Chapter III: Part 3

Multiply 8 by 8, þat wol be 64. Write þe 4 ou{er} 8, þat is to say, ou{er} þe hede of þe neþ{er} 8; & set 6, þe quych [*leaf 159a] is an articul, next aft{er}. And þen þou schalt haue such a nounb{r}e as is her{e},

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And þen worch forth.

¶ Si digitus t{amen} ponas ip{su}m sup{er} ip{s}am.

[Sidenote: What if it be a digit.]

¶ Her{e} is þe thryde case, þe quych is þis: yf hit happe þat of þi m{u}ltiplicaciouɳ come a digit, þ{o}u schalt write þe digit ou{er} þe hede of þe neþ{er} figur{e}, by the quych þou m{u}ltipliest þe hier{e} figur{e}, for þis nedes no Ensampul.

¶ Subdita m{u}ltiplica non hanc que [incidit] illi
Delet ea{m} penit{us} scribens quod p{ro}uenit inde.

[Sidenote: The fourth case of the craft.]

¶ Her{e} is þe 4 case, þe quych is: yf hit be happe þat þe neþ{er} figur{e} schal multiplye þat figur{e}, þe quych stondes ou{er} þat figures hede, þou schal do away þe hier figur{e} & sett þ{er}e þat þ{a}t comys of þ{a}t m{u}ltiplicacioɳ. As yf þ{er}e come of þat m{u}ltiplicacioɳ an articuls þou schalt write þere þe hier figur{e} stode a 0. ¶ And write þe articuls in þe lyft side, yf þat hit be a digit write þ{er}e a digit. yf þat h{i}t be a composit, write þe digit of þe composit. And þe articul in þe lyft side. al þis is lyȝt y-nowȝt, þ{er}e-for{e} þer nedes no Ensampul.

¶ S{ed} si m{u}ltiplicat alia{m} ponas sup{er} ip{s}am
Adiu{n}ges num{er}u{m} que{m} p{re}bet duct{us} ear{um}.

[Sidenote: The fifth case of the craft.]

¶ Her{e} is þe 5 case, þe quych is þis: yf [*leaf 159b] þe neþ{er} figur{e} schul m{u}ltiplie þe hier, and þat hier figur{e} is not recte ou{er} his hede. And þat neþ{er} figur{e} hase oþ{er} figures, or on figure ou{er} his hede by m{u}ltiplicacioɳ, þat hase be afor{e}, þou schalt write þat nounbre, þe quych comes of þat, ou{er} all{e} þe ylke figures hedes, as þus here:

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Multiply 2 by 2, þat wol be 4; set 4 ou{er} þe hede of þat 2. þen[{14}] m{u}ltiplies þe hier 2 by þe neþ{er} 3, þat wol be 6. set ou{er} his hede 6, multiplie þe hier 2 by þe neþ{er} 4, þat wol be 8. do away þe hier 2, þe quych stondes ou{er} þe hede of þe figur{e} of 4, and set þ{er}e 8. And þou schalt haue þis nounb{re} here

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And antery þi figur{e}s, þat is to say, set þi neþ{er} 4 vnd{er} þe hier 3, and set þi 2 other figures ner{e} hym, so þat þe neþ{er} 2 stonde vnd{ur} þe hier 6, þe quych 6 stondes in þe lyft side. And þat 3 þat stondes vndur 8, as þus aftur ȝe may se,

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Now worch forthermor{e}, And m{u}ltiplye þat hier 3 by 2, þat wol be 6, set þ{a}t 6 þe quych stondes ou{er} þe hede of þat 2, And þen worch as I taȝt þe afore.

[*leaf 160a]

¶ Si sup{ra}posita cifra debet m{u}ltiplicar{e}
Prorsus ea{m} deles & ibi scribi cifra debet.

[Sidenote: The sixth case of the craft.]

¶ Her{e} is þe 6 case, þe quych is þis: yf hit happe þat þe figur{e} by þe quych þou schal m{u}ltiplye þe hier figur{e}, þe quych stondes ryght ou{er} hym by a 0, þou schalt do away þat figur{e}, þe quych ou{er} þat cifre hede. ¶ And write þ{ere} þat nounbre þat comes of þe m{u}ltiplicacioɳ as þus, 23. do away 2 and sett þ{er}e a 0. vn{de} v{er}sus.

¶ Si cifra m{u}ltiplicat alia{m} posita{m} sup{er} ip{s}am
Sitq{ue} locus sup{ra} vacu{us} sup{er} hanc cifra{m} fiet.

[Sidenote: The seventh case of the craft.]

¶ Her{e} is þe 7 case, þe quych is þis: yf a 0 schal m{u}ltiply a figur{e}, þe quych stondes not recte ou{er} hym, And ou{er} þat 0 stonde no thyng, þou schalt write ou{er} þat 0 anoþ{er} 0 as þus:

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multiplye 2 be a 0, it wol be nothyng{e}. write þere a 0 ou{er} þe hede of þe neþ{er} 0, And þen worch forth til þou come to þe ende.

¶ Si sup{ra}[{15}] fuerit cifra sem{per} e{st} p{re}t{er}eunda.

[Sidenote: The eighth case of the craft.]

¶ Her{e} is þe 8 case, þe quych is þis: yf þ{ere} be a 0 or mony cifers in þe hier rewe, þ{o}u schalt not m{u}ltiplie hem, bot let hem stonde. And antery þe figures beneþe to þe next figur{e} sygnificatyf as þus:

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Ou{er}-lepe all{e} þese cifers & sett þat [*leaf 160b] neþ{er} 2 þat stondes toward þe ryght side, and sett hym vnd{ur} þe 3, and sett þe oþ{er} nether 2 nere hym, so þat he stonde vnd{ur} þe thrydde 0, þe quych stondes next 3. And þan worch. vnd{e} v{er}sus.

¶ Si dubites, an sit b{e}n{e} m{u}ltiplicac{i}o facta,
Diuide totalem nu{mer}u{m} p{er} multiplicante{m}.

[Sidenote: How to prove the multiplication.]

¶ Her{e} he teches how þou schalt know wheþ{er} þou hase wel I-do or no. And he says þat þou schalt deuide all{e} þe nounb{r}e þat comes of þe m{u}ltiplicacioɳ by þe neþ{er} figures. And þen þou schalt haue þe same nounbur þat þ{o}u hadyst in þe begynnyng{e}. but ȝet þou hast not þe craft of dyuisioɳ, but þ{o}u schalt haue hit aft{er}warde.

¶ P{er} num{er}u{m} si vis nu{mer}u{m} q{u}oq{ue} m{u}ltiplicar{e}
¶ T{antu}m p{er} normas subtiles absq{ue} figuris
Has normas pot{er}is p{er} v{er}sus scir{e} sequentes.

[Sidenote: Mental multiplication.]

¶ Her{e} he teches þe to m{u}ltiplie be þowȝt figures in þi mynde. And þe sentence of þis v{er}se is þis: yf þo{u} wel m{u}ltiplie on nounbre by anoþ{er} in þi mynde, þ{o}u schal haue þ{er}eto rewles in þe v{er}ses þat schal come aft{er}.

¶ Si tu p{er} digitu{m} digitu{m} vis m{u}ltiplicar{e}
Re{gula} p{re}cedens dat qualit{er} est op{er}andu{m}.

[Sidenote: Digit by digit is easy.]

¶ Her{e} he teches a rewle as þou hast afor{e} to m{u}ltiplie a digit be anoþ{er}, as yf þou wolde wete qwat is sex tymes 6. þou [*leaf 161a] schalt wete by þe rewle þat I taȝt þe befor{e}, yf þou haue mynde þ{er}of.

¶ Articulu{m} si p{er} reliquu{m} reliquu{m} vis m{u}lti{plica}r{e}
In p{ro}p{r}iu{m} digitu{m} debet vt{er}q{ue} resolui.
¶ Articul{us} digitos post se m{u}ltiplicantes
Ex digit{us} quociens retenerit m{u}ltipli{ca}r{i}
Articuli faciu{n}t tot centu{m} m{u}ltiplicati.

[Sidenote: The first case of the craft. Article by article; an
example: another example:]

[Headnote: How to work subtly without Figures.]

[Sidenote: Mental multiplication. Another example. Another example.
Notation. Notation again. Mental multiplication.]

¶ Her{e} he teches þe furst rewle, þe quych is þis: yf þou wel m{u}ltiplie an articul be anoþ{er}, so þat both þe articuls bene w{i}t{h}-Inne an hundreth, þus þ{o}u schalt do. take þe digit of bothe the articuls, for eu{er}y articul hase a digit, þen m{u}ltiplye þat on digit by þat oþ{er}, and loke how mony vnytes ben in þe nounbre þat comes of þe m{u}ltiplicacioɳ of þe 2 digittes, & so mony hundrythes ben in þe nounb{re} þat schal come of þe m{u}ltiplicacioɳ of þe ylke 2 articuls as þus. yf þ{o}u wold wete qwat is ten tymes ten. take þe digit of ten, þe quych is 1; take þe digit of þat oþ{er} ten, þe quych is on. ¶ Also m{u}ltiplie 1 be 1, as on tyme on þat is but 1. In on is but on vnite as þou wost welle, þ{er}efor{e} ten tymes ten is but a hundryth. ¶ 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. m{u}ltiplie 3 be 2, þat is 6. Now in 6 ben 6 vnites, ¶ And so mony hundrythes ben in 20 tymes 30[*leaf 161b], þ{ere}for{e} 20 tymes 30 is 6 hundryth eueɳ. loke & se. ¶ But yf it be so þat on{e} articul be w{i}t{h}-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 m{u}ltiplicacioɳ [{16}]And so mony tymes[{16}] of 2 digitt{es} of ylke articuls, so mony thowsant ben in þe nounbre, the qwych comes of þe m{u}ltiplicacioɳ. And so mony tymes ten thowsand schal be in þe nounbre þat comes of þe m{u}ltiplicacion of 2 articuls, as yf þ{o}u wold wete qwat is 4 hundryth tymes [two hundryth]. Multiply 4 be 2,[{17}] þat wol be 8. in 8 ben 8 vnites. ¶ And so mony tymes ten thousand be in 4 hundryth tymes [2][{17}] hundryth, þ{a}t is 80 thousand. Take hede, I schall telle þe a gen{e}rall{e} rewle whan þ{o}u hast 2 articuls, And þou wold wete qwat comes of þe m{u}ltiplicacioɳ of hem 2. m{u}ltiplie þe digit of þ{a}t on articuls, and kepe þat nounbre, þen loke how mony cifers schuld go befor{e} þat on articuls, and he wer{e} write. Als mony cifers schuld go befor{e} þat other, & he wer{e} write of cifers. And haue all{e} þe ylke cifers toged{ur} in þi mynde, [*leaf 162a] a-rowe ychoɳ aftur other, and in þe last plase set þe nounbre þat comes of þe m{u}ltiplicacioɳ of þe 2 digittes. And loke in þi mynde in what place he stondes, wher{e} in þe secunde, or in þe thryd, or in þe 4, or wher{e} ellis, and loke qwat þe figures by-token in þat place; & so mych is þe nounbre þat comes of þe 2 articuls y-m{u}ltiplied to-ged{ur} as þus: yf þ{o}u wold wete what is 20 thousant tymes 3 þowsande. m{u}ltiply þe digit of þat articull{e} þe quych is 2 by þe digitte of þat oþ{er} 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. c{er}tainly 4 cifers schuld go to 20 þowsant. ffor þis figure 2 in þe fyrst place betokenes twene. ¶ 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 h{i}t betokenes twenty þousant. þ{ere}for{e} he most haue 4 cifers a-for{e} hym þat he may sto{n}de 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 162b] goɳ 3 cifers afor{e}. Now cast ylke 4 cifers þat schuld go to twenty thousant, And thes 3 cifers þat schuld go afor{e} 3 thousant, & sette hem in rewe ychoɳ aft{er} oþ{er} in þi mynde, as þai schuld stonde in a tabull{e}. And þen schal þou haue 7 cifers; þen sett þat 6 þe quych comes of þe m{u}ltiplicacioɳ of þe 2 digitt{es} aft{u}r þe ylke cifers in þe 8 place as yf þat hit stode in a tabul. And loke qwat a figur{e} of 6 schuld betoken in þe 8 place. yf hit wer{e} 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. ¶ 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 thousant{es}. ¶ In þe 8 place sexty þowsant thousantes. þ{er}for{e} sett 6 in octauo loco, And he schal betoken sexty þowsant thousantes. And so mych is twenty þowsant tymes 3 thousant, ¶ And þis rewle is gen{er}all{e} for all{e} man{er} of articuls, Whethir þai be hundryth or þowsant; but þ{o}u most know well þe craft of þe wryrchyng{e} in þe tabull{e} [*leaf 163a] or þou know to do þus in þi mynde aftur þis rewle. Thou most þat þis rewle holdyþe note but wher{e} þ{ere} ben 2 articuls and no mo of þe quych ayther of hem hase but on figur{e} significatyf. As twenty tymes 3 thousant or 3 hundryth, and such oþ{ur}.

¶ Articulum digito si m{u}ltiplicare o{portet}
Articuli digit[i sumi quo multiplicate]
Debem{us} reliquu{m} quod m{u}ltiplicat{ur} ab ill{is}
P{er} reliq{u}o decuplu{m} sic su{m}ma{m} later{e} neq{ui}b{i}t.

[Sidenote: The third case of the craft; an example.]

¶ Her{e} he puttes þe thryde rewle, þe quych is þis. yf þ{o}u wel m{u}ltiply in þi mynde, And þe Articul be a digitte, þou schalt loke þat þe digitt be w{i}t{h}-Inne an hundryth, þen þou schalt m{u}ltiply the digitt of þe Articulle by þe oþer digitte. And eu{er}y vnite in þe nounbre þat schall{e} come þ{ere}-of schal betoken ten. As þus: yf þat þ{o}u wold wete qwat is twyes 40. m{u}ltiplie þe digitt{e} of 40, þe quych is 4, by þe oþ{er} diget, þe quych is 2. And þat wolle be 8. And in þe nombre of 8 ben 8 vnites, & eu{er}y of þe ylke vnites schuld stonde for 10. þ{ere}-fore þ{ere} 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 163b] worch as þo{u} dyddyst afor{e}, saue þ{o}u schalt rekene eu{er}y vnite for a hundryth.

¶ In nu{mer}u{m} mixtu{m} digitu{m} si ducer{e} cures
Articul{us} mixti sumat{ur} deinde resoluas
In digitu{m} post fac respectu de digitis
Articul{us}q{ue} docet excrescens in diriua{n}do
In digitu{m} mixti post ducas m{u}ltiplica{n}te{m}
¶ De digitis vt norma [{18}][docet] de [hunc]
Multiplica si{mu}l et sic postea summa patebit.

[Sidenote: The fourth case of the craft: Composite by digit. Mental
multiplication.]

Here he puttes þe 4 rewle, þe quych is þis: yf þou m{u}ltipliy on composit be a digit as 6 tymes 24, [{19}]þen take þe diget of þat composit, & m{u}ltiply þ{a}t digitt by þat oþ{er} diget, and kepe þe nomb{ur} þat comes þ{ere}-of. þen take þe digit of þat composit, & m{u}ltiply þat digit by anoþ{er} diget, by þe quych þ{o}u hast m{u}ltiplyed þe diget of þe articul, and loke qwat comes þ{ere}-of. þen take þ{o}u þat nounbur, & cast hit to þat other nounbur þat þ{o}u secheste as þus yf þou wel wete qwat comes of 6 tymes 4 & twenty. multiply þat articull{e} of þe composit by þe digit, þe quych is 6, as yn þe thryd rewle þ{o}u was tauȝt, And þat schal be 6 scor{e}. þen m{u}ltiply þe diget of þe {com}posit, [*leaf 164a] þe quych is 4, and m{u}ltiply þat by þat other diget, þe quych is 6, as þou wast tauȝt in þe first rewle, yf þ{o}u haue mynde þ{er}of, & þat wol be 4 & twenty. cast all ylke nounburs to-ged{ir}, & hit schal be 144. And so mych is 6 tymes 4 & twenty.

[Headnote: How to multiply without Figures.]

¶ Duct{us} in articulu{m} num{erus} si {com}posit{us} sit
Articulu{m} puru{m} comites articulu{m} q{u}o{que}
Mixti pro digit{is} post fiat [et articulus vt]
Norma iubet [retinendo quod extra dicta ab illis]
Articuli digitu{m} post tu mixtu{m} digitu{m} duc
Re{gula} de digitis nec p{re}cipit articul{us}q{ue}
Ex quib{us} exc{re}scens su{m}me tu iunge p{ri}ori
Sic ma{n}ifesta cito fiet t{ibi} su{m}ma petita.

[Sidenote: The fifth case of the craft: Article by Composite.
An example.]

¶ Her{e} he puttes þe 5 rewle, þe quych is þis: yf þ{o}u wel m{u}ltiply an Articul be a composit, m{u}ltiplie þat Articul by þe articul of þe composit, and worch as þou wos tauȝt in þe secunde rewle, of þe quych rewle þe v{er}se begynnes þus. ¶ Articulu{m} si p{er} Relicu{m} vis m{u}ltiplicare. þen m{u}ltiply þe diget of þe composit by þat oþ{ir} articul aft{ir} þe doctrine of þe 3 rewle. take þ{er}of gode hede, I p{ra}y þe as þus. Yf þ{o}u wel wete what is 24 tymes ten. Multiplie ten by 20, þat wel be 2 hundryth. þen m{u}ltiply þe diget of þe 10, þe quych is 1, by þe diget of þe composit, þe quych is 4, & þ{a}t [*leaf 164b] wol be 4. þen reken eu{er}y 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.

[Headnote: How to work without Figures.]

¶ Compositu{m} num{er}u{m} mixto si[c] m{u}ltiplicabis
Vndecies tredeci{m} sic e{st} ex hiis op{er}andum
In reliquu{m} p{rimu}m demu{m} duc post in eund{em}
Vnu{m} post den{u}m duc in t{ri}a dei{n}de p{er} vnu{m}
Multiplices{que} dem{u}m int{ra} o{mn}ia m{u}ltiplicata
In su{m}ma decies q{ua}m si fu{er}it t{ibi} doces
Multiplicandor{um} de normis sufficiunt h{ec}.

[Sidenote: The sixth case of the craft: Composite by Composite.
Mental multiplication. An example of the sixth case of the craft.]

¶ Here he puttes þe 6 rewle, & þe last of all{e} multiplicacioɳ, þe quych is þis: yf þ{o}u wel m{u}ltiplye a {com}posit by a-noþ{er} composit, þou schalt do þus. m{u}ltiplie þ{a}t on composit, qwych þ{o}u welt of the twene, by þe articul of þe toþ{er} composit, as þ{o}u wer{e} tauȝt in þe 5 rewle, þen m{u}ltiplie þ{a}t same composit, þe quych þou hast m{u}ltiplied by þe oþ{er} articul, by þe digit of þe oþ{er} composit, as þ{o}u was tauȝt in þe 4 rewle. As þus, yf þou wold wete what is 11 tymes 13, as þ{o}u was tauȝt in þe 5 rewle, & þat schal be an hundryth & ten, aft{er}warde m{u}ltiply þat same co{m}posit þ{a}t þ{o}u hast m{u}ltiplied, þe quych is a .11. And m{u}ltiplye hit be þe digit of þe oþ{er} 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 m{u}ltiplied by þe digit of þ{a}t oþ{er} {com}posit, [*leaf 165a] þe quych is a 11. ¶ Also of the quych 11 on is þe digit. m{u}ltiplie þat digitt by þe digett of þat oth{er} composit, þe quych diget is 3, as þ{o}u was tauȝt in þe first rewle i{n} þe begynnyng{e} of þis craft. þe quych rewle begynn{es} “In digitu{m} cures.” And of all{e} þe m{u}ltiplicacioɳ of þe 2 digitt comys thre, for onys 3 is but 3. Now cast all{e} þese nounbers toged{ur}, 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 aft{ur} towarde þe lyft side, And set aftur a hundryth as her{e} an Ensampull{e}, 143.

(Cetera desunt.)

FOOTNOTES (The Crafte of Nombrynge):

[1: In MS, ‘awiy.’]
[2: ‘ben’ repeated in MS.]
[3: In MS. ‘thausandes.’]
[4: Perhaps “So.”]
[5: ‘hali’ marked for erasure in MS.]
[6: ‘moy’ in MS.]
[7: ‘Subt{ra}has a{u}t addis a dext{ri}s {ve}l medi{a}b{is}’ added
on margin of MS.]
[8: After ‘craft’ insert ‘the .4. what is þe p{ro}fet of þis craft.’]
[9: After ‘sythes’ insert ‘& þis wordes fyue sithe & sex sythes.’]
[10: ‘t’l’ marked for erasure before ‘tyl’ in MS.]
[11: Here ‘of þe same rew’ is marked for erasure in MS.]
[12: ‘s{ed}’ deleted in MS.]
[13: 6883 in MS.]
[14: ‘þen’ overwritten on ‘þat’ marked for erasure.]
[15: ‘Supra’ inserted in MS. in place of ‘cifra’ marked for erasure.]
[16--16: Marked for erasure in MS.]
[17: 4 in MS.]
[18: docet. decet MS.]
[19: ‘4 times 4’ in MS.]

+The Art of Nombryng.+

A TRANSLATION OF

+John of Holywood’s De Arte Numerandi.+

[_Ashmole MS. 396, fol. 48._]

+Boys seying in the begynnyng of his Arsemetrik{e}:--All{e}
[*Fol. 48.] thynges that ben{e} fro the first begynnyng of thynges
have p{ro}ceded{e}, and come forth{e}, And by reso{u}n of nombre
ben formed{e}; And in wise as they ben{e}, So oweth{e} they to be
knowen{e}; wherfor in vniu{er}sall{e} knowlechyng of thynges the
Art of nombrynge is best, and most operatyf{e}.+

[Sidenote: The name of the art. Derivation of Algorism. Another.
Another. Kinds of numbers. The 9 rules of the Art.]

Therfore sithen the science of the whiche at this tyme we intenden{e} to write of standith{e} all{e} and about nombre: ffirst we most se, what is the p{ro}pre name therof{e}, and fro whens the name come: Afterward{e} what is nombre, And how manye spices of nombre ther ben. The name is cleped{e} Algorisme, had{e} out of Algor{e}, other of Algos, in grewe, That is clepid{e} in englissh{e} art other craft, And of Rithm{us} that is called{e} nombre. So algorisme is cleped{e} the art of nombryng, other it is had of{e} en or in, and gogos that is introduccio{u}n, and Rithm{us} nombre, that is to say Interduccio{u}n of nombre. And thirdly it is had{e} of the name of a kyng that is cleped{e} Algo and Rythm{us}; So called{e} Algorism{us}. Sothely .2. maner{e} of nombres ben notified{e}; Formall{e},[{1}] as nombr{e} i{s} vnitees gadred{e} to-gedres; Materiall{e},[{2}] as nombr{e} is a colleccio{u}n of vnitees. Other nombr{e} is a multitude had{e} out of vnitees, vnitee is that thynge wher-by eu{er}y thynge is called{e} oone, other o thynge. Of nombres, that one is cleped{e} digitall{e}, that other{e} Article, Another a nombre componed{e} oþ{er} myxt. Another digitall{e} is a nombre w{i}t{h}-in .10.; Article is þ{a}t nombre that may be dyvyded{e} in .10. p{ar}ties egally, And that there leve no residue; Componed{e} or medled{e} is that nombre that is come of a digite and of an article. And vndrestand{e} wele that all{e} nombres betwix .2. articles next is a nombr{e} componed{e}. Of this art ben{e} .9. spices, that is forto sey, num{er}acio{u}n, addicio{u}n, Subtraccio{u}n, Mediac{i}o{u}n, Duplacio{u}n, Multipliacio{u}n, Dyvysio{u}n, Progressio{u}n, And of Rootes the extraccio{u}n, and that may be had{e} in .2. maners, that is to sey in nombres quadrat, and in cubic{es}: Amonge the which{e}, ffirst of Num{er}acio{u}n, and aft{er}ward{e} of þe oþ{er}s by ordure, y entende to write.

[Headnote: Chapter I. Numeration.]

[*Fol. 48b]

+For-soth{e} num{er}acio{u}n is of eu{er}y numbre by
competent figures an artificiall{e} rep{re}sentacio{u}n.+

[Sidenote: Figures, differences, places, and limits. The 9 figures.
The cipher. The numeration of digits, of articles, of composites.
The value due to position. Numbers are written from right to left.]

Sothly figure, difference, places, and lynes supposen o thyng other the same, But they ben sette here for dyue{r}s resons. ffigure is cleped{e} for p{ro}traccio{u}n of figuracio{u}n; Difference is called{e} for therby is shewed{e} eu{er}y figure, how it hath{e} difference fro the figures before them: place by cause of space, where-in me writeth{e}: lynees, for that is ordeyned{e} for the p{re}sentacio{u}n of eu{er}y figure. And vnderstonde that ther ben .9. lymytes of figures that rep{re}senten the .9. digit{es} that ben these. 0. 9. 8. 7. 6. 5. 4. 3. 2. 1. The .10. is cleped{e} theta, or a cercle, other a cifre, other a figure of nought for nought it signyfieth{e}. Nathelesse she holdyng that place giveth{e} others for to signyfie; for with{e}-out cifre or cifres a pure article may not be writte. And sithen that by these .9. figures significatif{es} Ioyned{e} w{i}t{h} cifre or w{i}t{h} cifres all{e} nombres ben and may be rep{re}sented{e}, It was, nether is, no nede to fynde any more figures. And note wele that eu{er}y digite shall{e} be writte w{i}t{h} oo figure allone to it ap{ro}pred{e}. And all{e} articles by a cifre, ffor eu{er}y article is named{e} for oone of the digitis as .10. of 1.. 20. of. 2. and so of the others, &c. And all{e} nombres digitall{e} owen to be sette in the first difference: All{e} articles in the seconde. Also all{e} nombres fro .10. til an .100. [which] is excluded{e}, with .2. figures mvst be writte; And yf it be an article, by a cifre first put, and the figure y-writte toward{e} the lift hond{e}, that signifieth{e} the digit of the which{e} the article is named{e}; And yf it be a nombre componed{e}, ffirst write the digit that is a part of that componed{e}, and write to the lift side the article as it is seid{e} be-fore. All{e} nombre that is fro an hundred{e} tille a thousand{e} exclused{e}, owith{e} to be writ by .3. figures; and all{e} nombre that is fro a thousand{e} til .x. Mł. mvst be writ by .4. figures; And so forthe. And vnderstond{e} wele that eu{er}y figure sette in the first place signyfieth{e} his digit; In the second{e} place .10. tymes his digit; In the .3. place an hundred{e} so moche; In the .4. place a thousand{e} so moche; In the .5. place .x. thousand{e} so moch{e}; In the .6. place an hundred{e} thousand{e} so moch{e}; In the .7. place a thousand{e} thousand{e}. And so infynytly mvltiplying by [*Fol. 49.] these .3. 10, 100, 1000. And vnderstand{e} wele that competently me may sette vpon figure in the place of a thousand{e}, a prik{e} to shewe how many thousand{e} the last figure shall{e} rep{re}sent. We writen{e} in this art to the lift side-ward{e}, as arabien{e} writen{e}, that weren fynders of this science, other{e} for this reso{u}n, that for to kepe a custumable ordr{e} in redyng, Sette we all{e}-wey the more nombre before.

[Headnote: Chapter II. Addition.]

[Sidenote: Definition. How the numbers should be written. The method
of working. Begin at the right. The Sum is a digit, or an article,
or a composite.]

Addicio{u}n is of nombre other of nombres vnto nombre or to nombres aggregacio{u}n, that me may see that that is come therof as exc{re}ssent. In addicio{u}n, 2. ordres of figures and .2. nombres ben necessary, that is to sey, a nombre to be added{e} and the nombre wherto the addic{i}oun shold{e} be made to. The nombre to be added{e} is that þat shold{e} be added{e} therto, and shall{e} be vnderwriten; the nombre vnto the which{e} addicio{u}n shall{e} be made to is that nombre that resceyueth{e} the addicion of þat other, and shall{e} be writen above; and it is convenient that the lesse nombre be vnderwrit, and the more added{e}, than the contrary. But whether it happ{e} one other other, the same comyth{e} of, Therfor, yf þow wilt adde nombre to nombre, write the nombre wherto the addicio{u}n shall{e} be made in the omest ordre by his differences, so that the first of the lower ordre be vndre the first of the omyst ordre, and so of others. That done, adde the first of the lower ordre to the first of the omyst ordre. And of such{e} addicio{u}n, other þ{er}e grow{i}t{h} therof a digit, An article, other a composed{e}. If it be digit{us}, In the place of the omyst shalt thow write the digit excrescyng, as thus:--

+----------------------------+---+
|The resultant | 2 |
+----------------------------+---+
|To whom it shal be added{e} | 1 |
+----------------------------+---+
|The nombre to be added{e} | 1 |
+----------------------------+---+

If the article; in the place of the omyst put a-way by a cifre writte, and the digit transferred{e}, of þe which{e} the article toke his name, toward{e} the lift side, and be it added{e} to the next figure folowyng, yf ther be any figure folowyng; or no, and yf it be not, leve it [in the] void{e}, as thus:--

+---------------------------------+----+
| The resultant | 10 |
+---------------------------------+----+
| To whom it shall{e} be added{e} | 7 |
+---------------------------------+----+
| The nombre to be added{e} | 3 |
+---------------------------------+----+

+----------------------+---+---+---+---+---+
| Resultans | 2 | 7 | 8 | 2 | 7 |
+----------------------+---+---+---+---+---+
| Cui d{ebet} addi | 1 | 0 | 0 | 8 | 4 |
+----------------------+---+---+---+---+---+
| Num{erus} addend{us} | 1 | 7 | 7 | 4 | 3 |
+----------------------+---+---+---+---+---+

And yf it happe that the figure folowyng wherto the addicio{u}n shall{e} be made by [the cifre of] an article, it sette a-side; In his place write the [*Fol. 49b] [digit of the] Article as thus:--

+---------------------------------+----+
| The resultant | 17 |
+---------------------------------+----+
| To whom it shall{e} be added{e} | 10 |
+---------------------------------+----+
| The nombre to be added{e} | 7 |
+---------------------------------+----+

And yf it happe that a figure of .9. by the figure that me mvst adde [one] to, In the place of that 9. put a cifre {and} write þe article toward{e} þe lift hond{e} as bifore, and thus:--

+---------------------------------+----+
| The resultant | 10 |
+---------------------------------+----+
| To whom it shall{e} be added{e} | 9 |
+---------------------------------+----+
| The nombre to be added{e} | 1 |
+---------------------------------+----+

And yf[{3}] [therefrom grow a] nombre componed,[{4}] [in the place of the nombre] put a-way[{5}][let] the digit [be][{6}]writ þ{a}t is part of þ{a}t co{m}posid{e}, and þan put to þe lift side the article as before, and þus:--

+---------------------------------+----+
| The resultant | 12 |
+---------------------------------+----+
| To whom it shall{e} be added{e} | 8 |
+---------------------------------+----+
| The nombre to be added{e} | 4 |
+---------------------------------+----+

This done, adde the seconde to the second{e}, and write above oþ{er} as before.

[Sidenote: The translator’s note.]

Note wele þ{a}t in addic{i}ons and in all{e} spices folowyng, whan he seith{e} one the other shall{e} be writen aboue, and me most vse eu{er} figure, as that eu{er}y figure were sette by half{e}, and by hym-self{e}.

[Headnote: Chapter III. Subtraction.]

[Sidenote: Definition of Subtraction. How it may be done. What is
required. Write the greater number above. Subtract the first figure
if possible. If it is not possible ‘borrow ten,’ and then subtract.]

Subtraccio{u}n is of .2. p{ro}posed{e} nombres, the fyndyng of the excesse of the more to the lasse: Other subtraccio{u}n is ablacio{u}n of o nombre fro a-nother, that me may see a some left. The lasse of the more, or even of even, may be w{i}t{h}draw; The more fro the lesse may neu{er} be. And sothly that nombre is more that hath{e} more figures, So that the last be signyficatife{s}: And yf ther ben as many in that one as in that other, me most deme it by the last, other by the next last. More-ou{er} in w{i}t{h}-drawyng .2. nombres ben necessary; A nombre to be w{i}t{h}draw, And a nombre that me shall{e} w{i}t{h}-draw of. The nombre to be w{i}t{h}-draw shall{e} be writ in the lower ordre by his differences; The nombre fro the which{e} me shall{e} with{e}-draw in the omyst ordre, so that the first be vnder the first, the second{e} vnder the second{e}, And so of all{e} others. With{e}-draw therfor the first of the lower{e} ordre fro the first of the ordre above his hede, and that wolle be other more or lesse, oþ{er} egall{e}.

+---------------------------------+----+
| The remanent | 20 |
+---------------------------------+----+
| Wherof me shall{e} w{i}t{h}draw | 22 |
+---------------------------------+----+
| The nombre to be w{i}t{h}draw | 2 |
+---------------------------------+----+

yf it be egall{e} or even the figure sette beside, put in his place a cifre. And yf it be more put away þ{er}fro als many of vnitees the lower figure conteyneth{e}, and writ the residue as thus

+----------------------------------+---+---+
| The remanent | 2 | 2 |
+----------------------------------+---+---+
| Wherof me shall{e} w{i}t{h}-draw | 2 | 8 |
+----------------------------------+---+---+
| Þe nombre to be w{i}t{h}draw | | 6 |
+----------------------------------+---+---+

[*Fol. 50.]

+--------------------------+---+---+-----+---+---+---+---+---+---+
| Remane{n}s | 2 | 2 | 1 | 8 | 2 | 9 | 9 | 9 | 8 |
+--------------------------+---+---+-----+---+---+---+---+---+---+
| A quo sit subtraccio | 8 | 7 | 2 | 4 | 3 | 0 | 0 | 0 | 4 |
+--------------------------+---+---+-----+---+---+---+---+---+---+
| Numerus subt{ra}hend{us} | 6 | 5 |[{7}]|[6]| . | . | . | . | 6 |
+--------------------------+---+---+-----+---+---+---+---+---+---+

And yf it be lesse, by-cause the more may not be w{i}t{h}-draw ther-fro, borow an vnyte of the next figure that is worth{e} 10. Of that .10. and of the figure that ye wold{e} have w{i}t{h}-draw fro be-fore to-gedre Ioyned{e}, w{i}t{h}-draw þe figure be-nethe, and put the residue in the place of the figure put a-side as þ{us}:--

+----------------------------------+---+---+
| The remanent | 1 | 8 |
+----------------------------------+---+---+
| Wherof me shall{e} w{i}t{h}-draw | 2 | 4 |
+----------------------------------+---+---+
| The nombre to be w{i}t{h}-draw | 0 | 6 |
+----------------------------------+---+---+

[Sidenote: If the second figure is one.]

And yf the figure wherof me shal borow the vnyte be one, put it a-side, and write a cifre in the place þ{er}of, lest the figures folowing faile of thair{e} nombre, and þan worch{e} as it shew{i}t{h} in this figure here:--

+--------------------------------+---+---+------+
| The remanent | 3 | 0 |9[{8}]|
+--------------------------------+---+---+------+
| Wherof me shal w{i}t{h}-draw | 3 | 1 | 2 |
+--------------------------------+---+---+------+
| The nombre to be w{i}t{h}-draw | . | . | 3 |
+--------------------------------+---+---+------+

[Sidenote: If the second figure is a cipher.]

And yf the vnyte wherof me shal borow be a cifre, go ferther to the figure signyficatif{e}, and ther borow one, and reto{ur}nyng bak{e}, in the place of eu{er}y cifre þ{a}t ye passid{e} ou{er}, sette figures of .9. as here it is specified{e}:--

+----------------------------------+---+---+---+---+---+
| The remenaunt | 2 | 9 | 9 | 9 | 9 |
+----------------------------------+---+---+---+---+---+
| Wherof me shall{e} w{i}t{h}-draw | 3 | 0 | 0 | 0 | 3 |
+----------------------------------+---+---+---+---+---+
| The nombre to be w{i}t{h}-draw | | | | | 4 |
+----------------------------------+---+---+---+---+---+

[Sidenote: A justification of the rule given. Why it is better to
work from right to left. How to prove subtraction, and addition.]

And whan me cometh{e} to the nombre wherof me intendith{e}, there remayneth{e} all{e}-wayes .10. ffor þe which{e} .10. &c. The reson why þat for eu{er}y cifre left behynde me setteth figures ther of .9. this it is:--If fro the .3. place me borowed{e} an vnyte, that vnyte by respect of the figure that he came fro rep{re}sentith an .C., In the place of that cifre [passed over] is left .9., [which is worth ninety], and yit it remayneth{e} as .10., And the same reson{e} wold{e} be yf me had{e} borowed{e} an vnyte fro the .4., .5., .6., place, or ony other so vpward{e}. This done, withdraw the second{e} of the lower ordre fro the figure above his hede of þe omyst ordre, and wirch{e} as before. And note wele that in addicion or in subtracc{i}o{u}n me may wele fro the lift side begynne and ryn to the right side, But it wol be more p{ro}fitabler to be do, as it is taught. And yf thow wilt p{ro}ve yf thow have do wele or no, The figures that thow hast withdraw, adde them ayene to the omyst figures, and they wolle accorde w{i}t{h} the first that thow haddest yf thow have labored wele; and in like wise in addicio{u}n, whan thow hast added{e} all{e} thy figures, w{i}t{h}draw them that thow first [*Fol. 50b] addest, and the same wolle reto{ur}ne. The subtraccio{u}n is none other but a p{ro}uff{e} of the addicio{u}n, and the contrarye in like wise.

[Headnote: Chapter IV. Mediation.]

[Sidenote: Definition of mediation. Where to begin. If the first
figure is unity. What to do if it is not unity.]

Mediacio{u}n is the fyndyng of the halfyng of eu{er}y nombre, that it may be seyn{e} what and how moch{e} is eu{er}y half{e}. In halfyng ay oo order of figures and oo nombre is necessary, that is to sey the nombre to be halfed{e}. Therfor yf thow wilt half any nombre, write that nombre by his differences, and begynne at the right, that is to sey, fro the first figure to the right side, so that it be signyficatif{e} other rep{re}sent vnyte or eny other digitall{e} nombre. If it be vnyte write in his place a cifre for the figures folowyng, [lest they signify less], and write that vnyte w{i}t{h}out in the table, other resolue it in .60. mynvt{es} and sette a-side half of tho m{inutes} so, and reserve the remen{au}nt w{i}t{h}out in the table, as thus .30.; other sette w{i}t{h}out thus .{dī}: that kepeth{e} none ordre of place, Nathelesse it hath{e} signyficacio{u}n. And yf the other figure signyfie any other digital nombre fro vnyte forth{e}, oþ{er} the nombre is od{e} or even{e}. If it be even, write this half in this wise:--

+-----------------+---+---+
| Halfed{e} | 2 | 2 |
+-----------------+----+--+
| to be halfed{e} | 4 | 4 |
+-----------------+---+---+

And if it be odde, Take the next even vndre hym conteyned{e}, and put his half in the place of that odde, and of þe vnyte that remayneth{e} to be halfed{e} do thus:--

+-----------------+---+---+
| halfed{e} | 2 | 3 | [di]
+-----------------+---+---+
| To be halfed{e} | 4 | 7 |
+-----------------+---+---+

[Sidenote: Then halve the second figure. If it is odd, add 5 to the
figure before.]

This done, the second{e} is to be halfed{e}, yf it be a cifre put it be-side, and yf it be significatif{e}, other it is even or od{e}: If it be even, write in the place of þe nombres wiped{e} out the half{e}; yf it be od{e}, take the next even vnder it co{n}tenyth{e}, and in the place of the Impar sette a-side put half of the even: The vnyte that remayneth{e} to be halfed{e}, respect had{e} to them before, is worth{e} .10. Dyvide that .10. in .2., 5. is, and sette a-side that one, and adde that other to the next figure p{re}cedent as here:--

+-----------------+---+---+---+
| Halfed{e} | | | |
+-----------------+---+---+---+
| to be halfed{e} | | | |
+-----------------+---+---+---+

And yf þe addicio{u}n shold{e} be made to a cifre, sette it a-side, and write in his place .5. And vnder this fo{ur}me me shall{e} write and worch{e}, till{e} the totall{e} nombre be halfed{e}.

+------------------+---+---+---+---+---+----+----+---+
| doubled{e} | 2 | 6 | 8 | 9 | 0 | 10 | 17 | 4 |
+------------------+---+---+---+---+---+----+----+---+
| to be doubled{e} | 1 | 3 | 4 | 4 | 5 | 5 | 8 | 7 |
+------------------+---+---+---+---+---+----+----+---+

[Headnote: Chapter V. Duplation.]

[Sidenote: Definition of Duplation. Where to begin. Why. What to do
with the result.]

Duplicacio{u}n is ag{re}gacion of nombre [to itself] þat me may se the nombre growen. In doublyng{e} ay is but one ordre of figures necessarie. And me most be-gynne w{i}t{h} the lift side, other of the more figure, And after the nombre of the more figure rep{re}sentith{e}. [*Fol. 51.] In the other .3. before we begynne all{e} way fro the right side and fro the lasse nombre, In this spice and in all{e} other folowyng we wolle begynne fro the lift side, ffor and me bigon th{e} double fro the first, omwhile me myght double oo thynge twyes. And how be it that me myght double fro the right, that wold{e} be harder in techyng and in workyng. Therfor yf thow wolt double any nombre, write that nombre by his differences, and double the last. And of that doubly{n}g other growith{e} a nombre digital, article, or componed{e}. [If it be a digit, write it in the place of the first digit.] If it be article, write in his place a cifre and transferre the article toward{e} the lift, as thus:--

+------------------+----+
| double | 10 |
+------------------+----+
| to be doubled{e} | 5 |
+------------------+----+

And yf the nombre be componed{e}, write a digital that is part of his composicio{u}n, and sette the article to the lift hand{e}, as thus:--

+------------------+----+
| doubled{e} | 16 |
+------------------+----+
| to be doubled{e} | 8 |
+------------------+----+

That done, me most double the last save one, and what groweth{e} þ{er}of me most worche as before. And yf a cifre be, touch{e} it not. But yf any nombre shall{e} be added{e} to the cifre, in þe place of þe figure wiped{e} out me most write the nombre to be added{e}, as thus:--

+------------------+---+---+---+
| doubled{e} | 6 | 0 | 6 |
+------------------+---+---+---+
| to be doubled{e} | 3 | 0 | 3 |
+------------------+---+---+---+

[Sidenote: How to prove your answer.]

In the same wise me shall{e} wirch{e} of all{e} others. And this p{ro}bacio{u}n: If thow truly double the halfis, and truly half the doubles, the same nombre and figure shall{e} mete, such{e} as thow labo{ur}ed{e} vpon{e} first, And of the contrarie.

+------------------+---+---+---+
| Doubled{e} | 6 | 1 | 8 |
+------------------+---+---+---+
| to be doubled{e} | 3 | 0 | 9 |
+------------------+---+---+---+

[Headnote: Chapter VI. Multiplication.]

[Sidenote: Definition of Multiplication. Multiplier. Multiplicand.
Product.]

Multiplicacio{u}n of nombre by hym-self other by a-nother, w{i}t{h} p{ro}posid{e} .2. nombres, [is] the fyndyng of the third{e}, That so oft conteyneth{e} that other, as ther ben vnytes in the oþ{er}. In multiplicacio{u}n .2. nombres pryncipally ben necessary, that is to sey, the nombre multiplying and the nombre to be multiplied{e}, as here;--twies fyve. [The number multiplying] is designed{e} adu{er}bially. The nombre to be multiplied{e} resceyveth{e} a no{m}i{n}all{e} appellacio{u}n, as twies .5. 5. is the nombre multiplied{e}, and twies is the nombre to be multipliede.

+-----------------+-----+---+---++---+---+---+---+---+---+---+---+---+
| Resultans |[{9}]| 1 | 0 || 1 | 3 | 2 | 6 | 6 | 8 | 0 | 0 | 8 |
+-----------------+-----+---+---++---+---+---+---+---+---+---+---+---+
| Multiplicand{us}| . | . | 5 || . | . | 4 | . | 3 | 4 | 0 | 0 | 4 |
+-----------------+-----+---+---++---+---+---+---+---+---+---+---+---+
| Multiplicans | . | 2 | 2 || . | 3 | 3 | 2 | 2 | 2 | . | . | . |
+-----------------+-----+---+---++---+---+---+---+---+---+---+---+---+

Also me may thervpon{e} to assigne the. 3. nombre, the which{e} is [*Fol. 51b] cleped{e} p{ro}duct or p{ro}venient, of takyng out of one fro another: as twyes .5 is .10., 5. the nombre to be multiplied{e}, and .2. the multipliant, and. 10. as before is come therof. And vnderstonde wele, that of the multipliant may be made the nombre to be multiplied{e}, and of the contrarie, remaynyng eu{er} the same some, and herof{e} cometh{e} the comen speche, that seith{e} all nombre is converted{e} by Multiplying in hym-self{e}.

+----+----+----+----+----+--------+----+----+----+-----+
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
+----+----+----+----+----+--------+----+----+----+-----+
| 2 | 4 | 6 | 8 | 10 |10[{10}]| 14 | 16 | 18 | 20 |
+----+----+----+----+----+--------+----+----+----+-----+
| 3 | 6 | 9 | 12 | 15 | 18 | 21 | 24 | 27 | 30 |
+----+----+----+----+----+--------+----+----+----+-----+
| 4 | 8 | 12 | 16 | 20 | 24 | 28 | 32 | 36 | 40 |
+----+----+----+----+----+--------+----+----+----+-----+
| 5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 | 45 | 50 |
+----+----+----+----+----+--------+----+----+----+-----+
| 6 | 12 | 18 | 24 | 30 | 36 | 42 | 48 | 56 | 60 |
+----+----+----+----+----+--------+----+----+----+-----+
| 7 | 14 | 21 | 28 | 35 | 42 | 49 | 56 | 63 | 70 |
+----+----+----+----+----+--------+----+----+----+-----+
| 8 | 16 | 24 | 32 | 40 | 48 | 56 | 64 | 72 | 80 |
+----+----+----+----+----+--------+----+----+----+-----+
| 9 | 18 | 27 | 36 | 45 | 54 | 63 | 72 | 81 | 90 |
+----+----+----+----+----+--------+----+----+----+-----+
| 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | 100 |
+----+----+----+----+----+--------+----+----+----+-----+

[Headnote: The Cases of Multiplication.]

[Sidenote: There are 6 rules of Multiplication. (1) Digit by digit.
See the table above. (2) Digit by article. (3) Composite by digit.]

And ther ben .6 rules of Multiplicacio{u}n; ffirst, yf a digit multiplie a digit, considr{e} how many of vnytees ben betwix the digit by multiplying and his .10. beth{e} to-gedre accompted{e}, and so oft w{i}t{h}-draw the digit multiplying, vnder the article of his deno{m}i{n}acio{u}n. Example of grace. If thow wolt wete how moch{e} is .4. tymes .8., [{11}]se how many vnytees ben betwix .8.[{12}] and .10. to-geder rekened{e}, and it shew{i}t{h} that .2.: withdraw ther-for the quat{e}rnary, of the article of his deno{m}i{n}acion twies, of .40., And ther remayneth{e} .32., that is, to some of all{e} the multiplicacio{u}n. Wher-vpon for more evidence and declaracion the seid{e} table is made. Whan a digit multiplieth{e} an article, thow most bryng the digit into þe digit, of þe which{e} the article [has][{13}] his name, and eu{er}y vnyte shall{e} stond{e} for .10., and eu{er}y article an .100. Whan the digit multiplieth{e} a nombre componed{e}, þ{o}u most bryng the digit into aiþ{er} part of the nombre componed{e}, so þ{a}t digit be had into digit by the first rule, into an article by þe second{e} rule; and aft{er}ward{e} Ioyne the p{ro}duccio{u}n, and þ{er}e wol be the some totall{e}.

+----------------+---+---+--++---+---+--++---+---+--++---+---+---+---+
|Resultans | 1 | 2 | 6|| 7 | 3 | 6|| 1 | 2 | 0|| 1 | 2 | 0 | 8 |
+----------------+---+---+--++---+---+--++---+---+--++---+---+---+---+
|Multiplicand{us}| | | 2|| | 3 | 2|| | | 6|| | | | 4 |
+----------------+---+---+--++---+---+--++---+---+--++---+---+---+---+
|Multiplicans | | 6 | 3|| 2 | 3 | || | 2 | 0|| | 3 | 0 | 2 |
+----------------+---+---+--++---+---+--++---+---+--++---+---+---+---+

[Sidenote: (4) Article by article. (5) Composite by article.
(6) Composite by composite. How to set down your numbers. If the
result is a digit, an article, or a composite. Multiply next by
the last but one, and so on.]

Whan an article multiplieth{e} an article, the digit wherof he is named{e} is to be brought Into the digit wherof the oþ{er} is named{e}, and eu{er}y vnyte wol be worth{e} [*Fol. 52.] an .100., and eu{er}y article. a .1000. Whan an article multiplieth{e} a nombre componed{e}, thow most bryng the digit of the article into aither part of the nombre componed{e}; and Ioyne the p{ro}duccio{u}n, and eu{er}y article wol be worth{e} .100., and eu{er}y vnyte .10., and so woll{e} the some be open{e}. Whan a nombre componed{e} multiplieth{e} a nombre componed{e}, eu{er}y p{ar}t of the nombre multiplying is to be had{e} into eu{er}y p{ar}t of the nombre to be multiplied{e}, and so shall{e} the digit be had{e} twies, onys in the digit, that other in the article. The article also twies, ones in the digit, that other in the article. Therfor yf thow wilt any nombre by hym-self other by any other multiplie, write the nombre to be multiplied{e} in the ou{er} ordre by his differences, The nombre multiplying in the lower ordre by his differences, so that the first of the lower ordre be vnder the last of the ou{er} ordre. This done, of the multiplying, the last is to be had{e} into the last of the nombre to be multiplied{e}. Wherof than wolle grow a digit, an article, other a nombre componed{e}. If it be a digit, even above the figure multiplying is hede write his digit that come of, as it appereth{e} here:--

+-----------------------+---+
| The resultant | 6 |
+-----------------------+---+
| To be multiplied{e} | 3 |
+-----------------------+---+
| Þe nombre multipliyng | 2 |
+-----------------------+---+

And yf an article had be writ ou{er} the fig{ur}e multiplying his hede, put a cifre þ{er} and transferre the article toward{e} the lift hand{e}, as thus:--

+-------------------------+---+---+
| The resultant | 1 | 0 |
+-------------------------+---+---+
| to be multiplied{e} | | 5 |
+-------------------------+---+---+
| þe nombre m{u}ltipliyng | | 2 |
+-------------------------+---+---+

And yf a nombre componed{e} be writ ou{er} the figure multyplying is hede, write the digit in the nombre componed{e} is place, and sette the article to the lift hand{e}, as thus:--

+------------------------+---+---+
| Resultant | 1 | 2 |
+------------------------+---+---+
| to be multiplied{e} | | 4 |
+------------------------+---+---+
| the nombre multipliyng | | 3 |
+------------------------+---+---+

This done, me most bryng the last save one of the multipliyng into the last of þe nombre to be multiplied{e}, and se what comyth{e} therof as before, and so do w{i}t{h} all{e}, tille me come to the first of the nombre multiplying, that must be brought into the last of the nombre to be multiplied{e}, wherof growith{e} oþ{er} a digit, an article, [*Fol. 52b] other a nombre componed{e}. If it be a digit, In the place of the ou{er}er, sette a-side, as here:

+--------------------------+---+---+
| Resultant | 6 | 6 |
+--------------------------+---+---+
| to be multiplied{e} | | 3 |
+--------------------------+---+---+
| the nombre m{u}ltipliyng | 2 | 2 |
+--------------------------+---+---+

If an article happe, there put a cifre in his place, and put hym to the lift hand{e}, as here:

+-------------------------+---+---+---+
| The resultant | 1 | 1 | 0 |
+-------------------------+---+---+---+
| to be multiplied{e} | | | 5 |
+-------------------------+---+---+---+
| þe nombre m{u}ltiplying | | 2 | 2 |
+-------------------------+---+---+---+

If it be a nombre componed{e}, in the place of the ou{er}er sette a-side, write a digit that[{14}] is a p{ar}t of the componed{e}, and sette on the left hond{e} the article, as here:

+-----------------------------+---+-------+---+
| The resultant | 1 |3[{15}]| 2 |
+-----------------------------+---+-------+---+
| to be m{u}ltiplied{e} | | | 4 |
+-----------------------------+---+-------+---+
| þe nombr{e} m{u}ltiplia{n}t | | 3 | 3 |
+-----------------------------+---+-------+---+

[Sidenote: Then antery the multiplier one place. Work as before.
How to deal with ciphers.]

That done, sette forward{e} the figures of the nombre multiplying by oo difference, so that the first of the multipliant be vnder the last save one of the nombre to be multiplied{e}, the other by o place sette forward{e}. Than me shall{e} bryng{e} the last of the m{u}ltipliant in hym to be multiplied{e}, vnder the which{e} is the first multipliant. And than wolle growe oþ{er} a digit, an article, or a componed{e} nombre. If it be a digit, adde hym even above his hede; If it be an article, transferre hym to the lift side; And if it be a nombre componed{e}, adde a digit to the figure above his hede, and sette to the lift hand{e} the article. And all{e}-wayes eu{er}y figure of the nombre multipliant is to be brought to the last save one nombre to be multiplied{e}, til me come to the first of the multipliant, where me shall{e} wirche as it is seid{e} before of the first, and aft{er}ward{e} to put forward{e} the figures by o difference and one till{e} they all{e} be multiplied{e}. And yf it happe that the first figure of þe multipliant be a cifre, and boue it is sette the figure signyficatif{e}, write a cifre in the place of the figur{e} sette a-side, as thus, {et}c.:

+---------------------+---+---+---+
| The resultant | 1 | 2 | 0 |
+---------------------+---+---+---+
| to be multiplied{e} | | | 6 |
+---------------------+---+---+---+
| the multipliant | | 2 | 0 |
+---------------------+---+---+---+

[Sidenote: How to deal with ciphers.]

And yf a cifre happe in the lower order be-twix the first and the last, and even above be sette the fig{ur}e signyficatif, leve it vntouched{e}, as here:--

+---------------------+---+---+---+---+---+
| The resultant | 2 | 2 | 6 | 4 | 4 |
+---------------------+---+---+---+---+---+
| To be multiplied{e} | | | 2 | 2 | 2 |
+---------------------+---+---+---+---+---+
| The multipliant | 1 | 0 | 2 | | |
+---------------------+---+---+---+---+---+

And yf the space above sette be void{e}, in that place write thow a cifre. And yf the cifre happe betwix þe first and the last to be m{u}ltiplied{e}, me most sette forward{e} the ordre of the figures by thair{e} differences, for oft of duccio{u}n of figur{e}s in cifres nought is the resultant, as here,

+-----------------------+---+---+---+---+---+
| Resultant | 8 | 0 | 0 | 8 | |
+-----------------------+---+---+---+---+---+
| to be m{u}ltiplied{e} | 4 | 0 | 0 | 4 | |
+-----------------------+---+---+---+---+---+
| the m{u}ltipliant | 2 | . | . | . | |
+-----------------------+---+---+---+---+---+

[*Fol. 53.] wherof it is evident and open, yf that the first figure of the nombre be to be multiplied{e} be a cifre, vndir it shall{e} be none sette as here:--

+-----------------------+---+---+--------+
| Resultant | 3 | 2 |0[{16}] |
+-----------------------+---+---+--------+
| To be m{u}ltiplied{e} | | 8 | 0 |
+-----------------------+---+---+--------+
| The m{u}ltipliant | | 4 | |
+-----------------------+---+---+--------+

[Sidenote: Leave room between the rows of figures.]

Vnder[stand] also that in multiplicacio{u}n, divisio{u}n, and of rootis the extraccio{u}n, competently me may leve a mydel space betwix .2. ordres of figures, that me may write there what is come of addyng other with{e}-drawyng, lest any thynge shold{e} be ou{er}-hipped{e} and sette out of mynde.

[Headnote: Chapter VII. Division.]

[Sidenote: Definition of division. Dividend, Divisor, Quotient.
How to set down your Sum. An example. Examples.]

For to dyvyde oo nombre by a-nother, it is of .2. nombres p{ro}posed{e}, It is forto depart the moder nombre into as many p{ar}tis as ben of vnytees in the lasse nombre. And note wele that in makyng{e} of dyvysio{u}n ther ben .3. nombres necessary: that is to sey, the nombre to be dyvyded{e}; the nombre dyvydyng and the nombre exeant, other how oft, or quocient. Ay shall{e} the nombre that is to be dyvyded{e} be more, other at the lest even{e} w{i}t{h} the nombre the dyvysere, yf the nombre shall{e} be mad{e} by hole nombres. Therfor yf thow wolt any nombre dyvyde, write the nombre to be dyvyded{e} in þe ou{er}er bordur{e} by his differences, the dyviser{e} in the lower ordur{e} by his differences, so that the last of the dyviser be vnder the last of the nombre to be dyvyde, the next last vnder the next last, and so of the others, yf it may competently be done; as here:--

+------------------+---+---+---+
| The residue | | 2 | 7 |
+------------------+---+---+---+
| The quotient | | | 5 |
+------------------+---+---+---+
| To be dyvyded{e} | 3 | 4 | 2 |
+------------------+---+---+---+
| The dyvyser | | 6 | 3 |
+------------------+---+---+---+

+--------------+---+---+----+---+---++---+---+---++---+---+---+
| Residuu{m} | | | 8 || | || | 2 | 7 || | 2 | 6 |
+--------------+---+---+---++---+---++---+---+---++---+---+---+
| Quociens | | 2 | 1 || 2 | 2 || | | 5 || | | 9 |
+--------------+---+---+---++---+---++---+---+---++---+---+---+
| Diuidend{us} | 6 | 8 | 0 || 6 | 6 || 3 | 4 | 2 || 3 | 3 | 2 |
+--------------+---+---+---++---+---++---+---+---++---+---+---+
| Diuiser | 3 | 2 | || 3 | || | 6 | 3 || | 3 | 4 |
+--------------+---+---+---++---+---++---+---+---++---+---+---+

[Sidenote: When the last of the divisor must not be set below the
last of the dividend. How to begin.]