Egerton2622.leaf 136a.HEc algorismusars presens dicitur; in quaTalibusindorumfruimurbis quinquefiguris.A derivation of Algorism.This boke is called þe boke ofalgorym, or Augrym afterlewdervse. And þis boke tretys þeCraftof Nombryng, þe quych crafte is called also Algorym. Ther was a kyng of Inde, þe quichheythAlgor, & he made þis craft. And afterhis name he called hit algorym;Another derivation of the word.or els anoþercause is quy it is called Algorym, for þe latyn word of hits.Algorismuscomesof Algos, grece, quidestars, latine, craft oɳ englis, and rides, quidestnumerus, latine, A nomburoɳ englys, inde dicitur Algorismusperaddicionemhuiussillabe mus& subtraccionem d & e, quasi ars numerandi. ¶ fforthermoreȝemostvndirstonde þat in þis craft ben vsid teen figurys, as here benewriten for ensampul, φ 9 8 7 6 5 4 3 2 1. ¶Exponeþe too versus afore: this present craft ys called Algorismus, in þe quych we vse teen signys of Inde. Questio. ¶ Why teɳ fyguris of Inde? Solucio. for as I haue sayd afore þai werefonde fyrst in Inde of a kyngeof þat Cuntre, þat was called Algor.Notation and Numeration.versus[in margin].¶ Prima significat unum; duo vero secunda:¶ Tercia significat tria; sic procede sinistre.¶ Donec ad extremamvenias, que cifra vocatur.¶ Capitulum primum de significacione figurarum.Expositioversus.In þis verse is notifide þe significacion of þese figuris. And þus expone the verse.The meaning and place of the figures.Þe first signifiyth one, þe secundeleaf 136b.signi*fiyth tweyne, þe thryd signifiyth thre, & the fourte signifiyth 4. ¶ And so forthe towarde þe lyft syde of þe tabul or of þe boke þat þe figures benewritenein, til þat þou come to the last figure, þat iscalled acifre. ¶ Questio. In quych syde sittes þe first figure? Solucio, forsothe loke quich figure is first in þe ryȝt side of þe bok or of þe tabul, & þat same is þe first figure, for þou schal write bakeward, as here, 3. 2. 6. 4. 1. 2. 5.Which figure is read first.The figure of 5. was firstwrite, & he is þe first, for he sittes oɳ þe riȝt syde. And the figure of 3 is last. ¶ Neuer-þe-les wen he says ¶ Prima significat vnum&c., þat is to say, þe first betokenes one, þe secunde. 2. & fore-þer-more, he vndirstondesnoȝtof þe first figure of eueryrew. ¶ But he vndirstondes þe first figure þat is in þe nomburof þe forsayd teen figuris, þe quych is oneof þese. 1. And þe secunde 2. & so forth.versus[in margin].¶ Quelibetillarumsi primo limite ponas,¶ Simpliciterse significat: si vero secundo,Se decies: sursumprocedasmultiplicando.¶ Namquefigura sequens quamuis signat decies plus.¶ Ipsa locata loco quam significat pertinente.Expositio[in margin].¶ Expone þis verse þus. Euery of þese figuris bitokens hym selfe & no more, yf he stonde in þe first place of þerewele/ this worde Simpliciterin þat verse it is no more to say but þat, & no more.An explanation of the principles of notation.¶ If it stonde in the secunde place of þerewle, he betokens tenetymes hym selfe, as þis figure2 here 20 tokens ten tyme hym selfe,leaf 137a.*þat is twenty, for he hym selfe betokenes tweyne, & ten tymestweneis twenty. And for he stondis oɳ þe lyft side & in þe secunde place, he betokens ten tyme hymselfe. And so go forth. ¶ ffor eueryfigure, & he stonde aftura-noþertoward the lyft side, he schal betokeneten tymes as mich moreas he schul betoken & he stode in þe place þereþat þe figurea-forehym stondes.An example:loo an ensampulle. 9. 6. 3. 4. Þe figureof 4. þat hase þis schape{4}. betokens bot hymselfe, for he stondes in þe first place.units,The figureof 3. þat hase þis schape{3}. betokens ten tymes moreþen he schuld&he stde þereþat þe figureof 4. stondes, þat is thretty.tens,The figureof 6, þat hase þis schape{6}, betokens ten tymes moreþan he schuld & he stode þereas þe figureof{3}. stondes, for þerehe schuld tokynebot sexty, & now he betokens ten tymes more, þat is sex hundryth.hundreds,The figureof 9. þat hase þis schape{9}. betokens ten tymes moreþanehe schuld & he stode in þe place þereþe figureof sex stondes, for þen he schuld betoken to 9. hundryth, and in þe place þerehe stondes now he betokens 9. þousande.thousands.Al þeholenomburis 9 thousande sex hundryth & foure& thretty. ¶ fforthermore, whenþou schalt rede a nomburof figure,How to read the number.þou schalt begyneat þe last figurein the lyft side, & rede so forth to þe riȝt side as here9. 6. 3. 4. Thou schal begyn to rede at þe figureof 9. & rede forth þus. 9.leaf 137b.*thousand sex hundryth thritty & foure. But when þou schallewrite, þou schalt be-gynne to write at þe ryȝt side.¶ Nil cifra significat seddat signaresequenti.The meaning and use of the cipher.Expone þis verse. A cifre tokensnoȝt, bot he makes þe figureto betoken þat comes afturhym moreþan he schuld & he wereaway, as þus 1φ. hereþe figureof onetokens ten, & yf þe cifre wereaway1& no figureby-forehym he schuld token bot one, for þan he schuld stonde in þe first place. ¶ And þe cifre tokens nothyng hym selfe. for al þe nomburof þeylketoo figures is bot ten. ¶ Questio. Why says he þat a cifre makys a figuretosignifye (tyf)more&c. ¶ I speke for þis wordesignificatyf, ffor sothe it may happe aftura cifre schuld come a-noþurcifre, as þus 2φφ. And ȝet þe secunde cifre shuld token neuerþe moreexcephe schuld kepe þe orderof þe place. and a cifre is no figuresignificatyf.¶ Quam precedentes plus ultima significabit /The last figure means more than all the others, since it is of the highest value.Expone þis verse þus. Þe last figureschal token moreþan alleþe oþerafore,thouȝtþerewerea hundryth thousant figures afore, as þus, 16798. Þe last figureþat is 1. betokens ten thousant. And alleþe oþerfigures ben bot betokenebot sex thousant seuynehundryth nynty & 8. ¶ And ten thousant is moreþen alleþat nombur,ergo þe last figuretokens moreþan all þe nomburafore.The Three Kinds of Numbersleaf 138a.* ¶ Post predicta scias breuiterquodtres numerorumDistincte species sunt; nam quidam digiti sunt;Articuli quidam; quidam quoquecompositi sunt.¶ Capitulum 2mde triplice divisione numerorum.¶ The auctor of þis tretisdepartysþis worde a nomburinto 3 partes. Some nomburis called digituslatine, adigitin englys.Digits.Somme nomburis called articuluslatine. AnArticulin englys.Articles.Some nomburis called acomposytin englys.Composites.¶ Expone þis verse. know þou afturþe forsaydrewlesþat I sayd afore, þat þereben threspicesof nombur. Ooneis a digit, Anoþeris an Articul, & þe toþera Composyt. versus.Digits, Articles, and Composites.¶ Sunt digiti numeri qui citradenariumsunt.What are digits.¶ Herehe telles qwat is a digit, Expone versussic. Nomburs digitus beneallenomburs þat ben with-inne ten, as nyne, 8. 7. 6. 5. 4. 3. 2. 1.¶ Articupli decupli degitorum; compositi suntIlli qui constant ex articulis degitisque.¶ Herehe telles what is a composyt and what is anearticul. Expone sic versus.What are articles.¶ Articulis ben2alleþat may be deuidyt into nombursof ten & nothyngeleueouer, as twenty, thretty, fourty, a hundryth, a thousand, & such oþer, ffor twenty may be departyt in-to 2 nomburs of ten, fforty in to fourenomburs of ten, & so forth.leaf 138b.What numbers are composites.*Compositys beɳ nomburs þat bene componyt of a digyt & of an articulleas fouretene, fyftene, sextene, & such oþer. ffortene is componyd of foureþat is a digit & of ten þat is an articulle. ffiftene is componyd of 5 & ten, & so of all oþer, what þat þai ben. Short-lych euery nomburþat be-gynnes witha digit & endyth in a articulleis a composyt, as fortene bygennyngeby foureþat is a digit, & endes in ten.¶ Ergo, proposito numero tibi scribere, primoRespicias quid sit numerus; si digitus sitPrimo scribe loco digitum, si compositus sitPrimo scribe loco digitumpost articulum; sic.How to write a number,¶ here he telles how þou schaltwyrchwhan þou schalt write a nombur. Expone versumsic, & fac iuxta exponentis sentenciam; whan þou hast a nomburto write, loke fyrst what manernomburit ys þat þou schalt write, whether it be a digit or a composit or an Articul.if it is a digit;¶ If he be a digit, write a digit, as yf it be seuen, write seuen & write þat digit in þe first place toward þe ryght side.if it is a composite.If it be a composyt, write þe digit of þe composit in þe first place & write þe articul of þat digit in þe secunde place next toward þe lyft side. As yf þou schal write sex & twenty. write þe digit of þe nomburin þe first place þat is sex, and write þe articul next afturþat is twenty, as þus 26.How to read it.But whan þou schaltsowneor spekeleaf 139a.*or rede an Composyt þou schalt first sowne þe articul & afturþe digit, as þou seyst by þe comynespeche, Sex & twenty & nouȝt twenty & sex. versus.¶ Articulussi sit, in primo limite cifram,Articulum veroreliquis inscribe figuris.How to write Articles:¶ Here he tells how þou schal write when þe nombre þat þou hase to write is an Articul. Expone versus sic & fac secundum sentenciam. Ife þe nomburþat þou hast write be an Articul, write first a cifre & afturþe cifer write an Articulleþus. 2φ.tens,fforthermoreþou schalt vndirstonde yf þou haue an Articul, loke howmych he is, yf he be with-ynne an hundryth, þou schalt write bot onecifre, afore, as here.9φ.hundreds,If þe articullebe by hym-silfe & be an hundrid euene, þen schal þou write .1. & 2 cifers afore, þat he may stonde in þe thryd place, for euery figurein þe thryd place schal token a hundrid tymes hym selfe.thousands, &c.If þe articul be a thousant or thousandes3and he stonde by hymselfe, write afore3 cifers & so forþ of al oþer.¶ Quolibetin numero, si par sit prima figura,Par erit & totum, quicquid sibi continuatur;Imparsi fuerit, totumtunc fietetimpar.To tell an even number¶ Herehe teches a generallerewle þat yf þe first figurein þerewleof figures token a nomburþat is eueneal þat nomburof figurys in þat rewle schal be euene, as hereþou may see 6. 7. 3. 5. 4. Computa & proba.or an odd.¶ If þe firstleaf 139b.*figure token an nomburþat is ode, alleþat nomburin þat rewle schallebe ode, as here5 6 7 8 6 7. Computa & proba. versus.¶ Septemsunt partes, nonplures, istius artis;¶ Addere, subtrahere, duplare, dimidiare,Sextaquediuidere, sedquinta multiplicare;Radicemextraherepars septima dicitur esse.The Seven Rules of Arithmetic.The seven rules.¶ Heretelles þat þerbeɳ .7. spices or partes of þis craft. The first is called addicioñ, þe secunde is called subtraccioñ. The thryd is calledduplacioñ. The 4. is calleddimydicioñ. The 5. is called multiplicacioñ. The 6 is called diuisioñ. The 7. is called extraccioñ of þeRote. What all þese spices benehit schallebe tolde singillatiminherecaputule.¶ Subtrahis aut addis a dextris vel mediabis:Add, subtract, or halve, from right to left.Thou schal be-gynne in þe ryght side of þe boke or of a tabul. loke wereþou wul be-gynne to write latyn or englys in a boke, & þat schallebe called þe lyft side of the boke, þat þou writest toward þat side schal be called þe ryght side of þe boke. Versus.A leua dupla, diuide, multiplica.Here he telles þe in quych side of þe boke or of þe tabul þou schallebe-gyneto wyrch duplacioñ, diuisioñ, and multiplicacioñ.Multiply or divide from left to right.Thou schal begynetoworchin þe lyft side of þe boke or of þe tabul, but yn what wyse þou schal wyrch in hymdicetursingillatimin sequentibuscapituliset de vtilitate cuiuslibetartis& sic Completurleaf140.*prohemium& sequiturtractatus& primo de arte addicionis que prima ars est in ordine.The Craft of Addition.Adderesi numero numerumvis, ordine taliIncipe; scribe duas primo series numerorumPrimamsub prima recte ponendo figuram,Et sic de reliquis facias, si sint tibi plures.Four things must be known:¶ Hereby-gynnes þe craft of Addicioñ. In þis craft þou most knowe foure thynges. ¶ Fyrst þoumost know what is addicioñ. Next þou most know how mony rewles of figurys þou most haue. ¶ Next þou most know how mony diuers casys happes in þis craft of addicioñ. ¶ And next qwat is þe profet of þis craft.what it is;¶ As for þe first þou most know þat addicioñ is acastyngto-gedurof twoo nomburys in-to onenombre. As yf I aske qwat istwene& thre. Þou wylcastþese twene nombres to-gedur& say þat it is fyue.how many rows of figures;¶ As for þe secunde þou most know þat þou schallehaue tweyne rewes of figures, onevndur a-nother, as hereþou mayst se.12342168.how many cases;¶ As for þe thryd þou most know þat thereben foure diuerse cases.what is its result.As for þe forthe þou most know þat þe profet of þis craft is to telle what is þe hole nomburþat comes of diuerse nomburis. Now as to þe texte of oure verse, he teches therehow þou schalworchin þis craft. ¶ He says yf þou wilt cast onenomburto anoþernombur, þou most by-gynne on þis wyse.How to set down the sum.¶ ffyrst writeleaf 140b.*two rewes of figuris & nombris so þat þou write þe first figureof þe hyer nombureuenevndirthe first figureof þe nether nombur,123234.And þe secunde of þe nether nombureuenevndirþe secunde of þe hyer, & so forthe of euery figure of both þe rewes as þou mayst se.The Cases of the Craft of Addition.¶ Inde duas adde primas hac condicione:Si digitus crescat ex addicione priorum;Primo scribe loco digitum, quicunquesit ille.¶ Here he teches what þou schalt do when þou hast write too rewes of figurisonvnder an-oþer, as I sayd be-fore.Add the first figures;¶ He says þou schalt take þe first figure of þe heyer nombre& þe fyrst figureof þe neþernombre, & cast hem to-gedervp-on þis condicioɳ. Thou schal loke qweþerþe nomberþat comys þere-of be a digit or no.rub out the top figure;¶ If he be a digit þou schalt do away þe first figure of þe hyer nombre, and write þerein his stede þat he stode Inne þe digit, þat comes of þe ylke 2 figures, & sowrite the result in its place.wrichforth oɳ oþerfigures yf þerebe ony moo, til þou come to þe ende toward þe lyft side. Andledeþe nether figurestonde still euer-moretil þou haueydo. ffor þere-by þou schalwytewheþerþou hast donewel or no, as I schal tell þe afterward in þe ende of þis Chapter. ¶ And lokeallgateleaf 141a.þat þou be-gynne to worch in þis Craft of Addi*cioɳ in þe ryȝt side,Here is an example.here is an ensampul of þis case.12342142.Caste 2 to foure& þat wel be sex, do away 4. & write in þe same place þe figure of sex. ¶ And lete þe figure of 2 in þe nether rewe stonde stil. When þou hast do so, cast 3 & 4 to-gedurand þat wel be seuen þat is a digit. Do away þe 3, & set þereseueɳ, and lete þe neþerfigurestonde stille, & soworchforth bakward til þou hastydoall to-geder.Et si compositus, in limite scribe sequenteArticulum, primo digitum; quiasic iubet ordo.¶ Here is þe secunde case þat may happe in þis craft. And þe case is þis,Suppose it is a Composite, set down the digit, and carry the tens.yf of þe casting of 2 nomburis to-geder, as of þe figure of þe hyer rewe & of þe figure of þe neþerrewe come a Composyt, how schalt þouworch. Þusþou schalt worch. Thou shalt do away þe figure of þe hyer nomberþat was cast to þe figure of þe neþernomber. ¶ And write þereþe digit of þe Composyt. And set þe articul of þe composit next afterþe digit in þe same rewe, yf þerebe nomofigures after. But yf þerebe mo figuris afterþat digit. And þere he schall be rekend for hym selfe. And when þou schalt adde þat ylke figure þat berys þe articulleouerhis hed to þe figurevnderhym, þou schalt cast þat articul to þe figure þat hase hym ouerhis hed, & þereþat Articul schal tokeɳ hym selfe.Here is an example.lo an Ensampullleaf 141b.*of all.326216.Cast 6 to 6, & þere-of wil arise twelue. do away þe hyer 6 & write þere2, þat is þe digit of þis composit. And þenwrite þe articulleþat is ten ouerþe figuris hed of twene as þus.1322216.Now cast þe articulleþat standus vpon þe figuris of twene hed to þe same figure, & reken þat articul bot for one, and þan þerewil arise thre. Þan cast þat thre to þe neþerfigure, þat is one, & þat wul be foure. do away þe figure of 3, and write þerea figure of foure. and lete þe neþerfigure stonde stil, & þan worch forth. vndeversus.¶ Articulus si sit, in primo limite cifram,¶ Articulumvero reliquis inscribe figuris,Vel perse scribas si nulla figura sequatur.¶ Herehe puttes þe thryde case of þe craft of Addicioɳ. & þe case is þis.Suppose it is an Article, set down a cipher and carry the tens.yf of Addiciouɳ of 2 figuris a-ryse an Articulle, how schal þou do. thou most do away þeheerfigure þat was addid to þe neþer, & write þerea cifre, and sett þearticulson þe figuris hede, yf þat þerecome ony after. And wyrch þan as I haue tolde þe in þe secunde case. An ensampull.25.15Cast 5 to 5, þat wylle be ten. now do away þe hyer 5, & write þerea cifer. And sette ten vpon þe figuris hed of 2. And reken it but for on þus. loHere is an example.an Ensampulle12φ15Andleaf 142a.*þan worch forth. But yf þerecome no figure afterþe cifre, write þe articul next hym in þe same rewe as here55cast 5 to 5, and it wel be ten. do away 5. þat is þe hier 5. and write þerea cifre, & write afterhym þe articul as þus1φ5And þan þou hast done.¶ Si tibi cifra superueniens occurrerit, illamDele superpositam; fac illic scribe figuram,Postea procedas reliquas addendo figuras.What to do when you have a cipher in the top row.¶ Herehe puttesþe fourt case, & it is þis, þat yf þerecome a cifer in þe hier rewe, how þou schal do. þus þou schalt do. do away þe cifer, & sett þereþe digit þat comes of þe addiciounas þus1φφ84.17743An example of all the difficulties.In þis ensampul ben alleþe fourecases. Cast 3 to foure, þat wol be seueɳ. do away 4. & write þereseueɳ; þan cast 4 to þe figureof 8. þat wel be 12. do away 8, & sett þere2. þat is a digit, and sette þe articul of þe composit, þat is ten, vpon þe cifers hed, & reken it for hym selfe þat is on. þan cast oneto a cifer, & hit wullebe but on, for noȝt & on makes but one. þan cast 7. þat stondes vnderþat on to hym, & þat wel be 8. do away þe cifer & þat 1. & sette þere8. þan go forthermore. cast þe oþer7 to þe cifer þat stondes ouerhym. þat wul be bot seuen, for þe cifer betokens noȝt. do away þe cifer & sette þereseueɳ,leaf 142b.*& þen go forþermore& cast 1 to 1, & þat wel be 2. do away þe hier 1, & sette þere2. þan hast þou do. And yf þou haue wel ydo þis nomber þat is sett here-afterwel be þe nomber þat schallearyse of alleþe addicioɳ as here27827. ¶ Sequitur alia species.The Craft of Subtraction.A numero numerumsi sit tibi demerecuraScribe figurarumseries, vt in addicione.Four things to know about subtraction:¶ This is þe Chapterof subtraccioɳ, in the quych þou most know foure nessessary thynges. the first what is subtraccioɳ. þe secunde is how mony nombers þou most haue to subtraccioɳ, the thryd is how mony maners of cases þeremay happe in þis craft of subtraccioɳ. The fourte is qwat is þe profet of þis craft. ¶ As forthe first;þe first, þou most know þat subtraccioɳ is drawyngeof onenowmberoute of anoþernomber.the second;As for þe secunde, þou most knowe þat þou most haue two rewes of figuris onevnderanoþer, as þouaddystin addicioɳ.the third;As for þe thryd, þou moyst know þat fouremanerof diuerse casis mai happe in þis craft.the fourth.¶ As for þe fourt, þou most know þat þe profet of þis craft is whenne þou hasse taken þe lasse nomber out of þe moreto telle what þereleuesouerþat. & þou most be-gynne to wyrch in þiscraft in þe ryght side of þe boke, as þou diddyst in addicioɳ. Versus.¶ Maiori numero numerumsuppone minorem,¶ Siue pari numero supponaturnumerus par.leaf 143a.* ¶ Herehe telles þatPut the greater number above the less.þe hier nomber most be moreþen þe neþer, or els eueɳ as mych. but he may not be lasse. And þe case is þis, þou schalt drawe þe neþernomber out of þe hyer, & þou mayst not do þat yf þe hier nomber werelasse þan þat. ffor þou mayst not draw sex out of 2. But þou mast draw 2 out of sex. And þou maiste draw twene out of twene, for þou schal leue noȝt of þe hier twene vndeversus.The Cases of the Craft of Subtraction.¶ Postea si possis a prima subtrahe primamScribens quod remanet.The first case of subtraction.Hereis þe first case put of subtraccioɳ, & he says þou schalt begynne in þe ryght side, & draw þe first figureof þe neþerrewe out of þe first figureof þe hier rewe. qwether þe hier figure be moreþen þe neþer, or eueɳ as mych. And þat is notified in þe vers when he says “Si possis.” Whan þou has þus ydo, do away þe hiest figure & sett þereþatleuesof þe subtraccioɳ,Here is an example.lo an Ensampulle234122draw 2 out of 4. þan leues 2. do away 4 & write þere2, & latte þe neþerfigurestonde stille, & so gofor-byoþerfiguris till þou come to þe ende, þan hast þou do.¶ Cifram si nil remanebit.Put a cipher if nothing remains.¶ Herehe puttesþe secunde case, & hit is þis. yf it happe þat qwen þou hast draw on neþerfigureout of a hier, & þere leue noȝt afterþe subtraccioɳ, þusleaf 143b.*þou schalt do. þou schalledo away þe hier figure & write þerea cifer, asHere is an example.lo an Ensampull2424Take foureout of foureþanleusnoȝt. þereforedo away þe hier 4 & set þerea cifer, þan take 2 out of 2, þan leues noȝt. do away þe hier 2, & set þerea cifer, and so worch whareso euerþis happe.Sed si nonpossis a prima demere primamPrecedens vnumde limite deme sequente,Quod demptumprodenario reputabis ab illoSubtrahe totalem numerumquemproposuistiQuo facto scribe super quicquid remanebit.Suppose you cannot take the lower figure from the top one, borrow ten;Herehe puttes þe thryd case, þe quych is þis. yf it happe þat þe neþerfigure be moreþen þe hier figure þat he schallebe draw out of. how schalleþou do. þus þou schalledo. þou schalleborro.1. oute of þe next figure þat comes afterin þe same rewe, for þis case may neuerhapp but yf þerecome figures after. þan þou schalt settþat on ouerþe hier figureshed, of the quych þou woldist y-draw oute þe neyþerfigure yf þou haddysty-myȝt. Whane þou hase þus ydo þou schallerekene þat .1. for ten.take the lower number from ten;¶. And out of þat ten þou schal draw þe neyþermost figure, And alleþat leues þou schalleadd the answer to the top number.adde to þe figureon whos hed þat .1. stode. And þen þou schalledo away alleþat, & sett þereallethat arisys of the addicioɳ of þe ylke 2 figuris. And yf ytleaf 144a.*happe þat þe figure of þe quych þou schalt borro on be hym self but 1. If þou schalt þat one& sett it vppoɳ þe oþerfigurished, and sett in þat 1. place a cifer, yf þerecome mony figuresafter.Example.lo an Ensampul.21221134take 4 out of 2. it wyl not be, þerforeborro oneof þe next figure, þat is 2. and sett þat ouerþe hed of þe fyrst 2. & rekene it for ten. and þere þe secunde stondes write 1. for þou tokest on out of hym. þan take þe neþerfigure, þat is 4, out of ten. And þen leues 6. cast to 6 þe figure of þat 2 þat stode vnderþe hedde of 1. þat wasborwed& rekened for ten, and þat wylle be 8. do away þat 6 & þat 2, & sette þere8, & lette þe neþerfigure stonde stille. Whanne þou hast do þus, go to þe next figure þat is now bot 1. but first yt was 2, & þere-of wasborred1.How to ‘Pay back’ the borrowed ten.þan take out of þat þe figure vnderhym, þat is 3. hit wel not be. þer-foreboroweof the next figure, þe quych is bot 1. Also take & sett hym ouerþe hede of þe figureþat þou woldest haue y-draw oute of þe nether figure, þe quych was 3. & þou myȝt not, & rekene þat borwed 1 for ten & sett in þe same place, of þe quych place þou tokest hymof, a cifer, for he was bot 1. Whanne þou hast þusydo, take out of þat 1. þat is rekent for ten, þe neþerfigure of 3. And þereleues 7.leaf 144b.*cast þe ylke 7 to þe figure þat had þe ylke ten vpon his hed, þe quych figure was 1, & þat wol be 8. þan do away þat 1 and þat 7, & write þere8. & þan wyrch forth in oþerfiguris til þou come to þe ende, & þan þou hast þe do. Versus.¶ Facque nonenarios de cifris, cumremeabis¶ Occurrant si forte cifre; dum dempseris vnum¶ Postea procedas reliquas demendo figuras.A very hard case is put.¶ Herehe puttesþe fourte case, þe quych is þis, yf it happe þat þe neþerfigure, þe quych þou schalt draw out of þe hier figure be morepan þe hier figur ouerhym, & þe next figure of two or of thre or of foure, or how mony þerebe by cifers, how wold þou do. Þouwostwel þou most nede borow, & þou mayst not borow of þe cifers, for þai haue noȝt þat þai mayleneor spare. Ergo4howwoldest þou do. Certayɳ þus most þou do, þou most borow on of þe next figure significatyf in þat rewe, for þis case may not happe, but yf þerecome figures significatyf afterthe cifers. Whan þou hast borowede þat 1 of the next figure significatyf, sett þat on ouerþe hede of þat figure of þe quych þou wold haue draw þe neþerfigure out yf þou hadest myȝt, & reken it for ten as þoudiddest inþe oþercasehere-a-fore. Whaɳ þou hast þus y-do loke how mony cifers þerewerebye-twene þat figuresignificatyf, & þe figure of þe quych þou woldest haue y-draw theleaf 145a.*neþerfigure, and of euery of þe ylke cifers make a figureof 9.Here is an example.lo an Ensampulleafter.4000210004Take 4 out of 2. it wel not be. borow 1 out of be next figure significatyf, þe quych is 4, & þen leues 3. do away þat figureof 4 & write þere3. & sett þat 1 vppon þe figure of 2 hede, & þan take 4 out of ten, & þan þere leues 6. Cast 6 to the figure of 2, þat wol be 8. do away þat 6 & write þere 8. Whan þou hast þus y-do make of euery 0 betweyn 3 & 8 a figure of 9, & þan worch forth in goddes name.Sic.3999810004& yf þou hast wel y-do þou5schalt haue þis nomberHow to prove the Subtraction.
Egerton2622.leaf 136a.HEc algorismusars presens dicitur; in quaTalibusindorumfruimurbis quinquefiguris.A derivation of Algorism.This boke is called þe boke ofalgorym, or Augrym afterlewdervse. And þis boke tretys þeCraftof Nombryng, þe quych crafte is called also Algorym. Ther was a kyng of Inde, þe quichheythAlgor, & he made þis craft. And afterhis name he called hit algorym;Another derivation of the word.or els anoþercause is quy it is called Algorym, for þe latyn word of hits.Algorismuscomesof Algos, grece, quidestars, latine, craft oɳ englis, and rides, quidestnumerus, latine, A nomburoɳ englys, inde dicitur Algorismusperaddicionemhuiussillabe mus& subtraccionem d & e, quasi ars numerandi. ¶ fforthermoreȝemostvndirstonde þat in þis craft ben vsid teen figurys, as here benewriten for ensampul, φ 9 8 7 6 5 4 3 2 1. ¶Exponeþe too versus afore: this present craft ys called Algorismus, in þe quych we vse teen signys of Inde. Questio. ¶ Why teɳ fyguris of Inde? Solucio. for as I haue sayd afore þai werefonde fyrst in Inde of a kyngeof þat Cuntre, þat was called Algor.Notation and Numeration.versus[in margin].¶ Prima significat unum; duo vero secunda:¶ Tercia significat tria; sic procede sinistre.¶ Donec ad extremamvenias, que cifra vocatur.¶ Capitulum primum de significacione figurarum.Expositioversus.In þis verse is notifide þe significacion of þese figuris. And þus expone the verse.The meaning and place of the figures.Þe first signifiyth one, þe secundeleaf 136b.signi*fiyth tweyne, þe thryd signifiyth thre, & the fourte signifiyth 4. ¶ And so forthe towarde þe lyft syde of þe tabul or of þe boke þat þe figures benewritenein, til þat þou come to the last figure, þat iscalled acifre. ¶ Questio. In quych syde sittes þe first figure? Solucio, forsothe loke quich figure is first in þe ryȝt side of þe bok or of þe tabul, & þat same is þe first figure, for þou schal write bakeward, as here, 3. 2. 6. 4. 1. 2. 5.Which figure is read first.The figure of 5. was firstwrite, & he is þe first, for he sittes oɳ þe riȝt syde. And the figure of 3 is last. ¶ Neuer-þe-les wen he says ¶ Prima significat vnum&c., þat is to say, þe first betokenes one, þe secunde. 2. & fore-þer-more, he vndirstondesnoȝtof þe first figure of eueryrew. ¶ But he vndirstondes þe first figure þat is in þe nomburof þe forsayd teen figuris, þe quych is oneof þese. 1. And þe secunde 2. & so forth.versus[in margin].¶ Quelibetillarumsi primo limite ponas,¶ Simpliciterse significat: si vero secundo,Se decies: sursumprocedasmultiplicando.¶ Namquefigura sequens quamuis signat decies plus.¶ Ipsa locata loco quam significat pertinente.Expositio[in margin].¶ Expone þis verse þus. Euery of þese figuris bitokens hym selfe & no more, yf he stonde in þe first place of þerewele/ this worde Simpliciterin þat verse it is no more to say but þat, & no more.An explanation of the principles of notation.¶ If it stonde in the secunde place of þerewle, he betokens tenetymes hym selfe, as þis figure2 here 20 tokens ten tyme hym selfe,leaf 137a.*þat is twenty, for he hym selfe betokenes tweyne, & ten tymestweneis twenty. And for he stondis oɳ þe lyft side & in þe secunde place, he betokens ten tyme hymselfe. And so go forth. ¶ ffor eueryfigure, & he stonde aftura-noþertoward the lyft side, he schal betokeneten tymes as mich moreas he schul betoken & he stode in þe place þereþat þe figurea-forehym stondes.An example:loo an ensampulle. 9. 6. 3. 4. Þe figureof 4. þat hase þis schape{4}. betokens bot hymselfe, for he stondes in þe first place.units,The figureof 3. þat hase þis schape{3}. betokens ten tymes moreþen he schuld&he stde þereþat þe figureof 4. stondes, þat is thretty.tens,The figureof 6, þat hase þis schape{6}, betokens ten tymes moreþan he schuld & he stode þereas þe figureof{3}. stondes, for þerehe schuld tokynebot sexty, & now he betokens ten tymes more, þat is sex hundryth.hundreds,The figureof 9. þat hase þis schape{9}. betokens ten tymes moreþanehe schuld & he stode in þe place þereþe figureof sex stondes, for þen he schuld betoken to 9. hundryth, and in þe place þerehe stondes now he betokens 9. þousande.thousands.Al þeholenomburis 9 thousande sex hundryth & foure& thretty. ¶ fforthermore, whenþou schalt rede a nomburof figure,How to read the number.þou schalt begyneat þe last figurein the lyft side, & rede so forth to þe riȝt side as here9. 6. 3. 4. Thou schal begyn to rede at þe figureof 9. & rede forth þus. 9.leaf 137b.*thousand sex hundryth thritty & foure. But when þou schallewrite, þou schalt be-gynne to write at þe ryȝt side.¶ Nil cifra significat seddat signaresequenti.The meaning and use of the cipher.Expone þis verse. A cifre tokensnoȝt, bot he makes þe figureto betoken þat comes afturhym moreþan he schuld & he wereaway, as þus 1φ. hereþe figureof onetokens ten, & yf þe cifre wereaway1& no figureby-forehym he schuld token bot one, for þan he schuld stonde in þe first place. ¶ And þe cifre tokens nothyng hym selfe. for al þe nomburof þeylketoo figures is bot ten. ¶ Questio. Why says he þat a cifre makys a figuretosignifye (tyf)more&c. ¶ I speke for þis wordesignificatyf, ffor sothe it may happe aftura cifre schuld come a-noþurcifre, as þus 2φφ. And ȝet þe secunde cifre shuld token neuerþe moreexcephe schuld kepe þe orderof þe place. and a cifre is no figuresignificatyf.¶ Quam precedentes plus ultima significabit /The last figure means more than all the others, since it is of the highest value.Expone þis verse þus. Þe last figureschal token moreþan alleþe oþerafore,thouȝtþerewerea hundryth thousant figures afore, as þus, 16798. Þe last figureþat is 1. betokens ten thousant. And alleþe oþerfigures ben bot betokenebot sex thousant seuynehundryth nynty & 8. ¶ And ten thousant is moreþen alleþat nombur,ergo þe last figuretokens moreþan all þe nomburafore.The Three Kinds of Numbersleaf 138a.* ¶ Post predicta scias breuiterquodtres numerorumDistincte species sunt; nam quidam digiti sunt;Articuli quidam; quidam quoquecompositi sunt.¶ Capitulum 2mde triplice divisione numerorum.¶ The auctor of þis tretisdepartysþis worde a nomburinto 3 partes. Some nomburis called digituslatine, adigitin englys.Digits.Somme nomburis called articuluslatine. AnArticulin englys.Articles.Some nomburis called acomposytin englys.Composites.¶ Expone þis verse. know þou afturþe forsaydrewlesþat I sayd afore, þat þereben threspicesof nombur. Ooneis a digit, Anoþeris an Articul, & þe toþera Composyt. versus.Digits, Articles, and Composites.¶ Sunt digiti numeri qui citradenariumsunt.What are digits.¶ Herehe telles qwat is a digit, Expone versussic. Nomburs digitus beneallenomburs þat ben with-inne ten, as nyne, 8. 7. 6. 5. 4. 3. 2. 1.¶ Articupli decupli degitorum; compositi suntIlli qui constant ex articulis degitisque.¶ Herehe telles what is a composyt and what is anearticul. Expone sic versus.What are articles.¶ Articulis ben2alleþat may be deuidyt into nombursof ten & nothyngeleueouer, as twenty, thretty, fourty, a hundryth, a thousand, & such oþer, ffor twenty may be departyt in-to 2 nomburs of ten, fforty in to fourenomburs of ten, & so forth.leaf 138b.What numbers are composites.*Compositys beɳ nomburs þat bene componyt of a digyt & of an articulleas fouretene, fyftene, sextene, & such oþer. ffortene is componyd of foureþat is a digit & of ten þat is an articulle. ffiftene is componyd of 5 & ten, & so of all oþer, what þat þai ben. Short-lych euery nomburþat be-gynnes witha digit & endyth in a articulleis a composyt, as fortene bygennyngeby foureþat is a digit, & endes in ten.¶ Ergo, proposito numero tibi scribere, primoRespicias quid sit numerus; si digitus sitPrimo scribe loco digitum, si compositus sitPrimo scribe loco digitumpost articulum; sic.How to write a number,¶ here he telles how þou schaltwyrchwhan þou schalt write a nombur. Expone versumsic, & fac iuxta exponentis sentenciam; whan þou hast a nomburto write, loke fyrst what manernomburit ys þat þou schalt write, whether it be a digit or a composit or an Articul.if it is a digit;¶ If he be a digit, write a digit, as yf it be seuen, write seuen & write þat digit in þe first place toward þe ryght side.if it is a composite.If it be a composyt, write þe digit of þe composit in þe first place & write þe articul of þat digit in þe secunde place next toward þe lyft side. As yf þou schal write sex & twenty. write þe digit of þe nomburin þe first place þat is sex, and write þe articul next afturþat is twenty, as þus 26.How to read it.But whan þou schaltsowneor spekeleaf 139a.*or rede an Composyt þou schalt first sowne þe articul & afturþe digit, as þou seyst by þe comynespeche, Sex & twenty & nouȝt twenty & sex. versus.¶ Articulussi sit, in primo limite cifram,Articulum veroreliquis inscribe figuris.How to write Articles:¶ Here he tells how þou schal write when þe nombre þat þou hase to write is an Articul. Expone versus sic & fac secundum sentenciam. Ife þe nomburþat þou hast write be an Articul, write first a cifre & afturþe cifer write an Articulleþus. 2φ.tens,fforthermoreþou schalt vndirstonde yf þou haue an Articul, loke howmych he is, yf he be with-ynne an hundryth, þou schalt write bot onecifre, afore, as here.9φ.hundreds,If þe articullebe by hym-silfe & be an hundrid euene, þen schal þou write .1. & 2 cifers afore, þat he may stonde in þe thryd place, for euery figurein þe thryd place schal token a hundrid tymes hym selfe.thousands, &c.If þe articul be a thousant or thousandes3and he stonde by hymselfe, write afore3 cifers & so forþ of al oþer.¶ Quolibetin numero, si par sit prima figura,Par erit & totum, quicquid sibi continuatur;Imparsi fuerit, totumtunc fietetimpar.To tell an even number¶ Herehe teches a generallerewle þat yf þe first figurein þerewleof figures token a nomburþat is eueneal þat nomburof figurys in þat rewle schal be euene, as hereþou may see 6. 7. 3. 5. 4. Computa & proba.or an odd.¶ If þe firstleaf 139b.*figure token an nomburþat is ode, alleþat nomburin þat rewle schallebe ode, as here5 6 7 8 6 7. Computa & proba. versus.¶ Septemsunt partes, nonplures, istius artis;¶ Addere, subtrahere, duplare, dimidiare,Sextaquediuidere, sedquinta multiplicare;Radicemextraherepars septima dicitur esse.The Seven Rules of Arithmetic.The seven rules.¶ Heretelles þat þerbeɳ .7. spices or partes of þis craft. The first is called addicioñ, þe secunde is called subtraccioñ. The thryd is calledduplacioñ. The 4. is calleddimydicioñ. The 5. is called multiplicacioñ. The 6 is called diuisioñ. The 7. is called extraccioñ of þeRote. What all þese spices benehit schallebe tolde singillatiminherecaputule.¶ Subtrahis aut addis a dextris vel mediabis:Add, subtract, or halve, from right to left.Thou schal be-gynne in þe ryght side of þe boke or of a tabul. loke wereþou wul be-gynne to write latyn or englys in a boke, & þat schallebe called þe lyft side of the boke, þat þou writest toward þat side schal be called þe ryght side of þe boke. Versus.A leua dupla, diuide, multiplica.Here he telles þe in quych side of þe boke or of þe tabul þou schallebe-gyneto wyrch duplacioñ, diuisioñ, and multiplicacioñ.Multiply or divide from left to right.Thou schal begynetoworchin þe lyft side of þe boke or of þe tabul, but yn what wyse þou schal wyrch in hymdicetursingillatimin sequentibuscapituliset de vtilitate cuiuslibetartis& sic Completurleaf140.*prohemium& sequiturtractatus& primo de arte addicionis que prima ars est in ordine.The Craft of Addition.Adderesi numero numerumvis, ordine taliIncipe; scribe duas primo series numerorumPrimamsub prima recte ponendo figuram,Et sic de reliquis facias, si sint tibi plures.Four things must be known:¶ Hereby-gynnes þe craft of Addicioñ. In þis craft þou most knowe foure thynges. ¶ Fyrst þoumost know what is addicioñ. Next þou most know how mony rewles of figurys þou most haue. ¶ Next þou most know how mony diuers casys happes in þis craft of addicioñ. ¶ And next qwat is þe profet of þis craft.what it is;¶ As for þe first þou most know þat addicioñ is acastyngto-gedurof twoo nomburys in-to onenombre. As yf I aske qwat istwene& thre. Þou wylcastþese twene nombres to-gedur& say þat it is fyue.how many rows of figures;¶ As for þe secunde þou most know þat þou schallehaue tweyne rewes of figures, onevndur a-nother, as hereþou mayst se.12342168.how many cases;¶ As for þe thryd þou most know þat thereben foure diuerse cases.what is its result.As for þe forthe þou most know þat þe profet of þis craft is to telle what is þe hole nomburþat comes of diuerse nomburis. Now as to þe texte of oure verse, he teches therehow þou schalworchin þis craft. ¶ He says yf þou wilt cast onenomburto anoþernombur, þou most by-gynne on þis wyse.How to set down the sum.¶ ffyrst writeleaf 140b.*two rewes of figuris & nombris so þat þou write þe first figureof þe hyer nombureuenevndirthe first figureof þe nether nombur,123234.And þe secunde of þe nether nombureuenevndirþe secunde of þe hyer, & so forthe of euery figure of both þe rewes as þou mayst se.The Cases of the Craft of Addition.¶ Inde duas adde primas hac condicione:Si digitus crescat ex addicione priorum;Primo scribe loco digitum, quicunquesit ille.¶ Here he teches what þou schalt do when þou hast write too rewes of figurisonvnder an-oþer, as I sayd be-fore.Add the first figures;¶ He says þou schalt take þe first figure of þe heyer nombre& þe fyrst figureof þe neþernombre, & cast hem to-gedervp-on þis condicioɳ. Thou schal loke qweþerþe nomberþat comys þere-of be a digit or no.rub out the top figure;¶ If he be a digit þou schalt do away þe first figure of þe hyer nombre, and write þerein his stede þat he stode Inne þe digit, þat comes of þe ylke 2 figures, & sowrite the result in its place.wrichforth oɳ oþerfigures yf þerebe ony moo, til þou come to þe ende toward þe lyft side. Andledeþe nether figurestonde still euer-moretil þou haueydo. ffor þere-by þou schalwytewheþerþou hast donewel or no, as I schal tell þe afterward in þe ende of þis Chapter. ¶ And lokeallgateleaf 141a.þat þou be-gynne to worch in þis Craft of Addi*cioɳ in þe ryȝt side,Here is an example.here is an ensampul of þis case.12342142.Caste 2 to foure& þat wel be sex, do away 4. & write in þe same place þe figure of sex. ¶ And lete þe figure of 2 in þe nether rewe stonde stil. When þou hast do so, cast 3 & 4 to-gedurand þat wel be seuen þat is a digit. Do away þe 3, & set þereseueɳ, and lete þe neþerfigurestonde stille, & soworchforth bakward til þou hastydoall to-geder.Et si compositus, in limite scribe sequenteArticulum, primo digitum; quiasic iubet ordo.¶ Here is þe secunde case þat may happe in þis craft. And þe case is þis,Suppose it is a Composite, set down the digit, and carry the tens.yf of þe casting of 2 nomburis to-geder, as of þe figure of þe hyer rewe & of þe figure of þe neþerrewe come a Composyt, how schalt þouworch. Þusþou schalt worch. Thou shalt do away þe figure of þe hyer nomberþat was cast to þe figure of þe neþernomber. ¶ And write þereþe digit of þe Composyt. And set þe articul of þe composit next afterþe digit in þe same rewe, yf þerebe nomofigures after. But yf þerebe mo figuris afterþat digit. And þere he schall be rekend for hym selfe. And when þou schalt adde þat ylke figure þat berys þe articulleouerhis hed to þe figurevnderhym, þou schalt cast þat articul to þe figure þat hase hym ouerhis hed, & þereþat Articul schal tokeɳ hym selfe.Here is an example.lo an Ensampullleaf 141b.*of all.326216.Cast 6 to 6, & þere-of wil arise twelue. do away þe hyer 6 & write þere2, þat is þe digit of þis composit. And þenwrite þe articulleþat is ten ouerþe figuris hed of twene as þus.1322216.Now cast þe articulleþat standus vpon þe figuris of twene hed to þe same figure, & reken þat articul bot for one, and þan þerewil arise thre. Þan cast þat thre to þe neþerfigure, þat is one, & þat wul be foure. do away þe figure of 3, and write þerea figure of foure. and lete þe neþerfigure stonde stil, & þan worch forth. vndeversus.¶ Articulus si sit, in primo limite cifram,¶ Articulumvero reliquis inscribe figuris,Vel perse scribas si nulla figura sequatur.¶ Herehe puttes þe thryde case of þe craft of Addicioɳ. & þe case is þis.Suppose it is an Article, set down a cipher and carry the tens.yf of Addiciouɳ of 2 figuris a-ryse an Articulle, how schal þou do. thou most do away þeheerfigure þat was addid to þe neþer, & write þerea cifre, and sett þearticulson þe figuris hede, yf þat þerecome ony after. And wyrch þan as I haue tolde þe in þe secunde case. An ensampull.25.15Cast 5 to 5, þat wylle be ten. now do away þe hyer 5, & write þerea cifer. And sette ten vpon þe figuris hed of 2. And reken it but for on þus. loHere is an example.an Ensampulle12φ15Andleaf 142a.*þan worch forth. But yf þerecome no figure afterþe cifre, write þe articul next hym in þe same rewe as here55cast 5 to 5, and it wel be ten. do away 5. þat is þe hier 5. and write þerea cifre, & write afterhym þe articul as þus1φ5And þan þou hast done.¶ Si tibi cifra superueniens occurrerit, illamDele superpositam; fac illic scribe figuram,Postea procedas reliquas addendo figuras.What to do when you have a cipher in the top row.¶ Herehe puttesþe fourt case, & it is þis, þat yf þerecome a cifer in þe hier rewe, how þou schal do. þus þou schalt do. do away þe cifer, & sett þereþe digit þat comes of þe addiciounas þus1φφ84.17743An example of all the difficulties.In þis ensampul ben alleþe fourecases. Cast 3 to foure, þat wol be seueɳ. do away 4. & write þereseueɳ; þan cast 4 to þe figureof 8. þat wel be 12. do away 8, & sett þere2. þat is a digit, and sette þe articul of þe composit, þat is ten, vpon þe cifers hed, & reken it for hym selfe þat is on. þan cast oneto a cifer, & hit wullebe but on, for noȝt & on makes but one. þan cast 7. þat stondes vnderþat on to hym, & þat wel be 8. do away þe cifer & þat 1. & sette þere8. þan go forthermore. cast þe oþer7 to þe cifer þat stondes ouerhym. þat wul be bot seuen, for þe cifer betokens noȝt. do away þe cifer & sette þereseueɳ,leaf 142b.*& þen go forþermore& cast 1 to 1, & þat wel be 2. do away þe hier 1, & sette þere2. þan hast þou do. And yf þou haue wel ydo þis nomber þat is sett here-afterwel be þe nomber þat schallearyse of alleþe addicioɳ as here27827. ¶ Sequitur alia species.The Craft of Subtraction.A numero numerumsi sit tibi demerecuraScribe figurarumseries, vt in addicione.Four things to know about subtraction:¶ This is þe Chapterof subtraccioɳ, in the quych þou most know foure nessessary thynges. the first what is subtraccioɳ. þe secunde is how mony nombers þou most haue to subtraccioɳ, the thryd is how mony maners of cases þeremay happe in þis craft of subtraccioɳ. The fourte is qwat is þe profet of þis craft. ¶ As forthe first;þe first, þou most know þat subtraccioɳ is drawyngeof onenowmberoute of anoþernomber.the second;As for þe secunde, þou most knowe þat þou most haue two rewes of figuris onevnderanoþer, as þouaddystin addicioɳ.the third;As for þe thryd, þou moyst know þat fouremanerof diuerse casis mai happe in þis craft.the fourth.¶ As for þe fourt, þou most know þat þe profet of þis craft is whenne þou hasse taken þe lasse nomber out of þe moreto telle what þereleuesouerþat. & þou most be-gynne to wyrch in þiscraft in þe ryght side of þe boke, as þou diddyst in addicioɳ. Versus.¶ Maiori numero numerumsuppone minorem,¶ Siue pari numero supponaturnumerus par.leaf 143a.* ¶ Herehe telles þatPut the greater number above the less.þe hier nomber most be moreþen þe neþer, or els eueɳ as mych. but he may not be lasse. And þe case is þis, þou schalt drawe þe neþernomber out of þe hyer, & þou mayst not do þat yf þe hier nomber werelasse þan þat. ffor þou mayst not draw sex out of 2. But þou mast draw 2 out of sex. And þou maiste draw twene out of twene, for þou schal leue noȝt of þe hier twene vndeversus.The Cases of the Craft of Subtraction.¶ Postea si possis a prima subtrahe primamScribens quod remanet.The first case of subtraction.Hereis þe first case put of subtraccioɳ, & he says þou schalt begynne in þe ryght side, & draw þe first figureof þe neþerrewe out of þe first figureof þe hier rewe. qwether þe hier figure be moreþen þe neþer, or eueɳ as mych. And þat is notified in þe vers when he says “Si possis.” Whan þou has þus ydo, do away þe hiest figure & sett þereþatleuesof þe subtraccioɳ,Here is an example.lo an Ensampulle234122draw 2 out of 4. þan leues 2. do away 4 & write þere2, & latte þe neþerfigurestonde stille, & so gofor-byoþerfiguris till þou come to þe ende, þan hast þou do.¶ Cifram si nil remanebit.Put a cipher if nothing remains.¶ Herehe puttesþe secunde case, & hit is þis. yf it happe þat qwen þou hast draw on neþerfigureout of a hier, & þere leue noȝt afterþe subtraccioɳ, þusleaf 143b.*þou schalt do. þou schalledo away þe hier figure & write þerea cifer, asHere is an example.lo an Ensampull2424Take foureout of foureþanleusnoȝt. þereforedo away þe hier 4 & set þerea cifer, þan take 2 out of 2, þan leues noȝt. do away þe hier 2, & set þerea cifer, and so worch whareso euerþis happe.Sed si nonpossis a prima demere primamPrecedens vnumde limite deme sequente,Quod demptumprodenario reputabis ab illoSubtrahe totalem numerumquemproposuistiQuo facto scribe super quicquid remanebit.Suppose you cannot take the lower figure from the top one, borrow ten;Herehe puttes þe thryd case, þe quych is þis. yf it happe þat þe neþerfigure be moreþen þe hier figure þat he schallebe draw out of. how schalleþou do. þus þou schalledo. þou schalleborro.1. oute of þe next figure þat comes afterin þe same rewe, for þis case may neuerhapp but yf þerecome figures after. þan þou schalt settþat on ouerþe hier figureshed, of the quych þou woldist y-draw oute þe neyþerfigure yf þou haddysty-myȝt. Whane þou hase þus ydo þou schallerekene þat .1. for ten.take the lower number from ten;¶. And out of þat ten þou schal draw þe neyþermost figure, And alleþat leues þou schalleadd the answer to the top number.adde to þe figureon whos hed þat .1. stode. And þen þou schalledo away alleþat, & sett þereallethat arisys of the addicioɳ of þe ylke 2 figuris. And yf ytleaf 144a.*happe þat þe figure of þe quych þou schalt borro on be hym self but 1. If þou schalt þat one& sett it vppoɳ þe oþerfigurished, and sett in þat 1. place a cifer, yf þerecome mony figuresafter.Example.lo an Ensampul.21221134take 4 out of 2. it wyl not be, þerforeborro oneof þe next figure, þat is 2. and sett þat ouerþe hed of þe fyrst 2. & rekene it for ten. and þere þe secunde stondes write 1. for þou tokest on out of hym. þan take þe neþerfigure, þat is 4, out of ten. And þen leues 6. cast to 6 þe figure of þat 2 þat stode vnderþe hedde of 1. þat wasborwed& rekened for ten, and þat wylle be 8. do away þat 6 & þat 2, & sette þere8, & lette þe neþerfigure stonde stille. Whanne þou hast do þus, go to þe next figure þat is now bot 1. but first yt was 2, & þere-of wasborred1.How to ‘Pay back’ the borrowed ten.þan take out of þat þe figure vnderhym, þat is 3. hit wel not be. þer-foreboroweof the next figure, þe quych is bot 1. Also take & sett hym ouerþe hede of þe figureþat þou woldest haue y-draw oute of þe nether figure, þe quych was 3. & þou myȝt not, & rekene þat borwed 1 for ten & sett in þe same place, of þe quych place þou tokest hymof, a cifer, for he was bot 1. Whanne þou hast þusydo, take out of þat 1. þat is rekent for ten, þe neþerfigure of 3. And þereleues 7.leaf 144b.*cast þe ylke 7 to þe figure þat had þe ylke ten vpon his hed, þe quych figure was 1, & þat wol be 8. þan do away þat 1 and þat 7, & write þere8. & þan wyrch forth in oþerfiguris til þou come to þe ende, & þan þou hast þe do. Versus.¶ Facque nonenarios de cifris, cumremeabis¶ Occurrant si forte cifre; dum dempseris vnum¶ Postea procedas reliquas demendo figuras.A very hard case is put.¶ Herehe puttesþe fourte case, þe quych is þis, yf it happe þat þe neþerfigure, þe quych þou schalt draw out of þe hier figure be morepan þe hier figur ouerhym, & þe next figure of two or of thre or of foure, or how mony þerebe by cifers, how wold þou do. Þouwostwel þou most nede borow, & þou mayst not borow of þe cifers, for þai haue noȝt þat þai mayleneor spare. Ergo4howwoldest þou do. Certayɳ þus most þou do, þou most borow on of þe next figure significatyf in þat rewe, for þis case may not happe, but yf þerecome figures significatyf afterthe cifers. Whan þou hast borowede þat 1 of the next figure significatyf, sett þat on ouerþe hede of þat figure of þe quych þou wold haue draw þe neþerfigure out yf þou hadest myȝt, & reken it for ten as þoudiddest inþe oþercasehere-a-fore. Whaɳ þou hast þus y-do loke how mony cifers þerewerebye-twene þat figuresignificatyf, & þe figure of þe quych þou woldest haue y-draw theleaf 145a.*neþerfigure, and of euery of þe ylke cifers make a figureof 9.Here is an example.lo an Ensampulleafter.4000210004Take 4 out of 2. it wel not be. borow 1 out of be next figure significatyf, þe quych is 4, & þen leues 3. do away þat figureof 4 & write þere3. & sett þat 1 vppon þe figure of 2 hede, & þan take 4 out of ten, & þan þere leues 6. Cast 6 to the figure of 2, þat wol be 8. do away þat 6 & write þere 8. Whan þou hast þus y-do make of euery 0 betweyn 3 & 8 a figure of 9, & þan worch forth in goddes name.Sic.3999810004& yf þou hast wel y-do þou5schalt haue þis nomberHow to prove the Subtraction.
Egerton2622.
HEc algorismusars presens dicitur; in quaTalibusindorumfruimurbis quinquefiguris.
HEc algorismusars presens dicitur; in qua
Talibusindorumfruimurbis quinquefiguris.
A derivation of Algorism.This boke is called þe boke ofalgorym, or Augrym afterlewdervse. And þis boke tretys þeCraftof Nombryng, þe quych crafte is called also Algorym. Ther was a kyng of Inde, þe quichheythAlgor, & he made þis craft. And afterhis name he called hit algorym;Another derivation of the word.or els anoþercause is quy it is called Algorym, for þe latyn word of hits.Algorismuscomesof Algos, grece, quidestars, latine, craft oɳ englis, and rides, quidestnumerus, latine, A nomburoɳ englys, inde dicitur Algorismusperaddicionemhuiussillabe mus& subtraccionem d & e, quasi ars numerandi. ¶ fforthermoreȝemostvndirstonde þat in þis craft ben vsid teen figurys, as here benewriten for ensampul, φ 9 8 7 6 5 4 3 2 1. ¶Exponeþe too versus afore: this present craft ys called Algorismus, in þe quych we vse teen signys of Inde. Questio. ¶ Why teɳ fyguris of Inde? Solucio. for as I haue sayd afore þai werefonde fyrst in Inde of a kyngeof þat Cuntre, þat was called Algor.
Notation and Numeration.
¶ Prima significat unum; duo vero secunda:¶ Tercia significat tria; sic procede sinistre.¶ Donec ad extremamvenias, que cifra vocatur.
¶ Prima significat unum; duo vero secunda:
¶ Tercia significat tria; sic procede sinistre.
¶ Donec ad extremamvenias, que cifra vocatur.
Expositioversus.In þis verse is notifide þe significacion of þese figuris. And þus expone the verse.The meaning and place of the figures.Þe first signifiyth one, þe secundeleaf 136b.signi*fiyth tweyne, þe thryd signifiyth thre, & the fourte signifiyth 4. ¶ And so forthe towarde þe lyft syde of þe tabul or of þe boke þat þe figures benewritenein, til þat þou come to the last figure, þat iscalled acifre. ¶ Questio. In quych syde sittes þe first figure? Solucio, forsothe loke quich figure is first in þe ryȝt side of þe bok or of þe tabul, & þat same is þe first figure, for þou schal write bakeward, as here, 3. 2. 6. 4. 1. 2. 5.Which figure is read first.The figure of 5. was firstwrite, & he is þe first, for he sittes oɳ þe riȝt syde. And the figure of 3 is last. ¶ Neuer-þe-les wen he says ¶ Prima significat vnum&c., þat is to say, þe first betokenes one, þe secunde. 2. & fore-þer-more, he vndirstondesnoȝtof þe first figure of eueryrew. ¶ But he vndirstondes þe first figure þat is in þe nomburof þe forsayd teen figuris, þe quych is oneof þese. 1. And þe secunde 2. & so forth.
¶ Quelibetillarumsi primo limite ponas,¶ Simpliciterse significat: si vero secundo,Se decies: sursumprocedasmultiplicando.¶ Namquefigura sequens quamuis signat decies plus.¶ Ipsa locata loco quam significat pertinente.
¶ Quelibetillarumsi primo limite ponas,
¶ Simpliciterse significat: si vero secundo,
Se decies: sursumprocedasmultiplicando.
¶ Namquefigura sequens quamuis signat decies plus.
¶ Ipsa locata loco quam significat pertinente.
Expositio[in margin].¶ Expone þis verse þus. Euery of þese figuris bitokens hym selfe & no more, yf he stonde in þe first place of þerewele/ this worde Simpliciterin þat verse it is no more to say but þat, & no more.An explanation of the principles of notation.¶ If it stonde in the secunde place of þerewle, he betokens tenetymes hym selfe, as þis figure2 here 20 tokens ten tyme hym selfe,leaf 137a.*þat is twenty, for he hym selfe betokenes tweyne, & ten tymestweneis twenty. And for he stondis oɳ þe lyft side & in þe secunde place, he betokens ten tyme hymselfe. And so go forth. ¶ ffor eueryfigure, & he stonde aftura-noþertoward the lyft side, he schal betokeneten tymes as mich moreas he schul betoken & he stode in þe place þereþat þe figurea-forehym stondes.An example:loo an ensampulle. 9. 6. 3. 4. Þe figureof 4. þat hase þis schape{4}. betokens bot hymselfe, for he stondes in þe first place.units,The figureof 3. þat hase þis schape{3}. betokens ten tymes moreþen he schuld&he stde þereþat þe figureof 4. stondes, þat is thretty.tens,The figureof 6, þat hase þis schape{6}, betokens ten tymes moreþan he schuld & he stode þereas þe figureof{3}. stondes, for þerehe schuld tokynebot sexty, & now he betokens ten tymes more, þat is sex hundryth.hundreds,The figureof 9. þat hase þis schape{9}. betokens ten tymes moreþanehe schuld & he stode in þe place þereþe figureof sex stondes, for þen he schuld betoken to 9. hundryth, and in þe place þerehe stondes now he betokens 9. þousande.thousands.Al þeholenomburis 9 thousande sex hundryth & foure& thretty. ¶ fforthermore, whenþou schalt rede a nomburof figure,How to read the number.þou schalt begyneat þe last figurein the lyft side, & rede so forth to þe riȝt side as here9. 6. 3. 4. Thou schal begyn to rede at þe figureof 9. & rede forth þus. 9.leaf 137b.*thousand sex hundryth thritty & foure. But when þou schallewrite, þou schalt be-gynne to write at þe ryȝt side.
¶ Nil cifra significat seddat signaresequenti.
¶ Nil cifra significat seddat signaresequenti.
The meaning and use of the cipher.Expone þis verse. A cifre tokensnoȝt, bot he makes þe figureto betoken þat comes afturhym moreþan he schuld & he wereaway, as þus 1φ. hereþe figureof onetokens ten, & yf þe cifre wereaway1& no figureby-forehym he schuld token bot one, for þan he schuld stonde in þe first place. ¶ And þe cifre tokens nothyng hym selfe. for al þe nomburof þeylketoo figures is bot ten. ¶ Questio. Why says he þat a cifre makys a figuretosignifye (tyf)more&c. ¶ I speke for þis wordesignificatyf, ffor sothe it may happe aftura cifre schuld come a-noþurcifre, as þus 2φφ. And ȝet þe secunde cifre shuld token neuerþe moreexcephe schuld kepe þe orderof þe place. and a cifre is no figuresignificatyf.
¶ Quam precedentes plus ultima significabit /
¶ Quam precedentes plus ultima significabit /
The last figure means more than all the others, since it is of the highest value.Expone þis verse þus. Þe last figureschal token moreþan alleþe oþerafore,thouȝtþerewerea hundryth thousant figures afore, as þus, 16798. Þe last figureþat is 1. betokens ten thousant. And alleþe oþerfigures ben bot betokenebot sex thousant seuynehundryth nynty & 8. ¶ And ten thousant is moreþen alleþat nombur,ergo þe last figuretokens moreþan all þe nomburafore.
The Three Kinds of Numbers
leaf 138a.* ¶ Post predicta scias breuiterquodtres numerorumDistincte species sunt; nam quidam digiti sunt;Articuli quidam; quidam quoquecompositi sunt.
* ¶ Post predicta scias breuiterquodtres numerorum
Distincte species sunt; nam quidam digiti sunt;
Articuli quidam; quidam quoquecompositi sunt.
¶ The auctor of þis tretisdepartysþis worde a nomburinto 3 partes. Some nomburis called digituslatine, adigitin englys.Digits.Somme nomburis called articuluslatine. AnArticulin englys.Articles.Some nomburis called acomposytin englys.Composites.¶ Expone þis verse. know þou afturþe forsaydrewlesþat I sayd afore, þat þereben threspicesof nombur. Ooneis a digit, Anoþeris an Articul, & þe toþera Composyt. versus.
Digits, Articles, and Composites.
¶ Sunt digiti numeri qui citradenariumsunt.
¶ Sunt digiti numeri qui citradenariumsunt.
What are digits.¶ Herehe telles qwat is a digit, Expone versussic. Nomburs digitus beneallenomburs þat ben with-inne ten, as nyne, 8. 7. 6. 5. 4. 3. 2. 1.
¶ Articupli decupli degitorum; compositi suntIlli qui constant ex articulis degitisque.
¶ Articupli decupli degitorum; compositi sunt
Illi qui constant ex articulis degitisque.
¶ Herehe telles what is a composyt and what is anearticul. Expone sic versus.What are articles.¶ Articulis ben2alleþat may be deuidyt into nombursof ten & nothyngeleueouer, as twenty, thretty, fourty, a hundryth, a thousand, & such oþer, ffor twenty may be departyt in-to 2 nomburs of ten, fforty in to fourenomburs of ten, & so forth.
leaf 138b.What numbers are composites.*Compositys beɳ nomburs þat bene componyt of a digyt & of an articulleas fouretene, fyftene, sextene, & such oþer. ffortene is componyd of foureþat is a digit & of ten þat is an articulle. ffiftene is componyd of 5 & ten, & so of all oþer, what þat þai ben. Short-lych euery nomburþat be-gynnes witha digit & endyth in a articulleis a composyt, as fortene bygennyngeby foureþat is a digit, & endes in ten.
¶ Ergo, proposito numero tibi scribere, primoRespicias quid sit numerus; si digitus sitPrimo scribe loco digitum, si compositus sitPrimo scribe loco digitumpost articulum; sic.
¶ Ergo, proposito numero tibi scribere, primo
Respicias quid sit numerus; si digitus sit
Primo scribe loco digitum, si compositus sit
Primo scribe loco digitumpost articulum; sic.
How to write a number,¶ here he telles how þou schaltwyrchwhan þou schalt write a nombur. Expone versumsic, & fac iuxta exponentis sentenciam; whan þou hast a nomburto write, loke fyrst what manernomburit ys þat þou schalt write, whether it be a digit or a composit or an Articul.if it is a digit;¶ If he be a digit, write a digit, as yf it be seuen, write seuen & write þat digit in þe first place toward þe ryght side.if it is a composite.If it be a composyt, write þe digit of þe composit in þe first place & write þe articul of þat digit in þe secunde place next toward þe lyft side. As yf þou schal write sex & twenty. write þe digit of þe nomburin þe first place þat is sex, and write þe articul next afturþat is twenty, as þus 26.How to read it.But whan þou schaltsowneor spekeleaf 139a.*or rede an Composyt þou schalt first sowne þe articul & afturþe digit, as þou seyst by þe comynespeche, Sex & twenty & nouȝt twenty & sex. versus.
¶ Articulussi sit, in primo limite cifram,Articulum veroreliquis inscribe figuris.
¶ Articulussi sit, in primo limite cifram,
Articulum veroreliquis inscribe figuris.
How to write Articles:¶ Here he tells how þou schal write when þe nombre þat þou hase to write is an Articul. Expone versus sic & fac secundum sentenciam. Ife þe nomburþat þou hast write be an Articul, write first a cifre & afturþe cifer write an Articulleþus. 2φ.tens,fforthermoreþou schalt vndirstonde yf þou haue an Articul, loke howmych he is, yf he be with-ynne an hundryth, þou schalt write bot onecifre, afore, as here.9φ.hundreds,If þe articullebe by hym-silfe & be an hundrid euene, þen schal þou write .1. & 2 cifers afore, þat he may stonde in þe thryd place, for euery figurein þe thryd place schal token a hundrid tymes hym selfe.thousands, &c.If þe articul be a thousant or thousandes3and he stonde by hymselfe, write afore3 cifers & so forþ of al oþer.
¶ Quolibetin numero, si par sit prima figura,Par erit & totum, quicquid sibi continuatur;Imparsi fuerit, totumtunc fietetimpar.
¶ Quolibetin numero, si par sit prima figura,
Par erit & totum, quicquid sibi continuatur;
Imparsi fuerit, totumtunc fietetimpar.
To tell an even number¶ Herehe teches a generallerewle þat yf þe first figurein þerewleof figures token a nomburþat is eueneal þat nomburof figurys in þat rewle schal be euene, as hereþou may see 6. 7. 3. 5. 4. Computa & proba.or an odd.¶ If þe firstleaf 139b.*figure token an nomburþat is ode, alleþat nomburin þat rewle schallebe ode, as here5 6 7 8 6 7. Computa & proba. versus.
¶ Septemsunt partes, nonplures, istius artis;¶ Addere, subtrahere, duplare, dimidiare,Sextaquediuidere, sedquinta multiplicare;Radicemextraherepars septima dicitur esse.
¶ Septemsunt partes, nonplures, istius artis;
¶ Addere, subtrahere, duplare, dimidiare,
Sextaquediuidere, sedquinta multiplicare;
Radicemextraherepars septima dicitur esse.
The Seven Rules of Arithmetic.
The seven rules.¶ Heretelles þat þerbeɳ .7. spices or partes of þis craft. The first is called addicioñ, þe secunde is called subtraccioñ. The thryd is calledduplacioñ. The 4. is calleddimydicioñ. The 5. is called multiplicacioñ. The 6 is called diuisioñ. The 7. is called extraccioñ of þeRote. What all þese spices benehit schallebe tolde singillatiminherecaputule.
¶ Subtrahis aut addis a dextris vel mediabis:
¶ Subtrahis aut addis a dextris vel mediabis:
Add, subtract, or halve, from right to left.Thou schal be-gynne in þe ryght side of þe boke or of a tabul. loke wereþou wul be-gynne to write latyn or englys in a boke, & þat schallebe called þe lyft side of the boke, þat þou writest toward þat side schal be called þe ryght side of þe boke. Versus.
A leua dupla, diuide, multiplica.
A leua dupla, diuide, multiplica.
Here he telles þe in quych side of þe boke or of þe tabul þou schallebe-gyneto wyrch duplacioñ, diuisioñ, and multiplicacioñ.Multiply or divide from left to right.Thou schal begynetoworchin þe lyft side of þe boke or of þe tabul, but yn what wyse þou schal wyrch in hymdicetursingillatimin sequentibuscapituliset de vtilitate cuiuslibetartis& sic Completurleaf140.*prohemium& sequiturtractatus& primo de arte addicionis que prima ars est in ordine.
The Craft of Addition.
Adderesi numero numerumvis, ordine taliIncipe; scribe duas primo series numerorumPrimamsub prima recte ponendo figuram,Et sic de reliquis facias, si sint tibi plures.
Adderesi numero numerumvis, ordine tali
Incipe; scribe duas primo series numerorum
Primamsub prima recte ponendo figuram,
Et sic de reliquis facias, si sint tibi plures.
Four things must be known:¶ Hereby-gynnes þe craft of Addicioñ. In þis craft þou most knowe foure thynges. ¶ Fyrst þoumost know what is addicioñ. Next þou most know how mony rewles of figurys þou most haue. ¶ Next þou most know how mony diuers casys happes in þis craft of addicioñ. ¶ And next qwat is þe profet of þis craft.what it is;¶ As for þe first þou most know þat addicioñ is acastyngto-gedurof twoo nomburys in-to onenombre. As yf I aske qwat istwene& thre. Þou wylcastþese twene nombres to-gedur& say þat it is fyue.how many rows of figures;¶ As for þe secunde þou most know þat þou schallehaue tweyne rewes of figures, onevndur a-nother, as hereþou mayst se.12342168.how many cases;¶ As for þe thryd þou most know þat thereben foure diuerse cases.what is its result.As for þe forthe þou most know þat þe profet of þis craft is to telle what is þe hole nomburþat comes of diuerse nomburis. Now as to þe texte of oure verse, he teches therehow þou schalworchin þis craft. ¶ He says yf þou wilt cast onenomburto anoþernombur, þou most by-gynne on þis wyse.How to set down the sum.¶ ffyrst writeleaf 140b.*two rewes of figuris & nombris so þat þou write þe first figureof þe hyer nombureuenevndirthe first figureof þe nether nombur,123234.And þe secunde of þe nether nombureuenevndirþe secunde of þe hyer, & so forthe of euery figure of both þe rewes as þou mayst se.
The Cases of the Craft of Addition.
¶ Inde duas adde primas hac condicione:Si digitus crescat ex addicione priorum;Primo scribe loco digitum, quicunquesit ille.
¶ Inde duas adde primas hac condicione:
Si digitus crescat ex addicione priorum;
Primo scribe loco digitum, quicunquesit ille.
¶ Here he teches what þou schalt do when þou hast write too rewes of figurisonvnder an-oþer, as I sayd be-fore.Add the first figures;¶ He says þou schalt take þe first figure of þe heyer nombre& þe fyrst figureof þe neþernombre, & cast hem to-gedervp-on þis condicioɳ. Thou schal loke qweþerþe nomberþat comys þere-of be a digit or no.rub out the top figure;¶ If he be a digit þou schalt do away þe first figure of þe hyer nombre, and write þerein his stede þat he stode Inne þe digit, þat comes of þe ylke 2 figures, & sowrite the result in its place.wrichforth oɳ oþerfigures yf þerebe ony moo, til þou come to þe ende toward þe lyft side. Andledeþe nether figurestonde still euer-moretil þou haueydo. ffor þere-by þou schalwytewheþerþou hast donewel or no, as I schal tell þe afterward in þe ende of þis Chapter. ¶ And lokeallgateleaf 141a.þat þou be-gynne to worch in þis Craft of Addi*cioɳ in þe ryȝt side,Here is an example.here is an ensampul of þis case.12342142.Caste 2 to foure& þat wel be sex, do away 4. & write in þe same place þe figure of sex. ¶ And lete þe figure of 2 in þe nether rewe stonde stil. When þou hast do so, cast 3 & 4 to-gedurand þat wel be seuen þat is a digit. Do away þe 3, & set þereseueɳ, and lete þe neþerfigurestonde stille, & soworchforth bakward til þou hastydoall to-geder.
Et si compositus, in limite scribe sequenteArticulum, primo digitum; quiasic iubet ordo.
Et si compositus, in limite scribe sequente
Articulum, primo digitum; quiasic iubet ordo.
¶ Here is þe secunde case þat may happe in þis craft. And þe case is þis,Suppose it is a Composite, set down the digit, and carry the tens.yf of þe casting of 2 nomburis to-geder, as of þe figure of þe hyer rewe & of þe figure of þe neþerrewe come a Composyt, how schalt þouworch. Þusþou schalt worch. Thou shalt do away þe figure of þe hyer nomberþat was cast to þe figure of þe neþernomber. ¶ And write þereþe digit of þe Composyt. And set þe articul of þe composit next afterþe digit in þe same rewe, yf þerebe nomofigures after. But yf þerebe mo figuris afterþat digit. And þere he schall be rekend for hym selfe. And when þou schalt adde þat ylke figure þat berys þe articulleouerhis hed to þe figurevnderhym, þou schalt cast þat articul to þe figure þat hase hym ouerhis hed, & þereþat Articul schal tokeɳ hym selfe.Here is an example.lo an Ensampullleaf 141b.*of all.326216.Cast 6 to 6, & þere-of wil arise twelue. do away þe hyer 6 & write þere2, þat is þe digit of þis composit. And þenwrite þe articulleþat is ten ouerþe figuris hed of twene as þus.1322216.Now cast þe articulleþat standus vpon þe figuris of twene hed to þe same figure, & reken þat articul bot for one, and þan þerewil arise thre. Þan cast þat thre to þe neþerfigure, þat is one, & þat wul be foure. do away þe figure of 3, and write þerea figure of foure. and lete þe neþerfigure stonde stil, & þan worch forth. vndeversus.
¶ Articulus si sit, in primo limite cifram,¶ Articulumvero reliquis inscribe figuris,Vel perse scribas si nulla figura sequatur.
¶ Articulus si sit, in primo limite cifram,
¶ Articulumvero reliquis inscribe figuris,
Vel perse scribas si nulla figura sequatur.
¶ Herehe puttes þe thryde case of þe craft of Addicioɳ. & þe case is þis.Suppose it is an Article, set down a cipher and carry the tens.yf of Addiciouɳ of 2 figuris a-ryse an Articulle, how schal þou do. thou most do away þeheerfigure þat was addid to þe neþer, & write þerea cifre, and sett þearticulson þe figuris hede, yf þat þerecome ony after. And wyrch þan as I haue tolde þe in þe secunde case. An ensampull.25.15Cast 5 to 5, þat wylle be ten. now do away þe hyer 5, & write þerea cifer. And sette ten vpon þe figuris hed of 2. And reken it but for on þus. loHere is an example.an Ensampulle12φ15Andleaf 142a.*þan worch forth. But yf þerecome no figure afterþe cifre, write þe articul next hym in þe same rewe as here55cast 5 to 5, and it wel be ten. do away 5. þat is þe hier 5. and write þerea cifre, & write afterhym þe articul as þus1φ5And þan þou hast done.
¶ Si tibi cifra superueniens occurrerit, illamDele superpositam; fac illic scribe figuram,Postea procedas reliquas addendo figuras.
¶ Si tibi cifra superueniens occurrerit, illam
Dele superpositam; fac illic scribe figuram,
Postea procedas reliquas addendo figuras.
What to do when you have a cipher in the top row.¶ Herehe puttesþe fourt case, & it is þis, þat yf þerecome a cifer in þe hier rewe, how þou schal do. þus þou schalt do. do away þe cifer, & sett þereþe digit þat comes of þe addiciounas þus1φφ84.17743An example of all the difficulties.In þis ensampul ben alleþe fourecases. Cast 3 to foure, þat wol be seueɳ. do away 4. & write þereseueɳ; þan cast 4 to þe figureof 8. þat wel be 12. do away 8, & sett þere2. þat is a digit, and sette þe articul of þe composit, þat is ten, vpon þe cifers hed, & reken it for hym selfe þat is on. þan cast oneto a cifer, & hit wullebe but on, for noȝt & on makes but one. þan cast 7. þat stondes vnderþat on to hym, & þat wel be 8. do away þe cifer & þat 1. & sette þere8. þan go forthermore. cast þe oþer7 to þe cifer þat stondes ouerhym. þat wul be bot seuen, for þe cifer betokens noȝt. do away þe cifer & sette þereseueɳ,leaf 142b.*& þen go forþermore& cast 1 to 1, & þat wel be 2. do away þe hier 1, & sette þere2. þan hast þou do. And yf þou haue wel ydo þis nomber þat is sett here-afterwel be þe nomber þat schallearyse of alleþe addicioɳ as here27827. ¶ Sequitur alia species.
The Craft of Subtraction.
A numero numerumsi sit tibi demerecuraScribe figurarumseries, vt in addicione.
A numero numerumsi sit tibi demerecura
Scribe figurarumseries, vt in addicione.
Four things to know about subtraction:¶ This is þe Chapterof subtraccioɳ, in the quych þou most know foure nessessary thynges. the first what is subtraccioɳ. þe secunde is how mony nombers þou most haue to subtraccioɳ, the thryd is how mony maners of cases þeremay happe in þis craft of subtraccioɳ. The fourte is qwat is þe profet of þis craft. ¶ As forthe first;þe first, þou most know þat subtraccioɳ is drawyngeof onenowmberoute of anoþernomber.the second;As for þe secunde, þou most knowe þat þou most haue two rewes of figuris onevnderanoþer, as þouaddystin addicioɳ.the third;As for þe thryd, þou moyst know þat fouremanerof diuerse casis mai happe in þis craft.the fourth.¶ As for þe fourt, þou most know þat þe profet of þis craft is whenne þou hasse taken þe lasse nomber out of þe moreto telle what þereleuesouerþat. & þou most be-gynne to wyrch in þiscraft in þe ryght side of þe boke, as þou diddyst in addicioɳ. Versus.
¶ Maiori numero numerumsuppone minorem,¶ Siue pari numero supponaturnumerus par.
¶ Maiori numero numerumsuppone minorem,
¶ Siue pari numero supponaturnumerus par.
leaf 143a.* ¶ Herehe telles þatPut the greater number above the less.þe hier nomber most be moreþen þe neþer, or els eueɳ as mych. but he may not be lasse. And þe case is þis, þou schalt drawe þe neþernomber out of þe hyer, & þou mayst not do þat yf þe hier nomber werelasse þan þat. ffor þou mayst not draw sex out of 2. But þou mast draw 2 out of sex. And þou maiste draw twene out of twene, for þou schal leue noȝt of þe hier twene vndeversus.
The Cases of the Craft of Subtraction.
¶ Postea si possis a prima subtrahe primamScribens quod remanet.
¶ Postea si possis a prima subtrahe primam
Scribens quod remanet.
The first case of subtraction.Hereis þe first case put of subtraccioɳ, & he says þou schalt begynne in þe ryght side, & draw þe first figureof þe neþerrewe out of þe first figureof þe hier rewe. qwether þe hier figure be moreþen þe neþer, or eueɳ as mych. And þat is notified in þe vers when he says “Si possis.” Whan þou has þus ydo, do away þe hiest figure & sett þereþatleuesof þe subtraccioɳ,Here is an example.lo an Ensampulle234122draw 2 out of 4. þan leues 2. do away 4 & write þere2, & latte þe neþerfigurestonde stille, & so gofor-byoþerfiguris till þou come to þe ende, þan hast þou do.
¶ Cifram si nil remanebit.
¶ Cifram si nil remanebit.
Put a cipher if nothing remains.¶ Herehe puttesþe secunde case, & hit is þis. yf it happe þat qwen þou hast draw on neþerfigureout of a hier, & þere leue noȝt afterþe subtraccioɳ, þusleaf 143b.*þou schalt do. þou schalledo away þe hier figure & write þerea cifer, asHere is an example.lo an Ensampull2424Take foureout of foureþanleusnoȝt. þereforedo away þe hier 4 & set þerea cifer, þan take 2 out of 2, þan leues noȝt. do away þe hier 2, & set þerea cifer, and so worch whareso euerþis happe.
Sed si nonpossis a prima demere primamPrecedens vnumde limite deme sequente,Quod demptumprodenario reputabis ab illoSubtrahe totalem numerumquemproposuistiQuo facto scribe super quicquid remanebit.
Sed si nonpossis a prima demere primam
Precedens vnumde limite deme sequente,
Quod demptumprodenario reputabis ab illo
Subtrahe totalem numerumquemproposuisti
Quo facto scribe super quicquid remanebit.
Suppose you cannot take the lower figure from the top one, borrow ten;Herehe puttes þe thryd case, þe quych is þis. yf it happe þat þe neþerfigure be moreþen þe hier figure þat he schallebe draw out of. how schalleþou do. þus þou schalledo. þou schalleborro.1. oute of þe next figure þat comes afterin þe same rewe, for þis case may neuerhapp but yf þerecome figures after. þan þou schalt settþat on ouerþe hier figureshed, of the quych þou woldist y-draw oute þe neyþerfigure yf þou haddysty-myȝt. Whane þou hase þus ydo þou schallerekene þat .1. for ten.take the lower number from ten;¶. And out of þat ten þou schal draw þe neyþermost figure, And alleþat leues þou schalleadd the answer to the top number.adde to þe figureon whos hed þat .1. stode. And þen þou schalledo away alleþat, & sett þereallethat arisys of the addicioɳ of þe ylke 2 figuris. And yf ytleaf 144a.*happe þat þe figure of þe quych þou schalt borro on be hym self but 1. If þou schalt þat one& sett it vppoɳ þe oþerfigurished, and sett in þat 1. place a cifer, yf þerecome mony figuresafter.Example.lo an Ensampul.21221134take 4 out of 2. it wyl not be, þerforeborro oneof þe next figure, þat is 2. and sett þat ouerþe hed of þe fyrst 2. & rekene it for ten. and þere þe secunde stondes write 1. for þou tokest on out of hym. þan take þe neþerfigure, þat is 4, out of ten. And þen leues 6. cast to 6 þe figure of þat 2 þat stode vnderþe hedde of 1. þat wasborwed& rekened for ten, and þat wylle be 8. do away þat 6 & þat 2, & sette þere8, & lette þe neþerfigure stonde stille. Whanne þou hast do þus, go to þe next figure þat is now bot 1. but first yt was 2, & þere-of wasborred1.How to ‘Pay back’ the borrowed ten.þan take out of þat þe figure vnderhym, þat is 3. hit wel not be. þer-foreboroweof the next figure, þe quych is bot 1. Also take & sett hym ouerþe hede of þe figureþat þou woldest haue y-draw oute of þe nether figure, þe quych was 3. & þou myȝt not, & rekene þat borwed 1 for ten & sett in þe same place, of þe quych place þou tokest hymof, a cifer, for he was bot 1. Whanne þou hast þusydo, take out of þat 1. þat is rekent for ten, þe neþerfigure of 3. And þereleues 7.leaf 144b.*cast þe ylke 7 to þe figure þat had þe ylke ten vpon his hed, þe quych figure was 1, & þat wol be 8. þan do away þat 1 and þat 7, & write þere8. & þan wyrch forth in oþerfiguris til þou come to þe ende, & þan þou hast þe do. Versus.
¶ Facque nonenarios de cifris, cumremeabis¶ Occurrant si forte cifre; dum dempseris vnum¶ Postea procedas reliquas demendo figuras.
¶ Facque nonenarios de cifris, cumremeabis
¶ Occurrant si forte cifre; dum dempseris vnum
¶ Postea procedas reliquas demendo figuras.
A very hard case is put.¶ Herehe puttesþe fourte case, þe quych is þis, yf it happe þat þe neþerfigure, þe quych þou schalt draw out of þe hier figure be morepan þe hier figur ouerhym, & þe next figure of two or of thre or of foure, or how mony þerebe by cifers, how wold þou do. Þouwostwel þou most nede borow, & þou mayst not borow of þe cifers, for þai haue noȝt þat þai mayleneor spare. Ergo4howwoldest þou do. Certayɳ þus most þou do, þou most borow on of þe next figure significatyf in þat rewe, for þis case may not happe, but yf þerecome figures significatyf afterthe cifers. Whan þou hast borowede þat 1 of the next figure significatyf, sett þat on ouerþe hede of þat figure of þe quych þou wold haue draw þe neþerfigure out yf þou hadest myȝt, & reken it for ten as þoudiddest inþe oþercasehere-a-fore. Whaɳ þou hast þus y-do loke how mony cifers þerewerebye-twene þat figuresignificatyf, & þe figure of þe quych þou woldest haue y-draw theleaf 145a.*neþerfigure, and of euery of þe ylke cifers make a figureof 9.Here is an example.lo an Ensampulleafter.4000210004Take 4 out of 2. it wel not be. borow 1 out of be next figure significatyf, þe quych is 4, & þen leues 3. do away þat figureof 4 & write þere3. & sett þat 1 vppon þe figure of 2 hede, & þan take 4 out of ten, & þan þere leues 6. Cast 6 to the figure of 2, þat wol be 8. do away þat 6 & write þere 8. Whan þou hast þus y-do make of euery 0 betweyn 3 & 8 a figure of 9, & þan worch forth in goddes name.Sic.3999810004& yf þou hast wel y-do þou5schalt haue þis nomber
How to prove the Subtraction.