Chapter 2

In a composite treatise containing tracts written A.D. 1157 and 1208, on the calendar, the abacus, the manual calendar and the manual abacus, we have a number of the methods preserved. As an example we give the rule for multiplication (Claud. A. IV., f. 54 vo). “Si numerus multiplicat alium numerum auferatur differentia majoris a minore, et per residuum multiplicetur articulus, et una differentia per aliam, et summa proveniet.” Example, 8 × 7. The difference of 8 is 2, of 7 is 3, the next article being 10; 7 - 2 is 5. 5 × 10 = 50; 2 × 3 = 6. 50 + 6 = 56 answer. The rule will hold in such cases as 17 × 15 where the article next higher is the same for both, _i.e._, 20; but in such a case as 17 × 9 the difference for each number must be taken from the higher article, _i.e._, the difference of 9 will be 11.

THE ALGORISTS.

Algorism (augrim, augrym, algram, agram, algorithm), owes its name to the accident that the first arithmetical treatise translated from the Arabic happened to be one written by Al-Khowarazmi in the early ninth century, “de numeris Indorum,” beginning in its Latin form “Dixit Algorismi. . . .” The translation, of which only one MS. is known, was made about 1120 by Adelard of Bath, who also wrote on the Abacus and translated with a commentary Euclid from the Arabic. It is probable that another version was made by Gerard of Cremona (1114-1187); the number of important works that were not translated more than once from the Arabic decreases every year with our knowledge of medieval texts. A few lines of this translation, as copied by Halliwell, are given on p. 72, note 2. Another translation still seems to have been made by Johannes Hispalensis.

Algorism is distinguished from Abacist computation by recognising seven rules, Addition, Subtraction, Duplation, Mediation, Multiplication, Division, and Extraction of Roots, to which were afterwards added Numeration and Progression. It is further distinguished by the use of the zero, which enabled the computer to dispense with the columns of the Abacus. It obviously employs a board with fine sand or wax, and later, as a substitute, paper or parchment; slate and pencil were also used in the fourteenth century, how much earlier is unknown.[5*] Algorism quickly ousted the Abacus methods for all intricate calculations, being simpler and more easily checked: in fact, the astronomical revival of the twelfth and thirteenth centuries would have been impossible without its aid.

[Footnote 5*: Slates are mentioned by Chaucer, and soon after (1410) Prosdocimo de Beldamandi speaks of the use of a “lapis” for making notes on by calculators.]

The number of Latin Algorisms still in manuscript is comparatively large, but we are here only concerned with two--an Algorism in prose attributed to Sacrobosco (John of Holywood) in the colophon of a Paris manuscript, though this attribution is no longer regarded as conclusive, and another in verse, most probably by Alexander de Villedieu (Villa Dei). Alexander, who died in 1240, was teaching in Paris in 1209. His verse treatise on the Calendar is dated 1200, and it is to that period that his Algorism may be attributed; Sacrobosco died in 1256 and quotes the verse Algorism. Several commentaries on Alexander’s verse treatise were composed, from one of which our first tractate was translated, and the text itself was from time to time enlarged, sections on proofs and on mental arithmetic being added. We have no indication of the source on which Alexander drew; it was most likely one of the translations of Al-Khowarasmi, but he has also the Abacists in mind, as shewn by preserving the use of differences in multiplication. His treatise, first printed by Halliwell-Phillipps in his _Rara Mathematica_, is adapted for use on a board covered with sand, a method almost universal in the thirteenth century, as some passages in the algorism of that period already quoted show: “Est et alius modus qui utitur apud Indos, et doctor hujusmodi ipsos erat quidem nomine Algus. Et modus suus erat in computando per quasdam figuras scribendo in pulvere. . . .” “Si voluerimus depingere in pulvere predictos digitos secundum consuetudinem algorismi . . .” “et sciendum est quod in nullo loco minutorum sive secundorum . . . in pulvere debent scribi plusquam sexaginta.”

MODERN ARITHMETIC.

Modern Arithmetic begins with Leonardi Fibonacci’s treatise “de Abaco,” written in 1202 and re-written in 1228. It is modern rather in the range of its problems and the methods of attack than in mere methods of calculation, which are of its period. Its sole interest as regards the present work is that Leonardi makes use of the digital signs described in Record’s treatise on _The arte of nombrynge by the hand_ in mental arithmetic, calling it “modus Indorum.” Leonardo also introduces the method of proof by “casting out the nines.”

DIGITAL ARITHMETIC.

The method of indicating numbers by means of the fingers is of considerable age. The British Museum possesses two ivory counters marked on one side by carelessly scratched Roman numerals IIIV and VIIII, and on the other by carefully engraved digital signs for 8 and 9. Sixteen seems to have been the number of a complete set. These counters were either used in games or for the counting board, and the Museum ones, coming from the Hamilton collection, are undoubtedly not later than the first century. Frohner has published in the _Zeitschrift des Münchener Alterthumsvereins_ a set, almost complete, of them with a Byzantine treatise; a Latin treatise is printed among Bede’s works. The use of this method is universal through the East, and a variety of it is found among many of the native races in Africa. In medieval Europe it was almost restricted to Italy and the Mediterranean basin, and in the treatise already quoted (Sloane 3281) it is even called the Abacus, perhaps a memory of Fibonacci’s work.

Methods of calculation by means of these signs undoubtedly have existed, but they were too involved and liable to error to be much used.

THE USE OF “ARABIC” FIGURES.

It may now be regarded as proved by Bubnov that our present numerals are derived from Greek sources through the so-called Boethian “apices,” which are first found in late tenth century manuscripts. That they were not derived directly from the Arabic seems certain from the different shapes of some of the numerals, especially the 0, which stands for 5 in Arabic. Another Greek form existed, which was introduced into Europe by John of Basingstoke in the thirteenth century, and is figured by Matthew Paris (V. 285); but this form had no success. The date of the introduction of the zero has been hotly debated, but it seems obvious that the twelfth century Latin translators from the Arabic were perfectly well acquainted with the system they met in their Arabic text, while the earliest astronomical tables of the thirteenth century I have seen use numbers of European and not Arabic origin. The fact that Latin writers had a convenient way of writing hundreds and thousands without any cyphers probably delayed the general use of the Arabic notation. Dr. Hill has published a very complete survey of the various forms of numerals in Europe. They began to be common at the middle of the thirteenth century and a very interesting set of family notes concerning births in a British Museum manuscript, Harl. 4350 shows their extension. The first is dated Mij^c. lviii., the second Mij^c. lxi., the third Mij^c. 63, the fourth 1264, and the fifth 1266. Another example is given in a set of astronomical tables for 1269 in a manuscript of Roger Bacon’s works, where the scribe began to write MCC6. and crossed out the figures, substituting the “Arabic” form.

THE COUNTING BOARD.

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.]