The treatise on pp. 52-65 is the only one in English known on the
subject. It describes a method of calculation which, with slight
modifications, is current in Russia, China, and Japan, to-day, though it
went out of use in Western Europe by the seventeenth century. In Germany
the method is called “Algorithmus Linealis,” and there are several
editions of a tract under this name (with a diagram of the counting
board), printed at Leipsic at the end of the fifteenth century and the
beginning of the sixteenth. They give the nine rules, but “Capitulum de
radicum extractione ad algoritmum integrorum reservato, cujus species
per ciffrales figuras ostenduntur ubi ad plenum de hac tractabitur.” The
invention of the art is there attributed to Appulegius the philosopher.
The advantage of the counting board, whether permanent or constructed by
chalking parallel lines on a table, as shown in some sixteenth-century
woodcuts, is that only five counters are needed to indicate the number
nine, counters on the lines representing units, and those in the spaces
above representing five times those on the line below. The Russian
abacus, the “tchatui” or “stchota” has ten beads on the line; the
Chinese and Japanese “Swanpan” economises by dividing the line into two
parts, the beads on one side representing five times the value of those
on the other. The “Swanpan” has usually many more lines than the
“stchota,” allowing for more extended calculations, see Tylor,
_Anthropology_ (1892), p. 314.
Record’s treatise also mentions another method of counter notation
(p. 64) “merchants’ casting” and “auditors’ casting.” These were adapted
for the usual English method of reckoning numbers up to 200 by scores.
This method seems to have been used in the Exchequer. A counting board
for merchants’ use is printed by Halliwell in _Rara Mathematica_ (p. 72)
from Sloane MS. 213, and two others are figured in Egerton 2622 f. 82
and f. 83. The latter is said to be “novus modus computandi secundum
inventionem Magistri Thome Thorleby,” and is in principle, the same as
the “Swanpan.”
The Exchequer table is described in the _Dialogus de Scaccario_ (Oxford,
1902), p. 38.
+The Earliest Arithmetics in English.+
+The Crafte of Nombrynge+
_Egerton 2622._
[*leaf 136a]
Hec algorism{us} ars p{re}sens dicit{ur}; in qua
Talib{us} indor{um} fruim{ur} bis qui{n}q{ue} figuris.
[Sidenote: A derivation of Algorism. Another derivation of the word.]
This boke is called þe boke of algorym, or Augrym aft{er} lewd{er} vse.
And þis boke tretys þe Craft of Nombryng, þe quych crafte is called also
Algorym. Ther was a kyng of Inde, þe quich heyth Algor, & he made þis
craft. And aft{er} his name he called hit algory{m}; or els anoþ{er}
cause is quy it is called Algorym, for þe latyn word of hit s.
Algorism{us} com{es} of Algos, grece, q{uid} e{st} ars, latine, craft oɳ
englis, and rides, q{uid} e{st} {nu}me{rus}, latine, A nomb{ur} oɳ
englys, inde d{icitu}r Algorism{us} p{er} addic{i}one{m} hui{us} sillabe
m{us} & subtracc{i}onem d & e, q{ua}si ars num{er}andi. ¶ fforthermor{e}
ȝe most vnd{ir}stonde þ{a}t in þis craft ben vsid teen figurys, as here
ben{e} writen for ensampul, φ 9 8 7 6 5 4 3 2 1. ¶ Expone þe too
v{er}sus afor{e}: this p{re}sent craft ys called Algorism{us}, in þe
quych we vse teen signys of Inde. Questio. ¶ Why teɳ fyguris of Inde?
Solucio. for as I haue sayd afore þai wer{e} fonde fyrst in Inde of a
kyng{e} of þat Cuntre, þ{a}t was called Algor.
[Headnote: Notation and Numeration.]
[Sidenote: v{ersus} [in margin].]
¶ Prima sig{nifica}t unu{m}; duo ve{r}o s{e}c{un}da:
¶ Tercia sig{nifica}t tria; sic procede sinistre.
¶ Don{e}c ad extrema{m} venias, que cifra voca{tur}.
+¶ Cap{itulu}m primum de significac{i}o{n}e figurar{um}.+
[Sidenote: Expo{sitio} v{ersus}.]
[Sidenote: The meaning and place of the figures. Which figure is
read first.]
In þis verse is notifide þe significac{i}on of þese figur{is}. And þus
expone the verse. Þe first signifiyth on{e}, þe secu{n}de [*leaf 136b]
signi[*]fiyth tweyn{e}, þe thryd signifiyth thre, & the fourte
signifiyth 4. ¶ And so forthe towarde þe lyft syde of þe tabul or of þe
boke þ{a}t þe figures ben{e} writen{e} in, til þat þ{o}u come to the
last figure, þ{a}t is called a cifre. ¶ Questio. In quych syde sittes þe
first figur{e}? Soluc{io}, forsothe loke quich figure is first in þe
ryȝt side of þe bok or of þe tabul, & þ{a}t same is þe first figur{e},
for þ{o}u schal write bakeward, as here, 3. 2. 6. 4. 1. 2. 5. The
fig{ur}e of 5. was first write, & he is þe first, for he sittes oɳ þe
riȝt syde. And the fig{ur}e of 3 is last. ¶ Neu{er}-þe-les wen he says
¶ P{ri}ma sig{nifica}t vnu{m} &c., þat is to say, þe first betokenes
on{e}, þe secu{n}de. 2. & fore-þ{er}-mor{e}, he vnd{ir}stondes noȝt of
þe first fig{ur}e of eu{er}y rew. ¶ But he vnd{ir}stondes þe first
figure þ{a}t is in þe nomb{ur} of þe forsayd teen figuris, þe quych is
on{e} of þ{e}se. 1. And þe secu{n}de 2. & so forth.
[Sidenote: v{ersus} [in margin].]
¶ Quelib{et} illar{um} si pr{im}o limite ponas,
¶ Simplicite{r} se significat: si v{er}o se{cun}do,
Se decies: sursu{m} {pr}ocedas m{u}ltiplicando.
¶ Na{m}q{ue} figura seque{n}s q{uam}uis signat decies pl{us}.
¶ Ipsa locata loco quam sign{ific}at p{ertin}ente.
[Transcriber’s Note:
In the following section, numerals shown in +marks+ were printed in
a different font, possibly as facsimiles of the original MS form.]
[Sidenote: Expo{sitio} [in margin].]
[Sidenote: An explanation of the principles of notation. An example:
units, tens, hundreds, thousands. How to read the number.]
¶ Expone þis v{er}se þus. Eu{er}y of þese figuris bitokens hym selfe &
no mor{e}, yf he stonde in þe first place of þe rewele / this worde
Simplicit{er} in þat verse it is no more to say but þat, & no mor{e}.
¶ If it stonde in the secu{n}de place of þe rewle, he betokens ten{e}
tymes hym selfe, as þis figur{e} 2 here 20 tokens ten tyme hym selfe,
[*leaf 137a] þat is twenty, for he hym selfe betokenes twey{ne}, & ten
tymes twene is twenty. And for he stondis oɳ þe lyft side & in þe
secu{n}de place, he betokens ten tyme hy{m} selfe. And so go forth.
¶ ffor eu{er}y fig{ure}, & he stonde aft{ur} a-noþ{er} toward the lyft
side, he schal betoken{e} ten tymes as mich mor{e} as he schul betoken &
he stode in þe place þ{ere} þat þe fig{ure} a-for{e} hym stondes. loo an
ensampull{e}. 9. 6. 3. 4. Þe fig{ure} of 4. þ{a}t hase þis schape +4.+
betokens bot hymselfe, for he stondes in þe first place. The fig{ure} of
3. þat hase þis schape +3.+ betokens ten tymes mor{e} þen he schuld & he
stode þ{ere} þ{a}t þe fig{ure} of 4. stondes, þ{a}t is thretty. The
fig{ure} of 6, þ{a}t hase þis schape +6+, betokens ten tymes mor{e} þan
he schuld & he stode þ{ere} as þe fig{ure} of +3.+ stondes, for þ{ere}
he schuld tokyn{e} bot sexty, & now he betokens ten tymes mor{e}, þat is
sex hundryth. The fig{ure} of 9. þ{a}t hase þis schape +9.+ betokens ten
tymes mor{e} þan{e} he schuld & he stode in þe place þ{ere} þe fig{ure}
of sex stondes, for þen he schuld betoken to 9. hundryth, and in þe
place þ{ere} he stondes now he betokens 9. þousande. Al þe hole nomb{ur}
is 9 thousande sex hundryth & four{e} & thretty. ¶ fforthermor{e}, when
þ{o}u schalt rede a nomb{ur} of fig{ure}, þ{o}u schalt begyn{e} at þe
last fig{ure} in the lyft side, & rede so forth to þe riȝt side as
her{e} 9. 6. 3. 4. Thou schal begyn to rede at þe fig{ure} of 9. & rede
forth þus. 9. [*leaf 137b] thousand sex hundryth thritty & foure. But
when þ{o}u schall{e} write, þ{o}u schalt be-gynne to write at þe ryȝt
side.
¶ Nil cifra sig{nifica}t s{ed} dat signa{re} sequenti.
[Sidenote: The meaning and use of the cipher.]
Expone þis v{er}se. A cifre tokens noȝt, bot he makes þe fig{ure} to
betoken þat comes aft{ur} hym mor{e} þan he schuld & he wer{e} away, as
þus 1φ. her{e} þe fig{ure} of on{e} tokens ten, & yf þe cifre wer{e}
away[{1}] & no fig{ure} by-for{e} hym he schuld token bot on{e}, for
þan he sch{ul}d stonde in þe first place. ¶ And þe cifre tokens nothyng
hym selfe. for al þe nomb{ur} of þe ylke too fig{ure}s is bot ten.
¶ Questio. Why says he þat a cifre makys a fig{ure} to signifye (tyf)
mor{e} &c. ¶ I speke for þis worde significatyf, ffor sothe it may happe
aft{ur} a cifre schuld come a-noþ{ur} cifre, as þus 2φφ. And ȝet þe
secunde cifre shuld token neu{er} þe mor{e} excep he schuld kepe þe
ord{er} of þe place. and a cifre is no fig{ure} significatyf.
+¶ Q{ua}m p{re}cedentes plus ulti{m}a significabit+ /
[Sidenote: The last figure means more than all the others,
since it is of the highest value.]
Expone þis v{er}se þus. Þe last figu{re} schal token mor{e} þan all{e}
þe oþ{er} afor{e}, thouȝt þ{ere} wer{e} a hundryth thousant figures
afor{e}, as þus, 16798. Þe last fig{ure} þat is 1. betokens ten
thousant. And all{e} þe oþ{er} fig{ure}s b{e}n bot betoken{e} bot sex
thousant seuyn{e} h{u}ndryth nynty & 8. ¶ And ten thousant is mor{e} þen
all{e} þat nomb{ur}, {er}go þe last figu{re} tokens mor{e} þan all þe
nomb{ur} afor{e}.
[Headnote: The Three Kinds of Numbers]
[*leaf 138a]
¶ Post p{re}dicta scias breuit{er} q{uod} tres num{er}or{um}
Distincte species sunt; nam quidam digiti sunt;
Articuli quidam; quidam q{uoque} compositi sunt.
¶ Capit{ulu}m 2^m de t{ri}plice divisione nu{mer}or{um}.
[Sidenote: Digits. Articles. Composites.]
¶ The auctor of þis tretis dep{ar}tys þis worde a nomb{ur} into 3
p{ar}tes. Some nomb{ur} is called digit{us} latine, a digit in englys.
So{m}me nomb{ur} is called articul{us} latine. An Articul in englys.
Some nomb{ur} is called a composyt in englys. ¶ Expone þis v{er}se. know
þ{o}u aft{ur} þe forsayd rewles þ{a}t I sayd afore, þat þ{ere} ben thre
spices of nomb{ur}. Oon{e} is a digit, Anoþ{er} is an Articul, & þe
toþ{er} a Composyt. v{er}sus.
[Headnote: Digits, Articles, and Composites.]
¶ Sunt digiti num{er}i qui cit{ra} denariu{m} s{u}nt.
[Sidenote: What are digits.]
¶ Her{e} he telles qwat is a digit, Expone v{er}su{s} sic. Nomb{ur}s
digitus ben{e} all{e} nomb{ur}s þat ben w{i}t{h}-inne ten, as nyne,
8. 7. 6. 5. 4. 3. 2. 1.
¶ Articupli decupli degito{rum}; compositi s{u}nt
Illi qui constant ex articulis degitisq{ue}.
[Sidenote: What are articles.]
¶ Her{e} he telles what is a composyt and what is an{e} articul. Expone
sic v{er}sus. ¶ Articulis ben[{2}] all{e} þ{a}t may be deuidyt into
nomb{urs} of ten & nothyng{e} leue ou{er}, as twenty, thretty, fourty,
a hundryth, a thousand, & such oþ{er}, ffor twenty may be dep{ar}tyt
in-to 2 nomb{ur}s of ten, fforty in to four{e} nomb{ur}s of ten, & so
forth.
[Sidenote: What numbers are composites.]
[*leaf 138b] Compositys beɳ nomb{ur}s þat bene componyt of a digyt & of
an articull{e} as fouretene, fyftene, sextene, & such oþ{er}. ffortene
is co{m}ponyd of four{e} þat is a digit & of ten þat is an articull{e}.
ffiftene is componyd of 5 & ten, & so of all oþ{er}, what þat þai ben.
Short-lych eu{er}y nomb{ur} þat be-gynnes w{i}t{h} a digit & endyth in a
articull{e} is a composyt, as fortene bygennyng{e} by four{e} þat is a
digit, & endes in ten.
¶ Ergo, p{ro}posito nu{mer}o tibi scriber{e}, p{ri}mo
Respicias quid sit nu{merus}; si digitus sit
P{ri}mo scribe loco digitu{m}, si compositus sit
P{ri}mo scribe loco digitu{m} post articulu{m}; sic.
[Sidenote: How to write a number, if it is a digit; if it is a
composite. How to read it.]
¶ here he telles how þ{o}u schalt wyrch whan þ{o}u schalt write a
nomb{ur}. Expone v{er}su{m} sic, & fac iuxta expon{ent}is sentencia{m};
whan þ{o}u hast a nomb{ur} to write, loke fyrst what man{er} nomb{ur} it
ys þ{a}t þ{o}u schalt write, whether it be a digit or a composit or an
Articul. ¶ If he be a digit, write a digit, as yf it be seuen, write
seuen & write þ{a}t digit in þe first place toward þe ryght side. If it
be a composyt, write þe digit of þe composit in þe first place & write
þe articul of þat digit in þe secunde place next toward þe lyft side. As
yf þ{o}u schal write sex & twenty. write þe digit of þe nomb{ur} in þe
first place þat is sex, and write þe articul next aft{ur} þat is twenty,
as þus 26. But whan þ{o}u schalt sowne or speke [*leaf 139a] or rede an
Composyt þou schalt first sowne þe articul & aft{ur} þe digit, as þ{o}u
seyst by þe comyn{e} speche, Sex & twenty & nouȝt twenty & sex.
v{er}sus.
¶ Articul{us} si sit, in p{ri}mo limite cifram,
Articulu{m} {vero} reliq{ui}s insc{ri}be figur{is}.
[Sidenote: How to write Articles: tens, hundreds, thousands, &c.]
¶ Here he tells how þ{o}u schal write when þe nombre þ{a}t þ{o}u hase to
write is an Articul. Expone v{er}sus sic & fac s{ecundu}m sentenciam.
Ife þe nomb{ur} þ{a}t þ{o}u hast write be an Articul, write first a
cifre & aft{ur} þe cifer write an Articull{e} þus. 2φ. fforthermor{e}
þ{o}u schalt vnd{ir}stonde yf þ{o}u haue an Articul, loke how mych he
is, yf he be w{i}t{h}-ynne an hundryth, þ{o}u schalt write bot on{e}
cifre, afore, as her{e} .9φ. If þe articull{e} be by hym-silfe & be an
hundrid euen{e}, þen schal þ{o}u write .1. & 2 cifers afor{e}, þat he
may stonde in þe thryd place, for eu{er}y fig{ure} in þe thryd place
schal token a hundrid tymes hym selfe. If þe articul be a thousant or
thousandes[{3}] and he stonde by hy{m} selfe, write afor{e} 3 cifers &
so forþ of al oþ{er}.
¶ Quolib{et} in nu{mer}o, si par sit p{ri}ma figura,
Par erit & to{tu}m, quicquid sibi co{n}ti{nua}t{ur};
Imp{ar} si fu{er}it, totu{m} tu{n}c fiet {et} impar.
[Sidenote: To tell an even number or an odd.]
¶ Her{e} he teches a gen{er}all{e} rewle þ{a}t yf þe first fig{ure} in
þe rewle of fig{ure}s token a nomb{ur} þat is euen{e} al þ{a}t nomb{ur}
of fig{ur}ys in þat rewle schal be euen{e}, as her{e} þ{o}u may see 6.
7. 3. 5. 4. Computa & p{ro}ba. ¶ If þe first [*leaf 139b] fig{ur}e token
an nomb{ur} þat is ode, all{e} þat nomb{ur} in þat rewle schall{e} be
ode, as her{e} 5 6 7 8 6 7. Computa & p{ro}ba. v{er}sus.
¶ Septe{m} su{n}t partes, no{n} pl{u}res, istius artis;
¶ Adder{e}, subt{ra}her{e}, duplar{e}, dimidiar{e},
Sextaq{ue} diuider{e}, s{ed} qui{n}ta m{u}ltiplicar{e};
Radice{m} ext{ra}her{e} p{ar}s septi{m}a dicitur esse.
[Headnote: The Seven Rules of Arithmetic.]
[Sidenote: The seven rules.]
¶ Her{e} telles þ{a}t þ{er} beɳ .7. spices or p{ar}tes of þis craft.
The first is called addicioñ, þe secunde is called subtraccioñ. The
thryd is called duplacioñ. The 4. is called dimydicioñ. The 5. is called
m{u}ltiplicacioñ. The 6 is called diuisioñ. The 7. is called extraccioñ
of þe Rote. What all þese spices ben{e} hit schall{e} be tolde
singillati{m} in her{e} caputul{e}.
¶ Subt{ra}his aut addis a dext{ri}s vel mediabis:
[Sidenote: Add, subtract, or halve, from right to left.]
Thou schal be-gynne in þe ryght side of þe boke or of a tabul. loke
wer{e} þ{o}u wul be-gynne to write latyn or englys in a boke, & þ{a}t
schall{e} be called þe lyft side of the boke, þat þ{o}u writest toward
þ{a}t side schal be called þe ryght side of þe boke. V{er}sus.
A leua dupla, diuide, m{u}ltiplica.
[Sidenote: Multiply or divide from left to right.]
Here he telles þe in quych side of þe boke or of þe tabul þ{o}u
schall{e} be-gyn{e} to wyrch duplacioñ, diuisioñ, and m{u}ltiplicacioñ.
Thou schal begyn{e} to worch in þe lyft side of þe boke or of þe tabul,
but yn what wyse þ{o}u schal wyrch in hym +dicetur singillatim in
seque{n}tib{us} capi{tulis} et de vtilitate cui{us}li{bet} art{is} & sic
Completur [*leaf 140.] p{ro}hemi{um} & sequit{ur} tractat{us} & p{ri}mo
de arte addic{ion}is que p{ri}ma ars est in ordine.+
[Headnote: The Craft of Addition.]
++Adder{e} si nu{mer}o num{e}ru{m} vis, ordine tali
Incipe; scribe duas p{rim}o series nu{mer}or{um}
P{ri}ma{m} sub p{ri}ma recte pone{n}do figura{m},
Et sic de reliq{ui}s facias, si sint tibi plures.
[Sidenote: Four things must be known: what it is; how many rows of
figures; how many cases; what is its result. How to set down the sum.]
¶ Her{e} by-gynnes þe craft of Addicioñ. In þis craft þ{o}u most knowe
foure thyng{es}. ¶ Fyrst þ{ou} most know what is addicioñ. Next þ{o}u
most know how mony rewles of figurys þou most haue. ¶ Next þ{o}u most
know how mony diue{r}s casys happes in þis craft of addicioñ. ¶ And next
qwat is þe p{ro}fet of þis craft. ¶ As for þe first þou most know þat
addicioñ is a castyng to-ged{ur} of twoo nomburys in-to on{e} nombr{e}.
As yf I aske qwat is twene & thre. Þ{o}u wyl cast þese twene nomb{re}s
to-ged{ur} & say þ{a}t it is fyue. ¶ As for þe secunde þou most know
þ{a}t þou schall{e} haue tweyne rewes of figures, on{e} vndur a-nother,
as her{e} þ{o}u mayst se.
1234
2168.
¶ As for þe thryd þou most know þ{a}t ther{e} ben foure diu{er}se cases.
As for þe forthe þ{o}u most know þ{a}t þe p{ro}fet of þis craft is to
telle what is þe hole nomb{ur} þ{a}t comes of diu{er}se nomburis. Now as
to þe texte of oure verse, he teches ther{e} how þ{o}u schal worch in
þis craft. ¶ He says yf þ{o}u wilt cast on{e} nomb{ur} to anoþ{er}
nomb{ur}, þou most by-gynne on þis wyse. ¶ ffyrst write [*leaf 140b] two
rewes of figuris & nombris so þat þ{o}u write þe first figur{e} of þe
hyer nomb{ur} euen{e} vnd{ir} the first fig{ure} of þe nether nomb{ur},
And þe secunde of þe nether nomb{ur} euen{e} vnd{ir} þe secunde of þe
hyer, & so forthe of eu{er}y fig{ur}e of both þe rewes as þ{o}u
mayst se.
123
234.
[Headnote: The Cases of the Craft of Addition.]
¶ Inde duas adde p{ri}mas hac condic{i}one:
Si digitus crescat ex addic{i}one prior{um};
P{ri}mo scribe loco digitu{m}, quicu{n}q{ue} sit ille.
[Sidenote: Add the first figures; rub out the top figure;
write the result in its place. Here is an example.]
¶ Here he teches what þ{o}u schalt do when þ{o}u hast write too rewes of
figuris on vnder an-oþ{er}, as I sayd be-for{e}. ¶ He says þ{o}u schalt
take þe first fig{ur}e of þe heyer nomb{re} & þe fyrst figur{e} of þe
neþ{er} nombre, & cast hem to-ged{er} vp-on þis condicioɳ. Thou schal
loke qweþ{er} þe nombe{r} þat comys þ{ere}-of be a digit or no. ¶ If he
be a digit þ{o}u schalt do away þe first fig{ur}e of þe hyer nomb{re},
and write þ{ere} in his stede þat he stode Inne þe digit, þ{a}t comes of
þe ylke 2 fig{ur}es, & so wrich forth oɳ oþ{er} figures yf þ{ere} be ony
moo, til þ{o}u come to þe ende toward þe lyft side. And lede þe nether
fig{ure} stonde still eu{er}-mor{e} til þ{o}u haue ydo. ffor þ{ere}-by
þ{o}u schal wyte wheþ{er} þ{o}u hast don{e} wel or no, as I schal tell
þe aft{er}ward in þe ende of þis Chapt{er}. ¶ And loke allgate þat þou
be-gynne to worch in þis Craft of [*leaf 141a] Addi[*]cioɳ in þe ryȝt
side, here is an ensampul of þis case.
1234
2142.
Caste 2 to four{e} & þat wel be sex, do away 4. & write in þe same place
þe fig{ur}e of sex. ¶ And lete þe fig{ur}e of 2 in þe nether rewe stonde
stil. When þ{o}u hast do so, cast 3 & 4 to-ged{ur} and þat wel be seuen
þ{a}t is a digit. Do away þe 3, & set þ{ere} seueɳ, and lete þe neþ{er}
fig{ure} stonde still{e}, & so worch forth bakward til þ{o}u hast ydo
all to-ged{er}.
Et si composit{us}, in limite scribe seque{n}te
Articulum, p{ri}mo digitum; q{uia} sic iubet ordo.
[Sidenote: Suppose it is a Composite, set down the digit,
and carry the tens. Here is an example.]
¶ Here is þe secunde case þ{a}t may happe in þis craft. And þe case is
þis, yf of þe casting of 2 nomburis to-ged{er}, as of þe fig{ur}e of þe
hyer rewe & of þe figure of þe neþ{er} rewe come a Composyt, how schalt
þ{ou} worch. Þ{us} þ{o}u schalt worch. Thou shalt do away þe fig{ur}e of
þe hyer nomb{er} þat was cast to þe figure of þe neþ{er} nomber. ¶ And
write þ{ere} þe digit of þe Composyt. And set þe articul of þe composit
next aft{er} þe digit in þe same rewe, yf þ{ere} be no mo fig{ur}es
aft{er}. But yf þ{ere} be mo figuris aft{er} þat digit. And þere he
schall be rekend for hym selfe. And when þ{o}u schalt adde þ{a}t ylke
figure þ{a}t berys þe articull{e} ou{er} his hed to þe figur{e} vnd{er}
hym, þ{o}u schalt cast þat articul to þe figure þ{a}t hase hym ou{er}
his hed, & þ{ere} þat Articul schal tokeɳ hym selfe. lo an Ensampull
[*leaf 141b] of all.
326
216.
Cast 6 to 6, & þ{ere}-of wil arise twelue. do away þe hyer 6 & write
þ{ere} 2, þ{a}t is þe digit of þis composit. And þe{n} write þe
articull{e} þat is ten ou{er} þe figuris hed of twene as þ{us}.
1
322
216.
Now cast þe articull{e} þ{a}t standus vpon þe fig{ur}is of twene hed to
þe same fig{ur}e, & reken þat articul bot for on{e}, and þan þ{ere} wil
arise thre. Þan cast þat thre to þe neþ{er} figure, þat is on{e}, & þat
wul be four{e}. do away þe fig{ur}e of 3, and write þ{ere} a fig{ur}e of
foure. and lete þe neþ{er} fig{ur}e stonde stil, & þan worch forth.
vn{de} {ver}sus.
¶ Articulus 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: Suppose it is an Article, set down a cipher and carry
the tens. Here is an example.]
¶ Her{e} he puttes þe thryde case of þe craft of Addicioɳ. & þe case is
þis. yf of Addiciouɳ of 2 figuris a-ryse an Articull{e}, how schal þ{o}u
do. thou most do away þe heer fig{ur}e þ{a}t was addid to þe neþ{er},
& write þ{ere} a cifre, and sett þe articuls on þe figuris hede, yf
þ{a}t þ{ere} come ony aft{er}. And wyrch þan as I haue tolde þe in þe
secunde case. An ensampull.
25.
15
Cast 5 to 5, þat wylle be ten. now do away þe hyer 5, & write þ{ere} a
cifer. And sette ten vpon þe figuris hed of 2. And reken it but for on
þus.] lo an Ensampull{e}
+----+
| 1 |
| 2φ |
| 15 |
+----+
And [*leaf 142a] þan worch forth. But yf þ{ere} come no figure aft{er}
þe cifre, write þe articul next hym in þe same rewe as here
+---+
| 5 |
| 5 |
+---+
cast 5 to 5, and it wel be ten. do away 5. þat is þe hier 5. and write
þ{ere} a cifre, & write aft{er} hym þe articul as þus
+----+
| 1φ |
| 5 |
+----+
And þan þ{o}u hast done.
¶ Si tibi cifra sup{er}ueniens occurrerit, illa{m}
Dele sup{er}posita{m}; fac illic scribe figura{m},
Postea procedas reliquas addendo figuras.
[Sidenote: What to do when you have a cipher in the top row.
An example of all the difficulties.]
¶ Her{e} he putt{es} þe fourt case, & it is þis, þat yf þ{ere} come a
cifer in þe hier rewe, how þ{o}u schal do. þus þ{o}u schalt do. do away
þe cifer, & sett þ{ere} þe digit þ{a}t comes of þe addiciou{n} as þus
1φφ84.
17743
In þis ensampul ben all{e} þe four{e} cases. Cast 3 to foure, þ{a}t wol
be seueɳ. do away 4. & write þ{ere} seueɳ; þan cast 4 to þe figur{e} of
8. þ{a}t wel be 12. do away 8, & sett þ{ere} 2. þat is a digit, and
sette þe articul of þe composit, þat is ten, vpon þe cifers hed, & reken
it for hym selfe þat is on. þan cast on{e} to a cifer, & hit wull{e} be
but on, for noȝt & on makes but on{e}. þan cast 7. þ{a}t stondes vnd{er}
þat on to hym, & þat wel be 8. do away þe cifer & þat 1. & sette þ{ere}
8. þan go forthermor{e}. cast þe oþ{er} 7 to þe cifer þ{a}t stondes
ou{er} hy{m}. þ{a}t wul be bot seuen, for þe cifer betokens noȝt. do
away þe cifer & sette þ{ere} seueɳ, [*leaf 142b] & þen go forþ{er}mor{e}
& cast 1 to 1, & þat wel be 2. do away þe hier 1, & sette þ{ere} 2. þan
hast þ{o}u do. And yf þ{o}u haue wel ydo þis nomber þat is sett
her{e}-aft{er} wel be þe nomber þat schall{e} aryse of all{e} þe
addicioɳ as her{e} 27827. ¶ Sequi{tu}r alia sp{eci}es.
[Headnote: The Craft of Subtraction.]
++A nu{mer}o num{er}u{m} si sit tibi demer{e} cura
Scribe figurar{um} series, vt in addicione.
[Sidenote: Four things to know about subtraction: the first;
the second; the third; the fourth.]
¶ This is þe Chapt{er} of subtraccioɳ, in the quych þou most know foure
nessessary thyng{es}. the first what is subtraccioɳ. þe secunde is how
mony nombers þou most haue to subt{ra}ccioɳ, the thryd is how mony
maners of cases þ{ere} may happe in þis craft of subtraccioɳ. The fourte
is qwat is þe p{ro}fet of þis craft. ¶ As for þe first, þ{o}u most know
þ{a}t subtraccioɳ is drawyng{e} of on{e} nowmb{er} oute of anoþ{er}
nomber. As for þe secunde, þou most knowe þ{a}t þou most haue two rewes
of figuris on{e} vnd{er} anoþ{er}, as þ{o}u addyst in addicioɳ. As for
þe thryd, þ{o}u moyst know þ{a}t four{e} man{er} of diu{er}se casis mai
happe in þis craft. ¶ As for þe fourt, þou most know þ{a}t þe p{ro}fet
of þis craft is whenne þ{o}u hasse taken þe lasse nomber out of þe
mor{e} to telle what þ{ere} leues ou{er} þ{a}t. & þ{o}u most be-gynne to
wyrch in þ{is} craft in þe ryght side of þe boke, as þ{o}u diddyst in
addicioɳ. V{er}sus.
¶ Maiori nu{mer}o num{er}u{m} suppone minorem,
¶ Siue pari nu{mer}o supponat{ur} num{er}us par.
[Sidenote: Put the greater number above the less.]
[*leaf 143a] ¶ Her{e} he telles þat þe hier nomber most be mor{e} þen þe
neþ{er}, or els eueɳ as mych. but he may not be lasse. And þe case is
þis, þou schalt drawe þe neþ{er} nomber out of þe hyer, & þou mayst not
do þ{a}t yf þe hier nomber wer{e} lasse þan þat. ffor þ{o}u mayst not
draw sex out of 2. But þ{o}u mast draw 2 out of sex. And þou maiste draw
twene out of twene, for þou schal leue noȝt of þe hier twene vn{de}
v{er}sus.
[Headnote: The Cases of the Craft of Subtraction.]
¶ Postea si possis a prima subt{ra}he p{ri}ma{m}
Scribens quod remanet.
[Sidenote: The first case of subtraction. Here is an example.]
Her{e} is þe first case put of subtraccioɳ, & he says þou schalt begynne
in þe ryght side, & draw þe first fig{ure} of þe neþ{er} rewe out of þe
first fig{ure} of þe hier rewe. qwether þe hier fig{ur}e be mor{e} þen
þe neþ{er}, or eueɳ as mych. And þat is notified in þe vers when he says
“Si possis.” Whan þ{o}u has þus ydo, do away þe hiest fig{ur}e & sett
þ{ere} þat leues of þe subtraccioɳ, lo an Ensampull{e}
+-----+
| 234 |
| 122 |
+-----+
draw 2 out of 4. þan leues 2. do away 4 & write þ{ere} 2, & latte þe
neþ{er} figur{e} sto{n}de stille, & so go for-by oþ{er} figuris till
þ{o}u come to þe ende, þan hast þ{o}u do.
¶ Cifram si nil remanebit.
[Sidenote: Put a cipher if nothing remains. Here is an example.]
¶ Her{e} he putt{es} þe secunde case, & hit is þis. yf it happe þ{a}t
qwen þ{o}u hast draw on neþ{er} fig{ure} out of a hier, & þ{er}e leue
noȝt aft{er} þe subt{ra}ccioɳ, þus [*leaf 143b] þou schalt do. þ{o}u
schall{e} do away þe hier fig{ur}e & write þ{ere} a cifer, as lo an
Ensampull
+----+
| 24 |
| 24 |
+----+
Take four{e} out of four{e} þan leus noȝt. þ{er}efor{e} do away þe hier
4 & set þ{ere} a cifer, þan take 2 out of 2, þan leues noȝt. do away þe
hier 2, & set þ{ere} a cifer, and so worch whar{e} so eu{er} þis happe.
Sed si no{n} possis a p{ri}ma dem{er}e p{ri}ma{m}
P{re}cedens vnu{m} de limite deme seque{n}te,
Quod demptu{m} p{ro} denario reputabis ab illo
Subt{ra}he to{ta}lem num{er}u{m} qu{em} p{ro}posuisti
Quo facto sc{ri}be super quicquid remaneb{i}t.
[Sidenote: Suppose you cannot take the lower figure from the top one,
borrow ten; take the lower number from ten; add the answer to the top
number. How to ‘Pay back’ the borrowed ten. Example.]
Her{e} he puttes þe thryd case, þe quych is þis. yf it happe þat þe
neþ{er} fig{ur}e be mor{e} þen þe hier fig{ur}e þat he schall{e} be draw
out of. how schall{e} þou do. þus þ{o}u schall{e} do. þou schall{e}
borro .1. oute of þe next fig{ur}e þat comes aft{er} in þe same rewe,
for þis case may neu{er} happ but yf þ{ere} come figures aft{er}. þan
þ{o}u schalt sett þat on ou{er} þe hier figur{es} hed, of the quych þou
woldist y-draw oute þe neyþ{er} fig{ur}e yf þ{o}u haddyst y-myȝt. Whane
þou hase þus ydo þou schall{e} rekene þ{a}t .1. for ten. ¶. And out of
þat ten þ{o}u schal draw þe neyþermost fig{ur}e, And all{e} þ{a}t leues
þou schall{e} adde to þe figur{e} on whos hed þat .1. stode. And þen
þ{o}u schall{e} do away all{e} þat, & sett þ{ere} all{e} that arisys of
the addicioɳ of þe ylke 2 fig{ur}is. And yf yt [*leaf 144a] happe þat þe
fig{ur}e of þe quych þ{o}u schalt borro on be hym self but 1. If þ{o}u
schalt þat on{e} & sett it vppoɳ þe oþ{er} figur{is} hed, and sett in
þ{a}t 1. place a cifer, yf þ{ere} come mony figur{es} aft{er}. lo an
Ensampul.
+------+
| 2122 |
| 1134 |
+------+
take 4 out of 2. it wyl not be, þerfor{e} borro on{e} of þe next
figur{e}, þ{a}t is 2. and sett þat ou{er} þe hed of þe fyrst 2. & rekene
it for ten. and þere þe secunde stondes write 1. for þ{o}u tokest on out
of hy{m}. þan take þe neþ{er} fig{ur}e, þat is 4, out of ten. And þen
leues 6. cast to 6 þe fig{ur}e of þat 2 þat stode vnd{er} þe hedde of 1.
þat was borwed & rekened for ten, and þat wylle be 8. do away þ{a}t 6 &
þat 2, & sette þ{ere} 8, & lette þe neþ{er} fig{ur}e stonde stille.
Whanne þ{o}u hast do þus, go to þe next fig{ur}e þ{a}t is now bot 1. but
first yt was 2, & þ{ere}-of was borred 1. þan take out of þ{a}t þe
fig{ur}e vnd{er} hym, þ{a}t is 3. hit wel not be. þer-for{e} borowe of
the next fig{ur}e, þe quych is bot 1. Also take & sett hym ou{er} þe
hede of þe fig{ure} þat þou woldest haue y-draw oute of þe nether
figure, þe quych was 3. & þou myȝt not, & rekene þ{a}t borwed 1 for ten
& sett in þe same place, of þe quych place þ{o}u tokest hy{m} of,
a cifer, for he was bot 1. Whanne þ{o}u hast þ{us} ydo, take out of þat
1. þ{a}t is rekent for ten, þe neþ{er} figure of 3. And þ{ere} leues 7.
[*leaf 144b] cast þe ylke 7 to þe fig{ur}e þat had þe ylke ten vpon his
hed, þe quych fig{ur}e was 1, & þat wol be 8. þan do away þ{a}t 1 and
þ{a}t 7, & write þ{ere} 8. & þan wyrch forth in oþ{er} figuris til þ{o}u
come to þe ende, & þan þ{o}u hast þe do. V{er}sus.
¶ Facque nonenarios de cifris, cu{m} remeabis
¶ Occ{ur}rant si forte cifre; dum demps{er}is vnum
¶ Postea p{ro}cedas reliquas deme{n}do figuras.
[Sidenote: A very hard case is put. Here is an example.]
¶ Her{e} he putt{es} þe fourte case, þe quych is þis, yf it happe þat þe
neþ{er} fig{ur}e, þe quych þ{o}u schalt draw out of þe hier fig{ur}e be
mor{e} pan þe hier figur ou{er} hym, & þe next fig{ur}e of two or of
thre or of foure, or how mony þ{ere} be by cifers, how wold þ{o}u do.
Þ{o}u wost wel þ{o}u most nede borow, & þ{o}u mayst not borow of þe
cifers, for þai haue noȝt þat þai may lene or spar{e}. Ergo[{4}] how
woldest þ{o}u do. Certayɳ þus most þ{o}u do, þ{o}u most borow on of þe
next figure significatyf in þat rewe, for þis case may not happe, but yf
þ{ere} come figures significatyf aft{er} the cifers. Whan þ{o}u hast
borowede þ{a}t 1 of the next figure significatyf, sett þ{a}t on ou{er}
þe hede of þ{a}t fig{ur}e of þe quych þ{o}u wold haue draw þe neþ{er}
figure out yf þ{o}u hadest myȝt, & reken it for ten as þo{u} diddest
i{n} þe oþ{er} case her{e}-a-for{e}. Whaɳ þ{o}u hast þus y-do loke how
mony cifers þ{ere} wer{e} bye-twene þat figur{e} significatyf, & þe
fig{ur}e of þe quych þ{o}u woldest haue y-draw the [*leaf 145a] neþ{er}
figure, and of eu{er}y of þe ylke cifers make a figur{e} of 9. lo an
Ensampull{e} after.
+-----+
|40002|
|10004|
+-----+
Take 4 out of 2. it wel not be. borow 1 out of be next figure
significatyf, þe quych is 4, & þen leues 3. do away þ{a}t figur{e} of 4
& write þ{ere} 3. & sett þ{a}t 1 vppon þe fig{ur}e of 2 hede, & þan take
4 out of ten, & þan þere leues 6. Cast 6 to the fig{ur}e of 2, þ{a}t wol
be 8. do away þat 6 & write þ{er}e 8. Whan þ{o}u hast þus y-do make of
eu{er}y 0 betweyn 3 & 8 a figure of 9, & þan worch forth in goddes name.
& yf þ{o}u hast wel y-do þ{o}u[{5}] schalt haue þis nomb{er}
+-----+
|39998| Sic.
|10004|
+-----+
[Headnote: How to prove the Subtraction.]
¶ Si subt{ra}cc{i}o sit b{e}n{e} facta p{ro}bar{e} valebis
Quas s{u}btraxisti p{ri}mas addendo figuras.
[Sidenote: How to prove a subtraction sum. Here is an example.
He works his proof through, and brings out a result.]
¶ Her{e} he teches þe Craft how þ{o}u schalt know, whan þ{o}u hast
subt{ra}yd, wheþ{er} þou hast wel ydo or no. And þe Craft is þis, ryght
as þ{o}u subtrayd þe neþ{er} figures fro þe hier figures, ryȝt so adde
þe same neþ{er} figures to þe hier figures. And yf þ{o}u haue well
y-wroth a-for{e} þou schalt haue þe hier nombre þe same þ{o}u haddest or
þou be-gan to worch. as for þis I bade þou schulde kepe þe neþ{er}
figures stylle. lo an [*leaf 145b] Ensampull{e} of all{e} þe 4 cases
toged{re}. worche well{e} þis case
+--------+
|40003468|.
|20004664|
+--------+
And yf þou worch well{e} whan þou hast all{e} subtrayd þe þ{a}t hier
nombr{e} her{e}, þis schall{e} be þe nombre here foloyng whan þ{o}u hast
subtrayd.
+--------+
|39998804|. [Sidenote: Our author makes a slip here (3 for 1).]
|20004664|
+--------+
And þou schalt know þ{us}. adde þe neþ{er} rowe of þe same nombre to þe
hier rewe as þus, cast 4 to 4. þat wol be 8. do away þe 4 & write þ{ere}
8. by þe first case of addicioɳ. þan cast 6 to 0 þat wol be 6. do away
þe 0, & write þere 6. þan cast 6 to 8, þ{a}t wel be 14. do away 8 &
write þ{ere} a fig{ur}e of 4, þat is þe digit, and write a fig{ur}e of
1. þ{a}t schall be-token ten. þ{a}t is þe articul vpon þe hed of 8 next
aft{er}, þan reken þ{a}t 1. for 1. & cast it to 8. þat schal be 9. cast
to þat 9 þe neþ{er} fig{ur}e vnd{er} þat þe quych is 4, & þat schall{e}
be 13. do away þat 9 & sett þ{er}e 3, & sett a figure of 1. þ{a}t schall
be 10 vpon þe next figur{is} hede þe quych is 9. by þe secu{n}de case
þ{a}t þ{o}u hadest in addicioɳ. þan cast 1 to 9. & þat wol be 10. do
away þe 9. & þat 1. And write þ{ere} a cifer. and write þe articull{e}
þat is 1. betokenyng{e} 10. vpon þe hede of þe next figur{e} toward þe
lyft side, þe quych [*leaf 146a] is 9, & so do forth tyl þ{o}u come to
þe last 9. take þe figur{e} of þat 1. þe quych þ{o}u schalt fynde ou{er}
þe hed of 9. & sett it ou{er} þe next figures hede þat schal be 3.
¶ Also do away þe 9. & set þ{ere} a cifer, & þen cast þat 1 þat stondes
vpon þe hede of 3 to þe same 3, & þ{a}t schall{e} make 4, þen caste to
þe ylke 4 the figur{e} in þe neyþ{er} rewe, þe quych is 2, and þat
schall{e} be 6. And þen schal þ{o}u haue an Ensampull{e} aȝeyɳ, loke &
se, & but þ{o}u haue þis same þ{o}u hase myse-wroȝt.
+--------+
|60003468|
|20004664|
+--------+