,transcrib]'s ,note3 ,?is e-text includes *>act]s t w only display 9 ,,utf-#h 7,unicode7 text read]s3 1 7yogh1 long s7 1 7n ) curl1 cross$ l3 see bel[7 7,greek phi3 see bel[7 7small rais$ 8w07 ,if any ( ^! *>act]s d n display prop]ly1 or if ! apo/rophes & quotation m>ks 9 ?is p>agraph appe> z g>bage1 make sure y\r text read]'s 8"* set0 or 8file encod+0 is set to ,unicode 7,,utf-#h74 ,y may also ne$ to *ange ! default font4 ,9 ,! ,crafte ( ,nombrynge1 f9al 8n0 0 sometimes writt5 ) an extra curl4 ,x has be5 r5d]$ z = visual effect2 ! "* is n int5d$ to convey phonetic in=mation4 ,9 ! same selection1 ! num]al 8#j0 0 sometimes pr9t$ z ,greek 7phi72 ?is has be5 reta9$ = ! e-text4 ,d\ble 8l0 ) a l9e is %[n z 4 ,! fir/ few occurr5ces ( 8d0 7= 8p5ce07 7 pr9t$ ) a decorative curl4 ,! lett] is %[n ) ! same 8d'0 us$ 9 ! rema9d] ( ! text4 ,! ^w 8)draw0 or 8withdraw0 0 inconsi/5tly hyph5at$2 x 0 left z pr9t$1 & l9e-end hyph5s 7 reta9$4 ,sup]scripts are %[n ) c>ets z e4 ,except = ,illu/ration m>k]s & simil>1 all brackets are 9 ! orig9al4 ,individual lett]s 7 italiciz$ to %[ exp&$ abbreviations2 ^! are %[n 9 braces4 ,o!r italiciz$ words are %[n conv5tionally ) l9es1 boldface ) m>ks4 ,:5 a footnote call$ = add$ text1 ! addition is %[n 9 ! body text ) d\ble brackets4 ,! orig9al text conta9$ at lea/ five types ( m>g9al note4 ,details are giv5 at ! end ( ! e-text1 foll[$ by a li/+ ( typographical errors4 99 99 99 99 99 99 99 99 99 99 99 99 99 99 ,! ,e>lie/ ,ari?metics 9 ,engli% ,e>ly ,engli% ,text ,society4 ,extra ,s]ies1 ,no4 ,,cxviii4 #aibb 7= #aiaf74 ,,! ,,e>lie/ ,,ari?metics ,,9 ,,engli% ,edit$ ,) ,introduction by ,,rob]t ,,/eele ,london3 ,publi%$ = ! ,e>ly ,engli% ,text ,society ,by ,humphrey ,mil=d1 ,ox=d ,univ]sity ,press1 ,am5 ,corn]1 ,e4,c4 #d4 #aibb4 ,titles 7li/ add$ by transcrib]73 ,! ,crafte ( ,nombrynge ,! ,art ( ,nombryng ,accomptynge by ,c\nt]s ,! arte ( nombrynge by ! h&e ,,app4 ,i4 ,a ,treatise on ! ,num]ation ( ,algorism ,,app4 ,,ii4 ,c>m5 de ,algorismo ,,introduction ,! numb] ( ,engli% ari?metics be=e ! sixte5? c5tury is v small4 ,?is is h>dly to be wond]$ at1 z no "o requir+ to use ev5 ! simple/ op]ations ( ! art up to ! middle ( ! fifte5? c5tury 0 likely to be ignorant ( ,lat91 9 :i* language !re 7 sev]al treatises 9 a consid]able numb] ( manuscripts1 z %[n by ! quantity ( !m /ill 9 exi/5ce4 ,until mod]n comm]ce 0 fairly well e/abli%$1 few p]sons requir$ m ari?metic ?an addition & subtraction1 & ev5 9 ! ?irte5? c5tury1 sci5tific treatises address$ to advanc$ /ud5ts contemplat$ ! likelihood ( _! n be+ able to d simple division4 ,on ! o!r h&1 ! /udy ( a/ronomy necessitat$1 f its e>lie/ days z a sci5ce1 consid]able skill & accuracy 9 computation1 n only 9 ! calculation ( a/ronomical tables b 9 _! use1 a k ( :i* latt] 0 fairly common f ! ?irte5? to ! sixte5? c5turies4 ,! ari?metics 9 ,engli% kn[n to me are3-- 7#a7 ,bodl4 #gij ,g4 ,,vii4 7#bfec7 f4 #adf-aed 7#ae? c47 inc4 8,( angrym !r be ,,ix figures 9 numbray 4 4 40 ,a m]e unf9i%$ fragm5t1 only gett+ z f> z ,duplation4 7#b7 ,camb4 ,univ4 ,,li4 ,,iv4 #ad 7,,iii47 f4 #aba-adb 7#ae? c47 inc4 8,al man] ( ?yngis t pros$e? ffro ! fri/ begynnyng 4 4 40 7#c7 ,fragm5t>y passages or diagrams 9 ,sloane #bac f4 #abj-c 7a f\rte5?-c5tury c\nt+ bo>d71 ,eg]ton #bheb f4 #e-ac1 ,h>l4 #bah f4 #adg & 7#d7 ,! two ,,mss4 "h pr9t$2 ,eg4 #bfbb f4 #acf & ,a%mole #cif f4 #dh4 ,all ( ^!1 z ! language %[s1 are ( ! fifte5? c5tury4 ,! ,,crafte ,,( ,,nombrynge is "o ( a l>ge numb] ( sci5tific treatises1 mo/ly 9 ,lat91 b\nd up toge!r z ,eg]ton ,,ms4 #bfbb 9 ! ,briti% ,museum ,libr>y4 ,x measures #g0 #e01 #bi-cj l9es to ! page1 9 a r\gh h&4 ,! ,engli% is ,n4,e4 ,midl& 9 dialect4 ,x is a translation & amplification ( "o ( ! num]\s glosses on ! de algorismo ( ,alex&] de ,villa ,dei 7c4 #abbj71 su* z t ( ,?omas ( ,newm>ket conta9$ 9 ! ,briti% ,museum ,,ms4 ,reg4 #ab1 ,e4 #a4 ,a fragm5t ( ano!r translation ( ! same gloss 0 pr9t$ by ,halliwell 9 8 ,r>a ,ma!matica 7#ahce7 p4 #bi4a99 ,x corresponds1 z f> z p4 #ga1 l4 #b1 r\ghly to p4 #c ( our v]sion1 & f !nce to ! end p4 #b1 ll4 #af-dj4 ,footnote #a993 ,halliwell pr9t$ ! two sides ( 8 leaf 9 ! wrong ord]4 ,?is & "s obvi\s errors ( transcription-- #8f]ye' = #8f]!1' #8le/' = #8left1' etc41 h n be5 correct$ 9 ! repr9t on pp4 #gj-ga4 ,! ,,art ,,( ,,nombryng is "o ( ! treatises b\nd up 9 ! ,bodleian ,,ms4 ,a%mole #cif4 ,x measures #aa0 #ag01 & is writt5 ) ?irty-?ree l9es to ! page 9 a fifte5? c5tury h&4 ,x is a translation1 r lit]al1 ) amplifications ( ! de arte num]&i attribut$ to ,john ( ,holywood 7,sacrobosco7 & ! translator _h obvi\sly a poor ,,ms4 be=e him4 ,! de arte num]&i 0 pr9t$ 9 #adhh1 #adij 7s4n471 #aeja1 #aejc1 #aeaj1 #aeag1 #aeba1 #aebb1 #aebc1 #aehb1 & by ,halliwell sep>ately & 9 8 two editions ( ,r>a ,ma!matica1 #ahci & #ahda1 & repr9t$ by ,curze 9 #ahig4 ,bo? ^! tracts are "h pr9t$ = ! fir/ "t1 b ! fir/ hav+ be5 circulat$ 9 pro( a numb] ( ye>s ago1 9 an endeav\r to discov] o!r manuscripts or p>ts ( manuscripts ( x1 ,dr4 ,david ,eug5e ,smi?1 misund]/&+ ! position1 pr9t$ "s pages 9 a curi\s transcript ) f\r facsimiles 9 ! ,ar*iv fr die ,ges*i*te d] ,naturwiss5s*aft5 und d] ,te*nik1 #aiji1 & invit$ ! sci5tific _w to take up ! 8n unpleasant task0 ( edit+ x4 ,,accomptynge ,,by ,,c\nt]s is repr9t$ f ! #aedc edition ( ,rob]t ,record's ,ari?metic1 pr9t$ by ,r4 ,wolfe4 ,x has be5 repr9t$ )9 ! la/ few ye>s by ,mr4 ,f4 ,p4 ,b>n>d1 9 8 "w on ,ca/+ ,c\nt]s4 ,x is ! e>lie/ ,engli% treatise we h on ?is v>iety ( ! ,abacus 7!re are ,lat9 ones ( ! end ( ! fifte5? c5tury71 b !re is little d\bt 9 my m9d t ?is me?od ( p]=m+ ! simple op]ations ( ari?metic is mu* old] ?an any ( ! p5 me?ods4 ,at ! end ( ! treatise !re foll[s a note on m]*ants' & auditors' ways ( sett+ d[n sums1 & la/ly1 a sy/em ( digital num]ation :i* seems ( great antiquity & almo/ _w-wide ext5sion4 ,aft] ! fragm5t already ref]r$ to1 ,i pr9t z an app5dix ! #8,c>m5 de ,algorismo' ( ,alex&] de ,villa ,dei 9 an enl>g$ & correct$ =m4 ,x 0 pr9t$ = ! fir/ "t by ,halliwell 9 ,r>a ,ma!ma?ica1 b ,i h add$ a numb] ( /anzas f v>i\s manuscripts1 select+ v>i\s read+s on ! pr9ciple t ! v]ses 7 made to scan1 aid$ by ! advice ( my fri5d ,mr4 ,v]non ,r5dall1 :o is n responsible = ! few d\btful l9es ,i h cons]v$4 ,?is poem is at ! base ( all o!r treatises on ! subject 9 m$ieval times1 b ,i am unable to indicate its s\rces4 ,,! ,,subject ,,matt]4 ,anci5t & m$ieval writ]s obs]v$ a di/9ction betwe5 ! ,sci5ce & ! ,art ( ,ari?metic4 ,! classical treatises on ! subject1 ^? ( ,euclid among ! ,greeks & ,boe?ius among ! ,lat9s1 are devot$ to ! ,sci5ce ( ,ari?metic1 b x is obvi\s t coeval ) practical ,a/ronomy ! ,art ( ,calculation mu/ h exi/$ & h made consid]able progress4 ,if e>ly treatises on ?is art exi/$ at all !y mu/1 almo/ ( necessity1 h be5 9 ,greek1 :i* 0 ! language ( sci5ce = ! ,romans z long z ,lat9 civilisation exi/$4 ,b 9 _! abs5ce x is safe to say t no involv$ op]ations 7 or c\ld h be5 c>ri$ out by means ( ! alphabetic notation ( ! ,greeks & ,romans4 ,specim5 sums h inde$ be5 con/ruct$ by mod]ns :i* %[ its possibility1 b x is absurd to ?9k t m5 ( sci5ce1 acqua9t$ ) ,egyptian me?ods & 9 possession ( ! abacus1#b99 7 unable to devise me?ods = its use4 ,footnote #b993 ,= ,egyptian use see ,h]odotus1 ii4 #cf1 ,plato1 de ,legibus1 ,,vii4 ,,! ,,pre-,,m$ieval ,,in/rum5ts ,,us$ ,,9 ,,calculation4 ,! foll[+ are kn[n3-- 7#a7 ,a flat poli%$ surface or tablets1 /rewn ) s&1 on :i* figures 7 inscrib$ ) a /ylus4 7#b7 ,a poli%$ tablet divid$ longitud9ally into n9e columns 7or m7 gr\p$ 9 ?rees1 ) :i* c\nt]s 7 us$1 ei!r pla9 or m>k$ ) signs d5ot+ ! n9e num]als1 etc4 7#c7 ,tablets or boxes conta9+ n9e grooves or wires1 9 or on :i* ran beads4 7#d7 ,tablets on :i* n9e 7or m7 horizontal l9es 7 m>k$1 ea* ?ird be+ m>k$ (f4 ,! only ,greek c\nt+ bo>d we h is ( ! f\r? class & 0 discov]$ at ,salamis4 ,x 0 engrav$ on a block ( m>ble1 & measures #e feet by #b4 ,its *ief "p consi/s ( elev5 p>allel l9es1 ! #crd1 #f?1 & #i? be+ m>k$ ) a cross4 ,ano!r section consi/s ( five p>allel l9es1 & !re are ?ree r[s ( ari?metical symbols4 ,?is bo>d c\ld only h be5 us$ ) c\nt]s 7calculi71 pref]ably unm>k$1 z 9 our treatise ( ,accomptynge by ,c\nt]s4 ,,classical ,,roman ,,me?ods ,,( ,,calculation4 ,we h pro( ( two me?ods ( calculation 9 anci5t ,rome1 "o by ! fir/ me?od1 9 :i* ! surface ( s& 0 divid$ into columns by a /ylus or ! h&4 ,c\nt]s 7calculi1 or lapilli71 :i* 7 kept 9 boxes 7loculi71 7 us$ 9 calculation1 z we le>n f ,horace's s*oolboys 7,sat4 #a4 vi4 #gd74 ,= ! s& see ,p]sius ,i4 #aca1 8,nec qui abaco num]os et secto 9 pulv]e metas scit risisse10 ,apul4 ,apolog4 #af 7pulvisculo71 ,m>t4 ,capella1 lib4 vii4 #c1 #d1 etc4 ,cic]o says ( an exp]t calculator 8eruditum attigisse pulv]em10 7de nat4 ,deorum1 ii4 #ah74 ,t]tullian calls a tea*] ( ari?metic 8primus num]orum ar5>ius0 7de ,pallio1 9 f9e74 ,! c\nt]s 7 made ( v>i\s mat]ials1 ivory pr9cipally1 8,adeo nulla uncia nobis e/ eboris1 etc40 7,juv4 ,,xi4 #aca71 sometimes ( preci\s metals1 8,pro calculis albis et nigris aureos arg5teosque habebat d5>ios0 7,pet4 ,arb4 ,satyricon1 #cc74 ,!re are1 h[ev]1 /ill 9 exi/5ce f\r ,roman c\nt+ bo>ds ( a k9d :i* does n appe> to come into lit]ature4 ,a typical "o is ( ! ?ird class4 ,x consi/s ( a numb] ( transv]se wires1 brok5 at ! middle4 ,on ! left h& portion f\r beads are /rung1 on ! "r "o 7or two74 ,! left h& beads signify units1 ! "r h& "o five units4 ,?us any numb] up to n9e c be repres5t$4 ,?is in/rum5t is 9 all ess5tials ! same z ! ,swanpan or ,abacus 9 use ?r\gh\t ! ,f> ,ea/4 ,! ,russian /*ota 9 use ?r\gh\t ,ea/]n ,europe is simpl] /ill4 ,! me?od ( us+ ?is sy/em is exactly ! same z t ( ,accomptynge by ,c\nt]s1 ! "r-h& five bead replac+ ! c\nt] betwe5 ! l9es4 ,,! ,,boe?ian ,,abacus4 ,betwe5 classical times & ! t5? c5tury we h little or no guidance z to ! art ( calculation4 ,boe?ius 7fif? c5tury71 at ! end ( lib4 ,,ii4 ( 8 ,geometria gives u a figure ( an abacus ( ! second class ) a set ( c\nt]s arrang$ )9 x4 ,x has1 h[ev]1 be5 cont5d$ ) great probability t ! :ole passage is a t5? c5tury int]polation4 ,z no rules are giv5 = its use1 ! *ief value ( ! figure is t x gives ! signs ( ! n9e numb]s1 kn[n z ! ,boe?ian 8apices0 or 8notae0 7f :5ce our ^w 8notation074 ,to ^! we %all return lat] on4 ,,! ,,abaci/s4 ,x w\ld seem probable t writ]s on ! cal5d> l ,b$e 7,a4,d4 #gba7 & ,help]icus 7,a4,d4 #ijc7 7 able to p]=m simple calculations2 ?\gh we are unable to guess _! me?ods1 & = ! mo/ "p !y 7 dep5d5t on tables tak5 f ,greek s\rces4 ,we h no e>ly m$ieval treatises on ari?metic1 till t[>ds ! end ( ! t5? c5tury we f9d a revival ( ! /udy ( sci5ce1 c5tr+ = u r\nd ! "n ( ,g]b]t1 :o became ,pope z ,sylve/] ,,ii4 9 #iii4 ,8 treatise on ! use ( ! ,abacus 0 writt5 7c4 #ihj7 to a fri5d ,con/ant9e1 & 0 fir/ pr9t$ among ! works ( ,b$e 9 ! ,basle 7#aefc7 edition ( 8 works1 ,i4 #aei1 9 a some:at enl>g$ =m4 ,ano!r t5? c5tury treatise is t ( ,abbo ( ,fleury 7c4 #ihh71 pres]v$ 9 sev]al manuscripts4 ,v few treatises on ! use ( ! ,abacus c be c]ta9ly ascrib$ to ! elev5? c5tury1 b f ! beg9n+ ( ! twelf? c5tury _! numb]s increase rapidly1 to judge by ^? t h be5 pres]v$4 ,! ,abaci/s us$ a p]man5t bo>d usually divid$ into twelve columns2 ! columns 7 gr\p$ 9 ?rees1 ea* column be+ call$ an 8arcus10 & ! value ( a figure 9 x repres5t$ a t5? ( :at x w\ld h 9 ! column to ! left1 z 9 our ari?metic ( position4 ,) ?is bo>d c\nt]s or jetons 7 us$1 ei!r pla9 or1 m probably1 m>k$ ) num]ical signs1 :i* ) ! e>ly ,abaci/s 7 ! 8apices10 ?\gh c\nt]s f classical times 7 sometimes m>k$ on "o side ) ! digital signs1 on ! o!r ) ,roman num]als4 ,two ivory discs ( ?is k9d f ! ,hamilton collection may be se5 at ! ,briti% ,museum4 ,g]b]t is said by ,ri*] to h made = ! purpose ( computation a ?\s& c\nt]s ( horn2 ! usual numb] ( a set ( c\nt]s 9 ! sixte5? & sev5te5? c5turies 0 a hundr$4 ,treatises on ! ,abacus usually consi/ ( *apt]s on ,num]ation expla9+ ! notation1 & on ! rules = ,multiplication & ,division4 ,addition1 z f> z x requir$ any rules1 came naturally "u ,multiplication1 :ile ,subtraction 0 involv$ 9 ! process ( ,division4 ,^! rules 7 all t 7 ne$$ 9 ,we/]n ,europe 9 c5turies :5 comm]ce h>dly exi/$1 & a/ronomy 0 unpractis$1 & ev5 !y 7 only requir$ 9 ! prep>ation ( ! cal5d> & ! assignm5ts ( ! royal ex*equ]4 ,9 ,engl&1 = example1 :5 ! hide develop$ f ! normal hold+ ( a h\sehold into ! unit ( taxation1 ! calculation ( ! geldage 9 ea* %ire requir$ a sum 9 division2 z we "k f ! fact t "o ( ! ,abaci/s proposes ! sum3 8,if #bjj m>ks are levi$ on ! c\nty ( ,essex1 :i* conta9s accord+ to ,hugh ( ,bocl& #bejj hides1 h[ mu* does ea* hide pay80#c99 ,ex*equ] me?ods up to ! sixte5? c5tury 7 f\nd$ on ! abacus1 ?\gh :5 we h details lat] on1 a diff]5t & simpl] =m 0 us$4 ,footnote #c993 ,see on ?is ,dr4 ,poole1 ,! ,ex*equ] 9 ! ,twelf? ,c5tury1 ,*ap4 ,,iii41 & ,hask9s1 ,eng4 ,hi/4 ,review1 #bg1 #aja4 ,! hidage ( ,essex 9 #aacj 0 #bcfd hides4 ,! great difficulty ( ! e>ly ,abaci/s1 ow+ to ! abs5ce ( a figure repres5t+ z]o1 0 to place _! results & op]ations 9 ! prop] columns ( ! abacus1 especially :5 do+ a division sum4 ,! *ief diff]5ces noticeable 9 _! works are 9 ! me?ods = ?is rule4 ,division 0 ei!r done directly or by means ( diff]5ces betwe5 ! divisor & ! next high] multiple ( t5 to ! divisor4 ,lat] ,abaci/s made a di/9ction betwe5 8iron0 & 8gold50 me?ods ( division4 ,! foll[+ are examples tak5 f a twelf? c5tury treatise4 ,9 foll[+ ! op]ations x mu/ be rememb]$ t a figure a/]isk$ repres5ts a c\nt] tak5 f ! bo>d4 ,a z]o is obvi\sly n ne$$1 & ! result may be writt5 d[n 9 words4 7a7 ,,multiplication4 #dfjj #bc4 ---------------------- ,?\s&s ------------------ ,h ,t ,u ,h ,t ,u u e n u e n n n i n n i d s t d s t r s r s e e d d s s ------------------ #d #f ,multiplic&4 ------------------ #a #h #fjj #c4 #a #b #djjj #c4 #a #b #fjj #bj4 #h #djjj #bj4 ------------------ #a #e #h ,total product4 ------------------ #b #c ,multipli]4 ------------------ 7b7 ,,division3 ,,direct4 #ajj1jjj #bj1jbc4 ,"h ea* c\nt] 9 turn is a sep>ate divisor4 ---------------------- ,?\s&s ------------------ ,h4 ,t4 ,u4 ,h4 ,t4 ,u4 ------------------ #b #b #c ,divisors4 ------------------ #b ,place greate/ divisor to "r ( divid5d4 #a ,divid5d4 #b ,rema9d]4 #a #a #i #i ,ano!r =m ( same4 #h ,product ( #a/ ,quoti5t & #bj4 ------------------ #a #i #i #b ,rema9d]4 #a #b ,product ( #a/ ,quoti5t & #c4 ------------------ #a #i #i #h ,f9al rema9d]4 #d ,quoti5t4 ------------------ 7c7 ,,division ,,by ,,diff]5ces4 #ijj #h4 ,"h we divide by 7#aj-b74 -------------------- ,h4 ,t4 ,u4 -------------------- #b ,diff]5ce4 #h ,divisor4 -------------------- #d99i ,divid5d4 #d99a #h ,product ( diff]5ce by #a/ ,quoti5t 7#i74 #b ,product ( diff]5ce by #bnd ,quoti5t 7#a74 -------------------- #d99a ,sum ( #h & #b4 #b ,product ( diff]5ce by #crd ,quoti5t 7#a74 #d ,product ( diff]5ce by #d? ,quot4 7#b74 ,rema9d]4 -------------------- #b #d? ,quoti5t4 #a #crd ,quoti5t4 #a #bnd ,quoti5t4 #i #a/ ,quoti5t4 -------------------- #a #a #b ,quoti5t4 7,total ( all f\r47 -------------------- ,footnote #d993 ,^! figures are remov$ at ! next /ep4 ,,division4 #ghjj #aff4 ------------------------------ ,?\s&s -------------------------- ,h4 ,t4 ,u4 ,h4 ,t4 ,u4 -------------------------- #c #d ,diff]5ces 7mak+ #bjj trial divisor74 #a #f #f ,divisors4 -------------------------- #d99g #h ,divid5ds4 #a ,rema9d] ( greate/ divid5d4 #a #b ,product ( #a/ diff]5ce 7#d7 by #a/ ,quoti5t 7#c74 #i ,product ( #bnd diff]5ce 7#c7 by #a/ ,quoti5t 7#c74 -------------------------- #d99b #h #b ,new divid5ds4 #c #d ,product ( #a/ & #bnd diff]5ce by #bnd ,quoti5t 7#a74 -------------------------- #d99a #a #f ,new divid5ds4 #b ,product ( #a/ diff]5ce by #crd ,quoti5t 7#e74 #a #e ,product ( #bnd diff]5ce by #crd ,quoti5t 7#e74 -------------------------- #d99c #c ,new divid5ds4 #a ,rema9d] ( greate/ divid5d4 #c #d ,product ( #a/ & #bnd diff]5ce by #d? ,quoti5t 7#a74 -------------------------- #a #f #d ,rema9d] 7less ?an divisor74 #a #d? ,quoti5t4 #e #crd ,quoti5t4 #a #bnd ,quoti5t4 #c #a/ ,quoti5t4 -------------------------- #d #f ,quoti5t4 -------------------------- ,footnote #d993 ,^! figures are remov$ at ! next /ep4 ,,division4 #hjjj #fjf4 ------------------------ ,?\s&s -------------------- ,h4 ,t4 ,u4 ,h4 ,t4 ,u4 -------------------- #i ,diff]5ce 7mak+ #gjj trial divisor74 #d ,diff]5ce4 #f #f ,divisors4 -------------------- #d99h ,divid5d4 #a ,rema9d] ( divid5d4 #i #d ,product ( diff]5ce #a & #b ) #a/ ,quoti5t 7#a74 -------------------- #d99a #i #d ,new divid5ds4 #c ,rema9d] ( greate/ divid5d4 #i #d ,product ( diff]5ce #a & #b ) #bnd ,quoti5t 7#a74 -------------------- #d99a #c #c #d ,new divid5ds4 #c ,rema9d] ( greate/ divid5d4 #i #d ,product ( diff]5ce #a & #b ) #crd ,quoti5t 7#a74 -------------------- #g #b #h ,new divid5ds4 #f #f ,product ( divisors by #d? ,quoti5t 7#a74 -------------------- #a #b #b ,rema9d]4 #a #d? ,quoti5t4 #a #crd ,quoti5t4 #a #bnd ,quoti5t4 #a #a/ ,quoti5t4 -------------------- #a #c ,quoti5t4 -------------------- ,footnote #d993 ,^! figures are remov$ at ! next /ep4 ,! *ief ,abaci/s are ,g]b]t 7t5? c5tury71 ,abbo1 & ,h]mannus ,contractus 7#ajed71 :o are cr$it$ ) ! revival ( ! art1 ,b]nel9us1 ,g]l&1 & ,radulphus ( ,laon 7twelf? c5tury74 ,we "k z ,engli% ,abaci/s1 ,rob]t1 bi%op ( ,h]e=d1 #ajie1 8abacum et lun>em compotum et cele/ium cursum a/rorum rimatus10 ,tur*illus ,compoti/a 7,?urkil71 & ?r\gh him ( ,guilielmus ,r4 4 4 4 8! be/ ( liv+ comput]s10 ,gisleb]t1 & ,simonus de ,rotellis 7,simon ( ! ,rolls74 ,!y fl\ri%$ mo/ probably 9 ! fir/ qu>t] ( ! twelf? c5tury1 z ,?urkil's treatise deals also ) fractions4 ,wal*] ( ,durham1 ,?omas ( ,york1 & ,samson ( ,worce/] are also kn[n z ,abaci/s4 ,f9ally1 ! t]m ,abaci/s came to be appli$ to comput]s by manual ari?metic4 ,a ,,ms4 ,algori?m ( ! ?irte5? c5tury 7,sl4 #cbha1 f4 #f1 b71 conta9s ! foll[+ passage3 8,e/ et alius modus secundum op]atores sive practicos1 quorum unus appellatur ,abacus2 et modus ejus e/ 9 comput&o p] digitos et junctura manuum1 et i/e utitur ultra ,alpes40 ,9 a composite treatise conta9+ tracts writt5 ,a4,d4 #aaeg & #abjh1 on ! cal5d>1 ! abacus1 ! manual cal5d> & ! manual abacus1 we h a numb] ( ! me?ods pres]v$4 ,z an example we give ! rule = multiplication 7,claud4 ,a4 ,,iv41 f4 #ed vo74 8,si num]us multiplicat alium num]um auf]atur diff]5tia majoris a m9ore1 et p] residuum multiplicetur articulus1 et una diff]5tia p] aliam1 et summa prov5iet40 ,example1 #h #g4 ,! diff]5ce ( #h is #b1 ( #g is #c1 ! next article be+ #aj2 #g - #b is #e4 #e #aj #ej2 #b #c #f4 #ej #f #ef answ]4 ,! rule w hold 9 su* cases z #ag #ae :]e ! article next high] is ! same = bo?1 i4e41 #bj2 b 9 su* a case z #ag #i ! diff]5ce = ea* numb] mu/ be tak5 f ! high] article1 i4e41 ! diff]5ce ( #i w be #aa4 ,,! ,,algori/s4 ,algorism 7augrim1 augrym1 algram1 agram1 algori?m71 owes its "n to ! accid5t t ! fir/ ari?metical treatise translat$ f ! ,arabic happ5$ to be "o writt5 by ,al-,kh[>azmi 9 ! e>ly n9? c5tury1 8de num]is ,indorum10 beg9n+ 9 its ,lat9 =m 8,dixit ,algorismi4 4 4 40 ,! translation1 ( :i* only "o ,,ms4 is kn[n1 0 made ab\t #aabj by ,adel>d ( ,ba?1 :o also wrote on ! ,abacus & translat$ ) a comm5t>y ,euclid f ! ,arabic4 ,x is probable t ano!r v]sion 0 made by ,g]>d ( ,cremona 7#aaad-aahg72 ! numb] ( important works t 7 n translat$ m ?an once f ! ,arabic decreases e ye> ) our k ( m$ieval texts4 ,a few l9es ( ?is translation1 z copi$ by ,halliwell1 are giv5 on p4 #gb1 note #b4 ,ano!r translation /ill seems to h be5 made by ,johannes ,hispal5sis4 ,algorism is di/+ui%$ f ,abaci/ computation by recognis+ sev5 rules1 ,addition1 ,subtraction1 ,duplation1 ,m$iation1 ,multiplication1 ,division1 & ,extraction ( ,roots1 to :i* 7 aft]w>ds add$ ,num]ation & ,progression4 ,x is fur!r di/+ui%$ by ! use ( ! z]o1 :i* enabl$ ! comput] to disp5se ) ! columns ( ! ,abacus4 ,x obvi\sly employs a bo>d ) f9e s& or wax1 & lat]1 z a sub/itute1 pap] or p>*m5t2 slate & p5cil 7 also us$ 9 ! f\rte5? c5tury1 h[ mu* e>li] is unkn[n4#e99 ,algorism quickly ou/$ ! ,abacus me?ods = all intricate calculations1 be+ simpl] & m easily *eck$3 9 fact1 ! a/ronomical revival ( ! twelf? & ?irte5? c5turies w\ld h be5 impossible )\t its aid4 ,footnote #e993 ,slates are m5tion$ by ,*auc]1 & soon aft] 7#adaj7 ,prosdocimo de ,beldam&i speaks ( ! use ( a 8lapis0 = mak+ notes on by calculators4 ,! numb] ( ,lat9 ,algorisms /ill 9 manuscript is comp>atively l>ge1 b we are "h only conc]n$ ) two--an ,algorism 9 prose attribut$ to ,sacrobosco 7,john ( ,holywood7 9 ! colophon ( a ,p>is manuscript1 ?\gh ?is attribution is no long] reg>d$ z conclusive1 & ano!r 9 v]se1 mo/ probably by ,alex&] de ,vill$ieu 7,villa ,dei74 ,alex&]1 :o di$ 9 #abdj1 0 tea*+ 9 ,p>is 9 #abji4 ,8 v]se treatise on ! ,cal5d> is dat$ #abjj1 & x is to t p]iod t 8 ,algorism may be attribut$2 ,sacrobosco di$ 9 #abef & quotes ! v]se ,algorism4 ,sev]al comm5t>ies on ,alex&]'s v]se treatise 7 compos$1 f "o ( :i* our fir/ tractate 0 translat$1 & ! text itself 0 f "t to "t enl>g$1 sections on pro(s & on m5tal ari?metic be+ add$4 ,we h no indication ( ! s\rce on :i* ,alex&] drew2 x 0 mo/ likely "o ( ! translations ( ,al-,kh[>asmi1 b he has also ! ,abaci/s 9 m9d1 z %ewn by pres]v+ ! use ( diff]5ces 9 multiplication4 ,8 treatise1 fir/ pr9t$ by ,halliwell-,phillipps 9 8 ,r>a ,ma!matica1 is adapt$ = use on a bo>d cov]$ ) s&1 a me?od almo/ univ]sal 9 ! ?irte5? c5tury1 z "s passages 9 ! algorism ( t p]iod already quot$ %[3 8,e/ et alius modus qui utitur apud ,indos1 et doctor hujusmodi ipsos erat quidem nom9e ,algus4 ,et modus suus erat 9 comput&o p] quasdam figuras scrib5do 9 pulv]e4 4 4 40 8,si volu]imus dep+]e 9 pulv]e pr$ictos digitos secundum consuetud9em algorismi 4 4 40 8et sci5dum e/ quod 9 nullo loco m9utorum sive secundorum 4 4 4 9 pulv]e deb5t scribi plusquam sexag9ta40 ,,mod]n ,,ari?metic4 ,mod]n ,ari?metic beg9s ) ,leon>di ,fibonacci's treatise 8de ,abaco10 writt5 9 #abjb & re-writt5 9 #abbh4 ,x is mod]n r 9 ! range ( its problems & ! me?ods ( attack ?an 9 m]e me?ods ( calculation1 :i* are ( its p]iod4 ,its sole int]e/ z reg>ds ! pres5t "w is t ,leon>di makes use ( ! digital signs describ$ 9 ,record's treatise on ,! arte ( nombrynge by ! h& 9 m5tal ari?metic1 call+ x 8modus ,indorum40 ,leon>do also introduces ! me?od ( pro( by 8ca/+ out ! n9es40 ,,digital ,,ari?metic4 ,! me?od ( indicat+ numb]s by means ( ! f+]s is ( consid]able age4 ,! ,briti% ,museum possesses two ivory c\nt]s m>k$ on "o side by c>elessly scrat*$ ,roman num]als ,,iiiv & ,,viiii1 & on ! o!r by c>efully engrav$ digital signs = #h & #i4 ,sixte5 seems to h be5 ! numb] ( a complete set4 ,^! c\nt]s 7 ei!r us$ 9 games or = ! c\nt+ bo>d1 & ! ,museum ones1 com+ f ! ,hamilton collection1 are und\bt$ly n lat] ?an ! fir/ c5tury4 ,frohn] has publi%$ 9 ! ,zeits*rift des ,mn*5] ,alt]?umsv]e9s a set1 almo/ complete1 ( !m ) a ,byzant9e treatise2 a ,lat9 treatise is pr9t$ among ,b$e's works4 ,! use ( ?is me?od is univ]sal ?r\gh ! ,ea/1 & a v>iety ( x is f\nd among _m ( ! native races 9 ,africa4 ,9 m$ieval ,europe x 0 almo/ re/rict$ to ,italy & ! ,m$it]ranean bas91 & 9 ! treatise already quot$ 7,sloane #cbha7 x is ev5 call$ ! ,abacus1 p]haps a memory ( ,fibonacci's "w4 ,me?ods ( calculation by means ( ^! signs und\bt$ly h exi/$1 b !y 7 too involv$ & liable to error to be mu* us$4 ,,! ,,use ,,( 8,,arabic0 ,,figures4 ,x may n[ be reg>d$ z prov$ by ,bubnov t our pres5t num]als are d]iv$ f ,greek s\rces ?r\gh ! s-call$ ,boe?ian 8apices10 :i* are fir/ f\nd 9 late t5? c5tury manuscripts4 ,t !y 7 n d]iv$ directly f ! ,arabic seems c]ta9 f ! diff]5t %apes ( "s ( ! num]als1 especially ! #j1 :i* /&s = #e 9 ,arabic4 ,ano!r ,greek =m exi/$1 :i* 0 introduc$ into ,europe by ,john ( ,bas+/oke 9 ! ?irte5? c5tury1 & is figur$ by ,mat!w ,p>is 7,v4 #bhe72 b ?is =m _h no success4 ,! date ( ! introduction ( ! z]o has be5 hotly debat$1 b x seems obvi\s t ! twelf? c5tury ,lat9 translators f ! ,arabic 7 p]fectly well acqua9t$ ) ! sy/em !y met 9 _! ,arabic text1 :ile ! e>lie/ a/ronomical tables ( ! ?irte5? c5tury ,i h se5 use numb]s ( ,european & n ,arabic orig94 ,! fact t ,lat9 writ]s _h a conv5i5t way ( writ+ hundr$s & ?\s&s )\t any cyph]s probably delay$ ! g5]al use ( ! ,arabic notation4 ,dr4 ,hill has publi%$ a v complete survey ( ! v>i\s =ms ( num]als 9 ,europe4 ,!y began to be common at ! middle ( ! ?irte5? c5tury & a v int]e/+ set ( family notes conc]n+ bir?s 9 a ,briti% ,museum manuscript1 ,h>l4 #dcej %[s _! ext5sion4 ,! fir/ is dat$ ,mijc4 lviii41 ! second ,mijc4 lxi41 ! ?ird ,mijc4 #fc1 ! f\r? #abfd1 & ! fif? #abff4 ,ano!r example is giv5 9 a set ( a/ronomical tables = #abfi 9 a manuscript ( ,rog] ,bacon's works1 :]e ! scribe began to write ,,mcc#f4 & cross$ out ! figures1 sub/itut+ ! 8,arabic0 =m4 ,,! ,,c\nt+ ,,bo>d4 ,! treatise on pp4 #eb-fe is ! only "o 9 ,engli% kn[n on ! subject4 ,x describes a me?od ( calculation :i*1 ) slight modifications1 is curr5t 9 ,russia1 ,*9a1 & ,japan1 to-day1 ?\gh x w5t out ( use 9 ,we/]n ,europe by ! sev5te5? c5tury4 ,9 ,g]many ! me?od is call$ 8,algori?mus ,l9ealis10 & !re are sev]al editions ( a tract "u ?is "n 7) a diagram ( ! c\nt+ bo>d71 pr9t$ at ,leipsic at ! end ( ! fifte5? c5tury & ! beg9n+ ( ! sixte5?4 ,!y give ! n9e rules1 b 8,capitulum de radicum extractione ad algoritmum integrorum res]vato1 cujus species p] ciffrales figuras o/5duntur ubi ad pl5um de hac tractabitur40 ,! inv5tion ( ! art is !re attribut$ to ,appulegius ! philosoph]4 ,! advantage ( ! c\nt+ bo>d1 :e!r p]man5t or con/ruct$ by *alk+ p>allel l9es on a table1 z %[n 9 "s sixte5?-c5tury woodcuts1 is t only five c\nt]s are ne$$ to indicate ! numb] n9e1 c\nt]s on ! l9es repres5t+ units1 & ^? 9 ! spaces above repres5t+ five times ^? on ! l9e bel[4 ,! ,russian abacus1 ! 8t*atui0 or 8/*ota0 has t5 beads on ! l9e2 ! ,*9ese & ,japanese 8,swanpan0 economises by divid+ ! l9e into two p>ts1 ! beads on "o side repres5t+ five times ! value ( ^? on ! o!r4 ,! 8,swanpan0 has usually _m m l9es ?an ! 8/*ota10 all[+ = m ext5d$ calculations1 see ,tylor1 ,an?ropology 7#ahib71 p4 #cad4 ,record's treatise also m5tions ano!r me?od ( c\nt] notation 7p4 #fd7 8m]*ants' ca/+0 & 8auditors' ca/+40 ,^! 7 adapt$ = ! usual ,engli% me?od ( reckon+ numb]s up to #bjj by scores4 ,?is me?od seems to h be5 us$ 9 ! ,ex*equ]4 ,a c\nt+ bo>d = m]*ants' use is pr9t$ by ,halliwell 9 ,r>a ,ma!matica 7p4 #gb7 f ,sloane ,,ms4 #bac1 & two o!rs are figur$ 9 ,eg]ton #bfbb f4 #hb & f4 #hc4 ,! latt] is said to be 8novus modus comput&i secundum inv5tionem ,magi/ri ,?ome ,?orleby10 & is 9 pr9ciple1 ! same z ! 8,swanpan40 ,! ,ex*equ] table is describ$ 9 ! ,dialogus de ,scacc>io 7,ox=d1 #aijb71 p4 #ch4 ,! ,e>lie/ ,ari?metics 9 ,engli%4 ,! ,crafte ( ,nombrynge ,eg]ton #bfbb4 99leaf #acf;a ,hec algorismu ars pres5s dicitur2 9 qua ,talibu indorum fruimur bis quinque figuris4 ,sid5ote3 ,a d]ivation ( ,algorism4 ,ano!r d]ivation ( ! ^w4 ,?is boke is call$ e boke ( algorym1 or ,augrym after lewder vse4 ,& is boke tretys e ,craft ( ,nombryng1 e quy* crafte is call$ also ,algorym4 ,!r 0 a kyng ( ,inde1 e qui* hey? ,algor1 he made is craft4 ,& after 8 "n he call$ hit algorym2 or els anoer cause is quy x is call$ ,algorym1 = e latyn ^w ( hit s4 ,algorismu comes ( ,algos1 grece1 quid e/ ars1 lat9e1 craft o englis1 & rides1 quid e/ numerus1 lat9e1 ,a nombur o englys1 inde dicitur ,algorismu per addici"om huiu sillabe mu subtraccionem d e1 quasi ars numer&i4 f=!rmore e mo/ vndir/onde at 9 is craft b5 vsid te5 figurys1 z "h b5e writ5 = ensampul1 #i #h #g #f #e #d #c #b #a4 ,expone e too versus a=e3 ?is pres5t craft ys call$ ,algorismu1 9 e quy* we vse te5 signys ( ,inde4 ,que/io4 ,:y te fyguris ( ,inde8 ,solucio4 = z ,i haue sayd a=e ai w]e fonde fyr/ 9 ,inde ( a kynge ( at ,cuntre1 at 0 call$ ,algor4 ,headnote3 ,notation & ,num]ation4 ,sid5ote3 versus 9 m>g94 ,prima significat unum2 duo vero secunda3 ,t]cia significat tria2 sic proc$e s9i/re4 ,donec ad extremam v5ias1 que cifra vocatur4 ,capitulum primum de significacione figur>um4 ,sid5ote3 ,expositio versus4 ,sid5ote3 ,! mean+ & place ( ! figures4 ,:i* figure is read fir/4 ,9 is v]se is notifide e significacion ( ese figuris4 ,& u expone ! v]se4 e fir/ signifiy? one1 e secunde 99leaf #acf;b signi99fiy? tweyne1 e ?ryd signifiy? ?re1 ! f\rte signifiy? #d4 ,& s =! t[>de e lyft syde ( e tabul or ( e boke at e figures b5e writ5e 91 til at ou come to ! la/ figure1 at is call$ a cifre4 ,que/io4 ,9 quy* syde sittes e fir/ figure8 ,solucio1 =so! loke qui* figure is fir/ 9 e ryt side ( e bok or ( e tabul1 at same is e fir/ figure1 = ou s*al write bakew>d1 z "h1 #c4 #b4 #f4 #d4 #a4 #b4 #e4 ,! figure ( #e4 0 fir/ write1 he is e fir/1 = he sittes o e rit syde4 ,& ! figure ( #c is la/4 ,neuer-e-les w5 he says ,prima significat vnum c41 at is to say1 e fir/ betok5es one1 e secunde4 #b4 =e-er-more1 he vndir/ondes not ( e fir/ figure ( euery rew4 ,b he vndir/ondes e fir/ figure at is 9 e nombur ( e =sayd te5 figuris1 e quy* is one ( ese4 #a4 ,& e secunde #b4 s =?4 ,sid5ote3 versus 9 m>g94 ,quelibet ill>um si primo limite ponas1 ,simpliciter se significat3 si vero secund1 ,se decies3 sursum proc$as multiplic&o4 ,namque figura sequens quamuis signat decies plu4 ,ipsa locata loco quam significat pert9ente4 ,transcrib]'s ,note3 ,9 ! foll[+ section1 num]als %[n 9 m>ks 7 pr9t$ 9 a diff]5t font1 possibly z facsimiles ( ! orig9al ,,ms =m4 ,sid5ote3 ,expositio 9 m>g94 ,sid5ote3 ,an explanation ( ! pr9ciples ( notation4 ,an example3 units1 t5s1 hundr$s1 ?\s&s4 ,h[ to read ! numb]4 ,expone is verse u4 ,euery ( ese figuris bitok5s hym selfe no more1 yf he /onde 9 e fir/ place ( e rewele _/ ?is worde ,simpliciter 9 at v]se x is no m to say b at1 no more4 ,if x /onde 9 ! secunde place ( e rewle1 he betok5s t5e tymes hym selfe1 z is figure #b "h #bj tok5s t5 tyme hym selfe1 99leaf #acg;a at is tw5ty1 = he hym selfe betok5es tweyne1 t5 tymes tw5e is tw5ty4 ,& = he /ondis o e lyft side 9 e secunde place1 he betok5s t5 tyme hym selfe4 ,& s g =?4 f= euery figure1 he /onde aftur a-noer t[>d ! lyft side1 he s*al betok5e t5 tymes z mi* more z he s*ul betok5 he /ode 9 e place ere at e figure a-=e hym /ondes4 loo an ensampulle4 #i4 #f4 #c4 #d4 e figure ( #d4 at hase is s*ape #d4 betok5s bot hymselfe1 = he /ondes 9 e fir/ place4 ,! figure ( #c4 at hase is s*ape #c4 betok5s t5 tymes more en he s*uld he /ode ere at e figure ( #d4 /ondes1 at is ?retty4 ,! figure ( #f1 at hase is s*ape #f1 betok5s t5 tymes more an he s*uld he /ode ere z e figure ( #c4 /ondes1 = ere he s*uld tokyne bot sexty1 n[ he betok5s t5 tymes more1 at is sex hundry?4 ,! figure ( #i4 at hase is s*ape #i4 betok5s t5 tymes more ane he s*uld he /ode 9 e place ere e figure ( sex /ondes1 = en he s*uld betok5 to #i4 hundry?1 & 9 e place ere he /ondes n[ he betok5s #i4 ous&e4 ,al e hole nombur is #i ?\s&e sex hundry? f\re ?retty4 f=!rmore1 :5 ou s*alt r$e a nombur ( figure1 ou s*alt begyne at e la/ figure 9 ! lyft side1 r$e s =? to e rit side z h]e #i4 #f4 #c4 #d4 ,?\ s*al begyn to r$e at e figure ( #i4 r$e =? u4 #i4 99leaf #acg;b ?\s& sex hundry? ?ritty f\re4 ,b :5 ou s*alle write1 ou s*alt be-gynne to write at e ryt side4 ,nil cifra significat sed dat signare sequ5ti4 ,sid5ote3 ,! mean+ & use ( ! ciph]4 ,expone is verse4 ,a cifre tok5s not1 bot he makes e figure to betok5 at comes aftur hym more an he s*uld he w]e away1 z u #a4 h]e e figure ( one tok5s t51 yf e cifre w]e away#a no figure by-=e hym he s*uld tok5 bot one1 = an he s*uld /onde 9 e fir/ place4 ,& e cifre tok5s no?yng hym selfe4 = al e nombur ( e ylke too figures is bot t54 ,que/io4 ,:y says he at a cifre makys a figure to signifye 7tyf7 more c4 ,i speke = is worde significatyf1 f= so! x may happe aftur a cifre s*uld come a-nour cifre1 z u #b4 ,& et e secunde cifre %uld tok5 neuer e more excep he s*uld kepe e order ( e place4 & a cifre is no figure significatyf4 ,quam prec$5tes plus ultima significabit _/ ,sid5ote3 ,! la/ figure means m ?an all ! o!rs1 s9ce x is ( ! highe/ value4 ,expone is verse u4 e la/ figure s*al tok5 more an alle e oer a=e1 ?\t ere w]e a hundry? ?\sant figures a=e1 z u1 #afgih4 e la/ figure at is #a4 betok5s t5 ?\sant4 ,& alle e oer figures ben bot betok5e bot sex ?\sant seuyne hundry? nynty #h4 ,& t5 ?\sant is more en alle at nombur1 erg e la/ figure tok5s more an all e nombur a=e4 ,headnote3 ,! ,?ree ,k9ds ( ,numb]s 99leaf #ach;a ,po/ predicta scias breuiter quod tres numerorum ,di/9cte species sunt2 nam quidam digiti sunt2 ,articuli quidam2 quidam quoque compositi sunt4 ,capitulum #bm de triplice divisione num]orum4 ,sid5ote3 ,digits4 ,articles4 ,composites4 ,! auctor ( is tretis departys is worde a nombur into #c partes4 ,"s nombur is call$ digitu lat9e1 a digit 9 englys4 ,smme nombur is call$ articulu lat9e4 ,an ,articul 9 englys4 ,"s nombur is call$ a composyt 9 englys4 ,expone is verse4 "k ou aftur e =sayd rewles at ,i sayd a=e1 at ere b5 ?re spices ( nombur4 ,oone is a digit1 ,anoer is an ,articul1 e toer a ,composyt4 versus4 ,headnote3 ,digits1 ,articles1 & ,composites4 ,sunt digiti numeri qui citra d5>ium sunt4 ,sid5ote3 ,:at are digits4 ,h]e he telles qwat is a digit1 ,expone versus sic4 ,nomburs digitus b5e alle nomburs at b5 with-inne t51 z nyne1 #h4 #g4 #f4 #e4 #d4 #c4 #b4 #a4 ,articupli decupli degitorum2 compositi sunt ,illi qui con/ant ex articulis degitisque4 ,sid5ote3 ,:at are articles4 ,h]e he telles :at is a composyt & :at is ane articul4 ,expone sic versus4 ,articulis b5#b alle at may be deuidyt into nomburs ( t5 no?ynge leue ouer1 z tw5ty1 ?retty1 f\rty1 a hundry?1 a ?\s&1 su* oer1 f= tw5ty may be departyt 9-to #b nomburs ( t51 f=ty 9 to f\re nomburs ( t51 s =?4 ,sid5ote3 ,:at numb]s are composites4 99leaf #ach;b ,compositys be nomburs at b5e componyt ( a digyt ( an articulle z f\ret5e1 fyft5e1 sext5e1 su* oer4 f=t5e is componyd ( f\re at is a digit ( t5 at is an articulle4 ffift5e is componyd ( #e t51 s ( all oer1 :at at ai b54 ,%ort-ly* euery nombur at be-gynnes with a digit endy? 9 a articulle is a composyt1 z =t5e byg5nynge by f\re at is a digit1 endes 9 t54 ,ergo1 proposito num]o tibi scrib]e1 primo ,respicias quid sit num]us2 si digitus sit ,primo scribe loco digitum1 si compositus sit ,primo scribe loco digitum po/ articulum2 sic4 ,sid5ote3 ,h[ to write a numb]1 if x is a digit2 if x is a composite4 ,h[ to read x4 "h he telles h[ ou s*alt wyr* :an ou s*alt write a nombur4 ,expone versum sic1 fac iuxta exponentis s5t5ciam2 :an ou ha/ a nombur to write1 loke fyr/ :at maner nombur x ys at ou s*alt write1 :e!r x be a digit or a composit or an ,articul4 ,if he be a digit1 write a digit1 z yf x be seu51 write seu5 write at digit 9 e fir/ place t[>d e ryght side4 ,if x be a composyt1 write e digit ( e composit 9 e fir/ place write e articul ( at digit 9 e secunde place next t[>d e lyft side4 ,z yf ou s*al write sex tw5ty4 write e digit ( e nombur 9 e fir/ place at is sex1 & write e articul next aftur at is tw5ty1 z u #bf4 ,b :an ou s*alt s[ne or speke 99leaf #aci;a or r$e an ,composyt ou s*alt fir/ s[ne e articul aftur e digit1 z ou sey/ by e comyne spe*e1 ,sex tw5ty n\t tw5ty sex4 versus4 ,articulu si sit1 9 primo limite cifram1 ,articulum v]o reliquis inscribe figuris4 ,sid5ote3 ,h[ to write ,articles3 t5s1 hundr$s1 ?\s&s1 c4 ,"h he tells h[ ou s*al write :5 e nombre at ou hase to write is an ,articul4 ,expone versus sic fac secundum s5t5ciam4 ,ife e nombur at ou ha/ write be an ,articul1 write fir/ a cifre aftur e cif] write an ,articulle u4 #b4 f=!rmore ou s*alt vndir/onde yf ou haue an ,articul1 loke h[ my* he is1 yf he be with-ynne an hundry?1 ou s*alt write bot one cifre1 a=e1 z h]e 4#i4 ,if e articulle be by hym-silfe be an hundrid eu5e1 en s*al ou write 4#a4 #b cif]s a=e1 at he may /onde 9 e ?ryd place1 = euery figure 9 e ?ryd place s*al tok5 a hundrid tymes hym selfe4 ,if e articul be a ?\sant or ?\s&es#c & he /onde by hym selfe1 write a=e #c cif]s s = ( al oer4 ,quolibet 9 num]o1 si p> sit prima figura1 ,p> erit totum1 quicquid sibi continuatur2 ,impar si fuerx1 totum tunc fiet et imp>4 ,sid5ote3 ,to tell an ev5 numb] or an odd4 ,h]e he te*es a g5eralle rewle at yf e fir/ figure 9 e rewle ( figures tok5 a nombur at is eu5e al at nombur ( figurys 9 at rewle s*al be eu5e1 z h]e ou may see #f4 #g4 #c4 #e4 #d4 ,computa proba4 ,if e fir/ 99leaf #aci;b figure tok5 an nombur at is ode1 alle at nombur 9 at rewle s*alle be ode1 z h]e #e #f #g #h #f #g4 ,computa proba4 versus4 ,septem sunt p>tes1 non plures1 i/ius artis2 ,add]e1 subtrah]e1 dupl>e1 dimidi>e1 ,sextaque diuid]e1 sed quinta multiplic>e2 ,radicem extrah]e pars septima dicitur esse4 ,headnote3 ,! ,sev5 ,rules ( ,ari?metic4 ,sid5ote3 ,! sev5 rules4 ,h]e telles at er be 4#g4 spices or partes ( is craft4 ,! fir/ is call$ addicio1 e secunde is call$ subtraccio4 ,! ?ryd is call$ duplacio4 ,! #d4 is call$ dimydicio4 ,! #e4 is call$ multiplicacio4 ,! #f is call$ diuisio4 ,! #g4 is call$ extraccio ( e ,rote4 ,:at all ese spices b5e hit s*alle be tolde s+illatim 9 h]e caputule4 ,subtra8 aut addis a dextris vel m$iabis3 ,sid5ote3 ,add1 subtract1 or halve1 f "r to left4 ,?\ s*al be-gynne 9 e ryght side ( e boke or ( a tabul4 loke w]e ou wul be-gynne to write latyn or englys 9 a boke1 at s*alle be call$ e lyft side ( ! boke1 at ou write/ t[>d at side s*al be call$ e ryght side ( e boke4 ,versus4 ,a leua dupla1 diuide1 multiplica4 ,sid5ote3 ,multiply or divide f left to "r4 ,"h he telles e 9 quy* side ( e boke or ( e tabul ou s*alle be-gyne to wyr* duplacio1 diuisio1 & multiplicacio4 ,?\ s*al begyne to wor* 9 e lyft side ( e boke or ( e tabul1 b yn :at wyse ou s*al wyr* 9 hym dicetur s+illatim 9 sequentibu capitulis et de vtilitate cuiulibet artis sic ,completur 99leaf #adj4 prohemium sequitur tractatu primo de arte addicionis que prima ars e/ 9 ord9e4 ,headnote3 ,! ,craft ( ,addition4 ,add]e si num]o numerum vis1 ord9e tali ,incipe2 scribe duas primo s]ies num]orum ,primam sub prima recte ponend figuram1 ,et sic de reliquis facias1 si s9t tibi plures4 ,sid5ote3 ,f\r ?+s mu/ be kn[n3 :at x is2 h[ _m r[s ( figures2 h[ _m cases2 :at is its result4 ,h[ to set d[n ! sum4 ,h]e by-gynnes e craft ( ,addicio4 ,9 is craft ou mo/ kn[e f\re ?ynges4 ,fyr/ ou mo/ "k :at is addicio4 ,next ou mo/ "k h[ mony rewles ( figurys ou mo/ haue4 ,next ou mo/ "k h[ mony diuers casys happes 9 is craft ( addicio4 ,& next qwat is e profet ( is craft4 ,z = e fir/ ou mo/ "k at addicio is a ca/yng to-g$ur ( twoo nomburys 9-to one nombre4 ,z yf ,i aske qwat is tw5e ?re4 ou wyl ca/ ese tw5e nombres to-g$ur say at x is fyue4 ,z = e secunde ou mo/ "k at ou s*alle haue tweyne rewes ( figures1 one vndur a-no!r1 z h]e ou may/ se4 #abcd #bafh4 ,z = e ?ryd ou mo/ "k at !re b5 f\re diuerse cases4 ,z = e =! ou mo/ "k at e profet ( is craft is to telle :at is e hole nombur at comes ( diuerse nomburis4 ,n[ z to e texte ( oure v]se1 he te*es !re h[ ou s*al wor* 9 is craft4 ,he says yf ou wilt ca/ one nombur to anoer nombur1 ou mo/ by-gynne on is wyse4 ffyr/ write 99leaf #adj;b two rewes ( figuris nombris s at ou write e fir/ figure ( e hy] nombur eu5e vndir ! fir/ figure ( e ne!r nombur1 ,& e secunde ( e ne!r nombur eu5e vndir e secunde ( e hy]1 s =! ( euery figure ( bo? e rewes z ou may/ se4 #abc #bcd4 ,headnote3 ,! ,cases ( ! ,craft ( ,addition4 ,inde duas adde primas hac condici"o3 ,si digitus crescat ex addici"o priorum2 ,primo scribe loco digitum1 quicunque sit ille4 ,sid5ote3 ,add ! fir/ figures2 rub out ! top figure2 write ! result 9 its place4 ,"h is an example4 ,"h he te*es :at ou s*alt d :5 ou ha/ write too rewes ( figuris on vnd] an-oer1 z ,i sayd be-=e4 ,he says ou s*alt take e fir/ figure ( e hey] nombre e fyr/ figure ( e neer nombre1 ca/ hem to-g$er vp-on is condicio4 ,?\ s*al loke qweer e nomber at comys ere-( be a digit or no4 ,if he be a digit ou s*alt d away e fir/ figure ( e hy] nombre1 & write ere 9 8 /$e at he /ode ,inne e digit1 at comes ( e ylke #b figures1 s wri* =? o oer figures yf ere be ony moo1 til ou come to e ende t[>d e lyft side4 ,& l$e e ne!r figure /onde /ill euer-more til ou haue ydo4 f= ere-by ou s*al wyte :eer ou ha/ done wel or no1 z ,i s*al tell e afterw>d 9 e ende ( is ,*apter4 ,& loke allgate at ou be-gynne to wor* 9 is ,craft ( 99leaf #ada;a ,addi99cio 9 e ryt side1 "h is an ensampul ( is case4 #abcd #badb4 ,ca/e #b to f\re at wel be sex1 d away #d4 write 9 e same place e figure ( sex4 ,& lete e figure ( #b 9 e ne!r rewe /onde /il4 ,:5 ou ha/ d s1 ca/ #c #d to-g$ur & at wel be seu5 at is a digit4 ,d away e #c1 set ere seue1 & lete e neer figure /onde /ille1 s wor* =? bakw>d til ou ha/ ydo all to-g$er4 ,et si compositu1 9 limite scribe sequente ,articulum1 primo digitum2 quia sic iubet ordo4 ,sid5ote3 ,suppose x is a ,composite1 set d[n ! digit1 & c>ry ! t5s4 ,"h is an example4 ,"h is e secunde case at may happe 9 is craft4 ,& e case is is1 yf ( e ca/+ ( #b nomburis to-g$er1 z ( e figure ( e hy] rewe ( e figure ( e neer rewe come a ,composyt1 h[ s*alt ou wor*4 u ou s*alt wor*4 ,?\ %alt d away e figure ( e hy] nomber at 0 ca/ to e figure ( e neer nomb]4 ,& write ere e digit ( e ,composyt4 ,& set e articul ( e composit next after e digit 9 e same rewe1 yf ere be no mo figures after4 ,b yf ere be mo figuris after at digit4 ,& ere he s*all be rek5d = hym selfe4 ,& :5 ou s*alt adde at ylke figure at b]ys e articulle ouer 8 h$ to e figure vnder hym1 ou s*alt ca/ at articul to e figure at hase hym ouer 8 h$1 ere at ,articul s*al toke hym selfe4 lo an ,ensampull 99leaf #ada;b ( all4 #cbf #baf4 ,ca/ #f to #f1 ere-( wil arise twelue4 d away e hy] #f write ere #b1 at is e digit ( is composit4 ,& en write e articulle at is t5 ouer e figuris h$ ( tw5e z u4 #a #cbb #baf4 ,n[ ca/ e articulle at /&us vpon e figuris ( tw5e h$ to e same figure1 rek5 at articul bot = one1 & an ere wil arise ?re4 an ca/ at ?re to e neer figure1 at is one1 at wul be f\re4 d away e figure ( #c1 & write ere a figure ( f\re4 & lete e neer figure /onde /il1 an wor* =?4 vnde v]sus4 ,articulus si sit1 9 primo limite cifram1 ,articulum vero reliquis inscribe figuris1 ,vel per se scribas si nulla figura sequatur4 ,sid5ote3 ,suppose x is an ,article1 set d[n a ciph] & c>ry ! t5s4 ,"h is an example4 ,h]e he puttes e ?ryde case ( e craft ( ,addicio4 e case is is4 yf ( ,addici\ ( #b figuris a-ryse an ,articulle1 h[ s*al ou d4 ?\ mo/ d away e he] figure at 0 addid to e neer1 write ere a cifre1 & sett e articuls on e figuris h$e1 yf at ere come ony after4 ,& wyr* an z ,i haue tolde e 9 e secunde case4 ,an ensampull4 #be4 #ae ,ca/ #e to #e1 at wylle be t54 n[ d away e hy] #e1 write ere a cif]4 ,& sette t5 vpon e figuris h$ ( #b4 ,& rek5 x b = on u4 lo an ,ensampulle ---- #a #b #ae ---- ,& 99leaf #adb;a an wor* =?4 ,b yf ere come no figure after e cifre1 write e articul next hym 9 e same rewe z "h --- #e #e --- ca/ #e to #e1 & x wel be t54 d away #e4 at is e hi] #e4 & write ere a cifre1 write after hym e articul z u ---- #a #e ---- ,& an ou ha/ done4 ,si tibi cifra superu5i5s occurr]it1 illam ,dele superpositam2 fac illic scribe figuram1 ,po/ea proc$as reliquas add5do figuras4 ,sid5ote3 ,:at to d :5 y h a ciph] 9 ! top r[4 ,an example ( all ! difficulties4 ,h]e he puttes e f\rt case1 x is is1 at yf ere come a cif] 9 e hi] rewe1 h[ ou s*al d4 u ou s*alt d4 d away e cif]1 sett ere e digit at comes ( e addici\n z u #ahd4 #aggdc ,9 is ensampul b5 alle e f\re cases4 ,ca/ #c to f\re1 at wol be seue4 d away #d4 write ere seue2 an ca/ #d to e figure ( #h4 at wel be #ab4 d away #h1 sett ere #b4 at is a digit1 & sette e articul ( e composit1 at is t51 vpon e cif]s h$1 rek5 x = hym selfe at is on4 an ca/ one to a cif]1 hit wulle be b on1 = not on makes b one4 an ca/ #g4 at /ondes vnder at on to hym1 at wel be #h4 d away e cif] at #a4 sette ere #h4 an g =!rmore4 ca/ e oer #g to e cif] at /ondes ouer hym4 at wul be bot seu51 = e cif] betok5s not4 d away e cif] sette ere seue1 99leaf #adb;b en g =ermore ca/ #a to #a1 at wel be #b4 d away e hi] #a1 sette ere #b4 an ha/ ou d4 ,& yf ou haue wel ydo is nomb] at is sett h]e-after wel be e nomb] at s*alle aryse ( alle e addicio z h]e #bghbg4 ,sequitur alia species4 ,headnote3 ,! ,craft ( ,subtraction4 ,a num]o numerum si sit tibi dem]e cura ,scribe figur>um s]ies1 vt 9 addicione4 ,sid5ote3 ,f\r ?+s to "k ab\t subtraction3 ! fir/2 ! second2 ! ?ird2 ! f\r?4 ,?is is e ,*apter ( subtraccio1 9 ! quy* ou mo/ "k f\re nessess>y ?ynges4 ! fir/ :at is subtraccio4 e secunde is h[ mony nomb]s ou mo/ haue to subtraccio1 ! ?ryd is h[ mony man]s ( cases ere may happe 9 is craft ( subtraccio4 ,! f\rte is qwat is e profet ( is craft4 ,z = e fir/1 ou mo/ "k at subtraccio is drawynge ( one n[mber oute ( anoer nomb]4 ,z = e secunde1 ou mo/ kn[e at ou mo/ haue two rewes ( figuris one vnder anoer1 z ou addy/ 9 addicio4 ,z = e ?ryd1 ou moy/ "k at f\re maner ( diuerse casis mai happe 9 is craft4 ,z = e f\rt1 ou mo/ "k at e profet ( is craft is :5ne ou hasse tak5 e lasse nomb] out ( e more to telle :at ere leues ouer at4 ou mo/ be-gynne to wyr* 9 is craft 9 e ryght side ( e boke1 z ou diddy/ 9 addicio4 ,versus4 ,maiori num]o numerum suppone m9orem1 ,siue p>i num]o supponatur numeru p>4 ,sid5ote3 ,put ! great] numb] above ! less4 99leaf #adc;a ,h]e he telles at e hi] nomb] mo/ be more en e neer1 or els eue z my*4 b he may n be lasse4 ,& e case is is1 ou s*alt drawe e neer nomb] out ( e hy]1 ou may/ n d at yf e hi] nomb] w]e lasse an at4 f= ou may/ n draw sex out ( #b4 ,b ou ma/ draw #b out ( sex4 ,& ou mai/e draw tw5e out ( tw5e1 = ou s*al leue not ( e hi] tw5e vnde versus4 ,headnote3 ,! ,cases ( ! ,craft ( ,subtraction4 ,po/ea si possis a prima subtrahe primam ,scrib5s quod remanet4 ,sid5ote3 ,! fir/ case ( subtraction4 ,"h is an example4 ,h]e is e fir/ case put ( subtraccio1 he says ou s*alt begynne 9 e ryght side1 draw e fir/ figure ( e neer rewe out ( e fir/ figure ( e hi] rewe4 qwe!r e hi] figure be more en e neer1 or eue z my*4 ,& at is notifi$ 9 e v]s :5 he says 8,si possis40 ,:an ou has u ydo1 d away e hie/ figure sett ere at leues ( e subtraccio1 lo an ,ensampulle ----- #bcd #abb ----- draw #b out ( #d4 an leues #b4 d away #d write ere #b1 latte e neer figure /onde /ille1 s g =-by oer figuris till ou come to e ende1 an ha/ ou d4 ,cifram si nil remanebit4 ,sid5ote3 ,put a ciph] if no?+ rema9s4 ,"h is an example4 ,h]e he puttes e secunde case1 hit is is4 yf x happe at qw5 ou ha/ draw on neer figure out ( a hi]1 ere leue not after e subtraccio1 u 99leaf #adc;b ou s*alt d4 ou s*alle d away e hi] figure write ere a cif]1 z lo an ,ensampull ---- #bd #bd ---- ,take f\re out ( f\re an leus not4 ere=e d away e hi] #d set ere a cif]1 an take #b out ( #b1 an leues not4 d away e hi] #b1 set ere a cif]1 & s wor* :>e s euer is happe4 ,s$ si non possis a prima demere primam ,prec$5s vnum de limite deme sequente1 ,quod demptum pro d5>io reputabis ab illo ,subtrahe totalem numerum quem proposui/i ,quo facto scribe sup] quicquid remanebit4 ,sid5ote3 ,suppose y _c take ! l[] figure f ! top "o1 borr[ t52 take ! l[] numb] f t52 add ! answ] to ! top numb]4 ,h[ to #8,pay back' ! borr[$ t54 ,example4 ,h]e he puttes e ?ryd case1 e quy* is is4 yf x happe at e neer figure be more en e hi] figure at he s*alle be draw out (4 h[ s*alle ou d4 u ou s*alle d4 ou s*alle borro 4#a4 oute ( e next figure at comes after 9 e same rewe1 = is case may neuer happ b yf ere come figures after4 an ou s*alt sett at on ouer e hi] figures h$1 ( ! quy* ou woldi/ y-draw oute e neyer figure yf ou haddy/ y-myt4 ,:ane ou hase u ydo ou s*alle rek5e at 4#a4 = t54 4 ,& out ( at t5 ou s*al draw e neyermo/ figure1 ,& alle at leues ou s*alle adde to e figure on :os h$ at 4#a4 /ode4 ,& en ou s*alle d away alle at1 sett ere alle t arisys ( ! addicio ( e ylke #b figuris4 ,& yf yt 99leaf #add;a happe at e figure ( e quy* ou s*alt borro on be hym self b #a4 ,if ou s*alt at one sett x vppo e oer figuris h$1 & sett 9 at #a4 place a cif]1 yf ere come mony figures after4 lo an ,ensampul4 ------ #babb #aacd ------ take #d out ( #b4 x wyl n be1 er=e borro one ( e next figure1 at is #b4 & sett at ouer e h$ ( e fyr/ #b4 rek5e x = t54 & ere e secunde /ondes write #a4 = ou toke/ on out ( hym4 an take e neer figure1 at is #d1 out ( t54 ,& en leues #f4 ca/ to #f e figure ( at #b at /ode vnder e h$de ( #a4 at 0 borw$ rek5$ = t51 & at wylle be #h4 d away at #f at #b1 sette ere #h1 lette e neer figure /onde /ille4 ,:anne ou ha/ d u1 g to e next figure at is n[ bot #a4 b fir/ yt 0 #b1 ere-( 0 borr$ #a4 an take out ( at e figure vnder hym1 at is #c4 hit wel n be4 er-=e bor[e ( ! next figure1 e quy* is bot #a4 ,also take sett hym ouer e h$e ( e figure at ou wolde/ haue y-draw oute ( e ne!r figure1 e quy* 0 #c4 ou myt n1 rek5e at borw$ #a = t5 sett 9 e same place1 ( e quy* place ou toke/ hym (1 a cif]1 = he 0 bot #a4 ,:anne ou ha/ u ydo1 take out ( at #a4 at is rek5t = t51 e neer figure ( #c4 ,& ere leues #g4 99leaf #add;b ca/ e ylke #g to e figure at _h e ylke t5 vpon 8 h$1 e quy* figure 0 #a1 at wol be #h4 an d away at #a & at #g1 write ere #h4 an wyr* =? 9 oer figuris til ou come to e ende1 an ou ha/ e d4 ,versus4 ,facque non5>ios de cifris1 cum remeabis ,occurrant si =te cifre2 dum dempseris vnum ,po/ea proc$as reliquas demend figuras4 ,sid5ote3 ,a v h>d case is put4 ,"h is an example4 ,h]e he puttes e f\rte case1 e quy* is is1 yf x happe at e neer figure1 e quy* ou s*alt draw out ( e hi] figure be more pan e hi] figur ouer hym1 e next figure ( two or ( ?re or ( f\re1 or h[ mony ere be by cif]s1 h[ wold ou d4 ou wo/ wel ou mo/ n$e bor[1 ou may/ n bor[ ( e cif]s1 = ai haue not at ai may l5e or sp>e4 ,ergo#d h[ wolde/ ou d4 ,c]tay u mo/ ou d1 ou mo/ bor[ on ( e next figure significatyf 9 at rewe1 = is case may n happe1 b yf ere come figures significatyf after ! cif]s4 ,:an ou ha/ bor[$e at #a ( ! next figure significatyf1 sett at on ouer e h$e ( at figure ( e quy* ou wold haue draw e neer figure out yf ou hade/ myt1 rek5 x = t5 z ou didde/ in e oer case h]e-a-=e4 ,:a ou ha/ u y-d loke h[ mony cif]s ere w]e bye-tw5e at figure significatyf1 e figure ( e quy* ou wolde/ haue y-draw ! 99leaf #ade;a neer figure1 & ( euery ( e ylke cif]s make a figure ( #i4 lo an ,ensampulle aft]4 ----- #djjjb #ajjjd ----- ,take #d out ( #b4 x wel n be4 bor[ #a out ( be next figure significatyf1 e quy* is #d1 en leues #c4 d away at figure ( #d write ere #c4 sett at #a vppon e figure ( #b h$e1 an take #d out ( t51 an ere leues #f4 ,ca/ #f to ! figure ( #b1 at wol be #h4 d away at #f write ere #h4 ,:an ou ha/ u y-d make ( euery #j betweyn #c #h a figure ( #i1 an wor* =? 9 goddes "n4 yf ou ha/ wel y-d ou#e s*alt haue is nomber ----- #ciiih ,sic4 #ajjjd ----- ,headnote3 ,h[ to prove ! ,subtraction4 ,si subtraccio sit bene facta prob>e valebis ,quas subtraxi/i primas add5do figuras4 ,sid5ote3 ,h[ to prove a subtraction sum4 ,"h is an example4 ,he works 8 pro( ?r\gh1 & br+s out a result4 ,h]e he te*es e ,craft h[ ou s*alt "k1 :an ou ha/ subtrayd1 :eer ou ha/ wel ydo or no4 ,& e ,craft is is1 ryght z ou subtrayd e neer figures fro e hi] figures1 ryt s adde e same neer figures to e hi] figures4 ,& yf ou haue well y-wro? a-=e ou s*alt haue e hi] nombre e same ou hadde/ or ou be-gan to wor*4 z = is ,i bade ou s*ulde kepe e neer figures /ylle4 lo an 99leaf #ade;b ,ensampulle ( alle e #d cases tog$re4 wor*e welle is case -------- #djjjcdfh4 #bjjjdffd -------- ,& yf ou wor* welle :an ou ha/ alle subtrayd e at hi] nombre h]e1 is s*alle be e nombre "h foloyng :an ou ha/ subtrayd4 -------- #ciiihhjd4 ,sid5ote3 ,our au?or makes a slip "h 7#c = #a74 #bjjjdffd -------- ,& ou s*alt "k u4 adde e neer r[e ( e same nombre to e hi] rewe z u1 ca/ #d to #d4 at wol be #h4 d away e #d write ere #h4 by e fir/ case ( addicio4 an ca/ #f to #j at wol be #f4 d away e #j1 write ere #f4 an ca/ #f to #h1 at wel be #ad4 d away #h write ere a figure ( #d1 at is e digit1 & write a figure ( #a4 at s*all be-tok5 t54 at is e articul vpon e h$ ( #h next after1 an rek5 at #a4 = #a4 ca/ x to #h4 at s*al be #i4 ca/ to at #i e neer figure vnder at e quy* is #d1 at s*alle be #ac4 d away at #i sett ere #c1 sett a figure ( #a4 at s*all be #aj vpon e next figuris h$e e quy* is #i4 by e secunde case at ou hade/ 9 addicio4 an ca/ #a to #i4 at wol be #aj4 d away e #i4 at #a4 ,& write ere a cif]4 & write e articulle at is #a4 betok5ynge #aj4 vpon e h$e ( e next figure t[>d e lyft side1 e quy* 99leaf #adf;a is #i1 s d =? tyl ou come to e la/ #i4 take e figure ( at #a4 e quy* ou s*alt fynde ouer e h$ ( #i4 sett x ouer e next figures h$e at s*al be #c4 ,also d away e #i4 set ere a cif]1 en ca/ at #a at /ondes vpon e h$e ( #c to e same #c1 at s*alle make #d1 en ca/e to e ylke #d ! figure 9 e neyer rewe1 e quy* is #b1 & at s*alle be #f4 ,& en s*al ou haue an ,ensampulle aey1 loke se1 b ou haue is same ou hase myse-wrot4 -------- #fjjjcdfh #bjjjdffd -------- ,sequitur de duplaci"o ,headnote3 ,! ,craft ( ,duplation4 ,si vis dupl>e numerum1 sic incipe primo ,scribe figurarum s]iem quamcunque velis tu4 ,sid5ote3 ,f\r ?+s mu/ be kn[n 9 ,duplation4 ,"h !y are4 ,m9d :]e y beg94 ,rememb] y\r rules4 ,?is is ! ,*apture ( duplacio1 9 e quy* craft ou mo/ haue "k #d ?+es4 e fir/ at ou mo/ "k is :at is duplacio4 e secunde is h[ mony rewes ( figures ou mo/ haue to is craft4 e ?ryde is h[ _m cases may#f happe 9 is craft4 e f\rte is :at is e profet ( e craft4 ,z = e fir/4 duplacio is a d\blynge ( a nombre4 ,z = e secunde ou mo/ 99leaf #adf;b haue on nombre or on rewe ( figures1 ! quy* call$ num]us dupl&u4 ,z = e ?rid ou mo/ "k at #c diuerse cases may hap 9 is craft4 ,z = e f\rte4 qwat is e profet ( is craft1 at is to "k :at a-risyt ( a nombre ,i-d\blyde4 f=er-more1 ou mo/ "k take gode h$e 9 quy* side ou s*alle be-gyn 9 is craft1 or ellis ou may/ spyl alle i laber ere ab\te4 certeyn ou s*alt begy 9 ! lyft side 9 is ,craft4 !nke wel ouer is v]se4 #g;,a leua dupla1 diuide1 multiplica4#g ,subtrahas aut addis a dextris vel m$iabis ,! s5t5s ( es v]ses a=e1 z ou may see if ou take h$e4 ,z e text ( is v]se1 at is to say1 ,si vis dupl>e4 is is e s5t5ce4 ,if ou wel d\ble a nombre u ou mo/ be-gyn4 ,write a rewe ( figures ( :at nombre ou welt4 versus4 ,po/ea proc$as primam dupl&o figuram ,inde quod excrescit scribas vbi iuss]it ordo ,iuxta precepta tibi que dantur 9 addici"o4 ,sid5ote3 ,h[ to "w a sum4 ,h]e he telles h[ ou s*alt wor* 9 is ,craft4 he says1 fyr/1 :an ou ha/ writ5 e nombre ou s*alt be-gyn at e fir/ figure 9 ! lyft side1 d\bulle at figure1 e nombre at comes ere-( ou s*alt write z ou diddy/ 9 addicio1 z ,i s*al telle e 9 e case4 versus4 ,headnote3 ,! ,cases ( ! ,craft ( ,duplation4 99leaf #adg;a ,nam si sit digitus 9 primo limite scribas4 ,sid5ote3 ,if ! answ] is a digit1 write x 9 ! place ( ! top figure4 ,h]e is e fir/ case ( is craft1 e quy* is is4 yf ( duplacio ( a figure arise a digit4 :at s*al ou d4 u ou s*al d4 d away e figure at 0 d\bl$e1 sett ere e diget at comes ( e duplacio1 z u4 #bc4 d\ble #b1 at wel be #d4 d away e figure ( #b sett ere a figure ( #d1 s wor* =? tille ou come to e ende4 versus4 ,articulu si sit1 9 primo limite cifram1 ,articulum vero reliquis inscribe figuris2 ,vel per se scribas1 si nulla figura sequatur4 ,sid5ote3 ,if x is an article1 put a ciph] 9 ! place1 & #8c>ry' ! t5s4 ,if !re is no figure to #8c>ry' !m to1 write !m d[n4 ,"h is e secunde case1 e quy* is is yf ere come an articulle ( e duplacio ( a figure ou s*alt d ryt z ou diddy/ 9 addicio1 at is to wete at ou s*alt d away e figure at is d\blet sett ere a cif]1 write e articulle ouer e next figuris h$e1 yf ere be any after-w>de t[>d e lyft side z u4 #be4 begyn at ! lyft side1 & d\bulle #b4 at wel be #d4 d away at #b sett ere #d4 an d\bul #e4 at wel be #aj4 d away #e1 sett ere a #j1 sett #a vpon e next figuris h$e e quy* is #d4 en draw d[ne #a to #d at wolle be #e1 en d away at #d at #a1 sett ere #e4 = at #a s*al be rek5$ 9 e drawynge tog$re = #a4 w5 99leaf #adg;b ou ha/ ydon ou s*alt haue is nombre #ej4 yf ere come no figure after e figure at is addit1 ( e quy* addicio comes an articulle1 ou s*alt d away e figure at is d[blet sett ere a #j4 write e articul next by 9 e same rewe t[>d e lyft syde z u1 #ebc4 d\ble #e at woll be t54 d away e figure #e set ere a cif]1 sett e articul next after 9 e same rewe t[>d e lyft side1 ou s*alt haue is nombre #ajbc4 en g =? d\ble e oer nomb]s e quy* is lyt y-n[t to d4 versus4 ,compositus si sit1 9 limite scribe sequ5te ,articulum1 primo digitum2 quia sic iubet ordo3 ,et sic de reliquis faciens1 si s9t tibi plures4 ,sid5ote3 ,if x is a ,composite1 write d[n ! digit1 & #8c>ry' ! t5s4 ,"h is an example4 ,h]e he puttes e ,?ryd case1 e quy* is is1 yf ( duplacio ( a figure come a ,composit4 ou s*alt d away e figure at is d\blet set ere a digit ( e ,composit1 sett e articulle ouer e next figures h$e1 after draw hym d[ne with e figure ouer :os h$e he /ondes1 make ere-( an nombre z ou ha/ done a=e1 yf ere come no figure after at digit at ou ha/ y-write1 an set e articulle next after hym 9 e same rewe z u1 #fg3 d\ble #f at wel be #ab1 d away #f write ere e digit 99leaf #adh;a ( #ab1 e quy* is #b1 & set e articulle next after t[>d e lyft side 9 e same rewe1 = ere comes no figure after4 an d[ble at oer figure1 e quy* is #g1 at wel be #ad4 ! quy* is a ,composit4 en d away #g at ou d\blet sett e e diget ( hym1 ! quy* is #d1 sett e articulle ouer e next figures h$1 e quy* is #b1 en draw to hym at on1 make on nombre e quy* s*alle be #c4 ,& en yf ou haue wel y-d ou s*alle haue is nombre ( e duplacio1 #acd4 versus4 ,si sup] extremam nota sit monadem dat eidem ,quod tibi cont+at si primo dimidiabis4 ,sid5ote3 ,h[ to d\ble ! m>k = "o-half4 ,?is c only /& ov] ! fir/ figure4 ,h]e he says1 yf ouer e fyr/ figure 9 e ryt side be su* a m]ke z is h]e made1 1 ou s*alle fyr/ d\bulle e figure1 ! quy* /ondes vnder at m]ke1 en ou s*alt d\bul at m]ke e quy* /ondes = halu5del on4 = too halu$els makes on1 s at wol be on4 ca/ at on to at duplacio ( e figure ouer :os h$ /ode at m]ke1 write x 9 e same place ere at e figure e quy* 0 d\blet /ode1 z u #bc4 d\ble #c1 at wol be #f2 d\bul at halue on1 at wol be on4 ca/ on to #f1 at wel be #g4 d away #f at #a1 sett ere #g4 an hase ou d4 z = at figure1 an g 99leaf #adh;b to e oer figure wor* =?4 ou s*all neuer haue su* a m]k b ouer e h$ ( e fur/ figure 9 e ryght side4 ,& et x s*al n happe b yf x 7 y-halu$ a-=e1 u ou s*alt vnder/onde e v]se4 ,si super extremam c4 ,et nota1 talis figura significans m$ietatem1 unitatis v5iat1 i4e4 cont+at uel fiat super extremam1 i4e4 super primam figuram 9 extremo sic versus dextram ars dat3 i4e4 r$dit monadem4 i4e4 vnitatem eidem4 i4e4 eidem note decl9atur hec monos1 dis1 di1 dem1 c4 ,quod ergo totum hoc dabis monadem note cont+et4 i4e4 ev5iet tibi si dimidia/i1 i4e4 accipi/i uel subtuli/i m$ietatem alicuius unius1 9 cuius pr9cipio s9t figura num]um d5otans imp>em primo i4e4 pr9cipiis4 ,headnote3 ,! ,craft ( ,m$iation4 ,sequitur de m$iacione4 ,incipe sic1 si vis aliquem numerum m$i>e3 ,scribe figur>um s]iem solam1 velut ante4 ,sid5ote3 ,! f\r ?+s to be kn[n 9 m$iation3 ! fir/ ! second2 ! ?ird2 ! f\r?4 ,beg9 ?us4 ,9 is ,*apt] is tat e ,craft ( m$iaci\1 9 e quy* craft ou mo/ "k #d ?ynges4 ffur/ :at is m$iacio4 ! secunde h[ mony rewes ( figures ou mo/ haue 9 e wyr*ynge ( is craft4 e ?ryde h[ mony diuerse cases may happ 9 is craft4#h ! 4#d4 :at is e profet ( is craft4 ,z = e fur/1 ou s*alt vndur/onde at m$iacio is a takyng out ( halfe a nomb] out ( a holle nomb]1 99leaf #adi;a z yf ou wolde take #c out ( #f4 ,z = e secunde1 ou s*alt "k at ou mo/ haue one rewe ( figures1 no moo1 z ou hay/ 9 e craft ( duplacio4 ,z = ! ?ryd1 ou mo/ vnder/onde at #e cases may happe 9 is craft4 ,z = e f\rte1 ou s*alle "k at ! profet ( is craft is :5 ou ha/ take away e halu5del ( a nombre to telle qwat ere s*alle leue4 ,incipe sic1 c4 ,! s5t5ce ( is v]se is is4 yf ou wold m$ye1 at is to say1 take halfe out ( e holle1 or halfe out ( halfe1 ou mo/ begynne u4 ,write one rewe ( figures ( :at nombre ou wolte1 z ou dyddy/ be-=e 9 e ,craft ( duplacio4 versus4 ,po/ea proc$as m$ians1 si prima figura ,si p> aut imp> videas4 ,sid5ote3 ,see if ! numb] is ev5 or odd4 ,h]e he says1 :5 ou ha/ write a rewe ( figures1 ou s*alt take h$e :eer e fir/ figure be eue or odde 9 nombre1 vnder/onde at he spekes ( e fir/ figure 9 e ryt side4 ,& in ! ryght side ou s*alle begynne 9 is ,craft4 ,quia si fuerx p>1 ,dimidiabis eam1 scribens quicquid remanebit3 ,sid5ote3 ,if x is ev51 halve x1 & write ! answ] 9 its place4 ,h]e is ! fir/ case ( is craft1 e quy* is is1 yf e fir/ figure be eu54 ou s*al take away fro e figure eu5 halfe1 d away at figure & set ere at leues ouer1 z u1 #d4 take 99leaf #adi;b halfe out ( #d1 an ere leues #b4 d away #d sett ere #b4 is is lyght y-n[t4 versus4 ,headnote3 ,! ,m$iation ( an ,odd ,numb]4 ,imp> si fuerx vnum demas m$i>e ,quod non presumas1 sed quod supere/ m$iabis ,inde super tractum fac demptum quod notat vnum4 ,sid5ote3 ,if x is odd1 halve ! ev5 numb] less ?an x4 ,"h is an example4 ,!n write ! sign = "o-half ov] x4 ,put ! m>k only ov] ! fir/ figure4 ,h]e is e secunde case ( is craft1 ! quy* is is4 yf e fir/ figure betok5e a nombre at is odde1 ! quy* odde s*al n be m$iete1 en ou s*alt m$ye at nombre at leues1 :5 ! odde ( e same nombre is take away1 write at at leues z ou didde/ 9 e fir/ case ( is craft4 ,:a ou hay/ write at4 = at at leues1 write su* a m]ke z is h]e vpon 8 h$e1 e quy* m]ke s*al betoke halfe ( e odde at 0 take away4 lo an ,ensampull4 #bde4 ! fir/ figure h]e is betok5ynge odde nombre1 e quy* is #e1 = #e is odde2 ere-=e d away at at is odde1 e quy* is #a4 en leues #d4 en m$ye #d en leues #b4 d away #d4 sette ere #b1 make su* a m]ke ^u 8 h$e1 at is to say ouer 8 h$e ( #b z u4 #bdb4 ,& en wor* =? 9 e oer figures tyll ou come to e ende4 by e fur/ case z ou s*alt vnder/onde at ou s*alt 99leaf #aej;a neuer make su* a m]k b ouer e fir/ figure h$ 9 e rit side4 ,:eer e o!r figures at comy after hym be eue or odde4 versus4 ,headnote3 ,! ,cases ( ! ,craft ( ,m$iation4 ,si monos1 dele2 sit tibi cifra po/ nota supra4 ,sid5ote3 ,if ! fir/ figure is "o put a ciph]4 ,"h is e ?ryde case1 e quy* yf ! fir/ figure be a figure ( #a4 ou s*alt d away at #a set ere a cif]1 a m]ke ouer e cif] z u1 #bda4 d away #a1 sett ere a cif] with a m]ke ouer 8 h$e1 en ha/ ou ydo = at #j4 z u #j en wor* =? 9 e oer figurys till ou come to e ende1 = x is lyght z dy*e wat]4 vnde versus4 ,po/ea proc$as hac condici"o secunda3 ,impar si fuerx h9c vnum deme priori1 ,inscrib5s qu9que1 nam d5os significabit ,monos predictam4 ,sid5ote3 ,:at to d if any o!r figure is odd4 ,write a figure ( five ov] ! next l[] numb]'s head4 ,example4 ,h]e he puttes e f\rte case1 e quy* is is4 yf x happe ! secunde figure betok5 odde nombre1 ou s*al d away on ( at odde nombre1 e quy* is significatiue by at figure #a4 e quy* #a s*all be rek5de = #aj4 ,:an ou ha/ take away at #a out ( e nombre at is signifi$e by at figure1 ou s*alt m$ie at at leues ouer1 d away at figure at is m$i$1 sette 9 8 /yde halfe ( at nombre4 ,:an ou hase s done1 ou s*alt write 99leaf #aej;b a figure ( #e ouer e next figures h$e by-=e t[>d e ryt side1 = at #a1 e quy* made odd nombre1 s*all /onde = t51 #e is halfe ( #aj2 s ou mo/ write #e = 8 halu5delle4 lo an ,ensampulle1 #dfgh4 begy 9 e ryt side z ou mo/ n$es4 m$ie #h4 en ou s*alt leue #d4 d away at #h sette ere #d4 en out ( #g4 take away #a4 e quy* makes odde1 sett #e4 vpon e next figures h$e a=e t[>d e ryt side1 e quy* is n[ #d4 b a=e x 0 #h4 = at #a s*al be rek5et = #aj1 ( e quy* #aj1 #e is halfe1 z ou kn[e/ wel4 ,:an ou ha/ u ydo1 m$ye at e quy* leues after e taky+e away ( at at is odde1 e quy* leuynge s*alle be #c2 d away #f sette ere #c1 ou s*alt haue su* a nombre #e #dfcd4 after g =? to e next figure1 m$y at1 wor* =?1 = x is lyt ynovt to e certay4 ,si vero secunda dat vnum4 ,illa deleta1 scribatur cifra2 priori ,trad5do qu9que pro d5>io m$iato2 ,nec cifra scribatur1 nisi deinde figura sequatur3 ,po/ea proc$as reliquas m$i&o figuras ,vt supra docui1 si s9t tibi mille figure4 ,sid5ote3 ,if ! second figure is "o1 put a ciph]1 & write five ov] ! next figure4 ,h[ to halve f\rte54 ,h]e he puttes e #e case1 e quy* is 99leaf #aea;a is3 yf e secunde figure be ( #a1 z is is "h #ab1 ou s*alt d away at #a sett ere a cif]4 sett #e ouer e next figure h$e a=e t[>d e rit side1 z ou diddy/ a=e2 at #e s*al be haldel ( at #a1 e quy* #a is rek5t = #aj4 lo an ,ensampulle1 #bad4 m$ye #d4 at s*alle be #b4 d away #d sett ere #b4 en g =? to e next figure4 e quy* is bot #a4 d away at #a4 sett ere a cif]4 set #e vpon e figures h$ a=e1 e quy* is n[e #b1 en ou s*alt haue is nombre #e #bjb1 en wor* =? to e nex figure4 ,& also x is no may/ery yf ere come no figure aft] at on is m$yet1 ou s*alt write no #j4 ne n[t ellis1 b set #e ouer e next figure a=e t[>d e ryt1 z u #ad4 m$ie #d !n leues #b1 d away #d sett ere #b4 en m$ie #a4 e qui* is rek5de = t51 e haluendel ere-( wel be #e4 sett at #e vpon e h$e ( at figure1 e quy* is n[ #b1 d away at #a1 ou s*alt haue is nombre yf ou wor* wel1 #e #b4 vnde versus4 ,headnote3 ,h[ to prove ! ,m$iation4 ,si m$iacio sit bene facta prob>e valebis ,dupl&o numerum quem primo dimedia/i ,sid5ote3 ,h[ to prove y\r m$iation4 ,fir/ example4 ,! second4 ,! ?ird example4 ,! f\r? example4 ,! fif? example4 ,h]e he telles e h[ ou s*alt "k :eer ou hase wel ydo or no4 d\bul 99leaf #aea;b e nombre e quy* ou hase m$iet1 & yf ou haue wel y-m$yt aft] e dupleacio1 ou s*alt haue e same nombre at ou haddy/ 9 e tabulle or ou began to m$ye1 z u4 ,! fur/ ensampulle 0 is4 #d4 e quy* ,i-m$iet 0 laft #b1 e :y* #b 0 write 9 e place at #d 0 write a=e4 ,n[ d\bulle at #b1 ou s*al haue #d1 z ou hady/ a=e4 e secunde ,ensampulle 0 is1 #bde4 ,:5 ou haddy/ m$iet alle is nombre1 yf ou haue wel ydo ou s*alt haue ( at m$iacio is nombre1 #abb4 ,n[ d\bulle is nombre1 begyn 9 e lyft side2 d\bulle #a1 at s*al be #b4 d away at #a sett ere #b4 en d\bulle at oer #b sett ere #d1 en d\bulle at oer #b1 at wel be #d4 en d\bul at m]ke at /ondes = halue on4 at s*alle be #a4 ,ca/ at on to #d1 x s*alle be #e4 d away at #b at m]ke1 sette ere #e1 en ou s*al haue is nombre #bde4 is wos e same nombur at ou haddy/ or ou began to m$ye1 z ou may/ se yf ou take h$e4 ,! nombre e quy* ou haddi/ = an ,ensampul 9 e #c case ( m$iacio to be m$iet 0 is #bda4 :an ou haddi/ m$i$ alle is nombur truly 99leaf #aeb;a by euery figure1 ou s*all haue be at m$iacio is nombur #abj4 ,n[ d[bul is nombur1 begyn 9 e lyft side1 z ,i tolde e 9 e ,craft ( duplacio4 u d\bulle e figure ( #a1 at wel be #b4 d away at #a sett ere #b1 en d\bul e next figure a=e1 ! quy* is #b1 at wel be #d2 d away #b set ere #d4 en d\bul e cif]1 at wel be not1 = a #j is not4 ,& twyes not is b not4 ere=e d\bul ! m]ke ab\e e cif]s h$e1 e quy* betok5es e haluendel ( #a1 at s*al be #a4 d away e cif] e m]ke1 sett ere #a1 en ou s*alt haue is nombur #bda4 ,& is same nombur ou haddy/ a=e or ou began to m$y1 yf ou take gode h$e4 ,! next ensampul at _h 9 e #d case ( m$iacio 0 is #dfgh4 ,:an ou ha/ truly ym$it alle is nombur fro e begynnynge to e endynge1 ou s*alt haue ( e m$iacio is nombur #e #bccd4 ,n[ d\bul ?is nombur begyn 9 e lyft side1 d\bulle #b at s*al be #d4 d away #b & sette ere #d2 en d\bule #c1 at wol be #f2 d away #c sett ere #f1 en d\bul at oer #c1 at wel be #f2 d away #c set ere 99leaf #aeb;b #f1 en d\bul e #d1 at welle be #h2 en d\bul #e4 e quy* /ondes ouer e h$ ( #d1 at wol be #aj2 ca/ #aj to #h1 at s*al be #ah2 d away #d at #e1 sett ere #h1 sett t #a1 e quy* is an articul ( e ,composit e quy* is #ah1 ouer e next figures h$ t[>d e lyft side1 e quy* is #f4 drav at #a to #f1 e quy* #a 9 e dravyng s*al be rek5te bot = #a1 at #a at #f tog$ur wel be #g4 d away at #f at #a4 ! quy* /ondes ouer 8 h$e1 sett !r #g1 en ou s*alt haue is nombur #dfgh4 ,& is same nombur ou hady/ or ou began to m$ye1 z ou may/ see 9 e secunde ,ensampul at ou _h 9 e #d case ( m$iacio1 at 0 is3 :5 ou _h m$iet truly alle ! nombur1 a principio usque ad f9em4 ou s*alt haue ( at m$iacio is nombur #e #ajb4 ,n[ d\bul #a4 at wel be #b4 d away #a sett ere #b4 en d\bul #j4 at w be not4 ere=e take e #e1 e quy* /ondes ouer e next figures h$1 d\bul x1 at wol be #aj4 d away e #j at /ondes betw5e e two figuris1 sette ere 9 8 /id #a1 = at #a n[ s*al /onde 9 e secunde place1 :]e he s*al betok5 #aj2 en d\bul #b1 at wol be #d4 d away #b sett ere #d4 99leaf #aec;a ou s*al haue u nombur #bad4 is is e same numbur at ou hady/ or ou began to m$ye1 z ou may see4 ,& s d euer more1 yf ou wil kn[e :eer ou hase wel ym$yt or no4 4 d\bulle e numbur at comes after e m$iaci\1 ou s*al haue e same nombur at ou hady/ or ou began to m$ye1 yf ou haue welle ydo4 or els d\te e not1 b yf ou haue e same1 ou hase faylide 9 i ,craft4 ,sequitur de multiplicatione4 ,headnote3 ,! ,craft ( ,multiplication4 ,headnote3 ,to write d[n a ,multiplication ,sum4 ,si tu per numerum numerum vis multiplic>e ,scribe duas quascunque velis s]ies numerorum ,ordo servetur vt vltima multiplic&i ,ponatur super anteriorem multiplicantis ,a leua relique s9t scripte multiplicantes4 ,sid5ote3 ,f\r ?+s to be kn[n ( ,multiplication3 ! fir/3 ! second3 ! ?ird3 ! f\r?4 ,h[ to set d[n ! sum4 ,two sorts ( ,multiplication3 m5tally1 & on pap]4 ,h]e be-gynnes e ,*aptre ( multiplicatio1 9 e quy* ou mo/ "k #d ?ynges4 ,ffir/1 qwat is multiplicacio4 ,! secunde1 h[ mony cases may hap 9 multiplicacio4 ,! ?ryde1 h[ mony rewes ( figures ere mo/ be4 ,! #d4 :at is e profet ( is craft4 ,z = e fir/1 ou s*al vnder/onde at multiplicacio is a bryngynge to-g$er ( #b ?ynges 9 on nombur1 e quy* on nombur contynes s mony tymes on1 h[e 99leaf #aec;b mony tymes ere b5 vnytees 9 e n[mbre ( at #b1 z twyes #d is #h4 n[ h]e b5 e #b nombers1 ( e quy* too n[mbres on is betok5$ be an aduerbe1 e quy* is e worde twyes1 is worde ?ryes1 is worde f\re sy!s1#i is wordes fyue si! sex sy!s4 s fur? ( su* o!r lyke wordes4 ,& tweyn nombres s*al be tok5yde be a n[ne1 z is worde f\re %[ys es twey nombres y-bro? 9-to on hole nombur1 at is #h1 = twyes #d is #h1 z ou wo/ wel4 ,& es nombre #h conteynes z (t tymes #d z ere b5 vnites 9 at o!r nombre1 e quy* is #b1 = 9 #b b5 #b vnites1 s (t tymes #d b5 9 #h1 z ou wottys wel4 f= e secunde1 ou mo/ "k at ou mo/ haue too rewes ( figures4 ,z = e ?ryde1 ou mo/ "k at #h maner ( diuerse case may happe 9 is craft4 ,! profet ( is ,craft is to telle :5 a nombre is multiply$ be a noer1 qwat commys ere (4 f=!rmore1 z to e s5t5ce ( oure v]se1 yf ou wel multiply a nombur be a-noer nombur1 ou s*alt write 99leaf #aed;a a rewe ( figures ( :at nomburs s euer ou welt1 at s*al be call$ ,numerus multiplic&u1 ,anglice1 e nombur ! quy* to be multipli$4 en ou s*alt write a-no!r rewe ( figures1 by e quy* ou s*alt multiplie ! nombre at is to be multipli$1 ( e quy* nombur e fur/ figure s*al be write vnder e la/ figure ( e nombur1 e quy* is to be multipli$4 ,& s write =! t[>d e lyft side1 z h]e y may se1 ---------- #fgcbd #abcd ---------- ,& is one nombur s*alle be call$ num]us multiplicans4 ,anglice1 e nombur multipliynge1 = he s*alle multiply e hy] n\nbur1 z u one tyme #f4 ,& s =?1 z ,i s*al telle ! afterw>de4 ,& ou s*al begyn 9 e lyft side4 f=-ere-m ou s*alt vndur/onde at ere is two manurs ( multiplicacio2 "o ys ( e wyr*ynge ( e boke only 9 e mynde ( a mon4 fyr/ he te*es ( e fyr/ maner ( duplacio1 e quy* is be wyr*ynge ( tabuls4 ,afterw>de he wol te*e on e secunde maner4 vnde versus4 ,headnote3 ,to multiply "o ,digit by ano!r4 ,9 digitum cures digitum si ducere maior 99leaf #aed;b4 ,per quantum di/at a d5is respice debes ,namque suo decuplo toti5s del]e minorem ,sitque tibi num]us v5i5s ex9de patebit4 ,sid5ote3 ,h[ to multiply two digits4 ,subtract ! great] f t52 take ! less s _m times f t5 times itself4 ,example4 ,h]e he te*es a rewle1 h[ ou s*alt fynde e n\nbre at comes by e multiplicacio ( a digit be anoer4 loke h[ mony vnytes b54 bytw5e e more digit & #aj4 ,& rek5 t5 = on vnite4 ,& s (t d away e lasse n\nbre out ( 8 owne decuple1 at is to say1 fro at n\nbre at is t5 tymes s my* is e n\nbre at comes ( e multiplicacio4 ,z yf ou wol multiply #b be #d4 loke h[ mony vnitees b5 by-tw5e e quy* is e more n\nbre1 be-tw5e t54 ,cert5 ere wel be vj vnitees by-tw5e #d t54 yf ou rek5 ere with e t5 e vnite1 z ou may se4 s mony tymes take #b4 out ( 8 decuple1 e quy* is #bj4 = #bj is e decuple ( #b1 #aj is e decuple ( #a1 #cj is e decuple ( #c1 #dj is e decuple ( #d1 ,& e oer digetes til ou come to t52 :an ou ha/ y-take s mony tymes #b out ( tw5ty1 e quy* is sex tymes1 ou s*al leue #h z ou wo/ wel1 = #f times #b is twelue4 take #ab out ( tw5ty1 ere s*al leue #h4 bot yf bo! e digettes 99leaf #aee;a b5 y-lye* my* z h]e4 #bbb or too tymes tw5ty1 en x is no =s quy* ( hem tweyn ou take out ( "h decuple4 als mony tymes z at is fro #aj4 b neuer-e-lesse1 yf ou haue ha/ to wor*1 ou s*alt haue h]e a tabul ( figures1 :]e-by ou s*alt se a-non ryght :at is e n\nbre at comes ( e multiplicacio ( #b digittes4 u ou s*alt wor* 9 is figure4 ,sid5ote3 ,bett] use ?is table1 ?\gh4 ,h[ to use x4 ,! way to use ! ,multiplication table4 #a ----- #b #d -------- #c #f #i ----------- #d #habaf -------------- #eajaebjbe ----------------- #fabahbdcjcf -------------------- #gadbabhcedbdi ----------------------- #hafbdcbdjdheffd -------------------------- #iahbgcfdeedfcgbha ---------------------------- #a #b #c #d #e #f #g #h #i ---------------------------- yf e figure1 e quy* s*alle be multipli$1 be eu5e z my* z e diget be1 e quy* at oer figure s*al be multipli$1 z two tymes tway1 or ?re tymes #c4 or sy* o!r4 loke qw]e at figure sittes 9 e lyft side ( e triangle1 loke qw]e e diget sittes 9 e neer mo/ rewe ( e triangle4 g fro hym vpw>de 9 e same rewe1 e quy* rewe gose vpw>de til ou come agaynes e oer digette at sittes 9 e lyft side ( e triangle4 ,& at n\nbre1 e quy* ou 99leaf #aee;b fyn99des ere is e n\nbre at comes ( ! multiplicacio ( e #b digittes1 z yf ou wold wete qwat is #b tymes #b4 loke qu]e sittes #b 9 e lyft side in e fir/ rewe1 he sittes next #a 9 e lyft side al on hye1 z ou may se2 en loke qw]e sittes #b 9 e l[y/ rewe ( e triangle1 g fro hym vpw>de 9 e same rewe tylle ou come a-en5es #b 9 e hy] place1 er ou s*alt fynd ywrite #d1 at is e n\nbre at comes ( e multiplicacio ( two tymes tweyn is #d1 z ow wote/ welle4 yf e diget4 ! quy* is multipli$1 be more an e oer1 ou s*alt loke qw]e e more diget sittes 9 e l[e/ rewe ( e triangle1 g vpw>de 9 e same rewe tyl#aj ou come a-n5des e lasse diget 9 ! lyft side4 ,& ere ou s*alt fynde e nombre at comes ( e multiplicacio2 b ou s*alt vnder/onde at is rewle1 e quy* is 9 is verse4 ,9 digitum cures1 c41 noer is triangle s*alle n serue1 bot to fynde e n\nbres at comes ( ! multiplicacio at comes ( #b articuls or composites1 e n$es no craft b yf ou wolt multiply 9 i mynde4 ,& 99leaf #aef;a ere-to ou s*alt haue a craft afterw>de1 = ou s*all wyr* with digettes 9 e tables1 z ou s*alt "k afterw>de4 versus4 ,headnote3 ,to multiply "o ,composite by ano!r4 ,po/ea proc$as po/remam multiplicand ,recte multiplicans p] cunctas inf]iores ,condicionem tam5 tali quod multiplicantes ,scribas 9 capite quicquid process]it inde ,s$ po/quam fuit hec multiplicate figure ,ant]ior5tur s]ei multiplicantis ,et sic multiplica velut i/i multiplica/i ,qui sequitur num]um scriptum quiscunque figuris4 ,sid5ote3 ,h[ to multiply "o numb] by ano!r4 ,multiply ! #8la/' figure ( ! high] by ! #8fir/' ( ! l[] numb]4 ,set ! answ] ov] ! fir/ ( ! l[]3 !n multiply ! second ( ! l[]1 & s on4 ,!n ant]y ! l[] numb]3 z ?us4 ,n[ multiply by ! la/ b "o ( ! high]3 z ?us4 ,ant]y ! figures aga91 & multiply by five3 ,!n add all ! figures above ! l9e3 & y w h ! answ]4 ,h]e he te*es h[ ou s*alt wyr* 9 is craft4 ou s*alt multiplye e la/ figure ( e nombre1 & qu5 ou ha/ s ydo ou s*alt draw alle e figures ( e neer n\nbre more taw>d e ryt side1 s qwen ou ha/ multiply$ e la/ figure ( e hey] n\nbre by alle e neer figures4 ,& sette e n\nbir at comes er-( ouer e la/ figure ( e neer n\nbre1 en ou s*alt sette al e oer figures ( e neer n\nbre more n]e to e ryt side4 ,& :an ou ha/ multipli$ at figure at s*al be multipli$ e next after hym by al e neer figures4 ,& wor* z ou dyddy/ a=e til 99leaf #aef;b ou come to e ende4 ,& ou s*alt vnder/onde at euery figure ( e hi] n\nbre s*al be multipli$ be alle e figures ( ! neer n\nbre1 yf e hi] n\nbre be any figure en one4 lo an ,ensampul h]e fol[ynge4 ------ #bdfe4 #bcb ------ ou s*alt begyne to multiplye 9 e lyft side4 ,multiply #b be #b1 & twyes #b is #d4 set #d ouer e h$ ( at #b1 en multiplie e same hi] #b by #c ( e ne!r n\nbre1 z ?ryes #b at s*al be #f4 set #f ouer e h$ ( #c1 an multiplie e same hi] #b by at #b e quy* /ondes vnder hym1 at wol be #d2 d away e hi] #b sette ere #d4 ,n[ ou mo/ ant]y e ne!r n\nbre1 at is to say1 ou mo/ sett e neer n\nbre m t[>de e ryt side1 z u4 ,take e neer #b t[>d e ryt side1 sette x eue vnder e #d ( e hy] n\nbre1 antery alle e figures at comes after at #b1 z u2 sette #b vnder e #d4 en sett e figure ( #c ere at e figure ( #b /ode1 e quy* is n[ vndur at #d 9 e hi] n\nbre2 en sett e oer figure ( #b1 e quy* is e la/ figure t[>d e lyft side ( e neer nomber ere e figure ( #c /ode4 en ou s*alt haue su* a nombre4 ------ #dfddfe #bcb ------ 99leaf #aeg;a ,n[ multiply #d1 e quy* comes next after #f1 by e la/ #b ( e neer n\nbur t[>d e lyft side4 z #b tymes #d1 at wel be #h4 sette at #h ouer e figure ! quy* /ondes ouer e h$e ( at #b1 e quy* is e la/ figure ( e neer n\nbre2 an multiplie at same #d by #c1 at comes 9 e neer rewe1 at wol be #ab4 sette e digit ( e composyt ouer e figure e quy* /ondes ouer e h$ ( at #c1 sette e articule ( is composit ouer al e figures at /ondes ouer e neer #b h$e4 en multiplie e same #d by e #b 9 e ryt side 9 e neer n\nbur1 at wol be #h4 d away #d4 sette ere #h4 ,euer more qw5 ou multiplies e hi] figure by at figure e quy* /ondes vnder hym1 ou s*alt d away at hi] figure1 sett er at n\nbre e quy* comes ( multiplicacio ( ylke digittes4 ,:an ou ha/ done z ,i haue byde e1 ou s*alt haue suy* an order ( figure z is h]e1 -------- #a 4 #hb #dfdhfe #bcb -------- en take & antery i neer figures4 ,& sett e fyr/ figure ( e neer figures#aa vndre be figure ( #f4 ,& draw al e oer figures ( e same rewe to hym-w>de1 99leaf #aeg;b z ou diddy/ a=e4 en multiplye #f be #b1 sett at e quy* comes ouer ere-( ouer al e oer figures h$es at /ondes ouer at #b4 en multiply #f be #c1 sett alle at comes ere-( vpon alle e figures h$es at /&es ouer at #c2 an multiplye #f be #b1 e quy* /ondes vnder at #f1 en d away #f write ere e digitt ( e composit at s*al come ere(1 sette e articull ouer alle e figures at /ondes ouer e h$e ( at #c z h]e1 ------ #aa #aba #hbh #dfdhbe #bcb ------ en antery i figures z ou diddy/ a=e1 & multipli #e be #b1 at wol be #aj2 sett e #j ouer all e figures at /ond5 ouer at #b1 sett at #a4 ouer ! next figures h$es1 alle on hye t[>de e lyft side4 en multiplye #e be #c4 at wol be #ae1 write #e ouer e figures h$es at /ond5 ouer at #c1 sett at #a ouer e next figures h$es t[>d e lyft side4 en multiplye #e be #b1 at wol be #aj4 d away at #e sett ere a #j1 sett at #a ouer e figures h$es at /ond5 ouer #c4 ,& en ou s*alt haue su* a n\nbre z "h /ondes aftur499leaf #aeh;a ------ #aa #aaja #abae #hbhbj #dfdh #bcb ------ ,n[ draw alle ese figures d[ne tog$er z u1 #f4h4a4 #a draw to-g$ur2 at wolle be #af1 d away alle ese figures saue #f4 lat hym /onde1 = ow ou take hym away ou mo/ write er e same aene4 ere=e late hym /onde1 sett #a ouer e figure h$e ( #d t[>d e lyft side2 en draw on to #d1 at wolle be #e4 d away at #d at #a1 sette ere #e4 en draw #dbba #a tog$ur1 at wol be #aj4 d away alle at1 write ere at #d at #j1 sett at #a ouer e next figures h$e t[>d e lyft side1 e quy* is #f4 en draw at #f at #a tog$ur1 at wolle be #g2 d away #f sett ere #g1 en draw #hhaj #a1 at wel be #ah2 d away alle e figures at /ondes ouer e h$e ( at #h1 lette #h /onde /il1 write at #a ouer e next figuris h$e1 e quy* is a #j4 en d away at #j1 sett ere #a1 e quy* /ondes ouer e #j4 h$e4 en draw #b1 #e1 #a tog$ur1 at wolle be #h4 en d away alle at1 write ere #h4 ,& en ou s*alt haue is n\nbre1 #egahhj4 ,headnote3 ,! ,cases ( ?is ,craft4 99leaf #aeh;b ,sed cum multiplicabis1 primo sic e/ oper&um1 ,si dabit articulum tibi multiplicacio solum2 ,proposita cifra summam transf]re memento4 ,sid5ote3 ,:at to d if ! fir/ multiplication results 9 an article4 ,h]e he puttes e fyr/ case ( is craft1 e quy* is is3 yf ere come an articulle ( e multiplicacio ysette be=e ! articulle 9 e lyft side z u --- #ea4 #bc --- multiplye #e by #b1 at wol be #aj2 sette ouer e h$e ( at #b a #j1 sett at on1 at is e articul1 9 e lyft side1 at is next hym1 en ou s*alt haue is n\nbre ---- #ajea4 #bc ---- ,& en wor* =? z ou diddi/ a=e4 ,& ou s*alt vnder/onde at ou s*alt write no #j4 b :an at place :]e ou s*al write at #j has no figure a=e hym noer after4 versus4 ,si autem digitus excreuerx articuluque4 ,articulu#ab supraposito digito salit vltra4 ,sid5ote3 ,:at to d if ! result is a composite numb]4 ,h]e is e secunde case1 e quy* is is3 yf hit happe at ere come a composyt1 ou s*alt write e digitte ouer e h$e ( e neer figure by e quy* ou multiplie/ e hi] figure2 & sett e articulle next hym t[>d e lyft side1 z ou diddy/ a=e1 z u --- #hc4 #hc --- ,multiply #h by #h1 at wol be #fd4 ,write e #d ouer #h1 at is to say1 ouer e h$e ( e neer #h2 set #f1 e quy* 99leaf #aei;a is an articul1 next after4 ,& en ou s*alt haue su* a n\nbre z is h]e1 ----------- #fdhcac1 #hc ----------- ,& en wor* =?4 ,si digitus tam5 ponas ipsum super ipsam4 ,sid5ote3 ,:at if x be a digit4 ,h]e is e ?ryde case1 e quy* is is3 yf hit happe at ( i multiplicaci\ come a digit1 ou s*alt write e digit ouer e h$e ( e neer figure1 by ! quy* ou multiplie/ e hi]e figure1 = is n$es no ,ensampul4 ,subdita multiplica non hanc que incidit illi ,delet eam p5itu scrib5s quod prou5it inde4 ,sid5ote3 ,! f\r? case ( ! craft4 ,h]e is e #d case1 e quy* is3 yf hit be happe at e neer figure s*al multiplye at figure1 e quy* /ondes ouer at figures h$e1 ou s*al d away e hi] figure sett ere at at comys ( at multiplicacio4 ,z yf ere come ( at multiplicacio an articuls ou s*alt write ere e hi] figure /ode a #j4 ,& write e articuls 9 e lyft side1 yf at hit be a digit write ere a digit4 yf at hit be a composit1 write e digit ( e composit4 ,& e articul 9 e lyft side4 al is is lyt y-n[t1 ere-=e er n$es no ,ensampul4 ,sed si multiplicat aliam ponas super ipsam ,adiunges numerum quem prebet ductu e>um4 ,sid5ote3 ,! fif? case ( ! craft4 ,h]e is e #e case1 e quy* is is3 yf 99leaf #aei;b e neer figure s*ul multiplie e hi]1 & at hi] figure is n recte ouer 8 h$e4 ,& at neer figure hase oer figures1 or on figure ouer 8 h$e by multiplicacio1 at hase be a=e1 ou s*alt write at n\nbre1 e quy* comes ( at1 ouer alle e ylke figures h$es1 z u "h3 ----- #bcf #bcd ----- ,multiply #b by #b1 at wol be #d2 set #d ouer e h$e ( at #b4 en#ad multiplies e hi] #b by e neer #c1 at wol be #f4 set ouer 8 h$e #f1 multiplie e hi] #b by e neer #d1 at wol be #h4 d away e hi] #b1 e quy* /ondes ouer e h$e ( e figure ( #d1 & set ere #h4 ,& ou s*alt haue is n\nbre "h ------- #dfhcf #bcd ------- ,& ant]y i figures1 at is to say1 set i neer #d vnder e hi] #c1 & set i #b o!r figures n]e hym1 s at e neer #b /onde vndur e hi] #f1 e quy* #f /ondes 9 e lyft side4 ,& at #c at /ondes vndur #h1 z u aftur e may se1 ------- #dfhcf #bcd ------- ,n[ wor* =!rmore1 ,& multiplye at hi] #c by #b1 at wol be #f1 set at #f e quy* /ondes ouer e h$e ( at #b1 ,& en wor* z ,i tat e a=e4 99leaf #afj;a ,si supraposita cifra debet multiplic>e ,prorsus eam deles ibi scribi cifra debet4 ,sid5ote3 ,! six? case ( ! craft4 ,h]e is e #f case1 e quy* is is3 yf hit happe at e figure by e quy* ou s*al multiplye e hi] figure1 e quy* /ondes ryght ouer hym by a #j1 ou s*alt d away at figure1 e quy* ouer at cifre h$e4 ,& write ere at n\nbre at comes ( e multiplicacio z u1 #bc4 d away #b & sett ere a #j4 vnde versus4 ,si cifra multiplicat aliam positam super ipsam ,sitque locus supra vacuu super hanc cifram fiet4 ,sid5ote3 ,! sev5? case ( ! craft4 ,h]e is e #g case1 e quy* is is3 yf a #j s*al multiply a figure1 e quy* /ondes n recte ouer hym1 ,& ouer at #j /onde no ?yng1 ou s*alt write ouer at #j anoer #j z u3 ---- #bd #jc ---- multiplye #b be a #j1 x wol be no?ynge4 write ere a #j ouer e h$e ( e neer #j1 ,& en wor* =? til ou come to e ende4 ,si supra#ae fu]it cifra semp] e/ pretereunda4 ,sid5ote3 ,! eigh? case ( ! craft4 ,h]e is e #h case1 e quy* is is3 yf ere be a #j or mony cif]s 9 e hi] rewe1 ou s*alt n multiplie hem1 bot let hem /onde4 ,& ant]y e figures b5ee to e next figure sygnificatyf z u3 ----- #jjjcb4 #bb ----- ,ouer-lepe alle ese cif]s sett at 99leaf #afj;b neer #b at /ondes t[>d e ryght side1 & sett hym vndur e #c1 & sett e oer ne!r #b n]e hym1 s at he /onde vndur e ?rydde #j1 e quy* /ondes next #c4 ,& an wor*4 vnde versus4 ,si dubites1 an sit bene multiplicacio facta1 ,diuide totalem num]um per multiplicantem4 ,sid5ote3 ,h[ to prove ! multiplication4 ,h]e he te*es h[ ou s*alt "k :eer ou hase wel ,i-d or no4 ,& he says at ou s*alt deuide alle e n\nbre at comes ( e multiplicacio by e neer figures4 ,& en ou s*alt haue e same n\nbur at ou hady/ 9 e begynnynge4 b et ou ha/ n e craft ( dyuisio1 b ou s*alt haue hit afterw>de4 ,per numerum si vis num]um quoque multiplic>e ,tantum per normas subtiles absque figuris ,has normas poteris per versus scire sequ5tes4 ,sid5ote3 ,m5tal multiplication4 ,h]e he te*es e to multiplie be owt figures 9 i mynde4 ,& e s5t5ce ( is verse is is3 yf ou wel multiplie on n\nbre by anoer 9 i mynde1 ou s*al haue ereto rewles 9 e verses at s*al come after4 ,si tu per digitum digitum vis multiplic>e ,regula prec$5s dat qualiter e/ oper&um4 ,sid5ote3 ,digit by digit is easy4 ,h]e he te*es a rewle z ou ha/ a=e to multiplie a digit be anoer1 z yf ou wolde wete qwat is sex tymes #f4 ou 99leaf #afa;a s*alt wete by e rewle at ,i tat e be=e1 yf ou haue mynde er(4 ,articulum si per reliquum reliquum vis multiplicare ,9 proprium digitum debet vterque resolui4 ,articulu digitos po/ se multiplicantes ,ex digitu quoci5s ret5]it multiplicari ,articuli faciunt tot c5tum multiplicati4 ,sid5ote3 ,! fir/ case ( ! craft4 ,article by article2 an example3 ano!r example3 ,headnote3 ,h[ to "w subtly )\t ,figures4 ,sid5ote3 ,m5tal multiplication4 ,ano!r example4 ,ano!r example4 ,notation4 ,notation aga94 ,m5tal multiplication4 ,h]e he te*es e fur/ rewle1 e quy* is is3 yf ou wel multiplie an articul be anoer1 s at bo? e articuls b5e with-,inne an hundre?1 u ou s*alt d4 take e digit ( bo! ! articuls1 = euery articul hase a digit1 en multiplye at on digit by at oer1 & loke h[ mony vnytes b5 9 e n\nbre at comes ( e multiplicacio ( e #b digittes1 s mony hundry!s b5 9 e n\nbre at s*al come ( e multiplicacio ( e ylke #b articuls z u4 yf ou wold wete qwat is t5 tymes t54 take e digit ( t51 e quy* is #a2 take e digit ( at oer t51 e quy* is on4 ,also multiplie #a be #a1 z on tyme on at is b #a4 ,9 on is b on vnite z ou wo/ welle1 ere=e t5 tymes t5 is b a hundry?4 ,also yf ou wold wete :at is tw5ty tymes #cj4 take e digit ( tw5ty1 at is #b2 take e digitt ( ?rytty1 at is #c4 multiplie #c be #b1 at is #f4 ,n[ 9 #f b5 #f vnites1 ,& s mony hundry!s b5 9 #bj tymes #cj99leaf #afa;b1 ere=e #bj tymes #cj is #f hundry? eue4 loke se4 ,b yf x be s at one articul be with-,inne an hundry?1 or by-tw5e an hundry? & a ?[s&e1 s at x be n a ows&e fully4 en loke h[ mony vnytes b5 9 e n\nbur at comys ( e multiplicacio #af;,& s mony tymes#af ( #b digittes ( ylke articuls1 s mony ?[sant b5 9 e n\nbre1 ! qwy* comes ( e multiplicacio4 ,& s mony tymes t5 ?[s& s*al be 9 e n\nbre at comes ( e multiplicacion ( #b articuls1 z yf ou wold wete qwat is #d hundry? tymes two hundry?4 ,multiply #d be #b1ag at wol be #h4 9 #h b5 #h vnites4 ,& s mony tymes t5 ?\s& be 9 #d hundry? tymes #bag hundry?1 at is #hj ?\s&4 ,take h$e1 ,i s*all telle e a g5eralle rewle :an ou ha/ #b articuls1 ,& ou wold wete qwat comes ( e multiplicacio ( hem #b4 multiplie e digit ( at on articuls1 & kepe at n\nbre1 en loke h[ mony cif]s s*uld g be=e at on articuls1 & he w]e write4 ,als mony cif]s s*uld g be=e at o!r1 he w]e write ( cif]s4 ,& haue alle e ylke cif]s tog$ur 9 i mynde1 99leaf #afb;a a-r[e y*o aftur o!r1 & 9 e la/ plase set e n\nbre at comes ( e multiplicacio ( e #b digittes4 ,& loke 9 i mynde 9 :at place he /ondes1 :]e 9 e secunde1 or 9 e ?ryd1 or 9 e #d1 or :]e ellis1 & loke qwat e figures by-tok5 9 at place2 s my* is e n\nbre at comes ( e #b articuls y-multipli$ to-g$ur z u3 yf ou wold wete :at is #bj ?\sant tymes #c ows&e4 multiply e digit ( at articulle e quy* is #b by e digitte ( at oer articul e quy* is #c1 at wol be #f4 en loke h[ mony cif]s s*al g to #bj ?\sant z hit s*uld be write 9 a tabul4 certa9ly #d cif]s s*uld g to #bj owsant4 f= is figure #b 9 e fyr/ place betok5es tw5e4 ,9 e secunde place hit betok5es tw5ty4 ,9 e #c4 place hit betok5es #b hundry?4 44 ,9 e #d place #b ?\sant4 ,9 e #e place hit betok5es tw5ty ousant4 ere=e he mo/ haue #d cif]s a-=e hym at he may /onde 9 e #e place4 kepe ese #d cif]s 9 ?y mynde1 en loke h[ mony cif]s g to #c ?\sant4 ,c]tayn to #c ?\sante 99leaf #afb;b g #c cif]s a=e4 ,n[ ca/ ylke #d cif]s at s*uld g to tw5ty ?\sant1 ,& !s #c cif]s at s*uld g a=e #c ?\sant1 sette hem 9 rewe y*o after oer 9 i mynde1 z ai s*uld /onde 9 a tabulle4 ,& en s*al ou haue #g cif]s2 en sett at #f e quy* comes ( e multiplicacio ( e #b digittes aftur e ylke cif]s 9 e #h place z yf at hit /ode 9 a tabul4 ,& loke qwat a figure ( #f s*uld betok5 9 e #h place4 yf hit w]e 9 a tabul s my* x is4 yf at figure ( #f /onde 9 e fyr/ place he s*uld betok5 b #f4 ,9 e #b place he s*uld betok5 sexty4 ,9 ! #c place he s*uld betoke sex hundry?4 ,9 e #d place sex ?\sant4 ,9 e #e place sexty owsant4 ,9 e sext place sex hundry? owsant4 ,9 e #g place sex owsant ?\santes4 ,9 e #h place sexty owsant ?\santes4 er=e sett #f 9 octauo loco1 ,& he s*al betok5 sexty owsant ?\santes4 ,& s my* is tw5ty owsant tymes #c ?\sant1 ,& is rewle is g5eralle = alle maner ( articuls1 ,:e?ir ai be hundry? or owsant2 b ou mo/ "k well e craft ( e wryr*ynge 9 e tabulle 99leaf #afc;a or ou "k to d u 9 i mynde aftur is rewle4 ,?\ mo/ at is rewle holdye note b :]e ere b5 #b articuls & no mo ( e quy* ay!r ( hem hase b on figure significatyf4 ,z tw5ty tymes #c ?\sant or #c hundry?1 & su* our4 ,articulum digito si multiplic>e oportet ,articuli digiti sumi quo multiplicate ,debemu reliquum quod multiplicatur ab illis ,per reliquo decuplum sic summam lat]e nequibit4 ,sid5ote3 ,! ?ird case ( ! craft2 an example4 ,h]e he puttes e ?ryde rewle1 e quy* is is4 yf ou wel multiply 9 i mynde1 ,& e ,articul be a digitte1 ou s*alt loke at e digitt be with-,inne an hundry?1 en ou s*alt multiply ! digitt ( e ,articulle by e oer digitte4 ,& euery vnite 9 e n\nbre at s*alle come ere-( s*al betok5 t54 ,z u3 yf at ou wold wete qwat is twyes #dj4 multiplie e digitte ( #dj1 e quy* is #d1 by e oer diget1 e quy* is #b4 ,& at wolle be #h4 ,& 9 e nombre ( #h b5 #h vnites1 euery ( e ylke vnites s*uld /onde = #aj4 ere-=e ere s*al be #h tymes #aj1 at wol be #d score4 ,& s mony is twyes #dj4 ,if e articul be a hundry? or be #b hundry? ,& a owsant1 s at hit be notte a ?\sant1 99leaf #afc;b wor* z ou dyddy/ a=e1 saue ou s*alt rek5e euery vnite = a hundry?4 ,9 num]um mixtum digitum si duc]e cures ,articulu mixti sumatur de9de resoluas ,9 digitum po/ fac respectu de digitis ,articuluque docet excresc5s 9 diriuand ,9 digitum mixti po/ ducas multiplicantem ,de digitis vt norma #ah;docet de hunc ,multiplica simul et sic po/ea summa patebit4 ,sid5ote3 ,! f\r? case ( ! craft3 ,composite by digit4 ,m5tal multiplication4 ,"h he puttes e #d rewle1 e quy* is is3 yf ou multipliy on composit be a digit z #f tymes #bd1 #ai;en take e diget ( at composit1 multiply at digitt by at oer diget1 & kepe e nombur at comes ere-(4 en take e digit ( at composit1 multiply at digit by anoer diget1 by e quy* ou ha/ multiply$ e diget ( e articul1 & loke qwat comes ere-(4 en take ou at n\nbur1 ca/ hit to at o!r n\nbur at ou se*e/e z u yf ou wel wete qwat comes ( #f tymes #d tw5ty4 multiply at articulle ( e composit by e digit1 e quy* is #f1 z yn e ?ryd rewle ou 0 taut1 ,& at s*al be #f score4 en multiply e diget ( e composit1 99leaf #afd;a e quy* is #d1 & multiply at by at o!r diget1 e quy* is #f1 z ou wa/ taut 9 e fir/ rewle1 yf ou haue mynde er(1 at wol be #d tw5ty4 ca/ all ylke n\nburs to-g$ir1 hit s*al be #add4 ,& s my* is #f tymes #d tw5ty4 ,headnote3 ,h[ to multiply )\t ,figures4 ,ductu 9 articulum numerus si compositu sit ,articulum purum comites articulum quoque ,mixti pro digitis po/ fiat et articulus vt ,norma iubet ret95do quod extra dicta ab illis ,articuli digitum po/ tu mixtum digitum duc ,regula de digitis nec precipit articuluque ,ex quibu excresc5s summe tu iunge priori ,sic manife/a cito fiet tibi summa petita4 ,sid5ote3 ,! fif? case ( ! craft3 ,article by ,composite4 ,an example4 ,h]e he puttes e #e rewle1 e quy* is is3 yf ou wel multiply an ,articul be a composit1 multiplie at ,articul by e articul ( e composit1 & wor* z ou wos taut 9 e secunde rewle1 ( e quy* rewle e verse begynnes u4 ,articulum si per ,relicum vis multiplic>e4 en multiply e diget ( e composit by at oir articul aftir e doctr9e ( e #c rewle4 take er( gode h$e1 ,i pray e z u4 ,yf ou wel wete :at is #bd tymes t54 ,multiplie t5 by #bj1 at wel be #b hundry?4 en multiply e diget ( e #aj1 e quy* is #a1 by e diget ( e composit1 e quy* is #d1 at 99leaf #afd;b wol be #d4 en rek5 euery vnite at is 9 #d = #aj1 at s*al be #dj4 ,ca/ #dj to #b hundry?1 at wol be #b hundry? #dj4 ,& s my* is #bd tymes t54 ,headnote3 ,h[ to "w )\t ,figures4 ,compositum numerum mixto sic multiplicabis ,vndecies tr$ecim sic e/ ex hiis oper&um ,9 reliquum primum demum duc po/ 9 eundem ,vnum po/ d5um duc 9 tria deinde per vnum ,multiplicesque demum intra omnia multiplicata ,9 summa decies quam si fuerx tibi doces ,multiplic&orum de normis sufficiunt hec4 ,sid5ote3 ,! six? case ( ! craft3 ,composite by ,composite4 ,m5tal multiplication4 ,an example ( ! six? case ( ! craft4 ,"h he puttes e #f rewle1 e la/ ( alle multiplicacio1 e quy* is is3 yf ou wel multiplye a composit by a-noer composit1 ou s*alt d u4 multiplie at on composit1 qwy* ou welt ( ! tw5e1 by e articul ( e toer composit1 z ou w]e taut 9 e #e rewle1 en multiplie at same composit1 e quy* ou ha/ multipli$ by e oer articul1 by e digit ( e oer composit1 z ou 0 taut 9 e #d rewle4 ,z u1 yf ou wold wete :at is #aa tymes #ac1 z ou 0 taut 9 e #e rewle1 at s*al be an hundry? t51 afterw>de multiply at same composit at ou ha/ multipli$1 e quy* is a 4#aa4 ,& multiplye hit be e digit ( e oer composit1 e quy* is #c1 = #c is e digit ( #ac1 ,& at wel be #cj4 en take e digit ( at composit1 e quy* composit ou multipli$ by e digit ( at oer composit1 99leaf #afe;a e quy* is a #aa4 ,also ( ! quy* #aa on is e digit4 multiplie at digitt by e digett ( at o?er composit1 e quy* diget is #c1 z ou 0 taut 9 e fir/ rewle in e begynnynge ( is craft4 e quy* rewle begynnes 8,9 digitum cures40 ,& ( alle e multiplicacio ( e #b digitt comys ?re1 = onys #c is b #c4 ,n[ ca/ alle ese n\nb]s tog$ur1 ! quy* is is1 a hundry? t5 #cj #c4 ,& al at wel be #adc4 ,write #c fir/ 9 e ryght side4 ,& ca/ #aj to #cj1 at wol be #dj4 set #dj next aftur t[>de e lyft side1 ,& set aftur a hundry? z h]e an ,ensampulle1 #adc4 7,cet]a desunt47 ,,footnotes 7,! ,crafte ( ,nombrynge73 #a3 ,9 ,,ms1 #8awiy4' #b3 #8b5' repeat$ 9 ,,ms4 #c3 ,9 ,,ms4 #8?aus&es4' #d3 ,p]haps 8,s40 #e3 #8hali' m>k$ = erasure 9 ,,ms4 #f3 #8moy' 9 ,,ms4 #g3 #8,subtrahas aut addis a dextris vel m$iabis' add$ on m>g9 ( ,,ms4 #h3 ,aft] #8craft' ins]t #8! 4#d4 :at is e profet ( is craft4' #i3 ,aft] #8sy!s' ins]t #8 is wordes fyue si! sex sy!s4' #aj3 #8t'l' m>k$ = erasure be=e #8tyl' 9 ,,ms4 #aa3 ,"h #8( e same rew' is m>k$ = erasure 9 ,,ms4 #ab3 #8sed' delet$ 9 ,,ms4 #ac3 #fhhc 9 ,,ms4 #ad3 #8en' ov]writt5 on #8at' m>k$ = erasure4 #ae3 #8,supra' ins]t$ 9 ,,ms4 9 place ( #8cifra' m>k$ = erasure4 #af--af3 ,m>k$ = erasure 9 ,,ms4 #ag3 #d 9 ,,ms4 #ah3 docet4 decet ,,ms4 #ai3 #8#d times #d' 9 ,,ms4 ,! ,art ( ,nombryng4 ,a ,,translation ,,( ,john ( ,holywood's ,de ,arte ,num]&i4 ,a%mole ,,ms4 #cif1 fol4 #dh4 ,boys sey+ 9 ! begynnyng ( 8 ,arsemetrike3--,alle 99,fol4 #dh4 ?ynges t b5e fro ! fir/ begynnyng ( ?ynges h proc$$e1 & come =?e1 ,& by resoun ( nombre b5 =m$e2 ,& 9 wise z !y b5e1 ,s owe?e !y to be kn[5e2 :]= 9 vniuersalle kn[le*yng ( ?ynges ! ,art ( nombrynge is be/1 & mo/ op]atyfe4 ,sid5ote3 ,! "n ( ! art4 ,d]ivation ( ,algorism4 ,ano!r4 ,ano!r4 ,k9ds ( numb]s4 ,! #i rules ( ! ,art4 ,!r=e si!n ! sci5ce ( ! :i*e at ?is tyme we int5d5e to write ( /&i?e alle & ab\t nombre3 ffir/ we mo/ se1 :at is ! propre "n !r(e1 & fro :5s ! "n come3 ,aft]w>de :at is nombre1 ,& h[ manye spices ( nombre !r b54 ,! "n is clep$e ,algorisme1 _he out ( ,algore1 o!r ( ,algos1 9 grewe1 ,t is clepide 9 englis%e art o!r craft1 ,& ( ,ri?mu t is call$e nombre4 ,s algorisme is clep$e ! art ( nombryng1 o!r x is _h (e en or 91 & gogos t is introduccioun1 & ,ri?mu nombre1 t is to say ,int]duccioun ( nombre4 ,& ?irdly x is _he ( ! "n ( a kyng t is clep$e ,algo & ,ry?mu2 ,s call$e ,algorismu4 ,so!ly 4#b4 man]e ( nombres b5 notifi$e2 ,=malle1#a z nombre is vnitees gadr$e to-g$res2 ,mat]ialle1#b z nombre is a colleccioun ( vnitees4 ,o!r nombre is a multitude _he out ( vnitees1 vnitee is t ?ynge :]-by euery ?ynge is call$e oone1 o!r o ?ynge4 ,( nombres1 t "o is clep$e digitalle1 t o!re ,article1 ,ano!r a nombre compon$e oer myxt4 ,ano!r digitalle is a nombre with-9 4#aj42 ,article is at nombre t may be dyvyd$e 9 4#aj4 parties egally1 ,& t !re leve no residue2 ,compon$e or m$l$e is t nombre t is come ( a digite & ( an article4 ,& vndre/&e wele t alle nombres betwix 4#b4 articles next is a nombre compon$e4 ,( ?is art b5e 4#i4 spices1 t is =to sey1 numeracioun1 addicioun1 ,subtraccioun1 ,m$iacioun1 ,duplacioun1 ,multipliacioun1 ,dyvysioun1 ,progressioun1 ,& ( ,rootes ! extraccioun1 & t may be _he 9 4#b4 man]s1 t is to sey 9 nombres quadrat1 & 9 cubices3 ,amonge ! :i*e1 ffir/ ( ,numeracioun1 & afterw>de ( e oers by ordure1 y ent5de to write4 ,headnote3 ,*apt] ,i4 ,num]ation4 99,fol4 #dh;b ,=-so?e numeracioun is ( euery numbre by compet5t figures an artificialle repres5tacioun4 ,sid5ote3 ,figures1 diff]5ces1 places1 & limits4 ,! #i figures4 ,! ciph]4 ,! num]ation ( digits1 ( articles1 ( composites4 ,! value due to position4 ,numb]s are writt5 f "r to left4 ,so?ly figure1 diff]5ce1 places1 & lynes suppos5 o ?yng o!r ! same1 ,b !y b5 sette "h = dyuers resons4 ffigure is clep$e = protraccioun ( figuracioun2 ,diff]5ce is call$e = !rby is %ew$e euery figure1 h[ x ha?e diff]5ce fro ! figures be=e !m3 place by cause ( space1 :]e-9 me write?e3 lynees1 = t is ordeyn$e = ! pres5tacioun ( euery figure4 ,& vnd]/onde t !r b5 4#i4 lymytes ( figures t repres5t5 ! 4#i4 digites t b5 ^!4 #j4 #i4 #h4 #g4 #f4 #e4 #d4 #c4 #b4 #a4 ,! 4#aj4 is clep$e !ta1 or a c]cle1 o!r a cifre1 o!r a figure ( n\ght = n\ght x signyfie?e4 ,na!lesse %e holdyng t place give?e o!rs = to signyfie2 = )e-out cifre or cifres a pure article may n be writte4 ,& si!n t by ^! 4#i4 figures significatifes ,ioyn$e with cifre or with cifres alle nombres b5 & may be repres5t$e1 ,x 01 ne!r is1 no n$e to fynde any m figures4 ,& note wele t euery digite %alle be writte with oo figure allone to x apropr$e4 ,& alle articles by a cifre1 f= euery article is nam$e = oone ( ! digitis z 4#aj4 ( #a44 #bj4 (4 #b4 & s ( ! o!rs1 c4 ,& alle nombres digitalle ow5 to be sette 9 ! fir/ diff]5ce3 ,alle articles 9 ! seconde4 ,also alle nombres fro 4#aj4 til an 4#ajj4 :i* is exclud$e1 ) 4#b4 figures mv/ be writte2 ,& yf x be an article1 by a cifre fir/ put1 & ! figure y-writte t[>de ! lift honde1 t signifie?e ! digit ( ! :i*e ! article is nam$e2 ,& yf x be a nombre compon$e1 ffir/ write ! digit t is a "p ( t compon$e1 & write to ! lift side ! article z x is seide be-=e4 ,alle nombre t is fro an hundr$e tille a ?\s&e exclus$e1 o)e to be writ by 4#c4 figures2 & alle nombre t is fro a ?\s&e til 4x4 ,m4 mv/ be writ by 4#d4 figures2 ,& s =!4 ,& vnd]/onde wele t euery figure sette 9 ! fir/ place signyfie?e 8 digit2 ,9 ! seconde place 4#aj4 tymes 8 digit2 ,9 ! 4#c4 place an hundr$e s mo*e2 ,9 ! 4#d4 place a ?\s&e s mo*e2 ,9 ! 4#e4 place 4x4 ?\s&e s mo*e2 ,9 ! 4#f4 place an hundr$e ?\s&e s mo*e2 ,9 ! 4#g4 place a ?\s&e ?\s&e4 ,& s infynytly mvltiply+ by 99,fol4 #di4 ^! 4#c4 #aj1 #ajj1 #ajjj4 ,& vnd]/&e wele t compet5tly me may sette vpon figure 9 ! place ( a ?\s&e1 a prike to %ewe h[ _m ?\s&e ! la/ figure %alle repres5t4 ,we writ5e 9 ?is art to ! lift side-w>de1 z arabi5e writ5e1 t w]5 fynd]s ( ?is sci5ce1 o!re = ?is resoun1 t = to kepe a cu/umable ordre 9 r$yng1 ,sette we alle-wey ! m nombre be=e4 ,headnote3 ,*apt] ,,ii4 ,addition4 ,sid5ote3 ,def9ition4 ,h[ ! numb]s %\ld be writt54 ,! me?od ( work+4 ,beg9 at ! "r4 ,! ,sum is a digit1 or an article1 or a composite4 ,addicioun is ( nombre o!r ( nombres vnto nombre or to nombres aggregacioun1 t me may see t t is come !r( z excress5t4 ,9 addicioun1 #b4 ordres ( figures & 4#b4 nombres b5 necess>y1 t is to sey1 a nombre to be add$e & ! nombre :]to ! addicioun %olde be made to4 ,! nombre to be add$e is t at %olde be add$e !rto1 & %alle be vnd]writ52 ! nombre vnto ! :i*e addicioun %alle be made to is t nombre t resceyue?e ! addicion ( at o!r1 & %alle be writ5 above2 & x is conv5i5t t ! lesse nombre be vnd]writ1 & ! m add$e1 ?an ! contr>y4 ,b :e!r x happe "o o!r o!r1 ! same comy?e (1 ,!r=1 yf ow wilt adde nombre to nombre1 write ! nombre :]to ! addicioun %alle be made 9 ! ome/ ordre by 8 diff]5ces1 s t ! fir/ ( ! l[] ordre be vndre ! fir/ ( ! omy/ ordre1 & s ( o!rs4 ,t done1 adde ! fir/ ( ! l[] ordre to ! fir/ ( ! omy/ ordre4 ,& ( su*e addicioun1 o!r ere gr[ith !r( a digit1 ,an article1 o!r a compos$e4 ,if x be digitu1 ,9 ! place ( ! omy/ %alt ?[ write ! digit excrescyng1 z ?us3-- ------------------------------- ,! resultant #b ------------------------------- ,to :om x %al be add$e #a ------------------------------- ,! nombre to be add$e #a ------------------------------- ,if ! article2 9 ! place ( ! omy/ put a-way by a cifre writte1 & ! digit transf]r$e1 ( e :i*e ! article toke 8 "n1 t[>de ! lift side1 & be x add$e to ! next figure fol[yng1 yf !r be any figure fol[yng2 or no1 & yf x be n1 leve x 9 ! voide1 z ?us3-- ------------------------------------- ,! resultant #aj ------------------------------------- ,to :om x %alle be add$e #g ------------------------------------- ,! nombre to be add$e #c ------------------------------------- ------------------------------------- ,resultans #b #g #h #b #g ------------------------------------- ,cui debet addi #a #j #j #h #d ------------------------------------- ,numerus add5du #a #g #g #d #c ------------------------------------- ,& yf x happe t ! figure fol[yng :]to ! addicioun %alle be made by ! cifre ( an article1 x sette a-side2 ,9 8 place write ! 99,fol4 #di;b digit ( ! ,article z ?us3-- ------------------------------------- ,! resultant #ag ------------------------------------- ,to :om x %alle be add$e #aj ------------------------------------- ,! nombre to be add$e #g ------------------------------------- ,& yf x happe t a figure ( 4#i4 by ! figure t me mv/ adde "o to1 ,9 ! place ( t #i4 put a cifre & write e article t[>de e lift honde z bi=e1 & ?us3-- ------------------------------------- ,! resultant #aj ------------------------------------- ,to :om x %alle be add$e #i ------------------------------------- ,! nombre to be add$e #a ------------------------------------- ,& yf#c !refrom gr[ a nombre compon$1#d 9 ! place ( ! nombre put a-way#elet ! digit be#fwrit at is "p ( at composide1 & an put to e lift side ! article z be=e1 & u3-- ------------------------------------- ,! resultant #ab ------------------------------------- ,to :om x %alle be add$e #h ------------------------------------- ,! nombre to be add$e #d ------------------------------------- ,?is done1 adde ! seconde to ! seconde1 & write above oer z be=e4 ,sid5ote3 ,! translator's note4 ,note wele at 9 addicions & 9 alle spices fol[yng1 :an he sei?e "o ! o!r %alle be writ5 ab\e1 & me mo/ vse euer figure1 z t euery figure 7 sette by halfe1 & by hym-selfe4 ,headnote3 ,*apt] ,,iii4 ,subtraction4 ,sid5ote3 ,def9ition ( ,subtraction4 ,h[ x may be done4 ,:at is requir$4 ,write ! great] numb] above4 ,subtract ! fir/ figure if possible4 ,if x is n possible #8borr[ t51' & !n subtract4 ,subtraccioun is ( 4#b4 propos$e nombres1 ! fyndyng ( ! excesse ( ! m to ! lasse3 ,o!r subtraccioun is ablacioun ( o nombre fro a-no!r1 t me may see a "s left4 ,! lasse ( ! m1 or ev5 ( ev51 may be withdraw2 ,! m fro ! lesse may neuer be4 ,& so?ly t nombre is m t ha?e m figures1 ,s t ! la/ be signyficatifes3 ,& yf !r b5 z _m 9 t "o z 9 t o!r1 me mo/ deme x by ! la/1 o!r by ! next la/4 ,m-ouer 9 with-drawyng 4#b4 nombres b5 necess>y2 ,a nombre to be withdraw1 ,& a nombre t me %alle with-draw (4 ,! nombre to be with-draw %alle be writ 9 ! l[] ordre by 8 diff]5ces2 ,! nombre fro ! :i*e me %alle )e-draw 9 ! omy/ ordre1 s t ! fir/ be vnd] ! fir/1 ! seconde vnd] ! seconde1 ,& s ( alle o!rs4 ,)e-draw !r= ! fir/ ( ! l[]e ordre fro ! fir/ ( ! ordre above 8 h$e1 & t wolle be o!r m or lesse1 oer egalle4 ------------------------------------- ,! reman5t #bj ------------------------------------- ,:]( me %alle withdraw #bb ------------------------------------- ,! nombre to be withdraw #b ------------------------------------- yf x be egalle or ev5 ! figure sette beside1 put 9 8 place a cifre4 ,& yf x be m put away erfro als _m ( vnitees ! l[] figure conteyne?e1 & writ ! residue z ?us ---------------------------------------- ,! reman5t #b #b ---------------------------------------- ,:]( me %alle with-draw #b #h ---------------------------------------- e nombre to be withdraw #f ---------------------------------------- 99,fol4 #ej4 ---------------------------------------- --------------- ,remanens #b #b #a #h #b #i #i #i #h ---------------------------------------- --------------- ,a quo sit subtraccio #h #g #b #d #c #j #j #j #d ---------------------------------------- --------------- ,num]us subtrah5du #f #e #gf 4 4 4 4 #f ---------------------------------------- --------------- ,& yf x be lesse1 by-cause ! m may n be with-draw !r-fro1 bor[ an vnyte ( ! next figure t is wor?e #aj4 ,( t 4#aj4 & ( ! figure t ye wolde h with-draw fro be-=e to-g$re ,ioyn$e1 with-draw e figure be-ne!1 & put ! residue 9 ! place ( ! figure put a-side z u3-- ---------------------------------------- ,! reman5t #a #h ---------------------------------------- ,:]( me %alle with-draw #b #d ---------------------------------------- ,! nombre to be with-draw #j #f ---------------------------------------- ,sid5ote3 ,if ! second figure is "o4 ,& yf ! figure :]( me %al bor[ ! vnyte be "o1 put x a-side1 & write a cifre 9 ! place er(1 le/ ! figures fol[+ faile ( ?aire nombre1 & an wor*e z x %ewith 9 ?is figure "h3-- ---------------------------------------- ---- ,! reman5t #c #j #ih ---------------------------------------- ---- ,:]( me %al with-draw #c #a #b ---------------------------------------- ---- ,! nombre to be with-draw 4 4 #c ---------------------------------------- ---- ,sid5ote3 ,if ! second figure is a ciph]4 ,& yf ! vnyte :]( me %al bor[ be a cifre1 g f]!r to ! figure signyficatife1 & !r bor[ "o1 & retournyng bake1 9 ! place ( euery cifre at ye passide ouer1 sette figures ( 4#i4 z "h x is specifi$e3-- ---------------------------------------- --------- ,! rem5aunt #b #i #i #i #i ---------------------------------------- --------- ,:]( me %alle with-draw #c #j #j #j #c ---------------------------------------- --------- ,! nombre to be with-draw #d ---------------------------------------- --------- ,sid5ote3 ,a ju/ification ( ! rule giv54 ,:y x is bett] to "w f "r to left4 ,h[ to prove subtraction1 & addition4 ,& :an me come?e to ! nombre :]( me int5di?e1 !re remayne?e alle-wayes 4#aj4 f= e :i*e 4#aj4 c4 ,! reson :y at = euery cifre left behynde me sette? figures !r ( 4#i4 ?is x is3--,if fro ! 4#c4 place me bor[$e an vnyte1 t vnyte by respect ( ! figure t he came fro repres5ti? an 4,c41 ,9 ! place ( t cifre pass$ ov] is left 4#i41 :i* is wor? n9ety1 & yit x remayne?e z 4#aj41 ,& ! same resone wolde be yf me _he bor[$e an vnyte fro ! 4#d41 4#e41 4#f41 place1 or ony o!r s vpw>de4 ,?is done1 )draw ! seconde ( ! l[] ordre fro ! figure above 8 h$e ( e omy/ ordre1 & wir*e z be=e4 ,& note wele t 9 addicion or 9 subtraccioun me may wele fro ! lift side begynne & ryn to ! "r side1 ,b x wol be m profitabl] to be d1 z x is taught4 ,& yf ?[ wilt prove yf ?[ h d wele or no1 ,! figures t ?[ ha/ )draw1 adde !m ay5e to ! omy/ figures1 & !y wolle accorde with ! fir/ t ?[ hadde/ yf ?[ h labor$ wele2 & 9 l wise 9 addicioun1 :an ?[ ha/ add$e alle ?y figures1 withdraw !m t ?[ fir/ 99,fol4 #ej;b adde/1 & ! same wolle retourne4 ,! subtraccioun is none o!r b a prouffe ( ! addicioun1 & ! contr>ye 9 l wise4 ,headnote3 ,*apt] ,,iv4 ,m$iation4 ,sid5ote3 ,def9ition ( m$iation4 ,:]e to beg94 ,if ! fir/ figure is unity4 ,:at to d if x is n unity4 ,m$iacioun is ! fyndyng ( ! halfyng ( euery nombre1 t x may be seyne :at & h[ mo*e is euery halfe4 ,9 halfyng ay oo ord] ( figures & oo nombre is necess>y1 t is to sey ! nombre to be half$e4 ,!r= yf ?[ wilt half any nombre1 write t nombre by 8 diff]5ces1 & begynne at ! "r1 t is to sey1 fro ! fir/ figure to ! "r side1 s t x be signyficatife o!r repres5t vnyte or eny o!r digitalle nombre4 ,if x be vnyte write 9 8 place a cifre = ! figures fol[yng1 le/ !y signify less1 & write t vnyte without 9 ! table1 o!r resolue x 9 4#fj4 mynvtes & sette a-side half ( ?o minutes s1 & res]ve ! rem5aunt without 9 ! table1 z ?us 4#cj42 o!r sette without ?us 4d3 t kepe?e none ordre ( place1 ,na!lesse x ha?e signyficacioun4 ,& yf ! o!r figure signyfie any o!r digital nombre fro vnyte =?e1 oer ! nombre is ode or ev5e4 ,if x be ev51 write ?is half 9 ?is wise3-- ----------------------- ,half$e #b #b ----------------------- to be half$e #d #d ----------------------- ,& if x be odde1 ,take ! next ev5 vndre hym conteyn$e1 & put 8 half 9 ! place ( t odde1 & ( e vnyte t remayne?e to be half$e d ?us3-- ----------------------- half$e #b #c di ----------------------- ,to be half$e #d #g ----------------------- ,sid5ote3 ,!n halve ! second figure4 ,if x is odd1 add #e to ! figure be=e4 ,?is done1 ! seconde is to be half$e1 yf x be a cifre put x be-side1 & yf x be significatife1 o!r x is ev5 or ode3 ,if x be ev51 write 9 ! place ( e nombres wip$e out ! halfe2 yf x be ode1 take ! next ev5 vnd] x cont5y?e1 & 9 ! place ( ! ,imp> sette a-side put half ( ! ev53 ,! vnyte t remayne?e to be half$e1 respect _he to !m be=e1 is wor?e 4#aj4 ,dyvide t 4#aj4 9 4#b41 #e4 is1 & sette a-side t "o1 & adde t o!r to ! next figure prec$5t z "h3-- -------------------------- ,half$e -------------------------- to be half$e -------------------------- ,& yf e addicioun %olde be made to a cifre1 sette x a-side1 & write 9 8 place 4#e4 ,& vnd] ?is fourme me %alle write & wor*e1 tille ! totalle nombre be half$e4 ---------------------------------------- ---- d\bl$e #b #f #h #i #j #aj #ag #d ---------------------------------------- ---- to be d\bl$e #a #c #d #d #e #e #h #g ---------------------------------------- ---- ,headnote3 ,*apt] ,v4 ,duplation4 ,sid5ote3 ,def9ition ( ,duplation4 ,:]e to beg94 ,:y4 ,:at to d ) ! result4 ,duplicacioun is agregacion ( nombre to itself at me may se ! nombre gr[54 ,9 d\blynge ay is b "o ordre ( figures necess>ie4 ,& me mo/ be-gynne with ! lift side1 o!r ( ! m figure1 ,& aft] ! nombre ( ! m figure repres5ti?e4 99,fol4 #ea4 ,9 ! o!r 4#c4 be=e we begynne alle way fro ! "r side & fro ! lasse nombre1 ,9 ?is spice & 9 alle o!r fol[yng we wolle begynne fro ! lift side1 f= & me bigon ?e d\ble fro ! fir/1 om:ile me myght d\ble oo ?ynge twyes4 ,& h[ be x t me myght d\ble fro ! "r1 t wolde be h>d] 9 te*yng & 9 workyng4 ,!r= yf ?[ wolt d\ble any nombre1 write t nombre by 8 diff]5ces1 & d\ble ! la/4 ,& ( t d\blyng o!r gr[i?e a nombre digital1 article1 or compon$e4 ,if x be a digit1 write x 9 ! place ( ! fir/ digit4 ,if x be article1 write 9 8 place a cifre & transf]re ! article t[>de ! lift1 z ?us3-- ---------------------- d\ble #aj ---------------------- to be d\bl$e #e ---------------------- ,& yf ! nombre be compon$e1 write a digital t is "p ( 8 composicioun1 & sette ! article to ! lift h&e1 z ?us3-- ---------------------- d\bl$e #af ---------------------- to be d\bl$e #h ---------------------- ,t done1 me mo/ d\ble ! la/ save "o1 & :at gr[e?e er( me mo/ wor*e z be=e4 ,& yf a cifre be1 t\*e x n4 ,b yf any nombre %alle be add$e to ! cifre1 9 e place ( e figure wip$e out me mo/ write ! nombre to be add$e1 z ?us3-- --------------------------- d\bl$e #f #j #f --------------------------- to be d\bl$e #c #j #c --------------------------- ,sid5ote3 ,h[ to prove y\r answ]4 ,9 ! same wise me %alle wir*e ( alle o!rs4 ,& ?is probacioun3 ,if ?[ truly d\ble ! halfis1 & truly half ! d\bles1 ! same nombre & figure %alle mete1 su*e z ?[ labourede vpone fir/1 ,& ( ! contr>ie4 --------------------------- ,d\bl$e #f #a #h --------------------------- to be d\bl$e #c #j #i --------------------------- ,headnote3 ,*apt] ,,vi4 ,multiplication4 ,sid5ote3 ,def9ition ( ,multiplication4 ,multipli]4 ,multiplic&4 ,product4 ,multiplicacioun ( nombre by hym-self o!r by a-no!r1 with proposide 4#b4 nombres1 is ! fyndyng ( ! ?irde1 ,t s (t conteyne?e t o!r1 z !r b5 vnytes 9 ! oer4 ,9 multiplicacioun 4#b4 nombres pryncipally b5 necess>y1 t is to sey1 ! nombre multiply+ & ! nombre to be multipli$e1 z "h2--twies fyve4 ,! numb] multiply+ is design$e aduerbially4 ,! nombre to be multipli$e resceyve?e a nominalle appellacioun1 z twies 4#e4 #e4 is ! nombre multipli$e1 & twies is ! nombre to be multipli$e4 ---------------------------------------- --------------- ,resultans #i #a #j #a #c #b #f #f #h #j #j #h ---------------------------------------- --------------- ,multiplic&u 4 4 #e 4 4 #d 4 #c #d #j #j #d ---------------------------------------- --------------- ,multiplicans 4 #b #b 4 #c #c #b #b #b 4 4 4 ---------------------------------------- --------------- ,also me may !rvpone to assigne !4 #c4 nombre1 ! :i*e is 99,fol4 #ea;b clep$e product or prov5i5t1 ( takyng out ( "o fro ano!r3 z twyes 4#e is 4#aj41 #e4 ! nombre to be multipli$e1 & 4#b4 ! multipliant1 &4 #aj4 z be=e is come !r(4 ,& vnd]/onde wele1 t ( ! multipliant may be made ! nombre to be multipli$e1 & ( ! contr>ie1 remaynyng euer ! same "s1 & h](e come?e ! com5 spe*e1 t sei?e all nombre is conv]t$e by ,multiply+ 9 hym-selfe4 ---------------------------------------- ----- #a #b #c #d #e #f #g #h #i #aj ---------------------------------------- ----- #b #d #f #h #aj #ajaj #ad #af #ah #bj ---------------------------------------- ----- #c #f #i #ab #ae #ah #ba #bd #bg #cj ---------------------------------------- ----- #d #h #ab #af #bj #bd #bh #cb #cf #dj ---------------------------------------- ----- #e #aj #ae #bj #be #cj #ce #dj #de #ej ---------------------------------------- ----- #f #ab #ah #bd #cj #cf #db #dh #ef #fj ---------------------------------------- ----- #g #ad #ba #bh #ce #db #di #ef #fc #gj ---------------------------------------- ----- #h #af #bd #cb #dj #dh #ef #fd #gb #hj ---------------------------------------- ----- #i #ah #bg #cf #de #ed #fc #gb #ha #ij ---------------------------------------- ----- #aj #bj #cj #dj #ej #fj #gj #hj #ij #ajj ---------------------------------------- ----- ,headnote3 ,! ,cases ( ,multiplication4 ,sid5ote3 ,!re are #f rules ( ,multiplication4 7#a7 ,digit by digit4 ,see ! table above4 7#b7 ,digit by article4 7#c7 ,composite by digit4 ,& !r b5 4#f rules ( ,multiplicacioun2 ffir/1 yf a digit multiplie a digit1 considre h[ _m ( vnytees b5 betwix ! digit by multiply+ & 8 4#aj4 be?e to-g$re accompt$e1 & s (t with-draw ! digit multiply+1 vnd] ! article ( 8 d5ominacioun4 ,example ( grace4 ,if ?[ wolt wete h[ mo*e is 4#d4 tymes 4#h41 #aase h[ _m vnytees b5 betwix 4#h4ab & 4#aj4 to-g$] rek5$e1 & x %ewith t 4#b43 )draw !r-= ! quatern>y1 ( ! article ( 8 d5ominacion twies1 ( 4#dj41 ,& !r remayne?e 4#cb41 t is1 to "s ( alle ! multiplicacioun4 ,:]-vpon = m evid5ce & decl>acion ! seide table is made4 ,:an a digit multiplie?e an article1 ?[ mo/ bryng ! digit into e digit1 ( e :i*e ! article has#ac 8 "n1 & euery vnyte %alle /onde = 4#aj41 & euery article an 4#ajj4 ,:an ! digit multiplie?e a nombre compon$e1 ou mo/ bryng ! digit into aier "p ( ! nombre compon$e1 s at digit be _h into digit by ! fir/ rule1 into an article by e seconde rule2 & afterw>de ,ioyne ! produccioun1 & ere wol be ! "s totalle4 ---------------------------------------- ------------ ,resultans #a #b #f #g #c #f #a #b #j #a #b #j #h ---------------------------------------- ------------ ,multiplic&u #b #c #b #f #d ---------------------------------------- ------------ ,multiplicans #f #c #b #c #b #j #c #j #b ---------------------------------------- ------------ ,sid5ote3 7#d7 ,article by article4 7#e7 ,composite by article4 7#f7 ,composite by composite4 ,h[ to set d[n y\r numb]s4 ,if ! result is a digit1 an article1 or a composite4 ,multiply next by ! la/ b "o1 & s on4 ,:an an article multiplie?e an article1 ! digit :]( he is nam$e is to be br\ght ,into ! digit :]( ! oer is nam$e1 & euery vnyte wol be wor?e 99,fol4 #eb4 an 4#ajj41 & euery article4 a 4#ajjj4 ,:an an article multiplie?e a nombre compon$e1 ?[ mo/ bryng ! digit ( ! article into ai!r "p ( ! nombre compon$e2 & ,ioyne ! produccioun1 & euery article wol be wor?e 4#ajj41 & euery vnyte 4#aj41 & s wolle ! "s be op5e4 ,:an a nombre compon$e multiplie?e a nombre compon$e1 euery part ( ! nombre multiply+ is to be _he into euery part ( ! nombre to be multipli$e1 & s %alle ! digit be _he twies1 onys 9 ! digit1 t o!r 9 ! article4 ,! article also twies1 ones 9 ! digit1 t o!r 9 ! article4 ,!r= yf ?[ wilt any nombre by hym-self o!r by any o!r multiplie1 write ! nombre to be multipli$e 9 ! ouer ordre by 8 diff]5ces1 ,! nombre multiply+ 9 ! l[] ordre by 8 diff]5ces1 s t ! fir/ ( ! l[] ordre be vnd] ! la/ ( ! ouer ordre4 ,?is done1 ( ! multiply+1 ! la/ is to be _he into ! la/ ( ! nombre to be multipli$e4 ,:]( ?an wolle gr[ a digit1 an article1 o!r a nombre compon$e4 ,if x be a digit1 ev5 above ! figure multiply+ is h$e write 8 digit t come (1 z x app]e?e "h3-- -------------------------- ,! resultant #f -------------------------- ,to be multipli$e #c -------------------------- e nombre multipliyng #b -------------------------- ,& yf an article _h be writ ouer ! figure multiply+ 8 h$e1 put a cifre er & transf]re ! article t[>de ! lift h&e1 z ?us3-- ------------------------------- ,! resultant #a #j ------------------------------- to be multipli$e #e ------------------------------- e nombre multipliyng #b ------------------------------- ,& yf a nombre compon$e be writ ouer ! figure multyply+ is h$e1 write ! digit 9 ! nombre compon$e is place1 & sette ! article to ! lift h&e1 z ?us3-- ------------------------------ ,resultant #a #b ------------------------------ to be multipli$e #d ------------------------------ ! nombre multipliyng #c ------------------------------ ,?is done1 me mo/ bryng ! la/ save "o ( ! multipliyng into ! la/ ( e nombre to be multipli$e1 & se :at comy?e !r( z be=e1 & s d with alle1 tille me come to ! fir/ ( ! nombre multiply+1 t mu/ be br\ght into ! la/ ( ! nombre to be multipli$e1 :]( gr[i?e oer a digit1 an article1 99,fol4 #eb;b o!r a nombre compon$e4 ,if x be a digit1 ,9 ! place ( ! ouerer1 sette a-side1 z "h3 -------------------------------- ,resultant #f #f -------------------------------- to be multipli$e #c -------------------------------- ! nombre multipliyng #b #b -------------------------------- ,if an article happe1 !re put a cifre 9 8 place1 & put hym to ! lift h&e1 z "h3 ---------------------------------- ,! resultant #a #a #j ---------------------------------- to be multipli$e #e ---------------------------------- e nombre multiply+ #b #b ---------------------------------- ,if x be a nombre compon$e1 9 ! place ( ! ouerer sette a-side1 write a digit t#ad is a part ( ! compon$e1 & sette on ! left honde ! article1 z "h3 ---------------------------------------- -- ,! resultant #a #cae #b ---------------------------------------- -- to be multipli$e #d ---------------------------------------- -- e nombre multipliant #c #c ---------------------------------------- -- ,sid5ote3 ,!n ant]y ! multipli] "o place4 ,"w z be=e4 ,h[ to deal ) ciph]s4 ,t done1 sette =w>de ! figures ( ! nombre multiply+ by oo diff]5ce1 s t ! fir/ ( ! multipliant be vnd] ! la/ save "o ( ! nombre to be multipli$e1 ! o!r by o place sette =w>de4 ,?an me %alle brynge ! la/ ( ! multipliant 9 hym to be multipli$e1 vnd] ! :i*e is ! fir/ multipliant4 ,& ?an wolle gr[e oer a digit1 an article1 or a compon$e nombre4 ,if x be a digit1 adde hym ev5 above 8 h$e2 ,if x be an article1 transf]re hym to ! lift side2 ,& if x be a nombre compon$e1 adde a digit to ! figure above 8 h$e1 & sette to ! lift h&e ! article4 ,& alle-wayes euery figure ( ! nombre multipliant is to be br\ght to ! la/ save "o nombre to be multipli$e1 til me come to ! fir/ ( ! multipliant1 :]e me %alle wir*e z x is seide be=e ( ! fir/1 & afterw>de to put =w>de ! figures by o diff]5ce & "o tille !y alle be multipli$e4 ,& yf x happe t ! fir/ figure ( e multipliant be a cifre1 & b\e x is sette ! figure signyficatife1 write a cifre 9 ! place ( ! figure sette a-side1 z ?us1 etc43 ------------------------------ ,! resultant #a #b #j ------------------------------ to be multipli$e #f ------------------------------ ! multipliant #b #j ------------------------------ ,sid5ote3 ,h[ to deal ) ciph]s4 ,& yf a cifre happe 9 ! l[] ord] be-twix ! fir/ & ! la/1 & ev5 above be sette ! figure signyficatif1 leve x vnt\*$e1 z "h3-- ------------------------------------ ,! resultant #b #b #f #d #d ------------------------------------ ,to be multipli$e #b #b #b ------------------------------------ ,! multipliant #a #j #b ------------------------------------ ,& yf ! space above sette be voide1 9 t place write ?[ a cifre4 ,& yf ! cifre happe betwix e fir/ & ! la/ to be multipli$e1 me mo/ sette =w>de ! ordre ( ! figures by ?aire diff]5ces1 = (t ( duccioun ( figures 9 cifres n\ght is ! resultant1 z "h1 -------------------------------------- ,resultant #h #j #j #h -------------------------------------- to be multipli$e #d #j #j #d -------------------------------------- ! multipliant #b 4 4 4 -------------------------------------- 99,fol4 #ec4 :]( x is evid5t & op51 yf t ! fir/ figure ( ! nombre be to be multipli$e be a cifre1 vndir x %alle be none sette z "h3-- ------------------------------------- ,resultant #c #b #jaf ------------------------------------- ,to be multipli$e #h #j ------------------------------------- ,! multipliant #d ------------------------------------- ,sid5ote3 ,leave room betwe5 ! r[s ( figures4 ,vnd]/& also t 9 multiplicacioun1 divisioun1 & ( rootis ! extraccioun1 compet5tly me may leve a mydel space betwix 4#b4 ordres ( figures1 t me may write !re :at is come ( addyng o!r )e-drawyng1 le/ any ?ynge %olde be ouer-hipp$e & sette out ( mynde4 ,headnote3 ,*apt] ,,vii4 ,division4 ,sid5ote3 ,def9ition ( division4 ,divid5d1 ,divisor1 ,quoti5t4 ,h[ to set d[n y\r ,sum4 ,an example4 ,examples4 ,= to dyvyde oo nombre by a-no!r1 x is ( 4#b4 nombres propos$e1 ,x is =to dep>t ! mod] nombre into z _m partis z b5 ( vnytees 9 ! lasse nombre4 ,& note wele t 9 makynge ( dyvysioun !r b5 4#c4 nombres necess>y3 t is to sey1 ! nombre to be dyvyd$e2 ! nombre dyvydyng & ! nombre exeant1 o!r h[ (t1 or quoci5t4 ,ay %alle ! nombre t is to be dyvyd$e be m1 o!r at ! le/ ev5e with ! nombre ! dyvys]e1 yf ! nombre %alle be made by hole nombres4 ,!r= yf ?[ wolt any nombre dyvyde1 write ! nombre to be dyvyd$e 9 e ouerer bordure by 8 diff]5ces1 ! dyvis]e 9 ! l[] ordure by 8 diff]5ces1 s t ! la/ ( ! dyvis] be vnd] ! la/ ( ! nombre to be dyvyde1 ! next la/ vnd] ! next la/1 & s ( ! o!rs1 yf x may compet5tly be done2 z "h3-- --------------------------- ,! residue #b #g --------------------------- ,! quoti5t #e --------------------------- ,to be dyvyd$e #c #d #b --------------------------- ,! dyvys] #f #c --------------------------- ---------------------------------------- -------- ,residuum #h #b #g #b #f ---------------------------------------- ------- ,quoci5s #b #a #b #b #e #i ---------------------------------------- ------- ,diuid5du #f #h #j #f #f #c #d #b #c #c #b ---------------------------------------- ------- ,diuis] #c #b #c #f #c #c #d ---------------------------------------- ------- ,sid5ote3 ,:5 ! la/ ( ! divisor mu/ n be set bel[ ! la/ ( ! divid5d4 ,h[ to beg94 ,& !r b5 4#b4 causes :an ! la/ figure may n be sette vnd] ! la/1 o!r t ! la/ ( ! l[] nombre may n be with-draw ( ! la/ ( ! ouerer nombre = x is lasse ?an ! l[]1 o!r h[ be x1 t x myght be with-draw z = hym-self fro ! ouerer ! rem5aunt may n s (t ( !m above1 o!r yf e la/ ( ! l[] be ev5 to ! figure above 8 h$e1 & e next la/ oer ! figure be-=e at be m an ! figure above sette4 99,fol4 #ecb4 ,^! s ordeyn$e1 me mo/ wir*e f ! la/ figure ( e nombre ( ! dyvys]1 & se h[ (t x may be with-draw ( & fro ! figure ab\e 8 h$e1 namly s t ! rem5aunt may be take ( s (t1 & to se ! residue z "h3-- ,sid5ote3 ,an example4 --------------------------- ,! residue #b #f --------------------------- ,! quoci5t #i --------------------------- ,to be dyvyd$e #c #c #b --------------------------- ,! dyvys] #c #d --------------------------- ,sid5ote3 ,:]e to set ! quoti5te4 ,examples4 ,& note wele t me may n )e-draw m ?an 4#i4 tymes ne!r lasse ?an ones4 ,!r= se h[ (t e figures ( ! l[] ordre may be with-draw fro ! figures ( ! ouerer1 & ! nombre t %ewith e quoci5t mo/ be writ ouer ! h$e ( at figure1 vnd] ! :i*e ! fir/ figure is1 ( ! dyvis]2 ,& by t figure me mo/ )e-draw alle oer figures ( ! l[] ordir & t ( ! figures ab\e ?aire h$is4 ,?is s done1 me mo/ sette =w>de e figures ( ! diuis] by o diff]5ce t[>des ! "r honde & wor*e z be=e2 & ?us3-- ---------------------------------------- ------------- ,residuum 4 #a #b ---------------------------------------- ------------- quoci5s #f #e #d #b #j #j #d ---------------------------------------- ------------- ,diuid5du #c #e #e #a #b #b #h #h #f #c #g #j #d ---------------------------------------- ------------- ,diuisor #e #d #c #d #d #b #c ---------------------------------------- ------------- ------------------------------------ ,! quoci5t #f #e #d ------------------------------------ ,to be dyvyd$e #c #e #e #a #b #b ------------------------------------ ,! dyvys] #e #d #c ------------------------------------ ,sid5ote3 ,a special case4 ,& yf x happe aft] e settyng =w>de ( ! figures at e la/ ( ! divisor may n s (t be withdraw ( ! figure above 8 h$e1 above at figure vnd] ! :i*e ! fir/ ( ! diuis] is writ me mo/ sette a cifre 9 ordre ( ! nombre quoci5t1 & sette ! figures =w>de z be-=e be o diff]5ce alone1 & s me %alle d 9 alle nombres to be dyvid$e1 = :]e ! dyvis] may n be with-draw me mo/ sette !re a cifre1 & sette =w>de ! figures2 z "h3-- --------------------------------------- ,! residue #a #b --------------------------------------- ,! quoci5t #b #j #j #d --------------------------------------- ,to be dyvyd$e #h #h #f #c #g #j #d --------------------------------------- ,! dyvys] #d #d #b #c --------------------------------------- ,sid5ote3 ,ano!r example4 ,:at ! quoti5t %[s4 ,h[ to prove y\r division1 or multiplication4 ,& me %alle n cesse fro su*e settyng ( figures =w>de1 ne!r ( settynge ( e quoci5t into ! dyvis]1 neer ( subtraccioun ( ! dyvys]1 tille ! fir/ ( ! dyvys] be with-draw fro e fir/ to be divid$e4 ,! :i*e done1 or ought1#ag oer n\ght %alle remayne3 & yf x be ought1#ag kepe x 9 ! tables1 ,& euer vny x to e divis]4 ,& yf ou wilt wete h[ _m vnytees ( e divisioun 99,fol4 #ecc4 wol gr[e to ! nombre ( ! divis]e1 ! nombre quoci5t wol %ewe x3 & :an su*e divisioun is made1 & ou lu/ prove yf ?[ h wele done or no1 ,multiplie ! quoci5t by ! divis]1 ,& ! same figures wolle come ay5e t ?[ hadde/ bi=e & none o!r4 ,& yf ought be residue1 ?an with addicioun !r( %alle come ! same figures3 ,& s multiplicacioun provi?e divisioun1 & dyvisioun multiplicacioun3 z ?us1 yf multiplicacioun be made1 divide x by ! multipliant1 & ! nombre quoci5t wol %ewe ! nombre t 0 to be multipli$e1 etc4 ,headnote3 ,*apt] ,,viii4 ,progression4 ,sid5ote3 ,def9ition ( ,progression4 ,natural ,progression4 ,brok5 ,progression4 ,! #a/ rule = ,natural ,progression4 ,! second rule4 ,! fir/ rule ( ,brok5 ,progression4 ,! second rule4 ,progressioun is ( nombre aft] egalle excesse fro oone or tweyne take agregacioun4 ( progressioun "o is naturelle or contynuelle1 at oer brok5 & discontynuelle4 ,naturelle x is1 :an me begynne?e with "o1 & kepe?e ordure ouerlepyng "o2 z 4#a4 #b4 #c4 #d4 #e4 #f41 etc41 s at ! nombre fol[ynge passi?e ! o!r be-=e 9 "o4 ,brok5 x is1 :an me lepi?e fro o nombre tille ano!r1 & kepi?e n ! contynuel ordire2 z #a4 #c4 #e4 #g4 #i1 etc4 ,ay me may begynne with 4#b41 z u2 4#b4 #d4 #f4 #h41 etc41 & ! nombre fol[yng passe?e ! o!rs by-=e by 4#b4 ,& note wele1 t naturelle progressioun ay begynne?e with "o1 & ,intercise or brok5 progressioun1 om:ile begynny?e wi? "o1 om:ile with twayne4 ,( progressioun naturell 4#b4 rules !r be yove1 ( ! :i*e ! fir/ is ?is2 :an ! progressioun naturelle endi?e 9 ev5 nombre1 by ! half !r( multiplie e next totalle ouerere nombre2 ,example ( grace3 4#a4 #b4 #c4 #d4 ,multiplie 4#e4 by 4#b4 & s 4#aj4 come?e (1 t is ! totalle nombre er(4 ,! seconde rule is su*e1 :an ! progressioun naturelle endi?e 9 nombre ode4 ,take ! m porcioun ( ! oddes1 & multiplie !rby ! totalle nombre4 ,example ( grace #a4 #b4 #c4 #d4 #e41 multiplie 4#e4 by 4#c1 & ?ryes 4#e4 %alle be resultant4 s ! nombre totalle is 4#ae4 ,( progresioun intercise1 !r b5 also 4#b4ah rules2 & e fir/ is is3 ,:an ! ,intercise progression endi?e 9 ev5 nombre by half !r( multiplie ! next nombre to at halfe z 4#b4ah #d4 #f4 ,multiplie 4#d4 by 4#c4 s at is ?ryes 4#d41 & 4#ab4 ! nombre ( alle ! progressioun1 wolle fol[4 ,! seconde rule is ?is3 :an ! progressioun interscise endi?e 9 ode1 take e m porcioun ( alle e nombre1 99,fol4 #ecd4 & multiplie by hym-selfe2 z 4#a4 #c4 #e4 ,multiplie 4#c4 by hym-selfe1 & e "s ( alle wolle be 4#i41 etc4 ,headnote3 ,*apt] ,,ix4 ,extraction ( ,roots4 ,sid5ote3 ,! preamble ( ! extraction ( roots4 ,l9e>1 sup]ficial1 & solid numb]s4 ,sup]ficial numb]s4 ,squ>e numb]s4 ,! root ( a squ>e numb]4 ,notes ( "s examples ( squ>e roots "h int]polat$4 ,solid numb]s4 ,?ree dim5sions ( solids4 ,cubic numb]s4 ,all cubics are solid numb]s4 ,no numb] may be bo? l9e> & solid4 ,unity is n a numb]4 ,"h fol[i?e ! extraccioun ( rotis1 & fir/ 9 nombre quadrates4 ,:]= me %alle se :at is a nombre quadrat1 & :at is ! rote ( a nombre quadrat1 & :at x is to draw out ! rote ( a nombre4 ,& be=e o!r note ?is divisioun3 ,( nombres "o is lyneal1 anoer superficialle1 anoer quadrat1 anoer cubike or hoole4 lyneal is t at is considr$e aft] ! processe1 havynge no respect to ! direccioun ( nombre 9 nombre1 ,z a lyne ha?e b "o dym5sioun t is to sey aft] ! l5g?e4 ,nombre superficial is at come?e ( l$ynge ( oo nombre into a-no!r1 :]= x is call$e superficial1 = x ha?e 4#b4 nombres notyng or mesurynge hym1 z a superficialle ?ynge ha?e 4#b4 dim5sions1 at is to sey l5g?e & br$e4 ,& = bycause a nombre may be _he 9 a-no!r by 4#b4 maners1 at is to sey o!r 9 hym-selfe1 oer 9 anoer1 ,vnd]/onde yf x be _h 9 hym-self1 ,x is a quadrat4 f= dyvisioun write by vnytes1 ha?e 4#d4 sides ev5 z a quadrangille4 & yf ! nombre be _he 9 a-noer1 ! nombre is superficiel & n quadrat1 z 4#b4 _he 9 4#c4 make?e 4#f4 t is e fir/ nombre superficielle2 :]= x is op5 at alle nombre quadrat is superficiel1 & n conuertide4 ,! rote ( a nombre quadrat is at nombre t is _h ( hym-self1 z twies 4#b4 maki?e #d4 & 4#d4 is ! fir/ nombre quadrat1 & #b4 is 8 rote4 #i4 #h4 #g4 #f4 #e4 #d4 #c4 #b4 #a4 _/ ,! rote ( ! m quadrat 4#c4 #a4 #d4 #b4 #f4 ,! mo/ nombre quadrat #i4 #h4 #g4 #e4 #i4 #c4 #d4 #g4 #f4 _/ ! rem55t ouer ! quadrat 4#f4 #j4 #h4 #d4 #e4 _/ ,! fir/ caas ( nombre quadrat 4#e4 #d4 #g4 #e4 #f4 ,! rote 4#b4 #c4 #d4 ,! seconde caas 4#c4 #h4 #d4 #e4 ,! rote 4#f4 #b4 ,! ?irde caas 4#b4 #h4 #a4 #i4 ,! rote 4#e4 #c4 ,! 4#d4 caas 4#c4 #b4 #a4 ,! rote 4#a4 #g4 _/ ,! #e4 caas 4#i4 #a4 #b4 #j4 #d4 _/ ,! rote #c4 #j4 #b4 ,! solide nombre or cubike is at at comyte ( d\ble l$yng ( nombre 9 nombre2 ,& x is clep$e a solide body t ha?e er-9 4#c dim5sions at is to sey1 l5g?e1 br$e1 & ?iknesse4 s at nombre ha?e 4#c4 nombres to be br\ght =?e 9 hym4 ,b nombre may be _he twies 9 nombre1 = o!r x is _he 9 hym-selfe1 oer 9 a-noer4 ,if a nombre be _he twies 9 hym-self1 oer ones 9 8 quadrat1 at is ! same1 at a cubike 99,fol4 #ed4 is1 ,& is ! same t is solide4 ,& yf a nombre twies be _he 9 a-noer1 ! nombre is clep$e solide & n cubike1 z twies 4#c4 & at 4#b4 maki?e 4#ab4 ,:]= x is opyne t alle cubike nombre is solide1 & n conuertide4 ,cubike is at nombre at comy?e ( l$ynge ( hym-selfe twyes1 or ones 9 8 quadrat4 ,& "h-by x is op5 t o nombre is ! roote ( a quadrat & ( a cubike4 ,na!les ! same nombre is n quadrat & cubike4 ,opyne x is also t alle nombres may be a rote to a quadrat & cubike1 b n alle nombre quadrat or cubike4 ,!r= si!n e l$ynge ( vnyte 9 hym-self ones or twies n\ght come?e b vnytes1 ,sei?e ,boice 9 ,arsemetrike1 t vnyte pot5cially is al nombre1 & none 9 act4 ,& vndir/onde wele also t betwix eu]y 4#b4 quadrates !r is a me5e proporcionalle1 ,t is op5$e ?us2 l$e ! rote ( o quadrat into ! rote ( ! oer quadrat1 & an wolle e me5e %ew4 ,sid5ote3 ,examples ( squ>e roots4 ------------------------------------- ,residuum #j #d #j #j ------------------------------------- ,quadr&e #dcefcjbiagdbda #i #c #f ------------------------------------- ,duplum #ab #aj #b #f #hai ------------------------------------- ,subduplum #f #f #e #ea #c #b #d #d ------------------------------------- ,sid5ote3 ,a note on mean proportionals4 ,also betwix ! next 4#b4 cubikis1 me may fynde a d\ble me5e1 t is to sey a m me5e & a lesse4 ,! m me5e ?us1 z to brynge ! rote ( ! lesse into a quadrat ( ! m4 ,! lesse ?us1 ,if ! rote ( ! m be br\ght ,into ! quadrat ( ! lesse4 ,headnote3 ,*apt] ,x4 ,extraction ( ,squ>e ,root4 ,sid5ote3 ,to f9d a squ>e root4 ,beg9 ) ! la/ odd place4 ,f9d ! ne>e/ squ>e root ( t numb]1 subtract1 d\ble x1 & set ! d\ble "o to ! "r4 ,f9d ! second figure by division4 ,multiply ! d\ble by ! second figure1 & add aft] x ! squ>e ( ! second figure1 & subtract4 #bj,to draw a rote ( ! nombre quadrat x is ,:at-euer nombre be propos$e to fynde 8 rote & to se yf x be quadrat4 ,& yf x be n quadrat ! rote ( ! mo/ quadrat fynde out1 vnd] ! nombre propos$e4 ,!r= yf ?[ wilt ! rote ( any quadrat nombre draw out1 write ! nombre by 8 diff]5ces1 & compt ! nombre ( ! figures1 & wete yf x be ode or ev54 ,& yf x be ev51 ?an mo/ ?[ begynne wor*e vnd] ! la/ save "o4 ,& yf x be ode with ! la/2 & =to sey x %ortly1 al-weyes fro ! la/ ode me %alle begynne4 ,!r= vnd] ! la/ 9 an od place sette1 me mo/ fynde a digit1 ! :i*e lade 9 hym-selfe x putti?e away t1 at is ouer 8 h$e1 oer z neighe z me may3 su*e a digit f\nde & withdraw fro 8 ouerer1 me mo/ d\ble t digit & sette ! d\ble vnd] ! next figure t[>de ! "r honde1 & 8 vnd] d\ble vnd] hym4 ,t done1 ?an me mo/ fynde a-noer digit vnd] ! next figure bi=e ! d\bl$e1 ! :i*e 99,fol4 #ed;b br\ght 9 d\ble sette?e a-way alle t is ouer 8 h$e z to rew>de ( ! d\bl$e3 ,?an br\ght into hym-self setti?e all away 9 respect ( hym-self1 ,o!r d x z nye z x may be d3 o!r me may with-draw ! digit #bala/ f\nde1 & l$e hym 9 d\ble or d\ble hym1 & aft] 9 hym-selfe2 ,?an ,ioyne to-g$] ! produccione ( !m bo!1 ,s t ! fir/ figure ( ! la/ product be add$e be=e ! fir/ ( ! fir/ productes1 ! seconde ( ! fir/1 etc4 & s =?e1 subtrahe fro ! totalle nombre 9 respect ( e digit4 ,sid5ote3 ,examples4 ---------------------------------------- - ,! residue #e #d #cb ---------------------------------------- - ,to be quadr$e #dabjiaeaci #i #j #j #e #d #cb ---------------------------------------- - ,! d\ble #dj #b #d #f #j #j ---------------------------------------- - ,! vnd] d\ble #b #j #ca #b #cc #j #j #j ---------------------------------------- - ,sid5ote3 ,special cases4 ,! residue4 ,& if x hap at no digit may be f\nde1 ,?an sette a cifre vndre a cifre1 & cesse n tille ?[ fynde a digit2 & :an ?[ ha/ f\nde x to d\ble x1 neer to sette ! d\bl$e =w>de ne!r ! vnd] d\bl$e1 ,till ?[ fynde vndre ! fir/ figure a digit1 ! :i*e lade 9 alle d\ble1 settyng away alle t is ouer hym 9 respect ( ! d\bl$e3 ,?an l$e hym into hym-selfe1 & put a-way alle 9 reg>de ( hym1 o!r z nyghe z ?[ mai/4 ,t done1 o!r ought or n\ght wolle be ! residue4 ,if n\ght1 ?an x %e)e t a nombre compon$e 0 ! quadrat1 & 8 rote a digit la/ f\nde with vnd]e-d\ble o!r vndird\bles1 s t x be sette be-=e3 ,& yf ought#bb remayne1 t %ewith t ! nombre propos$e 0 n quadrat1#bc :]-vpone se ! table 9 ! next side ( ! next leefe4 b a digit la/ f\nd ) ! subduple or subduples is ,sid5ote3 ,?is table is con/ruct$ = use 9 cube root sums1 giv+ ! value ( ab4#b ---------------------------------------- ----------- #a #b #c #d #e #f #g #h #i ---------------------------------------- ----------- #b #h #ab #af #bj #bd #bh #cb #cf ---------------------------------------- ----------- #c #ah #bg #cf #de #ed #fc #gb #ha ---------------------------------------- ----------- #d #cb #dh #fd #hj #if #aabbd #abh #add ---------------------------------------- ----------- #e #ej #ge #ajj #abe #aej #age #bjj #bbe ---------------------------------------- ----------- #f #gb #ajh #add #ahj #baf #beb #bhh #cbd ---------------------------------------- ----------- #g #ih #adg #aif #bde #bid #cdc #cic #dda ---------------------------------------- ----------- #h #abh #aib #bef #cbj #chd #ddh #eab #egf ---------------------------------------- ----------- #i #afh #bdc #cbd #dje #dhf #efg #fdh #gbibe ---------------------------------------- ----------- ,sid5ote3 ,h[ to prove ! squ>e root )\t or ) a rema9d]4 ,! rote ( ! mo/ quadrat conteyn$e vndre ! nombre propos$e4 ,!r= yf ?[ wilt prove yf ?[ h wele d or no1 ,multiplie ! digit la/ f\nde with ! vnd]-d\ble oer vnd]-d\blis1 & ?[ %alt fynde ! same figures t ?[ hadde/ be=e2 ,& s t n\ght be ! 99,fol4 #ee4 residue4 ,& yf ?[ h any residue1 ?an with ! addicioun er( t is reseru$e with-out 9 ?y table1 ?[ %alt fynde ?i fir/ figures z ?[ hadde/ !m be=e1 etc4 ,headnote3 ,*apt] ,,xi4 ,extraction ( ,cube ,root4 ,sid5ote3 ,def9ition ( a cubic numb] & a cube root4 ,m>k (f ! places 9 ?rees4 ,f9d ! fir/ digit2 treble x & place x "u ! next b "o1 & multiply by ! digit4 ,!n f9d ! second digit4 ,multiply ! fir/ triplate & ! second digit1 twice by ?is digit4 ,subtract4 ,examples4 ,he]e fol[i?e ! extraccioun ( rotis 9 cubike nombres2 :]-= me mo/ se :at is a nombre cubike1 & :at is 8 roote1 ,& :at is ! extraccioun ( a rote4 ,a nombre cubike x is1 z x is be=e decl>$e1 t come?e ( l$yng ( any nombre twies 9 hym-selfe1 o!r ones 9 8 quadrat4 ,! rote ( a nombre cubike is ! nombre t is twies _he 9 hym-selfe1 or ones 9 8 quadrat4 ,:]-?urghe x is op51 t euery nombre quadrat or cubike h ! same rote1 z x is seide be=e4 ,& =to draw out ! rote ( a cubike1 ,x is fir/ to fynde e nombre propos$e yf x be a cubike2 ,& yf x be n1 ?an ?[ mo/ make extraccioun ( 8 rote ( ! mo/ cubike vndre ! nombre proposide 8 rote f\nde4 ,!r= propos$e "s nombre1 :os cubical rote ou wolde/ draw out2 ,fir/ ?[ mo/ compt ! figures by f\r!s1 t is to sey 9 ! place ( ?\s&es2 ,& vnd] ! la/ ?\s&e place1 ?[ mo/ fynde a digit1 ! :i*e lade 9 hym-self cubikly putti?e a-way t at is ouer 8 h$e z 9 respect ( hym1 o!r z nyghe z ?[ mai/4 ,t done1 ?[ mo/ trebille ! digit1 & t triplat is to be put vnd] ! 4#c4 next figure t[>de ! "r honde1 ,& ! vnd]-trebille vnd] ! trebille2 ,?an me mo/ fynde a digit vndre ! next figure bi=e ! triplat1 ! :i*e with 8 vnd]-trebille _h into a trebille1 afterw>de o!r vnd]trebille#bf _h 9 8 produccioun1 putte?e a-way alle t is ouer x 9 reg>de (#bg ! triplat4 ,!n lade 9 hymself putti! away t at is ov] 8 h$e z 9 respect ( hym1 o!r z nyghe z ?\ mai/3 ,t done1 ?[ mo/ trebille ! digit ay5e1 & ! triplat is to be sette vnd] ! next 4#c4 figure z be=e1 ,& ! vnd]-trebille vnd] ! trebille3 & ?an mo/ ?[ sette =w>de ! fir/ triplat with 8 vndre-trebille by 4#b4 diff]5ces4 ,& ?an mo/ ?[ fynde a digit vnd] ! next figure be=e ! triplat1 ! :i*e )e 8 vnd]-triplat _h 9 8 triplat aft]w>de1 o!r vnd]-treblis lad 9 product 99,fol4 #ee;b ,x sitte?e a-way a t is ouer 8 h$e 9 respect ( ! triplat ?an _h 9 hym-self cubikly1#bh x sette?e a-way alle 8 respect or z nyghe z ye may4 ---------------------------------------- ---- ,residuum #e #d #aja #i ---------------------------------------- ---- ,cubic&us #hcfedc #b #cjjgf #g #a #aff #g ---------------------------------------- ---- ,triplum #fj #ah #d ---------------------------------------- ----- ,subtriplum #b #j #c #f #g #b #b ---------------------------------------- ---- ,sid5ote3 ,cont9ue ?is process till ! fir/ figure is rea*$4 ,examples4 ,! residue4 ,special cases4 ,special case4 ,no!r me %alle n cesse ( ! fyndynge ( t digit1 nei!r ( 8 triplacioun1 neer ( ! triplat-is #bi;ant]ioracioun1 t is to sey1 settyng =w>de by 4#b4 diff]5ces1 ,ne !r( ! vndre-triple to be put vndre ! triple1 ,ne!r ( ! multiplicacioun er(1 ,nei!r ( ! subtraccioun1 tille x come to ! fir/ figure1 vnd] ! :i*e is a digitalle nombre to be f\nde1 ! :i*e )e 8 vndre-treblis mo/ be _he 9 tribles1 ,aft]-w>de without vnd]-treblis to be _he into produccioun1 settyng away alle t is ouer ! h$e ( ! triplat nombre1 ,aft] _h into hymselfe cubikly1 & sette alle-way t is ouer hym4 ---------------------------------------- ----- ,to be cubic$e #a #g #b #h #c #b #g #f #h ---------------------------------------- ----- ,! triple #c #b #i ---------------------------------------- ----- ,! vnd] triple #a #b #c #c #c ---------------------------------------- ----- ,also note wele t ! produccion comynge ( ! l$yng ( a digite f\nde#cj with an vndre-triple _/ o!r ( an vndre-triple 9 a triple or triplat is ,& aft]-w>de with out vndre-triple o!r vndre-triplis 9 ! product & ay5e t product t come?e ( ! l$ynge ( a digit f\nde 9 hym-selfe cubicalle me may adde to1 & also with-draw fro ( ! totalle nombre sette above t digit s f\nde4#ca z !r _h be a divisioun made z x is op5$e be=e ,t done ought or n\ght mo/ be ! residue4 ,if x be n\ght1 ,x is op5 t ! nombre propos$e 0 a cubike nombre1 ,& 8 rote a digit f\nde la/ with ! vnd]-triples3 ,if ! rote !r( wex bade 9 hym-selfe1 & aft]w>de product !y %alle make ! fir/ figures4 ,& yf ought be 9 residue1 kepe t without 9 ! table2 & x is op5e t ! nombre 0 n a cubike4 b a digit la/ f\nde with ! vndirtriplis is rote ( ! mo/ cubike vndre ! nombre propos$e conteyn$e1 ! :i*e rote yf x be _he 9 hym-selfe1 ,& afterw>de 9 a product ( t %alle gr[e ! mo/ cubike vndre ! nombre propos$e conteyn$e1 ,& yf t be add$e to a cubike ! residue reseru$e 9 ! table1 wolle make ! same figures t ye _he fir/4 99,fol4 #ef4 ,& yf no digit aft] ! ant]ioracioun#cb may n be f\nde1 ?an put !re a cifre vndre a cifre vndir ! ?irde figure1 ,& put =w>de e figures4 ,note also wele t yf 9 ! nombre propos$e !r b5 no place ( ?[s&es1 me mo/ begynne vnd] ! fir/ figure 9 ! extraccioun ( ! rote4 "s vs5 =to di/+ue ! nombre by ?rees1 & ay begynne =to wir*e vndre ! fir/ ( ! la/ tern>y o!r uncomplete nombre1 ! :i*e man] ( operacioun accorde?e with t be=e4 ,& ?is at ?is tyme suffise?e 9 extraccioun ( nombres quadrat or cubikes etc4 ,sid5ote3 ,examples4 ---------------------------------------- ------------ ,! residue #j #a #a ---------------------------------------- ------------ ,! cubic&u #h #j #j #j #j #j #j #h #b #d #b #d #a #i ---------------------------------------- ------------ ,! triple #cc #j #j #f ---------------------------------------- ------------ ,! vnd]triple #b #j #j #b #f #b ---------------------------------------- ------------ ,headnote3 ,table ( ,numb]s1 c4 ,sid5ote3 ,a table ( numb]s2 probably f ! ,abacus4 #a #b #c #d #e #f "o4 x4 an4 hundr$e_/ a ?[s&e_/ x4 ?[s&e_/ ,an hundr$e #g ?[s&e_/ ,a ?[s&e tymes a ?[s&e_/ x4 ?\s&e tymes a ?\s&e_/ ,an hundr$e ?\s&e tymes a ?\s&e ,a ?\s&e ?\s&e tymes a ?\s&e_/ ?is is ! x place etc4 ,ende4 ,,footnotes 7,! ,art ( ,nombryng73 #a3 ,,ms4 ,mat]ialle4 #b3 ,,ms4 ,=malle4 #c3 #8!' 9 ,,ms4 #d3 #8be' 9 ,,ms4 #e3 #8&' 9 ,,ms4 #f3 #8is' 9 ,,ms4 #g3 #f 9 ,,ms4 #h3 #j 9 ,,ms4 #i3 #b 9 ,,ms4 #aj3 sic4 #aa3 #8,&' ins]t$ 9 ,,ms4 #ab3 #8#d !' ins]t$ 9 ,,ms4 #ac3 #8to' 9 ,,ms4 #ad3 #8t' repeat$ 9 ,,ms4 #ae3 #8#a' 9 ,,ms4 #af3 ,blank 9 ,,ms4 #ag3 #8n\ght' 9 ,,ms4 #ah3 #c writt5 = #b 9 ,,ms4 #ai3 #g 9 ,,ms4 #bj3 runs on 9 ,,ms4 #ba3 #8s' 9 ,,ms4 #bb3 #8n\ght' 9 ,,ms4 #bc3 ,,ms4 adds "h3 #8:]-vpone se ! table 9 ! next side ( ! next leefe4' #bd3 #aaj 9 ,,ms4 #be3 #j 9 ,,ms4 #bf3 d\ble 9 ,,ms4 #bg3 #8x hym-selfe' 9 ,,ms4 #bh3 ,,ms4 adds "h3 #8x sette?e a-way alle 8 respect4' #bi3 #8auct]ioracioun' 9 ,,ms4 #cj3 ,,ms4 adds "h3 'with an vndre-triple _/ o!r ( an vndre-triple 9 a triple or triplat is ,& aft]-w>de with out vndre-triple o!r vndre-triplis 9 ! product & ay5e t product t come?e ( ! l$ynge ( a digit f\nde 9 hym-selfe cubicalle' _/ #ca3 ,,ms4 adds "h3 #8z !r _h be a divisioun made z x is op5$e be=e4' #cb3 ,,ms4 ant]iocacioun4 #cc3 #d 9 ,,ms4 ,accomptynge by c\nt]s4 ,transcrib]'s ,note3 ,! orig9al text 0 pr9t$ z a s+le cont9u\s p>agraph1 ) no break betwe5 speak]s2 all examples 7 %[n inl9e4 ,x has be5 brok5 up = ?is e-text4 99#aaf;b ,! seconde dialoge ( accomptynge by c\nt]s4 ,may/]4 ,n[e t y haue le>n$ ! comm5 kyndes ( ,ari?metyke ) ! p5ne1 y %all se ! same art 9 c\nt]s3 :i*e feate do? n only s]ue = !m t c n write & r$e1 b also = !m t c d bo!1 b haue n at "s tymes !yr p5ne or tables r$ye ) !m4 ,?is sorte is 9 two f\rmes comm5ly4 ,! "o by lynes1 & ! o!r )\t lynes3 9 t yt ha? lynes1 ! lynes d /&e = ! ord] ( places3 & 9 yt t ha? no lynes1 !re mu/ be sette 9 !yr /$e s _m c\nt]s z %all n$e1 = e*e lyne "o1 & !y %all supplye ! /$e ( ! lynes4 ,s4 ,by examples ,i %uld bett] perceaue y\r meanynge4 ,m4 ,= example ( ! 99#aag;a4 ly99nes3 ----#a-j-j-j-j-j-- ----#a-j-j-j-j---- -,x--#a-j-j-j------ ----#a-j-j-------- ----#a-j---------- ----#a------------ ,sid5ote3 ,num]ation4 ,lo "h y se 4vi4 lynes :i*e /&e = syxe places s t ! ne!rmo/ /&e? = ye fyr/ place1 & ! next ab\e x1 = ! second3 & s vpw>d tyll y come to ! hyghe/1 :i* is ! syxte lyne1 & /&e? = ! syxte place4 ,n[ :at is ! valewe ( eu]y place or lyne1 y may p]ceaue by ! figures :i*e ,i haue set on !m1 :i* is accordynge z y le>n$ be=e 9 ! ,num]ation ( figures by ! p5ne3 = ! fyr/e place is ! place ( vnities or ones1 & eu]y c\nt] set 9 t lyne betok5e? b "o3 & ! seconde lyne is ! place ( #aj1 = eu]y c\nt] !re1 /&e? = #aj4 ,! ?yrd lyne ! place ( hundr$es3 ! f\r? ( ?\s&es3 & s =?4 ,s4 ,syr ,i d p]ceaue t ! same ord] is "h ( lynes1 z 0 9 ! o!r figures 99#aag;b by places1 s t y %all n n$e long] to /&e ab\t ,num]ation1 excepte !re be any o!r diff]5ce4 ,m4 ,yf y d vnd]/ande x1 !n h[ wyll y set #aedc8 ,s4 ,?us1 z ,i suppose4 ------- -,x--#a-- ----#e-- ----#d-- ----#c-- ,m4 ,y haue set ye places truely1 b y\r figures be n mete = ?is vse3 = ! mete/ figure 9 ?is behalfe1 is ! figure ( a c\nt] r\nd1 z y se "h1 :]e ,i haue express$ t same summe4 ------------- -,x--o-------- o ------------- ----o-o-o-o-- ----o-o-o---- ,s4 ,s t y haue n "o figure = #b1 nor #c1 nor #d1 & s =?1 b z _m digettes z y haue1 y set 9 ! l[e/ lyne3 & = eu]y #aj y set "o 9 ! second l9e3 & s ( o!r4 ,b ,i "k n by :at reason y set t "o c\nt] = #ejj betw5e two lynes4 ,m4 y %all rememb] ?is1 t :5 s eu] y n$e to set d[ne #e1 #ej1 or #ejj1 or #ejjj1 or s =? any o!r nomb]1 ^: num]ator 99#aah;a is #e1 y %all set "o c\nt] = x1 9 ! next space ab\e ! lyne t x ha? 8 d5om9ation (1 z 9 ?is example ( t #ejj1 bycause ! num]ator is #e1 x mu/ be set 9 a voyd space3 & bycause ! d5om9ator is hundr$1 ,i kn[e t 8 place is ! voyde space next ab\e hundr$es1 t is to say1 ab\e ! ?yrd lyne4 ,& f>!r y %all m>ke1 t 9 all workynge by ?is sorte1 yf y %all sette d[ne any summe betw5e #d & #aj1 = ! fyr/e p>te ( t nomb] y %all set d[ne #e1 !n s _m c\nt]s m1 z !re re/e nomb]s ab\e #e4 ,& ?is is true bo! ( digettes & articles4 ,& = example ,i wyll set d[ne ?is summe #bhgife1 -,x----------- ------o-o---- o ------o-o-o-- o -,x----o-o---- o ----o-o-o-o-- o ----o-------- o ------------- :i* summe yf y m>ke well1 y n$e none o!r examples = to l]ne ! num]ation ( 99#aah;b ?is =me4 ,b ?is %al y m>ke1 t z y dyd 9 ! o!r kynde ( ari?metike1 set a pricke 9 ! places ( ?\sandes1 9 ?is worke y %all sette a />re1 z y se "h4 ,headnote3 ,addition on ! ,c\nt+ ,bo>d4 ,sid5ote3 ,addition4 ,s4 ,!n ,i p]ceave num]ation1 b ,i praye y1 h[e %all ,i d 9 ?is arte to adde two summes or m toge!r8 ,m4 ,! easye/ way 9 ?is arte is1 to adde b #b summes at ones toge!r3 h[ be x y may adde m1 z ,i wyll tell y anone4 ,!r=e :5 y wyll adde two summes1 y %all fyr/ set d[ne "o ( !m1 x =se? n :i*e1 & !n by x drawe a lyne crosse ! o!r lynes4 ,& aft]w>d set d[ne ! o!r summe1 s t t lyne may be betw5e !m1 z yf y wolde adde #bfei to #hcdb1 y mu/ set y\r summes z y se ------------------------ o -,x--o-o-o------o-o------ o ----o-o-o------o-------- o ----o-o-o-o------------- o ----o-o--------o-o-o-o-- "h4 ,& !n yf y ly/1 y 99#aai;a may adde ! "o to ! o!r 9 ! same place1 or els y may adde !m bo? toge!r 9 a newe place3 :i* waye1 bycause x is mo/e playne/1 ,i wyll %[e y fyr/4 ,!r=e wyl ,i begynne at ! vnites1 :i*e 9 ! fyr/ summe is b #b1 & 9 ye second summe #i1 t make? #aa1 ^? d ,i take vp1 & = !m ,i set #aa 9 ! new r\me1 ?us1 --------------------------- o -,x--o-o-o------o-o--------- o ----o-o-o------o----------- o ----o-o-o-o----------o----- ---------------------o----- ,!n d ,i take vp all ye articles vnd] a hundr$1 :i* 9 ! fyr/ summe are #dj1 & 9 ! second summe #ej1 t make? #ij3 or y may saye bett]1 t 9 ! fyr/e summe !re are #d articles ( #aj1 & 9 ! seconde summe #e1 :i* make #i1 b !n take h$e t y sette !m 9 !yr 99#aai;b ryght lynes z y se "h4 ---------------------------------- o -,x--o-o-o----o-o------------------ o ----o-o-o----o-------------------- o -----------------------o-o-o-o-o-- -----------------------o---------- ,:]e ,i haue tak5 awaye #dj from ! fyr/e summe1 & #ej f ye second1 & 9 !yr /$e ,i haue set #ij 9 ! ?yrde1 :i*e ,i haue set playnely yt y myght well p]ceaue x3 h[ be x seynge t #ij ) ! #aj t 0 9 ye ?yrd r\me all r$y1 do? make #ajj1 ,i myghte bett] = ^? #f c\nt]s set #a 9 ! ?yrde lyne1 ?us3 ---------- -,x-------- ----o----- ---------- ----o----- ,= x is all "o summe z y may se1 b x is be/e1 neu] to set #e c\nt]s 9 any l9e1 = t may be done ) #a c\nt] 9 a hygh] place4 ,s4 ,i iudge t good reason1 = _m are vnn$efull1 :]e "o wyll s]ue4 ,m4 ,well1 !n 99#abj;a wyll ,i adde =? ( hundr$es3 ,i fynde #c 9 ! fyr/e summe1 & #f 9 ! seconde1 :i*e make #ijj1 !m d ,i take vp & set 9 ! ?yrd r\me :]e is "o hundr$ all r$y1 to :i*e ,i put #ijj1 & x wyll be #ajjj1 !r=e ,i set "o c\nt] 9 ! f\r? lyne = !m all1 z y se "h4 -------------------------- o -,x--o-o-o----o-o----o----- -------------------------- -------------------------- --------------------o----- ,!n adde ,i ye ?\s&es toge!r1 :i*e 9 ! fyr/ summe are #hjjj1 & 9 ye second #bjjj1 t make? #ajjjj3 !m d ,i take vp from ^? two places1 & = !m ,i set "o c\nt] 9 ! fyfte lyne1 & !n app]e? z y se1 to be #aajja1 = s _m do? am\nt ( ! addition ( #hcdb to #bfei4 ----o----- -,x--o----- ---------- ---------- ----o----- 99#abj;b ,s4 ,syr1 ?is ,i d p]ceave3 b h[ %all ,i set "o summe to an o!r1 n *aungynge !m to a ?yrde place8 ,m4 ,m>ke well h[ ,i d x3 ,i wyll adde toge!r #fedcf1 & #cbde1 :i*e fyr/e ,i set d[ne ?us4 --------------------------- o ---------------o----------- o -,x--o-o-o------------------ ----o-o--------o-o-o-o----- ----o-o-o-o----o-o-o------- o o ---------------o----------- ,!n d ,i begynne ) ! smale/1 :i* 9 ! fyr/ summe is #e1 t d ,i take vp1 & wold put to ! o!r #e 9 ! seconde summe1 sauynge t two c\nt]s c n be set 9 a voyd place ( #e1 b = !m bo! ,i mu/ set #a 9 ! seconde lyne1 :i* is ! place ( #aj1 !r=e ,i take vp ! #e ( ! fyr/ summe1 & ! #e ( ! seconde1 & = !m ,i set #a 9 ! second lyne1 99#aba;a z y se "h4 --------------------------- o ---------------o----------- o -,x--o-o-o------------------ ----o-o--------o-o-o-o----- ----o-o-o-o----o-o-o-o----- ---------------o----------- ,!n d ,i lyke wayes take vp ! #d c\nt]s ( ! fyr/e summe & seconde lyne 7:i* make #dj7 & adde !m to ! #d c\nt]s ( ! same lyne1 9 ! second summe1 & x make? #hj1 ,b z ,i sayde ,i maye n conu5i5tly set ab\e #d c\nt]s 9 "o lyne1 !r=e to ^? #d t ,i toke vp 9 ! fyr/ summe1 ,i take "o also ( ! seconde summe1 & !n haue ,i tak5 vp #ej1 = :i*e #e c\nt]s ,i sette d[ne "o 9 ! space ou] ye second lyne1 z "h do? app]e4 ------------------------- o -------------o----------- o -,x--o-o-o---------------- ----o-o------o-o-o-o----- o -------------o-o-o------- -------------o----------- 99#aba;b4 & !n is !re #hj1 z well wt ^? #d c\nt]s1 z yf ,i _h set d[ne ye o!r #d also4 ,n[ d ,i take ! #bjj 9 ! fyr/e summe1 & adde !m to ! #djj 9 ! seconde summe1 & x make? #fjj1 !r=e ,i take vp ! #b c\nt]s 9 ! fyr/e summe1 & #c ( !m 9 ! seconde summe1 & = !m #e ,i set #a 9 ye space ab\e1 ?us4 ----------------------- o -------------o--------- o -,x--o-o-o-------------- o -------------o--------- o -------------o-o-o----- -------------o--------- ,!n ,i take ye #cjjj 9 ye fyr/e summe1 vnto :i*e !re are none 9 ! second summe agreynge1 !r=e ,i d onely rem\e ^? #c c\nt]s f ! fyr/e summe into ! seconde1 z "h do? app]e4 ----------------- o -------o--------- o -,x-----o-o-o----- o -----o----------- o -------o-o-o----- -------o--------- 99#abb;a4 ,& s y see ! hole summe1 t am\nte? ( ! addytion ( #fedcf ) #cbde to be #fhfha4 ,& yf y haue m>k$ ^! two examples well1 y n$e no f>!r en/ruction 9 ,addition ( #b only summes3 b yf y haue m !n two summes to adde1 y may adde !m ?us4 ,fyr/ adde two ( !m1 & !n adde ! ?yrde1 & ye f\r?1 or m yf !re be s _m3 z yf ,i wolde adde #bfgi ) #dbhf & #acia4 ,fyr/e ,i adde ! two fyr/e summes ?us4 -------------------------------------- o -,x--o-o--------o-o-o-o----o----------- o o ----o----------o-o--------o-o-o-o----- o o o ----o-o--------o-o-o------o----------- o o o ----o-o-o-o----o---------------------- 99#abb;b4 ,& !n ,i adde ! ?yrde !reto ?us4 ,& s ( m yf y haue !m4 ------------------------------------ o o -,x--o----------o----------o-o-o----- o ----o-o-o------o-o-o-o----o-o-o----- o o o ----o-o-o-o------o------------------ o o ----o---------------------o--------- ,headnote3 ,subtraction on ! ,c\nt+ ,bo>d4 ,sid5ote3 ,subtraction4 ,s4 ,n[e ,i ?ynke be/e t y passe =? to ,subtraction1 except !re be any wayes to examyn ?is man] ( ,addition1 !n ,i ?ynke t 7 good to be kn[5 nexte4 ,m4 ,!re is ! same pr(e "h t is 9 ! o!r ,addition by ! p5ne1 ,i meane ,subtraction1 = t onely is a sure waye3 b consyd]ynge t ,subtraction mu/ be fyr/e kn[51 ,i wyl fyr/e tea*e y ! arte ( ,subtraction1 & t by ?is example3 ,i wolde subtracte #bhib out ( #hgdf4 ,^! summes mu/ ,i set d[ne z ,i dyd 9 ,addition3 b "h x is be/ 99#aaf;a 7sic74 to set ! less] nomb] fyr/e1 ?us4 --------------------------- o -,x--o-o--------o-o-o------- o o ----o-o-o------o-o--------- o ----o-o-o-o----o-o-o-o----- o ----o-o--------o----------- ,!n %all ,i begynne to subtracte ! greate/ nombres fyr/e 7contr>y to ! vse ( ! p5ne7 yt is ! ?\s&es 9 ?is example3 !r=e ,i fynd amonge/ ! ?\s&es #b1 = :i* ,i )drawe s _m from ! seconde summe 7:]e are #h7 & s remayne? !re #f1 z ?is example %[e?4 --------------------------- o --------------o----------- o o ----o-o-o------o-o--------- o ----o-o-o-o----o-o-o-o----- o ----o-o--------o----------- ,!n d ,i lyke wayes ) ! hundr$es1 ( :i*e 9 ! fyr/e summe 99#aaf;b ,i fynde #h1 & is ! seconde summe b #g1 out ( :i*e ,i c n take #h1 !r=e ?us mu/e ,i d3 ,i mu/e loke h[ mo*e my summe dyff]e? f #aj1 :i*e ,i fynde "h to be #b1 !n mu/ ,i bate = my summe ( #hjj1 "o ?\s&e1 & set d[ne ! excesse ( hundr$es1 t is to saye #b1 = s mo*e #ajjj is m !n ,i %uld take vp4 ,!r=e from ! fyr/e summe ,i take t #hjj1 & f ! second summe :]e are #fjjj1 ,i take vp "o ?\s&e1 & leue #ejjj2 b !n set ,i d[ne ! #bjj unto ! #gjj yt are !re all r$ye1 & make !m #ijj ?us4 --------------------------- o -------------------------- o ---------------o-o-o-o----- o ----o-o-o-o----o-o-o-o----- o ----o-o--------o----------- ,!n come ,i to ! articles ( tennes :]e 9 ! fyr/e summe ,i fynde #ij1 99#aag;a & 9 ! seconde summe b only #dj3 ,n[ consyd]yng t #ij c n be bat$ f #dj1 ,i loke h[ mo*e yt #ij do? dyff] f ! next summe ab\e x1 t is #ajj 7or elles :i*e is all to "o effecte1 ,i loke h[ mo* #i do? dyff] from #aj7 & ,i fynd x to be #a1 !n 9 ! /$e ( t #ij1 ,i d take f ! second summe #ajj3 b consyd]ynge t x is #aj to mo*e1 ,i set d[ne #a 9 ye nexte lyne b5e? = x1 z y se "h4 --------------------- o -------------------- o -----------o-o-o----- o --------------------- o ----o-o----o--------- ,sauynge t "h ,i haue set "o c\nt] 9 ye space 9 /$e ( #e 9 ye nexte lyne4 ,& ?us haue ,i subtract$ all saue two1 :i* ,i mu/ bate f ! #f 9 ! second summe1 & !re wyll remayne #d1 ?us4 ------------------ o ----------------- o ------o-o-o------- o ------------------ ------o-o-o-o----- ,s yt yf ,i subtracte #bhib from #hgdf1 ! remayn] wyll be #ehed1 99#aag;b ,& t ?is is truely wr\ght1 y maye pr\e by ,addition3 = yf y adde to ?is remayn] ! same summe t y dyd subtracte1 !n wyll ! =m> summe #hgdf am\nt agayne4 ,s4 ,t wyll ,i pr\e3 & fyr/ ,i set ! summe t 0 subtract$1 :i* 0 #bhib1 & !n ! remayn] #ehed1 ?us4 ---------------------------- o ---o-o-------------------- o o -----o-o-o------o-o-o------- o o -----o-o-o-o---------------- -----o-o--------o-o-o-o----- ,!n d ,i adde fyr/ ye #b to #d1 :i*e make? #f1 s take ,i vp #e ( ^? c\nt]s1 & 9 !yr /$e ,i sette #a 9 ! space1 z "h app]e?4 -------------------------- o ---o-o------------------ o o -----o-o-o------o-o-o----- o o -----o-o-o-o-------------- o ----------------o--------- 99#aah;a ,!n d ,i adde ! #ij nexte ab\e to ! #ej1 & x make? #adj1 !r=e ,i take vp ^? #f c\nt]s1 & = !m ,i sette #a to ! hundr$es 9 ye ?yrde lyne1 & #d 9 ye second lyne1 ?us4 -------------------------- o ---o-o------------------ o o -----o-o-o----o-o-o-o----- --------------o-o-o-o----- o ----------------o--------- ,!n d ,i come to ! hundr$es1 ( :i*e ,i fynde #h 9 ! fyr/ summe1 & #i 9 ye second1 t make? #agjj1 !r=e ,i take vp ^? #i c\nt]s1 & 9 !yr /$e ,i sette #a 9 ! 4iiii4 lyne1 & #a 9 ! space nexte b5e?1 & #b 9 ! ?yrde lyne1 z y se "h4 ------------------------ o ---o-o----o----------- o ------------o-o--------- ------------o-o-o-o----- o ------------o----------- ,!n is !re lefte 9 ! fyr/e summe b only #bjjj1 :i*e ,i %all take vp f !nce1 & set 99#aah;b 9 ! same lyne 9 ye second summe1 to ye "o yt is !re all r$y3 & !n wyll ! hole summe app]e 7z y may wel se7 to be #hgdf1 :i* 0 ye fyr/ grosse summe1 & !r=e ,i d p]ceaue1 t ,i hadde well subtract$ be=e4 ,& ?us y may se h[ ,subtraction maye be try$ by ,addition4 ------------------ o -,x----o-o-o------- o ------o-o--------- ------o-o-o-o----- o --------o--------- ,s4 ,i p]ceaue ! same ord] "h wt c\nt]s1 yt ,i l]n$ be=e 9 figures4 ,m4 ,!n let me se h[e c y trye ,addition by ,subtraction4 ,s4 ,fyr/e ,i wyl set =? ?is example ( ,addition :]e ,i haue add$ #bahi to #dihh1 & ! hole summe app]e? to be #gagg1 ----------------------------------- o ---o-o--------o-o-o-o----o-o----- o -----o----------o-o-o-o----o------- o o o -----o-o-o------o-o-o------o-o----- o o o -----o-o-o-o----o-o-o------o-o----- 99#aai;a ,n[e to trye :e!r t summe be well add$ or no1 ,i wyll subtract "o ( ! fyr/ two summes f ! ?yrd1 & yf ,i haue well done ye remayn] wyll be lyke t o!r summe4 ,z = example3 ,i wyll subtracte ! fyr/e summe f ! ?yrde1 :i*e ,i set ?us 9 !yr ord]4 ------------------------ o ---o-o--------o-o----- -----o----------o------- o o -----o-o-o------o-o----- o o -----o-o-o-o----o-o----- ,!n d ,i subtract #bjjj ( ! fyr/e summe from ye second summe1 & !n remayne? !re #ejjj ?us4 ----------------------- o -,x---------------------- ----o----------o------- o o ----o-o-o------o-o----- o o ----o-o-o-o----o-o----- ,!n 9 ! ?yrd lyne1 ,i subtract ye #ajj ( ! fyr/e summe1 from ! second summe1 :]e is onely #ajj also1 & !n 9 ye ?yrde lyne re/e? no?yng4 ,!n 9 ! second lyne ) 8 space ou] hym1 ,i fynde #hj1 :i* ,i %uld subtract 99#aai;b f ! o!r summe1 !n seyng !re are b only #gj ,i mu/ take x out ( "s hygh] summe1 :i* is "h only #ejjj1 !r=e ,i take vp #ejjj1 & seyng t x is to mo* by #dibj1 ,i sette d[ne s _m 9 ! seconde r\me1 :i*e ) ! #gj beynge !re all r$y d make #diij1 !n ! summes do? /&e ?us4 ---------------------------- --------------o-o-o-o----- o ----------------o-o-o-o----- o ----------------o-o-o-o----- o o -----o-o-o-o----o-o--------- ,yet remayne? !re 9 ! fyr/ summe #i1 to be bat$ f ! second summe1 :]e 9 t place ( vnities do! app]e only #g1 !n ,i mu/e bate a hygh] summe1 t is to saye #aj1 b seynge t #aj is m !n #i 7:i* ,i %ulde abate7 by #a1 !r=e %all ,i take vp "o c\nt] f ! seconde lyne1 & set d[ne ! same 9 ! fyr/ 99#abj;a or l[e/ lyne1 z y se "h4 ------------------- -----o-o-o-o----- o -------o-o-o-o----- o -------o-o-o------- o -------o-o-o------- ,& s haue ,i end$ ?is worke1 & ! summe app]e? to be ye same1 :i*e 0 ye seconde summe ( my addition1 & !r=e ,i p]ceaue1 ,i haue wel done4 ,m4 ,to /&e long] ab\t ?is1 x is b folye3 excepte t ?is y maye also vnd]/&e1 t _m d begynne to subtracte ) c\nt]s1 n at ! hyghe/ summe1 z ,i haue taught y1 b at ! ne!rmo/e1 z !y d vse to adde3 & :5 ! summe to be abatyd1 9 any lyne appe>e? great] !n ! o!r1 !n d !y bor[e "o ( ! next hygh] r\me1 z = example3 yf !y %uld abate #ahdf f #bcgh1 !y set ye summes ?us4 -------------------------- ---o----------o-o------- o -----o-o-o------o-o-o----- o -----o-o-o-o----o-o------- o o -----o----------o-o-o----- 99#abj;b ,& fyr/e !y take #f :i*e is 9 ! l[] lyne1 & 8 space f #h 9 ! same r\mes1 9 ye second summe1 & yet !re remayne? #b c\nt]s 9 ! l[e/ lyne4 ,!n 9 ! second lyne mu/ #d be subtracte f #g1 & s remayne? !re #c4 ,!n #h 9 ! ?yrde lyne & 8 space1 f #c ( ! second summe c n be1 !r=e d !y bate x f a hygh] r\me1 t is1 f #ajjj1 & bycause t #ajjj is to mo* by #bjj1 !r=e mu/ ,i sette d[ne #bjj 9 ! ?yrde lyne1 aft] ,i haue tak5 vp #ajjj f ! f\r? lyne3 !n is !re yet #ajjj 9 ! f\r? lyne ( ! fyr/ summe1 :i*e yf ,i )drawe f ! seconde summe1 !n do? all ye figures /&e 9 ?is ord]4 ----------------- --------------- o ----------------- -------o-o-o----- -------o-o------- ,s t 7z y se7 x diff]e? n greatly :e!r y begynne subtraction at ! hygh] lynes1 or at 99#aba;a ! l[]4 ,h[ be x1 z "s m5ne lyke ! "o waye be/e1 s "s lyke ! o!r3 !r=e y n[ kn[yng bo!1 may vse :i*e y ly/4 ,headnote3 ,multiplication by ,c\nt]s4 ,sid5ote3 ,multiplication4 ,b n[e t\*ynge ,multiplication3 y %all set y\r nomb]s 9 two r\mes1 z y dyd 9 ^? two o!r kyndes1 b s t ! multipli] be set 9 ! fyr/e r\me4 ,!n %all y begyn ) ! hyghe/ nomb]s ( ye seconde r\me1 & multiply !m fyr/ aft] ?is sort4 ,take t ou]mo/ lyne 9 y\r fyr/ workynge1 z yf x 7 ! l[e/ lyne1 sett+ on x "s m\able m>ke1 z y ly/e1 & loke h[ _m c\nt]s be 9 hym1 take !m vp1 & = !m set d[ne ! hole multyply]1 s _m tymes z y toke vp c\nt]s1 reck5yng1 ,i saye t lyne = ! vnites3 & :5 y haue s done ) ! hyghee/ nomb] !n come to ! nexte lyne b5e?1 & d eu5 s ) x1 & s ) ye next1 tyll y haue done all4 ,& yf !re be any nomb] 9 a space1 !n = x 99#aba;b %all y take ye multiply] #e tymes1 & !n mu/ y reck5 t lyne = ! vnites :i*e is nexte b5e? t space3 or els aft] a %ort] way1 y %all take only halfe ! multyply]1 b !n %all y take ! lyne nexte ab\e t space1 = ! lyne ( vnites3 b 9 su*e workynge1 yf *aunce y\r multyply] be an odde nomb]1 s t y c n take ! halfe ( x iu/ly1 !n mu/e y take ! great] halfe1 & set d[ne t1 z if t x 7 ! iu/e halfe1 & f>!r y %all set "o c\nt] 9 ! space b5e? t l9e1 :i* y reck5 = ! lyne ( vnities1 or els only rem\e =w>d ! same t is to be multyply$4 ,s4 ,yf y set =? an example h]eto ,i ?ynke ,i %al p]ceaue y4 ,m4 ,take ?is example3 ,i wold multiply #aedb by #cfe1 !r=e ,i set ye nomb]s ?us4 -------------------------- ------------o----------- o -----o-o-o---------------- o -----o--------o-o-o-o----- o --------------o-o--------- 99#abb;a ,!n fyr/e ,i begynne at ! #ajjj 9 ye hyghe/ r\me1 z yf x 7 ye fyr/ place1 ,i take x vp1 settynge d[ne = x s (t5 7t is ones7 ! multyply]1 :i* is #cfe1 ?us1 z y se "h3 ---------------------------------- ------------------------o-o-o----- o ------------------------o--------- o -,x-------------------------------- - o ----o-o-o------------------------- o ----o--------o-o-o-o-------------- o -------------o-o------------------ :]e = ! "o c\nt] tak5 vp f ! f\r? lyne1 ,i haue sette d[ne o!r #f1 :i*e make ye summe ( ! multyply]1 reck5ynge t f\r? lyne1 z yf x 7 ! fyr/e3 :i*e ?yng ,i haue m>k$ by ! h& set at ! begynnyng ( ye same1 ,s4 ,i p]ceaue ?is well3 = 9 d$e1 ?is summe t y haue set d[ne is #cfejjj1 = s mo*e do? am\nt 99#abb;b ( #ajjj1 multiply$ by #cfe4 ,m4 ,well !n to g =?1 9 ! nexte space ,i fynde "o c\nt] :i* ,i rem\e =w>d b take n vp1 b d 7z 9 su* case ,i mu/7 set d[ne ! great] halfe ( my multipli] 7seyng x is an odde nomb]7 :i* is #ahb1 & "h ,i d /yll let t f\r? place /&1 z yf x 7 ye fyr/3 -------------------------o-o-o----o----- ---- o o -------------------------o--------o-o-o- ---- o --------------------------------o-o----- -- - o -----o-o-o------------------------------ ---- o -----o--------o-o-o-o------------------- ---- o --------------o-o----------------------- ---- z 9 ?is f\rme y se1 :]e ,i haue set ?is multiplycation ) ye o!r3 b = ! ease ( y\r vnd]/andynge1 ,i haue set a lytell lyne betw5e !m3 n[ %ulde !y bo? 9 "o summe /& ?us4 -------------------------o-o-o-o-o----- -------------------------o-o-o-o------- o -----------------------o-o----------- - o -----o-o-o----------------------------- o -----o--------o-o-o-o------------------ o --------------o-o---------------------- 99#abc;a ,h[e be x an o!r f\rme to multyplye su*e c\nt]s in space is ?is3 ,fyr/ to rem\e ! fyng] to ! lyne nexte b5e! ye space1 & !n to take vp ye c\nt]1 & to set d[ne ye multiply] 4v4 tymes1 z "h y se4 -------------------o-o-o---------------- ---------------- o -------------------o-----o-o-o-o-o-o-o-o -o-o-o-o-o-o-o-- o o o o o o -------------------------o-----------o-- ---o-----o------ o o o o o --,x-o-o-o------------------------------ ------------------ o ---o------o-o-o-o----------------------- ---------------- o ----------o-o--------------------------- ---------------- ,:i* summes yf y d adde toge!r into "o summe1 y %al perceaue t x wyll be ye same yt appe>e? ( ye o!r working be=e1 s t 99#abc;b bo! sortes are to "o ent5t1 b z ! o!r is mu* %ort]1 s ?is is playn] to reason1 = su*e z haue _h small ex]cyse 9 ?is arte4 ,n )/&ynge y maye adde !m 9 y\r mynde be=e y sette !m d[ne1 z 9 ?is example1 y myghte haue sayde #e tymes #cjj is #aejj1 & #e tymes #fj is #cjj1 also #e tymes #e is #be1 :i*e all put toge!r d make #ahbe1 :i* y maye at "o tyme set d[ne yf y ly/e4 ,b n[e to g =?1 ,i mu/ rem\e ! h& to ! nexte c\nt]s1 :i*e are 9 ! second lyne1 & !re mu/ ,i take vp ^? #d c\nt]s1 settynge d[ne = !m my multiply] #d tymes1 :i*e ?ynge o!r ,i maye d at #d tymes seu]ally1 or elles ,i may ga!r t hole summe 9 my mynde fyr/e1 & !n set x d[ne3 z to saye #d tymes #cjj is #abjj3 #d tymes #fj are #bdj3 & #d tymes #e make #bj3 yt is 9 all #adfj1 yt %all ,i set d[ne also3 z "h y se4 o ---------------------------------------- --- --------------------o-o-o-o----o-------- --- o -,x------------------o-o--------o-o-o-o- ---- o o ----o-o-o----------------------o-------- --- o - ----o----------------------------------- --- o -------------o-o------------------------ --- 99#abd;a :i*e yf ,i ioyne 9 "o summe ) ! =m> nomb]s1 x wyll appe>e ?us4 o -------------------------- o ------------------o------- ------------------o-o----- --o-o-o----------o-------- o - --o----------------------- o -----------o-o------------ ,!n to ende ?is multiplycation1 ,i rem\e ! fyng] to ! l[e/ lyne1 :]e are onely #b1 !m d ,i take vp1 & 9 !yr /$e d ,i set d[ne twyse #cfe1 t is #gcj1 = :i* ,i set 99#abd;b "o 9 ! space ab\e ! ?yrd lyne = #ejj1 & #b m 9 ! ?yrd lyne ) t "o t is !re all r$ye1 & ! re/e 9 !yr ord]1 & s haue ,i end$ ! hole summe ?us4 o -------------------------- o ----------------o--------- ----------------o-o------- o --o-o-o---------o-o-o----- o --o-------------o-o-o----- o -------------------------- ,:]by y se1 t #aedb 7:i* is ! nomb] ( ye>es sy? ,*ry/es inc>nation7 beyng multyply$ by #cfe 7:i* is ! nomb] ( dayes 9 "o ye>e7 do! am\nte vnto #efbhcj1 :i* decl>e? ye nomb] ( daies si? ,*ry/es inc>nation vnto ! ende ( #aedba ye>es4 7besyde #che dayes & #ab h\res = lepe ye>es74 ,s4 ,n[ wyll ,i pr\e by an o!r example1 z ?is3 #dj lab\r]s 7aft] #f d4 ye day = e*e man7 haue wr\ght #bh dayes1 ,i wold 99#abe;a "k :at !yr wages do? am\nt vnto3 ,9 ?is case mu/e ,i worke d\blely3 fyr/ ,i mu/ multyplye ! nomb] ( ! lab\r]s by ye wages ( a man = "o day1 s wyll ye *>ge ( "o daye am\nt3 !n second>ely %all ,i multyply t *>ge ( "o daye1 by ! hole nomb] ( dayes1 & s wyll ! hole summe appe>e3 fyr/ !re=e ,i %all set ! summes ?us4 -------------------- -------------------- -------------------- --------o-o-o-o----- o --o----------------- ,:]e 9 ! fyr/e space is ! multyply] 7yt is "o dayes wages = "o man7 & 9 ! second space is set ! nomb] ( ! worke m5 to be multyply$3 !n saye ,i1 #f tymes #d 7reck5ynge t second lyne z ! lyne ( vnites7 make? #bd1 = :i*e summe ,i %ulde set #b c\nt]s 9 ! ?yrde lyne1 & #d 9 ! seconde1 !r=e d ,i set #b 9 ! ?yrde lyne1 & let ! #d /& /yll 9 ! seconde lyne1 ?us499#abe;b ------------------- ------------------- -------o-o--------- -------o-o-o-o----- ------------------- ,s apw]e? ! hole dayes wages to be #bdj;d'4 t is #bj s4 ,!n d ,i multiply agayn ! same summe by ! nomb] ( dayes & fyr/e ,i sette ! nomb]s1 ?us4 ----------------------- ----------------------- -----------o-o--------- --o-o------o-o-o-o----- o --o-o-o--------------- ,!n bycause !re are c\nt]s 9 dyu]s lynes1 ,i %all begynne ) ! hyghe/1 & take !m vp1 settynge = !m ! multyply] s _m tymes1 z ,i toke vp c\nt]s1 yt is twyse1 !n wyll ye summe /&e ?us4 ------------------- o ------------------- o -------o----------- -------o-o-o-o----- ------------------- ,!n come ,i to ye seconde lyne1 & take vp ^? #d c\nt]s1 settynge = !m ! multiply] f\re tymes1 s wyll ! hole summe appe>e ?us499#abf;a --------------- o -------o------- o -------o-o----- -------o-o----- --------------- ,s is ! hole wages ( #dj workemen1 = #bh dayes 7aft] #f;d'4 e*e daye = a man7 #fgbj;d'4 t is #efj s4 or #bh l'i4 ,headnote3 ,division on ! ,c\nt+ ,bo>d4 ,sid5ote3 ,diuision4 ,m4 ,n[ if y wold pr\e ,multiplycation1 ! sure/ way is by ,dyuision3 !r=e wyll ,i ou] passe x tyll ,i haue taught y ye arte ( ,diuision1 :i*e y %all worke ?us4 ,fyr/e sette d[ne ! ,diuisor = fe>e ( =gettynge1 & !n set ! nomb] t %albe deuid$1 at ye ryghte syde1 s f>re f ! diuisor1 t ! quoti5t may be set betw5e !m3 z = example3 ,yf #bbe %epe co/ #de l'i4 :at dyd eu]y %epe co/8 ,to kn[e ?is1 ,i %ulde diuide ! hole summe1 t is #de l'i4 by #bbe1 b t c n be1 !r=e mu/ ,i fyr/e r$uce t #de l'i4 into a less] d5om9ation1 z into %yllynges3 !n ,i multiply #de by #bj1 & x is #ijj1 t summe %all ,i diuide by ! nomb] ( 99#abf;b %epe1 :i*e is #bbe1 ^! two nomb]s !r=e ,i sette ?us4 -------------------------- -------------------------- o --o-o---------o-o-o-o----- --o-o--------------------- o -------------------------- ,!n begynne ,i at ! hyghe/ lyne ( ! diuid5t1 & seke h[ (t5 ,i may haue ! diuisor !r91 & t maye ,i d #d tymes1 !n say ,i1 #d tymes #b are #h1 :y*e yf ,i take f #i1 !re re/e? b #a1 ?us -------------------------- -------------------------- --o-o---------------o----- --o-o--------------------- o ---------o-o-o-o---------- ,& bycause ,i f\nde ! diuisor #d tymes 9 ! diuid5te1 ,i haue set 7z y se7 #d 9 ! myddle r\me1 :i* 99#abg;a is ! place ( ! quoti5t3 b n[ mu/ ,i take ! re/e ( ! diuisor z (t5 out ( ! remayn]3 !r=e come ,i to ! seconde lyne ( ! diuisor1 say5g #b f\re tymes make #h1 take #h f #aj1 & !re re/e? #b1 ?us4 ------------------------------- ----------------------------- -----o-o----------------------- -----o-o---------------o-o----- o ------------o-o-o-o------------ ,!n come ,i to ! l[e/ nomb]1 :i* is #e1 & multyply x #d tymes1 s is x #bj1 t take ,i f #bj1 & !re remayne? no?ynge1 s t ,i se my quoti5t to be #d1 :i*e are 9 valewe %yllynges1 = s 0 ! diuid5t3 & !rby ,i kn[e1 t yf #bbe %epe dyd co/e #de l'i4 eu]y %epe co/e #d s4 ,s4 ,?is c ,i d1 z y %all p]ceaue by ?is example3 ,yf #afj s[ldy>s d sp5de eu]y mone? #fh l'i4 :at sp5de? e*e man8 ,fyr/ 99#abg;b bycause ,i c n diuide ! #fh by #afj1 !r=e ,i wyll turne ! p\ndes into p5nes by multiplicacion1 s %all !re be #afcbj d'4 ,n[e mu/e ,i diuide ?is summe by ! nomb] ( s[ldy>s1 !r=e ,i set !m in ord]1 ?us4 ----------------o--------- o --------------o--------- -----o----------o-o-o----- o -----o----------o-o------- -------------------------- ,!n begyn ,i at ! hyghe/ place ( ! diuid5te1 sekynge my diuisor !re1 :i*e ,i fynde ones1 ,!r=e set ,i #a 9 ! ne!r lyne4 ,m4 ,n 9 ! ne!r l9e ( ! hole summe1 b 9 ! ne!r lyne ( t worke1 :i*e is ! ?yrde lyne4 ,s4 ,s /&e? x ) reason4 ,m4 ,!n ?us d !y /&e499#abh;a -------------------------- ------------------------ -----o-----o----o-o-o----- o -----o----------o-o------- -------------------------- ,!n seke ,i agayne 9 ! re/e1 h[ (t5 ,i may fynde my diuisor1 & ,i se t 9 ! #cjj ,i myghte fynde #ajj ?re tymes1 b !n ! #fj wyll n be s (t5 f\nde 9 #bj1 !r=e ,i take #b = my quoti5t3 !n take ,i #ajj twyse f #cjj1 & !re re/e? #ajj1 out ( :i*e ) ! #bj 7t make? #abj7 ,i may take #fj also twyse1 & !n /&e? ! nomb]s ?us1 --------------------- ------------------- -----o-----o--------- o -----o--------------- -----------o-o------- 99#abh;b :]e ,i haue sette ! quoti5t #b 9 ! l[e/ lyne3 ,s is eu]y s[ldy>s portion #ajb d'4 t is #h s4 #f d'4 ,m4 ,b yet bycause y %all p]ceaue iu/ly ! reason ( ,diuision1 x %all be good t y d set y\r diuisor /yll agayn/ ^? nombres from :i*e y d take x3 z by ?is example ,i wyll decl>e4 ,yf ye pur*ace ( #bjj acres ( gr\nd dyd co/e #bij l'i4 :at dyd "o acre co/e8 ,fyr/ wyl ,i turne ! p\ndes into p5nes1 s wyll !re be #fifjj d' ,!n 9 settynge d[ne ^! nomb]s ,i %all d ?us4 ---------------------------- o ----o-o---------o----------- o -,x--------------o-o-o-o----- o ----------------o----------- ---------------------------- ---------------------------- ,fyr/ set ! diuid5t on ! ryghte h&e z x oughte1 & !n 99#abi;a ! diuisor on ! lefte h&e agayn/ ^? nomb]s1 from :i* ,i ent5de to take hym fyr/ z "h y se1 :] ,i haue set ! diuisor two lynes hygh] !n is !yr owne place4 ,s4 ,?is is lyke ! ord] ( diuision by ! p5ne4 ,m4 ,tru? y say1 & n[e mu/ ,i set ye quoti5t ( ?is worke 9 ! ?yrde lyne1 = t is ! lyne ( vnities 9 respecte to ! diuisor 9 ?is worke4 ,!n ,i seke h[e (t5 ! diuisor maye be f\nde 9 ! diuid5t1 & t ,i fynde #c tymes1 !n set ,i #c 9 ! ?yrde lyne = ! quoti5t1 & take awaye t #fjjjj from ! diuid5t1 & f>!r ,i d set ! diuisor "o l9e l[]1 z y[ se "h4 --------------------------------- o ---o-o-------------o-o-o-o----- o ------------o-o-o------o--------- --------------------------------- --------------------------------- 99#abi;b ,& !n seke ,i h[ (t5 ! diuisor wyll be tak5 f ! nomb] agayn/e x1 :i*e wyll be #d tymes & #a remaynynge4 ,s4 ,b :at yf x *aunce t :5 ! diuisor is s rem\$1 x c n be ones tak5 out ( ! diuid5t agayn/e x8 ,m4 ,!n mu/ ! diuisor be set 9 an o!r l9e l[]4 ,s4 ,s 0 x 9 diuision by ! p5ne1 & !r=e 0 !re a cyph] set 9 ! quoti5t3 b h[e %all t be not$ "h8 ,m4 ,"h n$e? no tok51 = ! lynes d repres5te ! places3 onely loke t y set y\r quoti5t 9 t place :i* /&e? = vnities 9 respecte ( ! diuisor3 b n[ to returne to ! example1 ,i fynde ! diuisor #d tymes 9 ! diuid5te1 & #a remaynynge1 = #d tymes #b make #h1 :i* ,i take f #i1 & !re re/e? #a1 z ?is figure %ewe?3 ------------------------------ ---o-o---------------o------ o ------------o-o-o------o------ ------------o-o-o-o----------- ------------------------------ & 9 ! myddle space = ! quoti5t ,i set #d 9 ! seconde lyne1 :i*e is 9 ?is worke ! place ( vnities499#acj;a ,!n rem\e ,i ye diuisor to ! next l[] l9e1 & seke h[ (t5 ,i may haue x 9 ! dyuid5t1 :i* ,i may d "h #h tymes iu/1 & no?ynge remayne1 z 9 ?is f\rme1 -------------------------- ---o-o------------------ ------------o-o-o--------- ------------o-o-o-o------- o ------------o-o-o--------- :]e y may se t ! hole quoti5t is #cdh d'1 t is #bi s4 :]by ,i kn[e t s mo*e co/e ! pur*ace ( "o ak]4 ,s4 ,n[ re/e? ! pr(es ( ,multiplycation1 & also ( ,diuision4 ,m4 ,!r be/ pr(es are e*e 99#acj;b "o by ! o!r1 = ,multyplication is pr\$ by ,diuision1 & ,diuision by ,multiplycation1 z 9 ! worke by ! p5ne y le>n$4 ,s4 ,yf t be all1 y %all n n$e to repete agayne t1 yt 0 sufficyently taughte all r$ye3 & excepte y wyll tea*e me any o!r feate1 "h maye y make an ende ( ?is arte ,i suppose4 ,m4 ,s wyll ,i d z t\*ynge hole nomb]1 & z = brok5 nomb]1 ,i wyll n tr\ble y\r wytte ) x1 tyll y haue practis$ ?is s well1 yt y be full p]fecte1 s t y n$e n to d\bte 9 any poynte t ,i haue taught y1 & !nne maye ,i boldly en/ructe y 9 ye arte ( fractions or brok5 nomb]1 :]9 ,i wyll also %[e y ! reasons ( all t y haue n[e le>n$4 ,b yet be=e ,i make an ende1 ,i wyll %[e y ! ord] ( comm5 ca/yng1 :] 9 are bo! p5nes1 %yllynges1 & p\ndes1 proc$ynge by no gr\nd$ reason1 b onely by a receau$ 99#aca;a f\rme1 & t dyu]sly ( dyu]s m53 = m>*auntes vse "o f\rme1 & auditors an o!r3 ,headnote3 ,m]*ants' ,ca/+ ,c\nt]s4 ,sid5ote3 ,m]*ants' ca/+4 ,b fyr/e = m>*auntes f\rme m>ke ?is example "h1 o o o o o o o o o o o o o o o o o o o o o o 9 :i* ,i haue express$ ?is summe #aih l'i4#b #ai s4 #aa d'4 ,s t y maye se t ! l[e/ lyne s]ue? = pennes1 ! next ab\e = %yllynges1 ! ?yrde = p\ndes1 & ! f\r? = scores ( p\ndes4 ,& f>!r y maye se1 t ! space betw5e p5nes & %yllynges may receaue b "o c\nt] 7z all o!r spaces lyke wayes d7 & t "o /&e? 9 t place = #f d'4 ,lyke wayes betw5e ! %yllynges & ! p\ndes1 "o c\nt] /&e? = #aj s4 ,& betw5e ! p\ndes & #bj l'i4 "o c\nt] /&e? = #aj p\ndes4 ,b besyde ^? y maye see at ! left syde ( %yllynges1 t "o c\nt] /&e? alone1 & betok5e? #e s4 99#aca;b ,s agayn/e ! p\ndes1 t "o c\nt] /&e? = #e l'i4 ,& agayn/ ! #bj p\ndes1 ! "o c\nt] /&e? = #e score p\ndes1 t is #ajj l'i4 s t eu]y syde c\nt] is #e tymes s mo* z "o ( !m agayn/ :i*e he /&e?4 ,sid5ote3 ,auditors' ca/+4 ,n[ = ! accompt ( auditors take ?is example4 o o o o o o o o o o o o o o o o o o o o o o :]e ,i haue express$ ye same summe #aih l'i4 #ai s4 #aa d'4 ,b "h y se ! pennes /&e t[>d ye ryght h&e1 & ! o!r encreasynge ord]ly t[>de ! lefte h&e4 ,agayne y maye se1 t audit\rs wyll make #b lynes 7yea & m7 = p5nes1 %yllynges1 & all o!r valewes1 yf !yr summes ext5de !rto4 ,also y se1 t !y set "o c\nt] at ! ryght ende ( e*e r[e1 :i*e s set !re /&e? = #e ( t r\me3 & on 99#acb;a ! lefte corn] ( ! r[e x /ande? = #aj1 ( ye same r[4 ,b n[ yf y wold adde o!r subtracte aft] any ( bo? ^? sortes1 yf y m>ke ye ord] ( yt o!r feate :i* ,i taught y1 y may easely d ! same "h )\t mo* tea*ynge3 = 9 ,addition y mu/ fyr/ set d[ne "o summe & to ! same set ! o!r ord]ly1 & lyke man] yf y haue _m3 b 9 ,subtraction y mu/ sette d[ne fyr/ ! greate/ summe1 & f x mu/ y abate t o!r eu]y d5om9ation f 8 dewe place4 ,s4 ,i d n d\bte b ) a lytell practise ,i %all attayne ^! bo!3 b h[ %all ,i multiply & diuide aft] ^! f\rmes8 ,m4 ,y c n duely d none ( bo? by ^! sortes1 !r=e 9 su*e case1 y mu/ resort to y\r o!r artes4 ,s4 ,syr1 yet ,i se n by ^! sortes h[ to expresse hundr$des1 yf !y exc$e "o hundr$1 no!r yet ?\s&es4 ,m4 ,!y t vse su* accomptes t x exc$e #bjj 99#acb;b 9 "o summe1 !y sette no #e at ! lefte h&e ( ! scores ( p\ndes1 b !y set all ! hundr$es 9 an o!r f>!r r[e & #ejj at ! lefte h& !r(1 & ! ?\s&es !y set 9 a f>!r r[e yet1 & at ! lefte syde !r( !y sette ! #ejjj1 & 9 ! space ou] !y sette ! #ajjjj1 & 9 a hygh] r[e #bjjjj1 :i*e all ,i haue express$ 9 ?is example1 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o :i* is #ighfi l'i4 #ab s4 #i d' ob4 q4 = ,i _h n told y be=e :]e1 no!r h[ y %uld set d[ne f>?ynges1 :i* 7z y se "h7 mu/ be set 9 a voyde space sydelynge b5e? ! p5nes3 = q "o c\nt]3 = ob4 #b c\nt]s3 = ob4 q4 #c c\nt]s3 & m !re c n be1 = #d f>?ynges 99#acc;a d make #a d'4 :i* mu/ be set 9 8 dewe place4 ,headnote3 ,auditors' ,ca/+ ,c\nt]s4 ,& yf y desyre ye same summe aft] audytors man]1 lo "h x is4 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o ,b 9 ?is ?yng1 y %all take ?is = suffycy5t1 & ! re/e y %all obs]ue z y maye se by ! work+ ( e*e sorte3 = ! dyu]s wittes ( m5 haue inu5t$ dyu]s & sundry wayes almo/ vnnum]able4 ,b "o feate ,i %all tea*e y1 :i*e n only = ! /raung5es & secretnes is mo*e pleasaunt1 b also = ! good commoditie ( x ryghte wor?y to be well m>k$4 ,?is feate ha? b5 vs$ ab\e #bjjj ye>es at ! lea/e1 & yet 0 x neu] comenly kn[51 especyally 9 ,englys%e x 0 neu] taughte yet4 ,?is is ! arte ( nombrynge on ! h&1 ) diu]s ge/ures ( ! fyng]s1 expressynge any summe conceau$ 9 ! 99#acc;b mynde4 ,& fyr/ to begynne1 yf y wyll expresse any summe vnd] #ajj1 y %all expresse x ) y\r lefte h&e3 & f #ajj vnto #ajjjj1 y %all expresse x ) y\r ryght h&e1 z "h ord]ly by ?is table fol[ynge y may p]ceaue4 ,"h fol[e? ! table ( ! arte ( ! h&e ,! arte ( nombrynge by ! h&e4 ,transcrib]'s ,note3 ,footnote #c reads3 8,bracket 77 d5otes new p>agraph 9 orig9al40 ,= ?is e-text1 ! brackets h be5 omitt$ 9 favor ( re/or+ ! p>agraph breaks4 ,*anges ( speak] 7,m1 ,s7 are also m>k$ by p>agraphs1 z 9 ! previ\s selection4 ,! illu/ration includes ! pr9t$ page numb] #acd2 !re is !re=e no sid5ote 99#acd;a4 ,! sid5ote = 8#d0 is miss+4 ,illu/ration3 7,numb]s z describ$ 9 text7 ,sid5ote3 #a 99#acd;b ,9 :i* z y may se #a is express$ by ye lyttle fyng] ( ye lefte h&e closely & h>de crok$4 ,sid5ote3 #b #cb is decl>$ by lyke b[ynge ( ! w$dynge fyng] 7:i*e is ! nexte to ! lyttell fyng]7 toge!r ) ! lytell fyng]4 ,sid5ote3 #c #c is signifi$ by ! myddle fyng] b[$ 9 lyke man]1 ) ^? o!r two4 #d is decl>$ by ! b[yng ( ! myddle fyng] & ! rynge fyng]1 or w$dynge fyng]1 ) ! o!r all /ret*$ =?4 ,sid5ote3 #e1 #f #e is repres5t$ by ! myddle fyng] onely b[$4 ,& #f by ! w$dynge fyng] only crook$3 & ?is y may m>ke 9 ^! a c]tayne ord]4 ,b n[ #g1 #h1 & #i1 are express$ with ! b[ynge ( ! same fyng]s z are #a1 #b1 & #c1 b aft] an o!r f\rme4 ,sid5ote3 #g ,= #g is decl>$ by ! b[ynge ( ! lytell fyng]1 z is #a1 saue t = #a ! fyng] is clasp$ 91 h>de & 99#ace;a r\nde1 b = to expresse #g1 y %all b[e ! myddle ioynte ( ! lytell fyng] only1 & holde ! o!r ioyntes /reyght4 ,s4 ,yf y wyll geue me leue to expresse x aft] my rude man]1 ?us ,i vnd]/& y\r meanyng3 t #a is express$ by crookynge 9 ! lyttell fyng] lyke ! head ( a bys%oppes bagle3 & #g is decl>$ by ! same fyng] b[$ lyke a gybbet4 ,m4 ,s ,i p]ceaue1 y vnd]/&e x4 ,sid5ote3 #h ,!n to expresse #h1 y %all b[e aft] ! same man] bo? ! lyttell fyng] & ! rynge fyng]4 ,sid5ote3 #i1 #aj ,& yf y b[e lyke wayes ) !m ! myddle fyng]1 !n do? x betok5 #i4 ,n[ to expresse #aj1 y %all b[e y\r =e fyng] r\nde1 & set ! ende ( x on ! hyghe/ ioynte ( ! ?ombe4 ,sid5ote3 #bj ,& = to expresse #bj1 y mu/ set y\r fyng]s /reyght1 & ! ende ( y\r ?ombe to ! p>tition ( ! 99#ace;b =e mo/e & myddle fyng]4 ,sid5ote3 #cj #cj is repres5t$ by ! ioynynge toge!r ( ye headdes ( ! =emo/ fyng] & ! ?ombe4 ,sid5ote3 #dj #dj is decl>$ by settynge ( ! ?ombe crossewayes on ! =emo/ fyng]4 ,sid5ote3 #ej #ej is signifi$ by ryght /ret*yng =? ( ! fyng]s ioyntly1 & apply5ge ( ! ?ombes ende to ! p>tition ( ! myddle fyng] & ! rynge fyng]1 or w$dynge fyng]4 ,sid5ote3 #fj #fj is =m$ by b5dynge ( ! ?ombe crok$ & crossynge x ) ! =e fyng]4 ,sid5ote3 #gj #gj is express$ by ! b[ynge ( ! =emo/ fyng]1 & settynge ! ende ( ! ?ombe betwe5 ! #b =emo/ or hyghe/ ioyntes ( x4 ,sid5ote3 #hj #hj is express$ by settynge ( ! =emo/ fyng] crossewayes on ! ?ombe1 s t #hj dyff]e? ?us from #dj1 t = #hj ! =efyng] is set crosse on ! ?ombe1 & = #dj ! ?ombe is set crosse ou] ye =ef+]4 ,sid5ote3 #ij 99#acf;a #ij is signifi$1 by b5dynge ! =e fyng]1 & settyng ! ende ( x 9 ! inn]mo/ ioynte ( ye ?ombe1 t is eu5 at ! foote ( x4 ,& ?us are all ! nomb]s end$ vnd] #ajj4 ,sid5ote3 #aa1 #ab1 #ac1 #ba1 #bb1 #bc ,s4 ,9 d$e ^! be all ! nomb]s from #a to #aj1 & !n all ! t5!s )9 #ajj1 b ?is teacy$ me n h[ to expresse #aa1 #ab1 #ac1 etc4 #ba1 #bb1 #bc1 etc4 & su* lyke4 ,m4 ,y c lytell vnd]/&e1 yf y c n d t )\t tea*ynge3 :at is #aa8 is x n #aj & #a8 !n expresse #aj z y 7 taught1 & #a also1 & t is #aa3 & = #ab expresse #aj & #b3 = #bc set #bj & #c3 & s = #fh y mu/e make #fj & !re to #h3 & s ( all o!r sortes4 ,sid5ote3 #ajj ,b n[ yf y wolde repres5te #ajj o!r any nomb] ab\e x1 y mu/e d t ) ! ryghte h&e1 aft] ?is man]4 ,y mu/ expresse #ajj 9 ! ryght h&1 ) ! lytell fyng] s b[$ z y dyd expresse #a 9 ! left h&4 ,sid5ote3 #bjj 99#acf;b ,& z y express$ #b 9 ! lefte h&e1 ! same fas%yon 9 ! ryght h&e do? decl>e #bjj4 ,sid5ote3 #cjj ,! f\rme ( #c 9 ! ryght h& /&e? = #cjj4 ,sid5ote3 #djj ,! f\rme ( #d1 = #djj4 ,sid5ote3 #ejj ,lykewayes ! f\rme ( #e1 = #ejj4 ,sid5ote3 #fjj ,! f\rme ( #f1 = #fjj4 ,& to be %orte3 loke h[ y did expresse s+le vnities & t5!s 9 ! lefte h&e1 s mu/ y expresse vnities & t5!s ( hundr$es1 9 ! ryghte h&e4 ,sid5ote3 #ijj ,s4 ,i vnd]/&e y ?us3 t yf ,i wold repres5t #ijj1 ,i mu/ s f\rme ! fyng]s ( my ryghte h&e1 z ,i %uld d 9 my left h& to expresse #i1 ,sid5ote3 #ajjj ,& z 9 my lefte h& ,i express$ #aj1 s 9 my ryght h&e mu/ ,i expresse #ajjj4 ,& s ! f\rme ( eu]y t5! 9 ! lefte h&e s]ue? to expresse lyke nomb] ( ?\sandes1 ,sid5ote3 #djjj s ye f\rme ( #dj /&e? = #djjj4 ,sid5ote3 #hjjj ,! f\rme ( #hj = #hjjj4 ,sid5ote3 #ijjj 99#acg;a ,& ! f\rme ( #ij 7:i*e is ! greate/7 = #ijjj1 & ab\e t ,i c n expresse any nomb]4 ,m4 ,no n ) "o fyng]3 h[ be x1 with dyu]s fyng]s y maye expresse #iiii1 & all at "o tyme1 & t lac ke? b #a ( #ajjjj4 ,s t vnd] #ajjjj y may by y\r fyng]s ex- presse any summe4 ,& ?is %al suf- fyce = ,num]ation on ! fyng]s4 ,& z = ,addition1 ,subtraction1 ,multiplication1 & ,diuision 7:i* yet 7 neu] taught by any man z f>re z ,i d kn[e7 ,i wyll en/ruct y aft] ! treatyse ( fractions4 ,& n[ = ?is tyme f>e well1 & loke t y cease n to practyse t y haue le> n$4 ,s4 ,syr1 ) mo/e h>ty mynde ,i ?anke y1 bo! = y\r good le>nyng1 & also y\r good c\nsel1 :i* 7god wyllyng7 ,i tru/e to fol[4 ,f9is4 ,,footnotes 7,accomptynge by c\nt]s & ,! arte ( nombrynge by ! h&e73 #a3 #acdb 9 orig9al4 #b3 #afh 9 orig9al4 #c3 ,bracket 77 d5otes new p>agraph 9 orig9al4 ,,app5dix ,i4 ,a ,treatise on ! ,num]ation ( ,algorism4 ,f a ,,ms4 ( ! #ad? ,c5tury4 ,to alle su*e ev5 nombrys ! mo/ h cifrys z to t54 tw5ty4 ?irtty4 an hundr$4 an ?\s& & su*e o!r4 b ye s*al vnd]/onde t a cifre tok5e? no?+e b he make? o!r ! m significatyf t comi? aft] hym4 ,also ye s*al vnd]/onde t 9 nombrys composyt & 9 alle o!r nombrys t b5 ( div]se figurys ye s*al begynne 9 ! ri?t syde & to rek5e backw>de & s he s*al be wryte z ?us--#ajjj4 ! sifre 9 ! ri?t side 0 fir/ wryte & yit he tok5e? no?+e to ! secunde no ! ?ridde b !i mak5 t figure ( #a ! m signyficatyf t comi? aft] hem by z mo*e z he born oute ( 8 fir/ place :]e he s*uld yf he /ode !r tok5e b "o4 ,& !re he /ondi? n[e 9 ! f]ye place he tok5e? a ?\s& z by ?is rewle4 ,9 ! fir/ place he tok5e? b hymself4 ,9 ! secunde place he tok5e? t5 times hymself4 ,9 ! ?ridde place he tok5e? an hundr$ tymes himself4 ,9 ! f]ye he tok5e? a ?\s& tymes himself4 ,9 ! fyftye place he tok5e? t5 ?\s& tymes himself4 ,9 ! sexte place he tok5e? an hundr$ ?\s& tymes hymself4 ,9 ! seve? place he tok5e? t5 hundr$ ?\s& tymes hymself1 c4 ,& ye s*al vnd]/ond t ?is worde nombre is p>tyd into ?re p>tyes4 ,somme is callyd nombre ( digitys = alle b5 digitys t b5 )9e t5 z ix1 viii1 vii1 vi1 v1 iv1 iii1 ii1 i4 ,articules b5 alle !i t m[ be devyd$ into nombrys ( t5 z xx1 xxx1 xl1 & su*e o!r4 ,composittys be alle nombrys t b5 componyd ( a digyt & ( an articule z f\rt5e fyft5e ?ritt5e & su*e o!r4 ,f\rt5e is componyd ( f\r t is a digyt & ( t5 t is an articule4 ,fyft5e is componyd ( fyve t is a digyt & ( t5 t is an articule & s ( o!rs 4 4 4 4 4 4 ,b z to ?is rewle4 ,9 ! fir/e place he tok5e? b himself t is to say he tok5e? b t & no m4 ,if t he /onde 9 ! secunde place he tok5e? t5 tymes himself z ?is figure #b "h #ba4 ?is is oon & tw5ty4 ,?is figure #b /ondi? 9 ! secunde place & !r= he tok5e? t5 tymes himself & t5 tymes #b is tw5ty & s =ye ( e figure & he /onde aft] ano!r t[>d ! le/ syde he s*al tok5e t5 tymes z mo*e m z he s*uld tok5 & he /ode 9 t place !r t ! figure a=e him /onde?3 lo an example z ?us #ifcd4 ,?is figure ( f\re t ha? ?is s*ape #d tok5e? b himself = he /onde? 9 ! fir/ place4 ,! figure ( ?re t ha? ?is s*ape #c tok5e? t5 tyme himself = he /onde? 9 ! secunde place & t is ?ritti4 ,! figure ( sexe t ha? ?is s*ape #f tok5e? t5 tyme m ?an he s*uld & he /ode 9 ! place y] ! figure ( ?re /onde? = !r he s*uld tok5e b sexty4 ,& n[ he tok5e? t5 tymes t is sexe hundrid4 ,! figure ( nyne t ha? ?is s*ape #i tok5e? t5 tymes m ?an he s*ulde & he /ode 9 ! place !r ! figure ( #f /onde? inne = ?anne he s*uld tok5e b nyne hundryd4 ,& 9 ! place t he /onde? inne n[e he tok5e? n9e ?\s&4 ,alle ! hole nombre ( ^! f\re figurys4 ,n9e ?\s& sexe hundrid & f\re & ?ritti4 ,,app5dix ,,ii4 ,c>m5 de ,algorismo4 ,f a ,b4,m4 ,,ms41 #h ,c4 iv41 ) additions f #ab ,e4 #a ,eg4 #bfbb4 ,hec algorismus ars pres5s dicitur#a2 9 qua ,talibus ,indorum#b fruimur 8 qu9que figuris4 #j4 #i4 #h4 #g4 #f4 #e4 #d4 #c4 #b4 #a4 ,prima significat unum3 duo v]o secunda3 ,t]cia significat tria3 sic proc$e s9i/re #d ,donec ad extremam v5ies1 qua cifra vocatur2 #c,que nil significat2 dat signific>e sequ5ti4 ,quelibet ill>um si primo limite ponas1 ,simplicit] se significat3 si v]o secundo1 #h ,se decies3 sursum proc$as multiplic&o4#d ,namque figura sequ5s quevis signat decies plus1 ,ipsa locata loco quam significet p]eunte3 #ab ,nam prec$5tes plus ultima significabit4 #e,po/ pr$icta scias quod tres breuit] num]orum ,di/9cte species sunt2 nam quidam digiti sunt2 ,articuli quidam2 quidam quoque compositi sunt4 #af ,sunt digiti num]i qui citra d5>ium sunt2 ,articuli decupli degitorum2 compositi sunt ,illi qui con/ant ex articulis digitisque4 ,ergo1 proposito num]o tibi scrib]e1 primo #bj ,respicias quis sit num]us2 quia si digitus sit1 #e,una figura satis sibi2 s$ si compositus sit1 ,primo scribe loco digitum po/ articulum fac ,articulus si sit1 cifram po/ articulum sit1 #bd ,articulum v]o reliqu5ti 9 scribe figure4 ,quolibet 9 num]o1 si p> sit prima figura1 ,p> erit et totum1 quicquid sibi cont9etur2 ,imp> si fu]it1 totum sibi fiet et imp>4 #bh ,septem#f sunt p>tes1 non plures1 i/ius artis2 ,add]e1 subtrah]e1 dupl>e1 dimidi>e2 ,sexta e/ diuid]e1 set qu9ta e/ multiplic>e2 ,radicem extrah]e p>s septima dicitur esse4 #cb ,subtrahis aut addis a dextris vel m$iabis2 ,a leua dupla1 diuide1 multiplicaque2 ,extrahe radicem semp] sub p>te s9i/ra4 ,sid5ote3 ,addition4 ,add]e si num]o num]um vis1 ord9e tali #cf ,incipe2 scribe duas primo s]ies num]orum ,prima sub prima recte pon5do figuram1 ,et sic de reliquis facias1 si s9t tibi plures4 ,inde duas adde primas hac condicione2 #dj ,si digitus crescat ex addicione priorum1 ,primo scribe loco digitum1 quicunque sit ille2 ,si sit compositus1 9 limite scribe sequ5ti ,articulum1 primo digitum2 quia sic iubet ordo4 #dd ,articulus si sit1 9 primo limite cifram1 ,articulum v]o reliquis inscribe figuris2 ,vel p] se scribas si nulla figura sequatur4 ,si tibi cifra sup]u5i5s occurr]it1 illam #dh ,deme suppositam2 po/ illic scribe figuram3 ,po/ea proc$as reliquas add5do figuras4 ,sid5ote3 ,subtraction4 ,a num]o num]um si sit tibi dem]e cura1 ,scribe figur>um s]ies1 vt 9 addicione2 #eb ,maiori num]o num]um suppone m9orem1 ,siue p>i num]o supponatur num]us p>4 ,po/ea si possis a prima subtrahe primam1 ,scrib5s quod remanet1 cifram si nil remanebit4 #ef ,set si non possis a prima dem]e primam2 ,proc$5s1 vnum de limite deme sequ5ti2 ,et demptum pro d5>io reputabis ab illo1 ,subtrahe totalit] num]um quem proposui/i4 #fj ,quo facto1 scribe supra quicquit remanebit1 ,facque nov5>ios de cifris1 cum remanebis1 ,occurrant si =te cifre1 dum dems]is vnum2 ,po/ea proc$as reliquas dem5do figuras4 #fd ,sid5ote3 ,pro(4 #g,si subtracio sit b5e facta prob>e valebis1 ,quas subtraxi/i primas add5do figuras4 ,nam1 subtractio si b5e sit1 primas ret9ebis1 ,et subtractio facta tibi probat additionem4 #fh ,sid5ote3 ,duplation4 ,si vis dupl>e num]um1 sic incipe2 solam ,scribe figur>um s]iem1 quamcumque voles que ,po/ea proc$as primam dupl&o figuram2 ,inde quod excrescet1 scrib5s1 vbi iuss]it ordo1 #gb ,juxta precepta que dantur 9 addicione4 ,nam si sit digitus1 9 primo limite scribe2 ,articulus si sit1 9 primo limite cifram1 ,articulum v]o reliquis inscribe figuris2 #gf ,vel p] se scribas1 si nulla figura sequatur3 ,compositus si sit1 9 limite scribe sequ5ti ,articulum primo1 digitum2 quia sic jubet ordo3 ,et sic de reliquis facias1 si s9t tibi plures4 #hj #h,si sup] extremam nota sit1 monadem dat eidem1 ,quod tibi cont+it1 si primo dimidiabis4 ,sid5ote3 ,m$iation4 ,incipe sic1 si vis aliquem num]um m$i>e3 ,scribe figur>um s]iem solam1 velud ante2 #hd ,po/ea proc$5s m$ias1 et prima figura ,si p> aut imp> videas2 quia si fu]it p>1 ,dimidiabis eam1 scrib5s quicquit remanebit2 ,imp> si fu]it1 vnum demas1 m$i>e1 #hh ,nonne presumas1 s$ quod sup]e/ m$iabis2 ,inde sup] tractum1 fac demptum quod notat unum2 ,si monos1 dele2 sit ibi cifra po/ nota supra4 ,po/ea proc$as hac condicione secunda3#i #ib ,imp>#aj si fu]it hic vnum deme priori1 ,inscrib5s qu9que1 nam d5os significabit ,monos prdictam3 si v]o secunda dat vnam1 ,illa deleta1 scribatur cifra2 priori #if ,trad5do qu9que pro d5>io m$iato2 ,nec cifra scribatur1 nisi inde figura sequatur3 ,po/ea proc$as reliquas m$i&o figuras1 ,qu9 supra docui1 si s9t tibi mille figure4 #ajj #aa,si m$iatio sit b5e facta prob>e valebis1 ,dupl&o num]um quem primo dimidia/i4 ,si sup] extremam nota sit monades dat eidem ,quod cont+at cum primo dimiabis ,atque figura prior nup] fu]it m$i&o4 ,sid5ote3 ,multiplication4 ,si tu p] num]um num]um vis multiplic>e1 ,scribe duas1 quascunque volis1 s]ies num]orum2 #ajd ,ordo tam5 s]uetur vt vltima multiplic&i ,ponatur sup] ant]iorem multiplicantis2 #ab;,a leua relique s9t scripte multiplicantes4 ,9 digitum cures digitum si duc]e1 major #ajh ,p] quantes di/at a d5is respice1 debes ,namque suo decuplo toci5s del]e m9orem2 ,sicque tibi num]us v5i5s ex9de patebit4 ,po/ea proc$as po/remam multiplic&o1 #aab ,ju/e multiplicans p] cunctas inf]iores1 ,condicione tam5 tali2 quod multiplicantis ,scribas 9 capite1 quicquid process]it inde2 ,set po/quam fu]it hec multiplicata1 figure #aaf ,ant]ior5tur s]iei multiplicantis2 ,et sic multiplica1 velut i/am multiplica/i1 ,qui sequitur num]um scriptum quicunque figuris4 ,set cum multiplicas1 primo sic e/ op]&um1 #abj ,si dabit articulum tibi multiplicacio solum2 ,proposita cifra1 summam transf]re mem5to4 ,s9 autem digitus excresc]it articulusque1 ,articulus supraposito digito salit ultra2 #abd ,si digitus tam51 ponas illum sup] ipsam1 ,subdita multiplicans hanc que sup] incidit illi ,delet eam p5itus1 scrib5s quod prov5it inde2 ,s$ si multiplices illam posite sup] ipsam1 #abh ,adiung5s num]um quem prebet ductus e>um2 ,si supraimpositam cifra debet multiplic>e1 ,prorsus eam delet1 scribi que loco cifra debet1 #ab,si cifra multiplicat aliam positam sup] ipsam1 #acb ,sitque locus supra vacuus sup] hanc cifra fiet2 ,si supra fu]it cifra semp] pret]eunda e/2 ,si dubites1 an sit b5e multiplic&o secunda1 ,diuide totalem num]um p] multiplicantem1 #acf ,et r$det num]us em]g5s inde priorem4 ,sid5ote3 ,m5tal ,multiplication4 #ac,p] num]um si vis num]um quoque multiplic>e ,tantum p] normas subtiles absque figuris ,has normas pot]is p] v]sus scire sequ5tes4 #adj ,si tu p] digitum digitum quilibet multiplicabis ,regula prec$5s dat qualit] e/ op]&um ,articulum si p] reliquum vis multiplic>e ,9 proprium digitum debebit ut]que resolvi #add ,articulus digitos po/ p] se multiplicantes ,ex digitis quoci5s t5]et multiplicatum ,articuli faciunt tot c5tum multiplicati4 ,articulum digito si multiplicamus oportet #adh ,articulum digitum sumi quo multiplic>e ,debemus reliquum quod multiplic>is ab illis ,p] reliquo decuplum sic omne lat]e nequibit ,9 num]um mixtum digitum si duc]e cures #aeb ,articulus mixti sumatur de9de resolvas ,9 digitum po/ hec fac ita de digitis nec ,articulusque docet excresc5s 9 det95do ,9 digitum mixti po/ ducas multiplicantem #aef ,de digitis ut norma docet sit juncta secundo ,multiplica summam et po/ea summa patebit ,junctus 9 articulum purum articulumque #ad;,articulum purum comittes articulum que #afj ,mixti pro digitis po/ fiat et articulus vt ,norma jubet ret95do quod egr$itur ab illis ,articuli digitum po/ 9 digitum mixti duc ,regula de digitis ut p]cipit articulusque #afd ,ex quibus excresc5s summe tu junge priori ,sic manife/a cito fiet tibi summa petita4 ,compositum num]um mixto sic multiplicabis ,vndecies tr$ecem sic e/ ex hiis op]&um #afh ,9 reliquum primum demum duc po/ 9 eundem ,unum po/ de9de duc 9 t]cia de9de p] unum ,multiplices t]cia demum tunc omnia multiplicata ,9 summa duces quam que fu]it te dices #agb ,hic ut hic mixtus int5tus e/ op]&um ,multiplic&orum de normis sufficiunt hec4 ,sid5ote3 ,division4 ,si vis divid]e num]um1 sic incipe primo2 ,scribe duas1 quascunque voles1 s]ies num]orum2 #agf ,majori num]o num]um suppone m9orem1 #ae,nam docet ut major t5eat bis t]ve m9orem2 ,et sub supprima supprimam pone figuram1 ,sic reliquis reliquas a dextra p>te locabis2 #ahj ,po/ea de prima primam sub p>te s9i/ra ,subtrahe1 si possis1 quoci5s potes adm9us i/ud1 ,scrib5s quod remanet sub tali conditione2 ,ut toti5s demas dem5das a reman5te1 #ahd ,que s]ie recte pon5tur 9 ant]iori1 ,unica si1 tantum sit ibi decet op]>i2 ,set si non possis a prima dem]e primam1 ,proc$as1 et eam num]o suppone sequ5ti2 #ahh ,hanc uno retrah5do gradu quo comites retrahantur1 ,et1 quoti5s pot]is1 ab eadem deme priorem1 ,ut toti5s demas dem5das a reman5ti1 ,nec plus quam novies quicquam tibi dem]e debes1 #aib ,nascitur h9c num]us quoci5s supraque sequ5tem ,hunc primo scribas1 retrahas ex9de figuras1 ,dum fu]it major supra positus inf]iori1 ,et rursum fiat divisio m priori2 #aif ,et num]um quoti5s supra scribas p]eunti1 ,si fiat sali5s retrah5do1 cifra locetur1 ,et p]eat num]o quoti5s1 proponas eidem ,cifram1 ne num]um p]eat vis1 dum locus illic #bjj ,re/at1 et expletis divisio non valet ultra3 ,dum fu]it num]us num]orum inf]iore seorsum ,illum s]vabis2 h9c multiplic&o probabis1 ,sid5ote3 ,pro(4 ,si b5e feci/i1 divisor multiplicetur #bjd ,p] num]um quoti5s2 cum multiplicav]is1 adde ,totali summ1 quod s]vatum fuit ante1 ,r$deturque tibi num]us quem proposui/i2 ,et si nil remanet1 hunc multiplic&o r$det1 #bjh ,sid5ote3 ,squ>e ,numb]s4 ,cum ducis num]um p] se1 qui prov5it inde ,sit tibi quadratus1 ductus radix erit hujus1 ,nec num]os omnes quadratos dic]e debes1 ,e/ autem omnis num]us radix alicujus4 #bab ,qu&o voles num]i radicem qu]]e1 scribi ,debet2 inde notes si sit locus ult]ius imp>1 ,e/que figura loco talis scrib5da sub illo1 ,que1 p] se dicta1 num]um tibi de/ruat illum1 #baf ,vel quantum pot]it ex inde delebis e&em2 ,vel retrah5do duples retrah5s dupl&o sub i/a ,que primo sequitur1 duplicatur p] duplacationem1 ,po/ p] se m9u5s pro posse quod e/ m9u5dum4 #bbj #af,po/ 8 propones digitum1 qui1 m priori ,p] prec$5tes1 po/ p] se multiplicatus1 ,de/ruat 9 quantum pot]it num]um reman5tem1 ,et sic proc$5s retrah5s dupl&o figuram1 #bbd ,prepon5do novam donec totum p]agatur1 ,subdupla propriis s]v>e docetque duplatis2 ,si det compositum num]um duplacio1 debet ,inscribi digitus a p>te dextra p>te prop9qua1 #bbh ,articulusque loco quo non duplicata resessit2 ,si dabit articulum1 sit cifra loco p]eunte ,articulusque locum t5et unum1 de duplicata resessit2 ,si donet digitum1 sub prima pone sequ5te1 #bcb ,si supraposita fu]it duplicata figura ,major proponi debet tantummodo cifra1 ,has retrah5s solito propones m figuram1 ,usque sub extrema ita fac retrah5do figuras1 #bcf ,si totum deles num]um quem proposui/i1 ,quadratus fu]it1 de dupla quod duplica/i1 ,sicque tibi radix illius c]ta patebit1 ,si de duplatis fit juncta supprima figura2 #bdj ,radicem p] se multiplices habeasque ,primo propositum1 b5e te fecisse proba/i2 ,non e/ quadratus1 si quis re/at1 s$ hab5tur ,radix quadrati qui /at major sub eadem2 #bdd ,vel quicquid remanet tabula s]v>e mem5to2 ,hoc casu radix p] se quoque multiplicetur1 ,vel sic quadratus sub primo major habetur1 ,h9c addas reman5s1 et prius debes hab]i2 #bdh ,si locus extremus fu]it p>1 scribe figuram ,sub p]eunte loco p] quam debes op]>i1 ,que quantum pot]it supprimas de/ruat ambas1 ,vel p5itus legem t5eas op]&o priorem1 #beb ,si suppositum digitus suo f9e rep]tus1 ,omn9o delet illic scribi cifra debet1 ,a leva si qua sit ei sociata figura2 ,si cifre reman5t 9 f9e p>es decet h>um #bef ,radices1 num]o m$iam propon]e p>tem1 ,tali quesita radix patet arte rep]ta4 ,p] num]um recte si no/i multiplic>e ,ejus quadratum1 num]us qui p]v5it inde #bfj ,dicetur cubicus2 primus radix erit ejus2 ,nec num]os omnes cubicatos dic]e debes1 ,e/ autem omnis num]us radix alicujus2 ,sid5ote3 ,cube ,root4 ,si curas cubici radicem qur]e1 primo #bfd ,inscriptum num]um di/+u]e p] loca debes2 ,que tibi mille notant a mille notante suprema ,initiam1 summa op]&i p>te s9i/ra1 ,illic sub scribas digitum1 qui multiplicatus #bfh ,9 semet cubice suprapositum sibi p]dat1 ,et si quid fu]it adjunctum p>te s9i/ra ,si non omn9o1 quantum pot]is m9u5do1 ,h9c triplans retrahe saltum1 faci5do sub illa #bgb ,que manet a digito deleto t]na1 figuram ,illi propones quo sub triplo asocietur1 ,ut cum subtriplo p] eam tripla multiplicatur2 ,h9c p] eam solam productum multiplicabis1 #bgf ,po/ea totalem num]um1 qui prov5it inde ,a suprapositis respectu tolle triplate ,addita supprimo cubice tunc multiplicetur1 ,respectu cujus1 num]us qui progr$ietur #bhj ,ex cubito ductu1 supra omnes adimetur2 ,tunc ipsam del5s triples saltum faci5do1 ,semp] sub t]nas1 retrah5s alias triplicatas ,ex h9c triplatis aliam propone figuram1 #bhd ,que p] triplatas ducatur m priori2 ,primo sub triplis sibi junctis1 po/ea p] se1 ,9 num]um ducta1 productum de triplicatis3 ,utque prius dixi num]us qui prov5it inde #bhh ,a suprapositis has respici5do trahatur1 ,huic cubice ductum sub primo multiplicabis1 ,respectumque sui1 removebis de reman5ti1 ,et sic proc$as retrah5do tripl&o figuram4 #bib ,et propon5do nonam1 donec totum p]agatur1 ,subtripla sub propriis s]v>e decet triplicatis2 ,si nil 9 f9e remanet1 num]us datus ante ,e/ cubicus2 cubicam radicem sub tripla preb5t1 #bif ,cum digito juncto quem supprimo posui/i1 ,hec cubice ducta1 num]um r$dant tibi primum4 ,si quid erit reman5s non e/ cubicus1 s$ habetur ,major sub primo qui /at radix cubicam1 #cjj ,s]v>i debet quicquid radice remansit1 ,extracto num]o1 decet hec addi cubicato4 ,quo facto1 num]us r$di debet tibi primus4 ,nam debes p] se radicem multiplic>e #cjd ,ex h9c 9 num]um duces1 qui prov5it inde ,sub primo cubicus major sic inv5ietur2 ,illi jungatur reman5s1 et primus habetur1 ,si p] triplatum num]um nequeas op]>i2 #cjh ,cifram propones1 nil v]o p] hanc op]>e ,set retrah5s illam cum saltu de9de triplata1 ,propones illi digitum sub lege priori1 ,cumque cifram retrahas sali5do1 non triplicabis1 #cab ,namque nihil cifre triplacio dicitur esse2 ,at tu cum cifram protrax]is aut triplicata1 ,hanc cum subtriplo semp] s]v>e mem5to3 ,si det compositum1 digiti triplacio debet #caf ,illius scribi1 digitus sali5do sub ipsam2 ,digito deleto1 que t]na dicitur esse2 ,jungitur articulus cum triplata p]eunte1 ,set facit hunc scribi p] se triplacio prima1 #cbj ,que si det digitum p] se scribi facit illum2 ,consumpto num]o1 si sole fuit tibi cifre ,triplato1 propone cifram saltum faci5do1 ,cumque cifram retrahe triplam1 scrib5do figuram1 #cbd ,preponas cifre1 sic proc$5s op]>e1 ,si tres vel duo s]ie 9 s9t1 pone sub yma1 ,a dextris digitum s]v&o prius docum5tum4 ,si sit cont9ua progressio t]m9us nup] #cbh ,p] majus m$ium totalem multiplicato2 ,si p>1 p] m$ium tunc multiplicato sequ5tem4 ,set si cont9ua non sit progressio f9is3 ,imp>1 tunc majus m$ium si multiplicabis1 #ccb ,si p> p] m$ium sibi multiplicato prop9quum4 #ccc ,,footnotes 7,app5dix ,,ii1 ,c>m5 de ,algorismo73 #a3 8,hec prs5s ars dicitur algorismus ab ,algore rege ejus inv5tore1 vel dicitur ab algos quod e/ ars1 et rodos quod e/ num]us2 qu e/ ars num]orum vel num]&i1 ad quam artem b5e sci5dum inv5iebantur apud ,indos bis qu9que 7id e/ decem7 figur40 --,comm5t4 ,?om de ,novo-,m]catu4 ,,ms4 ,bib4 ,reg4 ,mus4 ,brit4 #ab ,e4 #a4 #b3 8,h necess>i figur sunt ,indorum *>act]os40 ,,ms4 de num]atione4 ,bib4 ,sloan4 ,mus4 ,brit4 #eac1 fol4 #eh4 8,cum vidissem ,yndos con/ituisse ,,ix lit]as 9 univ]so num]o suo propt] dispositionem suam quam posu]unt1 volui patefac]e de op]e quod sit p] eas aliquidque esset levius disc5tibus1 si ,deus volu]it4 ,si autem ,indi hoc volu]unt et int5tio illorum nihil novem lit]is fuit1 causa que mihi potuit4 ,deus direxit me ad hoc4 ,si v]o alia dicam pret] eam quam ego exposui1 hoc fec]unt p] hoc quod ego exposui1 eadem tam c]tissime et absque ulla dubitatione pot]it inv5iri4 ,levitasque patebit aspici5tibus et disc5tibus40 ,,ms4 ,u4,l4,c41 ,ii4 vi4 #e1 f4 #ajb4 #c3 ,f ,eg4 #bfbb4 #d3 #h ,c4 iv4 ins]ts ,nullum cipa significat3 dat signific>e sequ5ti4 #e3 ,f #ab ,e4 #a4 #f3 ,en argorisme devon pr5dre ,vii especes 4 4 4 4 ,adision subtracion ,d\bloison m$iacion ,monteploie et division ,et de radix eu/racion ,a *ez vii especes savoir ,doit *ascun en memoire avoir ,letres qui figures sont dites ,et qui excell5s sont ecrites4 --,,ms4 ,seld4 ,ar*4 ,b4 #bf4 #g3 ,f #ab ,e4 #a4 #h3 ,f #ab ,e4 #a4 #i3 #h ,c4 iv4 ins]ts ,atque figura prior nup] fu]it m$i&o4 #aj3 ,i4e4 figura secundo loco posita4 #aa3 ,s #ab ,e4 #a2 #h ,c4 iv4 ins]ts-- #ab3 #ab ,e4 #a ins]ts4 #ac3 #ab ,e4 #a ins]ts to l4 #agd4 #ad3 #ab ,e4 #a omits1 ,eg4 #bfbb ins]ts4 #ae3 #ab ,e4 #a ins]ts4 #af3 #h ,c4 iv4 ins]ts-- ,h9c illam dele duplans sub ei psalli5do ,que sequitur retrah5s quicquid fu]it duplicatum4 ,,index ,,( ,,te*nical ,,t]ms#a99 ,footnote #a993 ,?is ,index has be5 k9dly prep>$ by ,pr(essor ,j4 ,b4 ,dale1 ( ,k+'s ,college1 ,univ]sity ( ,london1 & ! be/ ?anks ( ! ,society are due to him = 8 valuable contribution4 ,transcrib]'s ,note3 ,! ,te*nical ,t]ms & ,gloss>y 7foll[+7 ref] to page & l9e numb]s 9 ! pr9t$ book4 ,in=mation 9 d\ble brackets has be5 add$ by ! transcrib] to aid 9 text se>*+4 algorisme1 #cc_/ab2 algorym1 augrym1 #c_/c2 ! art ( comput+1 us+ ! s-call$ ,arabic num]als4 ,! ^w 9 its v>i\s =ms is d]iv$ f ! ,arabic al-,kh[>azmi 7i4e4 ! native ( ,khw>azm 7,khiva774 ,?is 0 ! surname ( ,ja'f> ,mohammad b5 ,musa1 :o wrote a treatise e>ly 9 ! #i? c5tury 7see p4 xiv74 ,! =m algori?m is also f\nd1 be+ sugge/$ by a suppos$ d]ivation f ! ,greek 7numb]74 ant]y1 #bd_/aa2 to move figures to ! "r ( ! position 9 :i* !y are fir/ writt54 ,?is op]ation is p]=m$ repeat$ly ^u ! multipli] 9 multiplication1 & ^u c]ta9 figures :i* arise 9 ! process ( root extraction4 ant]ioraci\n1 #ej_/e2 ! op]ation ( mov+ figures to ! "r4 writt5 ant]ioracioun or ant]ioracioun article1 #cd_/bc2 articul1 #e_/ca2 articuls1 #i_/cf1 #bi_/g1h2 a numb] divisible by t5 )\t rema9d]4 also articulle ca/1 #h_/ab2 to add "o numb] to ano!r4 #8,addition is a ca/+ toge!r ( two numb]s into "o numb]1' #h_/aj4 cifre1 #d_/a2 ! "n ( ! figure #j4 ,! ^w is d]iv$ f ! ,arabic sifr empty1 no?+4 ,h5ce z]o4 ,a ciph] is ! symbol ( ! abs5ce ( numb] or ( z]o quantity4 ,x may be us$ alone or 9 conjunction ) digits or o!r ciph]s1 & 9 ! latt] case1 accord+ to ! position :i* x occupies relative to ! o!r figures1 indicates ! abs5ce ( units1 or t5s1 or hundr$s1 etc4 ,! great sup]iority ( ! ,arabic to all o!r sy/ems ( notation resides 9 ! employm5t ( ?is symbol4 ,:5 ! ciph] is n us$1 ! place value ( digits has to be indicat$ by writ+ !m 9 assign$ r[s or columns4 ,ciph]s1 h[ev]1 may be int]polat$ among/ ! significant figures us$1 & z !y suffici5tly indicate ! positions ( ! empty r[s or columns1 ! latt] ne$ n be indicat$ 9 any o!r way4 ,! practical p]=mance ( calculations is ?us enorm\sly facilitat$ 7see p4 xvi74 compon$e1 #cc_/bd2 composyt1 #e_/ce2 ) ref]5ce to numb]s1 "o comp\nd$ ( a multiple ( t5 & a digit4 writt5 compon$e conu]tide conv]sely1 #df_/bi1 #dg_/i4 writt5 conuertide or conuertide cubic$e1 #ej_/ac2 to be c41 to h its cube root f\nd4 writt5 cubic$e cubike nombre1 #dg_/h2 a numb] =m$ by multiply+ a giv5 numb] twice by itself1 e4g4 #bg #c #c #c4 ,n[ call$ simply a cube4 writt5 cubike 444 decuple1 #bb_/ab2 ! product ( a numb] by t54 ,t5fold4 dep>tys divides1 #e_/bi4 writt5 departys digit1 #e_/cj2 digitalle1 #cc_/bd2 a numb] less ?an t51 repres5t$ by "o ( ! n9e ,arabic num]als4 writt5 digitalle dimydicion1 #g_/bc2 ! op]ation ( divid+ a numb] by two4 ,halv+4 writt5 dimydicio ducci\n1 multiplication1 #dc_/i4 writt5 duccioun duplacion1 #g_/bc1 #ad_/ae2 ! op]ation ( multiply+ a numb] by two4 ,d\bl+4 writt5 duplacio or duplacio ) fancy 8n0 i-m$iet halv$1 #ai_/bc4 int]cise brok51 #df_/b2 int]cise ,progression is ! "n giv5 to ei!r ( ! ,progressions #a1 #c1 #e1 #g1 etc42 #b1 #d1 #f1 #h1 etc41 9 :i* ! common diff]5ce is #b4 writt5 intercise l$e into1 multiply by1 #dg_/ah4 words always sep>at$1 z 8l$e 444 into0 lyneal nombre1 #df_/ad2 a numb] su* z t :i* expresses ! measure ( ! l5g? ( a l9e1 & !re=e is n necess>ily ! product ( two or m numb]s 7vide ,sup]ficial1 ,solid74 ,?is appe>s to be ! mean+ ( ! phrase z us$ 9 ,! ,art ( ,nombryng4 ,x is possible t ! numb]s s designat$ are ! prime numb]s1 t is1 numb]s n divisible by any o!r numb] except !mselves & unity1 b x is n cle> t ?is limitation is int5d$4 m$iaci\n1 #af_/cf1 #ch_/af2 divid+ by two 7see also dimydicion74 writt5 m$iacio ) fancy 8n01 g5]ally )\t 8u0 m$l$e nombre1 #cd_/a2 a numb] =m$ ( a multiple ( t5 & a digit 7vide compon$e1 composyt74 writt5 m$l$e 444 m$ye1 #ag_/h1 to halve2 m$iete1 halv$1 #ag_/cj2 ym$it1 #bj_/i4 naturelle progressi\n1 #de_/bb2 ! s]ies ( numb]s #a1 #b1 #c1 etc4 writt5 naturelle progressioun producci\n1 multiplication1 #ej_/aa4 writt5 produccioun quadrat nombre1 #df_/ab2 a numb] =m$ by multiply+ a giv5 numb] by itself1 e4g4 #i #c #c1 a squ>e4 rote1 #g_/be2 roote1 #dg_/aa2 root4 ,! roots ( squ>es & cubes are ! numb]s f :i* ! squ>es & cubes are d]iv$ by multiplication into !mselves4 significatyf1 significant1 #e_/ad2 ,! significant figures ( a numb] are1 /rictly speak+1 ^? o!r ?an z]o1 e4g4 9 #c #f #e #j #d #j #j1 ! significant figures are #c1 #f1 #e1 #d4 ,mod]n usage1 h[ev]1 reg>ds all figures betwe5 ! two extreme significant figures z significant1 ev5 :5 "s are z]o4 ,?us1 9 ! above example1 #c #f #e #j #d are consid]$ significant4 solide nombre1 #df_/cg2 a numb] :i* is ! product ( ?ree o!r numb]s1 e4g4 #ff #aa #b #c4 usually writt5 solide sup]ficial nombre1 #df_/ah2 a numb] :i* is ! product ( two o!r numb]s1 e4g4 #f #b #c4 writt5 superficial or superficialle t]n>y1 consi/+ ( ?ree digits1 #ea_/g4 writt5 tern>y vnd] d\ble1 a digit :i* has be5 d\bl$1 #dh_/c4 vnd]-trebille1 a digit :i* has be5 trebl$1 #di_/bh2 vnd]-triplat1 #di_/ci4 writt5 vnd]-trebille1 vnd]-triplat w1 a symbol us$ to d5ote half a unit1 #ag_/cc %[n 9 e-text z sup]script ,,gloss>y ,transcrib]'s ,note3 ,words ^: fir/ appe>ance is e>li] ?an ! page cit$ 9 ! ,gloss>y are id5tifi$ 9 d\ble-bracket$ notes4 ,to aid 9 text se>*+1 words writt5 ) int]nal italics are also not$1 & context is giv5 = common words4 ablaci\n1 tak+ away1 #cf_/ba writt5 ablacioun addy/1 hadde/1 #aj_/cg agregaci\n1 addition1 #de_/bb4 7,fir/ example 9 ,n4,e4,d41 #aedg47 writt5 agregacioun a-en5es1 aga9/1 #bc_/aj allgate1 always1 #h_/ci als1 z1 #bb_/bd &1 if1 #bi_/h2 1 #d_/bg2 yf1 #bj_/g a-n5des1 t[>ds1 #bc_/ae apropr$e1 appropriat$1 #cd_/bg writt5 apropr$e apw]e?1 appe>s1 #fa_/h a-risyt1 arises1 #ad_/bd a-r[e1 9 a r[1 #bi_/aj arsemetrike1 ari?metic1 #cc_/a writt5 arsemetrike ay5e1 aga91 #de_/ae bagle1 crozi]1 #fg_/ab bordure ordure1 r[1 #dc_/cj writt5 bordure borro1 inf4 borr[1 #aa_/ch2 imp4 s4 bor[e1 #ab_/bj2 pp4 borw$1 #ab_/ae2 borr$1 #ab_/ai b\e1 above1 #db_/cd caputule1 *apt]1 #g_/bf writt5 caputule c]tayn1 assur$ly1 #ah_/cd writt5 certay clep$e1 call$1 #dg_/g writt5 clep$e compet5tly1 conv5i5tly1 #ce_/h compt1 c\nt1 #dg_/bi contynes1 conta9s1 #ba_/ab2 writt5 contynes pp4 cont5y!1 #ch_/ci writt5 cont5y?e craft1 art1 #c_/d di/+ue1 divide1 #ea_/e egalle1 equal1 #de_/ba writt5 egalle excep1 except1 #e_/af exclus$e1 exclud$1 #cd_/cg writt5 exclus$e excress5t1 result+1 #ce_/af writt5 excress5t exeant1 result+1 #dc_/bf expone1 exp\nd1 #c_/bc f]ye f]e1 f\r?1 #gj_/ab figure figures1 #e_/a writt5 figure =-by1 pa/1 #ab_/aa =s2 no f41 no matt]1 #bb_/bd =se?1 matt]s1 #ec_/cj =ye =e1 =?1 #ga_/h fyftye fyfte1 fif?1 #gj_/af grewe1 ,greek1 #cc_/ac halu5del1 half1 #af_/af2 haldel1 #ai_/d2 pl4 halu$els1 #af_/af hay/1 ha/1 #ag_/c1 #cb ha/1 ha/e1 #bb_/be 9 8haue ha/ to0 he]1 high]1 #i_/ce "h1 _!1 #g_/bf 9 89 h]e caputule0 "h-a-=e1 h]et(ore1 #ac_/g writt5 h]e-a-=e hey?1 0 call$1 #c_/e hole1 :ole1 #d_/ci2 holle1 #ag_/a2 hoole1 ( ?ree dim5sions1 #df_/ae holdye1 holds good1 #cj_/e h[ be x t1 al?\gh1 #dd_/d l$e lete1 let1 #h_/cg l5e1 l5d1 #ab_/ci le/1 lea/1 #dc_/bg 9 8at ! le/0 le/ left1 #ga_/i 9 8! le/ syde0 leue1 leave1 #f_/e2 pr4 #c s4 leues1 rema9s1 #aa_/ai2 fir/ 9 #aj_/dj leus1 #aa_/bh2 pp4 laft1 left1 #ai_/bd lewd]1 m ignorant1 #c_/c writt5 lewder lu/1 desire/ to1 #de_/ac lyt1 easy1 #ae_/ca lymytes1 limits1 #cd_/ah2 lynes1 #cd_/ab2 lynees1 #cd_/ag2 ,lat4 limes1 pl4 limites4 may/]y1 a*ievem5t2 writt5 may/ery no m41 no a*ievem5t1 i4e4 easy1 #ai_/aj me1 indef4 pron4 "o1 #db_/a fir/ 9 #cd_/af mo1 m1 #i_/af mod] m 7,lat4 majorem71 #dc_/bb mo/1 mu/1 #cj_/c fir/ 9 #c_/ab & _m m multipli$e1 to be m4 multiply+1 #dj_/i mynvtes1 ! sixty p>ts into :i* a unit is divid$1 #ch_/be writt5 mynvtes myse-wrot1 mis-wr\ght1 #ad_/aa ne!r1 nor1 #cd_/be 9 8,x 01 ne!r is0 nex1 next1 #ai_/i not1 n\ght1 #e_/g fir/ 9 #d_/h note1 n1 #cj_/e oo1 "o1 #db_/bj2 o1 #db_/ba fir/ 9 #cd_/bg2 #cc_/bb ome/1 upp]mo/1 high]1 #ce_/bf2 omy/1 #ce_/bh om:ile1 sometimes1 #de_/ca fir/ 9 #ci_/ag on1 "o1 #h_/bi 9 8on vnd] an-oer0 opyne1 pla91 #dg_/h writt5 opyne or1 be=e1 #ac_/be 9 8or ou be-gan0 or e oer1 ! o!r1 #bh_/cd 9 8or by-tw5e0 ordure1 ord]1 #cd_/i2 r[1 #dc_/a ^w =m is 8ord]0 o!r1 or1 #cc_/ac1 #dc_/bf2 9 8art o!r craft0 on #cc_/ac1 8o!r h[ (t0 on #dc_/bf2 note also 8"o o!r o!r0 on #ce_/bd o!r 4 4 4 or1 ei!r 4 4 4 or1 #ch_/cg 9 8o!r x is ev5 or ode0 on #ch_/cg2 !re are e>li] occurr5ces ou]]1 upp]1 #db_/ae writt5 ouerer ou]-hipp$e1 pass$ ov]1 #dc_/ai writt5 ouer-hipp$e recte1 directly1 #bg_/bj 9 8/ondes n recte02 also on #bf_/ca 9 8recte ouer 8 h$e0 remayn]1 rema9d]1 #ef_/bh repres5ti!1 repres5t$1 #ci_/ad writt5 repres5ti?e re/e?1 rema9s1 #fc_/bi fir/ 9 #eg_/bi & o!rs rew>de1 reg>d1 #dh_/f writt5 rew>de rew1 r[1 #d_/h rewle1 r[1 #d_/bj1 #g_/ab2 9 8place ( e rewle01 8e rewle ( figures0 rewele1 #d_/ah2 rewles1 rules1 #e_/cc s4 scilicet1 #c_/h 9 8s4 ,algorismu0 s5t5s1 mean+1 #ad_/bi signifye7tyf71 #e_/ac4 ,! la/ ?ree lett]s are add$ above ! l9e1 evid5tly because ( ! ^w #8significatyf' 9 l4 #ad4 ,b ! #8,solucio1' :i* conta9$ ! ^w1 has be5 omitt$4 si!n1 s9ce1 #cc_/h "s1 sum1 result1 #dj_/ag1 #cb fir/ 9 #cf_/ba 9 8me may see a "s01 !n 9 8! same "s0 & 8to "s (0 s[ne1 pron\nce1 #f_/bi s+illatim1 s+ly1 #g_/be spices1 species1 k9ds1 #cd_/d fir/ 9 #e_/cd & o!rs spyl1 wa/e1 #ad_/bf /yde1 /ead1 #ah_/bj subtrahe1 subtract1 #dh_/ab2 pp4 subtrayd1 #ac_/ba sy!s1 times1 #ba_/af tat1 taught1 #af_/cf take1 pp4 tak52 t4 fro1 />t+ f1 #de_/bb 9 8fro oone or tweyne take0 taw>d1 t[>d1 #bc_/cd ?\t1 ?\gh1 #e_/bj trebille1 multiply by ?ree1 #di_/bf writt5 trebille tw5e1 two1 #h_/aa fir/ 9 #d_/bc ow1 ?\gh1 #be_/ae 9 8ow ou take0 owt1 ?\ght2 be 41 m5tally1 #bh_/d u is1 ?is1 #bj_/cc 9 8u nombur #bad0 vny1 unite1 #de_/aj wel1 wilt1 #ad_/ca 9 8,if ou wel0 wete1 wit1 #ae_/af2 wyte1 "k1 #h_/ch2 pr4 #b s4 wo/1 #ab_/ch wex1 become1 #ej_/ah :]e1 :e!r1 #bi_/ab writt5 :]e 9 8:]e 9 e secunde1 or0 :]-?urghe1 :5ce1 #di_/ae writt5 ,:]-?urghe wor*1 "w1 #h_/ai2 fir/ 9 #g_/ce wri*1 #h_/ce2 wyr*1 #f_/ai2 imp4 s4 wor*1 #ae_/i2 fir/ 9 #i_/f pp4 y-wro?1 #ac_/bd write1 writt51 #bi_/ai2 fir/ 9 #f_/cg 9 8ha/ write01 8be write0 y-write1 #af_/a wryr*ynge wyr*ynge1 work+1 #cj_/d writt5 wryr*ynge wt1 )1 #ee_/h y-bro?1 br\ght1 #ba_/ah y*on1 ea* "o1 #bi_/aj writt5 y*o ydo1 done1 add$1 #i_/f fir/ 9 #h_/cg 9 8haue ydo02 #i_/f 9 8ydo all to-g$er0 ylke1 same1 #e_/ab y-lye*1 alike1 #bb_/bc y-myt1 be5 able1 #ab_/b y-n[t1 en\gh1 #ae_/ca2 ynovt1 #ah_/cd yove1 giv51 #de_/cc yt1 t1 #eb_/h y-write1 v4 write4 y-wro?1 v4 wor*4 99 99 99 99 99 99 99 99 99 99 99 99 99 99 ,,m>g9al ,,notes3 ,headnotes h be5 mov$ to ! beg9n+ ( ! appropriate p>agraph4 ,headnotes 7 omitt$ f ! two ,app5dixes1 z sid5otes give ! same in=mation4 ,l9e ,numb]s are cit$ 9 ! ,index & ,gloss>y4 ,!y h be5 omitt$ f ! e-text except 9 ! "o v]se selection 7,app4 ,,ii1 ,c>m5 de ,algorismo74 ,in/ead1 ! ,index & ,gloss>y include supplem5tal in=mation to help locate ea* ^w4 ,numb]$ ,notes3 ,numb]$ sid5otes %[ page or leaf numb]s f ! orig9al ,,mss4 ,9 ! e-text1 ! page numb] is %[n z 99#abc;b inl9e2 mid-^w page breaks are m>k$ ) a supplem5tal a/]isk 994 ,numb]s are n us$4 ,footnotes give textual in=mation su* z v>iant read+s4 ,!y h be5 numb]$ sequ5tially )9 ea* title1 ) numb]s %[n z #a to avoid confusion ) brack$ text--includ+ s+le num]als--9 ! orig9al4 ,editorial notes are %[n z #a994 ,:5 a footnote calls = add$ text1 ! addition is %[n 9 ! body text ) d\ble brackets4 ,sid5otes giv+ a runn+ synopsis ( ! text h be5 mov$ to ! beg9n+ ( ea* p>agraph1 :]e !y are %[n z a s+le note4 ,,errors ,,& ,,anomalies 7,not$ by ,transcrib]73 ,introduction3 dat$ ,mijc ,9 ?is & ! rema9d] ( ! p>agraph1 ! lett] %[n z c is pr9t$ directly above ! prec$+ j4 ,! ,crafte ( ,nombrynge3 sursum proc$as multiplic&o ,italiciz$ z %[n3 error = 8proc$as08 ,sid5ote3 ,our au?or makes a slip "h ,else:]e 9 ! book1 num]ical errors are correct$ 9 ! body text1 ) a footnote giv+ ! orig9al =m4 t5 tymes s my* is e n\nbre text un*ang$3 error = 8z08 #f tymes #bd1 #ai;en take misplac$ footnote an*or 9 orig9al3 belongs ) 8#f times #bd0 ,fn4 #g3 #8,subtrahas aut addis a dextris op5 quote miss+ ,! ,art ( ,nombryng3 oone ( ! digitis z 4#aj4 ( #a44 #bj4 (4 #b4 text un*ang$3 error = 8z 4#aj4 ( 4#a4 #bj4 ( 4#b408 sette a-side half ( ?o minutes text un*ang$3 error = 8!08 &4 #aj4 z be=e is come !r( text un*ang$3 error = 8& 4#aj408 ,sid5ote3 ,:]e to set ! quoti5te spell+ 7#aibb7 un*ang$ ,sid5ote3 ,def9ition ( ,progression4 f 9 8(0 illegible ,sid5ote3 444 giv+ ! value ( ab4#b ,t is1 8a7b#b740 ,accomptynge by c\nt]s3 ,= example ( ! 99#aag;a4 ly99nes f9al 4 9 sid5ote miss+ or invisible also 9 #aba;b1 #abb;a :i* 9 ! fyr/ summe is #e invisible 8#e0 suppli$ by transcrib] 99#aaf;a 7sic74 ,editor's 8sic03 page numb]+ jumps back to #aaf in/ead ( ! expect$ #abc1 & cont9ues f #aaf4 99#abc;a 444 set d[ne ye multiply] 4v4 tymes1 z "h y se ,diagram %[n z pr9t$1 ) #ceejj = #cfejj 9 "o column1 & app>5t misplac$ 8?\s&s0 m>k] #cfe 7:i* is ! nomb] ( dayes 444 op5 7 miss+ ,! arte ( nombrynge by ! h&e3 = #a ! fyng] is clasp$ 9 ,9 at lea/ "o pr9t+ ( ! text1 8clasp$0 is mispr9t$ z 8elasp$0 b ?is teacy$ me n text un*ang$ ,app5dix ,i3 ,a ,treatise on ! ,num]ation ( ,algorism3 ,see ,introduction & ,gloss>y = 3f & 3y errors ,app5dix ,,ii3 ,c>m5 de ,algorismo3 ,9 ?is selection1 errors t are n expla9$ 9 footnotes 7 assum$ to be typographic4 l4 #ii ,po/ea proc$as procdeas l4 #afc ,articuli digitum po/ 9 digitum mixti duc po/ iu