[Headnote: The Craft of Duplation.]
++Si vis duplar{e} num{er}u{m}, sic i{n}cipe p{rim}o
Scribe fig{ur}ar{um} serie{m} q{ua}mcu{n}q{ue} vel{is} tu.
[Sidenote: Four things must be known in Duplation. Here they are.
Mind where you begin. Remember your rules.]
¶ This is the Chaptur{e} of duplacioɳ, in þe quych craft þ{o}u most haue
& know 4 thing{es}. ¶ Þe first þ{a}t þ{o}u most know is what is
duplacioɳ. þe secu{n}de is how mony rewes of fig{ur}es þ{o}u most haue
to þis craft. ¶ þe thryde is how many cases may[{6}] happe in þis craft.
¶ þe fourte is what is þe p{ro}fet of þe craft. ¶ As for þe first.
duplacioɳ is a doublyng{e} of a nombre. ¶ As for þe secu{n}de þ{o}u most
[*leaf 146b] haue on nombre or on rewe of figures, the quych called
nu{merus} dupland{us}. As for þe thrid þ{o}u most know þat 3 diu{er}se
cases may hap in þis craft. As for þe fourte. qwat is þe p{ro}fet of þis
craft, & þ{a}t is to know what a-risyȝt of a nombre I-doublyde.
¶ fforþ{er}-mor{e}, þ{o}u most know & take gode hede in quych side þ{o}u
schall{e} be-gyn in þis craft, or ellis þ{o}u mayst spyl all{e} þ{i}
lab{er} þ{er}e aboute. c{er}teyn þ{o}u schalt begyɳ in the lyft side in
þis Craft. thenke wel ou{er} þis verse. ¶ [{7}]A leua dupla, diuide,
m{u}ltiplica.[{7}] [[Subt{ra}has a{u}t addis a dext{ri}s {ve}l
medi{a}b{is}]] The sentens of þes verses afor{e}, as þ{o}u may see if
þ{o}u take hede. As þe text of þis verse, þat is to say, ¶ Si vis
duplare. þis is þe sentence. ¶ If þ{o}u wel double a nombre þus þ{o}u
most be-gynɳ. Write a rewe of figures of what nomb{re} þou welt.
v{er}sus.
Postea p{ro}cedas p{ri}ma{m} duplando figura{m}
Inde q{uo}d excrescit scribas vbi iusserit ordo
Iuxta p{re}cepta tibi que dant{ur} in addic{i}one.
[Sidenote: How to work a sum.]
¶ Her{e} he telles how þ{o}u schalt worch in þis Craft. he says, fyrst,
whan þ{o}u hast writen þe nombre þ{o}u schalt be-gyn at þe first
figur{e} in the lyft side, & doubull{e} þat fig{ur}e, & þe nombre þat
comes þ{ere}-of þ{o}u schalt write as þ{o}u diddyst in addicioɳ, as
¶ I schal telle þe in þe case. v{er}sus.
[Headnote: The Cases of the Craft of Duplation.]
[*leaf 147a]
¶ Nam si sit digitus in primo limite scribas.
[Sidenote: If the answer is a digit, write it in the place of the
top figure.]
¶ Her{e} is þe first case of þis craft, þe quych is þis. yf of duplacioɳ
of a figur{e} arise a digit. what schal þ{o}u do. þus þ{o}u schal do. do
away þe fig{ur}e þat was doublede, & sett þ{ere} þe diget þat comes of
þe duplacioɳ, as þus. 23. double 2, & þ{a}t wel be 4. do away þe
figur{e} of 2 & sett þ{ere} a figur{e} of 4, & so worch forth till{e}
þ{o}u come to þe ende. v{er}sus.
¶ Articul{us} si sit, in p{ri}mo limite cifram,
¶ Articulu{m} v{er}o reliquis inscribe figuris;
¶ Vel p{er} se scribas, si nulla figura sequat{ur}.
[Sidenote: If it is an article, put a cipher in the place, and
‘carry’ the tens. If there is no figure to ‘carry’ them to, write
them down.]
¶ Here is þe secunde case, þe quych is þis yf þ{ere} come an articull{e}
of þe duplacioɳ of a fig{ur}e þ{o}u schalt do ryȝt as þ{o}u diddyst in
addicioɳ, þat is to wete þat þ{o}u schalt do away þe figur{e} þat is
doublet & sett þ{ere} a cifer, & write þe articull{e} ou{er} þe next
figur{is} hede, yf þ{ere} be any aft{er}-warde toward þe lyft side as
þus. 25. begyn at the lyft side, and doubull{e} 2. þat wel be 4. do away
þat 2 & sett þere 4. þan doubul 5. þat wel be 10. do away 5, & sett
þ{ere} a 0, & sett 1 vpon þe next figur{is} hede þe quych is 4. & þen
draw downe 1 to 4 & þat woll{e} be 5, & þen do away þ{a}t 4 & þat 1,
& sett þ{ere} 5. for þat 1 schal be rekened in þe drawyng{e} toged{re}
for 1. wen [*leaf 147b] þou hast ydon þou schalt haue þis nomb{r}e 50.
yf þ{ere} come no figur{e} aft{er} þe fig{ur}e þ{a}t is addit, of þe
quych addicioɳ comes an articull{e}, þ{o}u schalt do away þe figur{e}
þ{a}t is dowblet & sett þ{ere} a 0. & write þe articul next by in þe
same rewe toward þe lyft syde as þus, 523. double 5 þat woll be ten. do
away þe figur{e} 5 & set þ{ere} a cifer, & sett þe articul next aft{er}
in þe same rewe toward þe lyft side, & þou schalt haue þis nombre 1023.
þen go forth & double þe oþ{er} nombers þe quych is lyȝt y-nowȝt to do.
v{er}sus.
¶ Compositus si sit, in limite sc{ri}be seq{uen}te
Articulu{m}, p{ri}mo digitu{m}; q{uia} sic iubet ordo:
Et sic de reliq{ui}s facie{n}s, si sint tibi plures.
[Sidenote: If it is a Composite, write down the digit, and ‘carry’
the tens. Here is an example.]
¶ Her{e} he putt{es} þe Thryd case, þe quych is þis, yf of duplacioɳ of
a fig{ur}e come a Composit. þ{o}u schalt do away þe fig{u}re þ{a}t is
doublet & set þ{ere} a digit of þe Composit, & sett þe articull{e}
ou{er} þe next figures hede, & aft{er} draw hym downe w{i}t{h} þe
figur{e} ou{er} whos hede he stondes, & make þ{ere}-of an nombre as
þ{o}u hast done afore, & yf þ{ere} come no fig{ur}e aft{er} þat digit
þat þ{o}u hast y-write, þa{n} set þe articull{e} next aft{er} hym in þe
same rewe as þus, 67: double 6 þat wel be 12, do away 6 & write þ{ere}
þe digit [*leaf 148a] of 12, þe quych is 2, and set þe articull{e} next
aft{er} toward þe lyft side in þe same rewe, for þ{ere} comes no
figur{e} aft{er}. þan dowble þat oþ{er} figur{e}, þe quych is 7, þat wel
be 14. the quych is a Composit. þen do away 7 þat þ{o}u doublet & sett
þe þe diget of hy{m}, the quych is 4, sett þe articull{e} ou{er} þe next
figur{es} hed, þe quych is 2, & þen draw to hym þat on, & make on nombre
þe quych schall{e} be 3. And þen yf þ{o}u haue wel y-do þ{o}u schall{e}
haue þis nombre of þe duplacioɳ, 134. v{er}sus.
¶ Si super ext{re}ma{m} nota sit monade{m} dat eid{em}
Quod t{ibi} {con}tingat si p{ri}mo dimidiabis.
[Sidenote: How to double the mark for one-half. This can only stand
over the first figure.]
¶ Her{e} he says, yf ou{er} þe fyrst fig{ur}e in þe ryȝt side be such a
merke as is her{e} made, ʷ, þ{o}u schall{e} fyrst doubull{e} þe
figur{e}, the quych stondes vnd{er} þ{a}t merke, & þen þou schalt doubul
þat merke þe quych stond{es} for haluendel on. for too haluedels makes
on, & so þ{a}t wol be on. cast þ{a}t on to þat duplacioɳ of þe figur{e}
ou{er} whos hed stode þat merke, & write it in þe same place þ{ere} þat
þe figur{e} þe quych was doublet stode, as þus 23ʷ. double 3, þat wol be
6; doubul þat halue on, & þat wol be on. cast on to 6, þ{a}t wel be 7.
do away 6 & þat 1, & sett þ{ere} 7. þan hase þou do. as for þat
figur{e}, þan go [*leaf 148b] to þe oþ{er} fig{ure} & worch forth.
& þ{o}u schall neu{er} haue such a merk but ou{er} þe hed of þe furst
figure in þe ryght side. And ȝet it schal not happe but yf it were
y-halued a-for{e}, þus þ{o}u schalt vnd{er}stonde þe verse. ¶ Si sup{er}
ext{re}ma{m} &c. Et nota, talis fig{ur}a ʷ significans medietate{m},
unitat{is} veniat, {i.e.} contingat u{e}l fiat sup{er} ext{re}ma{m},
{i.e.} sup{er} p{ri}ma{m} figura{m} in ext{re}mo sic v{er}sus dextram
ars dat: {i.e.} reddit monade{m}. {i.e.} vnitate{m} eide{m}. {i.e.}
eidem note & declina{tur} hec monos, d{i}s, di, dem, &c. ¶ Quod {er}g{o}
to{tum} ho{c} dabis monade{m} note {con}ting{et}. {i.e.} eveniet tibi si
dimidiasti, {i.e.} accipisti u{e}l subtulisti medietatem alicuius unius,
in cuius principio sint figura nu{mer}u{m} denotans i{m}pare{m} p{rim}o
{i.e.} principiis.
[Headnote: The Craft of Mediation.]
¶ Sequit{ur} de mediacione.
++Incipe sic, si vis alique{m} nu{me}ru{m} mediar{e}:
Sc{ri}be figurar{um} seriem sola{m}, velut an{te}.
[Sidenote: The four things to be known in mediation: the first the
second; the third; the fourth. Begin thus.]
¶ In þis Chapter is taȝt þe Craft of mediaciouɳ, in þe quych craft þ{o}u
most know 4 thynges. ffurst what is mediacioɳ. the secunde how mony
rewes of figur{es} þ{o}u most haue in þe wyrchyng{e} of þis craft. þe
thryde how mony diu{er}se cases may happ in þis craft.[{8}] [[the .4.
what is þe p{ro}fet of þis craft.]] ¶ As for þe furst, þ{o}u schalt
vndurstonde þat mediacioɳ is a takyng out of halfe a nomber out of a
holle nomber, [*leaf 149a] as yf þ{o}u wolde take 3 out of 6. ¶ As for
þe secunde, þ{o}u schalt know þ{a}t þ{o}u most haue on{e} rewe of
figures, & no moo, as þ{o}u hayst in þe craft of duplacioɳ. ¶ As for the
thryd, þou most vnd{er}stonde þat 5 cases may happe in þis craft. ¶ As
for þe fourte, þou schall{e} know þat the p{ro}fet of þis craft is when
þ{o}u hast take away þe haluendel of a nomb{re} to telle qwat þer{e}
schall{e} leue. ¶ Incipe sic, &c. The sentence of þis verse is þis. yf
þ{o}u wold medye, þat is to say, take halfe out of þe holle, or halfe
out of halfe, þou most begynne þ{us}. Write on{e} rewe of figur{es} of
what nombre þou wolte, as þ{o}u dyddyst be-for{e} in þe Craft of
duplacioɳ. v{er}sus.
¶ Postea p{ro}cedas medians, si p{ri}ma figura
Si par aut i{m}par videas.
[Sidenote: See if the number is even or odd.]
¶ Her{e} he says, when þ{o}u hast write a rewe of figures, þ{o}u schalt
take hede wheþ{er} þe first figur{e} be eueɳ or odde in nombre, &
vnd{er}stonde þ{a}t he spekes of þe first figure in þe ryȝt side. And
i{n} the ryght side þ{o}u schall{e} begynne in þis Craft.
¶ Quia si fu{er}it par,
Dimidiab{is} eam, scribe{n}s quicq{ui}d remanebit:
[Sidenote: If it is even, halve it, and write the answer in its
place.]
¶ Her{e} is the first case of þis craft, þe quych is þis, yf þe first
figur{e} be euen. þou schal take away fro þe figur{e} euen halfe, & do
away þat fig{ur}e and set þ{ere} þat leues ou{er}, as þus, 4. take
[*leaf 149b] halfe out of 4, & þan þ{ere} leues 2. do away 4 & sett
þ{ere} 2. þis is lyght y-nowȝt. v{er}sus.
[Headnote: The Mediation of an Odd Number.]
¶ Impar si fu{er}it vnu{m} demas mediar{e}
Quod no{n} p{re}sumas, s{ed} quod sup{er}est mediabis
Inde sup{er} tractu{m} fac demptu{m} quod no{ta}t vnu{m}.
[Sidenote: If it is odd, halve the even number less than it. Here is
an example. Then write the sign for one-half over it. Put the mark
only over the first figure.]
Her{e} is þe secunde case of þis craft, the quych is þis. yf þe first
figur{e} betoken{e} a nombre þat is odde, the quych odde schal not be
mediete, þen þ{o}u schalt medye þat nombre þat leues, when the odde of
þe same nomb{re} is take away, & write þat þ{a}t leues as þ{o}u diddest
in þe first case of þis craft. Whaɳ þ{o}u hayst write þat. for þ{a}t þat
leues, write such a merke as is her{e} ʷ vpon his hede, þe quych merke
schal betokeɳ halfe of þe odde þat was take away. lo an Ensampull. 245.
the first figur{e} her{e} is betokenyng{e} odde nombre, þe quych is 5,
for 5 is odde; þ{er}e-for{e} do away þat þ{a}t is odde, þe quych is 1.
þen leues 4. þen medye 4 & þen leues 2. do away 4. & sette þ{ere} 2,
& make such a merke ʷ upon his hede, þat is to say ou{er} his hede of 2
as þus. 242.ʷ And þen worch forth in þe oþ{er} figures tyll þ{o}u come
to þe ende. by þe furst case as þ{o}u schalt vnd{er}stonde þat þ{o}u
schalt [*leaf 150a] neu{er} make such a merk but ou{er} þe first
fig{ur}e hed in þe riȝt side. Wheþ{er} þe other fig{ur}es þat comyɳ
aft{er} hym be eueɳ or odde. v{er}sus.
[Headnote: The Cases of the Craft of Mediation.]
¶ Si monos, dele; sit t{ibi} cifra post no{ta} supra.
[Sidenote: If the first figure is one put a cipher.]
¶ Here is þe thryde case, þe quych yf the first figur{e} be a figur{e}
of 1. þ{o}u schalt do away þat 1 & set þ{ere} a cifer, & a merke ou{er}
þe cifer as þus, 241. do away 1, & sett þ{ere} a cifer w{i}t{h} a merke
ou{er} his hede, & þen hast þ{o}u ydo for þat 0. as þus 0ʷ þen worch
forth in þe oþer fig{ur}ys till þ{o}u come to þe ende, for it is lyght
as dyche water. vn{de} v{er}sus.
¶ Postea p{ro}cedas hac condic{i}one secu{n}da:
Imp{ar} si fu{er}it hinc vnu{m} deme p{ri}ori,
Inscribens quinque, nam denos significabit
Monos p{re}d{ict}am.
[Sidenote: What to do if any other figure is odd. Write a figure of
five over the next lower number’s head. Example.]
¶ Her{e} he putt{es} þe fourte case, þe quych is þis. yf it happeɳ the
secunde figur{e} betoken odde nombre, þou schal do away on of þat odde
nombre, þe quych is significatiue by þ{a}t figure 1. þe quych 1 schall
be rekende for 10. Whan þ{o}u hast take away þ{a}t 1 out of þe nombre
þ{a}t is signifiede by þat figur{e}, þ{o}u schalt medie þ{a}t þat leues
ou{er}, & do away þat figur{e} þat is medied, & sette in his styde halfe
of þ{a}t nombre. ¶ Whan þ{o}u hase so done, þ{o}u schalt write [*leaf
150b] a figure of 5 ou{er} þe next figur{es} hede by-for{e} toward þe
ryȝt side, for þat 1, þe quych made odd nombr{e}, schall stonde for ten,
& 5 is halfe of 10; so þ{o}u most write 5 for his haluendell{e}. lo an
Ensampull{e}, 4678. begyɳ in þe ryȝt side as þ{o}u most nedes. medie 8.
þen þ{o}u schalt leue 4. do away þat 8 & sette þ{ere} 4. þen out of 7.
take away 1. þe quych makes odde, & sett 5. vpon þe next figur{es} hede
afor{e} toward þe ryȝt side, þe quych is now 4. but afor{e} it was 8.
for þat 1 schal be rekenet for 10, of þe quych 10, 5 is halfe, as þou
knowest wel. Whan þ{o}u hast þus ydo, medye þ{a}t þe quych leues aft{er}
þe takying{e} away of þat þat is odde, þe quych leuyng{e} schall{e} be
3; do away 6 & sette þ{er}e 3, & þou schalt haue such a nombre
5
4634.
aft{er} go forth to þe next fig{ur}e, & medy þat, & worch forth, for it
is lyȝt ynovȝt to þe c{er}tayɳ.
¶ Si v{er}o s{e}c{un}da dat vnu{m}.
Illa deleta, sc{ri}bat{ur} cifra; p{ri}ori
¶ Tradendo quinque pro denario mediato;
Nec cifra sc{ri}batur, nisi dei{n}de fig{ur}a seq{u}at{ur}:
Postea p{ro}cedas reliq{ua}s mediando figuras
Vt sup{ra} docui, si sint tibi mille figure.
[Sidenote: If the second figure is one, put a cipher, and write
five over the next figure. How to halve fourteen.]
¶ Her{e} he putt{es} þe 5 case, þe quych is [*leaf 151a] þis: yf þe
secunde figur{e} be of 1, as þis is here 12, þou schalt do away þat 1 &
sett þ{ere} a cifer. & sett 5 ou{er} þe next fig{ur}e hede afor{e}
toward þe riȝt side, as þou diddyst afor{e}; & þat 5 schal be haldel of
þat 1, þe quych 1 is rekent for 10. lo an Ensampull{e}, 214. medye 4.
þ{a}t schall{e} be 2. do away 4 & sett þ{ere} 2. þe{n} go forth to þe
next figur{e}. þe quych is bot 1. do away þat 1. & sett þ{ere} a cifer.
& set 5 vpon þe figur{es} hed afor{e}, þe quych is nowe 2, & þen þou
schalt haue þis no{m}b{re}
5
202,
þen worch forth to þe nex fig{ur}e. And also it is no mayst{er}y yf
þ{ere} come no figur{e} after þat on is medyet, þ{o}u schalt write no 0.
ne nowȝt ellis, but set 5 ou{er} þe next fig{ur}e afor{e} toward þe
ryȝt, as þus 14. medie 4 then leues 2, do away 4 & sett þ{ere} 2. þen
medie 1. þe q{ui}ch is rekende for ten, þe halue{n}del þ{ere}-of wel be
5. sett þ{a}t 5 vpon þe hede of þ{a}t figur{e}, þe quych is now 2, & do
away þ{a}t 1, & þou schalt haue þis nombre yf þ{o}u worch wel,
5
2.
vn{de} v{er}sus.
[Headnote: How to prove the Mediation.]
¶ Si mediacio sit b{e}n{e} f{ac}ta p{ro}bar{e} valeb{is}
¶ Duplando num{er}u{m} que{m} p{ri}mo di{m}ediasti
[Sidenote: How to prove your mediation. First example. The second.
The third example. The fourth example. The fifth example.]
¶ Her{e} he telles þe how þou schalt know wheþ{er} þou hase wel ydo or
no. doubul [*leaf 151b] þe nombre þe quych þ{o}u hase mediet, and yf
þ{o}u haue wel y-medyt after þe dupleacioɳ, þou schalt haue þe same
nombre þat þ{o}u haddyst in þe tabull{e} or þ{o}u began to medye, as
þus. ¶ The furst ensampull{e} was þis. 4. þe quych I-mediet was laft 2,
þe whych 2 was write in þe place þ{a}t 4 was write afor{e}. Now
doubull{e} þat 2, & þ{o}u schal haue 4, as þ{o}u hadyst afor{e}. þe
secunde Ensampull{e} was þis, 245. When þ{o}u haddyst mediet all{e} þis
nomb{re}, yf þou haue wel ydo þou schalt haue of þ{a}t mediacioɳ þis
nombre, 122ʷ. Now doubull{e} þis nombre, & begyn in þe lyft side;
doubull{e} 1, þat schal be 2. do away þat 1 & sett þ{ere} 2. þen
doubull{e} þ{a}t oþ{er} 2 & sett þ{ere} 4, þen doubull{e} þat oþ{er} 2,
& þat wel be 4. þe{n} doubul þat merke þat stondes for halue on. & þat
schall{e} be 1. Cast þat on to 4, & it schall{e} be 5. do away þat 2 &
þat merke, & sette þ{ere} 5, & þen þ{o}u schal haue þis nombre 245. &
þis wos þe same nombur þ{a}t þ{o}u haddyst or þ{o}u began to medye, as
þ{o}u mayst se yf þou take hede. The nombre þe quych þou haddist for an
Ensampul in þe 3 case of mediacioɳ to be mediet was þis 241. whan þ{o}u
haddist medied all{e} þis nombur truly [*leaf 152a] by eu{er}y figur{e},
þou schall haue be þ{a}t mediacioɳ þis nombur 120ʷ. Now dowbul þis
nomb{ur}, & begyn in þe lyft side, as I tolde þe in þe Craft of
duplacioɳ. þus doubull{e} þe fig{ur}e of 1, þat wel be 2. do away þat 1
& sett þ{ere} 2, þen doubul þe next figur{e} afore, the quych is 2,
& þat wel be 4; do away 2 & set þ{ere} 4. þen doubul þe cifer, & þat wel
be noȝt, for a 0 is noȝt. And twyes noȝt is but noȝt. þ{ere}for{e}
doubul the merke aboue þe cifers hede, þe quych betokenes þe halue{n}del
of 1, & þat schal be 1. do away þe cifer & þe merke, & sett þ{ere} 1,
& þen þ{o}u schalt haue þis nombur 241. And þis same nombur þ{o}u
haddyst afore or þ{o}u began to medy, & yf þ{o}u take gode hede. ¶ The
next ensampul þat had in þe 4 case of mediacioɳ was þis 4678. Whan þ{o}u
hast truly ymedit all{e} þis nombur fro þe begynnyng{e} to þe endyng{e},
þ{o}u schalt haue of þe mediacioɳ þis nombur
5
2334.
Now doubul this nombur & begyn in þe lyft side, & doubull{e} 2 þat schal
be 4. do away 2 and sette þere 4; þen doubul{e} 3, þ{a}t wol be 6; do
away 3 & sett þ{ere} 6, þen doubul þat oþ{er} 3, & þat wel be 6; do away
3 & set þ{ere} [*leaf 152b] 6, þen doubul þe 4, þat welle be 8; þen
doubul 5. þe quych stondes ou{er} þe hed of 4, & þat wol be 10; cast 10
to 8, & þ{a}t schal be 18; do away 4 & þat 5, & sett þ{ere} 8, & sett
that 1, þe quych is an articul of þe Composit þe quych is 18, ou{er} þe
next figur{es} hed toward þe lyft side, þe quych is 6. drav þ{a}t 1 to
6, þe quych 1 in þe dravyng schal be rekente bot for 1, & þ{a}t 1 &
þ{a}t 6 togedur wel be 7. do away þat 6 & þat 1. the quych stondes
ou{er} his hede, & sett ther 7, & þen þou schalt haue þis nombur 4678.
And þis same nombur þ{o}u hadyst or þ{o}u began to medye, as þ{o}u mayst
see in þe secunde Ensampul þat þou had in þe 4 case of mediacioɳ, þat
was þis: when þ{o}u had mediet truly all{e} the nombur, a p{ri}ncipio
usque ad fine{m}. þ{o}u schalt haue of þat mediacioɳ þis nombur
5
102.
Now doubul 1. þat wel be 2. do away 1 & sett þ{ere} 2. þen doubul 0.
þ{a}t will be noȝt. þ{ere}for{e} take þe 5, þe quych stondes ou{er} þe
next figur{es} hed, & doubul it, & þat wol be 10. do away þe 0 þat
stondes betwene þe two fig{u}r{i}s, & sette þ{ere} in his stid 1, for
þ{a}t 1 now schal stonde in þe secunde place, wher{e} he schal betoken
10; þen doubul 2, þat wol be 4. do away 2 & sett þere 4. & [*leaf 153a]
þou schal haue þus nombur 214. þis is þe same nu{m}bur þat þ{o}u hadyst
or þ{o}u began to medye, as þ{o}u may see. And so do eu{er} mor{e}, yf
þ{o}u wil knowe wheþ{er} þou hase wel ymedyt or no. ¶. doubull{e} þe
nu{m}bur þat comes aft{er} þe mediaciouɳ, & þ{o}u schal haue þe same
nombur þ{a}t þ{o}u hadyst or þ{o}u began to medye, yf þ{o}u haue welle
ydo. or els doute þe noȝt, but yf þ{o}u haue þe same, þ{o}u hase faylide
in þ{i} Craft.
+Sequitur de multiplicatione.+
[Headnote: The Craft of Multiplication.]
[Headnote: To write down a Multiplication Sum.]
++Si tu p{er} num{er}u{m} num{er}u{m} vis m{u}ltiplicar{e}
Scribe duas q{ua}scu{nque} velis series nu{me}ror{um}
Ordo s{er}vet{ur} vt vltima m{u}ltiplicandi
Ponat{ur} sup{er} ant{er}iorem multiplicant{is}
A leua reliq{u}e sint scripte m{u}ltiplicantes.
[Sidenote: Four things to be known of Multiplication: the first:
the second: the third: the fourth. How to set down the sum. Two
sorts of Multiplication: mentally, and on paper.]
¶ Her{e} be-gynnes þe Chapt{r}e of m{u}ltiplicatioɳ, in þe quych þou
most know 4 thynges. ¶ Ffirst, qwat is m{u}ltiplicacioɳ. The secunde,
how mony cases may hap in multiplicacioɳ. The thryde, how mony rewes of
figur{es} þ{ere} most be. ¶ The 4. what is þe p{ro}fet of þis craft.
¶ As for þe first, þ{o}u schal vnd{er}stonde þat m{u}ltiplicacioɳ is a
bryngyng{e} to-ged{er} of 2 thyng{es} in on nombur, þe quych on nombur
{con}tynes so mony tymes on, howe [*leaf 153b] mony tymes þ{ere} ben
vnytees in þe nowmb{re} of þat 2, as twyes 4 is 8. now her{e} ben þe 2
nomb{er}s, of þe quych too nowmbr{e}s on is betokened be an adu{er}be,
þe quych is þe worde twyes, & þis worde thryes, & þis worde four{e}
sythes,[{9}] [[& þis wordes fyue sithe & sex sythes.]] & so furth of
such other lyke wordes. ¶ And tweyn nombres schal be tokenyde be a
nowne, as þis worde four{e} showys þes tweyɳ nombres y-broth in-to on
hole nombur, þat is 8, for twyes 4 is 8, as þ{o}u wost wel. ¶ And þes
nomb{re} 8 conteynes as oft tymes 4 as þ{ere} ben vnites in þ{a}t other
nomb{re}, þe quych is 2, for in 2 ben 2 vnites, & so oft tymes 4 ben in
8, as þ{o}u wottys wel. ¶ ffor þe secu{n}de, þ{o}u most know þat þ{o}u
most haue too rewes of figures. ¶ As for þe thryde, þ{o}u most know
þ{a}t 8 man{er} of diu{er}se case may happe in þis craft. The p{ro}fet
of þis Craft is to telle when a nomb{re} is m{u}ltiplyed be a noþ{er},
qwat co{m}mys þ{ere} of. ¶ fforthermor{e}, as to þe sentence of our{e}
verse, yf þ{o}u wel m{u}ltiply a nombur be a-noþ{er} nomb{ur}, þou
schalt write [*leaf 154a] a rewe of figures of what nomb{ur}s so eu{er}
þ{o}u welt, & þat schal be called Num{erus} m{u}ltiplicand{us}, Anglice,
þe nomb{ur} the quych to be m{u}ltiplied. þen þ{o}u schalt write
a-nother rewe of figur{e}s, by þe quych þ{o}u schalt m{u}ltiplie the
nombre þat is to be m{u}ltiplied, of þe quych nomb{ur} þe furst fig{ur}e
schal be write vnd{er} þe last figur{e} of þe nomb{ur}, þe quych is to
be m{u}ltiplied. And so write forthe toward þe lyft side, as her{e} you
may se,
+----------+
| 67324 |
| 1234 |
+----------+
And þis on{e} nomb{ur} schall{e} be called nu{meru}s m{u}ltiplicans.
An{gli}ce, þe nomb{ur} m{u}ltipliyng{e}, for he schall{e} m{u}ltiply þe
hyer nounb{ur}, as þus on{e} tyme 6. And so forth, as I schal telle the
aft{er}warde. And þou schal begyn in þe lyft side. ¶ ffor-þ{ere}-more
þou schalt vndurstonde þat þ{ere} is two man{ur}s of m{u}ltiplicacioɳ;
one ys of þe wyrchyng{e} of þe boke only in þe mynde of a mon. fyrst he
teches of þe fyrst man{er} of duplacioɳ, þe quych is be wyrchyng{e} of
tabuls. Aft{er}warde he wol teche on þe secunde man{er}. vn{de}
v{er}sus.
[Headnote: To multiply one Digit by another.]
In digitu{m} cures digitu{m} si duc{er}e ma{i}or
[*leaf 154b.]
P{er} qua{n}tu{m} distat a denis respice debes
¶ Namq{ue} suo decuplo totiens deler{e} mi{n}ore{m}
Sitq{ue} tibi nu{meru}s veniens exinde patebit.
[Sidenote: How to multiply two digits. Subtract the greater from ten;
take the less so many times from ten times itself. Example.]
¶ Her{e} he teches a rewle, how þ{o}u schalt fynde þe nounb{r}e þat
comes by þe m{u}ltiplicacioɳ of a digit be anoþ{er}. loke how mony
[vny]tes ben. bytwene þe mor{e} digit and 10. And reken ten for on
vnite. And so oft do away þe lasse nounbre out of his owne decuple, þat
is to say, fro þat nounb{r}e þat is ten tymes so mych is þe nounb{re}
þ{a}t comes of þe m{u}ltiplicacioɳ. As yf þ{o}u wol m{u}ltiply 2 be 4.
loke how mony vnitees ben by-twene þe quych is þe mor{e} nounb{re},
& be-twene ten. C{er}ten þ{ere} wel be vj vnitees by-twene 4 & ten.
yf þ{o}u reken þ{ere} w{i}t{h} þe ten þe vnite, as þou may se. so mony
tymes take 2. out of his decuple, þe quych is 20. for 20 is þe decuple
of 2, 10 is þe decuple of 1, 30 is þe decuple of 3, 40 is þe decuple of
4, And þe oþ{er} digetes til þ{o}u come to ten; & whan þ{o}u hast y-take
so mony tymes 2 out of twenty, þe quych is sex tymes, þ{o}u schal leue 8
as þ{o}u wost wel, for 6 times 2 is twelue. take [1]2 out of twenty,
& þ{ere} schal leue 8. bot yf bothe þe digett{es} [*leaf 155a] ben
y-lyech mych as her{e}. 222 or too tymes twenty, þen it is no fors quych
of hem tweyn þ{o}u take out of here decuple. als mony tymes as þ{a}t is
fro 10. but neu{er}-þe-lesse, yf þ{o}u haue hast to worch, þ{o}u schalt
haue her{e} a tabul of figures, wher{e}-by þ{o}u schalt se a-nonɳ ryght
what is þe nounbre þ{a}t comes of þe multiplicacioɳ of 2 digittes. þus
þ{o}u schalt worch in þis fig{ur}e.
[Sidenote: Better use this table, though. How to use it. The way to
use the Multiplication table.]
1|
-----
2| 4|
--------
3| 6| 9|
-----------
4| 8|12|16|
--------------
5|10|15|20|25|
-----------------
6|12|18|24|30|36|
--------------------
7|14|21|28|35|42|49|
-----------------------
8|16|24|32|40|48|56|64|
--------------------------
9|18|27|36|45|54|63|72|81|
----------------------------
1| 2| 3| 4| 5| 6| 7| 8| 9|
----------------------------
yf þe fig{ur}e, þe quych schall{e} be m{u}ltiplied, be euen{e} as mych
as þe diget be, þe quych þat oþ{er} figur{e} schal be m{u}ltiplied,
as two tymes twayɳ, or thre tymes 3. or sych other. loke qwer{e} þat
fig{ur}e sittes in þe lyft side of þe t{ri}angle, & loke qwer{e} þe
diget sittes in þe neþ{er} most rewe of þe triangle. & go fro hym
vpwarde in þe same rewe, þe quych rewe gose vpwarde til þ{o}u come
agaynes þe oþ{er} digette þat sittes in þe lyft side of þe t{ri}angle.
And þat nounbre, þe quych þou [*leaf 155b] fyn[*]des þ{ere} is þe
nounbre þat comes of the m{u}ltiplicacioɳ of þe 2 digittes, as yf þou
wold wete qwat is 2 tymes 2. loke quer{e} sittes 2 in þe lyft side i{n}
þe first rewe, he sittes next 1 in þe lyft side al on hye, as þ{o}u may
se; þe[{n}] loke qwer{e} sittes 2 in þe lowyst rewe of þe t{ri}angle,
& go fro hym vpwarde in þe same rewe tyll{e} þou come a-ȝenenes 2 in þe
hyer place, & þer þou schalt fynd ywrite 4, & þat is þe nounb{r}e þat
comes of þe multiplicacioɳ of two tymes tweyn is 4, as þow wotest
well{e}. yf þe diget. the quych is m{u}ltiplied, be mor{e} þan þe
oþ{er}, þou schalt loke qwer{e} þe mor{e} diget sittes in þe lowest rewe
of þe t{ri}angle, & go vpwarde in þe same rewe tyl[{10}] þ{o}u come
a-nendes þe lasse diget in the lyft side. And þ{ere} þ{o}u schalt fynde
þe no{m}b{r}e þat comes of þe m{u}ltiplicacioɳ; but þ{o}u schalt
vnd{er}stonde þat þis rewle, þe quych is in þis v{er}se. ¶ In digitu{m}
cures, &c., noþ{er} þis t{ri}angle schall{e} not s{er}ue, bot to fynde
þe nounbres þ{a}t comes of the m{u}ltiplicacioɳ þat comes of 2 articuls
or {com}posites, þe nedes no craft but yf þou wolt m{u}ltiply in þi
mynde. And [*leaf 156a] þere-to þou schalt haue a craft aft{er}warde,
for þou schall wyrch w{i}t{h} digettes in þe tables, as þou schalt know
aft{er}warde. v{er}sus.
[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 þ{o}u schalt wyrch in þis craft. þou schalt
m{ul}tiplye þe last figur{e} of þe nombre, and quen þ{o}u hast so ydo
þou schalt draw all{e} þe figures of þe neþ{er} nounbre mor{e} taward þe
ryȝt side, so qwe{n} þ{o}u hast m{u}ltiplyed þe last figur{e} of þe
heyer nounbre by all{e} þe neþ{er} figures. And sette þe nounbir þat
comes þer-of ou{er} þe last figur{e} of þe neþ{er} nounb{re}, & þen þou
schalt sette al þe oþ{er} fig{ur}es of þe neþ{er} nounb{re} mor{e}
ner{e} to þe ryȝt side. ¶ And whan þou hast m{u}ltiplied þat figur{e}
þat schal be m{u}ltiplied þe next aft{er} hym by al þe neþ{er} figures.
And worch as þou dyddyst afor{e} til [*leaf 156b] þou come to þe ende.
And þou schalt vnd{er}stonde þat eu{er}y figur{e} of þe hier nounb{re}
schal be m{u}ltiplied be all{e} þe figur{e}s of the neþ{er} nounbre,
yf þe hier nounb{re} be any figur{e} þen on{e}. lo an Ensampul her{e}
folowyng{e}.
+------+
| 2465|.
|232 |
+------+
þou schalt begyne to m{u}ltiplye in þe lyft side. M{u}ltiply 2 be 2, and
twyes 2 is 4. set 4 ou{er} þe hed of þ{a}t 2, þen m{u}ltiplie þe same
hier 2 by 3 of þe nether nounbre, as thryes 2 þat schal be 6. set 6
ou{er} þe hed of 3, þan m{u}ltiplie þe same hier 2 by þat 2 þe quych
stondes vnd{er} hym, þ{a}t wol be 4; do away þe hier 2 & sette þ{ere} 4.
¶ Now þ{o}u most antery þe nether nounbre, þat is to say, þ{o}u most
sett þe neþ{er} nounbre more towarde þe ryȝt side, as þus. Take þe
neþ{er} 2 toward þe ryȝt side, & sette it eueɳ vnd{er} þe 4 of þe hyer
nounb{r}e, & ant{er}y all{e} þe figures þat comes aft{er} þat 2, as þus;
sette 2 vnd{er} þe 4. þen sett þe figur{e} of 3 þ{ere} þat þe figure of
2 stode, þe quych is now vndur þ{a}t 4 in þe hier nounbre; þen sett þe
oþer figur{e} of 2, þe quych is þe last fig{ur}e toward þe lyft side of
þe neþ{er} nomb{er} þ{ere} þe figur{e} of 3 stode. þen þ{o}u schalt haue
such a nombre.
+------+
|464465|
| 232 |
+------+
[*leaf 157a] ¶ Now m{u}ltiply 4, þe quych comes next aft{er} 6, by þe
last 2 of þe neþ{er} nounbur toward þe lyft side. as 2 tymes 4, þat wel
be 8. sette þat 8 ou{er} þe figure the quych stondes ou{er} þe hede of
þat 2, þe quych is þe last figur{e} of þe neþ{er} nounbre; þan multiplie
þat same 4 by 3, þat comes in þe neþ{er} rewe, þat wol be 12. sette þe
digit of þe composyt ou{er} þe figure þe quych stondes ou{er} þe hed of
þat 3, & sette þe articule of þis co{m}posit ou{er} al þe figures þat
stondes ou{er} þe neþ{er} 2 hede. þen m{u}ltiplie þe same 4 by þe 2 in
þe ryȝt side in þe neþ{er} nounbur, þat wol be 8. do away 4. & sette
þ{ere} 8. Eu{er} mor{e} qwen þ{o}u m{u}ltiplies þe hier figur{e} by þat
figur{e} þe quych stondes vnd{er} hym, þou schalt do away þat hier
figur{e}, & sett þer þat nounbre þe quych comes of m{u}ltiplicacioɳ of
ylke digittes. Whan þou hast done as I haue byde þe, þ{o}u schalt haue
suych an ord{er} of figur{e} as is her{e},
+--------+
| 1 |.
| 82 |
|4648[65]|
| 232 |
+--------+
þen take and ant{er}y þi neþ{er} figures. And sett þe fyrst fig{ur}e of
þe neþ{er} figures[{11}] vndre be figur{e} of 6. ¶ And draw al þe oþ{er}
figures of þe same rewe to hym-warde, [*leaf 157b] as þ{o}u diddyst
afore. þen m{u}ltiplye 6 be 2, & sett þat þe quych comes ou{er}
þ{ere}-of ou{er} al þe oþ{er} figures hedes þat stondes ou{er} þat 2.
þen m{u}ltiply 6 be 3, & sett all{e} þat comes þ{ere}-of vpon all{e} þe
figur{e}s hedes þat standes ou{er} þat 3; þa{n} m{u}ltiplye 6 be 2, þe
quych stondes vnd{er} þat 6, þen do away 6 & write þ{ere} þe digitt of
þe composit þat schal come þ{ere}of, & sette þe articull ou{er} all{e}
þe figures þat stondes ou{er} þe hede of þat 3 as her{e},
+------+
| 11 |
| 121 |
| 828 |
|464825|
| 232 |
+------+
þen ant{er}y þi figures as þou diddyst afor{e}, and m{u}ltipli 5 be 2,
þat wol be 10; sett þe 0 ou{er} all þe figures þ{a}t stonden ou{er} þat
2, & sett þ{a}t 1. ou{er} the next figures hedes, all{e} on hye towarde
þe lyft side. þen m{u}ltiplye 5 be 3. þat wol be 15, write 5 ou{er} þe
figures hedes þat stonden ou{er} þ{a}t 3, & sett þat 1 ou{er} þe next
figur{e}s hedes toward þe lyft side. þen m{u}ltiplye 5 be 2, þat wol be
10. do away þat 5 & sett þ{ere} a 0, & sett þat 1 ou{er} þe figures
hedes þat stonden ou{er} 3. And þen þou schalt haue such a nounbre as
here stondes aftur.[*leaf 158a]
+------+
| 11 |
| 1101 |
| 1215 |
| 82820|
|4648 |
| 232|
+------+
¶ Now draw all{e} þese figures downe toged{er} as þus, 6.8.1. & 1 draw
to-gedur; þat wolle be 16, do away all{e} þese figures saue 6. lat hym
stonde, for þow þ{o}u take hym away þou most write þer þe same aȝene.
þ{ere}for{e} late hym stonde, & sett 1 ou{er} þe figur{e} hede of 4
toward þe lyft side; þen draw on to 4, þat woll{e} be 5. do away þat 4 &
þat 1, & sette þ{ere} 5. þen draw 4221 & 1 toged{ur}, þat wol be 10. do
away all{e} þat, & write þere þat 4 & þat 0, & sett þat 1 ou{er} þe next
figur{es} hede toward þe lyft side, þe quych is 6. þen draw þat 6 & þat
1 togedur, & þat wolle be 7; do away 6 & sett þ{ere} 7, þen draw 8810 &
1, & þat wel be 18; do away all{e} þe figures þ{a}t stondes ou{er} þe
hede of þat 8, & lette 8 stonde stil, & write þat 1 ou{er} þe next
fig{u}r{is} hede, þe quych is a 0. þen do away þat 0, & sett þ{ere} 1,
þe quych stondes ou{er} þe 0. hede. þen draw 2, 5, & 1 toged{ur}, þat
woll{e} be 8. þen do away all{e} þat, & write þ{ere} 8. ¶ And þen þou
schalt haue þis 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 þe fyrst case of þis craft, þe quych is þis: yf
þ{ere} come an articulle of þe m{u}ltiplicacioɳ ysette befor{e} the
articull{e} in þe lyft side as þus
+---+
| 51|.
|23 |
+---+
multiplye 5 by 2, þat wol be 10; sette ou{er} þe hede of þat 2 a 0,
& sett þat on, þat is þe articul, in þe lyft side, þat is next hym, þen
þ{o}u schalt haue þis nounbre
+----+
|1051|.
| 23 |
+----+
¶ And þen worch forth as þou diddist afore. And þ{o}u schalt
vnd{er}stonde þat þ{o}u schalt write no 0. but whan þat place where þou
schal write þat 0 has no figure afore hy{m} noþ{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 þe secunde case, þe quych is þis: yf hit happe þat þ{ere}
come a composyt, þou schalt write þe digitte ou{er} þe hede of þe
neþ{er} figur{e} by þe quych þ{o}u multipliest þe hier figure; and sett
þe articull{e} next hym toward þe lyft side, as þou diddyst afore, as
þ{us}
+---+
| 83|.
|83 |
+---+
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},
+-----------+
| 6483[{13}]|,
| 83 |
+-----------+
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:
+-----+
| 236|
|234 |
+-----+
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
+-------+
| 46836 |
| 234 |
+-------+
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,
+-------+
| 46836 |
| 234 |
+-------+
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:
+----+
| 24|
|03 |
+----+
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:
+-----+
|00032|.
|22 |
+-----+
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.+