biblio!que ,nationale4 ,! num]al =ms are symbols1 #bj be+ indicat$ by symbol ) dot & #bjj by symbol ) #b dots4 ,?is s*eme ( z]o dots 0 also adopt$ by ! ,byzant9e ,greeks1 = a manuscript ( ,planudes 9 ! ,biblio!que ,nationale has numb]s l pi alpha ) #d dots = #h1ajj1jjj1jjj4 ,see ,g]h>dt1 ,etudes1 p4 #ai4 ,pihan1 ,expose etc41 p4 #bjh1 gives two =ms1 ,asiatic & ,maghrebian1 ( ,ghobar num]als4 #bef ,see ,*ap4 ,,iv4 #beg ,possibly z e>ly z ! ?ird c5tury ,a4,d41 b probably ( ! eigh? or n9?4 ,see ,cantor1 ,i 7#c71 p4 #eih4 #beh ,ascrib$ by ! ,arabic writ] to ,india4 #bei ,see ,woepcke's description ( a manuscript 9 ! ,*asles libr>y1 ,re*]*es sur l'hi/oire des sci5ces ma!matiques *ez les ori5taux1 ,j\rnal ,asiatique1 ,,iv 7#e71 #ahei1 p4 #ceh1 note4 #bfj ,p4 #ef4 #bfa ,re9aud1 ,memoire sur l',inde1 p4 #cii4 ,9 ! f\rte5? c5tury "o ,sihab al-,din wrote a "w on :i*1 a s*olia/ to ! ,bodleian manuscript rem>ks3 ,! sci5ce is call$ ,algob> because ! inv5tor _h ! habit ( writ+ ! figures on a tablet cov]$ ) s&4 ,g]h>dt1 ,etudes1 p4 #aa1 note4 #bfb ,g]h>dt1 ,ent/ehung etc41 p4 #bj4 #bfc ,h4 ,sut]1 ,das ,re*5bu* des ,abu ,zak>ija el-,h4as4s4ar1 ,biblio!ca ,ma!matica1 ,vol4 ,,ii 7#c71 p4 #ae4 #bfd ,a4 ,dev\lx1 ,les *iffres arabes1 ,revue ,africa9e1 ,vol4 ,,xvi1 pp4 #dee-deh4 #bfe ,kitab al-,fihri/1 ,g4 ,fluegel1 ,leipzig1 ,vol4 ,i1 #ahga1 & ,vol4 ,,ii1 #ahgb4 ,?is "w 0 publi%$ aft] ,pr(essor ,fluegel's dea? by ,j4 ,ro$ig] & ,a4 ,muell]4 ,! fir/ volume conta9s ! ,arabic text & ! second volume conta9s critical notes ^u x4 #bff ,l ^? ( l9e #e 9 ! illu/ration on page #fi4 #bfg ,woepcke1 ,re*]*es sur l'hi/oire des sci5ces ma!matiques *ez les ori5taux1 loc4 cit42 ,propagation1 p4 #eg4 #bfh ,al-,h4as4s4ar's =ms1 ,sut]1 ,biblio!ca ,ma!matica1 ,vol4 ,,ii 7#c71 p4 #ae4 #bfi ,woepcke1 ,sur une donnee hi/orique1 etc41 loc4 cit4 ,! "n 4gobar is n us$ 9 ! text4 ,! manuscript f :i* ^! are tak5 is ! olde/ 7#igj ,a4,d47 ,arabic docum5t kn[n to conta9 all ( ! num]als4 #bgj ,silve/re de ,sacy1 loc4 cit4 ,he gives ! ord9>y mod]n ,arabic =ms1 call+ !m ,indi54 #bga ,woepcke1 ,introduction au calcul ,gobari et ,hawai1 ,atti dell' accademia pontificia dei nuovi ,l9cei1 ,vol4 ,,xix4 ,! adjective appli$ to ! =ms 9 #e is gobari & to ^? 9 #f indi5ne4 ,?is is ! direct opposite ( ,woepcke's use ( ^! adjectives 9 ! ,re*]*es sur l'hi/oire cit$ above1 9 :i* ! ord9>y ,arabic =ms 7l ^? 9 r[ #e7 are call$ indi5s4 ,^! =ms are usually writt5 f "r to left4 #bgb ,j4 ,g4 ,wilk9son1 ,! ,mann]s & ,cu/oms ( ! ,anci5t ,egyptians1 revis$ by ,s4 ,bir*1 ,london1 #ahgh1 ,vol4 ,,ii1 p4 #dic1 plate ,,xvi4 #bgc ,!re is an ext5sive lit]ature on ?is ,boe?ius-,frage4 ,! read] :o c>es to g fully into x %\ld consult ! v>i\s volumes ( ! ,jahrbu* ueb] die ,=ts*ritte d] ,ma!matik4 #bgd ,?is title 0 fir/ appli$ to ,roman emp]ors 9 po/hum\s co9s ( ,julius ,caes>4 ,subsequ5tly ! emp]ors assum$ x dur+ _! own lifetimes1 ?us deify+ !mselves4 ,see ,f4 ,gnec*i1 ,monete romane1 #b;d ed41 ,milan1 #aijj1 p4 #bii4 #bge ,?is is ! common spell+ ( ! "n1 al?\gh ! m correct ,lat9 =m is ,boetius4 ,see ,h>p]'s ,dict4 ( ,class4 ,lit4 & ,antiq41 ,new ,york1 #ahig1 ,vol4 ,i1 p4 #bac4 ,!re is mu* unc]ta9ty z to 8 life4 ,a good summ>y ( ! evid5ce is giv5 9 ! la/ two editions ( ! ,encyclopa$ia ,britannica4 #bgf ,8 fa!r1 ,flavius ,manlius ,boe?ius1 0 consul 9 #dhg4 #bgg ,!re is1 h[ev]1 no good hi/oric evid5ce ( ?is soj\rn 9 ,a!ns4 #bgh ,8 ari?metic is d$icat$ to ,symma*us3 ,dom9o suo patricio ,symma*o ,boetius4 ,fri$le9 ed41 p4 #c4 #bgi ,x 0 :ile "h t he wrote ,de consolatione philosophiae4 #bhj ,x is sometimes giv5 z #ebe4 #bha ,!re 0 a m$ieval tradition t he 0 execut$ because ( a "w on ! ,tr9ity4 #bhb ,h5ce ! ,divus 9 8 "n4 #bhc ,?us ,dante1 speak+ ( 8 burial place 9 ! mona/]y ( ,/4 ,pietro 9 ,ciel d',oro1 at ,pavia1 says3 ,! sa9tly s\l1 t %[s ,! _w's deceitfulness1 to all :o he> him1 ,is1 ) ! sight ( all ! good t is1 ,ble/ !re4 ,! limbs1 :5ce x 0 driv51 lie ,d[n 9 ,cieldauro2 & f m>tyrdom ,& exile came x "h4--,p>adiso1 ,canto ,x4 #bhd ,n1 h[ev]1 9 ! m]cantile s*ools4 ,! ari?metic ( ,boe?ius w\ld h be5 ab\t ! la/ book to be ?\ght ( 9 su* in/itutions4 ,:ile ref]r$ to by ,ba$a 7#fgb-gce7 & ,hrabanus ,maurus 7c4 #ggf-hef71 x 0 only aft] ,g]b]t's "t t ! ,boetii de in/itutione ari?metica libri duo 0 really a common "w4 #bhe ,also spell$ ,cassiodorius4 #bhf ,z a matt] ( fact1 ,boe?ius c\ld n h translat$ any "w by ,py?agoras on music1 because !re 0 no su* "w1 b he did make ! !ories ( ! ,py?agoreans kn[n4 ,nei!r did he translate ,nicoma*us1 al?\gh he embodi$ _m ( ! ideas ( ! ,greek writ] 9 8 own ari?metic4 ,gibbon foll[s ,cassiodorus 9 ^! /atem5ts 9 8 ,decl9e & ,fall ( ! ,roman ,empire1 *ap4 xxxix4 ,m>t9 po9t$ out ) positiv5ess ! simil>ity ( ! fir/ book ( ,boe?ius to ! fir/ five books ( ,nicoma*us4 ,les signes num]aux etc41 repr9t1 p4 #d4 #bhg ,! g5]al idea goes back to ,py?agoras1 h[ev]4 #bhh ,j4 ,c4 ,scalig] 9 8 ,poetice also said ( him3 ,boe?ii ,sev]9i +5ium1 eruditio1 ars1 sapi5tia facile provocat omnes auctores1 sive illi ,graeci s9t1 sive ,lat9i ,heilbronn]1 ,hi/4 ma?4 univ41 p4 #chg4 ,libri1 speak+ ( ! "t ( ,boe?ius1 rem>ks3 ,n\s voyons du temps de ,!odoric1 les lettres repr5dre une n\velle vie en ,italie1 les ecoles florissantes et les savans honores4 ,et c]tes les ouvrages de ,boece1 de ,cassiodore1 de ,symmaque1 surpass5t de beauc\p t\tes les productions du siecle prec$5t4 ,hi/oire des ma!matiques1 ,vol4 ,i1 p4 #gh4 #bhi ,c>ra de ,vaux1 ,avic5ne1 ,p>is1 #aijj2 ,woepcke1 ,sur l'introduction1 etc42 ,g]h>dt1 ,ent/ehung etc41 p4 #bj4 ,avic5na is a corruption f ,ibn ,sina1 z po9t$ out by ,wue/5feld1 ,ges*i*te d] arabis*5 ,a]zte und ,natur=s*]1 ,goett+51 #ahdj4 ,8 full "n is ,abu ,ali al-,h4ose9 ibn ,sina4 ,= notes on ,avic5na's ari?metic1 see ,woepcke1 ,propagation1 p4 #ejb4 #bij ,on ! e>ly travel betwe5 ! ,ea/ & ! ,we/ ! foll[+ works may be consult$3 ,a4 ,hillebr&t1 ,alt-,indi51 conta9+ ,*9esis*e ,reis5de 9 ,indi51 ,breslau1 #ahii1 p4 #agi2 ,c4 ,a4 ,skeel1 ,travel 9 ! ,fir/ ,c5tury aft] ,*ri/1 ,cambridge1 #aija1 p4 #adb2 ,m4 ,re9aud1 ,relations politiques et comm]ciales de l'empire roma9 avec l',asie ori5tale1 9 ! ,j\rnal ,asiatique1 ,m>s-,avril1 #ahfc1 ,vol4 ,i 7#f71 p4 #ic2 ,beazley1 ,dawn ( ,mod]n ,geography1 a ,hi/ory ( ,exploration & ,geographical ,sci5ce f ! ,conv]sion ( ! ,roman ,empire to ,a4,d4 #adbj1 ,london1 #ahig-aijf1 #c vols42 ,heyd1 ,ges*i*te des ,levan?&els im ,mittelalt]1 ,/uttg>t1 #ahig2 ,j4 ,keane1 ,! ,evolution ( ,geography1 ,london1 #ahii1 p4 #ch2 ,a4 ,cunn+ham1 ,corpus inscriptionum ,indic>um1 ,calcutta1 #ahgg1 ,vol4 ,i2 ,a4 ,ne&]1 ,g5]al ,hi/ory ( ! ,*ri/ian ,religion & ,*ur*1 #e? ,am]ican ed41 ,bo/on1 #ahee1 ,vol4 ,,iii1 p4 #hi2 ,r4 ,c4 ,dutt1 ,a ,hi/ory ( ,civilization 9 ,anci5t ,india1 ,vol4 ,,ii1 ,bk4 ,v1 *ap1 ii2 ,e4 ,c4 ,bayley1 loc4 cit41 p4 #bh et seq42 ,a4 ,c4 ,burnell1 loc4 cit41 p4 #c2 ,j4 ,e4 ,t5n5t1 ,ceylon1 ,london1 #ahei1 ,vol4 ,i1 p4 #aei2 ,geo4 ,turn\r1 ,epitome ( ! ,hi/ory ( ,ceylon1 ,london1 n4d41 preface2 ,philale!s1 ,hi/ory ( ,ceylon1 ,london1 #ahaf1 *ap1 i2 ,h4 ,c4 ,sirr1 ,ceylon & ! ,c+alese1 ,london1 #ahej1 ,vol4 ,i1 *ap4 ix4 ,on ! ,h9du k ( ! ,nile see ,f4 ,wil=d1 ,asiatick ,rese>*es1 ,vol4 ,,iii1 p4 #bie1 ,calcutta1 #agib4 #bia ,g4 ,opp]t1 ,on ! ,anci5t ,comm]ce ( ,india1 ,madras1 #ahgi1 p4 #h4 #bib ,g]h>dt1 ,etudes etc41 pp4 #h1 #aa4 #bic ,see ,smi?'s ,diction>y ( ,greek & ,roman ,biography & ,my?ology4 #bid ,p4 ,m4 ,sykes1 ,t5 ,?\s& ,miles 9 ,p]sia1 or ,eight ,ye>s 9 ,iran1 ,london1 #aijb1 p4 #afg4 ,sykes 0 ! fir/ ,european to foll[ ! c\rse ( ,alex&]'s army across ea/]n ,p]sia4 #bie ,buehl]1 ,indian ,brahma ,alphabet1 note1 p4 #bg2 ,palaeographie1 p4 #b2 ,h]odoti ,halic>nassei hi/oria1 ,am/]dam1 #agfc1 ,bk4 ,,iv1 p4 #cjj2 ,isaac ,vossius1 ,p]iplus ,scylacis ,c>y&5sis1 #afci4 ,x is d\btful :e!r ! "w attribut$ to ,scylax 0 writt5 by him1 b 9 any case ! "w dates back to ! f\r? c5tury ,b4,c4 ,see ,smi?'s ,diction>y ( ,greek & ,roman ,biography4 #bif ,h]odotus1 ,bk4 ,,iii4 #big ,rameses ,,ii7871 ! ,sesoosis ( ,diodorus ,siculus4 #bih ,indian ,antiqu>y1 ,vol4 ,i1 p4 #bbi2 ,f4 ,b4 ,jevons1 ,manual ( ,greek ,antiquities1 ,london1 #ahie1 p4 #chf4 ,on ! relations1 political & comm]cial1 betwe5 ,india & ,egypt c4 #gb ,b4,c41 "u ,ptolemy ,auletes1 see ! ,j\rnal ,asiatique1 #ahfc1 p4 #big4 #bii ,sik&>1 z ! "n /ill rema9s 9 nor!rn ,india4 #cjj ,h>p]'s ,classical ,dict41 ,new ,york1 #ahig1 ,vol4 ,i1 p4 #gbd2 ,f4 ,b4 ,jevons1 loc4 cit41 p4 #chi2 ,j4 ,c4 ,m>%man1 ,abridgm5t ( ! ,hi/ory ( ,india1 *aps4 i & ii4 #cja ,opp]t1 loc4 cit41 p4 #aa4 ,x 0 at or ne> ?is place t ! fir/ great ,indian ma!matician1 ,aryabhat4a1 0 born 9 #dgf ,a4,d4 #cjb ,buehl]1 ,palaeographie1 p4 #b1 speaks ( ,greek co9s ( a p]iod ant]ior to ,alex&]1 f\nd 9 nor!rn ,india4 ,m complete in=mation may be f\nd 9 ,indian ,co9s1 by ,e4 ,j4 ,rapson1 ,/rassburg1 #ahih1 pp4 #c-g4 #cjc ,opp]t1 loc4 cit41 p4 #ad2 & to him is due o!r simil> in=mation4 #cjd ,j4 ,belo*1 ,grie*is*e ,ges*i*te1 ,vol4 ,,iii1 ,/rassburg1 #aijd1 pp4 #cj-ca4 #cje ,e4g41 ! d5>ius1 ! words = h\r & m9ute 7,greek3 hora1 lepton71 & possibly ! signs ( ! zodiac4 ,r4 ,caldwell1 ,comp>ative ,gramm> ( ! ,dravidian ,languages1 ,london1 #ahef1 p4 #dch4 ,on ! probable ,*9ese orig9 ( ! zodiac see ,s*legel1 loc4 cit4 #cjf ,m>ie1 ,vol4 ,,ii1 p4 #gc2 ,r4 ,caldwell1 loc4 cit4 #cjg ,a4 ,cunn+ham1 loc4 cit41 p4 #ej4 #cjh ,c4 ,a4 ,j4 ,skeel1 ,travel1 loc4 cit41 p4 #ad4 #cji ,in*iv]1 f in*i1 ! gre5 root4 ,indian ,antiqu>y1 ,vol4 ,i1 p4 #ceb4 #caj ,9 ,*9a dat+ only f ! second c5tury ,a4,d41 h[ev]4 #caa ,! ,italian morra4 #cab ,j4 ,b[r+1 ,! ,decimal ,sy/em1 ,london1 #ahed1 p4 #b4 #cac ,h4 ,a4 ,giles1 lecture at ,columbia ,univ]sity1 ,m>* #ab1 #aijb1 on ,*9a & ,anci5t ,greece4 #cad ,giles1 loc4 cit4 #cae ,e4g41 ! names = grape1 radi% 7la-po1 ,greek3 rhaphe71 wat]-lily 7si-kua1 we/ g\rds2 ,greek3 sikua1 g\rds71 are mu* alike4 ,giles1 loc4 cit4 #caf ,epi/les1 ,i1 #a1 #de-df4 ,on ! ,roman trade r\tes1 see ,beazley1 loc4 cit41 ,vol4 ,i1 p4 #agi4 #cag ,am4 ,j\rn4 ( ,ar*eol41 ,vol4 ,,iv1 p4 #cff4 #cah ,m4 ,p]rot gives ?is conjectural re/oration ( 8 words3 ,ad me ex ,india regum legationes saepe missi sunt numquam antea visae apud quemquam pr9cipem ,romanorum4 ,m4 ,re9aud1 ,relations politiques et comm]ciales de l'empire roma9 avec l',asie ori5tale1 ,j\rn4 ,asiat41 ,vol4 ,i 7#f71 p4 #ic4 #cai ,re9aud1 loc4 cit41 p4 #ahi4 ,florus1 ,,ii1 #cd 7,,iv1 #ab71 ref]s to x3 ,s]es etiam habitantesque sub ipso sole ,indi1 cum gemmis et m>g>itis elephantes quoque int] mun]a trah5tes nihil magis quam long9quitatem viae imputabant4 ,horace %[s 8 geographical k by say+3 ,n ^? :o dr9k ( ! deep ,danube %all n[ break ! ,julian edicts2 n ! ,getae1 n ! ,s]es1 nor ! p]fidi\s ,p]sians1 nor ^? born on ! riv] ,tanais4 ,odes1 ,bk4 ,,iv1 ,ode #ae1 #ba-bd4 #cbj ,qua virtutis mod]ationisque fama ,indos etiam ac ,scy?as auditu modo cognitos pellexit ad amicitiam suam populique ,romani ultro p] legatos pet5dam4 ,re9aud1 loc4 cit41 p4 #ahj4 #cba ,re9aud1 loc4 cit41 p4 #ahj4 #cbb ,georgics1 ,,ii1 #agj-agb4 ,s ,prop]tius 7,elegies1 ,,iii1 #d73 ,arma deus ,caes> dites m$itatur ad ,indos ,et freta gemmif]i f9d]e classe m>is4 ,! div9e ,caes> m$itat$ c>ry+ arms aga9/ opul5t ,india1 & ) 8 %ips to cut ! gem-be>+ seas4 #cbc ,heyd1 loc4 cit41 ,vol4 ,i1 p4 #d4 #cbd ,re9aud1 loc4 cit41 p4 #cic4 #cbe ,! title page ( ,cal&ri 7#adia71 = example1 repres5ts ,py?agoras ) ^! num]als be=e him4 ,smi?1 ,r>a ,ari?metica1 p4 #df4 ,isaacus ,vossius1 ,obs]vationes ad ,pomponium ,melam de situ orbis1 #afeh1 ma9ta9$ t ! ,arabs d]iv$ ^! num]als f ! we/4 ,a le>n$ diss]tation to ?is effect1 b d]iv+ !m f ! ,romans in/ead ( ! ,greeks1 0 writt5 by ,g9anni 9 #agec 7,diss]tatio ma!matica critica de num]alium not>um m9uscul>um orig9e1 ,v5ice1 #agec74 ,see also ,mann]t1 ,de num]orum quos arabicos vocant v]a orig9e ,py?agorica1 ,nu]nb]g1 #ahja4 ,ev5 z late z #ahbg ,romagnosi 79 8 supplem5t to ,ric]*e /ori*e sull' ,india etc41 by ,rob]tson1 ,vol4 ,,ii1 p4 #ehj1 #ahbg7 ass]t$ t ,py?agoras orig9at$ !m4 ,r4 ,bombelli1 ,l'antica num]azione italica1 ,rome1 #ahgf1 p4 #ei4 ,g[ 7,hi/4 ( ,greek ,ma?41 p4 #ih7 ?9ks t ,iambli*us mu/ h kn[n a simil> sy/em 9 ord] to h work$ out c]ta9 ( 8 !orems1 b ?is is an unw>rant$ d$uction f ! passage giv54 #cbf ,a4 ,hillebr&t1 ,alt-,indi51 p4 #agi4 #cbg ,j4 ,c4 ,m>%man1 loc4 cit41 *aps4 i & ii4 #cbh ,he reign$ #fca-egi ,a4,d42 call$ ,nu'sirwan1 ! holy "o4 #cbi ,j4 ,keane1 ,! ,evolution ( ,geography1 ,london1 #ahii1 p4 #ch4 #ccj ,! ,arabs :o liv$ 9 & ab\t ,mecca4 #cca ,s4 ,guy>d1 9 ,encyc4 ,brit41 #i? ed41 ,vol4 ,,xvi1 p4 #eig4 #ccb ,opp]t1 loc4 cit41 p4 #bi4 #ccc ,at non cr$5dum e/ id 9 ,autographis contigisse1 aut vetu/ioribus ,codd4 ,,mss4 ,wallis1 ,op]a omnia1 ,vol4 ,,ii1 p4 #aa4 #ccd ,9 ,obs]vationes ad ,pomponium ,melam de situ orbis4 ,! "q 0 next tak5 up 9 a l>ge way by ,weidl]1 loc4 cit41 ,de *>act]ibus etc41 #agbg1 & 9 ,spicilegium etc41 #agee4 #cce ,! be/ edition ( ^! works is t ( ,g4 ,fri$le91 ,anicii ,manlii ,torquati ,sev]9i ,boetii de in/itutione ari?metica libri duo1 de in/itutione musica libri qu9que4 ,acc$it geometria quae f]tur ,boetii4444 ,leipzig4444 ,,mdccclxvii4 #ccf ,see also ,p4 ,tann]y1 ,notes sur la pseudo-geometrie de ,boece1 9 ,biblio!ca ,ma!matica1 ,vol4 ,i 7#c71 p4 #ci4 ,?is is n ! geometry 9 two books 9 :i* are m5tion$ ! num]als4 ,!re is a manuscript ( ?is pseudo-geometry ( ! n9? c5tury1 b ! e>lie/ "o ( ! o!r "w is ( ! elev5? c5tury 7,tann]y71 unless ! ,vatican codex is ( ! t5? c5tury z ,fri$le9 7p4 #cgb7 ass]ts4 #ccg ,fri$le9 feels t x is p>tly spuri\s1 b he says3 ,eorum librorum1 quos ,boetius de geometria scripsisse dicitur1 inve/ig>e v]am inscriptionem nihil aliud esset nisi op]am et tempus p]d]e4 ,preface1 p4 v4 ,n4 ,bubnov 9 ! ,russian ,j\rnal ( ! ,m9i/ry ( ,public ,in/ruction1 #aijg1 9 an article ( :i* a synopsis is giv5 9 ! ,jahrbu* ueb] die ,=ts*ritte d] ,ma!matik = #aijg1 ass]ts t ! geometry 0 writt5 9 ! elev5? c5tury4 #cch ,! mo/ notewor?y ( ^! 0 = a long "t ,cantor 7,ges*i*te1 ,vol4 ,i41 #c;d ed41 pp4 #ehg-ehh71 :o 9 8 e>li] days ev5 believ$ t ,py?agoras _h kn[n !m4 ,cantor says 7,die roemis*5 ,agrim5sor51 ,leipzig1 #ahge1 p4 #acj73 ,uns also1 wir wi$]hol5 es1 i/ die ,geometrie des ,boetius e*t1 dieselbe ,s*rift1 wel*e er na* ,euklid be>beitete1 von wel*] e9 ,codex b]eits 9 ,jahre #hba im ,klo/] ,rei*5au vorh&5 w>1 von wel*] e9 &]es ,exempl> im ,jahre #ihb zu ,mantua 9 die ,ha5de ,g]b]t's gelangte1 von wel*] mannigfa*e ,h&s*rift5 no* heute vorh&5 s9d4 ,b aga9/ ?is op9ion ( ! antiquity ( ,,mss4 conta9+ ^! num]als is ! important /atem5t ( ,p4 ,tann]y1 p]haps ! mo/ critical ( mod]n hi/orians ( ma!matics1 t none exi/s e>li] ?an ! elev5? c5tury4 ,see also ,j4 ,l4 ,heib]g 9 ,philologus1 ,zeits*rift f4 d4 klass4 ,alt]tum1 ,vol4 ,,xliii1 p4 #ejh4 ,( ,cantor's pr$ecessors1 ,?4 ,h4 ,m>t9 0 "o ( ! mo/ prom95t1 8 argum5t = au!nticity appe>+ 9 ! ,revue ,ar*eologique = #ahef-aheg1 & 9 8 treatise ,les signes num]aux etc4 ,see also ,m4 ,*asles1 ,de la connaissance qu'ont eu les anci5s d'une num]ation decimale ecrite qui fait usage de neuf *iffres pr5ant les valeurs de position1 ,comptes r5dus1 ,vol4 ,,vi1 pp4 #fgh-fhj2 ,sur l'orig9e de notre sy/eme de num]ation1 ,comptes r5dus1 ,vol4 ,,viii1 pp4 #gb-ha2 & note ,sur le passage du premi] livre de la geometrie de ,boece1 relatif a un n\veau sy/eme de num]ation1 9 8 "w ,ap]cu hi/orique sur l'orig9e et le developpem5t des me?odes en geometrie1 ( :i* ! fir/ edition appe>$ 9 #ahcg4 #cci ,j4 ,l4 ,heib]g places ! book 9 ! elev5? c5tury on philological gr\nds1 ,philologus1 loc4 cit42 ,woepcke1 9 ,propagation1 p4 #dd2 ,blume1 ,la*mann1 & ,rudorff1 ,die ,s*rift5 d] roemis*5 ,feldmess]1 ,b]l91 #ahdh2 ,boeckh1 ,de abaco graecorum1 ,b]l91 #ahda2 ,fri$le91 9 8 ,leipzig edition ( #ahfg2 ,weiss5born1 ,abh&lung51 ,vol4 ,,ii1 p4 #ahe1 8 ,g]b]t1 pp4 #a1 #bdg1 & 8 ,ges*i*te d] ,e9fuehrung d] jetzig5 ,ziff]n 9 ,europa dur* ,g]b]t1 ,b]l91 #ahib1 p4 #aa2 ,bayley1 loc4 cit41 p4 #ei2 ,g]h>dt1 ,etudes1 p4 #ag1 ,ent/ehung und ,ausbreitung1 p4 #ad2 ,nagl1 ,g]b]t1 p4 #eg2 ,bubnov1 loc4 cit4 ,see also ! discussion by ,*asles1 ,halliwell1 & ,libri1 9 ! ,comptes r5dus1 #ahci1 ,vol4 ,,ix1 p4 #ddg1 & 9 ,vols4 ,,viii1 ,,xvi1 ,,xvii ( ! same j\rnal4 #cdj ,j4 ,m>qu>dt1 ,la vie privee des ,roma9s1 ,vol4 ,,ii 7,fr5* trans471 p4 #eje1 ,p>is1 #ahic4 #cda ,9 a ,plimpton manuscript ( ! ari?metic ( ,boe?ius ( ! ?irte5? c5tury1 = example1 ! ,roman num]als are all replac$ by ! ,arabic1 & ! same is true 9 ! fir/ pr9t$ edition ( ! book4 7,see ,smi?'s ,r>a ,ari?metica1 pp4 #dcd1 #be-bg47 ,d4 ,e4 ,smi? also copi$ f a manuscript ( ! ari?metic 9 ! ,laur5tian libr>y at ,flor5ce1 ( #acgj1 ! foll[+ =ms1 ,=g$ num]als #cdb ,halliwell1 9 8 ,r>a ,ma!matica1 p4 #ajg1 /ates t ! disput$ passage is n 9 a manuscript belong+ to ,mr4 ,ames1 nor 9 "o at ,tr9ity ,college4 ,see also ,woepcke1 9 ,propagation1 pp4 #cg & #db4 ,x 0 ! evid5t corruption ( ! texts 9 su* editions ( ,boe?ius z ^? ( ,v5ice1 #adii1 ,basel1 #aedf & #aegj1 t l$ ,woepcke to publi% 8 "w ,sur l'introduction de l'ari?metique indi5ne en ,occid5t4 #cdc ,!y are f\nd 9 none ( ! v anci5t manuscripts1 z1 = example1 9 ! n9?-c5tury 787 codex 9 ! ,laur5tian libr>y :i* "o ( ! au?ors has exam9$4 ,x %\ld be said1 h[ev]1 t ! disput$ passage 0 writt5 aft] ! ari?metic1 = x conta9s a ref]5ce to t "w4 ,see ! ,fri$le9 ed41 p4 #cig4 #cdd ,smi?1 ,r>a ,ari?metica1 p4 #ff4 #cde ,j4 ,l4 ,heib]g1 ,philologus1 ,vol4 ,,xliii1 p4 #ejg4 #cdf ,nosse autem huius artis dispici5tem1 quid s9t digiti1 quid articuli1 quid compositi1 quid incompositi num]i4 ,fri$le9 ed41 p4 #cie4 #cdg ,de ratione abaci4 ,9 ?is he describes qu&am =mulam1 quam ob honorem sui praeceptoris m5sam ,py?agoream nom9abant 444 a po/]ioribus appellabatur abacus4 ,?is1 z pictur$ 9 ! text1 is ! common ,g]b]t abacus4 ,9 ! edition 9 ,migne's ,patrologia ,lat9a1 ,vol4 ,,lxiii1 an ord9>y multiplication table 7sometimes call$ ,py?agorean abacus7 is giv5 9 ! illu/ration4 #cdh ,habebant enim div]se =matos apices vel c>act]es4 ,see ! ref]5ce to ,g]b]t on p4 #aag4 #cdi ,c4 ,h5ry1 ,sur l'orig9e de quelques notations ma!matiques1 ,revue ,ar*eologique1 #ahgi1 d]ives ^! f ! initial lett]s us$ z abbreviations = ! names ( ! num]als1 a !ory t f9ds few support]s4 #cej ,e4g41 x appe>s 9 ,s*on]us1 ,algori?mus ,demon/ratus1 ,nu]nb]g1 #aecd1 f4 ,a#d4 ,9 ,engl& x appe>$ 9 ! e>lie/ ,engli% ari?metical manuscript kn[n1 ,! ,crafte ( ,nombrynge3 f=!rmore ye mo/ vndir/onde t 9 ?is craft b5 vsid te5 figurys1 z "h b5e writ5 = ensampul1 ,num]als 444 9 ! quy* we vse te5 figurys ( ,inde4 ,que/io4 :y t5 fyguris ( ,inde8 ,solucio4 = z ,i h sayd a=e !i 7 fonde fyr/ 9 ,inde ( a kynge ( t ,cuntre1 t 0 call$ ,algor4 ,see ,smi?1 ,an ,e>ly ,engli% ,algorism1 loc4 cit4 #cea ,fri$le9 ed41 p4 #cig4 #ceb ,c>lsruhe codex ( ,g]l&o4 #cec ,muni* codex ( ,g]l&o4 #ced ,c>lsruhe codex ( ,b]nel9us4 #cee ,muni* codex ( ,b]nel9us4 #cef ,tur*ill1 c4 #abjj4 #ceg ,anon4 ,,ms41 ?irte5? c5tury1 ,alex&rian ,libr>y1 ,rome4 #ceh ,twelf?-c5tury ,boe?ius1 ,fri$le91 p4 #cif4 #cei ,vatican codex1 t5? c5tury1 ,boe?ius4 #cfj a1 h1 i1 are f ! ,fri$le9 ed42 ! orig9al 9 ! manuscript f :i* a is tak5 conta9s a z]o symbol1 z d all ( ! six plates giv5 by ,fri$le94 b-e f ! ,boncompagni ,bullet9o1 ,vol4 ,x1 p4 #eif2 f ibid41 ,vol4 ,,xv1 p4 #ahf2 g ,memorie della classe di sci41 ,reale ,acc4 dei ,l9cei1 ,an4 ,,cclxxiv 7#ahgf-ahgg71 ,april1 #ahgg4 ,a twelf?-c5tury ari?metician1 possibly ,john ( ,luna 7,hispal5sis1 ( ,seville1 c4 #aaej71 speaks ( ! great div]sity ( ^! =ms ev5 9 8 day1 say+3 ,e/ autem 9 aliquibus figuram i/>um apud multos diu]sitas4 ,quidam enim septimam hanc figuram repres5tant ,symbol alii autem sic ,symbol1 uel sic ,symbol4 ,quidam v]o qu>tam sic ,symbol4 ,boncompagni1 ,trattati1 ,vol4 ,,ii1 p4 #bh4 #cfa ,loc4 cit41 p4 #ei4 #cfb ,ibid41 p4 #aja4 #cfc ,loc4 cit41 p4 #cif4 #cfd ,khosru ,i1 :o began to reign 9 #eca ,a4,d4 ,see ,w4 ,s4 ,w ,vaux1 ,p]sia1 ,london1 #ahge1 p4 #afi2 ,?4 ,noeldeke1 ,aufsaetze zur p]si*5 ,ges*i*te1 ,leipzig1 #ahhg1 p4 #aac1 & 8 article 9 ! n9? edition ( ! ,encyclopa$ia ,britannica4 #cfe ,colebrooke1 ,essays1 ,vol4 ,,ii1 p4 #ejd1 on ! au?ority ( ,ibn al-,adami1 a/ronom]1 9 a "w publi%$ by 8 cont9uator ,al-,qasim 9 #ibj ,a4,d42 ,al-,biruni1 ,india1 ,vol4 ,,ii1 p4 #ae4 #cff ,h4 ,sut]1 ,die ,ma!matik] etc41 pp4 #d-e1 /ates t ,al-,fazari di$ betwe5 #gif & #hjf4 #cfg ,sut]1 loc4 cit41 p4 #fc4 #cfh ,sut]1 loc4 cit41 p4 #gd4 #cfi ,sut]1 ,das ,ma!matik]-,v]zei*niss im ,fihri/4 ,! ref]5ces to ,sut]1 unless o!rwise /at$1 are to 8 lat] "w ,die ,ma!matik] und ,a/ronom5 d] ,arab] etc4 #cgj ,sut]1 ,fihri/1 p4 #cg1 no date4 #cga ,sut]1 ,fihri/1 p4 #ch1 no date4 #cgb ,possibly late t5?1 s9ce he ref]s to "o ari?metical "w :i* is entitl$ ,book ( ! ,cyph]s 9 8 ,*ronology1 ,engli% ed41 p4 #acb4 ,sut]1 ,die ,ma!matik] etc41 pp4 #ih-ajj1 does n m5tion ?is "w2 see ! ,na*traege und ,b]i*tigung51 pp4 #agj-agb4 #cgc ,sut]1 pp4 #if-ig4 #cgd ,sut]1 p4 #aaa4 #cge ,sut]1 p4 #abd4 ,z ! "n %[s1 he came f ! ,we/4 #cgf ,sut]1 p4 #ach4 #cgg ,hankel1 ,zur ,ges*i*te d] ,ma!matik1 p4 #bef1 ref]s to him z writ+ on ! ,h9du art ( reckon+2 ,sut]1 p4 #afb4 #cgh ,greek3 ,psephophoria kat' ,ind\s1 ,greek ed41 ,c4 ,i4 ,g]h>dt1 ,halle1 #ahfe2 & ,g]man translation1 ,das ,re*5bu* des ,maximus ,planudes1 ,h4 ,waes*ke1 ,halle1 #ahgh4 #cgi ,sur une donnee hi/orique relative a l'emploi des *iffres indi5s p> les ,arabes1 ,tortol9i's ,annali di sci5ze mat4 e fis41 #ahee4 #chj ,sut]1 p4 #hj4 #cha ,sut]1 p4 #fh4 #chb ,spr5g] also calls att5tion to ?is fact1 9 ! ,zeits*rift d4 deuts*5 morg5la5d4 ,gesells*aft1 ,vol4 ,,xlv1 p4 #cfg4 #chc ,libri1 ,hi/oire des ma!matiques1 ,vol4 ,i1 p4 #adg4 #chd ,dictant la paix a l'emp]eur de ,con/ant9ople1 l',arabe victorieux dem&ait des manuscrits et des savans4 ,libri1 loc4 cit41 p4 #ajh4 #che ,p]sian bagadata1 ,god-giv54 #chf ,"o ( ! ,abbassides1 ! 7at lea/ pret5d$7 desc5dants ( ,al-,abbas1 uncle & advis] ( ,moh4amm$4 #chg ,e4 ,reclus1 ,asia1 ,am]ican ed41 ,n4 ,y41 #ahia1 ,vol4 ,,iv1 p4 #bbg4 #chh ,hi/orical ,sket*es1 ,vol4 ,,iii1 *ap4 iii4 #chi ,on its prom95ce at t p]iod see ,villicus1 p4 #gj4 #cij ,see pp4 #d-e4 #cia ,smi?1 ,d4 ,e41 9 ! ,cantor ,fe/s*rift1 #aiji1 note pp4 #aj-aa4 ,see also ,f4 ,woepcke1 ,propagation4 #cib ,ene/roem1 9 ,biblio!ca ,ma!matica1 ,vol4 ,i 7#c71 p4 #dii2 ,cantor1 ,ges*i*te1 ,vol4 ,i 7#c71 p4 #fga4 #cic ,cit$ 9 ,*apt] ,i4 ,x beg9s3 ,dixit algoritmi3 laudes deo rectori no/ro atque def5sori dicamus dignas4 ,x is devot$ entirely to ! fundam5tal op]ations & conta9s no applications4 #cid ,m4 ,/e9s*neid]1 ,die ,ma!matik bei d5 ,jud51 ,biblio!ca ,ma!matica1 ,vol4 ,,viii 7#b71 p4 #ii4 ,see also ! ref]5ce to ?is writ] 9 ,*apt] ,i4 #cie ,"p ( ?is "w has be5 translat$ f a ,leyd5 ,,ms4 by ,f4 ,woepcke1 ,propagation1 & m rec5tly by ,h4 ,sut]1 ,biblio!ca ,ma!matica1 ,vol4 ,,vii 7#c71 pp4 #aac-aai4 #cif ,a4 ,ne&]1 ,g5]al ,hi/ory ( ! ,*ri/ian ,religion & ,*ur*1 #e? ,am]ican ed41 ,bo/on1 #ahee1 ,vol4 ,,iii1 p4 #cce4 #cig ,beazley1 loc4 cit41 ,vol4 ,i1 p4 #di4 #cih ,beazley1 loc4 cit41 ,vol4 ,i1 pp4 #ej1 #dfj4 #cii ,see pp4 #g-h4 #djj ,! "n also appe>s z ,moh4amm$ ,abu'l-,qasim1 & ,ibn ,hauqal4 ,beazley1 loc4 cit41 ,vol4 ,i1 p4 #de4 #dja ,kitab al-masalik wa'l-mamalik4 #djb ,re9aud1 ,mem4 sur l',inde2 9 ,g]h>dt1 ,etudes1 p4 #ah4 #djc ,born at ,%iraz 9 #aaic4 ,he himself _h travel$ f ,india to ,europe4 #djd ,guli/an 7,rose ,g>d571 ,gateway ! ?ird1 ,,xxii4 ,sir ,edw9 ,arnold's translation1 ,n4 ,y41 #ahii1 p4 #agg4 #dje ,cunn+ham1 loc4 cit41 p4 #ha4 #djf ,putnam1 ,books1 ,vol4 ,i1 p4 #bbg3 ,non semel ext]nas p]egr9o tramite t]ras ,jam p]agravit ovans1 sophiae d$uctus amore1 ,si quid =te novi librorum seu /udiorum ,quod secum f]ret1 t]ris rep]iret 9 illis4 ,hic quoque ,romuleum v5it devotus ad urbem4 7,m ?an once he has travel$ joyfully ?r\gh remote regions & by /range roads1 l$ on by 8 zeal = k & seek+ to discov] 9 =eign l&s novelties 9 books or 9 /udies :i* he c\ld take back ) him4 ,& ?is zeal\s /ud5t j\rney$ to ! city ( ,romulus47 #djg ,a4 ,ne&]1 ,g5]al ,hi/ory ( ! ,*ri/ian ,religion & ,*ur*1 #e? ,am]ican ed41 ,bo/on1 #ahee1 ,vol4 ,,iii1 p4 #hi1 note #d2 ,libri1 ,hi/oire1 ,vol4 ,i1 p4 #adc4 #djh ,cunn+ham1 loc4 cit41 p4 #ha4 #dji ,heyd1 loc4 cit41 ,vol4 ,i1 p4 #d4 #daj ,ibid41 p4 #e4 #daa ,ibid41 p4 #ba4 #dab ,ibid41 p4 #bc4 #dac ,libri1 ,hi/oire1 ,vol4 ,i1 p4 #afg4 #dad ,picavet1 ,g]b]t1 un pape philosophe1 d'apres l'hi/oire et d'apres la leg5de1 ,p>is1 #ahig1 p4 #ai4 #dae ,beazley1 loc4 cit41 ,vol4 ,i1 *ap1 i1 & p4 #ed seq4 #daf ,ibid41 p4 #eg4 #dag ,libri1 ,hi/oire1 ,vol4 ,i1 p4 #aaj1 n41 cit+ au?orities1 & p4 #aeb4 #dah ,possibly ! old tradition1 ,prima d$it nautis usum magnetis ,amalphis1 is true s f> z x means ! mod]n =m ( compass c>d4 ,see ,beazley1 loc4 cit41 ,vol4 ,,ii1 p4 #cih4 #dai ,r4 ,c4 ,dutt1 loc4 cit41 ,vol4 ,,ii1 p4 #cab4 #dbj ,e4 ,j4 ,payne1 9 ,! ,cambridge ,mod]n ,hi/ory1 ,london1 #aijb1 ,vol4 ,i1 *ap4 i4 #dba ,geo4 ,phillips1 ,! ,id5tity ( ,m>co ,polo's ,zaitun ) ,*ang*au1 9 ,t'oung pao1 ,ar*ives p\r s]vir a l'etude de l'hi/oire de l',asie ori5tale1 ,leyd51 #ahij1 ,vol4 ,i1 p4 #bah4 ,w4 ,heyd1 ,ges*i*te des ,levan?&els im ,mittelalt]1 ,vol4 ,,ii1 p4 #baf4 ,! ,palazzo dei ,poli1 :]e ,m>co 0 born & di$1 /ill /&s 9 ! ,corte del ,milione1 9 ,v5ice4 ,! be/ description ( ! ,polo travels1 & ( o!r travels ( ! lat] ,middle ,ages1 is f\nd 9 ,c4 ,r4 ,beazley's ,dawn ( ,mod]n ,geography1 ,vol4 ,,iii1 *ap1 ii1 & ,"p ,,ii4 #dbb ,heyd1 loc4 cit41 ,vol4 ,,ii1 p4 #bbj2 ,h4 ,yule1 9 ,encyclopa$ia ,britannica1 #i? 7#aj?7 or #aa? ed41 article ,*9a4 ,! h&book cit$ is ,pegolotti's ,libro di divisam5ti di paesi1 *apt]s i-ii1 :]e x is impli$ t #fj1jjj w\ld be a likely am\nt = a m]*ant go+ to ,*9a to inve/ 9 8 trip4 #dbc ,cunn+ham1 loc4 cit41 p4 #aid4 #dbd ,i4e4 a commission h\se4 #dbe ,cunn+ham1 loc4 cit41 p4 #ahf4 #dbf ,j4 ,r4 ,gre51 ,%ort ,hi/ory ( ! ,engli% ,p1 ,new ,york1 #ahij1 p4 #ff4 #dbg ,w4 ,besant1 ,london1 ,new ,york1 #ahib1 p4 #dc4 #dbh ,baldak91 baldek91 balda*9o4 #dbi ,italian ,baldacco4 #dcj ,j4 ,k4 ,mum=d1 ,ori5tal ,rugs1 ,new ,york1 #aija1 p4 #ah4 #dca ,or ,girb]t1 ! ,lat9 =ms ,g]b]tus & ,girb]tus appe>+ indiff]5tly 9 ! docum5ts ( 8 "t4 #dcb ,see1 = example1 ,j4 ,c4 ,heilbronn]1 ,hi/oria ma!seos univ]sae1 p4 #gdj4 #dcc ,obscuro loco natum1 z an old *ronicle ( ,aurillac has x4 #dcd ,n4 ,bubnov1 ,g]b]ti po/ea ,silve/ri ,,ii papae op]a ma!matica1 ,b]l91 #ahii1 is ! mo/ complete & reliable s\rce ( in=mation2 ,picavet1 loc4 cit41 ,g]b]t etc42 ,oll]is1 ,oeuvres de ,g]b]t1 ,p>is1 #ahfg2 ,havet1 ,lettres de ,g]b]t1 ,p>is1 #ahhi 2 ,h4 ,weiss5born1 ,g]b]t2 ,beitraege zur ,k5ntnis d] ,ma!matik des ,mittelalt]s1 ,b]l91 #ahhh1 & ,zur ,ges*i*te d] ,e9fuehrung d] jetzig5 ,ziff]n 9 ,europa dur* ,g]b]t1 ,b]l91 #ahib2 ,bu$+]1 ,ueb] ,g]b]ts wiss5s*aftli*e und politis*e ,/ellung1 ,cassel1 #ahea2 ,ri*]1 ,hi/ori>um lib] ,,iii1 9 ,bubnov1 loc4 cit41 pp4 #cgf-cha2 ,nagl1 ,g]b]t und die ,re*5kun/ des #aj4 ,jahrhund]ts1 ,vi5na1 #ahhh4 #dce ,ri*] tells ( ! visit to ,aurillac by ,borel1 a ,spani% nobleman1 j z ,g]b]t 0 ent]+ into "y manhood4 ,he relates h[ affectionately ! abbot receiv$ him1 ask+ if !re 7 m5 9 ,spa9 well v]s$ 9 ! arts4 ,^u ,borel's reply 9 ! affirmative1 ! abbot ask$ t "o ( 8 "y m5 might accompany him ^u 8 return1 t he might c>ry on 8 /udies !re4 #dcf ,vicus ,ausona4 ,hatto also appe>s z ,atton & ,hatton4 #dcg ,?is is all t we "k ( 8 soj\rn 9 ,spa91 & ?is comes f 8 pupil ,ri*]4 ,! /ories told by ,adhem> ( ,*abanois1 an app>5tly ignorant & c]ta9ly untru/wor?y contempor>y1 ( 8 go+ to ,cordova1 are unsupport$4 7,see e4g4 ,picavet1 p4 #cd47 ,nev]!less ?is te/imony is /ill accept$3 ,k4 von ,raum]1 = example 7,ges*i*te d] ,pa$agogik1 #f? ed41 #ahij1 ,vol4 ,i1 p4 #f71 says ,ma!matik /udi]te man im ,mittelalt] bei d5 ,arab]n 9 ,spani54 ,zu ihn5 gi5g ,g]b]t1 na*malig] ,pab/ ,sylve/] ,,ii4 #dch ,?us 9 a lett] to ,aldab]on he says3 ,quos po/ repp]imus sp]etis1 id e/ ,,viii volum9a ,boeti de a/rologia1 praecl>issima quoque figur>um geometriae1 aliaque non m9us admir&a 7,epi/4 #h74 ,also 9 a lett] to ,ra9>d 7,epi/4 #acj71 he says3 ,ex tuis sumptibus fac ut mi*i scribantur ,m4 ,manlius 7,manilius 9 "o ,,ms47 de a/rologia4 #dci ,picavet1 loc4 cit41 p4 #ca4 #ddj ,picavet1 loc4 cit41 p4 #cf4 #dda ,havet1 loc4 cit41 p4 vii4 #ddb ,picavet1 loc4 cit41 p4 #cg4 #ddc ,con s9i/re arti conseguri la dignita del ,pontificato4444 ,lasciato poi l' abito1 e 'l mona/]io1 e datosi tutto 9 pot]e del diavolo4 ,quot$ 9 ,bombelli1 ,l'antica num]azione ,italica1 ,rome1 #ahgf1 p4 #da n4 #ddd ,he writes f ,rheims 9 #ihd to "o ,lupitus1 9 ,b>celona1 say+3 ,itaque librum de a/rologia translatum a te mi*i pet5ti dirige1 presumably ref]r+ to "s ,arabic treatise4 ,epi/4 no4 #bd ( ! ,havet collection1 p4 #ai4 #dde ,see ,bubnov1 loc4 cit41 p4 x4 #ddf ,oll]is1 loc4 cit41 p4 #cfa1 l4 #ae1 = ,b]nel9us2 & ,bubnov1 loc4 cit41 p4 #cha1 l4 #d1 = ,ri*]4 #ddg ,woepcke f\nd ?is 9 a ,p>is ,,ms4 ( ,radulph ( ,laon1 c4 #aajj4 ,propagation1 p4 #bdf4 ,et prima quidem trium spaciorum sup]ductio unitatis c>act]e inscribitur1 qui *aldeo nom9e dicitur ig94 ,see also ,alfr$ ,nagl1 ,d] ari?metis*e ,tractat des ,radulph von ,laon 7,abh&lung5 zur ,ges*i*te d] ,ma!matik1 ,vol4 ,v1 pp4 #he-acc71 p4 #ig4 #ddh ,weiss5born1 loc4 cit41 p4 #bci4 ,:5 ,oll]is 7,oeuvres de ,g]b]t1 ,p>is1 #ahfg1 p4 cci7 says1 ,c'e/ a lui et non po9t aux ,arabes1 que l',europe doit son sy/eme et ses signes de num]ation1 he exagg]ates1 s9ce ! evid5ce is all aga9/ 8 kn[+ ! place value4 ,fri$le9 emphasizes ?is 9 ! ,zeits*rift fu] ,ma!matik und ,physik1 ,vol4 ,,xii 7#ahfg71 ,lit]aturzeitung1 p4 #gj3 ,fu] das ,sy/em uns]] ,num]ation i/ die ,null das wes5tli*/e ,m]kmal1 und diese kannte ,g]b]t ni*t4 ,er selb/ s*rieb alle ,zahl5 mit d5 roemis*5 ,zahlzei*5 und man kann ihm also ni*t v]dank51 0 er selb/ ni*t kannte4 #ddi ,e4g41 ,*asles1 ,bu$+]1 ,g]h>dt1 & ,ri*]4 ,s ,m>t9 7,re*]*es n\velles etc47 believes t ,g]b]t receiv$ !m f ,boe?ius or 8 foll[]s4 ,see ,woepcke1 ,propagation1 p4 #da4 #dej ,bu$+]1 loc4 cit41 p4 #aj4 ,nev]!less1 9 ,g]b]t's "t "o ,al-,mans4ur1 gov]n+ ,spa9 "u ! "n ( ,hi%am 7#igf-ajjb71 call$ f ! ,ori5t ,al-,be4gani to tea* 8 son1 s t s*ol>s 7 recogniz$4 ,picavet1 p4 #cf4 #dea ,weiss5born1 loc4 cit41 p4 #bce4 #deb ,ibid41 p4 #bcd4 #dec ,^! lett]s1 ( ! p]iod #ihc-iig1 7 edit$ by ,havet1 loc4 cit41 &1 less completely1 by ,oll]is1 loc4 cit4 ,^? t\*+ ma!matical topics 7 edit$ by ,bubnov1 loc4 cit41 pp4 #ih-ajf4 #ded ,he publi%$ x 9 ! ,monum5ta ,g]maniae hi/orica1 ,scriptores1 ,vol4 ,,iii1 & at lea/ ?ree o!r editions h s9ce appe>$1 viz4 ^? by ,guadet 9 #ahde1 by ,po9signon 9 #ahee1 & by ,waitz 9 #ahgg4 #dee ,dom9o ac beatissimo ,patri ,g]b]to1 ,remorum ar*iepiscopo1 ,ri*]us ,mon*us1 ,gallorum congressibus 9 volum9e reg]5dis1 imp]ii tui1 pat] sanctissime ,g]b]te1 auctoritas sem9>ium d$it4 #def ,9 epi/le #ag 7,havet collection7 he speaks ( ! ,de multiplicatione et divisione num]orum libellum a ,joseph ,ispano editum abbas ,w>n]ius 7a p]son o!rwise unkn[n74 ,9 epi/le #be he says3 ,de multiplicatione et divisione num]orum1 ,joseph ,sapi5s s5t5tias quasdam edidit4 #deg ,h4 ,sut]1 ,zur ,frage ueb] d5 ,josephus ,sapi5s1 ,biblio!ca ,ma!matica1 ,vol4 ,,viii 7#b71 p4 #hd2 ,weiss5born1 ,e9fuehrung1 p4 #ad2 also 8 ,g]b]t2 ,m4 ,/e9s*neid]1 9 ,biblio!ca ,ma!matica1 #ahic1 p4 #fh4 ,wallis 7,algebra1 #afhe1 *ap4 #ad7 w5t ov] ! li/ ( ,spani% ,josephs v c>efully1 b c\ld f9d no?+ save t ,josephus ,hispanus seu ,josephus sapi5s videtur aut ,maurus fuisse aut alius quis 9 ,hispania4 #deh ,p4 ,ewald1 ,mit!ilung51 ,neues ,ar*iv d4 ,gesells*aft fu] aelt]e deuts*e ,ges*i*tskunde1 ,vol4 ,,viii1 #ahhc1 pp4 #ced-cfd4 ,"o ( ! manuscripts is ( #igf ,a4,d4 & ! o!r ( #iib ,a4,d4 ,see also ,franz ,/eff5s1 ,late9is*e ,palaeographie1 ,freiburg 7,s*weiz71 #aijc1 pp4 xxxix-xl4 ,! =ms are reproduc$ 9 ! plate on page #adj4 #dei ,x is entitl$ ,con/ant9o suo ,g]b]tus scola/icus1 because x 0 address$ to ,con/ant9e1 a monk ( ! ,abbey ( ,fleury4 ,! text ( ! lett] to ,con/ant9e1 prec$+ ! treatise on ! ,abacus1 is giv5 9 ! ,comptes r5dus1 ,vol4 ,,xvi 7#ahdc71 p4 #bie4 ,?is book seems to h be5 writt5 c4 #ihj ,a4,d4 ,bubnov1 loc4 cit41 p4 #f4 #dfj ,hi/oire de l',ari?metique1 ,comptes r5dus1 ,vol4 ,,xvi 7#ahdc71 pp4 #aef1 #bha4 #dfa ,loc4 cit41 ,g]b]ti ,op]a etc4 #dfb ,fri$le9 ?\ght x spuri\s4 ,see ,zeits*rift fu] ,ma!matik und ,physik1 ,vol4 ,,xii 7#ahfg71 ,hi/4-lit4 suppl41 p4 #gd4 ,x 0 discov]$ 9 ! libr>y ( ! ,b5$ict9e mona/ry ( ,/4 ,pet]1 at ,salzburg1 & 0 publi%$ by ,pet] ,b]nh>d ,pez 9 #agba4 ,d\bt 0 fir/ ca/ ^u x 9 ! ,oll]is edition 7,oeuvres de ,g]b]t74 ,see ,weiss5born1 ,g]b]t1 pp4 #b1 #f1 #afh1 & ,picavet1 p4 #ha4 ,hock1 ,cantor1 & ,?4 ,m>t9 place ! composition ( ! "w at c4 #iif :5 ,g]b]t 0 9 ,g]many1 :ile ,oll]is & ,picavet ref] x to ! p]iod :5 he 0 at ,rheims4 #dfc ,picavet1 loc4 cit41 p4 #ahb4 #dfd ,:o wrote aft] ,g]b]t became pope1 = he uses1 9 8 preface1 ! words1 a dom9o pape ,g]b]to4 ,he 0 q c]ta9ly n lat] ?an ! elev5? c5tury2 we d n h exact in=mation ab\t ! "t 9 :i* he liv$4 #dfe ,picavet1 loc4 cit41 p4 #ahb4 ,weiss5born1 ,g]b]t1 p4 #bbg4 ,9 ,oll]is1 ,lib] ,abaci 7( ,b]nel9us71 p4 #cfa4 #dff ,ri*]1 9 ,bubnov1 loc4 cit41 p4 #cha4 #dfg ,weiss5born1 ,g]b]t1 p4 #bda4 #dfh ,writ]s on numismatics are q unc]ta9 z to _! use4 ,see ,f4 ,gnec*i1 ,monete ,romane1 #b;d ed41 ,milan1 #aijj1 cap4 ,,xxxvii4 ,= pictures ( old ,greek tess]ae ( ,s>matia1 see ,s4 ,ambrosoli1 ,monete ,gre*e1 ,milan1 #ahii1 p4 #bjb4 #dfi ,?us ,tzwivel's ari?metic ( #aejg1 fol4 #b1 v41 speaks ( ! t5 figures z *>act]es sive num]orum apices a diuo ,seu]9o ,boetio4 #dgj ,weiss5born uses sipos = #j4 ,x is n giv5 by ,b]nel9us1 & appe>s 9 ,radulph ( ,laon1 9 ! twelf? c5tury4 ,see ,gu5!r's ,ges*i*te1 p4 #ih1 n42 ,weiss5born1 p4 #aa2 ,pihan1 ,expose etc41 pp4 xvi-xxii4 ,9 ,fri$le9's ,boetius1 p4 #cif1 ! plate %[s t all ( ! six important manuscripts f :i* ! illu/rations are tak5 conta9 ! symbol1 :ile f\r out ( five :i* give ! words use ! ^w sipos = #j4 ,! names appe> 9 a twelf?-c5tury anonym\s manuscript 9 ! ,vatican1 9 a passage beg9n+ ,ord9e primig5o sibi nom5 possidet ig94 ,&ras ecce locum mox u5dicat ipse secundum ,ormis po/ num]os incompositus sibi primus4 ,boncompagni ,buttet9o1 ,,xv1 p4 #acb4 ,tur*ill 7twelf? c5tury7 gives ! names ,ig91 &ras1 hormis1 arbas1 quimas1 caletis1 z5is1 tem5ias1 cel5tis1 say+3 ,has autem figuras1 ut donnus dom9us ,gvillelmus ,rx te/atur1 a pytagoricis habemus1 nom9a u]o ab arabibus4 7,:o ! ,william ,r4 0 is n kn[n4 ,boncompagni ,bullet9o ,,xv1 p4 #acf47 ,radulph ( ,laon 7d4 #aaca7 ass]t$ t !y 7 ,*aldean 7,propagation1 p4 #dh n474 ,a discussion ( ! :ole "q is also giv5 9 ,e4 ,c4 ,bayley1 loc4 cit4 ,huet1 writ+ 9 #afgi1 ass]t$ t !y 7 ( ,semitic orig91 z did ,nesselmann 9 spite ( 8 despair ov] ormis1 calctis1 & cel5tis2 see ,woepcke1 ,propagation1 p4 #dh4 ,! names 7 us$ z late z ! fifte5? c5tury1 )\t ! z]o1 b ) ! sup]script dot = #aj's1 two dots = #ajj's1 etc41 z among ! e>ly ,arabs4 ,g]h>dt m5tions hav+ se5 a f\rte5? or fifte5? c5tury manuscript 9 ! ,biblio!ca ,amploniana ) ! names ,+n91 &ras1 armis1 arbas1 qu9as1 calctis1 z5cis1 zem5ias1 zcel5tis1 & ! /atem5t ,si unum punctum sup] +n9 ponitur1 ,x significat4444 ,si duo puncta sup] 444 figuras sup]ponunt]1 fiet decuplim illius quod cum uno puncto significabatur1 9 ,monatsb]i*te d] ,k4 ,p4 ,akad4 d4 ,wiss41 ,b]l91 #ahfg1 p4 #dj4 #dga ,a *>t ( t5 num]als 9 #bjj tongues1 by ,rev4 ,r4 ,patrick1 ,london1 #ahab4 #dgb ,num]atio figuralis e/ cuiusuis num]i p] notas1 et figuras num]ates descriptio4 ,cli*toveus1 edition ( c4 #aejg1 fol4 ,c ii1 v4 ,ari/oteles enim uoces r]um ,greek3 sumbola uocat3 id translatum1 sonat notas4 ,noviomagus1 ,de ,num]is ,libri ,,ii1 cap4 vi4 ,alphabetum decem not>um4 ,s*on]us1 notes to ,ramus1 #aehf1 p4 #c seq4 ,ri*] says3 novem num]o notas omnem num]um significantes4 ,bubnov1 loc4 cit41 p4 #cha4 #dgc ,il y a dix ,*>act]es1 autrem5t ,figures1 ,notes1 ou ,elem5ts4 ,peleti]1 edition ( #afjg1 p4 #ac4 ,num]orum notas alij figuras1 alij signa1 alij *>act]es uocant4 ,gl>eanus1 #aede edition1 f4 #i1 r4 ,p] figuras 7quas zyphras uocant7 assignationem1 quales sunt hae notulae1 #a4 #b4 #c4 #d4444 ,noviomagus1 ,de ,num]is ,libri ,,ii1 cap4 vi4 ,gemma ,frisius also uses elem5ta & ,c>dan uses lit]ae4 ,9 ! fir/ ari?metic by an ,am]ican 7,gre5wood1 #agbi7 ! au?or speaks ( a few ,arabian ,*>ect]s or ,num]al ,figures1 call$ ,digits 7p4 #a71 & z late z #agij1 9 ! ?ird edition ( ,j4 ,j4 ,blassi]e's ari?metic 7#a/ ed4 #agfi71 ! "n *>act]s is /ill 9 use1 bo? = de ,latyns*e en de ,arabis*e 7p4 #d71 z is also ! t]m ,cyff]lett]s 7p4 #f1 n474 ,ziff]1 ! mod]n ,g]man =m ( ciph]1 0 commonly us$ to designate any ( ! n9e figures1 z by ,boes*5/e9 & ,riese1 al?\gh o!rs1 l ,koebel1 us$ x only = ! z]o4 ,s zifre appe>s 9 ! ari?metic by ,borgo1 #aeej ed4 ,9 a ,muni* codex ( ! twelf? c5tury1 attribut$ to ,g]l&1 !y are call$ *>act]s only3 ,usque ad ,,viiii4 enim porrigitur omnis num]us et qui sup]crescit eisdem designator ,k>act]ibus4 ,boncompagni ,bullet9o1 ,vol4 ,x4 p4 #fjg4 #dgd ,! title ( 8 "w is ,prologus ,n4 ,ocreati 9 ,helceph 7,arabic al-qeif1 inve/igation or memoir7 ad ,adel>dum ,bat5sem magi/rum suum4 ,! "w 0 made kn[n by ,c4 ,h5ry1 9 ! ,zeits*rift fu] ,ma!matik und ,physik1 ,vol4 ,,xxv1 p4 #abi1 & 9 ! ,abh&lung5 zur ,ges*i*te d] ,ma!matik1 ,vol4 ,,iii2 ,weiss5born1 ,g]b]t1 p4 #ahh4 #dge ,! z]o is indicat$ by a vacant column4 #dgf ,leo ,jordan1 loc4 cit41 p4 #agj4 ,*ifre en augorisme is ! expression us$1 :ile a c5tury lat] giffre en argorisme & cyffres d'augorisme are simil>ly us$4 #dgg ,! ,works ( ,ge(frey ,*auc]1 edit$ by ,w4 ,w4 ,skeat1 ,vol4 ,,iv1 ,ox=d1 #ahid1 p4 #ib4 #dgh ,loc4 cit41 ,vol4 ,,iii1 pp4 #agi & #ahj4 #dgi ,9 ,book ,,ii1 *ap1 vii1 ( ,! ,te/am5t ( ,love1 pr9t$ ) ,*auc]'s ,works1 loc4 cit41 ,vol4 ,,vii1 ,london1 #ahig4 #dhj ,lib] ,abacci1 publi%$ 9 ,oll]is1 ,oeuvres de ,g]b]t1 pp4 #ceg-djj4 #dha ,g4 ,r4 ,kaye1 ,! ,use ( ! ,abacus 9 ,anci5t ,india1 ,j\rnal & ,proce$+s ( ! ,asiatic ,society ( ,b5gal1 #aijh1 pp4 #bic-big4 #dhb ,lib] ,abbaci1 by ,leon>do ,pisano1 loc4 cit41 p4 #a4 #dhc ,fri$le91 ,die ,entwickelung des ,re*n5s mit ,column51 ,zeits*rift fu] ,ma!matik und ,physik1 ,vol4 ,x1 p4 #bdg4 #dhd ,! divisor #f or #af be+ increas$ by ! diff]5ce #d1 to #aj or #bj respectively4 #dhe ,e4g4 ,cantor1 ,vol4 ,i1 p4 #hhb4 #dhf ,fri$le91 loc4 cit42 ,fri$le91 ,g]b]t's ,regeln d] ,division & ,das ,re*n5 mit ,column5 vor dem #aj4 ,jahrhund]t1 ,zeits*rift fu] ,ma!matik und ,physik1 ,vol4 ,,ix2 ,bubnov1 loc4 cit41 pp4 #aig-bde2 ,m4 ,*asles1 ,hi/oire de l'ari?metique4 ,re*]*es des traces du sy/eme de l'abacus1 apres que cette me?ode a pris le nom d',algorisme4--,preuves qu'a t\tes les epoques1 jusq'au ,,xvie siecle1 on a su que l'ari?metique vulgaire avait p\r orig9e cette me?ode anci5ne1 ,comptes r5dus1 ,vol4 ,,xvii1 pp4 #adc-aed1 also ,regles de l'abacus1 ,comptes r5dus1 ,vol4 ,,xvi1 pp4 #bah-bdf1 & ,analyse et explication du traite de ,g]b]t1 ,comptes r5dus1 ,vol4 ,,xvi1 pp4 #bha-bii4 #dhg ,bubnov1 loc4 cit41 pp4 #bjc-bjd1 ,abbonis abacus4 #dhh ,regulae de num]orum abaci rationibus1 9 ,bubnov1 loc4 cit41 pp4 #bje-bbe4 #dhi ,p4 ,treutle91 ,intorno ad alcuni scritti in$iti relativi al calcolo dell' abaco1 ,bullet9o di bibliografia e di /oria delle sci5ze matemati*e e fisi*e1 ,vol4 ,x1 pp4 #ehi-fdg4 #dij ,intorno ad uno scritto in$ito di ,adelh>do di ,ba? intitolato ',regulae ,abaci1' ,b4 ,boncompagni1 9 8 ,bullet9o1 ,vol4 ,,xiv1 pp4 #a-acd4 #dia ,treutle91 loc4 cit42 ,boncompagni1 ,intorno al ,tractatus de ,abaco di ,g]l&o1 ,bullet9o1 ,vol4 ,x1 pp4 #fdh-fef4 #dib ,e4 ,n>ducci1 ,intorno a due trattati in$iti d'abaco cont5uti 9 due codici ,vaticani del secolo ,,xii1 ,boncompagni ,bullet9o1 ,vol4 ,,xv1 pp4 #aaa-afb4 #dic ,see ,mol9i]1 ,les s\rces de l'hi/oire de ,france1 ,vol4 ,,ii1 ,p>is1 #aijb1 pp4 #b1 #c4 #did ,cantor1 ,ges*i*te1 ,vol4 ,i1 p4 #gfb4 ,a4 ,nagl 9 ! ,abh&lung5 zur ,ges*i*te d] ,ma!matik1 ,vol4 ,v1 p4 #he4 #die #ajcj-aaag4 #dif ,abh&lung5 zur ,ges*i*te d] ,ma!matik1 ,vol4 ,v1 pp4 #he-acc4 ,! "w beg9s ,incipit ,lib] ,radulfi laudun5sis de abaco4 #dig ,mat]iali5 zur ,ges*i*te d] arabis*5 ,zahlzei*5 9 ,frankrei*1 loc4 cit4 #dih ,:o di$ 9 #abjb4 #dii ,cantor1 ,ges*i*te1 ,vol4 ,i 7#c71 pp4 #hjj-hjc2 ,boncompagni1 ,trattati1 ,"p ,,ii4 ,m4 ,/e9s*neid] 7,die ,ma!matik bei d5 ,jud51 ,biblio!ca ,ma!matica1 ,vol4 ,x 7#b71 p4 #gi7 +5i\sly d]ives ano!r "n by :i* he is call$ 7,ab5deu?7 f ,ibn ,daud 7,son ( ,david74 ,see also ,abh&lung51 ,vol4 ,,iii1 p4 #aaj4 #ejj ,john is said to h di$ 9 #aaeg4 #eja ,= x says1 ,incipit prologus 9 libro algho>ismi de practica arismetrice4 ,qui editus e/ a magi/ro ,johanne yspal5si4 ,x is publi%$ 9 full 9 ! second "p ( ,boncompagni's ,trattati d'aritmetica4 #ejb ,possibly1 inde$1 ! mean+ ( libro algho>ismi is n to ,al-,kh[arazmi's book1 b to a book ( algorism4 ,john ( ,luna says ( x3 ,hoc idem e/ illud etiam quod 444 alcorismus dic]e videtur4 ,trattati1 p4 #fh4 #ejc ,= a resume1 see ,cantor1 ,vol4 ,i 7#c71 pp4 #hjj-hjc4 ,z to ! au?or1 see ,ene/roem 9 ! ,biblio!ca ,ma!matica1 ,vol4 ,,vi 7#c71 p4 #aad1 & ,vol4 ,,ix 7#c71 p4 #b4 #ejd ,born at ,cremona 7al?\gh "s h ass]t$ at ,c>mona1 9 ,&alusia7 9 #aaad2 di$ at ,tol$o 9 #aahg4 ,cantor1 loc4 cit42 ,boncompagni1 ,atti d4 ,r4 ,accad4 d4 n4 ,l9cei1 #ahea4 #eje ,see ,abh&lung5 zur ,ges*i*te d] ,ma!matik1 ,vol4 ,,xiv1 p4 #adi2 ,biblio!ca ,ma!matica1 ,vol4 ,,iv 7#c71 p4 #bjf4 ,boncompagni _h a f\rte5?-c5tury manuscript ( 8 "w1 ,g]>di ,cremon5sis artis metrice practice4 ,see also ,t4 ,l4 ,hea?1 ,! ,?irte5 ,books ( ,euclid's ,elem5ts1 #c vols41 ,cambridge1 #aijh1 ,vol4 ,i1 pp4 #ib-id 2 ,a4 ,a4 ,bjo]nbo1 ,g]h>d von ,cremonas ,ueb]setzung von ,alkw>izmis ,algebra und von ,euklids ,elem5t51 ,biblio!ca ,ma!matica1 ,vol4 ,,vi 7#c71 pp4 #bci-bdh4 #ejf ,wallis1 ,algebra1 #afhe1 p4 #ab seq4 #ejg ,cantor1 ,ges*i*te1 ,vol4 ,i 7#c71 p4 #ijf2 ,a4 ,a4 ,bjo]nbo1 ,al-,*warizmi's trigonometriske ,tavl]1 ,fe/skrift til ,h4 ,g4 ,zeu!n1 ,cop5hag51 #aiji1 pp4 #a-ag4 #ejh ,hea?1 loc4 cit41 pp4 #ic-if4 #eji ,m4 ,/e9s*neid]1 ,zeits*rift d] deuts*5 morg5la5dis*5 ,gesells*aft1 ,vol4 ,,xxv1 #ahga1 p4 #ajd1 & ,zeits*rift fu] ,ma!matik und ,physik1 ,vol4 ,,xvi1 #ahga1 pp4 #cib-cic2 ,m4 ,curtze1 ,c5tralblatt fu] ,biblio!kswes51 #ahii1 p4 #bhi2 ,e4 ,wappl]1 ,zur ,ges*i*te d] deuts*5 ,algebra im #ae4 ,jahrhund]t1 ,programm1 ,zwickau1 #ahhg2 ,l4 ,c4 ,k>p9ski1 ,rob]t ( ,*e/]'s ,translation ( ! ,algebra ( ,al-,kh[arazmi1 ,biblio!ca ,ma!matica1 ,vol4 ,,xi 7#c71 p4 #abe4 ,he is also kn[n z ,rob]tus ,ret95sis1 or ,rob]t ( ,read+4 #eaj ,nagl1 ,a41 ,ueb] e9e ,algorismus-,s*rift des ,,xii4 ,jahrhund]ts und ueb] die ,v]breitung d] indis*-arabis*5 ,re*5kun/ und ,zahlzei*5 im *ri/l4 ,ab5dl&e1 9 ! ,zeits*rift fu] ,ma!matik und ,physik1 ,hi/4-lit4 ,ab?41 ,vol4 ,,xxxiv1 p4 #abi4 ,curtze1 ,abh&lung5 zur ,ges*i*te d] ,ma!matik1 ,vol4 ,,viii1 pp4 #a-bg4 #eaa ,see l9e a 9 ! plate on p4 #adc4 #eab ,sef] ha-,misp>1 ,das ,bu* d] ,zahl1 e9 hebraeis*-ari?metis*es ,w]k des ,r4 ,abraham ibn ,esra1 ,moritz ,silb]b]g1 ,frankfurt a4 ,m41 #ahie4 #eac ,br[n+'s ,rabbi b5 ,ezra4 #ead ,d>um hab5 au* die ,weis5 ,indi5s all ihre ,zahl5 dur* neun bezei*net und ,=m5 fu] die #i ,ziff]n gebildet4 ,sef] ha-,misp>1 loc4 cit41 p4 #b4 #eae ,f4 ,bona9i1 ,memoria unica s9crona di ,leon>do ,fibonacci1 ,pisa1 #aheh1 republi%$ 9 #ahfg1 & appe>+ 9 ! ,giornale ,arcadico1 ,vol4 ,,cxcvii 7,n4,s4 ,,lii72 ,gaetano ,milanesi1 ,docum5to in$ito e sconosciuto a ,lion>do ,fibonacci1 ,roma1 #ahfg2 ,guglielm9i1 ,elogio di ,lion>do ,pisano1 ,bologna1 #ahab1 p4 #ce2 ,libri1 ,hi/oire des sci5ces ma!matiques1 ,vol4 ,,ii1 p4 #be2 ,d4 ,m>t9es1 ,orig9e e progressi dell' aritmetica1 ,mess9a1 #ahfe1 p4 #dg2 ,lucas1 9 ,boncompagni ,bullet9o1 ,vol4 ,x1 pp4 #abi1 #bci2 ,besagne1 ibid41 ,vol4 ,,ix1 p4 #ehc2 ,boncompagni1 ?ree works z cit$ 9 ,*ap4 ,i2 ,g4 ,ene/roem1 ,ueb] zwei angebli*e ma!matis*e ,s*ul5 im *ri/li*5 ,mittelalt]1 ,biblio!ca ,ma!matica1 ,vol4 ,,viii 7#c71 pp4 #beb-bfb2 ,boncompagni1 ,della vita e delle op]e di ,leon>do ,pisano1 loc4 cit4 #eaf ,! date is purely conjectural4 ,see ! ,biblio!ca ,ma!matica1 ,vol4 ,,iv 7#c71 p4 #bae4 #eag ,an old *ronicle relates t 9 #ajfc ,pisa f\ght a great battle ) ! ,s>ac5s at ,pal]mo1 captur+ six %ips1 "o be+ full ( wondr\s treasure1 & ?is 0 devot$ to build+ ! ca!dral4 #eah ,heyd1 loc4 cit41 ,vol4 ,i1 p4 #adi4 #eai ,ibid41 p4 #baa4 #ebj ,j4 ,a4 ,symonds1 ,r5aissance 9 ,italy4 ,! ,age ( ,despots4 ,new ,york1 #ahhc1 p4 #fb4 #eba ,symonds1 loc4 cit41 p4 #gi4 #ebb ,j4 ,a4 ,fr\de1 ,! ,sci5ce ( ,hi/ory1 ,london1 #ahfd4 ,un brevet d'apo?icaire n'empe*a pas ,dante d'etre le plus gr& poete de l',italie1 et ce fut un petit m>*& de ,pise qui donna l'algebre aux ,*reti5s4 ,libri1 ,hi/oire1 ,vol4 ,i1 p4 xvi4 #ebc ,a docum5t ( #abbf1 f\nd & publi%$ 9 #aheh1 reads3 ,leon>do bigollo quondam ,guilielmi4 #ebd ,bonacc+o g]mano suo4 #ebe ,e4g4 ,libri1 ,guglielm9i1 ,tirabos*i4 #ebf ,lat91 ,bonaccius4 #ebg ,boncompagni & ,milanesi4 #ebh ,repr9t1 p4 #e4 #ebi ,:5ce ! ,fr5* "n = c&le4 #ecj ,n[ "p ( ,algi]s4 #eca ,e4 ,reclus1 ,africa1 ,new ,york1 #ahic1 ,vol4 ,,ii1 p4 #bec4 #ecb ,s$ hoc totum et algorismum atque arcus pictagore quasi errorem computavi respectu modi indorum4 ,woepcke1 ,propagation etc41 reg>ds ?is z ref]r+ to two diff]5t sy/ems1 b ! expression may v well mean algorism z p]=m$ ^u ! ,py?agorean arcs 7or table74 #ecc ,book ( ! ,abacus1 ?is t]m !n be+ us$1 & long aft]w>ds 9 ,italy1 to mean m]ely ! ari?metic ( computation4 #ecd ,incipit lib] ,abaci a ,leon>do filio ,bonacci compositus anno #abjb et correctus ab eodem anno #abbh4 ,?ree ,,mss4 ( ! ?irte5? c5tury are kn[n1 viz4 at ,milan1 at ,si5a1 & 9 ! ,vatican libr>y4 ,! "w 0 fir/ pr9t$ by ,boncompagni 9 #aheg4 #ece ,i4e4 9 relation to ! quadrivium4 ,non legant 9 fe/ivis diebus1 nisi ,philosophos et rhetoricas et quadrivalia et b>b>ismum et e?icam1 si placet4 ,sut]1 ,die ,ma!matik auf d5 ,univ]sitaet5 des ,mittelalt]s1 ,zu]i*1 #ahhg1 p4 #ef4 ,rog] ,bacon gives a /ill m gloomy view ( ,ox=d 9 8 "t 9 8 ,opus m9us1 9 ! ,r]um ,britannic>um m$ii aevi scriptores1 ,london1 #ahei1 ,vol4 ,i1 p4 #cbg4 ,= a picture ( ,cambridge at ?is "t consult ,f4 ,w4 ,newman1 ,! ,engli% ,univ]sities1 translat$ f ! ,g]man ( ,v4 ,a4 ,hub]1 ,london1 #ahdc1 ,vol4 ,i1 p4 #fa2 ,w4 ,w4 ,r4 ,ball1 ,hi/ory ( ,ma!matics at ,cambridge1 #ahhi2 ,s4 ,gu5!r1 ,ges*i*te des ma!matis*5 ,unt]ri*ts im deuts*5 ,mittelalt] bis zum ,jahre #aebe1 ,b]l91 #ahhg1 be+ ,vol4 ,,iii ( ,monum5ta ,g]maniae pa$agogica4 #ecf ,on ! comm]cial activity ( ! p]iod1 x is kn[n t bills ( ex*ange pass$ betwe5 ,mess9a & ,con/ant9ople 9 #aafa1 & t a bank 0 f\nd$ at ,v5ice 9 #aagj1 ! ,bank ( ,san ,m>co be+ e/abli%$ 9 ! foll[+ ye>4 ,! activity ( ,pisa 0 v manife/ at ?is "t4 ,heyd1 loc4 cit41 ,vol4 ,,ii1 p4 #e2 ,v4 ,casagr&i1 ,/oria e cronologia1 #c;d ed41 ,milan1 #aija1 p4 #ef4 #ecg ,j4 ,a4 ,symonds1 loc4 cit41 ,vol4 ,,ii1 p4 #abg4 #ech ,i4 ,taylor1 ,! ,alphabet1 ,london1 #ahhc1 ,vol4 ,,ii1 p4 #bfc4 #eci ,cit$ by ,ung]'s ,hi/ory1 p4 #ae4 ,! ,arabic num]als appe> 9 a ,reg5sburg *ronicle ( #aafg & 9 ,silesia 9 #acdj4 ,see ,s*midt's ,encyclopa$ie d] ,erziehung1 ,vol4 ,,vi1 p4 #gbf2 ,a4 ,kuckuk1 ,die ,re*5kun/ im se*zehnt5 ,jahrhund]t1 ,fe/s*rift zur dritt5 ,saecul>fei] des ,b]l9is*5 ,gymnasiums zum grau5 ,klo/]1 ,b]l91 #ahgd1 p4 #d4 #edj ,! text is giv5 9 ,halliwell1 ,r>a ,ma!matica1 ,london1 #ahci4 #eda ,sev5 are giv5 9 ,a%mole's ,catalogue ( ,manuscripts 9 ! ,ox=d ,libr>y1 #ahde4 #edb ,maximilian ,curtze1 ,petri ,philom5i de ,dacia 9 ,algorismum ,vulg>em ,johannis de ,sacrobosco comm5t>ius1 una cum ,algorismo ipso1 ,cop5hag51 #ahig2 ,l4 ,c4 ,k>p9ski1 ,jordanus ,nemor>ius & ,john ( ,halifax1 ,am]ican ,ma!matical ,mon?ly1 ,vol4 ,,xvii1 pp4 #ajh-aac4 #edc ,j4 ,as*ba*1 ,ges*i*te d] ,wi5] ,univ]sitaet im er/5 ,jahrhund]te ihres ,be/eh5s1 ,wi51 #ahfe1 p4 #ic4 #edd ,curtze1 loc4 cit41 gives ! text4 #ede ,curtze1 loc4 cit41 f\nd "s =ty-five copies ( ! ,algorismus 9 ?ree libr>ies ( ,muni*1 ,v5ice1 & ,erfurt 7,amploniana74 ,exam9ation ( two manuscripts f ! ,plimpton collection & ! ,columbia libr>y %[s su* m>k$ div]g5ce f ea* o!r & f ! text publi%$ by ,curtze t ! conclusion seems legitimate t ^! 7 /ud5ts' lecture notes4 ,! %or?& "* ( ! writ+ fur!r confirms ?is view1 z x %[s t !y 7 writt5 l>gely = ! p]sonal use ( ! writ]s4 #edf ,quidam philosophus edidit nom9e ,algus1 unde et ,algorismus nuncupatur4 ,curtze1 loc4 cit41 p4 #a4 #edg ,s9i/rorsum aut]a scribimus 9 hac arte m arabico sive iudaico1 huius sci5tiae inv5torum4 ,curtze1 loc4 cit41 p4 #g4 ,! ,plimpton manuscript omits ! words sive iudaico4 #edh ,non enim omnis num]us p] quascumque figuras ,indorum repraes5tatur1 s$ tantum det]m9atus p] det]m9atam1 ut #d non p] #e1444 ,curtze1 loc4 cit41 p4 #be4 #edi ,c4 ,h5ry1 ,sur les deux plus anci5s traites francais d'algorisme et de geometrie1 ,boncompagni ,bullet9o1 ,vol4 ,,xv1 p4 #di2 ,victor ,mortet1 ,le plus anci5 traite francais d'algorisme1 loc4 cit4 #eej ,l',etat des sci5ces en ,france1 depute la mort du ,roy ,rob]t1 arrivee en #ajca1 jusqu'a celle de ,philippe le ,bel1 arrivee en #acad1 ,p>is1 #agda4 #eea ,disc\rs sur l'etat des lettres en ,france au ,,xiiie siecle1 ,p>is1 #ahbd4 #eeb ,ap]cu hi/orique1 ,p>is1 #ahgf ed41 p4 #dfd4 #eec ,ranulf ,higd51 a native ( ! we/ ( ,engl&1 ent]$ ,/4 ,w]burgh's mona/]y at ,*e/] 9 #abii4 ,he 0 a ,b5$ict9e monk & *ronicl]1 & di$ 9 #acfd4 ,8 ,poly*ronicon1 a hi/ory 9 sev5 books1 0 pr9t$ by ,caxton 9 #adhj4 #eed ,trevisa's translation1 ,higd5 hav+ writt5 9 ,lat94 #eee ,an illu/ration ( ?is feel+ is se5 9 ! writ+s ( ,prosdocimo de' ,beldom&i 7b4 c4 #acgj-achj1 d4 #adbh73 ,inv5i 9 quam pluribus libris algorismi nuncupatis mores circa num]os op]&i satis v>ios atque div]sos1 qui licet boni exi/]5t atque v]i erant1 tam5 fa/idiosi1 tum propt] ips>um regul>um multitud9em1 tum propt] e>um deleationes1 tum etiam propt] ips>um op]ationum probationes1 utrum si bone fu]9t vel ne4 ,erant et etiam i/i modi int]im fa/idiosi1 quod si 9 aliquo calculo a/roloico error contigisset1 calculatorem op]ationem suam a capite incip]e oportebat1 dato quod error suus adhuc satis prop9quus exi/]et2 et hoc propt] figuras 9 sua op]atione deletas4 ,indigebat etiam calculator semp] aliquo lapide vel sibi con=mi1 sup] quo scrib]e atque facilit] del]e posset figuras cum quibus op]abatur 9 calculo suo4 ,et quia haec omnia satis fa/idiosa atque laboriosa mihi visa sunt1 disposui libellum ed]e 9 quo omnia i/a abic]5tur3 qui etiam algorismus sive lib] de num]is d5om9>i pot]it4 ,scias tam5 quod 9 hoc libello pon]e non int5do nisi ea quae ad calculum necess>ia sunt1 alia quae 9 aliis libris practice arismetrice tanguntur1 ad calculum non necess>ia1 propt] brevitatem dimit5do4 ,quot$ by ,a4 ,nagl1 ,zeits*rift fu] ,ma!matik und ,physik1 ,hi/4-lit4 ,ab?41 ,vol4 ,,xxxiv1 p4 #adc2 ,smi?1 ,r>a ,ari?metica1 p4 #ad1 9 facsimile4 #eef ,p4 ,ewald1 loc4 cit42 ,franz ,/eff5s1 ,late9is*e ,palaeographie1 pp4 xxxix-xl4 ,we are indebt$ to ,pr(essor ,j4 ,m4 ,burnam = a photograph ( ?is r>e manuscript4 #eeg ,see ! plate ( =ms on p4 #hh4 #eeh ,k>abacek1 loc4 cit41 p4 #ef2 ,k>p9ski1 ,h9du ,num]als 9 ! ,fihri/1 ,biblio!ca ,ma!matica1 ,vol4 ,,xi 7#c71 p4 #aba4 #eei ,woepcke1 ,sur une donnee hi/orique1 etc41 loc4 cit41 & ,essai d'une re/itution de travaux p]dus d',apollonius sur les quantites irrationnelles1 d'apres des indications tirees d'un manuscrit arabe1 ,tome ,,xiv des ,memoires pres5tes p> div]s savants a l',academie des sci5ces1 ,p>is1 #ahef1 note1 pp4 #f-ad4 #efj ,ar*eological ,report ( ! ,egypt ,exploration ,fund = #aijh-aiji1 ,london1 #aiaj1 p4 #ah4 #efa ,!re 0 a set ( a/ronomical tables 9 ,boncompagni's libr>y be>+ ?is date3 ,nota quod anno dni nri ihu xpi4 #abfd4 p]fecto4 ,see ,n>ducci's ,catalogo1 p4 #acj4 #efb ,on ! ,e>ly use ( ,arabic ,num]als 9 ,europe1 read be=e ! ,society ( ,antiqu>ies ,april #ad1 #aiaj1 & publi%$ 9 ,ar*aeologia 9 ! same ye>4 #efc ,ibid41 p4 #h1 n4 ,! date is "p ( an ,arabic inscription4 #efd ,o4 ,codr+ton1 ,a ,manual ( ,musalman ,numismatics1 ,london1 #aijd4 #efe ,see ,arbu?not1 ,! ,my/]ies ( ,*ronology1 ,london1 #aijj1 pp4 #ge1 #gh1 #ih2 ,f4 ,pi*l]1 ,rep]torium d] /ei]is*5 ,mu5zkunde1 ,graetz1 #ahge1 :]e ! claim is made ( an ,au/rian co9 ( #adeh2 ,biblio!ca ,ma!matica1 ,vol4 ,x 7#b71 p4 #abj1 & ,vol4 ,,xii 7#b71 p4 #abj4 ,!re is a ,brabant piece ( #adgh 9 ! collection ( ,d4 ,e4 ,smi?4 #eff ,a specim5 is 9 ! ,briti% ,museum4 ,arbu?not1 p4 #gi4 #efg ,ibid41 p4 #gi4 #efh ,lib] de ,rem$iis utriusque =tunae ,coloniae4 #efi ,fr4 ,wal!rn et ,hans ,hurn+1 ,no]dl+54 #egj ,ars ,memor&i1 "o ( ! olde/ ,european block-books4 #ega ,eusebius ,caes>i5sis1 ,de praep>atione evangelica1 ,v5ice1 ,j5son1 #adgj4 ,! above /atem5t holds = copies 9 ! ,a/or ,libr>y & 9 ! ,h>v>d ,univ]sity ,libr>y4 #egb ,francisco de ,retza1 ,come/orium vitiorum1 ,nu]nb]g1 #adgj4 ,! copy ref]r$ to is 9 ! ,a/or ,libr>y4 #egc ,see ,mau*1 ,ueb] d5 ,gebrau* arabis*] ,ziff]n und die ,v]a5d]ung5 d]selb51 ,anzeig] fu] ,kunde d] deuts*5 ,vorzeit1 #ahfa1 columns #df1 #ha1 #aaf1 #aea1 #ahi1 #bbi1 & #bfh2 ,calmet1 ,re*]*es sur l'orig9e des *iffres d'ari?metique1 plate1 loc4 cit4 #egd ,gu5!r1 ,ges*i*te1 p4 #age1 n42 ,mau*1 loc4 cit4 #ege ,^! are giv5 by ,w4 ,r4 ,le?aby1 f draw+s by ,j4 ,t4 ,irv9e1 9 ! ,proce$+s ( ! ,society ( ,antiqu>ies1 #aijf1 p4 #bjj4 #egf ,!re are "s ill-tabulat$ =ms to be f\nd 9 ,j4 ,b[r+1 ,! ,decimal ,sy/em1 ,london1 #ahed1 pp4 #bc1 #be1 & 9 ,l4 ,a4 ,*assant1 ,dictionnaire des abreviations lat9es et francaises 444 du moy5 age1 ,p>is1 ,,mdccclxvi1 p4 #aac4 ,! be/ s\rces we h at pres5t1 aside f ! ,hill monograph1 are ,p4 ,treutle91 ,ges*i*te uns]] ,zahlzei*51 ,k>lsruhe1 #ahge2 ,cantor's ,ges*i*te1 ,vol4 ,i1 table2 ,m4 ,pr\1 ,manuel de paleographie lat9e et francaise1 #b;d ed41 ,p>is1 #ahib1 p4 #afd2 ,a4 ,cappelli1 ,dizion>io di abbreviature lat9e ed italiane1 ,milan1 #ahii4 ,an int]e/+ e>ly s\rce is f\nd 9 ! r>e ,caxton "w ( #adhj1 ,! ,myrr\r ( ! ,_w4 ,9 ,*ap4 ,x is a cut ) ! v>i\s num]als1 ! *apt] beg9n+ ,! f\r? scy5ce is call$ arsmetrique4 ,two ( ! fifte5 extant copies ( ?is "w are at pres5t 9 ! libr>y ( ,mr4 ,j4 ,p4 ,morgan1 9 ,new ,york4 #egg ,f ! twelf?-c5tury manuscript on ari?metic1 ,curtze1 loc4 cit41 ,abh&lung51 & ,nagl1 loc4 cit4 ,! =ms are copi$ f ,plate ,,vii 9 ,zeits*rift fu] ,ma!matik und ,physik1 ,vol4 ,,xxxiv4 #egh ,f ! ,reg5sburg *ronicle4 ,plate conta9+ "s ( ^! num]als 9 ,monum5ta ,g]maniae hi/orica1 ,scriptores ,vol4 ,,xvii1 plate to p4 #ahd2 ,watt5ba*1 ,anleitung zur late9is*5 ,palaeographie1 ,leipzig1 #ahhf1 p4 #ajb2 ,boehm]1 ,fontes r]um ,g]manic>um1 ,vol4 ,,iii1 ,/uttg>t1 #aheb1 p4 lxv4 #egi ,fr5* ,algorismus ( #abge2 f an unpubli%$ photograph ( ! orig9al1 9 ! possession ( ,d4 ,e4 ,smi?4 ,see also p4 #ace4 #ehj ,f a manuscript ( ,boe?ius c4 #abid1 9 ,mr4 ,plimpton's libr>y4 ,smi?1 ,r>a ,ari?metica1 ,plate ,i4 #eha ,num]als 9 a #acjc manuscript 9 ,sigm>+51 copi$ f ,watt5ba*1 loc4 cit41 p4 #ajb4 #ehb ,f a manuscript1 ,add4 ,manuscript #bg1ehi1 ,briti% ,museum1 #acfj ,a4,d4 ,! "w is a computus 9 :i* ! date #acfj appe>s1 assign$ 9 ! ,briti% ,museum catalogue to ! ?irte5? c5tury4 #ehc ,f ! copy ( ,sacrabosco's ,algorismus 9 ,mr4 ,plimpton's libr>y4 ,date c4 #addb4 ,see ,smi?1 ,r>a ,ari?metica1 p4 #dej4 #ehd ,see ,r>a ,ari?metica1 pp4 #ddf-ddg4 #ehe ,ibid41 pp4 #dfi-dgj4 #ehf ,ibid41 pp4 #dgg-dgh4 #ehg ,! i is us$ = "o 9 ! ,treviso ari?metic 7#adgh71 ,cli*toveus 7c4 #aejg ed41 :]e bo? i & j are s us$71 ,*i>9i 7#adha71 ,sacrobosco 7#adhh ed471 & ,tzwivel 7#aejg ed41 :]e jj & jz are us$ = #aa & #ab74 ,?is 0 n univ]sal1 h[ev]1 = ! ,algori?mus l9ealis ( c4 #adhh has a special type = #a4 ,9 a /ud5t's notebook ( lectures tak5 at ! ,univ]sity ( ,wu]zburg 9 #affj1 9 ,mr4 ,plimpton's libr>y1 ! ones are all 9 ! =m ( i4 #ehh ,?us ! date ,num]als #aehj1 = #aehj1 appe>s 9 a ,,ms4 9 ! ,laur5tian libr>y at ,flor5ce4 ,! second & ! foll[+ five *>act]s are tak5 f ,cappelli's ,dizion>io1 p4 #chj1 & are f manuscripts ( ! twelf?1 ?irte5?1 f\rte5?1 sixte5?1 sev5te5?1 & eighte5? c5turies1 respectively4 #ehi ,e4g4 ,*i>9i's "w ( #adha2 ,cli*toveus 7c4 #aejg74 #eij ,! fir/ is f an algorismus ( ! ?irte5? c5tury1 9 ! ,hannov] ,libr>y4 ,see ,g]h>dt1 ,ueb] die ,ent/ehung und ,ausbreitung des dekadis*5 ,zahl5sy/ems1 loc4 cit41 p4 #bh4 ,! second "* is f a ,fr5* algorismus1 c4 #abge4 ,boncompagni ,bullet9o1 ,vol4 ,,xv1 p4 #ea4 ,! ?ird & ! foll[+ sixte5 *>act]s are giv5 by ,cappelli1 loc4 cit41 & are f manuscripts ( ! twelf? 7#a71 ?irte5? 7#b71 f\rte5? 7#g71 fifte5? 7#c71 sixte5? 7#a71 sev5te5? 7#b71 & eighte5? 7#a7 c5turies1 respectively4 #eia ,?us ,*i>9i 7#adha7 has ,symbol = #bc4 #eib ,! fir/ ( ^! is f a ,fr5* algorismus1 c4 #abge4 ,! second & ! foll[+ eight *>act]s are giv5 by ,cappelli1 loc4 cit41 & are f manuscripts ( ! twelf? 7#b71 ?irte5?1 f\rte5?1 fifte5? 7#c71 sev5te5?1 & eighte5? c5turies1 respectively4 #eic ,see ,nagl1 loc4 cit4 #eid ,hannov] algorismus1 ?irte5? c5tury4 #eie ,see ! ,dagom>i manuscript1 9 ,r>a ,ari?metica1 pp4 #dce1 #dcg-ddj4 #eif ,b 9 ! woodcuts ( ! ,m>g>ita ,philosophica 7#aejc7 ! old =ms are us$1 al?\gh ! new ones appe> 9 ! text4 ,9 ,caxton's ,myrr\r ( ! ,_w 7#adhj7 ! old =m is us$4 #eig ,cappelli1 loc4 cit4 ,!y are p>tly f manuscripts ( ! t5?1 twelf?1 ?irte5? 7#c71 f\rte5? 7#g71 fifte5? 7#f71 & eighte5? c5turies1 respectively4 ,^? 9 ! ?ird l9e are f ,*assant's ,dictionnaire1 p4 #aac1 )\t m5tion ( dates4 #eih ,! fir/ is f ! ,hannov] algorismus1 ?irte5? c5tury4 ,! second is tak5 f ! ,roll&us manuscript1 #adbd4 ,! o!rs 9 ! fir/ two l9es are f ,cappelli1 twelf? 7#c71 f\rte5? 7#f71 fifte5? 7#ac7 c5turies1 respectively4 ,! ?ird l9e is f ,*assant1 loc4 cit41 p4 #aac1 no m5tion ( dates4 #eii ,! fir/ ( ^! =ms is f ! ,hannov] algorismus1 ?irte5? c5tury4 ,! foll[+ are f ,cappelli1 f\rte5? 7#c71 fifte5?1 sixte5? 7#b71 & eighte5? c5turies1 respectively4 #fjj ,! fir/ ( ^! is tak5 f ! ,hannov] algorismus1 ?irte5? c5tury4 ,! foll[+ =ms are f ,cappelli1 twelf?1 ?irte5?1 f\rte5? 7#e71 fifte5? 7#b71 sev5te5?1 & eighte5? c5turies1 respectively4 #fja ,all ( ^! are giv5 by ,cappelli1 ?irte5?1 f\rte5?1 fifte5? 7#b71 & sixte5? c5turies1 respectively4 #fjb ,smi?1 ,r>a ,ari?metica1 p4 #dhi4 ,?is is also se5 9 sev]al ( ! ,plimpton manuscripts1 z 9 "o writt5 at ,ancona 9 #afhd4 ,see also ,cappelli1 loc4 cit4 #fjc ,fr5* algorismus1 c4 #abge1 = ! fir/ ( ^! =ms4 ,cappelli1 ?irte5?1 f\rte5?1 fifte5? 7#c71 & sev5te5? c5turies1 respectively4 ,! la/ ?ree are tak5 f ,byzant9is*e ,analekt51 ,j4 ,l4 ,heib]g1 be+ =ms ( ! fifte5? c5tury1 b n at all common4 ,symbol3 ,qoppa 0 ! old ,greek symbol = #ij4 #fjd ,= ! fir/ ( ^! ! read] is ref]r$ to ! =ms ascrib$ to ,boe?ius1 9 ! illu/ration on p4 #hh2 = ! second1 to ,radulph ( ,laon1 see p4 #fj4 ,! ?ird is us$ occasionally 9 ! ,roll&us 7#adbd7 manuscript1 9 ,mr4 ,plimpton's libr>y4 ,! rema9+ ?ree are f ,cappelli1 f\rte5? 7#b7 & sev5te5? c5turies4 #fje ,smi?1 ,an ,e>ly ,engli% ,algorism4 #fjf ,kuckuck1 p4 #e4 #fjg ,a4 ,cappelli1 loc4 cit41 p4 #cgb4 #fjh ,smi?1 ,r>a ,ari?metica1 p4 #ddc4 #fji ,curtze1 ,petri ,philom5i de ,dacia etc41 p4 ,,ix4 #faj ,cappelli1 loc4 cit41 p4 #cgf4 #faa ,curtze1 loc4 cit41 pp4 ,,viii-,,ix1 note4 #fab ,edition ( #aedd-aede1 f4 #eb4 #fac ,de num]is libri ,,ii1 #aedd ed41 cap4 ,,xv4 ,heilbronn]1 loc4 cit41 p4 #gcf1 also gives !m1 & comp>es ?is ) o!r sy/ems4 #fad ,noviomagus says ( !m3 ,de quibusdam ,a/rologicis1 sive ,*aldaicis num]orum notis4444 ,sunt aliae qua$am notae1 quibus ,*aldaei ,a/rologii quemlibet num]um artificiose argute describunt1 scitu p]iucundae1 quas nobis communicauit ,rodolphus ,paludanus ,n\iomagus4