[Headnote: To multiply one Composite by another.]

Postea p{ro}cedas postrema{m} m{u}ltiplica{n}do [Recte multiplicans per cu{n}ctas i{n}feriores] Condic{i}onem tamen t{a}li q{uod} m{u}ltiplicant{es} Scribas in capite quicq{ui}d p{ro}cesserit inde Sed postq{uam} fuit hec m{u}ltiplicate fig{ur}e Anteriorent{ur} serei m{u}ltiplica{n}t{is} Et sic m{u}ltiplica velut isti m{u}ltiplicasti Qui sequit{ur} nu{mer}u{m} sc{ri}ptu{m} quiscu{n}q{ue} figur{is}.

[Sidenote: How to multiply one number by another. Multiply the 'last' figure of the higher by the 'first' of the lower number. Set the answer over the first of the lower: then multiply the second of the lower, and so on. Then antery the lower number: as thus. Now multiply by the last but one of the higher: as thus. Antery the figures again, and multiply by five: Then add all the figures above the line: and you will have the answer.]

Her{e} he teches how {o}u schalt wyrch in is craft. ou schalt m{ul}tiplye e last figur{e} of e nombre, and quen {o}u hast so ydo ou schalt draw all{e} e figures of e ne{er} nounbre mor{e} taward e ry[gh]t side, so qwe{n} {o}u hast m{u}ltiplyed e last figur{e} of e heyer nounbre by all{e} e ne{er} figures. And sette e nounbir at comes er-of ou{er} e last figur{e} of e ne{er} nounb{re}, &en ou schalt sette al e o{er} fig{ur}es of e ne{er} nounb{re} mor{e} ner{e} to e ry[gh]t side. And whan ou hast m{u}ltiplied at figur{e} at schal be m{u}ltiplied e next aft{er} hym by al e ne{er} figures. And worch as ou dyddyst afor{e} til [*leaf 156b] ou come to e ende. And ou schalt vnd{er}stonde at eu{er}y figur{e} of e hier nounb{re} schal be m{u}ltiplied be all{e} e figur{e}s of the ne{er} nounbre, yf e hier nounb{re} be any figur{e} en on{e}. lo an Ensampul her{e} folowyng{e}.

2465 . 232

ou schalt begyne to m{u}ltiplye in e lyft side. M{u}ltiply 2 be 2, and twyes 2 is 4. set 4 ou{er} e hed of {a}t 2, en m{u}ltiplie e same hier 2 by 3 of e nether nounbre, as thryes 2 at schal be 6. set 6 ou{er} e hed of 3, an m{u}ltiplie e same hier 2 by at 2 e quych stondes vnd{er} hym, {a}t wol be 4; do away e hier 2 & sette {ere} 4. Now {o}u most antery e nether nounbre, at is to say, {o}u most sett e ne{er} nounbre more towarde e ry[gh]t side, as us. Take e ne{er} 2 toward e ry[gh]t side, &sette it euen vnd{er} e 4 of e hyer nounb{r}e, & ant{er}y all{e} e figures at comes aft{er} at 2, as us; sette 2 vnd{er} e 4. en sett e figur{e} of 3 {ere} at e figure of 2 stode, e quych is now vndur {a}t 4 in e hier nounbre; en sett e oer figur{e} of 2, e quych is e last fig{ur}e toward e lyft side of e ne{er} nomb{er} {ere} e figur{e} of 3 stode. en {o}u schalt haue such a nombre.

464465 232

[*leaf 157a] Now m{u}ltiply 4, e quych comes next aft{er} 6, by e last 2 of e ne{er} nounbur toward e lyft side. as 2 tymes 4, at wel be 8. sette at 8 ou{er} e figure the quych stondes ou{er} e hede of at 2, e quych is e last figur{e} of e ne{er} nounbre; an multiplie at same 4 by 3, at comes in e ne{er} rewe, at wol be 12. sette e digit of e composyt ou{er} e figure e quych stondes ou{er} e hed of at 3, &sette e articule of is co{m}posit ou{er} al e figures at stondes ou{er} e ne{er} 2 hede. en m{u}ltiplie e same 4 by e 2 in e ry[gh]t side in e ne{er} nounbur, at wol be 8. do away 4. & sette {ere} 8. Eu{er} mor{e} qwen {o}u m{u}ltiplies e hier figur{e} by at figur{e} e quych stondes vnd{er} hym, ou schalt do away at hier figur{e}, & sett er at nounbre e quych comes of m{u}ltiplicacion of ylke digittes. Whan ou hast done as I haue byde e, {o}u schalt haue suych an ord{er} of figur{e} as is her{e},

1 . 82 4648[65] 232

en take and ant{er}y i ne{er} figures. And sett e fyrst fig{ur}e of e ne{er} figures[{11}] vndre be figur{e} of 6. And draw al e o{er} figures of e same rewe to hym-warde, [*leaf 157b] as {o}u diddyst afore. en m{u}ltiplye 6 be 2, &sett at e quych comes ou{er} {ere}-of ou{er} al e o{er} figures hedes at stondes ou{er} at 2. en m{u}ltiply 6 be 3, &sett all{e} at comes {ere}-of vpon all{e} e figur{e}s hedes at standes ou{er} at 3; a{n} m{u}ltiplye 6 be 2, e quych stondes vnd{er} at 6, en do away 6 & write {ere} e digitt of e composit at schal come {ere}of, &sette e articull ou{er} all{e} e figures at stondes ou{er} e hede of at 3 as her{e},

11 121 828 464825 232

en ant{er}y i figures as ou diddyst afor{e}, and m{u}ltipli 5 be 2, at wol be 10; sett e 0 ou{er} all e figures {a}t stonden ou{er} at 2, &sett {a}t 1. ou{er} the next figures hedes, all{e} on hye towarde e lyft side. en m{u}ltiplye 5 be 3. at wol be 15, write 5 ou{er} e figures hedes at stonden ou{er} {a}t 3, & sett at 1 ou{er} e next figur{e}s hedes toward e lyft side. en m{u}ltiplye 5 be 2, at wol be 10. do away at 5 & sett {ere} a 0, & sett at 1 ou{er} e figures hedes at stonden ou{er} 3. And en ou schalt haue such a nounbre as here stondes aftur.[*leaf 158a]

11 1101 1215 82820 4648 232

Now draw all{e} ese figures downe toged{er} as us, 6.8.1. & 1 draw to-gedur; at wolle be 16, do away all{e} ese figures saue 6. lat hym stonde, for ow {o}u take hym away ou most write er e same a[gh]ene. {ere}for{e} late hym stonde, &sett 1 ou{er} e figur{e} hede of 4 toward e lyft side; en draw on to 4, at woll{e} be 5. do away at 4 & at 1, &sette {ere} 5. en draw 4221 & 1 toged{ur}, at wol be 10. do away all{e} at, &write ere at 4 & at 0, &sett at 1 ou{er} e next figur{es} hede toward e lyft side, e quych is 6. en draw at 6 & at 1 togedur, &at wolle be 7; do away 6 & sett {ere} 7, en draw 8810 & 1, &at wel be 18; do away all{e} e figures {a}t stondes ou{er} e hede of at 8, &lette 8 stonde stil, &write at 1 ou{er} e next fig{u}r{is} hede, e quych is a 0. en do away at 0, &sett {ere} 1, e quych stondes ou{er} e 0. hede. en draw 2, 5, &1 toged{ur}, at woll{e} be 8. en do away all{e} at, &write {ere}8. And en ou schalt haue is nounbre, 571880.

[Headnote: The Cases of this Craft.]

[*leaf 158b]

S{ed} cu{m} m{u}ltiplicabis, p{ri}mo sic e{st} op{er}andu{m}, Si dabit articulu{m} tibi m{u}ltiplicacio solu{m}; P{ro}posita cifra su{m}ma{m} t{ra}nsferre meme{n}to.

[Sidenote: What to do if the first multiplication results in an article.]

Her{e} he puttes e fyrst case of is craft, e quych is is: yf {ere} come an articulle of e m{u}ltiplicacion ysette befor{e} the articull{e} in e lyft side as us

- 51 . 23 -

multiplye 5 by 2, at wol be 10; sette ou{er} e hede of at 2 a 0, &sett at on, at is e articul, in e lyft side, at is next hym, en {o}u schalt haue is nounbre

1051 . 23

And en worch forth as ou diddist afore. And {o}u schalt vnd{er}stonde at {o}u schalt write no 0. but whan at place where ou schal write at 0 has no figure afore hy{m} no{er} aft{er}. v{er}sus.

Si aut{em} digitus excreu{er}it articul{us}q{ue}. Articul{us}[{12}] sup{ra}p{osit}o digito salit vltra.

[Sidenote: What to do if the result is a composite number.]

Her{e} is e secunde case, e quych is is: yf hit happe at {ere} come a composyt, ou schalt write e digitte ou{er} e hede of e ne{er} figur{e} by e quych {o}u multipliest e hier figure; and sett e articull{e} next hym toward e lyft side, as ou diddyst afore, as {us}

- 83 . 83 -

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

- 6483[{13}] , 83 -

And en 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 e thryde case, e quych is is: yf hit happe at of i m{u}ltiplicacioun come a digit, {o}u schalt write e digit ou{er} e hede of e ne{er} figur{e}, by the quych ou m{u}ltipliest e hier{e} figur{e}, for is 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 e 4 case, e quych is: yf hit be happe at e ne{er} figur{e} schal multiplye at figur{e}, e quych stondes ou{er} at figures hede, ou schal do away e hier figur{e} & sett {er}e at {a}t comys of {a}t m{u}ltiplicacion. As yf {er}e come of at m{u}ltiplicacion an articuls ou schalt write ere e hier figur{e} stode a 0. And write e articuls in e lyft side, yf at hit be a digit write {er}e a digit. yf at h{i}t be a composit, write e digit of e composit. And e articul in e lyft side. al is is ly[gh]t y-now[gh]t, {er}e-for{e} er 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 e 5 case, e quych is is: yf [*leaf 159b] e ne{er} figur{e} schul m{u}ltiplie e hier, and at hier figur{e} is not recte ou{er} his hede. And at ne{er} figur{e} hase o{er} figures, or on figure ou{er} his hede by m{u}ltiplicacion, at hase be afor{e}, ou schalt write at nounbre, e quych comes of at, ou{er} all{e} e ylke figures hedes, as us here:

- 236 234 -

Multiply 2 by 2, at wol be 4; set 4 ou{er} e hede of at 2. en[{14}] m{u}ltiplies e hier 2 by e ne{er} 3, at wol be 6. set ou{er} his hede 6, multiplie e hier 2 by e ne{er} 4, at wol be 8. do away e hier 2, e quych stondes ou{er} e hede of e figur{e} of4, and set {er}e 8. And ou schalt haue is nounb{re} here

- 46836 234 -

And antery i figur{e}s, at is to say, set i ne{er} 4 vnd{er} e hier 3, and set i 2 other figures ner{e} hym, so at e ne{er} 2 stonde vnd{ur} e hier 6, e quych 6 stondes in e lyft side. And at 3 at stondes vndur 8, as us aftur [gh]e mayse,

- 46836 234 -

Now worch forthermor{e}, And m{u}ltiplye at hier 3 by 2, at wol be 6, set {a}t 6 e quych stondes ou{er} e hede of at 2, And en worch as I ta[gh]t e 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 e 6 case, e quych is is: yf hit happe at e figur{e} by e quych ou schal m{u}ltiplye e hier figur{e}, e quych stondes ryght ou{er} hym by a 0, ou schalt do away at figur{e}, e quych ou{er} at cifre hede. And write {ere} at nounbre at comes of e m{u}ltiplicacion as us, 23. do away 2 and sett {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 e 7 case, e quych is is: yf a 0 schal m{u}ltiply a figur{e}, e quych stondes not recte ou{er} hym, And ou{er} at 0 stonde no thyng, ou schalt write ou{er} at 0 ano{er} 0 as us:

24 03

multiplye 2 be a 0, it wol be nothyng{e}. write ere a 0 ou{er} e hede of e ne{er} 0, And en worch forth til ou come to e 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 e 8 case, e quych is is: yf {ere} be a 0 or mony cifers in e hier rewe, {o}u schalt not m{u}ltiplie hem, bot let hem stonde. And antery e figures benee to e next figur{e} sygnificatyf as us:

- 00032 . 22 -

Ou{er}-lepe all{e} ese cifers & sett at [*leaf 160b] ne{er} 2 at stondes toward e ryght side, and sett hym vnd{ur} e 3, and sett e o{er} nether 2 nere hym, so at he stonde vnd{ur} e thrydde 0, e quych stondes next 3. And an 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 ou schalt know whe{er} ou hase wel I-do or no. And he says at ou schalt deuide all{e} e nounb{r}e at comes of e m{u}ltiplicacion by e ne{er} figures. And en ou schalt haue e same nounbur at {o}u hadyst in e begynnyng{e}. but [gh]et ou hast not e craft of dyuision, but {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 e to m{u}ltiplie be ow[gh]t figures in i mynde. And e sentence of is v{er}se is is: yf o{u} wel m{u}ltiplie on nounbre by ano{er} in i mynde, {o}u schal haue {er}eto rewles in e v{er}ses at 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 ou hast afor{e} to m{u}ltiplie a digit be ano{er}, as yf ou wolde wete qwat is sex tymes 6. ou [*leaf 161a] schalt wete by e rewle at I ta[gh]t e befor{e}, yf ou haue mynde {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 e furst rewle, e quych is is: yf ou wel m{u}ltiplie an articul be ano{er}, so at both e articuls bene w{i}t{h}-Inne an hundreth, us {o}u schalt do. take e digit of bothe the articuls, for eu{er}y articul hase a digit, en m{u}ltiplye at on digit by at o{er}, and loke how mony vnytes ben in e nounbre at comes of e m{u}ltiplicacion of e 2 digittes, &so mony hundrythes ben in e nounb{re} at schal come of e m{u}ltiplicacion of e ylke 2 articuls as us. yf {o}u wold wete qwat is ten tymes ten. take e digit of ten, e quych is 1; take e digit of at o{er} ten, e quych is on. Also m{u}ltiplie 1 be 1, as on tyme on at is but 1. In on is but on vnite as ou wost welle, {er}efor{e} ten tymes ten is but a hundryth. Also yf ou wold wete what is twenty tymes 30. take e digit of twenty, at is 2; & take e digitt of thrytty, at is 3. m{u}ltiplie 3 be 2, at is 6. Now in 6 ben 6 vnites, And so mony hundrythes ben in 20 tymes 30[*leaf 161b], {ere}for{e} 20 tymes 30 is 6 hundryth euen. loke & se. But yf it be so at on{e} articul be w{i}t{h}-Inne an hundryth, or by-twene an hundryth and a thowsande, so at it be not a owsande fully. en loke how mony vnytes ben in e nounbur at comys of e m{u}ltiplicacion [{16}]And so mony tymes[{16}] of 2 digitt{es} of ylke articuls, so mony thowsant ben in e nounbre, the qwych comes of e m{u}ltiplicacion. And so mony tymes ten thowsand schal be in e nounbre at comes of e m{u}ltiplicacion of 2 articuls, as yf {o}u wold wete qwat is 4 hundryth tymes [two hundryth]. Multiply 4 be 2,[{17}] at wol be 8. in 8 ben 8 vnites. And so mony tymes ten thousand be in 4 hundryth tymes [2][{17}] hundryth, {a}t is 80 thousand. Take hede, Ischall telle e a gen{e}rall{e} rewle whan {o}u hast 2 articuls, And ou wold wete qwat comes of e m{u}ltiplicacion of hem 2. m{u}ltiplie e digit of {a}t on articuls, and kepe at nounbre, en loke how mony cifers schuld go befor{e} at on articuls, and he wer{e} write. Als mony cifers schuld go befor{e} at other, &he wer{e} write of cifers. And haue all{e} e ylke cifers toged{ur} in i mynde, [*leaf 162a] a-rowe ychon aftur other, and in e last plase set e nounbre at comes of e m{u}ltiplicacion of e 2 digittes. And loke in i mynde in what place he stondes, wher{e} in e secunde, or in e thryd, or in e 4, or wher{e} ellis, and loke qwat e figures by-token in at place; & so mych is e nounbre at comes of e 2 articuls y-m{u}ltiplied to-ged{ur} as us: yf {o}u wold wete what is 20 thousant tymes 3 owsande. m{u}ltiply e digit of at articull{e} e quych is 2 by e digitte of at o{er} articul e quych is 3, at wol be 6. en 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 owsant. ffor is figure 2 in e fyrst place betokenes twene. In e secunde place hit betokenes twenty. In e 3. place hit betokenes 2 hundryth. .. In e 4 place 2 thousant. In e 5 place h{i}t betokenes twenty ousant. {ere}for{e} he most haue 4 cifers a-for{e} hym at he may sto{n}de in e 5 place. kepe ese 4 cifers in thy mynde, en loke how mony cifers gon to 3 thousant. Certayn to 3 thousante [*leaf 162b] gon 3 cifers afor{e}. Now cast ylke 4 cifers at schuld go to twenty thousant, And thes 3 cifers at schuld go afor{e} 3 thousant, &sette hem in rewe ychon aft{er} o{er} in i mynde, as ai schuld stonde in a tabull{e}. And en schal ou haue 7 cifers; en sett at 6 e quych comes of e m{u}ltiplicacion of e 2 digitt{es} aft{u}r e ylke cifers in e 8 place as yf at hit stode in a tabul. And loke qwat a figur{e} of 6 schuld betoken in e 8 place. yf hit wer{e} in a tabul & so mych it is. & yf at figure of 6 stonde in e fyrst place he schuld betoken but 6. In e 2 place he schuld betoken sexty. In the 3 place he schuld betoken sex hundryth. In e 4 place sex thousant. In e 5 place sexty owsant. In e sext place sex hundryth owsant. In e 7 place sex owsant thousant{es}. In e 8 place sexty owsant thousantes. {er}for{e} sett 6 in octauo loco, And he schal betoken sexty owsant thousantes. And so mych is twenty owsant tymes 3 thousant, And is rewle is gen{er}all{e} for all{e} man{er} of articuls, Whethir ai be hundryth or owsant; but {o}u most know well e craft of e wryrchyng{e} in e tabull{e} [*leaf 163a] or ou know to do us in i mynde aftur is rewle. Thou most at is rewle holdye note but wher{e} {ere} ben 2 articuls and no mo of e quych ayther of hem hase but on figur{e} significatyf. As twenty tymes 3 thousant or 3 hundryth, and such o{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 e thryde rewle, e quych is is. yf {o}u wel m{u}ltiply in i mynde, And e Articul be a digitte, ou schalt loke at e digitt be w{i}t{h}-Inne an hundryth, en ou schalt m{u}ltiply the digitt of e Articulle by e oer digitte. And eu{er}y vnite in e nounbre at schall{e} come {ere}-of schal betoken ten. As us: yf at {o}u wold wete qwat is twyes 40. m{u}ltiplie e digitt{e} of 40, e quych is 4, by e o{er} diget, e quych is 2. And at wolle be 8. And in e nombre of 8 ben 8 vnites, &eu{er}y of e ylke vnites schuld stonde for 10. {ere}-fore {ere} schal be 8 tymes 10, at wol be 4 score. And so mony is twyes 40. If e articul be a hundryth or be 2 hundryth And a owsant, so at hit be notte a thousant, [*leaf 163b] worch as o{u} dyddyst afor{e}, saue {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 e 4 rewle, e quych is is: yf ou m{u}ltipliy on composit be a digit as 6 tymes 24, [{19}]en take e diget of at composit, & m{u}ltiply {a}t digitt by at o{er} diget, and kepe e nomb{ur} at comes {ere}-of. en take e digit of at composit, &m{u}ltiply at digit by ano{er} diget, by e quych {o}u hast m{u}ltiplyed e diget of e articul, and loke qwat comes {ere}-of. en take {o}u at nounbur, & cast hit to at other nounbur at {o}u secheste as us yf ou wel wete qwat comes of 6 tymes 4 & twenty. multiply at articull{e} of e composit by e digit, e quych is 6, as yn e thryd rewle {o}u was tau[gh]t, And at schal be 6 scor{e}. en m{u}ltiply e diget of e {com}posit, [*leaf 164a] e quych is 4, and m{u}ltiply at by at other diget, e quych is 6, as ou wast tau[gh]t in e first rewle, yf {o}u haue mynde {er}of, &at 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 e 5 rewle, e quych is is: yf {o}u wel m{u}ltiply an Articul be a composit, m{u}ltiplie at Articul by e articul of e composit, and worch as ou wos tau[gh]t in e secunde rewle, of e quych rewle e v{er}se begynnes us. Articulu{m} si p{er} Relicu{m} vis m{u}ltiplicare. en m{u}ltiply e diget of e composit by at o{ir} articul aft{ir} e doctrine of e 3 rewle. take {er}of gode hede, Ip{ra}y e as us. Yf {o}u wel wete what is 24 tymes ten. Multiplie ten by 20, at wel be 2 hundryth. en m{u}ltiply e diget of e 10, e quych is 1, by e diget of e composit, e quych is 4, & {a}t [*leaf 164b] wol be 4. en reken eu{er}y vnite at is in 4 for 10, &at schal be 40. Cast 40 to 2 hundryth, &at 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 e 6 rewle, &e last of all{e} multiplicacion, e quych is is: yf {o}u wel m{u}ltiplye a {com}posit by a-no{er} composit, ou schalt do us. m{u}ltiplie {a}t on composit, qwych {o}u welt of the twene, by e articul of e to{er} composit, as {o}u wer{e} tau[gh]t in e 5 rewle, en m{u}ltiplie {a}t same composit, e quych ou hast m{u}ltiplied by e o{er} articul, by e digit of e o{er} composit, as {o}u was tau[gh]t in e 4 rewle. As us, yf ou wold wete what is 11 tymes 13, as {o}u was tau[gh]t in e 5 rewle, &at schal be an hundryth & ten, aft{er}warde m{u}ltiply at same co{m}posit {a}t {o}u hast m{u}ltiplied, e quych is a .11. And m{u}ltiplye hit be e digit of e o{er} composit, e quych is 3, for 3 is e digit of 13, And at wel be 30. en take e digit of at composit, e quych composit ou m{u}ltiplied by e digit of {a}t o{er} {com}posit, [*leaf 165a] e quych is a 11. Also of the quych 11 on is e digit. m{u}ltiplie at digitt by e digett of at oth{er} composit, e quych diget is 3, as {o}u was tau[gh]t in e first rewle i{n} e begynnyng{e} of is craft. e quych rewle begynn{es} "In digitu{m} cures." And of all{e} e m{u}ltiplicacion of e 2 digitt comys thre, for onys 3 is but 3. Now cast all{e} ese nounbers toged{ur}, the quych is is, ahundryth & ten & 30 & 3. And al at wel be 143. Write 3 first in e ryght side. And cast 10 to 30, at wol be 40. set 40 next aft{ur} towarde e 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 ofMS.] [8: After 'craft' insert 'the .4. what is e p{ro}fet of is craft.'] [9: After 'sythes' insert '& is wordes fyue sithe & sex sythes.'] [10: 't'l' marked for erasure before 'tyl' in MS.] [11: Here 'of e same rew' is marked for erasure in MS.] [12: 's{ed}' deleted in MS.] [13: 6883 in MS.] [14: 'en' overwritten on 'at' 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} o{er} myxt. Another digitall{e} is a nombre w{i}t{h}-in .10.; Article is {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 e o{er}s by ordure, yentende 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. Ml. 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}, aprik{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, anombre 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 at 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 at 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 ow 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 {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 e 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 e article toward{e} e 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 {a}t is part of {a}t co{m}posid{e}, and an put to e lift side the article as before, and us:—

- + 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 o{er} as before.

[Sidenote: The translator's note.]

Note wele {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; Anombre 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, o{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 {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 - - e 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 e figure be-nethe, and put the residue in the place of the figure put a-side as {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 {er}of, lest the figures folowing faile of thair{e} nombre, and an 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 {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: Ajustification 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 e which{e} .10. &c. The reson why at 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 e 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 .{dɨ}: 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}, o{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 e 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 e 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 e 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] at 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} {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 e place of e 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 o{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 e digit, of e 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}, {o}u most bryng the digit into ai{er} part of the nombre componed{e}, so {a}t digit be had into digit by the first rule, into an article by e second{e} rule; and aft{er}ward{e} Ioyne the p{ro}duccio{u}n, and {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 o{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 - -+ e nombre multipliyng 2 + - -

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

- - - The resultant 1 0 - - - to be multiplied{e} 5 - - - e 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 e 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} o{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 - - - -+ e 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 - - - -+ e 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 o{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 e 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 e 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 e 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.]

And ther ben .2. causes whan the last figure may not be sette vnder the last, other that the last of the lower nombre may not be w{i}t{h}-draw of the last of the ou{er}er nombre for it is lasse than the lower, other how be it, that it myght be w{i}t{h}-draw as for hym-self fro the ou{er}er the remenaunt may not so oft of them above, other yf e last of the lower be even to the figure above his hede, and e next last o{er} the figure be-fore {a}t be more an the figure above sette. [*Fol. 53^2.] These so ordeyned{e}, me most wirch{e} from the last figure of e nombre of the dyvyser, and se how oft it may be w{i}t{h}-draw of and fro the figure aboue his hede, namly so that the remen{au}nt may be take of so oft, and to se the residue as here:—

[Sidenote: An example.]

+ + -+ -+ -+ The residue 2 6 + + -+ -+ -+ The quocient 9 + + -+ -+ -+ To be dyvyded{e} 3 3 2 + + -+ -+ -+ The dyvyser 3 4 + + -+ -+ -+

[Sidenote: Where to set the quotiente. Examples.]

And note wele that me may not with{e}-draw more than .9. tymes nether lasse than ones. Therfor se how oft e figures of the lower ordre may be w{i}t{h}-draw fro the figures of the ou{er}er, and the nombre that shew{i}t{h} e q{u}ocient most be writ ou{er} the hede of at figure, vnder the which{e} the first figure is, of the dyviser; And by that figure me most with{e}-draw all{e} o{er} figures of the lower ordir and that of the figures aboue thair{e} hedis. This so don{e}, me most sette forward{e} e figures of the diuiser by o difference toward{es} the right hond{e} and worch{e} as before; and thus:—

+ + -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ Residuu{m} . 1 2 + + -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ quo{ciens} 6 5 4 2 0 0 4 + + -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ Diuidend{us} 3 5 5 1 2 2 8 8 6 3 7 0 4 + + -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ Diuisor 5 4 3 4 4 2 3 + + -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+

- - - - - - The quocient 6 5 4 - - - - - - To be dyvyded{e} 3 5 5 1 2 2 - - - - - - The dyvyser 5 4 3 - - - - - -

[Sidenote: A special case.]

And yf it happ{e} after e settyng forward{e} of the fig{ur}es {a}t e last of the divisor may not so oft be w{i}t{h}draw of the fig{ur}e above his hede, above at fig{ur}e vnder the which{e} the first of the diuiser is writ me most sette a cifre in ordre of the nombre quocient, and sette the fig{ur}es forward{e} as be-fore be o difference alone, and so me shall{e} do in all{e} nombres to be dyvided{e}, for where the dyviser may not be w{i}t{h}-draw me most sette there a cifre, and sette forward{e} the figures; as here:—

- - - - - - -+ The residue 1 2 + - - - - - - -+ The quocient 2 0 0 4 + - - - - - - -+ To be dyvyded{e} 8 8 6 3 7 0 4 + - - - - - - -+ The dyvyser 4 4 2 3 + - - - - - - -

[Sidenote: Another example. What the quotient shows. How to prove your division, or multiplication.]

And me shall{e} not cesse fro such{e} settyng of fig{ur}es forward{e}, nether of settyng{e} of e quocient into the dyviser, ne{er} of subt{ra}ccio{u}n of the dyvyser, till{e} the first of the dyvyser be w{i}t{h}-draw fro e first to be divided{e}. The which{e} don{e}, or ought,[{17}] o{er} nought shall{e} remayne: and yf it be ought,[{17}] kepe it in the tables, And eu{er} vny it to e diviser. And yf {o}u wilt wete how many vnytees of e divisio{u}n [*Fol. 53^3.] wol growe to the nombre of the diviser{e}, the nombre quocient wol shewe it: and whan such{e} divisio{u}n is made, and {o}u lust p{ro}ve yf thow have wele done or no, Multiplie the quocient by the diviser, And the same fig{ur}es wolle come ayene that thow haddest bifore and none other. And yf ought be residue, than w{i}t{h} addicio{u}n therof shall{e} come the same figures: And so multiplicacio{u}n p{ro}vith{e} divisio{u}n, and dyvisio{u}n multiplicacio{u}n: as thus, yf multiplicacio{u}n be made, divide it by the multipliant, and the nombre quocient wol shewe the nombre that was to be multiplied{e}, {et}c.

[Headnote: Chapter VIII. Progression.]

[Sidenote: Definition of Progression. Natural Progression. Broken Progression. The 1st rule for Natural Progression. The second rule. The first rule of Broken Progression. The second rule.]

Progressio{u}n is of nombre after egall{e} excesse fro oone or tweyn{e} take ag{r}egacio{u}n. of p{ro}gressio{u}n one is naturell{e} or co{n}tynuell{e}, {a}t o{er} broken and discontynuell{e}. Naturell{e} it is, whan me begynneth{e} w{i}t{h} one, and kepeth{e} ordure ou{er}lepyng one; as .1. 2. 3. 4. 5. 6., {et}c., so {a}t the nombre folowyng{e} passith{e} the other be-fore in one. Broken it is, whan me lepith{e} fro o nombre till{e} another, and kepith{e} not the contynuel ordir{e}; as 1. 3. 5. 7. 9, {et}c. Ay me may begynne w{i}t{h} .2., as us; .2. 4. 6. 8., {et}c., and the nombre folowyng passeth{e} the others by-fore by .2. And note wele, that naturell{e} p{ro}gressio{u}n ay begynneth{e} w{i}t{h} one, and Int{er}cise or broken p{ro}gressio{u}n, omwhile begynnyth{e} w{i}th one, omwhile w{i}t{h} twayn{e}. Of p{ro}gressio{u}n naturell .2. rules ther be yove, of the which{e} the first is this; whan the p{ro}gressio{u}n naturell{e} endith{e} in even nombre, by the half therof multiplie e next totall{e} ou{er}er{e} nombre; Example of grace: .1. 2. 3. 4. Multiplie .5. by .2. and so .10. cometh{e} of, that is the totall{e} nombre {er}of. The second{e} rule is such{e}, whan the p{ro}gressio{u}n naturell{e} endith{e} in nombre od{e}. Take the more porcio{u}n of the oddes, and multiplie therby the totall{e} nombre. Example of grace 1. 2. 3. 4. 5., multiplie .5. by .3, and thryes .5. shall{e} be resultant. so the nombre totall{e} is .15. Of p{ro}gresio{u}n int{er}cise, ther ben also .2.[{18}] rules; and e first is is: Whan the Int{er}cise p{ro}gression endith{e} in even nombre by half therof multiplie the next nombre to at half{e} as .2.[{18}] 4. 6. Multiplie .4. by .3. so at is thryes .4., and .12. the nombre of all{e} the p{ro}gressio{u}n, woll{e} folow. The second{e} rule is this: whan the p{ro}gressio{u}n int{er}scise endith{e} in od{e}, take e more porcio{u}n of all{e} e nombre, [*Fol. 53^4.] and multiplie by hym-self{e}; as .1. 3. 5. Multiplie .3. by hym-self{e}, and e some of all{e} wolle be .9., {et}c.

[Headnote: Chapter IX. Extraction of Roots.]

[Sidenote: The preamble of the extraction of roots. Linear, superficial, and solid numbers. Superficial numbers. Square numbers. The root of a square number. Notes of some examples of square roots here interpolated. Solid numbers. Three dimensions of solids. Cubic numbers. All cubics are solid numbers. No number may be both linear and solid. Unity is not a number.]

Here folowith{e} the extraccio{u}n of rotis, and first in nombre q{ua}drat{es}. Wherfor me shall{e} se what is a nombre quadrat, and what is the rote of a nombre quadrat, and what it is to draw out the rote of a nombre. And before other note this divisio{u}n: Of nombres one is lyneal, ano{er} sup{er}ficiall{e}, ano{er} quadrat, ano{er} cubik{e} or hoole. lyneal is that at is considred{e} after the p{ro}cesse, havyng{e} no respect to the direccio{u}n of nombre in nombre, As a lyne hath{e} but one dymensio{u}n that is to sey after the length{e}. Nombre sup{er}ficial is {a}t cometh{e} of ledyng{e} of oo nombre into a-nother, wherfor it is called{e} sup{er}ficial, for it hath{e} .2. nombres notyng or mesuryng{e} hym, as a sup{er}ficiall{e} thyng{e} hath{e} .2. dimensions, {a}t is to sey length{e} and brede. And for bycause a nombre may be had{e} in a-nother by .2. man{er}s, {a}t is to sey other in hym-self{e}, o{er} in ano{er}, Vnderstond{e} yf it be had in hym-self, It is a quadrat. ffor dyvisio{u}n write by vnytes, hath{e} .4. sides even as a quadrangill{e}. and yf the nombre be had{e} in a-no{er}, the nombre is sup{er}ficiel and not quadrat, as .2. had{e} in .3. maketh{e} .6. that is e first nombre sup{er}ficiell{e}; wherfor it is open at all{e} nombre quadrat is sup{er}ficiel, and not co{n}u{er}tid{e}. The rote of a nombre quadrat is at nombre that is had of hym-self, as twies .2. makith{e} 4. and .4. is the first nombre quadrat, and 2. is his rote. 9. 8. 7. 6. 5. 4. 3. 2. 1. / The rote of the more quadrat .3. 1. 4. 2. 6. The most nombre quadrat 9. 8. 7. 5. 9. 3. 4. 7. 6. / the remenent ou{er} the quadrat .6. 0. 8. 4. 5. / The first caas of nombre quadrat .5. 4. 7. 5. 6. The rote .2. 3. 4. The second{e} caas .3. 8. 4. 5. The rote .6. 2. The third{e} caas .2. 8. 1. 9. The rote .5. 3. The .4. caas .3. 2. 1. The rote .1. 7. / The 5. caas .9. 1. 2. 0. 4. / The rote 3. 0. 2. The solid{e} nombre or cubik{e} is at {a}t comytħe of double ledyng of nombre in nombre; And it is cleped{e} a solid{e} body that hath{e} {er}-in .3 [dimensions] at is to sey, length{e}, brede, and thiknesse. so {a}t nombre hath{e} .3. nombres to be brought forth{e} in hym. But nombre may be had{e} twies in nombre, for other it is had{e} in hym-self{e}, o{er} in a-no{er}. If a nombre be had{e} twies in hym-self, o{er} ones in his quadrat, {a}t is the same, {a}t a cubik{e} [*Fol. 54.] is, And is the same that is solide. And yf a nombre twies be had{e} in a-no{er}, the nombre is cleped{e} solide and not cubik{e}, as twies .3. and {a}t .2. makith{e} .12. Wherfor it is opyn{e} that all{e} cubik{e} nombre is solid{e}, and not {con}u{er}tid{e}. Cubik{e} is {a}t nombre at comyth{e} of ledyng{e} of hym-self{e} twyes, or ones in his quadrat. And here-by it is open that o nombre is the roote of a quadrat and of a cubik{e}. Natheles the same nombre is not q{ua}drat and cubik{e}. Opyn{e} it is also that all{e} nombres may be a rote to a q{ua}drat and cubik{e}, but not all{e} nombre quadrat or cubik{e}. Therfor sithen e ledyng{e} of vnyte in hym-self ones or twies nought cometh{e} but vnytes, Seith{e} Boice in Arsemetrik{e}, that vnyte potencially is al nombre, and none in act. And vndirstond{e} wele also that betwix euery .2. quadrat{es} ther is a meene p{ro}porcionall{e}, That is opened{e} thus; lede the rote of o quadrat into the rote of the o{er} quadrat, and an wolle e meene shew.

[Sidenote: Examples of square roots.]

+ -+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ -+ +-+ Residuu{m} 0 4 0 0 + -+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ -+ +-+ Quadrand{e} 4 3 5 6 3 0 2 9 1 7 4 2 4 1 9 3 6 + -+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ -+ +-+ Duplum 1 2 1 0 2 6 [8] [{19}] + -+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ -+ +-+ Subduplu{m} 6 6 5 5 1 3 2 4 4 + -+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ -+ +-+

[Sidenote: A note on mean proportionals.]

Also betwix the next .2. cubikis, me may fynde a double meene, that is to sey a more meene and a lesse. The more meene thus, as to bryng{e} the rote of the lesse into a quadrat of the more. The lesse thus, If the rote of the more be brought Into the quadrat of the lesse.

[Headnote: Chapter X. Extraction of Square Root.]

[Sidenote: To find a square root. Begin with the last odd place. Find the nearest square root of that number, subtract, double it, and set the double one to the right. Find the second figure by division. Multiply the double by the second figure, and add after it the square of the second figure, and subtract.]

[{20}]To draw a rote of the nombre quadrat it is What-eu{er} nombre be p{ro}posed{e} to fynde his rote and to se yf it be quadrat. And yf it be not quadrat the rote of the most quadrat fynde out, vnder the nombre p{ro}posed{e}. Therfor yf thow wilt the rote of any quadrat nombre draw out, write the nombre by his differences, and compt the nombre of the figures, and wete yf it be od{e} or even. And yf it be even, than most thow begynne worche vnder the last save one. And yf it be od{e} w{i}t{h} the last; and forto sey it shortly, al-weyes fro the last od{e} me shall{e} begynne. Therfor vnder the last in an od place sette, me most fynd{e} a digit, the which{e} lad{e} in hym-self{e} it puttith{e} away that, at is ou{er} his hede, o{er} as neigh{e} as me may: suche a digit found{e} and w{i}t{h}draw fro his ou{er}er, me most double that digit and sette the double vnder the next figure toward{e} the right hond{e}, and his vnder double vnder hym. That done, than me most fy{n}d{e} a-no{er} digit vnder the next figure bifore the doubled{e}, the which{e} [*Fol. 54b] brought in double setteth{e} a-way all{e} that is ou{er} his hede as to reward{e} of the doubled{e}: Than brought into hym-self settith{e} all away in respect of hym-self, Other do it as nye as it may be do: other me may w{i}t{h}-draw the digit [{21}][last] found{e}, and lede hym in double or double hym, and after in hym-self{e}; Than Ioyne to-geder the p{ro}duccion{e} of them bothe, So that the first figure of the last p{ro}duct be added{e} before the first of the first p{ro}duct{es}, the second{e} of the first, {et}c. and so forth{e}, subtrahe fro the totall{e} nombre in respect of e digit.

[Sidenote: Examples.]

+ +-+-+-+-+-+-+-+-+-+-+ -+-+ -+-+ -+-+-+ The residue 5 4 3 2 + +-+-+-+-+-+-+-+-+-+-+ -+-+ -+-+ -+-+-+ To be quadred{e} 4 1 2 0 9 1 5 1 3 9 9 0 0 5 4 3 2 + +-+-+-+-+-+-+-+-+-+-+ -+-+ -+-+ -+-+-+ The double 4 0 2 4 6 0 0 + +-+-+-+-+-+-+-+-+-+-+ -+-+ -+-+ -+-+-+ The vnder double 2 0 3 1 2 3 [3] [0] [0] 0 + +-+-+-+-+-+-+-+-+-+-+ -+-+ -+-+ -+-+-+

[Sidenote: Special cases. The residue.]

And if it hap {a}t no digit may be found{e}, Than sette a cifre vndre a cifre, and cesse not till{e} thow fynde a digit; and whan thow hast founde it to double it, ne{er} to sette the doubled{e} forward{e} nether the vnder doubled{e}, Till thow fynde vndre the first figure a digit, the which{e} lad{e} in all{e} double, settyng away all{e} that is ou{er} hym in respect of the doubled{e}: Than lede hym into hym-self{e}, and put a-way all{e} in regard{e} of hym, other as nygh{e} as thow maist. That done, other ought or nought wolle be the residue. If nought, than it shewith{e} that a nombre componed{e} was the quadrat, and his rote a digit last found{e} w{i}t{h} vnder{e}-double other vndirdoubles, so that it be sette be-fore: And yf ought[{22}] remayn{e}, that shew{i}t{h} that the nombre p{ro}posed{e} was not quadrat,[{23}] [[wher-vpon{e} se the table in the next side of the next leef{e}.]] but a digit [last found with the subduple or subduples is]

[Sidenote: This table is constructed for use in cube root sums, giving the value of ab.^2]

- - - - - - - - - 1 2 3 4 5 6 7 8 9 - - - - - - - - - 2 8 12 16 20 24 28 32 36 - - - - - - - - - 3 18 27 36 45 54 63 72 81 - - - - - - - - - 4 32 48 64 80 96 112[{24}] 128 144 - - - - - - - - - 5 50 75 100 125 150 175 200 225 - - - - - - - - - 6 72 108 144 180 216 252 288 324 - - - - - - - - - 7 98 147 196 245 294 343 393 441 - - - - - - - - - 8 128 192 256 320 384 448 512 576 - - - - - - - - - 9 168 243 324 405 486 567 648 729[{25}] - - - - - - - - -

[Sidenote: How to prove the square root without or with a remainder.]

The rote of the most quadrat conteyned{e} vndre the nombre p{ro}posed{e}. Therfor yf thow wilt p{ro}ve yf thow have wele do or no, Multiplie the digit last found{e} w{i}t{h} the vnder-double o{er} vnder-doublis, and thow shalt fynde the same figures that thow haddest before; And so that nought be the [*Fol. 55.] residue. And yf thow have any residue, than w{i}t{h} the addicio{u}n {er}of that is res{er}ued{e} w{i}t{h}-out in thy table, thow shalt fynd{e} thi first figures as thow haddest them before, {et}c.

[Headnote: Chapter XI. Extraction of Cube Root.]

[Sidenote: Definition of a cubic number and a cube root. Mark off the places in threes. Find the first digit; treble it and place it under the next but one, and multiply by the digit. Then find the second digit. Multiply the first triplate and the second digit, twice by this digit. Subtract. Examples.]

Heere folowith{e} the extraccio{u}n of rotis in cubik{e} nombres; wher-for me most se what is a nombre cubik{e}, and what is his roote, And what is the extraccio{u}n of a rote. Anombre cubik{e} it is, as it is before declared{e}, that cometh{e} of ledyng of any nombre twies in hym-self{e}, other ones in his quadrat. The rote of a nombre cubik{e} is the nombre that is twies had{e} in hy{m}-self{e}, or ones in his quadrat. Wher-thurgh{e} it is open, that eu{er}y nombre quadrat or cubik{e} have the same rote, as it is seid{e} before. And forto draw out the rote of a cubik{e}, It is first to fynd{e} e nombr{e} p{ro}posed{e} yf it be a cubik{e}; And yf it be not, than thow most make extraccio{u}n of his rote of the most cubik{e} vndre the nombre p{ro}posid{e} his rote found{e}. Therfor p{ro}posed{e} some nombre, whos cubical rote {o}u woldest draw out; First thow most compt the figures by fourthes, that is to sey in the place of thousand{es}; And vnder the last thousand{e} place, thow most fynde a digit, the which{e} lad{e} in hym-self cubikly puttith{e} a-way that at is ou{er} his hede as in respect of hym, other as nygh{e} as thow maist. That done, thow most trebill{e} the digit, and that triplat is to be put vnder the .3. next figure toward{e} the right hond{e}, And the vnder-trebill{e} vnder the trebill{e}; Than me most fynd{e} a digit vndre the next figure bifore the triplat, the which{e} w{i}t{h} his vnder-trebill{e} had into a trebill{e}, aft{er}warde other vnder[trebille][{26}] had in his p{ro}duccio{u}n, putteth{e} a-way all{e} that is ou{er} it in regard{e} of[{27}] [the triplat. Then lade in hymself puttithe away that at is over his hede as in respect of hym, other as nyghe as thou maist:] That done, thow most trebill{e} the digit ayene, and the triplat is to be sette vnder the next .3. figure as before, And the vnder-trebill{e} vnder the trebill{e}: and than most thow sette forward{e} the first triplat w{i}t{h} his vndre-trebill{e} by .2. differences. And than most thow fynde a digit vnder the next figure before the triplat, the which{e} with{e} his vnder-t{r}iplat had in his triplat afterward{e}, other vnder-treblis lad in p{ro}duct [*Fol. 55b] It sitteth{e} a-way a[l~l] that is ou{er} his hede in respect of the triplat than had in hym-self cubikly,[{28}] [[it setteth{e} a-way all{e} his respect]] or as nygh{e} as ye may.

+ + +-+-+-+-+-+ -+ +-+-+-+-+ + +-+ +-+ + Residuu{m} 5 4 1 0 1 9 + + +-+-+-+-+-+ -+ +-+-+-+-+ + +-+ +-+ + Cubicandu{s} 8 3 6 5 4 3 2 3 0 0 7 6 7 1 1 6 6 7 + + +-+-+-+-+-+ -+ +-+-+-+-+ + +-+ +-+ + Triplum 6 0 1 8 4 + + +-+-+-+-+-+ -+ +-+-+ -+ + +-+ +-+ + Subt{r}iplu{m} 2 0 [3] 6 7 2 2 + + +-+-+-+-+-+ -+ +-+-+-+-+ + +-+ +-+ +

[Sidenote: Continue this process till the first figure is reached. Examples. The residue. Special cases. Special case.]

Nother me shall{e} not cesse of the fyndyng{e} of that digit, neither of his triplacio{u}n, ne{er} of the triplat-is [{29}]anteriorac{i}o{u}n, that is to sey, settyng forward{e} by .2. differences, Ne therof the vndre-triple to be put vndre the triple, Nether of the multiplicacio{u}n {er}of, Neither of the subtraccio{u}n, till{e} it come to the first figure, vnder the which{e} is a digitall{e} nombre to be found{e}, the which{e} with{e} his vndre-treblis most be had{e} in tribles, After-ward{e} w{i}t{h}out vnder-treblis to be had{e} into produccio{u}n, settyng away all{e} that is ou{er} the hed{e} of the triplat nombre, After had into hymself{e} cubikly, and sette all{e}-way that is ou{er} hym.

- - - - - - - - -+ To be cubiced{e} 1 7 2 8 3 2 7 6 8 + - - - - - - - - - The triple 3 2 9 - - - - - - - - -+ The vnder triple 1 2 [3] 3 3 + - - - - - - - - -

Also note wele that the p{ro}ducc{i}on comyng{e} of the ledyng of a digite found{e}[{30}] [[w{i}t{h} an vndre-triple / other of an vndre-triple in a triple or triplat is And after-ward{e} w{i}t{h} out vndre-triple other vndre-triplis in the p{ro}duct and ayene that p{ro}duct that cometh{e} of the ledyng{e} of a digit found{e} in hym-self{e} cubicall{e}]] me may adde to, and also w{i}t{h}-draw fro of the totall{e} nombre sette above that digit so found{e}.[{31}] [[as ther had be a divisio{u}n made as it is opened{e} before]] That done ought or nought most be the residue. If it be nought, It is open that the nombre p{ro}posed{e} was a cubik{e} nombre, And his rote a digit founde last w{i}t{h} the vnder-triples: If the rote therof wex bad{e} in hym-self{e}, and afterward{e} p{ro}duct they shall{e} make the first fig{ur}es. And yf ought be in residue, kepe that w{i}t{h}out in the table; and it is open{e} that the nombre was not a cubik{e}. but a digit last founde w{i}t{h} the vndirtriplis is rote of the most cubik{e} vndre the nombre p{ro}posed{e} conteyned{e}, the which{e} rote yf it be had{e} in hym-self{e}, And aft{er}ward{e} in a p{ro}duct of that shall{e} growe the most cubik{e} vndre the nombre p{ro}posed{e} conteyned{e}, And yf that be added{e} to a cubik{e} the residue res{er}ued{e} in the table, woll{e} make the same figures that ye had{e} first. [*Fol. 56.] And yf no digit after the anterioracio{u}n[{32}] may not be found{e}, than put ther{e} a cifre vndre a cifre vndir the third{e} figure, And put forward{e} e fig{ur}es. Note also wele that yf in the nombre p{ro}posed{e} ther ben no place of thowsand{es}, me most begynne vnder the first figure in the extraccio{u}n of the rote. some vsen forto distingue the nombre by threes, and ay begynne forto wirch{e} vndre the first of the last t{er}nary other unco{m}plete nombre, the which{e} maner of op{er}acio{u}n accordeth{e} w{i}t{h} that before. And this at this tyme suffiseth{e} in extraccio{u}n of nombres quadrat or cubik{es} {et}c.

[Sidenote: Examples.]

- - The residue 0 1 1 - - The cubicand{us} 8 0 0 0 0 0 0 8 2 4 2 4 1 9 - - The triple [{33}] 0 0 6 - - The vndert{r}iple [2] 0 0 2 6 2 - -

[Headnote: Table of Numbers, &c.]

[Sidenote: A table of numbers; probably from the Abacus.]

1 2 3 4 5 6 one. x. an. hundred{e}/ a thowsand{e}/ x. thowsand{e}/ An hundred{e} 7 thowsand{e}/ A thowsand{e} tymes a thowsand{e}/ x. thousand{e} tymes

a thousand{e}/ An hundred{e} thousand{e} tymes a thousand{e} A

thousand{e} thousand{e} tymes a thousand{e}/ this is the x place

{et}c.

[Ende.]

FOOTNOTES (The Art of Nombryng):

[1: MS. Materiall{e}.] [2: MS. Formall{e}.] [3: 'the' in MS.] [4: 'be' in MS.] [5: 'and' in MS.] [6: 'is' in MS.] [7: 6 in MS.] [8: 0 in MS.] [9: 2 in MS.] [10: sic.] [11: 'And' inserted in MS.] [12: '4 the' inserted in MS.] [13: 'to' in MS.] [14: 'that' repeated in MS.] [15: '1' in MS.] [16: Blank in MS.] [17: 'nought' in MS.] [18: 3 written for 2 in MS.] [19: 7 in MS.] [20: runs on in MS.] [21: 'so' in MS.] [22: 'nought' in MS.] [23: MS. adds here: 'wher-vpon{e} se the table in the next side of the next leef{e}.'] [24: 110 in MS.] [25: 0 in MS.] [26: double in MS.] [27: 'it hym-self{e}' in MS.] [28: MS. adds here: 'it setteth{e} a-way all{e} his respect.'] [29: 'aucterioracio{u}n' in MS.] [30: MS. adds here: 'w{i}t{h} an vndre-triple / other of an vndre-triple in a triple or triplat is And after-ward{e} w{i}t{h} out vndre-triple other vndre-triplis in the p{ro}duct and ayene that p{ro}duct that cometh{e} of the ledyng{e} of a digit found{e} in hym-self{e} cubicall{e}' /] [31: MS. adds here: 'as ther had be a divisio{u}n made as it is opened{e} before.'] [32: MS. anteriocacio{u}n.] [33: 4 in MS.]