Chapter 3

¶ Si subtraccio sit benefacta probarevalebisQuas subtraxisti primas addendo figuras.How to prove a subtraction sum.¶ Herehe teches þe Craft how þou schalt know, whan þou hast subtrayd, wheþerþou hast wel ydo or no. And þe Craft is þis, ryght as þousubtraydþe neþerfigures fro þe hier figures, ryȝt so adde þe same neþerfigures to þe hier figures. And yf þou haue welly-wrotha-foreþou schalt haue þe hier nombre þe same þou haddestorþou be-gan to worch. as for þis I bade þou schulde kepe þe neþerfigures stylle.Here is an example.lo anleaf 145b.*Ensampulleof alleþe 4 cases togedre. worche welleþis case4000346820004664And yf þou worch wellewhan þou hast allesubtrayd þe þat hier nombrehere, þis schallebe þe nombre here foloyng whan þou hast subtrayd.3999880420004664Our author makes a slip here (3 for 1).And þou schalt know þus. adde þe neþerrowe of þe same nombre to þe hier rewe as þus, cast 4 to 4. þat wol be 8. do away þe 4 & write þere8. by þe first case of addicioɳ. þan cast 6 to 0 þat wol be 6. do away þe 0, & write þere 6. þan cast 6 to 8, þat wel be 14. do away 8 & write þerea figure of 4, þat is þe digit, and write a figure of 1. þat schall be-token ten. þat is þe articul vpon þe hed of 8 next after, þan reken þat 1. for 1. & cast it to 8. þat schal be 9. cast to þat 9 þe neþerfigure vnderþat þe quych is 4, & þat schallebe 13. do away þat 9 & sett þere 3, & sett a figure of 1. þat schall be 10 vpon þe next figurishede þequych is 9. by þe secunde case þat þou hadest in addicioɳ. þan cast 1 to 9. & þat wol be 10. do away þe 9. & þat 1. And write þerea cifer. and write þe articulleþat is 1. betokenynge10. vpon þe hede of þe next figuretoward þe lyft side, þe quychleaf 146a.*is 9, & so do forth tyl þou come to þe last 9.He works his proof through,take þe figureof þat 1. þe quych þou schalt fynde ouerþe hed of 9. & sett it ouerþe next figures hede þat schal be 3. ¶ Also do away þe 9. & set þerea cifer, & þen cast þat 1 þat stondes vpon þe hede of 3 to þe same 3, & þat schallemake 4, þen caste to þe ylke 4 the figurein þe neyþerrewe, þe quych is 2, and þat schallebe 6.and brings out a result.6000346820004664And þen schal þou haue an Ensampulleaȝeyɳ, loke & se, & but þou haue þis same þou hasemyse-wroȝt.The Craft of Duplation.Sequiturde duplacioneSi vis duplarenumerum, sic incipe primoScribe figurarumseriemquamcunquevelistu.Four things must be known in Duplation.¶ This is the Chaptureofduplacioɳ, in þe quych craft þou most haue & know 4 thinges. ¶ Þe first þat þou most know is what is duplacioɳ. þe secunde is how mony rewes of figures þou most haue to þis craft. ¶ þe thryde is how many cases may6happe in þis craft. ¶ þe fourte is what is þe profet of þe craft.Here they are.¶ As for þe first. duplacioɳ is a doublyngeof a nombre. ¶ As for þe secunde þou mostleaf 146b.*haue on nombre or on rewe of figures, the quych called numerusduplandus. As for þe thrid þou most know þat 3 diuerse cases may hap in þis craft. As for þe fourte. qwat is þe profet of þis craft, & þat is to know whata-risyȝtof a nombre I-doublyde.Mind where you begin.¶ fforþer-more, þou most know & take gode hede in quych side þou schallebe-gyn in þis craft, or ellis þou maystspylalleþilaberþere aboute. certeyn þou schalt begyɳ in the lyft side in þis Craft. thenke wel ouerþis verse. ¶7A leua dupla, diuide, multiplica.7Thesentensof þes verses afore, as þou may see if þou take hede.Remember your rules.As þe text of þis verse, þat is to say, ¶ Si vis duplare. þis is þe sentence. ¶ If þouweldouble a nombre þus þou most be-gynɳ. Write a rewe of figures of what nombreþou welt. versus.Postea procedas primamduplando figuramInde quod excrescit scribas vbi iusserit ordoIuxta precepta tibi que danturin addicione.How to work a sum.¶ Herehe telles how þou schalt worch in þis Craft. he says, fyrst, whan þou hast writen þe nombre þou schalt be-gyn at þe firstfigurein the lyft side, & doubulleþat figure, & þe nombre þat comes þere-of þou schalt write as þou diddyst in addicioɳ, as ¶ I schal telle þe in þe case. versus.The Cases of the Craft of Duplation.leaf 147a.* ¶ Nam si sit digitus in primo limite scribas.If the answer is a digit,¶ Hereis þe first case of þis craft, þe quych is þis. yf of duplacioɳ of a figurearise a digit. what schal þou do. þus þou schal do.write it in the place of the top figure.do away þe figure þat was doublede, & sett þereþe diget þat comes of þe duplacioɳ, as þus. 23. double 2, & þat wel be 4. do away þe figureof 2 & sett þerea figureof 4, & soworchforth tilleþou come to þe ende. versus.¶ Articulussi sit, in primo limite cifram,¶ Articulumvero reliquis inscribe figuris;¶ Vel perse scribas, si nulla figura sequatur.If it is an article,¶ Here is þe secunde case, þe quych is þis yf þerecome an articulleof þe duplacioɳ of a figure þou schalt do ryȝt as þou diddyst in addicioɳ, þat is toweteþat þou schalt do away þe figureþat is doublet &put a cipher in the place, and ‘carry’ the tens.sett þerea cifer, & write þe articulleouerþe next figurishede, yf þerebe any after-warde toward þe lyft side as þus. 25. begyn at the lyft side, and doubulle2. þat wel be 4. do away þat 2 & sett þere 4. þan doubul 5. þat wel be 10. do away 5, & sett þerea 0, & sett 1 vpon þe next figurishede þe quych is 4. & þen draw downe 1 to 4 & þat wollebe 5, & þen do away þat 4 & þat 1, & sett þere5. for þat 1 schal be rekened in þe drawyngetogedrefor 1. wenleaf 147b.*þou hast ydon þou schalt haue þis nombre 50.If there is no figure to ‘carry’ them to, write them down.yf þerecome no figureafterþe figure þat is addit, of þe quych addicioɳ comes an articulle, þou schalt do away þe figureþat is dowblet & sett þerea 0. & write þe articul next by in þe same rewe toward þe lyft syde as þus, 523. double 5 þat woll be ten. do away þe figure5 & set þerea cifer, & sett þe articul next afterin þe same rewe toward þe lyft side, & þou schalt haue þis nombre 1023. þen go forth & double þe oþernombers þe quych islyȝty-nowȝtto do. versus.¶ Compositus si sit, in limite scribe sequenteArticulum, primo digitum; quiasic iubet ordo:Et sic de reliquis faciens, si sint tibi plures.If it is a Composite,¶ Herehe puttesþe Thryd case, þe quych is þis, yf of duplacioɳ of a figure come a Composit. þou schalt do away þe figure þat is doublet & set þerea digit of þe Composit,write down the digit, and ‘carry’ the tens.& sett þe articulleouerþe next figures hede, & afterdraw hym downe withþe figureouerwhos hede he stondes, & make þere-of an nombre as þou hast doneafore, & yf þerecome no figure afterþat digit þat þou hasty-write, þanset þe articullenext afterhym in þe same rewe as þus, 67: double 6 þat wel be 12, do away 6 & write þereþe digitleaf 148a.*of 12, þe quych is 2,Here is an example.and set þe articullenext aftertoward þe lyft side in þe same rewe, for þerecomes no figureafter. þan dowble þat oþerfigure, þe quych is 7, þat wel be 14. the quych is a Composit. þen do away 7 þat þou doublet & sett þe þe diget of hym, the quych is 4, sett þe articulleouerþe next figureshed, þe quych is 2, & þen draw to hym þat on, & make on nombre þe quych schallebe 3. And þen yf þou haue wel y-do þou schallehaue þis nombre of þe duplacioɳ, 134. versus.¶ Si super extremamnota sit monademdat eidemQuod tibicontingat si primo dimidiabis.How to double the mark for one-half.¶ Herehe says, yf ouerþe fyrst figure in þe ryȝt side be such a merke as is heremade,w, þou schallefyrst doubulleþe figure, the quych stondes vnderþat merke, & þen þou schalt doubul þat merke þe quych stondesforhaluendelon. for toohaluedelsmakes on, & so þat wol be on. cast þat on to þat duplacioɳ of þe figureouerwhos hed stode þat merke, & write it in þe same place þereþat þe figureþe quych was doublet stode, as þus 23w. double 3, þat wol be 6; doubul þat halue on, & þat wol be on. cast on to 6, þat wel be 7. do away 6 & þat 1, & sett þere7. þan hase þou do. as for þat figure, þan goleaf 148b.*to þe oþerfigure& worch forth.This can only stand over the first figure.& þou schall neuerhaue such a merk but ouerþe hed of þe furst figure in þe ryght side. And ȝet it schal not happe but yf it were y-halued a-fore, þus þou schalt vnderstonde þe verse. ¶ Si superextremam&c. Et nota, talis figurawsignificans medietatem, unitatisveniat,i.e.contingat uel fiat superextremam,i.e.superprimamfiguramin extremo sic versus dextram ars dat:i.e.reddit monadem.i.e.vnitatemeidem.i.e.eidem note & declinaturhec monos, dis, di, dem, &c. ¶ Quodergototumhocdabis monademnotecontinget.i.e.eveniet tibi si dimidiasti,i.e.accipisti uel subtulisti medietatem alicuius unius, in cuius principio sint figura numerumdenotans imparemprimoi.e.principiis.The Craft of Mediation.¶ Sequiturde mediacione.Incipe sic, si vis aliquemnumerummediare:Scribe figurarumseriem solam, velut ante.The four things to be known in mediation:¶ In þis Chapter istaȝtþe Craft ofmediaciouɳ, in þe quych craft þou most know 4 thynges. ffurst what is mediacioɳ. the secunde how mony rewes of figuresþou most haue in þe wyrchyngeof þis craft. þe thryde how mony diuerse cases may happ in þis craft.8the first¶ As for þe furst, þou schalt vndurstonde þat mediacioɳ is atakyng out of halfe a nomber out of ahollenomber,leaf 149a.*as yf þouthe second;wolde take 3 out of 6. ¶ As for þe secunde, þou schalt know þat þou most haue onerewe of figures, & no moo, as þouhaystin þethe third;craft of duplacioɳ. ¶ As for the thryd, þou most vnderstonde þatthe fourth.5 cases may happe in þis craft. ¶ As for þe fourte, þou schalleknow þat the profet of þis craft is when þou hast take away þe haluendel of a nombreto telle qwat þereschalleleue. ¶ Incipe sic, &c. The sentence of þis verse is þis. yf þou woldmedye, þat is to say, take halfe out of þe holle, or halfe out of halfe, þou most begynne þus.Begin thus.Write onerewe of figuresof what nombre þou wolte, as þou dyddyst be-forein þe Craft of duplacioɳ. versus.¶ Postea procedas medians, si prima figuraSi par aut impar videas.¶ Herehe says, when þou hast write a rewe of figures, þou schaltSee if the number is even or odd.take hede wheþerþe first figurebe eueɳ or odde in nombre, & vnderstonde þat he spekes of þe first figure in þe ryȝt side. And inthe ryght side þou schallebegynne in þis Craft.¶ Quia si fuerit par,Dimidiabiseam, scribens quicquid remanebit:If it is even, halve it, and write the answer in its place.¶ Hereis the first case of þis craft, þe quych is þis, yf þe first figurebe euen. þou schal take away fro þe figureeuen halfe, & do away þat figure and set þereþat leues ouer, as þus, 4. takeleaf 149b.*halfe out of 4, & þan þereleues 2. do away 4 & sett þere2. þis is lyght y-nowȝt. versus.The Mediation of an Odd Number.¶ Impar si fuerit vnumdemas mediareQuod nonpresumas, sedquod superest mediabisInde supertractumfac demptumquod notat vnum.If it is odd, halve the even number less than it.Hereis þe secunde case of þis craft, the quych is þis. yf þe first figurebetokenea nombre þat is odde, the quych odde schal not bemediete, þen þou schalt medye þat nombre þat leues, when the odde of þe same nombreis take away, & write þat þat leues as þou diddest in þe first case of þis craft. Whaɳ þouhaystwrite þat. for þat þat leues,Then write the sign for one-half over it.write such a merke as is herewvpon his hede, þe quych merke schal betokeɳ halfe of þe odde þat was take away.Here is an example.lo an Ensampull. 245. the first figurehereis betokenyngeodde nombre, þe quych is 5, for 5 is odde; þere-foredo away þat þat is odde, þe quych is 1. þen leues 4. þen medye 4 & þen leues 2. do away 4. & sette þere2, & make such a merkewupon his hede, þat is to say ouerhis hede of 2 as þus. 242.wAnd þen worch forth in þe oþerfigures tyll þou come to þe ende. by þe furst case as þou schaltvnderstonde þatPut the mark only over the first figure.þou schaltleaf 150a.*neuermake such a merk but ouerþe first figure hed in þe riȝt side. Wheþerþe other figures þat comyɳ afterhym be eueɳ or odde. versus.The Cases of the Craft of Mediation.¶ Si monos, dele; sit tibicifra post notasupra.If the first figure is one put a cipher.¶ Here is þe thryde case, þe quych yf the first figurebe a figureof 1. þou schalt do away þat 1 & set þerea cifer, & a merke ouerþe cifer as þus, 241. do away 1, & sett þerea cifer witha merke ouerhis hede, & þen hast þou ydo for þat 0. as þus 0wþen worch forth in þe oþer figurys till þou come to þe ende, for it is lyght as dyche water. vndeversus.¶ Postea procedas hac condicione secunda:Imparsi fuerit hinc vnumdeme priori,Inscribens quinque, nam denos significabitMonos predictam.What to do if any other figure is odd.¶ Herehe puttesþe fourte case, þe quych is þis. yf it happeɳ the secunde figurebetoken odde nombre, þou schal do away on of þat odde nombre, þe quych is significatiue by þat figure 1. þe quych 1 schall be rekende for 10. Whan þou hast take away þat 1 out of þe nombre þat is signifiede by þat figure, þou schalt medie þat þat leues ouer, & do away þat figureþat is medied, & sette in hisstydehalfe of þat nombre.Write a figure of five over the next lower number’s head.¶ Whan þou hase so done, þou schalt writeleaf 150b.*a figure of 5 ouerþe next figureshede by-foretoward þe ryȝt side, for þat 1, þe quych made odd nombre, schall stonde for ten, & 5 is halfe of 10; so þou most write 5 for his haluendelle.Example.lo an Ensampulle, 4678. begyɳ in þe ryȝt side as þou most nedes. medie 8. þen þou schalt leue 4. do away þat 8 & sette þere4. þen out of 7. take away 1. þe quych makes odde, & sett 5. vpon þe next figureshede aforetoward þe ryȝt side, þe quych is now 4. but aforeit was 8. for þat 1 schal be rekenet for 10, of þe quych 10, 5 is halfe, as þou knowest wel. Whan þou hast þus ydo, medye þat þe quych leues afterþe takyingeaway of þat þat is odde, þe quych leuyngeschallebe 3;54634.do away 6 & sette þere 3, & þou schalt haue such a nombre aftergo forth to þe next figure, & medy þat, & worch forth, for it is lyȝtynovȝtto þecertayɳ.¶ Si vero secunda dat vnum.Illa deleta, scribaturcifra; priori¶ Tradendo quinque pro denario mediato;Nec cifra scribatur, nisi deinde figura sequatur:Postea procedas reliquas mediando figurasVt supradocui, si sint tibi mille figure.¶ Herehe puttesþe 5 case, þe quych isleaf 151a.*þis:If the second figure is one, put a cipher, and write five over the next figure.yf þe secunde figurebe of 1, as þis is here 12, þou schalt do away þat 1 & sett þerea cifer. & sett 5 ouerþe next figure hede aforetoward þe riȝt side, as þou diddyst afore; & þat 5 schal behaldelof þat 1, þe quych 1 is rekent for 10. lo an Ensampulle, 214. medye 4. þat schallebe 2. do away 4 & sett þere2. þengo forth to þe next figure. þe quych is bot 1. do away þat 1. & sett þerea cifer. & set 5 vpon þe figureshed afore, þe quych is nowe 2, & þen þou schalt haue þis nombre5202,þen worch forth to þenexfigure. And also it is nomaysteryyf þerecome no figureafter þat on is medyet, þou schalt write no 0. ne nowȝt ellis, but set 5 ouerþe next figure aforetoward þe ryȝt, as þus 14.How to halve fourteen.medie 4 then leues 2, do away 4 & sett þere2. þen medie 1. þe quich is rekende for ten, þe haluendel þere-of wel be 5. sett þat 5 vpon þe hede of þat figure, þe quych is now 2,52,& do away þat 1, & þou schalt haue þis nombre yf þou worch wel, vndeversus.How to prove the Mediation.¶ Si mediacio sit benefacta probarevalebis¶ Duplando numerumquemprimo dimediastiHow to prove your mediation.¶ Herehe telles þe how þou schalt know wheþerþou hase wel ydo or no. doubulleaf 151b.*þe nombre þe quych þou hase mediet, and yf þou haue wel y-medyt after þe dupleacioɳ, þou schalt haue þe same nombre þat þou haddyst in þe tabulleor þou began to medye, as þus.First example.¶ The furst ensampullewas þis. 4. þe quychI-medietwaslaft2, þe whych 2 was write in þe place þat 4 was write afore. Now doubulleþat 2, & þou schal haue 4, as þou hadyst afore.The second.þe secunde Ensampullewas þis, 245. When þou haddyst mediet alleþis nombre, yf þou haue wel ydo þou schalt haue of þat mediacioɳ þis nombre, 122w. Now doubulleþis nombre, & begyn in þe lyft side; doubulle1, þat schal be 2. do away þat 1 & sett þere2. þen doubulleþat oþer2 & sett þere4, þen doubulleþat oþer2, & þat wel be 4. þendoubul þat merke þat stondes for halue on. & þat schallebe 1. Cast þat on to 4, & it schallebe 5. do away þat 2 & þat merke, & sette þere5, & þen þou schal haue þis nombre 245. & þis wos þe same nombur þat þou haddyst or þou began to medye, as þou mayst se yf þou take hede.The third example.The nombre þe quych þou haddist for an Ensampul in þe 3 case of mediacioɳ to be mediet was þis 241. whan þou haddist medied alleþis nombur trulyleaf 152a.*by euery figure, þou schall haue be þat mediacioɳ þis nombur 120w. Now dowbul þis nombur, & begyn in þe lyft side, as I tolde þe in þe Craft of duplacioɳ. þus doubulleþe figure of 1, þat wel be 2. doaway þat 1 & sett þere2, þen doubul þe next figureafore, the quych is 2, & þat wel be 4; do away 2 & set þere4. þen doubul þe cifer, & þat wel be noȝt, for a 0 is noȝt. And twyes noȝt is but noȝt. þereforedoubul the merke aboue þe cifers hede, þe quych betokenes þe haluendel of 1, & þat schal be 1. do away þe cifer & þe merke, & sett þere1, & þen þou schalt haue þis nombur 241. And þis same nombur þou haddyst afore or þou began to medy,& yfþou take gode hede.The fourth example.¶ The next ensampul þat had in þe 4 case of mediacioɳ was þis 4678. Whan þou hast trulyymeditalleþis nombur fro þe begynnyngeto þe endynge, þou schalt haue of þe mediacioɳ þis nombur52334.Now doubul this nombur & begyn in þe lyft side, & doubulle2 þat schal be 4. do away 2 and sette þere 4; þen doubule3, þat wol be 6; do away 3 & sett þere6, þen doubul þat oþer3, & þat wel be 6; do away 3 & set þereleaf 152b.*6, þen doubul þe 4, þat welle be 8; þen doubul 5. þe quych stondes ouerþe hed of 4, & þat wol be 10; cast 10 to 8, & þat schal be 18; do away 4 & þat 5, & sett þere8, & sett that 1, þe quych is an articul of þe Composit þe quych is 18, ouerþe next figureshed toward þe lyft side, þe quych is 6. drav þat 1 to 6, þe quych 1 in þe dravyng schal be rekente bot for 1, & þat 1 & þat 6 togedur wel be 7. do away þat 6 & þat 1. the quych stondes ouerhis hede, & sett ther 7, & þen þou schalt haue þis nombur 4678. And þis same nombur þou hadyst or þou began to medye, as þou mayst see in þe secunde Ensampul þat þou had in þe 4 case of mediacioɳ, þat was þis:The fifth example.when þou had mediet truly allethe nombur, a principio usque ad finem. þou schalt haue of þat mediacioɳ þis nombur5102.Now doubul 1. þat wel be 2. do away 1 & sett þere2. þen doubul 0. þat will be noȝt. þereforetake þe 5, þe quych stondes ouerþe next figureshed, & doubul it, & þat wol be 10. do away þe 0 þat stondes betwene þe two figuris, & sette þerein his stid 1, for þat 1 now schal stonde in þe secunde place, wherehe schal betoken 10; þen doubul 2, þat wol be 4. do away 2 & sett þere 4. &leaf 153a.*þou schal haueþusnombur 214. þis is þe same numbur þat þou hadyst or þou began to medye, as þou may see. And so do euermore, yf þou wil knowe wheþerþou hase wel ymedyt or no. ¶. doubulleþe numbur þat comes afterþe mediaciouɳ, & þou schal haue þe same nombur þat þou hadyst or þou began to medye, yf þou haue welle ydo. or els doute þe noȝt, but yf þou haue þe same, þou hase faylide in þiCraft.The Craft of Multiplication.Sequitur de multiplicatione.To write down a Multiplication Sum.Si tu pernumerumnumerumvis multiplicareScribe duas quascunquevelis series numerorumOrdo serveturvt vltima multiplicandiPonatursuperanteriorem multiplicantisA leua relique sint scripte multiplicantes.Four things to be known of Multiplication:¶ Herebe-gynnes þe Chaptre of multiplicatioɳ, in þe quych þou most know 4 thynges. ¶ Ffirst, qwat is multiplicacioɳ. The secunde, how mony cases may hap in multiplicacioɳ. The thryde, how mony rewes of figuresþeremost be. ¶ The 4. what is þe profet of þis craft.the first:¶ As for þe first, þou schal vnderstonde þat multiplicacioɳ is a bryngyngeto-gederof 2 thyngesin on nombur, þe quych on nomburcontynesso mony tymes on, howeleaf 153b.*mony tymes þereben vnytees in þe nowmbreof þat 2, as twyes 4 is 8. now hereben þe 2 nombers, of þe quych too nowmbres on is betokened be an aduerbe, þe quych is þe worde twyes, & þis worde thryes, & þis worde fouresythes,9& so furth of such other lyke wordes. ¶ And tweyn nombres schal be tokenyde be a nowne, as þis worde foureshowys þes tweyɳ nombresy-brothin-to on hole nombur, þat is 8, for twyes 4 is 8, as þou wost wel. ¶ And þes nombre8 conteynes as oft tymes 4 as þereben vnites in þat other nombre, þe quych is 2, for in 2 ben 2 vnites, & so oft tymes 4 ben in 8, as þou wottys wel.the second:¶ ffor þe secunde, þou most know þat þou most haue too rewes of figures.the third:¶ As for þe thryde, þou most know þat 8 manerof diuerse case may happe in þis craft.the fourth.The profet of þis Craft is to telle when a nombreis multiplyed be a noþer, qwat commys þereof. ¶ fforthermore, as to þe sentence of oureverse, yf þou wel multiply a nombur be a-noþernombur, þou schalt writeleaf 154a.*a rewe of figures of what nomburs so euerþou welt,The multiplicand.& þat schal be called Numerusmultiplicandus, Anglice, þe nomburthe quych to be multiplied. þen þou schalt write a-nother rewe of figures, by þe quych þou schalt multiplie the nombre þat is to be multiplied, of þe quych nomburþe furst figure schal be write vnderþe last figureof þe nombur, þe quych is to be multiplied.How to set down the sum.And so write forthe toward þe lyft side, as hereyou may se,673241234And þis onenomburschallebe called numerus multiplicans. Anglice, þe nomburmultipliynge, for he schallemultiply þe hyer nounbur, as þus onetyme 6. And so forth, as I schal telle the afterwarde. And þou schal begyn in þe lyft side.Two sorts of Multiplication: mentally,¶ ffor-þere-more þou schalt vndurstonde þat þereis two manurs of multiplicacioɳ; one ys of þe wyrchyngeof þe boke only in þe mynde of a mon. fyrst heteches of þe fyrst manerof duplacioɳ, þe quych is be wyrchyngeof tabuls.and on paper.Afterwarde he wol teche on þe secunde maner. vndeversus.To multiply one Digit by another.In digitumcures digitumsi ducere maiorleaf 154b.* Perquantumdistat a denis respice debes¶ Namquesuo decuplo totiens delereminoremSitquetibi numerus veniens exinde patebit.How to multiply two digits.¶ Herehe teches a rewle, how þou schalt fynde þe nounbre þat comes by þe multiplicacioɳ of a digit be anoþer. loke how mony [vny]tes ben. bytwene þe moredigit and 10. And reken ten for on vnite.Subtract the greater from ten;And so oft do away þe lasse nounbre out of his ownedecuple, þat is to say, fro þat nounbre þat is ten tymes so mychisþe nounbreþat comes of þe multiplicacioɳ. As yf þou wol multiply 2 be 4. loke how mony vnitees ben by-twene þe quych is þe morenounbre, & be-twene ten. Certen þerewel be vj vnitees by-twene 4 & ten. yf þou reken þerewithþe ten þe vnite, as þou may se.take the less so many times from ten times itself.so mony tymes take 2. out of his decuple, þe quych is 20. for 20 is þe decuple of 2, 10 is þe decuple of 1, 30 is þe decuple of 3, 40 is þe decuple of 4, And þe oþerdigetes til þou come to ten; & whan þouExample.hast y-take so mony tymes 2 out of twenty, þe quych is sex tymes, þou schal leue 8 as þou wost wel, for 6 times 2 is twelue. take [1]2 out of twenty, & þereschal leue 8. bot yf bothe þe digettesleaf 155a.*beny-lyechmych as here. 222 or too tymes twenty, þen it isno forsquych of hem tweyn þou take out of here decuple.alsmonyBetter use this table, though.tymes as þat is fro 10. but neuer-þe-lesse, yf þou hauehastto worch, þou schalt haue herea tabul of figures, where-by þou schalt se a-nonɳ ryght what is þe nounbre þat comes of þe multiplicacioɳ of 2 digittes. þus þou schalt worch in þis figure.12436948121651015202561218243036714212835424981624324048566491827364554637281123456789How to use it.yf þe figure, þe quych schallebe multiplied, be eueneas mych as þe diget be, þe quych þat oþerfigureschal be multiplied, as two tymes twayɳ, or thre tymes 3. or sych other.The way to use the Multiplication table.loke qwereþat figure sittes inþe lyft side of þe triangle, & loke qwereþe diget sittes in þe neþermost rewe of þe triangle. & go fro hym vpwarde in þe same rewe, þe quych rewe gose vpwarde til þou come agaynes þe oþerdigette þat sittes in þe lyft side of þe triangle. And þat nounbre, þe quych þouleaf 155b.fyn*des þereis þe nounbre þat comes of the multiplicacioɳ of þe 2 digittes, as yf þou wold wete qwat is 2 tymes 2. loke queresittes 2 in þe lyft side inþe first rewe, he sittes next 1 in þe lyft side al on hye, as þou may se; þe[n] loke qweresittes 2 in þe lowyst rewe of þe triangle, & go fro hym vpwarde in þe same rewe tylleþou comea-ȝenenes2 in þe hyer place, & þer þou schalt fynd ywrite 4, & þat is þe nounbre þat comes of þe multiplicacioɳ of two tymes tweyn is 4, as þow wotest welle. yf þe diget. the quych is multiplied, be moreþan þe oþer, þou schalt loke qwereþe morediget sittes in þe lowest rewe of þe triangle, & go vpwarde in þe same rewe tyl10þou comea-nendesþe lasse diget in the lyft side. And þereþou schalt fynde þe nombre þat comes of þe multiplicacioɳ; but þou schalt vnderstonde þat þis rewle, þe quych is in þis verse. ¶ In digitumcures, &c., noþerþis triangle schallenot serue, bot to fynde þe nounbres þat comes of the multiplicacioɳ þat comes of 2 articuls orcomposites, þe nedes no craft but yf þou wolt multiply in þi mynde. Andleaf 156a.*þere-to þou schalt haue a craft afterwarde, for þou schall wyrch withdigettes in þe tables, as þou schalt know afterwarde. versus.To multiply one Composite by another.¶ Postea procedas postremammultiplicando[Recte multiplicans per cunctas inferiores]Condicionem tamen tali quodmultiplicantesScribas in capite quicquid processerit indeSed postquamfuit hec multiplicate figureAnteriorenturserei multiplicantisEt sic multiplica velut isti multiplicastiQui sequiturnumerumscriptumquiscunquefiguris.How to multiply one number by another.¶ Herehe teches how þou schalt wyrch in þis craft. þou schalt multiplye þe last figureof þe nombre, and quen þou hast so ydo þou schalt draw alleþe figures of þe neþernounbre moretawardþe ryȝt side, so qwenþou hast multiplyed þe last figureof þe heyer nounbre by alleþe neþerfigures.Multiply the ‘last’ figure of the higher by the ‘first’ of the lower number.And sette þe nounbir þat comes þer-of ouerþe last figureof þe neþernounbre, & þen þou schalt sette al þe oþerfigures of þe neþernounbremorenereto þe ryȝt side. ¶ And whan þou hast multiplied þat figureþat schal be multiplied þe next afterhym by al þe neþerfigures. And worch as þou dyddyst aforetilleaf 156b.*þou come to þe ende. And þou schalt vnderstonde þat euery figureof þe hier nounbreschal be multiplied be alleþe figures of the neþernounbre, yf þe hier nounbrebe any figureþen one.Set the answer over the first of the lower:lo an Ensampul herefolowynge.2465.232þou schalt begyne to multiplye in þe lyft side. Multiply 2 be 2, and twyes 2 is 4. set 4then multiply the second of the lower, and so on.ouerþe hed of þat 2, þen multiplie þe same hier 2 by 3 of þe nether nounbre, as thryes 2 þat schal be 6. set 6 ouerþe hed of 3, þan multiplie þe same hier 2 by þat 2 þe quych stondes vnderhym, þat wol be 4; do away þe hier 2 & sette þere4.Then antery the lower number:¶ Now þou mostanteryþe nether nounbre, þat is to say, þou most sett þe neþernounbre more towarde þe ryȝt side, as þus. Take þe neþer2 toward þe ryȝt side, & sette it eueɳ vnderþe 4 of þe hyer nounbre, & antery alleþe figures þat comes afterþat 2, as þus; sette 2 vnderþe 4. þen sett þe figureof 3 þereþat þe figure of 2 stode, þe quych is now vndur þat 4 in þe hier nounbre; þen sett þe oþer figureof 2, þe quych is þe last figure toward þe lyft side of þe neþernomberþereþe figureof 3 stode.as thus.þen þou schalt haue such a nombre.464465232leaf 157a.* ¶ Now multiply 4, þe quych comes next after6, by þe last 2 of þe neþernounbur toward þe lyft side. as 2 tymes 4, þat wel be 8. sette þat 8 ouerþe figure the quych stondes ouerþe hede of þat 2, þe quych is þe last figureof þe neþernounbre; þan multiplie þat same 4 by 3, þat comes in þe neþerrewe, þat wol be 12. sette þe digit of þe composyt ouerþe figure þe quych stondes ouerþe hed of þat 3, & sette þe articule of þis composit oueral þe figures þat stondes ouerþe neþer2 hede.Now multiply by the last but one of the higher:þen multiplie þe same 4 by þe 2 in þe ryȝt side in þe neþernounbur, þat wol be 8. do away 4. & sette þere8. Euermoreqwen þou multiplies þe hier figureby þat figureþe quych stondes vnderhym, þou schalt do away þat hier figure, & sett þer þat nounbre þe quych comes of multiplicacioɳ of ylke digittes.as thus.Whan þou hast done as I haue byde þe, þou schalt haue suych an orderof figureas is here,1824648[65]232.þen take and antery þi neþerfigures. And sett þe fyrst figure of þe neþerfigures11vndre be figureof 6. ¶ And draw al þe oþerfigures of þe same rewe to hym-warde,leaf 157b.*as þou diddyst afore. þen multiplye 6 be 2, & sett þat þe quych comes ouerþere-of oueral þe oþerfigures hedes þat stondes ouerþat 2. þen multiply 6 be 3, & sett alleþat comes þere-of vpon alleþe figures hedes þat standes ouerþat 3; þanmultiplye 6 be 2, þe quychstondes vnderþat 6, þen do away 6 & write þereþe digitt of þe composit þat schal come þereof, & sette þe articull oueralleþe figures þat stondes ouerþe hede of þat 3 as here,11121828464825232Antery the figures again, and multiply by five:þen antery þi figures as þou diddyst afore, and multipli 5 be 2, þat wol be 10; sett þe 0 ouerall þe figures þat stonden ouerþat 2, & sett þat 1. ouerthe next figures hedes, alleon hye towarde þe lyft side. þen multiplye 5 be 3. þat wol be 15, write 5 ouerþe figures hedes þat stonden ouerþat 3, & sett þat 1 ouerþe next figures hedes toward þe lyft side. þen multiplye 5 be 2, þat wol be 10. do away þat 5 & sett þerea 0, & sett þat 1 ouerþe figures hedes þat stonden ouer3. And þenleaf 158a.þou schalt haue such a nounbre as here stondes aftur.*1111011215828204648232¶ Now draw alleþese figures downe togederas þus, 6.8.1. & 1 draw to-gedur; þat wolle be 16, do away alleþese figures saue 6. lat hym stonde, forþowþou take hym away þou most write þer þe same aȝene. þereforelate hym stonde, & sett 1 ouerþe figurehede of 4 toward þe lyft side; þen draw on to 4, þat wollebe 5.Then add all the figures above the line:do away þat 4 & þat 1, & sette þere5. þen draw 4221 & 1 togedur, þat wol be 10. do away alleþat, & write þere þat 4 & þat 0, & sett þat 1 ouerþe next figureshede toward þe lyft side, þe quych is 6. þen draw þat 6 & þat 1 togedur, & þat wolle be 7; do away 6 & sett þere7, þen draw 8810 & 1, & þat wel be 18; do away alleþe figures þat stondes ouerþe hede of þat 8, & lette 8 stonde stil, & write þat 1 ouerþe next figurishede, þe quych is a 0. þen do away þat 0, & sett þere1, þe quych stondes ouerþe 0. hede. þen draw 2, 5, & 1 togedur, þat wollebe 8. þen do away alleþat, & write þere8.and you will have the answer.¶ And þen þou schalt haue þis nounbre, 571880.The Cases of this Craft.leaf 158b.

¶ Si subtraccio sit benefacta probarevalebisQuas subtraxisti primas addendo figuras.

¶ Si subtraccio sit benefacta probarevalebis

Quas subtraxisti primas addendo figuras.

How to prove a subtraction sum.¶ Herehe teches þe Craft how þou schalt know, whan þou hast subtrayd, wheþerþou hast wel ydo or no. And þe Craft is þis, ryght as þousubtraydþe neþerfigures fro þe hier figures, ryȝt so adde þe same neþerfigures to þe hier figures. And yf þou haue welly-wrotha-foreþou schalt haue þe hier nombre þe same þou haddestorþou be-gan to worch. as for þis I bade þou schulde kepe þe neþerfigures stylle.Here is an example.lo anleaf 145b.*Ensampulleof alleþe 4 cases togedre. worche welleþis case4000346820004664And yf þou worch wellewhan þou hast allesubtrayd þe þat hier nombrehere, þis schallebe þe nombre here foloyng whan þou hast subtrayd.3999880420004664Our author makes a slip here (3 for 1).And þou schalt know þus. adde þe neþerrowe of þe same nombre to þe hier rewe as þus, cast 4 to 4. þat wol be 8. do away þe 4 & write þere8. by þe first case of addicioɳ. þan cast 6 to 0 þat wol be 6. do away þe 0, & write þere 6. þan cast 6 to 8, þat wel be 14. do away 8 & write þerea figure of 4, þat is þe digit, and write a figure of 1. þat schall be-token ten. þat is þe articul vpon þe hed of 8 next after, þan reken þat 1. for 1. & cast it to 8. þat schal be 9. cast to þat 9 þe neþerfigure vnderþat þe quych is 4, & þat schallebe 13. do away þat 9 & sett þere 3, & sett a figure of 1. þat schall be 10 vpon þe next figurishede þequych is 9. by þe secunde case þat þou hadest in addicioɳ. þan cast 1 to 9. & þat wol be 10. do away þe 9. & þat 1. And write þerea cifer. and write þe articulleþat is 1. betokenynge10. vpon þe hede of þe next figuretoward þe lyft side, þe quychleaf 146a.*is 9, & so do forth tyl þou come to þe last 9.He works his proof through,take þe figureof þat 1. þe quych þou schalt fynde ouerþe hed of 9. & sett it ouerþe next figures hede þat schal be 3. ¶ Also do away þe 9. & set þerea cifer, & þen cast þat 1 þat stondes vpon þe hede of 3 to þe same 3, & þat schallemake 4, þen caste to þe ylke 4 the figurein þe neyþerrewe, þe quych is 2, and þat schallebe 6.and brings out a result.6000346820004664And þen schal þou haue an Ensampulleaȝeyɳ, loke & se, & but þou haue þis same þou hasemyse-wroȝt.

The Craft of Duplation.

Si vis duplarenumerum, sic incipe primoScribe figurarumseriemquamcunquevelistu.

Si vis duplarenumerum, sic incipe primo

Scribe figurarumseriemquamcunquevelistu.

Four things must be known in Duplation.¶ This is the Chaptureofduplacioɳ, in þe quych craft þou most haue & know 4 thinges. ¶ Þe first þat þou most know is what is duplacioɳ. þe secunde is how mony rewes of figures þou most haue to þis craft. ¶ þe thryde is how many cases may6happe in þis craft. ¶ þe fourte is what is þe profet of þe craft.Here they are.¶ As for þe first. duplacioɳ is a doublyngeof a nombre. ¶ As for þe secunde þou mostleaf 146b.*haue on nombre or on rewe of figures, the quych called numerusduplandus. As for þe thrid þou most know þat 3 diuerse cases may hap in þis craft. As for þe fourte. qwat is þe profet of þis craft, & þat is to know whata-risyȝtof a nombre I-doublyde.Mind where you begin.¶ fforþer-more, þou most know & take gode hede in quych side þou schallebe-gyn in þis craft, or ellis þou maystspylalleþilaberþere aboute. certeyn þou schalt begyɳ in the lyft side in þis Craft. thenke wel ouerþis verse. ¶7A leua dupla, diuide, multiplica.7

Thesentensof þes verses afore, as þou may see if þou take hede.Remember your rules.As þe text of þis verse, þat is to say, ¶ Si vis duplare. þis is þe sentence. ¶ If þouweldouble a nombre þus þou most be-gynɳ. Write a rewe of figures of what nombreþou welt. versus.

Postea procedas primamduplando figuramInde quod excrescit scribas vbi iusserit ordoIuxta precepta tibi que danturin addicione.

Postea procedas primamduplando figuram

Inde quod excrescit scribas vbi iusserit ordo

Iuxta precepta tibi que danturin addicione.

How to work a sum.¶ Herehe telles how þou schalt worch in þis Craft. he says, fyrst, whan þou hast writen þe nombre þou schalt be-gyn at þe firstfigurein the lyft side, & doubulleþat figure, & þe nombre þat comes þere-of þou schalt write as þou diddyst in addicioɳ, as ¶ I schal telle þe in þe case. versus.

The Cases of the Craft of Duplation.

* ¶ Nam si sit digitus in primo limite scribas.

If the answer is a digit,¶ Hereis þe first case of þis craft, þe quych is þis. yf of duplacioɳ of a figurearise a digit. what schal þou do. þus þou schal do.write it in the place of the top figure.do away þe figure þat was doublede, & sett þereþe diget þat comes of þe duplacioɳ, as þus. 23. double 2, & þat wel be 4. do away þe figureof 2 & sett þerea figureof 4, & soworchforth tilleþou come to þe ende. versus.

¶ Articulussi sit, in primo limite cifram,¶ Articulumvero reliquis inscribe figuris;¶ Vel perse scribas, si nulla figura sequatur.

¶ Articulussi sit, in primo limite cifram,

¶ Articulumvero reliquis inscribe figuris;

¶ Vel perse scribas, si nulla figura sequatur.

If it is an article,¶ Here is þe secunde case, þe quych is þis yf þerecome an articulleof þe duplacioɳ of a figure þou schalt do ryȝt as þou diddyst in addicioɳ, þat is toweteþat þou schalt do away þe figureþat is doublet &put a cipher in the place, and ‘carry’ the tens.sett þerea cifer, & write þe articulleouerþe next figurishede, yf þerebe any after-warde toward þe lyft side as þus. 25. begyn at the lyft side, and doubulle2. þat wel be 4. do away þat 2 & sett þere 4. þan doubul 5. þat wel be 10. do away 5, & sett þerea 0, & sett 1 vpon þe next figurishede þe quych is 4. & þen draw downe 1 to 4 & þat wollebe 5, & þen do away þat 4 & þat 1, & sett þere5. for þat 1 schal be rekened in þe drawyngetogedrefor 1. wenleaf 147b.*þou hast ydon þou schalt haue þis nombre 50.If there is no figure to ‘carry’ them to, write them down.yf þerecome no figureafterþe figure þat is addit, of þe quych addicioɳ comes an articulle, þou schalt do away þe figureþat is dowblet & sett þerea 0. & write þe articul next by in þe same rewe toward þe lyft syde as þus, 523. double 5 þat woll be ten. do away þe figure5 & set þerea cifer, & sett þe articul next afterin þe same rewe toward þe lyft side, & þou schalt haue þis nombre 1023. þen go forth & double þe oþernombers þe quych islyȝty-nowȝtto do. versus.

¶ Compositus si sit, in limite scribe sequenteArticulum, primo digitum; quiasic iubet ordo:Et sic de reliquis faciens, si sint tibi plures.

¶ Compositus si sit, in limite scribe sequente

Articulum, primo digitum; quiasic iubet ordo:

Et sic de reliquis faciens, si sint tibi plures.

If it is a Composite,¶ Herehe puttesþe Thryd case, þe quych is þis, yf of duplacioɳ of a figure come a Composit. þou schalt do away þe figure þat is doublet & set þerea digit of þe Composit,write down the digit, and ‘carry’ the tens.& sett þe articulleouerþe next figures hede, & afterdraw hym downe withþe figureouerwhos hede he stondes, & make þere-of an nombre as þou hast doneafore, & yf þerecome no figure afterþat digit þat þou hasty-write, þanset þe articullenext afterhym in þe same rewe as þus, 67: double 6 þat wel be 12, do away 6 & write þereþe digitleaf 148a.*of 12, þe quych is 2,Here is an example.and set þe articullenext aftertoward þe lyft side in þe same rewe, for þerecomes no figureafter. þan dowble þat oþerfigure, þe quych is 7, þat wel be 14. the quych is a Composit. þen do away 7 þat þou doublet & sett þe þe diget of hym, the quych is 4, sett þe articulleouerþe next figureshed, þe quych is 2, & þen draw to hym þat on, & make on nombre þe quych schallebe 3. And þen yf þou haue wel y-do þou schallehaue þis nombre of þe duplacioɳ, 134. versus.

¶ Si super extremamnota sit monademdat eidemQuod tibicontingat si primo dimidiabis.

¶ Si super extremamnota sit monademdat eidem

Quod tibicontingat si primo dimidiabis.

How to double the mark for one-half.¶ Herehe says, yf ouerþe fyrst figure in þe ryȝt side be such a merke as is heremade,w, þou schallefyrst doubulleþe figure, the quych stondes vnderþat merke, & þen þou schalt doubul þat merke þe quych stondesforhaluendelon. for toohaluedelsmakes on, & so þat wol be on. cast þat on to þat duplacioɳ of þe figureouerwhos hed stode þat merke, & write it in þe same place þereþat þe figureþe quych was doublet stode, as þus 23w. double 3, þat wol be 6; doubul þat halue on, & þat wol be on. cast on to 6, þat wel be 7. do away 6 & þat 1, & sett þere7. þan hase þou do. as for þat figure, þan goleaf 148b.*to þe oþerfigure& worch forth.This can only stand over the first figure.& þou schall neuerhaue such a merk but ouerþe hed of þe furst figure in þe ryght side. And ȝet it schal not happe but yf it were y-halued a-fore, þus þou schalt vnderstonde þe verse. ¶ Si superextremam&c. Et nota, talis figurawsignificans medietatem, unitatisveniat,i.e.contingat uel fiat superextremam,i.e.superprimamfiguramin extremo sic versus dextram ars dat:i.e.reddit monadem.i.e.vnitatemeidem.i.e.eidem note & declinaturhec monos, dis, di, dem, &c. ¶ Quodergototumhocdabis monademnotecontinget.i.e.eveniet tibi si dimidiasti,i.e.accipisti uel subtulisti medietatem alicuius unius, in cuius principio sint figura numerumdenotans imparemprimoi.e.principiis.

The Craft of Mediation.

Incipe sic, si vis aliquemnumerummediare:Scribe figurarumseriem solam, velut ante.

Incipe sic, si vis aliquemnumerummediare:

Scribe figurarumseriem solam, velut ante.

The four things to be known in mediation:¶ In þis Chapter istaȝtþe Craft ofmediaciouɳ, in þe quych craft þou most know 4 thynges. ffurst what is mediacioɳ. the secunde how mony rewes of figuresþou most haue in þe wyrchyngeof þis craft. þe thryde how mony diuerse cases may happ in þis craft.8the first¶ As for þe furst, þou schalt vndurstonde þat mediacioɳ is atakyng out of halfe a nomber out of ahollenomber,leaf 149a.*as yf þouthe second;wolde take 3 out of 6. ¶ As for þe secunde, þou schalt know þat þou most haue onerewe of figures, & no moo, as þouhaystin þethe third;craft of duplacioɳ. ¶ As for the thryd, þou most vnderstonde þatthe fourth.5 cases may happe in þis craft. ¶ As for þe fourte, þou schalleknow þat the profet of þis craft is when þou hast take away þe haluendel of a nombreto telle qwat þereschalleleue. ¶ Incipe sic, &c. The sentence of þis verse is þis. yf þou woldmedye, þat is to say, take halfe out of þe holle, or halfe out of halfe, þou most begynne þus.Begin thus.Write onerewe of figuresof what nombre þou wolte, as þou dyddyst be-forein þe Craft of duplacioɳ. versus.

¶ Postea procedas medians, si prima figuraSi par aut impar videas.

¶ Postea procedas medians, si prima figura

Si par aut impar videas.

¶ Herehe says, when þou hast write a rewe of figures, þou schaltSee if the number is even or odd.take hede wheþerþe first figurebe eueɳ or odde in nombre, & vnderstonde þat he spekes of þe first figure in þe ryȝt side. And inthe ryght side þou schallebegynne in þis Craft.

¶ Quia si fuerit par,Dimidiabiseam, scribens quicquid remanebit:

¶ Quia si fuerit par,

Dimidiabiseam, scribens quicquid remanebit:

If it is even, halve it, and write the answer in its place.¶ Hereis the first case of þis craft, þe quych is þis, yf þe first figurebe euen. þou schal take away fro þe figureeuen halfe, & do away þat figure and set þereþat leues ouer, as þus, 4. takeleaf 149b.*halfe out of 4, & þan þereleues 2. do away 4 & sett þere2. þis is lyght y-nowȝt. versus.

The Mediation of an Odd Number.

¶ Impar si fuerit vnumdemas mediareQuod nonpresumas, sedquod superest mediabisInde supertractumfac demptumquod notat vnum.

¶ Impar si fuerit vnumdemas mediare

Quod nonpresumas, sedquod superest mediabis

Inde supertractumfac demptumquod notat vnum.

If it is odd, halve the even number less than it.Hereis þe secunde case of þis craft, the quych is þis. yf þe first figurebetokenea nombre þat is odde, the quych odde schal not bemediete, þen þou schalt medye þat nombre þat leues, when the odde of þe same nombreis take away, & write þat þat leues as þou diddest in þe first case of þis craft. Whaɳ þouhaystwrite þat. for þat þat leues,Then write the sign for one-half over it.write such a merke as is herewvpon his hede, þe quych merke schal betokeɳ halfe of þe odde þat was take away.Here is an example.lo an Ensampull. 245. the first figurehereis betokenyngeodde nombre, þe quych is 5, for 5 is odde; þere-foredo away þat þat is odde, þe quych is 1. þen leues 4. þen medye 4 & þen leues 2. do away 4. & sette þere2, & make such a merkewupon his hede, þat is to say ouerhis hede of 2 as þus. 242.wAnd þen worch forth in þe oþerfigures tyll þou come to þe ende. by þe furst case as þou schaltvnderstonde þatPut the mark only over the first figure.þou schaltleaf 150a.*neuermake such a merk but ouerþe first figure hed in þe riȝt side. Wheþerþe other figures þat comyɳ afterhym be eueɳ or odde. versus.

The Cases of the Craft of Mediation.

¶ Si monos, dele; sit tibicifra post notasupra.

If the first figure is one put a cipher.¶ Here is þe thryde case, þe quych yf the first figurebe a figureof 1. þou schalt do away þat 1 & set þerea cifer, & a merke ouerþe cifer as þus, 241. do away 1, & sett þerea cifer witha merke ouerhis hede, & þen hast þou ydo for þat 0. as þus 0wþen worch forth in þe oþer figurys till þou come to þe ende, for it is lyght as dyche water. vndeversus.

¶ Postea procedas hac condicione secunda:Imparsi fuerit hinc vnumdeme priori,Inscribens quinque, nam denos significabitMonos predictam.

¶ Postea procedas hac condicione secunda:

Imparsi fuerit hinc vnumdeme priori,

Inscribens quinque, nam denos significabit

Monos predictam.

What to do if any other figure is odd.¶ Herehe puttesþe fourte case, þe quych is þis. yf it happeɳ the secunde figurebetoken odde nombre, þou schal do away on of þat odde nombre, þe quych is significatiue by þat figure 1. þe quych 1 schall be rekende for 10. Whan þou hast take away þat 1 out of þe nombre þat is signifiede by þat figure, þou schalt medie þat þat leues ouer, & do away þat figureþat is medied, & sette in hisstydehalfe of þat nombre.Write a figure of five over the next lower number’s head.¶ Whan þou hase so done, þou schalt writeleaf 150b.*a figure of 5 ouerþe next figureshede by-foretoward þe ryȝt side, for þat 1, þe quych made odd nombre, schall stonde for ten, & 5 is halfe of 10; so þou most write 5 for his haluendelle.Example.lo an Ensampulle, 4678. begyɳ in þe ryȝt side as þou most nedes. medie 8. þen þou schalt leue 4. do away þat 8 & sette þere4. þen out of 7. take away 1. þe quych makes odde, & sett 5. vpon þe next figureshede aforetoward þe ryȝt side, þe quych is now 4. but aforeit was 8. for þat 1 schal be rekenet for 10, of þe quych 10, 5 is halfe, as þou knowest wel. Whan þou hast þus ydo, medye þat þe quych leues afterþe takyingeaway of þat þat is odde, þe quych leuyngeschallebe 3;54634.do away 6 & sette þere 3, & þou schalt haue such a nombre aftergo forth to þe next figure, & medy þat, & worch forth, for it is lyȝtynovȝtto þecertayɳ.

¶ Si vero secunda dat vnum.Illa deleta, scribaturcifra; priori¶ Tradendo quinque pro denario mediato;Nec cifra scribatur, nisi deinde figura sequatur:Postea procedas reliquas mediando figurasVt supradocui, si sint tibi mille figure.

¶ Si vero secunda dat vnum.

Illa deleta, scribaturcifra; priori

¶ Tradendo quinque pro denario mediato;

Nec cifra scribatur, nisi deinde figura sequatur:

Postea procedas reliquas mediando figuras

Vt supradocui, si sint tibi mille figure.

¶ Herehe puttesþe 5 case, þe quych isleaf 151a.*þis:If the second figure is one, put a cipher, and write five over the next figure.yf þe secunde figurebe of 1, as þis is here 12, þou schalt do away þat 1 & sett þerea cifer. & sett 5 ouerþe next figure hede aforetoward þe riȝt side, as þou diddyst afore; & þat 5 schal behaldelof þat 1, þe quych 1 is rekent for 10. lo an Ensampulle, 214. medye 4. þat schallebe 2. do away 4 & sett þere2. þengo forth to þe next figure. þe quych is bot 1. do away þat 1. & sett þerea cifer. & set 5 vpon þe figureshed afore, þe quych is nowe 2, & þen þou schalt haue þis nombre5202,þen worch forth to þenexfigure. And also it is nomaysteryyf þerecome no figureafter þat on is medyet, þou schalt write no 0. ne nowȝt ellis, but set 5 ouerþe next figure aforetoward þe ryȝt, as þus 14.How to halve fourteen.medie 4 then leues 2, do away 4 & sett þere2. þen medie 1. þe quich is rekende for ten, þe haluendel þere-of wel be 5. sett þat 5 vpon þe hede of þat figure, þe quych is now 2,52,& do away þat 1, & þou schalt haue þis nombre yf þou worch wel, vndeversus.

How to prove the Mediation.

¶ Si mediacio sit benefacta probarevalebis¶ Duplando numerumquemprimo dimediasti

¶ Si mediacio sit benefacta probarevalebis

¶ Duplando numerumquemprimo dimediasti

How to prove your mediation.¶ Herehe telles þe how þou schalt know wheþerþou hase wel ydo or no. doubulleaf 151b.*þe nombre þe quych þou hase mediet, and yf þou haue wel y-medyt after þe dupleacioɳ, þou schalt haue þe same nombre þat þou haddyst in þe tabulleor þou began to medye, as þus.First example.¶ The furst ensampullewas þis. 4. þe quychI-medietwaslaft2, þe whych 2 was write in þe place þat 4 was write afore. Now doubulleþat 2, & þou schal haue 4, as þou hadyst afore.The second.þe secunde Ensampullewas þis, 245. When þou haddyst mediet alleþis nombre, yf þou haue wel ydo þou schalt haue of þat mediacioɳ þis nombre, 122w. Now doubulleþis nombre, & begyn in þe lyft side; doubulle1, þat schal be 2. do away þat 1 & sett þere2. þen doubulleþat oþer2 & sett þere4, þen doubulleþat oþer2, & þat wel be 4. þendoubul þat merke þat stondes for halue on. & þat schallebe 1. Cast þat on to 4, & it schallebe 5. do away þat 2 & þat merke, & sette þere5, & þen þou schal haue þis nombre 245. & þis wos þe same nombur þat þou haddyst or þou began to medye, as þou mayst se yf þou take hede.The third example.The nombre þe quych þou haddist for an Ensampul in þe 3 case of mediacioɳ to be mediet was þis 241. whan þou haddist medied alleþis nombur trulyleaf 152a.*by euery figure, þou schall haue be þat mediacioɳ þis nombur 120w. Now dowbul þis nombur, & begyn in þe lyft side, as I tolde þe in þe Craft of duplacioɳ. þus doubulleþe figure of 1, þat wel be 2. doaway þat 1 & sett þere2, þen doubul þe next figureafore, the quych is 2, & þat wel be 4; do away 2 & set þere4. þen doubul þe cifer, & þat wel be noȝt, for a 0 is noȝt. And twyes noȝt is but noȝt. þereforedoubul the merke aboue þe cifers hede, þe quych betokenes þe haluendel of 1, & þat schal be 1. do away þe cifer & þe merke, & sett þere1, & þen þou schalt haue þis nombur 241. And þis same nombur þou haddyst afore or þou began to medy,& yfþou take gode hede.The fourth example.¶ The next ensampul þat had in þe 4 case of mediacioɳ was þis 4678. Whan þou hast trulyymeditalleþis nombur fro þe begynnyngeto þe endynge, þou schalt haue of þe mediacioɳ þis nombur52334.Now doubul this nombur & begyn in þe lyft side, & doubulle2 þat schal be 4. do away 2 and sette þere 4; þen doubule3, þat wol be 6; do away 3 & sett þere6, þen doubul þat oþer3, & þat wel be 6; do away 3 & set þereleaf 152b.*6, þen doubul þe 4, þat welle be 8; þen doubul 5. þe quych stondes ouerþe hed of 4, & þat wol be 10; cast 10 to 8, & þat schal be 18; do away 4 & þat 5, & sett þere8, & sett that 1, þe quych is an articul of þe Composit þe quych is 18, ouerþe next figureshed toward þe lyft side, þe quych is 6. drav þat 1 to 6, þe quych 1 in þe dravyng schal be rekente bot for 1, & þat 1 & þat 6 togedur wel be 7. do away þat 6 & þat 1. the quych stondes ouerhis hede, & sett ther 7, & þen þou schalt haue þis nombur 4678. And þis same nombur þou hadyst or þou began to medye, as þou mayst see in þe secunde Ensampul þat þou had in þe 4 case of mediacioɳ, þat was þis:The fifth example.when þou had mediet truly allethe nombur, a principio usque ad finem. þou schalt haue of þat mediacioɳ þis nombur5102.Now doubul 1. þat wel be 2. do away 1 & sett þere2. þen doubul 0. þat will be noȝt. þereforetake þe 5, þe quych stondes ouerþe next figureshed, & doubul it, & þat wol be 10. do away þe 0 þat stondes betwene þe two figuris, & sette þerein his stid 1, for þat 1 now schal stonde in þe secunde place, wherehe schal betoken 10; þen doubul 2, þat wol be 4. do away 2 & sett þere 4. &leaf 153a.*þou schal haueþusnombur 214. þis is þe same numbur þat þou hadyst or þou began to medye, as þou may see. And so do euermore, yf þou wil knowe wheþerþou hase wel ymedyt or no. ¶. doubulleþe numbur þat comes afterþe mediaciouɳ, & þou schal haue þe same nombur þat þou hadyst or þou began to medye, yf þou haue welle ydo. or els doute þe noȝt, but yf þou haue þe same, þou hase faylide in þiCraft.

The Craft of Multiplication.

To write down a Multiplication Sum.

Si tu pernumerumnumerumvis multiplicareScribe duas quascunquevelis series numerorumOrdo serveturvt vltima multiplicandiPonatursuperanteriorem multiplicantisA leua relique sint scripte multiplicantes.

Si tu pernumerumnumerumvis multiplicare

Scribe duas quascunquevelis series numerorum

Ordo serveturvt vltima multiplicandi

Ponatursuperanteriorem multiplicantis

A leua relique sint scripte multiplicantes.

Four things to be known of Multiplication:¶ Herebe-gynnes þe Chaptre of multiplicatioɳ, in þe quych þou most know 4 thynges. ¶ Ffirst, qwat is multiplicacioɳ. The secunde, how mony cases may hap in multiplicacioɳ. The thryde, how mony rewes of figuresþeremost be. ¶ The 4. what is þe profet of þis craft.the first:¶ As for þe first, þou schal vnderstonde þat multiplicacioɳ is a bryngyngeto-gederof 2 thyngesin on nombur, þe quych on nomburcontynesso mony tymes on, howeleaf 153b.*mony tymes þereben vnytees in þe nowmbreof þat 2, as twyes 4 is 8. now hereben þe 2 nombers, of þe quych too nowmbres on is betokened be an aduerbe, þe quych is þe worde twyes, & þis worde thryes, & þis worde fouresythes,9& so furth of such other lyke wordes. ¶ And tweyn nombres schal be tokenyde be a nowne, as þis worde foureshowys þes tweyɳ nombresy-brothin-to on hole nombur, þat is 8, for twyes 4 is 8, as þou wost wel. ¶ And þes nombre8 conteynes as oft tymes 4 as þereben vnites in þat other nombre, þe quych is 2, for in 2 ben 2 vnites, & so oft tymes 4 ben in 8, as þou wottys wel.the second:¶ ffor þe secunde, þou most know þat þou most haue too rewes of figures.the third:¶ As for þe thryde, þou most know þat 8 manerof diuerse case may happe in þis craft.the fourth.The profet of þis Craft is to telle when a nombreis multiplyed be a noþer, qwat commys þereof. ¶ fforthermore, as to þe sentence of oureverse, yf þou wel multiply a nombur be a-noþernombur, þou schalt writeleaf 154a.*a rewe of figures of what nomburs so euerþou welt,The multiplicand.& þat schal be called Numerusmultiplicandus, Anglice, þe nomburthe quych to be multiplied. þen þou schalt write a-nother rewe of figures, by þe quych þou schalt multiplie the nombre þat is to be multiplied, of þe quych nomburþe furst figure schal be write vnderþe last figureof þe nombur, þe quych is to be multiplied.How to set down the sum.And so write forthe toward þe lyft side, as hereyou may se,673241234And þis onenomburschallebe called numerus multiplicans. Anglice, þe nomburmultipliynge, for he schallemultiply þe hyer nounbur, as þus onetyme 6. And so forth, as I schal telle the afterwarde. And þou schal begyn in þe lyft side.Two sorts of Multiplication: mentally,¶ ffor-þere-more þou schalt vndurstonde þat þereis two manurs of multiplicacioɳ; one ys of þe wyrchyngeof þe boke only in þe mynde of a mon. fyrst heteches of þe fyrst manerof duplacioɳ, þe quych is be wyrchyngeof tabuls.and on paper.Afterwarde he wol teche on þe secunde maner. vndeversus.

To multiply one Digit by another.

In digitumcures digitumsi ducere maiorleaf 154b.* Perquantumdistat a denis respice debes¶ Namquesuo decuplo totiens delereminoremSitquetibi numerus veniens exinde patebit.

In digitumcures digitumsi ducere maior

* Perquantumdistat a denis respice debes

¶ Namquesuo decuplo totiens delereminorem

Sitquetibi numerus veniens exinde patebit.

How to multiply two digits.¶ Herehe teches a rewle, how þou schalt fynde þe nounbre þat comes by þe multiplicacioɳ of a digit be anoþer. loke how mony [vny]tes ben. bytwene þe moredigit and 10. And reken ten for on vnite.Subtract the greater from ten;And so oft do away þe lasse nounbre out of his ownedecuple, þat is to say, fro þat nounbre þat is ten tymes so mychisþe nounbreþat comes of þe multiplicacioɳ. As yf þou wol multiply 2 be 4. loke how mony vnitees ben by-twene þe quych is þe morenounbre, & be-twene ten. Certen þerewel be vj vnitees by-twene 4 & ten. yf þou reken þerewithþe ten þe vnite, as þou may se.take the less so many times from ten times itself.so mony tymes take 2. out of his decuple, þe quych is 20. for 20 is þe decuple of 2, 10 is þe decuple of 1, 30 is þe decuple of 3, 40 is þe decuple of 4, And þe oþerdigetes til þou come to ten; & whan þouExample.hast y-take so mony tymes 2 out of twenty, þe quych is sex tymes, þou schal leue 8 as þou wost wel, for 6 times 2 is twelue. take [1]2 out of twenty, & þereschal leue 8. bot yf bothe þe digettesleaf 155a.*beny-lyechmych as here. 222 or too tymes twenty, þen it isno forsquych of hem tweyn þou take out of here decuple.alsmonyBetter use this table, though.tymes as þat is fro 10. but neuer-þe-lesse, yf þou hauehastto worch, þou schalt haue herea tabul of figures, where-by þou schalt se a-nonɳ ryght what is þe nounbre þat comes of þe multiplicacioɳ of 2 digittes. þus þou schalt worch in þis figure.

How to use it.yf þe figure, þe quych schallebe multiplied, be eueneas mych as þe diget be, þe quych þat oþerfigureschal be multiplied, as two tymes twayɳ, or thre tymes 3. or sych other.The way to use the Multiplication table.loke qwereþat figure sittes inþe lyft side of þe triangle, & loke qwereþe diget sittes in þe neþermost rewe of þe triangle. & go fro hym vpwarde in þe same rewe, þe quych rewe gose vpwarde til þou come agaynes þe oþerdigette þat sittes in þe lyft side of þe triangle. And þat nounbre, þe quych þouleaf 155b.fyn*des þereis þe nounbre þat comes of the multiplicacioɳ of þe 2 digittes, as yf þou wold wete qwat is 2 tymes 2. loke queresittes 2 in þe lyft side inþe first rewe, he sittes next 1 in þe lyft side al on hye, as þou may se; þe[n] loke qweresittes 2 in þe lowyst rewe of þe triangle, & go fro hym vpwarde in þe same rewe tylleþou comea-ȝenenes2 in þe hyer place, & þer þou schalt fynd ywrite 4, & þat is þe nounbre þat comes of þe multiplicacioɳ of two tymes tweyn is 4, as þow wotest welle. yf þe diget. the quych is multiplied, be moreþan þe oþer, þou schalt loke qwereþe morediget sittes in þe lowest rewe of þe triangle, & go vpwarde in þe same rewe tyl10þou comea-nendesþe lasse diget in the lyft side. And þereþou schalt fynde þe nombre þat comes of þe multiplicacioɳ; but þou schalt vnderstonde þat þis rewle, þe quych is in þis verse. ¶ In digitumcures, &c., noþerþis triangle schallenot serue, bot to fynde þe nounbres þat comes of the multiplicacioɳ þat comes of 2 articuls orcomposites, þe nedes no craft but yf þou wolt multiply in þi mynde. Andleaf 156a.*þere-to þou schalt haue a craft afterwarde, for þou schall wyrch withdigettes in þe tables, as þou schalt know afterwarde. versus.

To multiply one Composite by another.

¶ Postea procedas postremammultiplicando[Recte multiplicans per cunctas inferiores]Condicionem tamen tali quodmultiplicantesScribas in capite quicquid processerit indeSed postquamfuit hec multiplicate figureAnteriorenturserei multiplicantisEt sic multiplica velut isti multiplicastiQui sequiturnumerumscriptumquiscunquefiguris.

¶ Postea procedas postremammultiplicando

[Recte multiplicans per cunctas inferiores]

Condicionem tamen tali quodmultiplicantes

Scribas in capite quicquid processerit inde

Sed postquamfuit hec multiplicate figure

Anteriorenturserei multiplicantis

Et sic multiplica velut isti multiplicasti

Qui sequiturnumerumscriptumquiscunquefiguris.

How to multiply one number by another.¶ Herehe teches how þou schalt wyrch in þis craft. þou schalt multiplye þe last figureof þe nombre, and quen þou hast so ydo þou schalt draw alleþe figures of þe neþernounbre moretawardþe ryȝt side, so qwenþou hast multiplyed þe last figureof þe heyer nounbre by alleþe neþerfigures.Multiply the ‘last’ figure of the higher by the ‘first’ of the lower number.And sette þe nounbir þat comes þer-of ouerþe last figureof þe neþernounbre, & þen þou schalt sette al þe oþerfigures of þe neþernounbremorenereto þe ryȝt side. ¶ And whan þou hast multiplied þat figureþat schal be multiplied þe next afterhym by al þe neþerfigures. And worch as þou dyddyst aforetilleaf 156b.*þou come to þe ende. And þou schalt vnderstonde þat euery figureof þe hier nounbreschal be multiplied be alleþe figures of the neþernounbre, yf þe hier nounbrebe any figureþen one.Set the answer over the first of the lower:lo an Ensampul herefolowynge.2465.232þou schalt begyne to multiplye in þe lyft side. Multiply 2 be 2, and twyes 2 is 4. set 4then multiply the second of the lower, and so on.ouerþe hed of þat 2, þen multiplie þe same hier 2 by 3 of þe nether nounbre, as thryes 2 þat schal be 6. set 6 ouerþe hed of 3, þan multiplie þe same hier 2 by þat 2 þe quych stondes vnderhym, þat wol be 4; do away þe hier 2 & sette þere4.Then antery the lower number:¶ Now þou mostanteryþe nether nounbre, þat is to say, þou most sett þe neþernounbre more towarde þe ryȝt side, as þus. Take þe neþer2 toward þe ryȝt side, & sette it eueɳ vnderþe 4 of þe hyer nounbre, & antery alleþe figures þat comes afterþat 2, as þus; sette 2 vnderþe 4. þen sett þe figureof 3 þereþat þe figure of 2 stode, þe quych is now vndur þat 4 in þe hier nounbre; þen sett þe oþer figureof 2, þe quych is þe last figure toward þe lyft side of þe neþernomberþereþe figureof 3 stode.as thus.þen þou schalt haue such a nombre.464465232leaf 157a.* ¶ Now multiply 4, þe quych comes next after6, by þe last 2 of þe neþernounbur toward þe lyft side. as 2 tymes 4, þat wel be 8. sette þat 8 ouerþe figure the quych stondes ouerþe hede of þat 2, þe quych is þe last figureof þe neþernounbre; þan multiplie þat same 4 by 3, þat comes in þe neþerrewe, þat wol be 12. sette þe digit of þe composyt ouerþe figure þe quych stondes ouerþe hed of þat 3, & sette þe articule of þis composit oueral þe figures þat stondes ouerþe neþer2 hede.Now multiply by the last but one of the higher:þen multiplie þe same 4 by þe 2 in þe ryȝt side in þe neþernounbur, þat wol be 8. do away 4. & sette þere8. Euermoreqwen þou multiplies þe hier figureby þat figureþe quych stondes vnderhym, þou schalt do away þat hier figure, & sett þer þat nounbre þe quych comes of multiplicacioɳ of ylke digittes.as thus.Whan þou hast done as I haue byde þe, þou schalt haue suych an orderof figureas is here,1824648[65]232.þen take and antery þi neþerfigures. And sett þe fyrst figure of þe neþerfigures11vndre be figureof 6. ¶ And draw al þe oþerfigures of þe same rewe to hym-warde,leaf 157b.*as þou diddyst afore. þen multiplye 6 be 2, & sett þat þe quych comes ouerþere-of oueral þe oþerfigures hedes þat stondes ouerþat 2. þen multiply 6 be 3, & sett alleþat comes þere-of vpon alleþe figures hedes þat standes ouerþat 3; þanmultiplye 6 be 2, þe quychstondes vnderþat 6, þen do away 6 & write þereþe digitt of þe composit þat schal come þereof, & sette þe articull oueralleþe figures þat stondes ouerþe hede of þat 3 as here,11121828464825232Antery the figures again, and multiply by five:þen antery þi figures as þou diddyst afore, and multipli 5 be 2, þat wol be 10; sett þe 0 ouerall þe figures þat stonden ouerþat 2, & sett þat 1. ouerthe next figures hedes, alleon hye towarde þe lyft side. þen multiplye 5 be 3. þat wol be 15, write 5 ouerþe figures hedes þat stonden ouerþat 3, & sett þat 1 ouerþe next figures hedes toward þe lyft side. þen multiplye 5 be 2, þat wol be 10. do away þat 5 & sett þerea 0, & sett þat 1 ouerþe figures hedes þat stonden ouer3. And þenleaf 158a.þou schalt haue such a nounbre as here stondes aftur.*1111011215828204648232¶ Now draw alleþese figures downe togederas þus, 6.8.1. & 1 draw to-gedur; þat wolle be 16, do away alleþese figures saue 6. lat hym stonde, forþowþou take hym away þou most write þer þe same aȝene. þereforelate hym stonde, & sett 1 ouerþe figurehede of 4 toward þe lyft side; þen draw on to 4, þat wollebe 5.Then add all the figures above the line:do away þat 4 & þat 1, & sette þere5. þen draw 4221 & 1 togedur, þat wol be 10. do away alleþat, & write þere þat 4 & þat 0, & sett þat 1 ouerþe next figureshede toward þe lyft side, þe quych is 6. þen draw þat 6 & þat 1 togedur, & þat wolle be 7; do away 6 & sett þere7, þen draw 8810 & 1, & þat wel be 18; do away alleþe figures þat stondes ouerþe hede of þat 8, & lette 8 stonde stil, & write þat 1 ouerþe next figurishede, þe quych is a 0. þen do away þat 0, & sett þere1, þe quych stondes ouerþe 0. hede. þen draw 2, 5, & 1 togedur, þat wollebe 8. þen do away alleþat, & write þere8.and you will have the answer.¶ And þen þou schalt haue þis nounbre, 571880.

The Cases of this Craft.

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