The Earliest Arithmetics in English

---

Egerton 2622.

leaf 136 a

A derivation of Algorism. This boke is called þe boke of algorym, or Augrym after lewder vse. And þis boke tretys þe Craft of Nombryng, þe quych crafte is called also Algorym. Ther was a kyng of Inde, þe quich heyth Algor, & he made þis craft. And after his name he called hit algorym; Another derivation of the word. or els anoþer cause is quy it is called Algorym, for þe latyn word of hit s. Algorismus comes of Algos, grece, quid est ars, latine, craft oɳ englis, and rides, quid est numerus, latine, A nombur oɳ englys, inde dicitur Algorismus per addicionem huius sillabe mus & subtraccionem d & e, quasi ars numerandi.¶ fforthermore ȝe most vndirstonde þat in þis craft ben vsid teen figurys, as here bene writen for ensampul, φ 9 8 7 6 5 4 3 2 1.  ¶ Expone þe 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 were fonde fyrst in Inde of a kynge of þat Cuntre, þat was called Algor.

Notation and Numeration.

versus [in margin].

¶ Capitulum primum de significacione figurarum

Expositio versus 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 secunde leaf 136 b 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 bene writene in, til þat þou come to the last figure, þat is called a cifre¶ 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 first write, & 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 vndirstondes noȝt of þe first figure of euery rew¶ But he vndirstondes þe first figure þat is in þe nombur of þe forsayd teen figuris, þe quych is one of þese.1.And þe secunde 2.& so forth.

versus [in margin].

Expositio [in margin]. ¶ Expone þis verse þus.Euery of þese figuris bitokens hym selfe & no more, yf he stonde in þe first place of þe rewele / this worde Simpliciter in 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 þe rewle, he betokens tene tymes hym selfe, as þis figure 2 here 20 tokens ten tyme hym selfe, leaf 137 a *þat is twenty, for he hym selfe betokenes tweyne, & ten tymes twene is twenty. And for he stondis oɳ þe lyft side & in þe secunde place, he betokens ten tyme hym selfe. And so go forth. ¶ ffor euery figure, & he stonde aftur a-noþer toward the lyft side, he schal betokene ten tymes as mich more as he schul betoken & he stode in þe place þere þat þe figure a-fore hym stondes. An example: loo an ensampulle9.6.3.4.Þe figure of 4. þat hase þis schape . betokens bot hymselfe, for he stondes in þe first place. units, The figure of 3. þat hase þis schape betokens ten tymes more þen he schuld & he stde þere þat þe figure of 4. stondes, þat is thretty. tens, The figure of 6, þat hase þis schape , betokens ten tymes more þan he schuld & he stode þere as þe figure of stondes, for þere he schuld tokyne bot sexty, & now he betokens ten tymes more, þat is sex hundryth. hundreds, The figure of 9. þat hase þis schape betokens ten tymes more þane he schuld & he stode in þe place þere þe figure of sex stondes, for þen he schuld betoken to 9. hundryth, and in þe place þere he stondes now he betokens 9. þousande. thousands. Al þe hole nombur is 9 thousande sex hundryth & foure & thretty. ¶ fforthermore, when þou schalt rede a nombur of figure, How to read the number. þou schalt begyne at þe last figure in the lyft side, & rede so forth to þe riȝt side as here 9. 6. 3. 4. Thou schal begyn to rede at þe figure of 9. & rede forth þus. 9. leaf 137 b *thousand sex hundryth thritty & foure. But when þou schalle write, þou schalt be-gynne to write at þe ryȝt side.

The meaning and use of the cipher. Expone þis verse. A cifre tokens noȝt, bot he makes þe figure to betoken þat comes aftur hym more þan he schuld & he were away, as þus 1φ. here þe figure of one tokens ten, & yf þe cifre were away1 & no figure by-fore hym he schuld token bot one, for þan he schuld stonde in þe first place.¶ And þe cifre tokens nothyng hym selfe.for al þe nombur of þe ylke too figures is bot ten.¶ Questio.Why says he þat a cifre makys a figure to signifye (tyf) more &c. ¶ I speke for þis worde significatyf, ffor sothe it may happe aftur a cifre schuld come a-noþur cifre, as þus 2φφ. And ȝet þe secunde cifre shuld token neuer þe more excep he schuld kepe þe order of þe place. and a cifre is no figure significatyf.

The last figure means more than all the others, since it is of the highest value. Expone þis verse þus.Þe last figure schal token more þan alle þe oþer afore, thouȝt þere were a hundryth thousant figures afore, as þus, 16798.Þe last figure þat is 1. betokens ten thousant. And alle þe oþer figures ben bot betokene bot sex thousant seuyne hundryth nynty & 8.¶ And ten thousant is more þen alle þat nombur, ergo þe last figure tokens more þan all þe nombur afore

The Three Kinds of Numbers

¶ Capitulum 2^m de triplice divisione numerorum

¶ The auctor of þis tretis departys þis worde a nombur into 3 partes.Some nombur is called digitus latine, a digit in englys. Digits. Somme nombur is called articulus latine. An Articul in englys. Articles. Some nombur is called a composyt in englys. Composites. ¶ Expone þis verse.know þou aftur þe forsayd rewles þat I sayd afore, þat þere ben thre spices of nomburOone is a digit, Anoþer is an Articul, & þe toþer a Composyt. versus.

Digits, Articles, and Composites.

What are digits. ¶ Here he telles qwat is a digit, Expone versus sic. Nomburs digitus bene alle nomburs þat ben with-inne ten, as nyne, 8.7.6.5.4.3.2.1.

¶ Here he telles what is a composyt and what is ane articul. Expone sic versus. What are articles. ¶ Articulis ben2 alle þat may be deuidyt into nomburs of ten & nothynge leue ouer, as twenty, thretty, fourty, a hundryth, a thousand, & such oþer, ffor twenty may be departyt in-to 2 nomburs of ten, fforty in to foure nomburs of ten, & so forth.

leaf 138 b What numbers are composites. *Compositys beɳ nomburs þat bene componyt of a digyt & of an articulle as fouretene, fyftene, sextene, & such oþerffortene is componyd of foure þat is a digit & of ten þat is an articulleffiftene is componyd of 5 & ten, & so of all oþer, what þat þai ben.Short-lych euery nombur þat be-gynnes with a digit & endyth in a articulle is a composyt, as fortene bygennynge by foure þat is a digit, & endes in ten.

How to write a number, ¶ here he telles how þou schalt wyrch whan þou schalt write a nomburExpone versum sic, & fac iuxta exponentis sentenciam; whan þou hast a nombur to write, loke fyrst what maner nombur it 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 nombur in þe first place þat is sex, and write þe articul next aftur þat is twenty, as þus 26. How to read it. But whan þou schalt sowne or speke leaf 139 a *or rede an Composyt þou schalt first sowne þe articul & aftur þe digit, as þou seyst by þe comyne speche, Sex & twenty & nouȝt twenty & sex. versus.

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 nombur þat þou hast write be an Articul, write first a cifre & aftur þe cifer write an Articulle þus. 2φ. tens, fforthermore þou schalt vndirstonde yf þou haue an Articul, loke how mych he is, yf he be with-ynne an hundryth, þou schalt write bot one cifre, afore, as here . 9φ. hundreds, If þe articulle be 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 figure in þe thryd place schal token a hundrid tymes hym selfe. thousands, &c. If þe articul be a thousant or thousandes3 and he stonde by hym selfe, write afore 3 cifers & so forþ of al oþer

To tell an even number ¶ Here he teches a generalle rewle þat yf þe first figure in þe rewle of figures token a nombur þat is euene al þat nombur of figurys in þat rewle schal be euene, as here þou may see 6.7.3.5.4.Computa & proba. or an odd. ¶ If þe first leaf 139 b *figure token an nombur þat is ode, alle þat nombur in þat rewle schalle be ode, as here 5 6 7 8 6 7. Computa & proba.versus.

The Seven Rules of Arithmetic.

The seven rules. ¶ Here telles þat þer beɳ . 7. spices or partes of þis craft. The first is called addicioñ, þe secunde is called subtraccioñ. The thryd is called duplacioñ. The 4. is called dimydicioñThe 5.is called multiplicacioñ. The 6 is called diuisioñ. The 7. is called extraccioñ of þe RoteWhat all þese spices bene hit schalle be tolde singillatim in here caputule

Add, subtract, or halve, from right to left. Thou schal be-gynne in þe ryght side of þe boke or of a tabul. loke were þou wul be-gynne to write latyn or englys in a boke, & þat schalle be called þe lyft side of the boke, þat þou writest toward þat side schal be called þe ryght side of þe boke.Versus.

Here he telles þe in quych side of þe boke or of þe tabul þou schalle be-gyne to wyrch duplacioñ, diuisioñ, and multiplicacioñ. Multiply or divide from left to right. Thou schal begyne to worch in þe lyft side of þe boke or of þe tabul, but yn what wyse þou schal wyrch in hym dicetur singillatim in sequentibus capitulis et de vtilitate cuiuslibet artis & sic Completur leaf 140 *prohemium & sequitur tractatus & primo de arte addicionis que prima ars est in ordine.

The Craft of Addition.

Four things must be known: ¶ Here by-gynnes þe craft of Addicioñ. In þis craft þou most knowe foure thynges¶ Fyrst þou most 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 a castyng to-gedur of twoo nomburys in-to one nombre. As yf I aske qwat is twene & thre. Þou wyl cast þese twene nombres to-gedur & say þat it is fyue. how many rows of figures; ¶ As for þe secunde þou most know þat þou schalle haue tweyne rewes of figures, one vndur a-nother, as here þou mayst se. 1234
2168. how many cases; ¶ As for þe thryd þou most know þat there ben 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 nombur þat comes of diuerse nomburis.Now as to þe texte of oure verse, he teches there how þou schal worch in þis craft. ¶ He says yf þou wilt cast one nombur to anoþer nombur, þou most by-gynne on þis wyse. How to set down the sum. ¶ ffyrst write leaf 140 b *two rewes of figuris & nombris so þat þou write þe first figure of þe hyer nombur euene vndir the first figure of þe nether nombur, 123
234. And þe secunde of þe nether nombur euene vndir þe secunde of þe hyer, & so forthe of euery figure of both þe rewes as þou mayst se.

The Cases of the Craft of Addition.

¶ Here he teches what þou schalt do when þou hast write too rewes of figuris on vnder 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 figure of þe neþer nombre, & cast hem to-geder vp-on þis condicioɳ. Thou schal loke qweþer þe nomber þat 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 þere in his stede þat he stode Inne þe digit, þat comes of þe ylke 2 figures, & so write the result in its place. wrich forth oɳ oþer figures yf þere be ony moo, til þou come to þe ende toward þe lyft side. And lede þe nether figure stonde still euer-more til þou haue ydoffor þere-by þou schal wyte wheþer þou hast done wel or no, as I schal tell þe afterward in þe ende of þis Chapter. ¶ And loke allgate leaf 141 a 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. 1234
2142. 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-gedur and þat wel be seuen þat is a digit.Do away þe 3, & set þere seueɳ, and lete þe neþer figure stonde stille, & so worch forth bakward til þou hast ydo all to-geder

¶ 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 neþer rewe come a Composyt, how schalt þou worch. Þus þou schalt worch.Thou shalt do away þe figure of þe hyer nomber þat was cast to þe figure of þe neþer nomber. ¶ And write þere þe digit of þe Composyt. And set þe articul of þe composit next after þe digit in þe same rewe, yf þere be no mo figures afterBut yf þere be mo figuris after þat digit. And þere he schall be rekend for hym selfe. And when þou schalt adde þat ylke figure þat berys þe articulle ouer his hed to þe figure vnder hym, þou schalt cast þat articul to þe figure þat hase hym ouer his hed, & þere þat Articul schal tokeɳ hym selfe. Here is an example. lo an Ensampull leaf 141 b *of all. 326
216. Cast 6 to 6, & þere-of wil arise twelue.do away þe hyer 6 & write þere 2, þat is þe digit of þis composit.And þen write þe articulle þat is ten ouer þe figuris hed of twene as þus.   1
322
216. Now cast þe articulle þat standus vpon þe figuris of twene hed to þe same figure, & reken þat articul bot for one, and þan þere wil arise thre. Þan cast þat thre to þe neþer figure, þat is one, & þat wul be fouredo away þe figure of 3, and write þere a figure of foure.and lete þe neþer figure stonde stil, & þan worch forth.vnde versus.

¶ Here he 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 þe heer figure þat was addid to þe neþer, & write þere a cifre, and sett þe articuls on þe figuris hede, yf þat þere come ony after. And wyrch þan as I haue tolde þe in þe secunde case. An ensampull. 25.
15 Cast 5 to 5, þat wylle be ten. now do away þe hyer 5, & write þere a cifer. And sette ten vpon þe figuris hed of 2. And reken it but for on þus. lo Here is an example. an Ensampulle 1  
2φ
15 And leaf 142 a *þan worch forth. But yf þere come no figure after þe cifre, write þe articul next hym in þe same rewe as here 5
5 cast 5 to 5, and it wel be ten. do away 5. þat is þe hier 5. and write þere a cifre, & write after hym þe articul as þus 1φ
  5 And þan þou hast done.

What to do when you have a cipher in the top row. ¶ Here he puttes þe fourt case, & it is þis, þat yf þere come a cifer in þe hier rewe, how þou schal do.þus þou schalt do.do away þe cifer, & sett þere þe digit þat comes of þe addicioun as þus 1φφ84.
17743 An example of all the difficulties. In þis ensampul ben alle þe foure cases. Cast 3 to foure, þat wol be seueɳ.do away 4.& write þere seueɳ; þan cast 4 to þe figure of 8. þat wel be 12.do away 8, & sett þere 2. 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 one to a cifer, & hit wulle be but on, for noȝt & on makes but oneþan cast 7.þat stondes vnder þat on to hym, & þat wel be 8. do away þe cifer & þat 1. & sette þere 8. þan go forthermorecast þe oþer 7 to þe cifer þat stondes ouer hymþat wul be bot seuen, for þe cifer betokens noȝt.do away þe cifer & sette þere seueɳ, leaf 142 b *& þen go forþermore & cast 1 to 1, & þat wel be 2. do away þe hier 1, & sette þere 2. þan hast þou do.And yf þou haue wel ydo þis nomber þat is sett here-after wel be þe nomber þat schalle aryse of alle þe addicioɳ as here 27827. ¶ Sequitur alia species.

The Craft of Subtraction.

Four things to know about subtraction: ¶ This is þe Chapter of subtraccioɳ, in the quych þou most know foure nessessary thyngesthe first what is subtraccioɳ.þe secunde is how mony nombers þou most haue to subtraccioɳ, the thryd is how mony maners of cases þere may happe in þis craft of subtraccioɳ. The fourte is qwat is þe profet of þis craft. ¶ As for the first; þe first, þou most know þat subtraccioɳ is drawynge of one nowmber oute of anoþer nomber. the second; As for þe secunde, þou most knowe þat þou most haue two rewes of figuris one vnder anoþer, as þou addyst in addicioɳ. the third; As for þe thryd, þou moyst know þat foure maner of 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 more to telle what þere leues ouer þat.& þou most be-gynne to wyrch in þis craft in þe ryght side of þe boke, as þou diddyst in addicioɳ.Versus.

leaf 143 a * ¶ Here he telles þat Put the greater number above the less. þe hier nomber most be more þen þe neþer, or els eueɳ as mych.but he may not be lasse.And þe case is þis, þou schalt drawe þe neþer nomber out of þe hyer, & þou mayst not do þat yf þe hier nomber were lasse 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 vnde versus.

The Cases of the Craft of Subtraction.

The first case of subtraction. Here is þe first case put of subtraccioɳ, & he says þou schalt begynne in þe ryght side, & draw þe first figure of þe neþer rewe out of þe first figure of þe hier rewe. qwether þe hier figure be more þen 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 þere þat leues of þe subtraccioɳ, Here is an example. lo an Ensampulle 234
122 draw 2 out of 4. þan leues 2. do away 4 & write þere 2, & latte þe neþer figure stonde stille, & so go for-by oþer figuris till þou come to þe ende, þan hast þou do.

Put a cipher if nothing remains. ¶ Here he puttes þe secunde case, & hit is þis. yf it happe þat qwen þou hast draw on neþer figure out of a hier, & þere leue noȝt after þe subtraccioɳ, þus leaf 143 b *þou schalt do. þou schalle do away þe hier figure & write þere a cifer, as Here is an example. lo an Ensampull 24
24 Take foure out of foure þan leus noȝt. þerefore do away þe hier 4 & set þere a cifer, þan take 2 out of 2, þan leues noȝt. do away þe hier 2, & set þere a cifer, and so worch whare so euer þis happe.

Suppose you cannot take the lower figure from the top one, borrow ten; Here he puttes þe thryd case, þe quych is þis. yf it happe þat þe neþer figure be more þen þe hier figure þat he schalle be draw out of. how schalle þou do. þus þou schalle do. þou schalle borro . 1. oute of þe next figure þat comes after in þe same rewe, for þis case may neuer happ but yf þere come figures afterþan þou schalt sett þat on ouer þe hier figures hed, of the quych þou woldist y-draw oute þe neyþer figure yf þou haddyst y-myȝtWhane þou hase þus ydo þou schalle rekene þat . 1. for ten. take the lower number from ten; ¶. And out of þat ten þou schal draw þe neyþermost figure, And alle þat leues þou schalle add the answer to the top number. adde to þe figure on whos hed þat . 1. stode. And þen þou schalle do away alle þat, & sett þere alle that arisys of the addicioɳ of þe ylke 2 figuris. And yf yt leaf 144 a *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 oþer figuris hed, and sett in þat 1.place a cifer, yf þere come mony figures after. Example. lo an Ensampul. 2122
1134 take 4 out of 2. it wyl not be, þerfore borro one of þe next figure, þat is 2.and sett þat ouer þe hed of þe fyrst 2. & rekene it for ten. and þere þe secunde stondes write 1. for þou tokest on out of hymþan take þe neþer figure, þat is 4, out of ten.And þen leues 6.cast to 6 þe figure of þat 2 þat stode vnder þe hedde of 1. þat was borwed & rekened for ten, and þat wylle be 8. do away þat 6 & þat 2, & sette þere 8, & lette þe neþer figure stonde stille.Whanne þou hast do þus, go to þe next figure þat is now bot 1.but first yt was 2, & þere-of was borred 1.   How to ‘Pay back’ the borrowed ten. þan take out of þat þe figure vnder hym, þat is 3.hit wel not be.þer-fore borowe of the next figure, þe quych is bot 1.Also take & sett hym ouer þe hede of þe figure þat 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 hym of, a cifer, for he was bot 1. Whanne þou hast þus ydo, take out of þat 1. þat is rekent for ten, þe neþer figure of 3. And þere leues 7. leaf 144 b *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 þere 8. & þan wyrch forth in oþer figuris til þou come to þe ende, & þan þou hast þe do.Versus.

A very hard case is put. ¶ Here he puttes þe fourte case, þe quych is þis, yf it happe þat þe neþer figure, þe quych þou schalt draw out of þe hier figure be more pan þe hier figur ouer hym, & þe next figure of two or of thre or of foure, or how mony þere be by cifers, how wold þou do.Þou wost wel þou most nede borow, & þou mayst not borow of þe cifers, for þai haue noȝt þat þai may lene or spareErgo4 how woldest 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 þere come figures significatyf after the cifers. Whan þou hast borowede þat 1 of the next figure significatyf, sett þat on ouer þe hede of þat figure of þe quych þou wold haue draw þe neþer figure out yf þou hadest myȝt, & reken it for ten as þou diddest in þe oþer case here-a-foreWhaɳ þou hast þus y-do loke how mony cifers þere were bye-twene þat figure significatyf, & þe figure of þe quych þou woldest haue y-draw the leaf 145 a *neþer figure, and of euery of þe ylke cifers make a figure of 9. Here is an example. lo an Ensampulle after. 40002
10004 Take 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 figure of 4 & write þere 3. & 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. 39998
10004 & yf þou hast wel y-do þou5 schalt haue þis nomber

How to prove the Subtraction.

How to prove a subtraction sum. ¶ Here he teches þe Craft how þou schalt know, whan þou hast subtrayd, wheþer þou hast wel ydo or no. And þe Craft is þis, ryght as þou subtrayd þe neþer figures fro þe hier figures, ryȝt so adde þe same neþer figures to þe hier figures. And yf þou haue well y-wroth a-fore þou schalt haue þe hier nombre þe same þou haddest or þou be-gan to worch. as for þis I bade þou schulde kepe þe neþer figures stylle. Here is an example. lo an leaf 145 b *Ensampulle of alle þe 4 cases togedreworche welle þis case 40003468
20004664 And yf þou worch welle whan þou hast alle subtrayd þe þat hier nombre here, þis schalle be þe nombre here foloyng whan þou hast subtrayd. 39998804
20004664 Our author makes a slip here (3 for 1). And þou schalt know þusadde þe neþer rowe of þe same nombre to þe hier rewe as þus, cast 4 to 4. þat wol be 8. do away þe 4 & write þere 8. 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 þere a 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 neþer figure vnder þat þe quych is 4, & þat schalle be 13. do away þat 9 & sett þere 3, & sett a figure of 1.þat schall be 10 vpon þe next figuris hede þe quych 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 þere a cifer. and write þe articulle þat is 1. betokenynge 10. vpon þe hede of þe next figure toward þe lyft side, þe quych leaf 146 a *is 9, & so do forth tyl þou come to þe last 9. He works his proof through, take þe figure of þat 1. þe quych þou schalt fynde ouer þe hed of 9. & sett it ouer þe next figures hede þat schal be 3. ¶ Also do away þe 9. & set þere a cifer, & þen cast þat 1 þat stondes vpon þe hede of 3 to þe same 3, & þat schalle make 4, þen caste to þe ylke 4 the figure in þe neyþer rewe, þe quych is 2, and þat schalle be 6. and brings out a result. 60003468
20004664 And þen schal þou haue an Ensampulle aȝeyɳ, loke & se, & but þou haue þis same þou hase myse-wroȝt

The Craft of Duplation.

Sequitur de duplacione

Four things must be known in Duplation. ¶ This is the Chapture of duplacioɳ, 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 may6 happe in þis craft. ¶ þe fourte is what is þe profet of þe craft. Here they are. ¶ As for þe first. duplacioɳ is a doublynge of a nombre. ¶ As for þe secunde þou most leaf 146 b *haue on nombre or on rewe of figures, the quych called numerus duplandusAs 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 what a-risyȝt of a nombre I-doublyde. Mind where you begin. ¶ fforþer-more, þou most know & take gode hede in quych side þou schalle be-gyn in þis craft, or ellis þou mayst spyl alle þi laber þere aboute.certeyn þou schalt begyɳ in the lyft side in þis Craft.thenke wel ouer þis verse.  ¶ 7A leua dupla, diuide, multiplica.7

The sentens of 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 þou wel double a nombre þus þou most be-gynɳ.Write a rewe of figures of what nombre þou welt. versus.

How to work a sum. ¶ Here he telles how þou schalt worch in þis Craft.he says, fyrst, whan þou hast writen þe nombre þou schalt be-gyn at þe first figure in the lyft side, & doubulle þat 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 147 a

If the answer is a digit, ¶ Here is þe first case of þis craft, þe quych is þis. yf of duplacioɳ of a figure arise 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 þere þe diget þat comes of þe duplacioɳ, as þus. 23. double 2, & þat wel be 4.do away þe figure of 2 & sett þere a figure of 4, & so worch forth tille þou come to þe ende.versus.

If it is an article, ¶ Here is þe secunde case, þe quych is þis yf þere come an articulle of þe duplacioɳ of a figure þou schalt do ryȝt as þou diddyst in addicioɳ, þat is to wete þat þou schalt do away þe figure þat is doublet & put a cipher in the place, and ‘carry’ the tens. sett þere a cifer, & write þe articulle ouer þe next figuris hede, yf þere be any after-warde toward þe lyft side as þus.25.begyn at the lyft side, and doubulle 2. þat wel be 4. do away þat 2 & sett þere 4. þan doubul 5. þat wel be 10. do away 5, & sett þere a 0, & sett 1 vpon þe next figuris hede þe quych is 4. & þen draw downe 1 to 4 & þat wolle be 5, & þen do away þat 4 & þat 1, & sett þere 5. for þat 1 schal be rekened in þe drawynge togedre for 1. wen leaf 147 b *þou hast ydon þou schalt haue þis nombre 50. If there is no figure to ‘carry’ them to, write them down. yf þere come no figure after þe figure þat is addit, of þe quych addicioɳ comes an articulle, þou schalt do away þe figure þat is dowblet & sett þere a 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 figure 5 & set þere a cifer, & sett þe articul next after in þe same rewe toward þe lyft side, & þou schalt haue þis nombre 1023. þen go forth & double þe oþer nombers þe quych is lyȝt y-nowȝt to do. versus.

If it is a Composite, ¶ Here he puttes þe 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 þere a digit of þe Composit, write down the digit, and ‘carry’ the tens. & sett þe articulle ouer þe next figures hede, & after draw hym downe with þe figure ouer whos hede he stondes, & make þere-of an nombre as þou hast done afore, & yf þere come no figure after þat digit þat þou hast y-write, þan set þe articulle next after hym in þe same rewe as þus, 67: double 6 þat wel be 12, do away 6 & write þere þe digit leaf 148 a *of 12, þe quych is 2, Here is an example. and set þe articulle next after toward þe lyft side in þe same rewe, for þere comes no figure afterþan dowble þat oþer figure, 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 articulle ouer þe next figures hed, þe quych is 2, & þen draw to hym þat on, & make on nombre þe quych schalle be 3. And þen yf þou haue wel y-do þou schalle haue þis nombre of þe duplacioɳ, 134. versus.

How to double the mark for one-half. ¶ Here he says, yf ouer þe fyrst figure in þe ryȝt side be such a merke as is here made, ^w, þou schalle fyrst doubulle þe figure, the quych stondes vnder þat merke, & þen þou schalt doubul þat merke þe quych stondes for haluendel on. for too haluedels makes on, & so þat wol be on.cast þat on to þat duplacioɳ of þe figure ouer whos hed stode þat merke, & write it in þe same place þere þat þe figure þe quych was doublet stode, as þus 23^wdouble 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 þere 7. þan hase þou do. as for þat figure, þan go leaf 148 b *to þe oþer figure & worch forth. This can only stand over the first figure. & þou schall neuer haue such a merk but ouer þe 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 super extremam &c. Et nota, talis figura ^w significans medietatem, unitatis veniat, i.e. contingat uel fiat super extremam, i.e. super primam figuram in extremo sic versus dextram ars dat: i.e. reddit monadem. i.e. vnitatem eidem. i.e. eidem note & declinatur hec monos, dis, di, dem, &c. ¶ Quod ergo totum hoc dabis monadem note continget. i.e. eveniet tibi si dimidiasti, i.e. accipisti uel subtulisti medietatem alicuius unius, in cuius principio sint figura numerum denotans imparem primo i.e. principiis.

The Craft of Mediation.

¶ Sequitur de mediacione.

The four things to be known in mediation: ¶ In þis Chapter is taȝt þe Craft of mediaciouɳ, in þe quych craft þou most know 4 thynges.ffurst what is mediacioɳ.the secunde how mony rewes of figures þou most haue in þe wyrchynge of þis craft. þe thryde how mony diuerse cases may happ in þis craft.8 the first ¶ As for þe furst, þou schalt vndurstonde þat mediacioɳ is a takyng out of halfe a nomber out of a holle nomber, leaf 149 a *as yf þou the second; wolde take 3 out of 6. ¶ As for þe secunde, þou schalt know þat þou most haue one rewe of figures, & no moo, as þou hayst in þe the third; craft of duplacioɳ. ¶ As for the thryd, þou most vnderstonde þat the fourth. 5 cases may happe in þis craft. ¶ As for þe fourte, þou schalle know þat the profet of þis craft is when þou hast take away þe haluendel of a nombre to telle qwat þere schalle leue. ¶ Incipe sic, &c. The sentence of þis verse is þis. yf þou wold medye, þat is to say, take halfe out of þe holle, or halfe out of halfe, þou most begynne þus. Begin thus. Write one rewe of figures of what nombre þou wolte, as þou dyddyst be-fore in þe Craft of duplacioɳ. versus.

¶ Here he says, when þou hast write a rewe of figures, þou schalt See if the number is even or odd. take hede wheþer þe first figure be eueɳ or odde in nombre, & vnderstonde þat he spekes of þe first figure in þe ryȝt side.And in the ryght side þou schalle begynne in þis Craft.

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

The Mediation of an Odd Number.

If it is odd, halve the even number less than it. Here is þe secunde case of þis craft, the quych is þis. yf þe first figure betokene a nombre þat is odde, the quych odde schal not be mediete, þen þou schalt medye þat nombre þat leues, when the odde of þe same nombre is take away, & write þat þat leues as þou diddest in þe first case of þis craft.Whaɳ þou hayst write þat. for þat þat leues, Then write the sign for one-half over it. write such a merke as is here ^w vpon 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 figure here is betokenynge odde nombre, þe quych is 5, for 5 is odde; þere-fore do away þat þat is odde, þe quych is 1.þen leues 4.þen medye 4 & þen leues 2.do away 4.& sette þere 2, & make such a merke ^w upon his hede, þat is to say ouer his hede of 2 as þus. 242.^w And þen worch forth in þe oþer figures tyll þou come to þe ende.by þe furst case as þou schalt vnderstonde þat Put the mark only over the first figure. þou schalt leaf 150 a *neuer make such a merk but ouer þe first figure hed in þe riȝt side.Wheþer þe other figures þat comyɳ after hym be eueɳ or odde. versus.

The Cases of the Craft of Mediation.

If the first figure is one put a cipher. ¶ Here is þe thryde case, þe quych yf the first figure be a figure of 1. þou schalt do away þat 1 & set þere a cifer, & a merke ouer þe cifer as þus, 241. do away 1, & sett þere a cifer with a merke ouer his hede, & þen hast þou ydo for þat 0.as þus 0^w þen worch forth in þe oþer figurys till þou come to þe ende, for it is lyght as dyche water.vnde versus.

What to do if any other figure is odd. ¶ Here he puttes þe fourte case, þe quych is þis. yf it happeɳ the secunde figure betoken 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 figure þat is medied, & sette in his styde halfe of þat nombre. Write a figure of five over the next lower number’s head. ¶ Whan þou hase so done, þou schalt write leaf 150 b *a figure of 5 ouer þe next figures hede by-fore toward 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 þere 4. þen out of 7. take away 1. þe quych makes odde, & sett 5. vpon þe next figures hede afore toward þe ryȝt side, þe quych is now 4. but afore it 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 after þe takyinge away of þat þat is odde, þe quych leuynge schalle be 3;       5
4634. do away 6 & sette þere 3, & þou schalt haue such a nombre after go forth to þe next figure, & medy þat, & worch forth, for it is lyȝt ynovȝt to þe certayɳ

¶ Here he puttes þe 5 case, þe quych is leaf 151 a *þis: If the second figure is one, put a cipher, and write five over the next figure. yf þe secunde figure be of 1, as þis is here 12, þou schalt do away þat 1 & sett þere a cifer. & sett 5 ouer þe next figure hede afore toward þe riȝt side, as þou diddyst afore; & þat 5 schal be haldel of þat 1, þe quych 1 is rekent for 10. lo an Ensampulle, 214.medye 4.þat schalle be 2. do away 4 & sett þere 2. þen go forth to þe next figureþe quych is bot 1.do away þat 1.& sett þere a cifer. & set 5 vpon þe figures hed afore, þe quych is nowe 2, & þen þou schalt haue þis nombre     5
202, þen worch forth to þe nex figure. And also it is no maystery yf þere come no figure after þat on is medyet, þou schalt write no 0.ne nowȝt ellis, but set 5 ouer þe next figure afore toward þe ryȝt, as þus 14. How to halve fourteen. medie 4 then leues 2, do away 4 & sett þere 2. 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,  5
2, & do away þat 1, & þou schalt haue þis nombre yf þou worch wel, vnde versus.

How to prove the Mediation.

How to prove your mediation. ¶ Here he telles þe how þou schalt know wheþer þou hase wel ydo or no. doubul leaf 151 b *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 tabulle or þou began to medye, as þus. First example. ¶ The furst ensampulle was þis. 4. þe quych I-mediet was laft 2, þe whych 2 was write in þe place þat 4 was write aforeNow doubulle þat 2, & þou schal haue 4, as þou hadyst afore. The second. þe secunde Ensampulle was þis, 245. When þou haddyst mediet alle þis nombre, yf þou haue wel ydo þou schalt haue of þat mediacioɳ þis nombre, 122^wNow doubulle þis nombre, & begyn in þe lyft side; doubulle 1, þat schal be 2. do away þat 1 & sett þere 2. þen doubulle þat oþer 2 & sett þere 4, þen doubulle þat oþer 2, & þat wel be 4. þen doubul þat merke þat stondes for halue on. & þat schalle be 1. Cast þat on to 4, & it schalle be 5. do away þat 2 & þat merke, & sette þere 5, & 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 alle þis nombur truly leaf 152 a *by euery figure, þou schall haue be þat mediacioɳ þis nombur 120^wNow dowbul þis nombur, & begyn in þe lyft side, as I tolde þe in þe Craft of duplacioɳ.þus doubulle þe figure of 1, þat wel be 2. do away þat 1 & sett þere 2, þen doubul þe next figure afore, the quych is 2, & þat wel be 4; do away 2 & set þere 4. þen doubul þe cifer, & þat wel be noȝt, for a 0 is noȝt. And twyes noȝt is but noȝt. þerefore doubul the merke aboue þe cifers hede, þe quych betokenes þe haluendel of 1, & þat schal be 1.do away þe cifer & þe merke, & sett þere 1, & þen þou schalt haue þis nombur 241.And þis same nombur þou haddyst afore or þou began to medy, & yf þou take gode hede. The fourth example. ¶ The next ensampul þat had in þe 4 case of mediacioɳ was þis 4678. Whan þou hast truly ymedit alle þis nombur fro þe begynnynge to þe endynge, þou schalt haue of þe mediacioɳ þis nombur      5
2334. Now doubul this nombur & begyn in þe lyft side, & doubulle 2 þat schal be 4. do away 2 and sette þere 4; þen doubule 3, þat wol be 6; do away 3 & sett þere 6, þen doubul þat oþer 3, & þat wel be 6; do away 3 & set þere leaf 152 b *6, þen doubul þe 4, þat welle be 8; þen doubul 5. þe quych stondes ouer þe hed of 4, & þat wol be 10; cast 10 to 8, & þat schal be 18; do away 4 & þat 5, & sett þere 8, & sett that 1, þe quych is an articul of þe Composit þe quych is 18, ouer þe next figures hed 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 ouer his 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 alle the nombur, a principio usque ad finemþou schalt haue of þat mediacioɳ þis nombur     5
102. Now doubul 1. þat wel be 2. do away 1 & sett þere 2. þen doubul 0. þat will be noȝt.þerefore take þe 5, þe quych stondes ouer þe next figures hed, & doubul it, & þat wol be 10. do away þe 0 þat stondes betwene þe two figuris, & sette þere in his stid 1, for þat 1 now schal stonde in þe secunde place, where he schal betoken 10; þen doubul 2, þat wol be 4. do away 2 & sett þere 4. & leaf 153 a *þou schal haue þus nombur 214. þis is þe same numbur þat þou hadyst or þou began to medye, as þou may see.And so do euer more, yf þou wil knowe wheþer þou hase wel ymedyt or no. ¶. doubulle þe numbur þat comes after þe 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 þi Craft.

The Craft of Multiplication.

Sequitur de multiplicatione.

To write down a Multiplication Sum.

Four things to be known of Multiplication: ¶ Here be-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 figures þere most be. ¶ The 4. what is þe profet of þis craft. the first: ¶ As for þe first, þou schal vnderstonde þat multiplicacioɳ is a bryngynge to-geder of 2 thynges in on nombur, þe quych on nombur contynes so mony tymes on, howe leaf 153 b *mony tymes þere ben vnytees in þe nowmbre of þat 2, as twyes 4 is 8. now here ben 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 foure sythes,9 & so furth of such other lyke wordes. ¶ And tweyn nombres schal be tokenyde be a nowne, as þis worde foure showys þes tweyɳ nombres y-broth in-to on hole nombur, þat is 8, for twyes 4 is 8, as þou wost wel.¶ And þes nombre 8 conteynes as oft tymes 4 as þere ben 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 maner of diuerse case may happe in þis craft. the fourth. The profet of þis Craft is to telle when a nombre is multiplyed be a noþer, qwat commys þere of. ¶ fforthermore, as to þe sentence of oure verse, yf þou wel multiply a nombur be a-noþer nombur, þou schalt write leaf 154 a *a rewe of figures of what nomburs so euer þou welt, The multiplicand. & þat schal be called Numerus multiplicandus, Anglice, þe nombur the 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 nombur þe furst figure schal be write vnder þe last figure of þe nombur, þe quych is to be multiplied. How to set down the sum. And so write forthe toward þe lyft side, as here you may se,      67324
1234 And þis one nombur schalle be called numerus multiplicans.Anglice, þe nombur multipliynge, for he schalle multiply þe hyer nounbur, as þus one tyme 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 þere is two manurs of multiplicacioɳ; one ys of þe wyrchynge of þe boke only in þe mynde of a mon. fyrst he teches of þe fyrst maner of duplacioɳ, þe quych is be wyrchynge of tabuls. and on paper. Afterwarde he wol teche on þe secunde manervnde versus.

To multiply one Digit by another.

How to multiply two digits. ¶ Here he teches a rewle, how þou schalt fynde þe nounbre þat comes by þe multiplicacioɳ of a digit be anoþerloke how mony [vny]tes ben.bytwene þe more digit and 10. And reken ten for on vnite. Subtract the greater from ten; And so oft do away þe lasse nounbre out of his owne decuple, þat is to say, fro þat nounbre þat is ten tymes so mych is þe nounbre þat comes of þe multiplicacioɳ.As yf þou wol multiply 2 be 4.loke how mony vnitees ben by-twene þe quych is þe more nounbre, & be-twene ten.Certen þere wel be vj vnitees by-twene 4 & ten. yf þou reken þere with þe 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 oþer digetes til þou come to ten; & whan þou Example. 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, & þere schal leue 8. bot yf bothe þe digettes leaf 155 a *ben y-lyech mych as here. 222 or too tymes twenty, þen it is no fors quych of hem tweyn þou take out of here decuple. als mony Better use this table, though. tymes as þat is fro 10.but neuer-þe-lesse, yf þou haue hast to worch, þou schalt haue here a 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.

1
2	4
3	6	9
4	8	12	16
5	10	15	20	25
6	12	18	24	30	36
7	14	21	28	35	42	49
8	16	24	32	40	48	56	64
9	18	27	36	45	54	63	72	81
1	2	3	4	5	6	7	8	9

How to use it. yf þe figure, þe quych schalle be multiplied, be euene as mych as þe diget be, þe quych þat oþer figure schal be multiplied, as two tymes twayɳ, or thre tymes 3. or sych other. The way to use the Multiplication table. loke qwere þat figure sittes in þe lyft side of þe triangle, & loke qwere þe diget sittes in þe neþer most rewe of þe triangle. & go fro hym vpwarde in þe same rewe, þe quych rewe gose vpwarde til þou come agaynes þe oþer digette þat sittes in þe lyft side of þe triangle. And þat nounbre, þe quych þou leaf 155 b fyn*des þere is þe nounbre þat comes of the multiplicacioɳ of þe 2 digittes, as yf þou wold wete qwat is 2 tymes 2.loke quere sittes 2 in þe lyft side in þe first rewe, he sittes next 1 in þe lyft side al on hye, as þou may se; þe[n] loke qwere sittes 2 in þe lowyst rewe of þe triangle, & go fro hym vpwarde in þe same rewe tylle þou come a-ȝenenes 2 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 welleyf þe diget.the quych is multiplied, be more þan þe oþer, þou schalt loke qwere þe more diget sittes in þe lowest rewe of þe triangle, & go vpwarde in þe same rewe tyl10 þou come a-nendes þe lasse diget in the lyft side. And þere þou schalt fynde þe nombre þat comes of þe multiplicacioɳ; but þou schalt vnderstonde þat þis rewle, þe quych is in þis verse.¶ In digitum cures, &c. , noþer þis triangle schalle not serue, bot to fynde þe nounbres þat comes of the multiplicacioɳ þat comes of 2 articuls or composites, þe nedes no craft but yf þou wolt multiply in þi mynde. And leaf 156 a *þere-to þou schalt haue a craft afterwarde, for þou schall wyrch with digettes in þe tables, as þou schalt know afterwarde.versus.

To multiply one Composite by another.

How to multiply one number by another. ¶ Here he teches how þou schalt wyrch in þis craft.þou schalt multiplye þe last figure of þe nombre, and quen þou hast so ydo þou schalt draw alle þe figures of þe neþer nounbre more taward þe ryȝt side, so qwen þou hast multiplyed þe last figure of þe heyer nounbre by alle þe neþer figures. Multiply the ‘last’ figure of the higher by the ‘first’ of the lower number. And sette þe nounbir þat comes þer-of ouer þe last figure of þe neþer nounbre, & þen þou schalt sette al þe oþer figures of þe neþer nounbre more nere to þe ryȝt side. ¶ And whan þou hast multiplied þat figure þat schal be multiplied þe next after hym by al þe neþer figures. And worch as þou dyddyst afore til leaf 156 b *þou come to þe ende. And þou schalt vnderstonde þat euery figure of þe hier nounbre schal be multiplied be alle þe figures of the neþer nounbre, yf þe hier nounbre be any figure þen one. Set the answer over the first of the lower: lo an Ensampul here folowynge.     2465.
232 þou schalt begyne to multiplye in þe lyft side.Multiply 2 be 2, and twyes 2 is 4. set 4 then multiply the second of the lower, and so on. ouer þe 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 ouer þe hed of 3, þan multiplie þe same hier 2 by þat 2 þe quych stondes vnder hym, þat wol be 4; do away þe hier 2 & sette þere 4. Then antery the lower number: ¶ Now þou most antery þe nether nounbre, þat is to say, þou most sett þe neþer nounbre more towarde þe ryȝt side, as þus. Take þe neþer 2 toward þe ryȝt side, & sette it eueɳ vnder þe 4 of þe hyer nounbre, & antery alle þe figures þat comes after þat 2, as þus; sette 2 vnder þe 4. þen sett þe figure of 3 þere þat þe figure of 2 stode, þe quych is now vndur þat 4 in þe hier nounbre; þen sett þe oþer figure of 2, þe quych is þe last figure toward þe lyft side of þe neþer nomber þere þe figure of 3 stode. as thus. þen þou schalt haue such a nombre. 464465
  232 leaf 157 a * ¶ Now multiply 4, þe quych comes next after 6, by þe last 2 of þe neþer nounbur toward þe lyft side. as 2 tymes 4, þat wel be 8. sette þat 8 ouer þe figure the quych stondes ouer þe hede of þat 2, þe quych is þe last figure of þe neþer nounbre; þan multiplie þat same 4 by 3, þat comes in þe neþer rewe, þat wol be 12. sette þe digit of þe composyt ouer þe figure þe quych stondes ouer þe hed of þat 3, & sette þe articule of þis composit ouer al þe figures þat stondes ouer þe neþer 2 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 neþer nounbur, þat wol be 8. do away 4. & sette þere 8. Euer more qwen þou multiplies þe hier figure by þat figure þe quych stondes vnder hym, 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 order of figure as is here,   1
  82
4648[65]
  232. þen take and antery þi neþer figures. And sett þe fyrst figure of þe neþer figures11 vndre be figure of 6. ¶ And draw al þe oþer figures of þe same rewe to hym-warde, leaf 157 b *as þou diddyst afore.þen multiplye 6 be 2, & sett þat þe quych comes ouer þere-of ouer al þe oþer figures hedes þat stondes ouer þat 2. þen multiply 6 be 3, & sett alle þat comes þere-of vpon alle þe figures hedes þat standes ouer þat 3; þan multiplye 6 be 2, þe quych stondes vnder þat 6, þen do away 6 & write þere þe digitt of þe composit þat schal come þereof, & sette þe articull ouer alle þe figures þat stondes ouer þe hede of þat 3 as here,   11
  121
  828
464825
    232 Antery 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 ouer all þe figures þat stonden ouer þat 2, & sett þat 1.ouer the next figures hedes, alle on hye towarde þe lyft side. þen multiplye 5 be 3.þat wol be 15, write 5 ouer þe figures hedes þat stonden ouer þat 3, & sett þat 1 ouer þe next figures hedes toward þe lyft side.þen multiplye 5 be 2, þat wol be 10.do away þat 5 & sett þere a 0, & sett þat 1 ouer þe figures hedes þat stonden ouer 3. And þen leaf 158 a þou schalt haue such a nounbre as here stondes aftur. *     11
  1101
  1215
  82820
4648
      232 ¶ Now draw alle þese figures downe togeder as þus, 6.8. 1. & 1 draw to-gedur; þat wolle be 16, do away alle þese figures saue 6. lat hym stonde, for þow þou take hym away þou most write þer þe same aȝene.þerefore late hym stonde, & sett 1 ouer þe figure hede of 4 toward þe lyft side; þen draw on to 4, þat wolle be 5. Then add all the figures above the line: do away þat 4 & þat 1, & sette þere 5. þen draw 4221 & 1 togedur, þat wol be 10.do away alle þat, & write þere þat 4 & þat 0, & sett þat 1 ouer þe next figures hede toward þe lyft side, þe quych is 6. þen draw þat 6 & þat 1 togedur, & þat wolle be 7; do away 6 & sett þere 7, þen draw 8810 & 1, & þat wel be 18; do away alle þe figures þat stondes ouer þe hede of þat 8, & lette 8 stonde stil, & write þat 1 ouer þe next figuris hede, þe quych is a 0. þen do away þat 0, & sett þere 1, þe quych stondes ouer þe 0. hede. þen draw 2, 5, & 1 togedur, þat wolle be 8. þen do away alle þat, & write þere 8.   and you will have the answer. ¶ And þen þou schalt haue þis nounbre, 571880.

The Cases of this Craft.

leaf 158 b

What to do if the first multiplication results in an article. ¶ Here he puttes þe fyrst case of þis craft, þe quych is þis: yf þere come an articulle of þe multiplicacioɳ ysette before the articulle in þe lyft side as þus   51
23. multiplye 5 by 2, þat wol be 10; sette ouer þe hede of þat 2 a 0, & sett þat on, þat is þe articul, in þe lyft side, þat is next hym, þen þou schalt haue þis nounbre 1051.
  23 ¶ And þen worch forth as þou diddist afore. And þou schalt vnderstonde þat þou schalt write no 0.but whan þat place where þou schal write þat 0 has no figure afore hym noþer afterversus.

What to do if the result is a composite number. ¶ Here is þe secunde case, þe quych is þis: yf hit happe þat þere come a composyt, þou schalt write þe digitte ouer þe hede of þe neþer figure by þe quych þou multipliest þe hier figure; and sett þe articulle next hym toward þe lyft side, as þou diddyst afore, as þus   83.
83 Multiply 8 by 8, þat wol be 64. Write þe 4 ouer 8, þat is to say, ouer þe hede of þe neþer 8; & set 6, þe quych leaf 159 a *is an articul, next afterAnd þen þou schalt haue such a nounbre as is here, 648313,
  83 And þen worch forth.

What if it be a digit. ¶ Here is þe thryde case, þe quych is þis: yf hit happe þat of þi multiplicaciouɳ come a digit, þou schalt write þe digit ouer þe hede of þe neþer figure, by the quych þou multipliest þe hiere figure, for þis nedes no Ensampul.

The fourth case of the craft. ¶ Here is þe 4 case, þe quych is: yf hit be happe þat þe neþer figure schal multiplye þat figure, þe quych stondes ouer þat figures hede, þou schal do away þe hier figure & sett þere þat þat comys of þat multiplicacioɳ.As yf þere come of þat multiplicacioɳ an articuls þou schalt write þere þe hier figure stode a 0. ¶ And write þe articuls in þe lyft side, yf þat hit be a digit write þere a digit.yf þat hit be a composit, write þe digit of þe composit.And þe articul in þe lyft side.al þis is lyȝt y-nowȝt, þere-fore þer nedes no Ensampul.

leaf 159 b

The fifth case of the craft. ¶ Here is þe 5 case, þe quych is þis: yf *þe neþer figure schul multiplie þe hier, and þat hier figure is not recte ouer his hede. And þat neþer figure hase oþer figures, or on figure ouer his hede by multiplicacioɳ, þat hase be afore, þou schalt write þat nounbre, þe quych comes of þat, ouer alle þe ylke figures hedes, as þus here:     236
234 Multiply 2 by 2, þat wol be 4; set 4 ouer þe hede of þat 2. þen14 multiplies þe hier 2 by þe neþer 3, þat wol be 6. set ouer his hede 6, multiplie þe hier 2 by þe neþer 4, þat wol be 8. do away þe hier 2, þe quych stondes ouer þe hede of þe figure of 4, and set þere 8.And þou schalt haue þis nounbre here 46836
234 And antery þi figures, þat is to say, set þi neþer 4 vnder þe hier 3, and set þi 2 other figures nere hym, so þat þe neþer 2 stonde vndur þe hier 6, þe quych 6 stondes in þe lyft side. And þat 3 þat stondes vndur 8, as þus aftur ȝe may se, 46836
  234 Now worch forthermore, And multiplye þat hier 3 by 2, þat wol be 6, set þat 6 þe quych stondes ouer þe hede of þat 2, And þen worch as I taȝt þe afore.

leaf 160 a

The sixth case of the craft. ¶ Here is þe 6 case, þe quych is þis: yf hit happe þat þe figure by þe quych þou schal multiplye þe hier figure, þe quych stondes ryght ouer hym by a 0, þou schalt do away þat figure, þe quych ouer þat cifre hede. ¶ And write þere þat nounbre þat comes of þe multiplicacioɳ as þus, 23.do away 2 and sett þere a 0.vnde versus.

The seventh case of the craft. ¶ Here is þe 7 case, þe quych is þis: yf a 0 schal multiply a figure, þe quych stondes not recte ouer hym, And ouer þat 0 stonde no thyng, þou schalt write ouer þat 0 anoþer 0 as þus:     24
03 multiplye 2 be a 0, it wol be nothyngewrite þere a 0 ouer þe hede of þe neþer 0, And þen worch forth til þou come to þe ende.

The eighth case of the craft. ¶ Here is þe 8 case, þe quych is þis: yf þere be a 0 or mony cifers in þe hier rewe, þou schalt not multiplie hem, bot let hem stonde.And antery þe figures beneþe to þe next figure sygnificatyf as þus: 00032.
22 Ouer-lepe alle þese cifers & sett þat leaf 160 b *neþer 2 þat stondes toward þe ryght side, and sett hym vndur þe 3, and sett þe oþer nether 2 nere hym, so þat he stonde vndur þe thrydde 0, þe quych stondes next 3. And þan worch. vnde versus.

How to prove the multiplication. ¶ Here he teches how þou schalt know wheþer þou hase wel I-do or no. And he says þat þou schalt deuide alle þe nounbre þat comes of þe multiplicacioɳ by þe neþer figures. And þen þou schalt haue þe same nounbur þat þou hadyst in þe begynnyngebut ȝet þou hast not þe craft of dyuisioɳ, but þou schalt haue hit afterwarde.

Mental multiplication. ¶ Here he teches þe to multiplie be þowȝt figures in þi mynde. And þe sentence of þis verse is þis: yf þou wel multiplie on nounbre by anoþer in þi mynde, þou schal haue þereto rewles in þe verses þat schal come after

Digit by digit is easy. ¶ Here he teches a rewle as þou hast afore to multiplie a digit be anoþer, as yf þou wolde wete qwat is sex tymes 6. þou leaf 161 a *schalt wete by þe rewle þat I taȝt þe before, yf þou haue mynde þerof.

The first case of the craft. ¶ Here he teches þe furst rewle, þe quych is þis: yf þou wel multiplie an articul be anoþer, so þat both þe articuls bene with-Inne an hundreth, þus þou schalt do. Article by article; take þe digit of bothe the articuls, for euery articul hase a digit, þen multiplye þat on digit by þat oþer, and loke how mony vnytes ben in þe nounbre þat comes of þe multiplicacioɳ of þe 2 digittes, & so mony hundrythes ben in þe nounbre þat schal come of þe multiplicacioɳ of þe ylke 2 articuls as þus. an example: yf þou wold wete qwat is ten tymes ten.take þe digit of ten, þe quych is 1; take þe digit of þat oþer ten, þe quych is on. ¶ Also multiplie 1 be 1, as on tyme on þat is but 1.In on is but on vnite as þou wost welle, þerefore ten tymes ten is but a hundryth. another example: ¶ Also yf þou wold wete what is twenty tymes 30. take þe digit of twenty, þat is 2; & take þe digitt of thrytty, þat is 3. multiplie 3 be 2, þat is 6. Now in 6 ben 6 vnites, ¶ And so mony hundrythes ben in 20 tymes 30*, leaf 161 b þerefore 20 tymes 30 is 6 hundryth eueɳ. loke & se. ¶ But yf it be so þat one articul be with-Inne an hundryth, or by-twene an hundryth and a thowsande, so þat it be not a þowsande fully. þen loke how mony vnytes ben in þe nounbur þat comys of þe multiplicacioɳ 16And so mony tymes16 of 2 digittes of ylke articuls, so mony thowsant ben in þe nounbre, the qwych comes of þe multiplicacioɳ.And so mony tymes ten thowsand schal be in þe nounbre þat comes of þe multiplicacion of 2 articuls, as yf þou wold wete qwat is 4 hundryth tymes [two hundryth].Multiply 4 be 2,17 þat wol be 8. in 8 ben 8 vnites. How to work subtly without Figures. Mental multiplication. ¶ And so mony tymes ten thousand be in 4 hundryth tymes [2]17 hundryth, þat is 80 thousand. Take hede, I schall telle þe a Another example. generalle rewle whan þou hast 2 articuls, And þou wold wete qwat comes of þe multiplicacioɳ of hem 2.multiplie þe digit of þat on articuls, and kepe þat nounbre, þen loke how mony cifers schuld go before þat on articuls, and he were writeAls mony cifers schuld go before þat other, & he were write of cifers. And haue alle þe ylke cifers togedur in þi mynde, leaf 162 a *a-rowe ychoɳ aftur other, and in þe last plase set þe nounbre þat comes of þe multiplicacioɳ of þe 2 digittes. And loke in þi mynde in what place he stondes, where in þe secunde, or in þe thryd, or in þe 4, or where ellis, and loke qwat þe figures by-token in þat place; & so mych is þe nounbre þat Another example. comes of þe 2 articuls y-multiplied to-gedur as þus: yf þou wold wete what is 20 thousant tymes 3 þowsande.multiply þe digit of þat articulle þe quych is 2 by þe digitte of þat oþer articul þe quych is 3, þat wol be 6. þen loke how mony cifers schal go to 20 thousant as hit schuld be write in a tabul. certainly 4 cifers schuld go to 20 þowsant. ffor þis figure 2 in þe fyrst place betokenes twene. Notation. ¶ In þe secunde place hit betokenes twenty. ¶ In þe 3. place hit betokenes 2 hundryth. . ¶. In þe 4 place 2 thousant. ¶ In þe 5 place hit betokenes twenty þousant.þerefore he most haue 4 cifers a-fore hym þat he may stonde in þe 5 place. kepe þese 4 cifers in thy mynde, þen loke how mony cifers goɳ to 3 thousant. Certayn to 3 thousante leaf 162 b *goɳ 3 cifers aforeNow cast ylke 4 cifers þat schuld go to twenty thousant, And thes 3 cifers þat schuld go afore 3 thousant, & sette hem in rewe ychoɳ after oþer in þi mynde, as þai schuld stonde in a tabulleAnd þen schal þou haue 7 cifers; þen sett þat 6 þe quych comes of þe multiplicacioɳ of þe 2 digittes aftur þe ylke cifers in þe 8 place as yf þat hit stode in a tabul.And loke qwat a figure of 6 schuld betoken in þe 8 place. yf hit were in a tabul & so mych it is. & yf þat figure of 6 stonde in þe fyrst place he schuld betoken but 6. ¶ In þe 2 place he schuld betoken sexty. ¶ In the 3 place he schuld betokeɳ sex hundryth. Notation again. ¶ In þe 4 place sex thousant. ¶ In þe 5 place sexty þowsant. ¶ In þe sext place sex hundryth þowsant. ¶ In þe 7 place sex þowsant thousantes¶ In þe 8 place sexty þowsant thousantes.þerfore sett 6 in octauo loco, And he schal betoken sexty þowsant thousantes. Mental multiplication. And so mych is twenty þowsant tymes 3 thousant, ¶ And þis rewle is generalle for alle maner of articuls, Whethir þai be hundryth or þowsant; but þou most know well þe craft of þe wryrchynge in þe tabulle leaf 163 a *or þou know to do þus in þi mynde aftur þis rewle. Thou most þat þis rewle holdyþe note but where þere ben 2 articuls and no mo of þe quych ayther of hem hase but on figure significatyf. As twenty tymes 3 thousant or 3 hundryth, and such oþur

The third case of the craft; ¶ Here he puttes þe thryde rewle, þe quych is þis. yf þou wel multiply in þi mynde, And þe Articul be a digitte, þou schalt loke þat þe digitt be with-Inne an hundryth, þen þou schalt multiply the digitt of þe Articulle by þe oþer digitte.And euery vnite in þe nounbre þat schalle come þere-of schal betoken ten. As þus: an example. yf þat þou wold wete qwat is twyes 40.multiplie þe digitte of 40, þe quych is 4, by þe oþer diget, þe quych is 2. And þat wolle be 8. And in þe nombre of 8 ben 8 vnites, & euery of þe ylke vnites schuld stonde for 10.þere-fore þere schal be 8 tymes 10, þat wol be 4 score. And so mony is twyes 40. ¶ If þe articul be a hundryth or be 2 hundryth And a þowsant, so þat hit be notte a thousant, leaf 163 b *worch as þou dyddyst afore, saue þou schalt rekene euery vnite for a hundryth.

The fourth case of the craft: Here he puttes þe 4 rewle, þe quych is þis: yf þou multipliy on composit be a digit as 6 tymes 24, 19þen take þe diget of þat composit, & multiply þat digitt by þat oþer diget, and kepe þe nombur þat comes þere-of.þen take þe digit of þat composit, & multiply þat digit by anoþer diget, by þe quych þou hast multiplyed þe diget of þe articul, and loke qwat comes þere-of. Composite by digit. þen take þou þat nounbur, & cast hit to þat other nounbur þat þou secheste as þus yf þou wel wete qwat comes of 6 tymes 4 & twenty. Mental multiplication. multiply þat articulle of þe composit by þe digit, þe quych is 6, as yn þe thryd rewle þou was tauȝt, And þat schal be 6 scoreþen multiply þe diget of þe composit, leaf 164 a *þe quych is 4, and multiply þat by þat other diget, þe quych is 6, as þou wast tauȝt in þe first rewle, yf þou haue mynde þerof, & þat wol be 4 & twenty.cast all ylke nounburs to-gedir, & hit schal be 144.And so mych is 6 tymes 4 & twenty.

How to multiply without Figures.

The fifth case of the craft: ¶ Here he puttes þe 5 rewle, þe quych is þis: yf þou wel multiply an Articul be a composit, multiplie þat Articul by þe articul of þe composit, and worch as þou wos tauȝt in þe secunde rewle, of þe quych rewle þe verse begynnes þus. Article by Composite. ¶ Articulum si per Relicum vis multiplicare.þen multiply þe diget of þe composit by þat oþir articul aftir þe doctrine of þe 3 rewle. take þerof gode hede, I pray þe as þus.Yf þou wel wete what is 24 tymes ten. An example. Multiplie ten by 20, þat wel be 2 hundryth. þen multiply þe diget of þe 10, þe quych is 1, by þe diget of þe composit, þe quych is 4, & þat leaf 164 b *wol be 4. þen reken euery vnite þat is in 4 for 10, & þat schal be 40.Cast 40 to 2 hundryth, & þat wol be 2 hundryth & 40.And so mych is 24 tymes ten.

How to work without Figures.

The sixth case of the craft: ¶ Here he puttes þe 6 rewle, & þe last of alle multiplicacioɳ, þe quych is þis: yf þou wel multiplye a composit by a-noþer composit, þou schalt do þus. Composite by Composite. multiplie þat on composit, qwych þou welt of the twene, by þe articul of þe toþer composit, as þou were tauȝt in þe 5 rewle, þen multiplie þat same composit, þe quych þou hast multiplied by þe oþer articul, by þe digit of þe oþer composit, Mental multiplication. as þou was tauȝt in þe 4 rewle. An example As þus, yf þou wold wete what is 11 tymes 13, as þou was tauȝt in þe 5 rewle, & þat schal be an hundryth & ten, afterwarde multiply þat same composit þat þou hast multiplied, þe quych is a .11.And multiplye hit be þe digit of þe oþer composit, þe quych is 3, for 3 is þe digit of 13, And þat wel be 30. þen take þe digit of þat composit, þe quych composit þou multiplied by þe digit of þat oþer composit, leaf 165 a *þe quych is a 11. of the sixth case of the craft. ¶ Also of the quych 11 on is þe digit. multiplie þat digitt by þe digett of þat other composit, þe quych diget is 3, as þou was tauȝt in þe first rewle in þe begynnynge of þis craft. þe quych rewle begynnes “In digitum cures.” And of alle þe multiplicacioɳ of þe 2 digitt comys thre, for onys 3 is but 3.Now cast alle þese nounbers togedur, the quych is þis, a hundryth & ten & 30 & 3.And al þat wel be 143.Write 3 first in þe ryght side.And cast 10 to 30, þat wol be 40.set 40 next aftur towarde þe lyft side, And set aftur a hundryth as here an Ensampulle, 143.

(Cetera desunt.)

---