* ¶ Sedcummultiplicabis, primo sic estoperandum,Si dabit articulumtibi multiplicacio solum;Proposita cifra summamtransferre memento.What to do if the first multiplication results in an article.¶ Herehe puttes þe fyrst case of þis craft, þe quych is þis: yf þerecome an articulle of þe multiplicacioɳ ysette beforethe articullein þe lyft side as þus5123.multiplye 5 by 2, þat wol be 10; sette ouerþe hede of þat 2 a 0, & sett þat on, þat is þe articul, in þe lyft side, þat is next hym, þen þou schalt haue þis nounbre1051.23¶ And þen worch forth as þou diddist afore. And þou schalt vnderstonde þat þou schalt write no 0. but whan þat place where þou schal write þat 0 has no figure afore hymnoþerafter. versus.¶ Si autemdigitus excreuerit articulusque.Articulus12supraposito digito salit vltra.What to do if the result is a composite number.¶ Hereis þe secunde case, þe quych is þis: yf hit happe þat þerecome a composyt, þou schalt write þe digitte ouerþe hede of þe neþerfigureby þe quych þou multipliest þe hier figure; and sett þe articullenext hym toward þe lyft side, as þou diddyst afore, as þus83.83Multiply 8 by 8, þat wol be 64. Write þe 4 ouer8, þat is to say, ouerþe hede of þe neþer8; & set 6, þe quychleaf 159a.*is an articul, next after. And þen þou schalt haue such a nounbre as is here,648313,83And þen worch forth.¶ Si digitus tamenponas ipsum superipsam.What if it be a digit.¶ Hereis þe thryde case, þe quych is þis: yf hit happe þat of þi multiplicaciouɳ come a digit, þou schalt write þe digit ouerþe hede of þe neþerfigure, by the quych þou multipliest þe hierefigure, for þis nedes no Ensampul.¶ Subdita multiplica non hanc que [incidit] illiDelet eampenitusscribens quod prouenit inde.The fourth case of the craft.¶ Hereis þe 4 case, þe quych is: yf hit be happe þat þe neþerfigureschal multiplye þat figure, þe quych stondes ouerþat figures hede, þou schal do away þe hier figure& sett þere þat þat comys of þat multiplicacioɳ. As yf þere come of þat multiplicacioɳ an articuls þou schalt write þere þe hier figurestode a 0. ¶ And write þe articuls in þe lyft side, yf þat hit be a digit write þere a digit. yf þat hit be a composit, write þe digit of þe composit. And þe articul in þe lyft side. al þis is lyȝt y-nowȝt, þere-foreþer nedes no Ensampul.¶ Sedsi multiplicat aliamponas superipsamAdiunges numerumquemprebet ductusearum.leaf 159b.The fifth case of the craft.¶ Hereis þe 5 case, þe quych is þis: yf *þe neþerfigureschul multiplie þe hier, and þat hier figureis notrecteouerhis hede. And þat neþerfigurehase oþerfigures, or on figure ouerhis hede by multiplicacioɳ, þat hase be afore, þou schalt write þat nounbre, þe quych comes of þat, oueralleþe ylke figures hedes, as þus here:236234Multiply 2 by 2, þat wol be 4; set 4 ouerþe hede of þat 2. þen14multiplies þe hier 2 by þe neþer3, þat wol be 6. set ouerhis hede 6, multiplie þe hier 2 by þe neþer4, þat wol be 8. do away þe hier 2, þe quych stondes ouerþe hede of þe figureof 4,and set þere 8. And þou schalt haue þis nounbrehere46836234And antery þi figures, þat is to say, set þi neþer4 vnderþe hier 3, and set þi 2 other figures nerehym, so þat þe neþer2 stonde vndurþe hier 6, þe quych 6 stondes in þe lyft side. And þat 3 þat stondes vndur 8, as þus aftur ȝe may se,46836234Now worch forthermore, And multiplye þat hier 3 by 2, þat wol be 6, set þat 6 þe quych stondes ouerþe hede of þat 2, And þen worch as I taȝt þe afore.leaf 160a.* ¶ Si supraposita cifra debet multiplicareProrsus eamdeles & ibi scribi cifra debet.The sixth case of the craft.¶ Hereis þe 6 case, þe quych is þis: yf hit happe þat þe figureby þe quych þou schal multiplye þe hier figure, þe quych stondes ryght ouerhym by a 0, þou schalt do away þat figure, þe quych ouerþat cifre hede. ¶ And write þereþat nounbre þat comes of þe multiplicacioɳ as þus, 23. do away 2 and sett þere a 0. vndeversus.¶ Si cifra multiplicat aliampositamsuperipsamSitquelocus supravacuussuperhanc ciframfiet.The seventh case of the craft.¶ Hereis þe 7 case, þe quych is þis: yf a 0 schal multiply a figure, þe quych stondes notrecteouerhym, And ouerþat 0 stonde no thyng, þou schalt write ouerþat 0 anoþer0 as þus:2403multiplye 2 be a 0, it wol be nothynge. write þere a 0 ouerþe hede of þe neþer0, And þen worch forth til þou come to þe ende.¶ Si supra15fuerit cifra semperestpretereunda.The eighth case of the craft.¶ Hereis þe 8 case, þe quych is þis: yf þerebe a 0 or mony cifers in þe hier rewe, þou schalt not multiplie hem, bot let hem stonde. And antery þe figures beneþe to þe next figuresygnificatyf as þus:00032.22Ouer-lepe alleþese cifers & sett þatleaf 160b.*neþer2 þat stondes toward þe ryght side, and sett hym vndurþe 3, and sett þe oþernether 2 nere hym, so þat he stonde vndurþe thrydde 0, þe quych stondes next 3. And þan worch. vndeversus.¶ Si dubites, an sit benemultiplicacio facta,Diuide totalem numerumpermultiplicantem.How to prove the multiplication.¶ Herehe teches how þou schalt know wheþerþou hase wel I-do or no. And he says þat þou schalt deuide alleþe nounbre þat comes of þe multiplicacioɳ by þe neþerfigures. And þen þou schalt haue þe same nounbur þat þou hadyst in þe begynnynge. but ȝet þou hast not þe craft of dyuisioɳ, but þou schalt haue hit afterwarde.¶ Pernumerumsi vis numerumquoquemultiplicare¶ Tantum pernormas subtiles absquefigurisHas normas poteris perversus sciresequentes.Mental multiplication.¶ Herehe teches þe to multipliebe þowȝtfigures in þi mynde. And þe sentence of þis verse is þis: yf þouwel multiplie on nounbre by anoþerin þi mynde, þou schal haue þereto rewles in þe verses þat schal come after.¶ Si tu perdigitumdigitumvis multiplicareRegulaprecedens dat qualiterest operandum.Digit by digit is easy.¶ Herehe teches a rewle as þou hast aforeto multiplie a digit be anoþer, as yf þou wolde wete qwat is sex tymes 6. þouleaf 161a.*schalt wete by þe rewle þat I taȝt þe before, yf þou haue mynde þerof.¶ Articulumsi perreliquumreliquumvis multiplicareIn propriumdigitumdebet vterqueresolui.¶ Articulusdigitos post se multiplicantesEx digitusquociens retenerit multiplicariArticuli faciunt tot centummultiplicati.The first case of the craft.¶ Herehe teches þe furst rewle, þe quych is þis: yf þou wel multiplie an articul be anoþer, so þat both þe articuls bene with-Inne an hundreth, þus þou schalt do.Article by article;take þe digit of bothe the articuls, for euery articul hase a digit, þen multiplye þat on digit by þat oþer, and loke how mony vnytes ben in þe nounbre þat comes of þe multiplicacioɳ of þe 2 digittes, & so mony hundrythes ben in þe nounbreþat schal come of þe multiplicacioɳ of þe ylke 2 articuls as þus.an example:yf þou wold wete qwat is ten tymes ten. take þe digit of ten, þe quych is 1; take þe digit of þat oþerten, þe quych is on. ¶ Also multiplie 1 be 1, as on tyme on þat is but 1. In on is but on vnite as þou wost welle, þereforeten tymes ten is but a hundryth.another example:¶ Also yf þou wold wete what is twenty tymes 30. take þe digit of twenty, þat is 2; & take þe digitt of thrytty, þat is 3. multiplie 3 be 2, þat is 6. Now in 6 ben 6 vnites, ¶ And so mony hundrythes ben in 20 tymes 30*,leaf 161b.þerefore20 tymes 30 is 6 hundryth eueɳ. loke & se. ¶ But yf it be so þat onearticul be with-Inne an hundryth,orby-twene an hundryth and a thowsande, so þat it be not a þowsande fully. þen loke how mony vnytes ben in þe nounbur þat comys of þe multiplicacioɳ16And so mony tymes16of 2 digittesof ylke articuls, so mony thowsant ben in þe nounbre, the qwych comes of þe multiplicacioɳ. And so mony tymes ten thowsand schal be in þe nounbre þat comes of þe multiplicacion of2 articuls, as yf þou wold wete qwat is 4 hundryth tymes [two hundryth]. Multiply 4 be 2,17þat wol be 8. in 8 ben 8 vnites.How to work subtly without Figures.Mental multiplication.¶ And so mony tymes ten thousand be in 4 hundryth tymes [2]17hundryth, þat is 80 thousand. Take hede, I schall telle þe aAnother example.generallerewle whan þou hast 2 articuls, And þou wold wete qwat comes of þe multiplicacioɳ of hem 2. multiplie þe digit of þat onarticuls, and kepe þat nounbre, þen loke how mony cifers schuld go beforeþat on articuls,andhe werewrite. Als mony cifers schuld go beforeþat other, & he werewrite of cifers. And haue alleþe ylke cifers togedurin þi mynde,leaf 162a.*a-roweychoɳaftur other, and in þe last plase set þe nounbre þat comes of þe multiplicacioɳ of þe 2 digittes. And loke in þi mynde in what place he stondes,wherein þe secunde, or in þe thryd, or in þe 4, or whereellis, and loke qwat þe figures by-token in þat place; & so mych is þe nounbre þatAnother example.comes of þe 2 articuls y-multiplied to-geduras þus: yf þou wold wete what is 20 thousant tymes 3 þowsande. multiply þe digit of þat articulleþe quych is 2 by þe digitte of þat oþerarticul þe quych is 3, þat wol be 6. þen loke how mony cifers schal go to 20 thousant as hit schuld be write in a tabul. certainly 4 cifers schuld go to 20 þowsant. ffor þis figure 2 in þe fyrst place betokenes twene.Notation.¶ In þe secunde place hit betokenes twenty. ¶ In þe 3. place hit betokenes 2 hundryth. .¶. In þe 4 place 2 thousant. ¶ In þe 5 place hit betokenes twenty þousant. þereforehe most haue 4 cifers a-forehym þat he may stonde in þe 5 place. kepe þese 4 cifers in thy mynde, þen loke how mony cifers goɳ to 3 thousant. Certayn to 3 thousanteleaf 162b.*goɳ 3 cifers afore. Now cast ylke 4 cifers þat schuld go to twenty thousant, And thes 3 cifers þat schuld go afore3 thousant, & sette hem in rewe ychoɳ afteroþerin þi mynde, as þai schuld stonde in a tabulle. And þen schal þou haue 7 cifers; þen sett þat 6 þe quych comes of þe multiplicacioɳ of þe 2 digittesaftur þe ylke cifers in þe 8 place as yf þat hit stode in a tabul. And loke qwat a figureof 6 schuld betoken in þe 8 place. yf hit werein a tabul & so mych it is. & yf þat figure of 6 stonde in þe fyrst place he schuld betoken but 6. ¶ In þe 2 place he schuld betoken sexty. ¶ In the 3 place he schuld betokeɳ sex hundryth.Notation again.¶ In þe 4 place sex thousant. ¶ In þe 5 place sexty þowsant. ¶ In þe sext place sex hundryth þowsant. ¶ In þe 7 place sex þowsant thousantes. ¶ In þe 8 place sexty þowsant thousantes. þerforesett 6 in octauo loco, And he schal betoken sexty þowsantthousantes.Mental multiplication.And so mych is twenty þowsant tymes 3 thousant, ¶ And þis rewle is generallefor allemanerof articuls, Whethir þai be hundryth or þowsant; but þoumostknow well þe craft of þewryrchyngein þe tabulleleaf 163a.*or þou know to do þus in þi mynde aftur þis rewle. Thou most þat þis rewleholdyþenotebut whereþereben 2 articuls and no mo of þe quych ayther of hem hase but on figuresignificatyf. As twenty tymes 3 thousant or 3 hundryth, and such oþur.¶ Articulum digito si multiplicare oportetArticuli digit[i sumi quo multiplicate]Debemusreliquumquod multiplicaturab illisPerreliquo decuplumsic summamlaterenequibit.The third case of the craft;¶ Herehe puttes þe thryde rewle, þe quych is þis. yf þou wel multiply in þi mynde, And þe Articul be a digitte, þou schalt loke þat þe digitt be with-Inne an hundryth, þen þou schalt multiply the digitt of þe Articulle by þe oþer digitte. And euery vnite in þe nounbre þat schallecome þere-of schal betoken ten. As þus:an example.yf þat þou wold wete qwat is twyes 40. multiplie þe digitteof 40, þe quych is 4, by þe oþerdiget, þe quych is 2. And þat wolle be 8. And in þe nombre of 8 ben 8 vnites, & euery of þe ylke vnites schuld stonde for 10. þere-fore þereschal be 8 tymes 10, þat wol be 4 score. And so mony is twyes 40. ¶ If þe articul be a hundryth or be 2 hundryth And a þowsant, so þat hit be notte a thousant,leaf 163b.*worch as þoudyddyst afore, saue þou schalt rekene euery vnite for a hundryth.¶ In numerummixtumdigitumsi ducerecuresArticulusmixti sumaturdeinde resoluasIn digitumpost fac respectu de digitisArticulusquedocet excrescens in diriuandoIn digitummixti post ducas multiplicantem¶ De digitis vt norma18[docet] de [hunc]Multiplica simul et sic postea summa patebit.The fourth case of the craft:Here he puttes þe 4 rewle, þe quych is þis: yf þou multipliy on composit be a digit as 6 tymes 24,19þen take þe diget of þat composit, & multiply þat digitt by þat oþerdiget, and kepe þe nomburþat comes þere-of. þen take þe digit of þat composit, & multiply þat digit by anoþerdiget, by þe quych þou hast multiplyed þe diget of þe articul, and loke qwat comes þere-of.Composite by digit.þen take þou þat nounbur, & cast hit to þat other nounbur þat þou secheste as þus yf þou welwete qwat comes of 6 tymes 4 & twenty.Mental multiplication.multiply þat articulleof þe composit by þe digit, þe quych is 6, as yn þe thryd rewle þou was tauȝt, And þat schal be 6 score. þen multiply þe diget of þecomposit,leaf 164a.*þe quych is 4, and multiply þat by þat other diget, þe quych is 6, as þou wast tauȝt in þe first rewle, yf þou haue mynde þerof, & þat wol be 4 & twenty. cast all ylke nounburs to-gedir, & hit schal be 144. And so mych is 6 tymes 4 & twenty.How to multiply without Figures.¶ Ductusin articulumnumerussicompositussitArticulumpurumcomites articulumquoqueMixti pro digitispost fiat [et articulus vt]Norma iubet [retinendo quod extra dicta ab illis]Articuli digitumpost tu mixtumdigitumducRegulade digitis nec precipit articulusqueEx quibusexcrescens summe tu iunge prioriSic manifesta cito fiet tibisumma petita.The fifth case of the craft:¶ Herehe puttes þe 5 rewle, þe quych is þis: yf þou wel multiply an Articul be a composit, multiplie þat Articul by þe articul of þe composit, and worch as þou wos tauȝt in þe secunde rewle, of þe quych rewle þe verse begynnes þus.Article by Composite.¶ Articulumsi perRelicumvis multiplicare. þen multiply þe diget of þe composit by þat oþirarticul aftirþe doctrine of þe 3 rewle. take þerof gode hede, I pray þe as þus. Yf þou wel wete what is 24 tymes ten.An example.Multiplie ten by 20, þat wel be 2 hundryth. þen multiply þe diget of þe 10, þe quych is 1, by þe diget of þe composit, þe quych is 4, & þatleaf 164b.*wol be 4. þen reken euery vnite þat is in 4 for 10, & þat schal be 40. Cast 40 to 2 hundryth, & þat wol be 2 hundryth & 40. And so mych is 24 tymes ten.How to work without Figures.¶ Compositumnumerummixto si[c] multiplicabisVndecies tredecimsic estex hiis operandumIn reliquumprimum demumduc post in eundemVnumpost denum duc in tria deinde pervnumMultiplicesquedemum intraomnia multiplicataIn summa decies quam si fuerit tibidocesMultiplicandorumde normis sufficiunt hec.The sixth case of the craft:¶ Here he puttes þe 6 rewle, & þe last of allemultiplicacioɳ, þe quych is þis: yf þou wel multiplye acomposit by a-noþercomposit, þou schalt do þus.Composite by Composite.multiplie þat on composit, qwych þou welt of the twene, by þe articul of þe toþercomposit, as þou weretauȝt in þe 5 rewle, þen multiplie þat same composit, þe quych þou hast multiplied by þe oþerarticul, by þe digit of þe oþercomposit,Mental multiplication.asþou was tauȝt in þe 4 rewle.An exampleAs þus, yf þou wold wete what is 11 tymes 13, as þou was tauȝt in þe 5 rewle, & þat schal be an hundryth & ten, afterwarde multiply þat same composit þat þou hast multiplied, þe quych is a .11. And multiplye hit be þe digit of þe oþercomposit, þe quych is 3, for 3 is þe digit of 13, And þat wel be 30. þen take þe digit of þat composit, þe quych composit þou multiplied by þe digit of þat oþercomposit,leaf 165a.*þe quych is a 11.of the sixth case of the craft.¶ Also of the quych 11 on is þe digit. multiplie þat digitt by þe digett of þat othercomposit, þe quych diget is 3, as þou was tauȝt in þe first rewle inþe begynnyngeof þis craft. þe quych rewle begynnes“In digitumcures.” And of alleþe multiplicacioɳ of þe 2 digitt comys thre, for onys 3 is but 3. Now cast alleþese nounbers togedur, the quych is þis, a hundryth & ten & 30 & 3. And al þat wel be 143. Write 3 first in þe ryght side. And cast 10 to 30, þat wol be 40. set 40 next afturtowarde þe lyft side, And set aftur a hundryth as herean Ensampulle, 143.(Cetera desunt.)1.In MS, ‘awiy.’2.‘ben’ repeated in MS.3.In MS. ‘thausandes.’4.Perhaps “So.”5.‘hali’ marked for erasure in MS.6.‘moy’ in MS.7.‘Subtrahas aut addis a dextrisvel mediabis’ added on margin of MS.8.After ‘craft’ insert ‘the .4. what is þe profet of þis craft.’9.After ‘sythes’ insert ‘& þis wordes fyue sithe & sex sythes.’10.‘t’l’ marked for erasure before ‘tyl’ in MS.11.Here ‘of þe same rew’ is marked for erasure in MS.12.‘sed’ deleted in MS.13.6883 in MS.14.‘þen’ overwritten on ‘þat’ marked for erasure.15.‘Supra’ inserted in MS. in place of ‘cifra’ marked for erasure.16–16.Marked for erasure in MS.17.4 in MS.18.docet. decet MS.19.‘4 times 4’ in MS.
* ¶ Sedcummultiplicabis, primo sic estoperandum,Si dabit articulumtibi multiplicacio solum;Proposita cifra summamtransferre memento.What to do if the first multiplication results in an article.¶ Herehe puttes þe fyrst case of þis craft, þe quych is þis: yf þerecome an articulle of þe multiplicacioɳ ysette beforethe articullein þe lyft side as þus5123.multiplye 5 by 2, þat wol be 10; sette ouerþe hede of þat 2 a 0, & sett þat on, þat is þe articul, in þe lyft side, þat is next hym, þen þou schalt haue þis nounbre1051.23¶ And þen worch forth as þou diddist afore. And þou schalt vnderstonde þat þou schalt write no 0. but whan þat place where þou schal write þat 0 has no figure afore hymnoþerafter. versus.¶ Si autemdigitus excreuerit articulusque.Articulus12supraposito digito salit vltra.What to do if the result is a composite number.¶ Hereis þe secunde case, þe quych is þis: yf hit happe þat þerecome a composyt, þou schalt write þe digitte ouerþe hede of þe neþerfigureby þe quych þou multipliest þe hier figure; and sett þe articullenext hym toward þe lyft side, as þou diddyst afore, as þus83.83Multiply 8 by 8, þat wol be 64. Write þe 4 ouer8, þat is to say, ouerþe hede of þe neþer8; & set 6, þe quychleaf 159a.*is an articul, next after. And þen þou schalt haue such a nounbre as is here,648313,83And þen worch forth.¶ Si digitus tamenponas ipsum superipsam.What if it be a digit.¶ Hereis þe thryde case, þe quych is þis: yf hit happe þat of þi multiplicaciouɳ come a digit, þou schalt write þe digit ouerþe hede of þe neþerfigure, by the quych þou multipliest þe hierefigure, for þis nedes no Ensampul.¶ Subdita multiplica non hanc que [incidit] illiDelet eampenitusscribens quod prouenit inde.The fourth case of the craft.¶ Hereis þe 4 case, þe quych is: yf hit be happe þat þe neþerfigureschal multiplye þat figure, þe quych stondes ouerþat figures hede, þou schal do away þe hier figure& sett þere þat þat comys of þat multiplicacioɳ. As yf þere come of þat multiplicacioɳ an articuls þou schalt write þere þe hier figurestode a 0. ¶ And write þe articuls in þe lyft side, yf þat hit be a digit write þere a digit. yf þat hit be a composit, write þe digit of þe composit. And þe articul in þe lyft side. al þis is lyȝt y-nowȝt, þere-foreþer nedes no Ensampul.¶ Sedsi multiplicat aliamponas superipsamAdiunges numerumquemprebet ductusearum.leaf 159b.The fifth case of the craft.¶ Hereis þe 5 case, þe quych is þis: yf *þe neþerfigureschul multiplie þe hier, and þat hier figureis notrecteouerhis hede. And þat neþerfigurehase oþerfigures, or on figure ouerhis hede by multiplicacioɳ, þat hase be afore, þou schalt write þat nounbre, þe quych comes of þat, oueralleþe ylke figures hedes, as þus here:236234Multiply 2 by 2, þat wol be 4; set 4 ouerþe hede of þat 2. þen14multiplies þe hier 2 by þe neþer3, þat wol be 6. set ouerhis hede 6, multiplie þe hier 2 by þe neþer4, þat wol be 8. do away þe hier 2, þe quych stondes ouerþe hede of þe figureof 4,and set þere 8. And þou schalt haue þis nounbrehere46836234And antery þi figures, þat is to say, set þi neþer4 vnderþe hier 3, and set þi 2 other figures nerehym, so þat þe neþer2 stonde vndurþe hier 6, þe quych 6 stondes in þe lyft side. And þat 3 þat stondes vndur 8, as þus aftur ȝe may se,46836234Now worch forthermore, And multiplye þat hier 3 by 2, þat wol be 6, set þat 6 þe quych stondes ouerþe hede of þat 2, And þen worch as I taȝt þe afore.leaf 160a.* ¶ Si supraposita cifra debet multiplicareProrsus eamdeles & ibi scribi cifra debet.The sixth case of the craft.¶ Hereis þe 6 case, þe quych is þis: yf hit happe þat þe figureby þe quych þou schal multiplye þe hier figure, þe quych stondes ryght ouerhym by a 0, þou schalt do away þat figure, þe quych ouerþat cifre hede. ¶ And write þereþat nounbre þat comes of þe multiplicacioɳ as þus, 23. do away 2 and sett þere a 0. vndeversus.¶ Si cifra multiplicat aliampositamsuperipsamSitquelocus supravacuussuperhanc ciframfiet.The seventh case of the craft.¶ Hereis þe 7 case, þe quych is þis: yf a 0 schal multiply a figure, þe quych stondes notrecteouerhym, And ouerþat 0 stonde no thyng, þou schalt write ouerþat 0 anoþer0 as þus:2403multiplye 2 be a 0, it wol be nothynge. write þere a 0 ouerþe hede of þe neþer0, And þen worch forth til þou come to þe ende.¶ Si supra15fuerit cifra semperestpretereunda.The eighth case of the craft.¶ Hereis þe 8 case, þe quych is þis: yf þerebe a 0 or mony cifers in þe hier rewe, þou schalt not multiplie hem, bot let hem stonde. And antery þe figures beneþe to þe next figuresygnificatyf as þus:00032.22Ouer-lepe alleþese cifers & sett þatleaf 160b.*neþer2 þat stondes toward þe ryght side, and sett hym vndurþe 3, and sett þe oþernether 2 nere hym, so þat he stonde vndurþe thrydde 0, þe quych stondes next 3. And þan worch. vndeversus.¶ Si dubites, an sit benemultiplicacio facta,Diuide totalem numerumpermultiplicantem.How to prove the multiplication.¶ Herehe teches how þou schalt know wheþerþou hase wel I-do or no. And he says þat þou schalt deuide alleþe nounbre þat comes of þe multiplicacioɳ by þe neþerfigures. And þen þou schalt haue þe same nounbur þat þou hadyst in þe begynnynge. but ȝet þou hast not þe craft of dyuisioɳ, but þou schalt haue hit afterwarde.¶ Pernumerumsi vis numerumquoquemultiplicare¶ Tantum pernormas subtiles absquefigurisHas normas poteris perversus sciresequentes.Mental multiplication.¶ Herehe teches þe to multipliebe þowȝtfigures in þi mynde. And þe sentence of þis verse is þis: yf þouwel multiplie on nounbre by anoþerin þi mynde, þou schal haue þereto rewles in þe verses þat schal come after.¶ Si tu perdigitumdigitumvis multiplicareRegulaprecedens dat qualiterest operandum.Digit by digit is easy.¶ Herehe teches a rewle as þou hast aforeto multiplie a digit be anoþer, as yf þou wolde wete qwat is sex tymes 6. þouleaf 161a.*schalt wete by þe rewle þat I taȝt þe before, yf þou haue mynde þerof.¶ Articulumsi perreliquumreliquumvis multiplicareIn propriumdigitumdebet vterqueresolui.¶ Articulusdigitos post se multiplicantesEx digitusquociens retenerit multiplicariArticuli faciunt tot centummultiplicati.The first case of the craft.¶ Herehe teches þe furst rewle, þe quych is þis: yf þou wel multiplie an articul be anoþer, so þat both þe articuls bene with-Inne an hundreth, þus þou schalt do.Article by article;take þe digit of bothe the articuls, for euery articul hase a digit, þen multiplye þat on digit by þat oþer, and loke how mony vnytes ben in þe nounbre þat comes of þe multiplicacioɳ of þe 2 digittes, & so mony hundrythes ben in þe nounbreþat schal come of þe multiplicacioɳ of þe ylke 2 articuls as þus.an example:yf þou wold wete qwat is ten tymes ten. take þe digit of ten, þe quych is 1; take þe digit of þat oþerten, þe quych is on. ¶ Also multiplie 1 be 1, as on tyme on þat is but 1. In on is but on vnite as þou wost welle, þereforeten tymes ten is but a hundryth.another example:¶ Also yf þou wold wete what is twenty tymes 30. take þe digit of twenty, þat is 2; & take þe digitt of thrytty, þat is 3. multiplie 3 be 2, þat is 6. Now in 6 ben 6 vnites, ¶ And so mony hundrythes ben in 20 tymes 30*,leaf 161b.þerefore20 tymes 30 is 6 hundryth eueɳ. loke & se. ¶ But yf it be so þat onearticul be with-Inne an hundryth,orby-twene an hundryth and a thowsande, so þat it be not a þowsande fully. þen loke how mony vnytes ben in þe nounbur þat comys of þe multiplicacioɳ16And so mony tymes16of 2 digittesof ylke articuls, so mony thowsant ben in þe nounbre, the qwych comes of þe multiplicacioɳ. And so mony tymes ten thowsand schal be in þe nounbre þat comes of þe multiplicacion of2 articuls, as yf þou wold wete qwat is 4 hundryth tymes [two hundryth]. Multiply 4 be 2,17þat wol be 8. in 8 ben 8 vnites.How to work subtly without Figures.Mental multiplication.¶ And so mony tymes ten thousand be in 4 hundryth tymes [2]17hundryth, þat is 80 thousand. Take hede, I schall telle þe aAnother example.generallerewle whan þou hast 2 articuls, And þou wold wete qwat comes of þe multiplicacioɳ of hem 2. multiplie þe digit of þat onarticuls, and kepe þat nounbre, þen loke how mony cifers schuld go beforeþat on articuls,andhe werewrite. Als mony cifers schuld go beforeþat other, & he werewrite of cifers. And haue alleþe ylke cifers togedurin þi mynde,leaf 162a.*a-roweychoɳaftur other, and in þe last plase set þe nounbre þat comes of þe multiplicacioɳ of þe 2 digittes. And loke in þi mynde in what place he stondes,wherein þe secunde, or in þe thryd, or in þe 4, or whereellis, and loke qwat þe figures by-token in þat place; & so mych is þe nounbre þatAnother example.comes of þe 2 articuls y-multiplied to-geduras þus: yf þou wold wete what is 20 thousant tymes 3 þowsande. multiply þe digit of þat articulleþe quych is 2 by þe digitte of þat oþerarticul þe quych is 3, þat wol be 6. þen loke how mony cifers schal go to 20 thousant as hit schuld be write in a tabul. certainly 4 cifers schuld go to 20 þowsant. ffor þis figure 2 in þe fyrst place betokenes twene.Notation.¶ In þe secunde place hit betokenes twenty. ¶ In þe 3. place hit betokenes 2 hundryth. .¶. In þe 4 place 2 thousant. ¶ In þe 5 place hit betokenes twenty þousant. þereforehe most haue 4 cifers a-forehym þat he may stonde in þe 5 place. kepe þese 4 cifers in thy mynde, þen loke how mony cifers goɳ to 3 thousant. Certayn to 3 thousanteleaf 162b.*goɳ 3 cifers afore. Now cast ylke 4 cifers þat schuld go to twenty thousant, And thes 3 cifers þat schuld go afore3 thousant, & sette hem in rewe ychoɳ afteroþerin þi mynde, as þai schuld stonde in a tabulle. And þen schal þou haue 7 cifers; þen sett þat 6 þe quych comes of þe multiplicacioɳ of þe 2 digittesaftur þe ylke cifers in þe 8 place as yf þat hit stode in a tabul. And loke qwat a figureof 6 schuld betoken in þe 8 place. yf hit werein a tabul & so mych it is. & yf þat figure of 6 stonde in þe fyrst place he schuld betoken but 6. ¶ In þe 2 place he schuld betoken sexty. ¶ In the 3 place he schuld betokeɳ sex hundryth.Notation again.¶ In þe 4 place sex thousant. ¶ In þe 5 place sexty þowsant. ¶ In þe sext place sex hundryth þowsant. ¶ In þe 7 place sex þowsant thousantes. ¶ In þe 8 place sexty þowsant thousantes. þerforesett 6 in octauo loco, And he schal betoken sexty þowsantthousantes.Mental multiplication.And so mych is twenty þowsant tymes 3 thousant, ¶ And þis rewle is generallefor allemanerof articuls, Whethir þai be hundryth or þowsant; but þoumostknow well þe craft of þewryrchyngein þe tabulleleaf 163a.*or þou know to do þus in þi mynde aftur þis rewle. Thou most þat þis rewleholdyþenotebut whereþereben 2 articuls and no mo of þe quych ayther of hem hase but on figuresignificatyf. As twenty tymes 3 thousant or 3 hundryth, and such oþur.¶ Articulum digito si multiplicare oportetArticuli digit[i sumi quo multiplicate]Debemusreliquumquod multiplicaturab illisPerreliquo decuplumsic summamlaterenequibit.The third case of the craft;¶ Herehe puttes þe thryde rewle, þe quych is þis. yf þou wel multiply in þi mynde, And þe Articul be a digitte, þou schalt loke þat þe digitt be with-Inne an hundryth, þen þou schalt multiply the digitt of þe Articulle by þe oþer digitte. And euery vnite in þe nounbre þat schallecome þere-of schal betoken ten. As þus:an example.yf þat þou wold wete qwat is twyes 40. multiplie þe digitteof 40, þe quych is 4, by þe oþerdiget, þe quych is 2. And þat wolle be 8. And in þe nombre of 8 ben 8 vnites, & euery of þe ylke vnites schuld stonde for 10. þere-fore þereschal be 8 tymes 10, þat wol be 4 score. And so mony is twyes 40. ¶ If þe articul be a hundryth or be 2 hundryth And a þowsant, so þat hit be notte a thousant,leaf 163b.*worch as þoudyddyst afore, saue þou schalt rekene euery vnite for a hundryth.¶ In numerummixtumdigitumsi ducerecuresArticulusmixti sumaturdeinde resoluasIn digitumpost fac respectu de digitisArticulusquedocet excrescens in diriuandoIn digitummixti post ducas multiplicantem¶ De digitis vt norma18[docet] de [hunc]Multiplica simul et sic postea summa patebit.The fourth case of the craft:Here he puttes þe 4 rewle, þe quych is þis: yf þou multipliy on composit be a digit as 6 tymes 24,19þen take þe diget of þat composit, & multiply þat digitt by þat oþerdiget, and kepe þe nomburþat comes þere-of. þen take þe digit of þat composit, & multiply þat digit by anoþerdiget, by þe quych þou hast multiplyed þe diget of þe articul, and loke qwat comes þere-of.Composite by digit.þen take þou þat nounbur, & cast hit to þat other nounbur þat þou secheste as þus yf þou welwete qwat comes of 6 tymes 4 & twenty.Mental multiplication.multiply þat articulleof þe composit by þe digit, þe quych is 6, as yn þe thryd rewle þou was tauȝt, And þat schal be 6 score. þen multiply þe diget of þecomposit,leaf 164a.*þe quych is 4, and multiply þat by þat other diget, þe quych is 6, as þou wast tauȝt in þe first rewle, yf þou haue mynde þerof, & þat wol be 4 & twenty. cast all ylke nounburs to-gedir, & hit schal be 144. And so mych is 6 tymes 4 & twenty.How to multiply without Figures.¶ Ductusin articulumnumerussicompositussitArticulumpurumcomites articulumquoqueMixti pro digitispost fiat [et articulus vt]Norma iubet [retinendo quod extra dicta ab illis]Articuli digitumpost tu mixtumdigitumducRegulade digitis nec precipit articulusqueEx quibusexcrescens summe tu iunge prioriSic manifesta cito fiet tibisumma petita.The fifth case of the craft:¶ Herehe puttes þe 5 rewle, þe quych is þis: yf þou wel multiply an Articul be a composit, multiplie þat Articul by þe articul of þe composit, and worch as þou wos tauȝt in þe secunde rewle, of þe quych rewle þe verse begynnes þus.Article by Composite.¶ Articulumsi perRelicumvis multiplicare. þen multiply þe diget of þe composit by þat oþirarticul aftirþe doctrine of þe 3 rewle. take þerof gode hede, I pray þe as þus. Yf þou wel wete what is 24 tymes ten.An example.Multiplie ten by 20, þat wel be 2 hundryth. þen multiply þe diget of þe 10, þe quych is 1, by þe diget of þe composit, þe quych is 4, & þatleaf 164b.*wol be 4. þen reken euery vnite þat is in 4 for 10, & þat schal be 40. Cast 40 to 2 hundryth, & þat wol be 2 hundryth & 40. And so mych is 24 tymes ten.How to work without Figures.¶ Compositumnumerummixto si[c] multiplicabisVndecies tredecimsic estex hiis operandumIn reliquumprimum demumduc post in eundemVnumpost denum duc in tria deinde pervnumMultiplicesquedemum intraomnia multiplicataIn summa decies quam si fuerit tibidocesMultiplicandorumde normis sufficiunt hec.The sixth case of the craft:¶ Here he puttes þe 6 rewle, & þe last of allemultiplicacioɳ, þe quych is þis: yf þou wel multiplye acomposit by a-noþercomposit, þou schalt do þus.Composite by Composite.multiplie þat on composit, qwych þou welt of the twene, by þe articul of þe toþercomposit, as þou weretauȝt in þe 5 rewle, þen multiplie þat same composit, þe quych þou hast multiplied by þe oþerarticul, by þe digit of þe oþercomposit,Mental multiplication.asþou was tauȝt in þe 4 rewle.An exampleAs þus, yf þou wold wete what is 11 tymes 13, as þou was tauȝt in þe 5 rewle, & þat schal be an hundryth & ten, afterwarde multiply þat same composit þat þou hast multiplied, þe quych is a .11. And multiplye hit be þe digit of þe oþercomposit, þe quych is 3, for 3 is þe digit of 13, And þat wel be 30. þen take þe digit of þat composit, þe quych composit þou multiplied by þe digit of þat oþercomposit,leaf 165a.*þe quych is a 11.of the sixth case of the craft.¶ Also of the quych 11 on is þe digit. multiplie þat digitt by þe digett of þat othercomposit, þe quych diget is 3, as þou was tauȝt in þe first rewle inþe begynnyngeof þis craft. þe quych rewle begynnes“In digitumcures.” And of alleþe multiplicacioɳ of þe 2 digitt comys thre, for onys 3 is but 3. Now cast alleþese nounbers togedur, the quych is þis, a hundryth & ten & 30 & 3. And al þat wel be 143. Write 3 first in þe ryght side. And cast 10 to 30, þat wol be 40. set 40 next afturtowarde þe lyft side, And set aftur a hundryth as herean Ensampulle, 143.(Cetera desunt.)1.In MS, ‘awiy.’2.‘ben’ repeated in MS.3.In MS. ‘thausandes.’4.Perhaps “So.”5.‘hali’ marked for erasure in MS.6.‘moy’ in MS.7.‘Subtrahas aut addis a dextrisvel mediabis’ added on margin of MS.8.After ‘craft’ insert ‘the .4. what is þe profet of þis craft.’9.After ‘sythes’ insert ‘& þis wordes fyue sithe & sex sythes.’10.‘t’l’ marked for erasure before ‘tyl’ in MS.11.Here ‘of þe same rew’ is marked for erasure in MS.12.‘sed’ deleted in MS.13.6883 in MS.14.‘þen’ overwritten on ‘þat’ marked for erasure.15.‘Supra’ inserted in MS. in place of ‘cifra’ marked for erasure.16–16.Marked for erasure in MS.17.4 in MS.18.docet. decet MS.19.‘4 times 4’ in MS.
* ¶ Sedcummultiplicabis, primo sic estoperandum,Si dabit articulumtibi multiplicacio solum;Proposita cifra summamtransferre memento.
* ¶ Sedcummultiplicabis, primo sic estoperandum,
Si dabit articulumtibi multiplicacio solum;
Proposita cifra summamtransferre memento.
What to do if the first multiplication results in an article.¶ Herehe puttes þe fyrst case of þis craft, þe quych is þis: yf þerecome an articulle of þe multiplicacioɳ ysette beforethe articullein þe lyft side as þus5123.multiplye 5 by 2, þat wol be 10; sette ouerþe hede of þat 2 a 0, & sett þat on, þat is þe articul, in þe lyft side, þat is next hym, þen þou schalt haue þis nounbre1051.23¶ And þen worch forth as þou diddist afore. And þou schalt vnderstonde þat þou schalt write no 0. but whan þat place where þou schal write þat 0 has no figure afore hymnoþerafter. versus.
¶ Si autemdigitus excreuerit articulusque.Articulus12supraposito digito salit vltra.
¶ Si autemdigitus excreuerit articulusque.
Articulus12supraposito digito salit vltra.
What to do if the result is a composite number.¶ Hereis þe secunde case, þe quych is þis: yf hit happe þat þerecome a composyt, þou schalt write þe digitte ouerþe hede of þe neþerfigureby þe quych þou multipliest þe hier figure; and sett þe articullenext hym toward þe lyft side, as þou diddyst afore, as þus83.83Multiply 8 by 8, þat wol be 64. Write þe 4 ouer8, þat is to say, ouerþe hede of þe neþer8; & set 6, þe quychleaf 159a.*is an articul, next after. And þen þou schalt haue such a nounbre as is here,648313,83And þen worch forth.
¶ Si digitus tamenponas ipsum superipsam.
¶ Si digitus tamenponas ipsum superipsam.
What if it be a digit.¶ Hereis þe thryde case, þe quych is þis: yf hit happe þat of þi multiplicaciouɳ come a digit, þou schalt write þe digit ouerþe hede of þe neþerfigure, by the quych þou multipliest þe hierefigure, for þis nedes no Ensampul.
¶ Subdita multiplica non hanc que [incidit] illiDelet eampenitusscribens quod prouenit inde.
¶ Subdita multiplica non hanc que [incidit] illi
Delet eampenitusscribens quod prouenit inde.
The fourth case of the craft.¶ Hereis þe 4 case, þe quych is: yf hit be happe þat þe neþerfigureschal multiplye þat figure, þe quych stondes ouerþat figures hede, þou schal do away þe hier figure& sett þere þat þat comys of þat multiplicacioɳ. As yf þere come of þat multiplicacioɳ an articuls þou schalt write þere þe hier figurestode a 0. ¶ And write þe articuls in þe lyft side, yf þat hit be a digit write þere a digit. yf þat hit be a composit, write þe digit of þe composit. And þe articul in þe lyft side. al þis is lyȝt y-nowȝt, þere-foreþer nedes no Ensampul.
¶ Sedsi multiplicat aliamponas superipsamAdiunges numerumquemprebet ductusearum.
¶ Sedsi multiplicat aliamponas superipsam
Adiunges numerumquemprebet ductusearum.
The fifth case of the craft.¶ Hereis þe 5 case, þe quych is þis: yf *þe neþerfigureschul multiplie þe hier, and þat hier figureis notrecteouerhis hede. And þat neþerfigurehase oþerfigures, or on figure ouerhis hede by multiplicacioɳ, þat hase be afore, þou schalt write þat nounbre, þe quych comes of þat, oueralleþe ylke figures hedes, as þus here:236234Multiply 2 by 2, þat wol be 4; set 4 ouerþe hede of þat 2. þen14multiplies þe hier 2 by þe neþer3, þat wol be 6. set ouerhis hede 6, multiplie þe hier 2 by þe neþer4, þat wol be 8. do away þe hier 2, þe quych stondes ouerþe hede of þe figureof 4,and set þere 8. And þou schalt haue þis nounbrehere46836234And antery þi figures, þat is to say, set þi neþer4 vnderþe hier 3, and set þi 2 other figures nerehym, so þat þe neþer2 stonde vndurþe hier 6, þe quych 6 stondes in þe lyft side. And þat 3 þat stondes vndur 8, as þus aftur ȝe may se,46836234Now worch forthermore, And multiplye þat hier 3 by 2, þat wol be 6, set þat 6 þe quych stondes ouerþe hede of þat 2, And þen worch as I taȝt þe afore.
* ¶ Si supraposita cifra debet multiplicareProrsus eamdeles & ibi scribi cifra debet.
* ¶ Si supraposita cifra debet multiplicare
Prorsus eamdeles & ibi scribi cifra debet.
The sixth case of the craft.¶ Hereis þe 6 case, þe quych is þis: yf hit happe þat þe figureby þe quych þou schal multiplye þe hier figure, þe quych stondes ryght ouerhym by a 0, þou schalt do away þat figure, þe quych ouerþat cifre hede. ¶ And write þereþat nounbre þat comes of þe multiplicacioɳ as þus, 23. do away 2 and sett þere a 0. vndeversus.
¶ Si cifra multiplicat aliampositamsuperipsamSitquelocus supravacuussuperhanc ciframfiet.
¶ Si cifra multiplicat aliampositamsuperipsam
Sitquelocus supravacuussuperhanc ciframfiet.
The seventh case of the craft.¶ Hereis þe 7 case, þe quych is þis: yf a 0 schal multiply a figure, þe quych stondes notrecteouerhym, And ouerþat 0 stonde no thyng, þou schalt write ouerþat 0 anoþer0 as þus:2403multiplye 2 be a 0, it wol be nothynge. write þere a 0 ouerþe hede of þe neþer0, And þen worch forth til þou come to þe ende.
¶ Si supra15fuerit cifra semperestpretereunda.
¶ Si supra15fuerit cifra semperestpretereunda.
The eighth case of the craft.¶ Hereis þe 8 case, þe quych is þis: yf þerebe a 0 or mony cifers in þe hier rewe, þou schalt not multiplie hem, bot let hem stonde. And antery þe figures beneþe to þe next figuresygnificatyf as þus:00032.22Ouer-lepe alleþese cifers & sett þatleaf 160b.*neþer2 þat stondes toward þe ryght side, and sett hym vndurþe 3, and sett þe oþernether 2 nere hym, so þat he stonde vndurþe thrydde 0, þe quych stondes next 3. And þan worch. vndeversus.
¶ Si dubites, an sit benemultiplicacio facta,Diuide totalem numerumpermultiplicantem.
¶ Si dubites, an sit benemultiplicacio facta,
Diuide totalem numerumpermultiplicantem.
How to prove the multiplication.¶ Herehe teches how þou schalt know wheþerþou hase wel I-do or no. And he says þat þou schalt deuide alleþe nounbre þat comes of þe multiplicacioɳ by þe neþerfigures. And þen þou schalt haue þe same nounbur þat þou hadyst in þe begynnynge. but ȝet þou hast not þe craft of dyuisioɳ, but þou schalt haue hit afterwarde.
¶ Pernumerumsi vis numerumquoquemultiplicare¶ Tantum pernormas subtiles absquefigurisHas normas poteris perversus sciresequentes.
¶ Pernumerumsi vis numerumquoquemultiplicare
¶ Tantum pernormas subtiles absquefiguris
Has normas poteris perversus sciresequentes.
Mental multiplication.¶ Herehe teches þe to multipliebe þowȝtfigures in þi mynde. And þe sentence of þis verse is þis: yf þouwel multiplie on nounbre by anoþerin þi mynde, þou schal haue þereto rewles in þe verses þat schal come after.
¶ Si tu perdigitumdigitumvis multiplicareRegulaprecedens dat qualiterest operandum.
¶ Si tu perdigitumdigitumvis multiplicare
Regulaprecedens dat qualiterest operandum.
Digit by digit is easy.¶ Herehe teches a rewle as þou hast aforeto multiplie a digit be anoþer, as yf þou wolde wete qwat is sex tymes 6. þouleaf 161a.*schalt wete by þe rewle þat I taȝt þe before, yf þou haue mynde þerof.
¶ Articulumsi perreliquumreliquumvis multiplicareIn propriumdigitumdebet vterqueresolui.¶ Articulusdigitos post se multiplicantesEx digitusquociens retenerit multiplicariArticuli faciunt tot centummultiplicati.
¶ Articulumsi perreliquumreliquumvis multiplicare
In propriumdigitumdebet vterqueresolui.
¶ Articulusdigitos post se multiplicantes
Ex digitusquociens retenerit multiplicari
Articuli faciunt tot centummultiplicati.
The first case of the craft.¶ Herehe teches þe furst rewle, þe quych is þis: yf þou wel multiplie an articul be anoþer, so þat both þe articuls bene with-Inne an hundreth, þus þou schalt do.Article by article;take þe digit of bothe the articuls, for euery articul hase a digit, þen multiplye þat on digit by þat oþer, and loke how mony vnytes ben in þe nounbre þat comes of þe multiplicacioɳ of þe 2 digittes, & so mony hundrythes ben in þe nounbreþat schal come of þe multiplicacioɳ of þe ylke 2 articuls as þus.an example:yf þou wold wete qwat is ten tymes ten. take þe digit of ten, þe quych is 1; take þe digit of þat oþerten, þe quych is on. ¶ Also multiplie 1 be 1, as on tyme on þat is but 1. In on is but on vnite as þou wost welle, þereforeten tymes ten is but a hundryth.another example:¶ Also yf þou wold wete what is twenty tymes 30. take þe digit of twenty, þat is 2; & take þe digitt of thrytty, þat is 3. multiplie 3 be 2, þat is 6. Now in 6 ben 6 vnites, ¶ And so mony hundrythes ben in 20 tymes 30*,leaf 161b.þerefore20 tymes 30 is 6 hundryth eueɳ. loke & se. ¶ But yf it be so þat onearticul be with-Inne an hundryth,orby-twene an hundryth and a thowsande, so þat it be not a þowsande fully. þen loke how mony vnytes ben in þe nounbur þat comys of þe multiplicacioɳ16And so mony tymes16of 2 digittesof ylke articuls, so mony thowsant ben in þe nounbre, the qwych comes of þe multiplicacioɳ. And so mony tymes ten thowsand schal be in þe nounbre þat comes of þe multiplicacion of2 articuls, as yf þou wold wete qwat is 4 hundryth tymes [two hundryth]. Multiply 4 be 2,17þat wol be 8. in 8 ben 8 vnites.How to work subtly without Figures.Mental multiplication.¶ And so mony tymes ten thousand be in 4 hundryth tymes [2]17hundryth, þat is 80 thousand. Take hede, I schall telle þe aAnother example.generallerewle whan þou hast 2 articuls, And þou wold wete qwat comes of þe multiplicacioɳ of hem 2. multiplie þe digit of þat onarticuls, and kepe þat nounbre, þen loke how mony cifers schuld go beforeþat on articuls,andhe werewrite. Als mony cifers schuld go beforeþat other, & he werewrite of cifers. And haue alleþe ylke cifers togedurin þi mynde,leaf 162a.*a-roweychoɳaftur other, and in þe last plase set þe nounbre þat comes of þe multiplicacioɳ of þe 2 digittes. And loke in þi mynde in what place he stondes,wherein þe secunde, or in þe thryd, or in þe 4, or whereellis, and loke qwat þe figures by-token in þat place; & so mych is þe nounbre þatAnother example.comes of þe 2 articuls y-multiplied to-geduras þus: yf þou wold wete what is 20 thousant tymes 3 þowsande. multiply þe digit of þat articulleþe quych is 2 by þe digitte of þat oþerarticul þe quych is 3, þat wol be 6. þen loke how mony cifers schal go to 20 thousant as hit schuld be write in a tabul. certainly 4 cifers schuld go to 20 þowsant. ffor þis figure 2 in þe fyrst place betokenes twene.Notation.¶ In þe secunde place hit betokenes twenty. ¶ In þe 3. place hit betokenes 2 hundryth. .¶. In þe 4 place 2 thousant. ¶ In þe 5 place hit betokenes twenty þousant. þereforehe most haue 4 cifers a-forehym þat he may stonde in þe 5 place. kepe þese 4 cifers in thy mynde, þen loke how mony cifers goɳ to 3 thousant. Certayn to 3 thousanteleaf 162b.*goɳ 3 cifers afore. Now cast ylke 4 cifers þat schuld go to twenty thousant, And thes 3 cifers þat schuld go afore3 thousant, & sette hem in rewe ychoɳ afteroþerin þi mynde, as þai schuld stonde in a tabulle. And þen schal þou haue 7 cifers; þen sett þat 6 þe quych comes of þe multiplicacioɳ of þe 2 digittesaftur þe ylke cifers in þe 8 place as yf þat hit stode in a tabul. And loke qwat a figureof 6 schuld betoken in þe 8 place. yf hit werein a tabul & so mych it is. & yf þat figure of 6 stonde in þe fyrst place he schuld betoken but 6. ¶ In þe 2 place he schuld betoken sexty. ¶ In the 3 place he schuld betokeɳ sex hundryth.Notation again.¶ In þe 4 place sex thousant. ¶ In þe 5 place sexty þowsant. ¶ In þe sext place sex hundryth þowsant. ¶ In þe 7 place sex þowsant thousantes. ¶ In þe 8 place sexty þowsant thousantes. þerforesett 6 in octauo loco, And he schal betoken sexty þowsantthousantes.Mental multiplication.And so mych is twenty þowsant tymes 3 thousant, ¶ And þis rewle is generallefor allemanerof articuls, Whethir þai be hundryth or þowsant; but þoumostknow well þe craft of þewryrchyngein þe tabulleleaf 163a.*or þou know to do þus in þi mynde aftur þis rewle. Thou most þat þis rewleholdyþenotebut whereþereben 2 articuls and no mo of þe quych ayther of hem hase but on figuresignificatyf. As twenty tymes 3 thousant or 3 hundryth, and such oþur.
¶ Articulum digito si multiplicare oportetArticuli digit[i sumi quo multiplicate]Debemusreliquumquod multiplicaturab illisPerreliquo decuplumsic summamlaterenequibit.
¶ Articulum digito si multiplicare oportet
Articuli digit[i sumi quo multiplicate]
Debemusreliquumquod multiplicaturab illis
Perreliquo decuplumsic summamlaterenequibit.
The third case of the craft;¶ Herehe puttes þe thryde rewle, þe quych is þis. yf þou wel multiply in þi mynde, And þe Articul be a digitte, þou schalt loke þat þe digitt be with-Inne an hundryth, þen þou schalt multiply the digitt of þe Articulle by þe oþer digitte. And euery vnite in þe nounbre þat schallecome þere-of schal betoken ten. As þus:an example.yf þat þou wold wete qwat is twyes 40. multiplie þe digitteof 40, þe quych is 4, by þe oþerdiget, þe quych is 2. And þat wolle be 8. And in þe nombre of 8 ben 8 vnites, & euery of þe ylke vnites schuld stonde for 10. þere-fore þereschal be 8 tymes 10, þat wol be 4 score. And so mony is twyes 40. ¶ If þe articul be a hundryth or be 2 hundryth And a þowsant, so þat hit be notte a thousant,leaf 163b.*worch as þoudyddyst afore, saue þou schalt rekene euery vnite for a hundryth.
¶ In numerummixtumdigitumsi ducerecuresArticulusmixti sumaturdeinde resoluasIn digitumpost fac respectu de digitisArticulusquedocet excrescens in diriuandoIn digitummixti post ducas multiplicantem¶ De digitis vt norma18[docet] de [hunc]Multiplica simul et sic postea summa patebit.
¶ In numerummixtumdigitumsi ducerecures
Articulusmixti sumaturdeinde resoluas
In digitumpost fac respectu de digitis
Articulusquedocet excrescens in diriuando
In digitummixti post ducas multiplicantem
¶ De digitis vt norma18[docet] de [hunc]
Multiplica simul et sic postea summa patebit.
The fourth case of the craft:Here he puttes þe 4 rewle, þe quych is þis: yf þou multipliy on composit be a digit as 6 tymes 24,19þen take þe diget of þat composit, & multiply þat digitt by þat oþerdiget, and kepe þe nomburþat comes þere-of. þen take þe digit of þat composit, & multiply þat digit by anoþerdiget, by þe quych þou hast multiplyed þe diget of þe articul, and loke qwat comes þere-of.Composite by digit.þen take þou þat nounbur, & cast hit to þat other nounbur þat þou secheste as þus yf þou welwete qwat comes of 6 tymes 4 & twenty.Mental multiplication.multiply þat articulleof þe composit by þe digit, þe quych is 6, as yn þe thryd rewle þou was tauȝt, And þat schal be 6 score. þen multiply þe diget of þecomposit,leaf 164a.*þe quych is 4, and multiply þat by þat other diget, þe quych is 6, as þou wast tauȝt in þe first rewle, yf þou haue mynde þerof, & þat wol be 4 & twenty. cast all ylke nounburs to-gedir, & hit schal be 144. And so mych is 6 tymes 4 & twenty.
How to multiply without Figures.
¶ Ductusin articulumnumerussicompositussitArticulumpurumcomites articulumquoqueMixti pro digitispost fiat [et articulus vt]Norma iubet [retinendo quod extra dicta ab illis]Articuli digitumpost tu mixtumdigitumducRegulade digitis nec precipit articulusqueEx quibusexcrescens summe tu iunge prioriSic manifesta cito fiet tibisumma petita.
¶ Ductusin articulumnumerussicompositussit
Articulumpurumcomites articulumquoque
Mixti pro digitispost fiat [et articulus vt]
Norma iubet [retinendo quod extra dicta ab illis]
Articuli digitumpost tu mixtumdigitumduc
Regulade digitis nec precipit articulusque
Ex quibusexcrescens summe tu iunge priori
Sic manifesta cito fiet tibisumma petita.
The fifth case of the craft:¶ Herehe puttes þe 5 rewle, þe quych is þis: yf þou wel multiply an Articul be a composit, multiplie þat Articul by þe articul of þe composit, and worch as þou wos tauȝt in þe secunde rewle, of þe quych rewle þe verse begynnes þus.Article by Composite.¶ Articulumsi perRelicumvis multiplicare. þen multiply þe diget of þe composit by þat oþirarticul aftirþe doctrine of þe 3 rewle. take þerof gode hede, I pray þe as þus. Yf þou wel wete what is 24 tymes ten.An example.Multiplie ten by 20, þat wel be 2 hundryth. þen multiply þe diget of þe 10, þe quych is 1, by þe diget of þe composit, þe quych is 4, & þatleaf 164b.*wol be 4. þen reken euery vnite þat is in 4 for 10, & þat schal be 40. Cast 40 to 2 hundryth, & þat wol be 2 hundryth & 40. And so mych is 24 tymes ten.
How to work without Figures.
¶ Compositumnumerummixto si[c] multiplicabisVndecies tredecimsic estex hiis operandumIn reliquumprimum demumduc post in eundemVnumpost denum duc in tria deinde pervnumMultiplicesquedemum intraomnia multiplicataIn summa decies quam si fuerit tibidocesMultiplicandorumde normis sufficiunt hec.
¶ Compositumnumerummixto si[c] multiplicabis
Vndecies tredecimsic estex hiis operandum
In reliquumprimum demumduc post in eundem
Vnumpost denum duc in tria deinde pervnum
Multiplicesquedemum intraomnia multiplicata
In summa decies quam si fuerit tibidoces
Multiplicandorumde normis sufficiunt hec.
The sixth case of the craft:¶ Here he puttes þe 6 rewle, & þe last of allemultiplicacioɳ, þe quych is þis: yf þou wel multiplye acomposit by a-noþercomposit, þou schalt do þus.Composite by Composite.multiplie þat on composit, qwych þou welt of the twene, by þe articul of þe toþercomposit, as þou weretauȝt in þe 5 rewle, þen multiplie þat same composit, þe quych þou hast multiplied by þe oþerarticul, by þe digit of þe oþercomposit,Mental multiplication.asþou was tauȝt in þe 4 rewle.An exampleAs þus, yf þou wold wete what is 11 tymes 13, as þou was tauȝt in þe 5 rewle, & þat schal be an hundryth & ten, afterwarde multiply þat same composit þat þou hast multiplied, þe quych is a .11. And multiplye hit be þe digit of þe oþercomposit, þe quych is 3, for 3 is þe digit of 13, And þat wel be 30. þen take þe digit of þat composit, þe quych composit þou multiplied by þe digit of þat oþercomposit,leaf 165a.*þe quych is a 11.of the sixth case of the craft.¶ Also of the quych 11 on is þe digit. multiplie þat digitt by þe digett of þat othercomposit, þe quych diget is 3, as þou was tauȝt in þe first rewle inþe begynnyngeof þis craft. þe quych rewle begynnes“In digitumcures.” And of alleþe multiplicacioɳ of þe 2 digitt comys thre, for onys 3 is but 3. Now cast alleþese nounbers togedur, the quych is þis, a hundryth & ten & 30 & 3. And al þat wel be 143. Write 3 first in þe ryght side. And cast 10 to 30, þat wol be 40. set 40 next afturtowarde þe lyft side, And set aftur a hundryth as herean Ensampulle, 143.
(Cetera desunt.)
1.In MS, ‘awiy.’2.‘ben’ repeated in MS.3.In MS. ‘thausandes.’4.Perhaps “So.”5.‘hali’ marked for erasure in MS.6.‘moy’ in MS.7.‘Subtrahas aut addis a dextrisvel mediabis’ added on margin of MS.8.After ‘craft’ insert ‘the .4. what is þe profet of þis craft.’9.After ‘sythes’ insert ‘& þis wordes fyue sithe & sex sythes.’10.‘t’l’ marked for erasure before ‘tyl’ in MS.11.Here ‘of þe same rew’ is marked for erasure in MS.12.‘sed’ deleted in MS.13.6883 in MS.14.‘þen’ overwritten on ‘þat’ marked for erasure.15.‘Supra’ inserted in MS. in place of ‘cifra’ marked for erasure.16–16.Marked for erasure in MS.17.4 in MS.18.docet. decet MS.19.‘4 times 4’ in MS.
1.In MS, ‘awiy.’
2.‘ben’ repeated in MS.
3.In MS. ‘thausandes.’
4.Perhaps “So.”
5.‘hali’ marked for erasure in MS.
6.‘moy’ in MS.
7.‘Subtrahas aut addis a dextrisvel mediabis’ added on margin of MS.
8.After ‘craft’ insert ‘the .4. what is þe profet of þis craft.’
9.After ‘sythes’ insert ‘& þis wordes fyue sithe & sex sythes.’
10.‘t’l’ marked for erasure before ‘tyl’ in MS.
11.Here ‘of þe same rew’ is marked for erasure in MS.
12.‘sed’ deleted in MS.
13.6883 in MS.
14.‘þen’ overwritten on ‘þat’ marked for erasure.
15.‘Supra’ inserted in MS. in place of ‘cifra’ marked for erasure.
16–16.Marked for erasure in MS.
17.4 in MS.
18.docet. decet MS.
19.‘4 times 4’ in MS.