The Earliest Arithmetics in English

8/13/2019 The Earliest Arithmetics in English

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Earliest Arithmetics in English

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Contents(added by transcriber)

Introduction v

The Crafte of Nombrynge 3

The Art of Nombryng 33

Accomptynge by Counters 52

The arte of nombrynge by the hande 66

APP. I. A Treatise on the Numeration of Algorism 70

APP. II. Carmen de Algorismo 72

Index of Technical Terms 81

Glossary 83

The Earliest Arithmeticsin English

EDITED WITH INTRODUCTIONBY

ROBERT STEELE


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LONDON:PUBLISHED FOR THE EARLY ENGLISH TEXT SOCIETY

BY HUMPHREY MILFORD, OXFORD UNIVERSITY PRESS,AMEN CORNER, E.C. 4.

1922.

INTRODUCTION

THEnumber of English arithmetics before the sixteenth century is very small. This ishardly to be wondered at, as no one requiring to use even the simplest operations ofthe art up to the middle of the fifteenth century was likely to be ignorant of Latin, inwhich language there were several treatises in a considerable number of manuscripts,as shown by the quantity of them still in existence. Until modern commerce was fairly

well established, few persons required more arithmetic than addition and subtraction,and even in the thirteenth century, scientific treatises addressed to advanced studentscontemplated the likelihood of their not being able to do simple division. On the otherhand, the study of astronomy necessitated, from its earliest days as a science,considerable skill and accuracy in computation, not only in the calculation ofastronomical tables but in their use, a knowledge of which latter was fairly commonfrom the thirteenth to the sixteenth centuries.

The arithmetics in English known to me are:

(1) Bodl. 790 G. VII. (2653) f. 146-154 (15th c.) inc.Of angrym ther be IX figures innumbray . . . A mere unfinished fragment, only getting as far as Duplation.

(2) Camb. Univ. LI. IV. 14 (III.) f. 121-142 (15th c.) inc.Al maner of thyngis thatprosedeth ffro the frist begynnyng . . .

(3) Fragmentary passages or diagrams in Sloane 213 f. 120-3 (a fourteenth-centurycounting board), Egerton 2852 f. 5-13, Harl. 218 f. 147 and

(4) The two MSS. here printed; Eg. 2622 f. 136 and Ashmole 396 f. 48. All of these,as the language shows, are of the fifteenth century.

The CRAFTEOFNOMBRYNGEis one of a large number of scientific treatises, mostly in Latin,bound up together as Egerton MS. 2622 in the British Museum Library. It measures7 5, 29-30 lines to the page, in a rough hand. The English is N.E. Midland in dialect.

It is a translation and amplification of one of the numerous glosses on the de algorismoof Alexander de Villa Dei (c. 1220), such as that of Thomas of Newmarket contained inthe British Museum MS. Reg. 12, E. 1. A fragment of another translation of the samegloss was printed by Halliwell in his Rara Mathematica(1835) p. 29. 1It corresponds, asfar as p. 71, l. 2, roughly to p. 3 of our version, and from thence to the end p. 2, ll. 16-40.

The ARTOFNOMBRYNGis one of the treatises bound up in the Bodleian MS. Ashmole 396.It measures 11 17, and is written with thirty-three lines to the page in afifteenth century hand. It is a translation, rather literal, with amplifications of the de artenumerandiattributed to John of Holywood (Sacrobosco) and the translator had


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threes, with which counters were used, either plain or marked with signs denoting thenine numerals, etc.

(3) Tablets or boxes containing nine grooves or wires, in or on which ran beads.

(4) Tablets on which nine (or more) horizontal lines were marked, each third beingmarked off.

The only Greek counting board we have is of the fourth class and was discovered atSalamis. It was engraved on a block of marble, and measures 5 feet by 2. Its chief

part consists of eleven parallel lines, the 3rd, 6th, and 9th being marked with a cross.Another section consists of five parallel lines, and there are three rows of arithmeticalsymbols. This board could only have been used with counters (calculi), preferablyunmarked, as in our treatise ofAccomptynge by Counters.

CLASSICALROMANMETHODSOFCALCULATION.

We have proof of two methods of calculation in ancient Rome, one by the first method,in which the surface of sand was divided into columns by a stylus or the hand. Counters(calculi, or lapilli), which were kept in boxes ( loculi), were used in calculation, as we

learn from Horaces schoolboys (Sat. 1. vi. 74). For the sand see Persius I. 131, Nec quiabaco numeros et secto in pulvere metas scit risisse, Apul. Apolog. 16 (pulvisculo),Mart. Capella, lib. vii. 3, 4, etc. Cicero says of an expert calculator eruditum attigissepulverem, (de nat. Deorum, ii. 18). Tertullian calls a teacher of arithmetic primusnumerorum arenarius (de Pallio, in fine). The counters were made of various materials,ivory principally, Adeo nulla uncia nobis est eboris, etc. (Juv. XI. 131), sometimes ofprecious metals, Pro calculis albis et nigris aureos argenteosque habebat denarios(Pet. Arb. Satyricon, 33).

There are, however, still in existence four Roman counting boards of a kind which doesnot appear to come into literature. A typical one is of the third class. It consists of anumber of transverse wires, broken at the middle. On the left hand portion four beads

are strung, on the right one (or two). The left hand beads signify units, the right handone five units. Thus any number up to nine can be represented. This instrument is in allessentials the same as the Swanpan or Abacus in use throughout the Far East. TheRussian stchota in use throughout Eastern Europe is simpler still. The method of usingthis system is exactly the same as that ofAccomptynge by Counters, the right-hand fivebead replacing the counter between the lines.

THEBOETHIANABACUS.

Between classical times and the tenth century we have little or no guidance as to the

art of calculation. Boethius (fifth century), at the end of lib. II. of his Geometriagives usa figure of an abacus of the second class with a set of counters arranged within it. Ithas, however, been contended with great probability that the whole passage is a tenthcentury interpolation. As no rules are given for its use, the chief value of the figure isthat it gives the signs of the nine numbers, known as the Boethian apices or notae(from whence our word notation). To these we shall return later on.

THEABACISTS.

It would seem probable that writers on the calendar like Bede (A.D. 721) and Helpericus


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(A.D. 903) were able to perform simple calculations; though we are unable to guesstheir methods, and for the most part they were dependent on tables taken from Greeksources. We have no early medieval treatises on arithmetic, till towards the end of thetenth century we find a revival of the study of science, centring for us round the nameof Gerbert, who became Pope as Sylvester II. in 999. His treatise on the use of the

Abacus was written (c. 980) to a friend Constantine, and was first printed among theworks of Bede in the Basle (1563) edition of his works, I. 159, in a somewhat enlargedform. Another tenth century treatise is that of Abbo of Fleury (c. 988), preserved in

several manuscripts. Very few treatises on the use of the Abacus can be certainlyascribed to the eleventh century, but from the beginning of the twelfth century theirnumbers increase rapidly, to judge by those that have been preserved.

The Abacists used a permanent board usually divided into twelve columns; the columnswere grouped in threes, each column being called an arcus, and the value of a figurein it represented a tenth of what it would have in the column to the left, as in ourarithmetic of position. With this board counters or jetons were used, either plain or,more probably, marked with numerical signs, which with the early Abacists were the

apices, though counters from classical times were sometimes marked on one side withthe digital signs, on the other with Roman numerals. Two ivory discs of this kind fromthe Hamilton collection may be seen at the British Museum. Gerbert is said by Richer to

have made for the purpose of computation a thousand counters of horn; the usualnumber of a set of counters in the sixteenth and seventeenth centuries was a hundred.

Treatises on the Abacus usually consist of chapters on Numeration explaining thenotation, and on the rules for Multiplication and Division. Addition, as far as it requiredany rules, came naturally under Multiplication, while Subtraction was involved in theprocess of Division. These rules were all that were needed in Western Europe incenturies when commerce hardly existed, and astronomy was unpractised, and eventhey were only required in the preparation of the calendar and the assignments of theroyal exchequer. In England, for example, when the hide developed from the normalholding of a household into the unit of taxation, the calculation of the geldage in each

shire required a sum in division; as we know from the fact that one of the Abacistsproposes the sum: If 200 marks are levied on the county of Essex, which containsaccording to Hugh of Bocland 2500 hides, how much does each hide pay?3Exchequermethods up to the sixteenth century were founded on the abacus, though when wehave details later on, a different and simpler form was used.

The great difficulty of the early Abacists, owing to the absence of a figure representingzero, was to place their results and operations in the proper columns of the abacus,especially when doing a division sum. The chief differences noticeable in their works arein the methods for this rule. Division was either done directly or by means of differencesbetween the divisor and the next higher multiple of ten to the divisor. Later Abacists

made a distinction between iron and golden methods of division. The following areexamples taken from a twelfth century treatise. In following the operations it must beremembered that a figure asterisked represents a counter taken from the board. A zerois obviously not needed, and the result may be written down in words.

(a) MULTIPLICATION. 4600 23.

Thousands

Hun T

U

Hun T

U


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dreds

ens

nits

dreds

ens

nits

4 6 Multiplicand.

1 8 600 3.

1 2 4000 3.

1 2 600 20. 8 4000 20.

1 5 8 Total product.

2 3 Multiplier.

(b) DIVISION: DIRECT. 100,000 20,023. Here each counter in turn is a separatedivisor.

H. T. U. H. T. U.

2 2 3 Divisors.

2 Place greatest divisor to right of dividend.

1 Dividend.

2 Remainder.

1

1 9 9 Another form of same.

8 Product of 1st Quotient and 20.

1 9 9 2 Remainder.

1 2 Product of 1st Quotient and 3.

1 9 9 8 Final remainder.

4 Quotient.

(c) DIVISIONBYDIFFERENCES. 900 8. Here we divide by (10-2).

H. T. U.

2 Difference.

8 Divisor.

49 Dividend.

41 8 Product of difference by 1st Quotient (9).

2 Product of difference by 2nd Quotient (1). 41 Sum of 8 and 2.

2 Product of difference by 3rd Quotient (1).

4 Product of difference by 4th Quot. (2). Remainder.

2 4th Quotient.

1 3rd Quotient.

1 2nd Quotient.

9 1st Quotient.

1 1 2 Quotient.(Total of all four.)


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DIVISION. 7800 166.

Thousands

H. T. U. H. T. U.

3 4 Differences (making 200 trial divisor).

1 6 6 Divisors.

47 8 Dividends.

1 Remainder of greatest dividend.

1 2 Product of 1st difference (4) by 1st Quotient (3).

9 Product of 2nd difference (3) by 1st Quotient (3).

42 8 2 New dividends.

3 4 Product of 1st and 2nd difference by 2nd Quotient (1).


2 Product of 1st difference by 3rd Quotient (5).

1 5 Product of 2nd difference by 3rd Quotient (5).

43 3 New dividends.


3 4 Product of 1st and 2nd difference by 4th Quotient (1).

1 6 4 Remainder(less than divisor).

1 4th Quotient.

5 3rd Quotient.

1 2nd Quotient.

3 1st Quotient.

4 6 Quotient.

DIVISION. 8000 606.

Thousands

H. T. U. H. T. U.

9 Difference (making 700 trial divisor).

4 Difference.

6 6 Divisors.

48 Dividend.

1 Remainder of dividend.

9 4 Product of difference 1 and 2 with 1st Quotient (1). 41 9 4 New dividends.


9 4 Product of difference 1 and 2 with 2nd Quotient (1).

41 3 3 4 New dividends.


9 4 Product of difference 1 and 2 with 3rd Quotient (1).


6 6 Product of divisors by 4th Quotient (1).


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1 2 2 Remainder.

1 4th Quotient.

1 3rd Quotient.

1 2nd Quotient.

1 1st Quotient.

1 3 Quotient.

The chief Abacists are Gerbert (tenth century), Abbo, and Hermannus Contractus(1054), who are credited with the revival of the art, Bernelinus, Gerland, and Radulphusof Laon (twelfth century). We know as English Abacists, Robert, bishop of Hereford,1095, abacum et lunarem compotum et celestium cursum astrorum rimatus, TurchillusCompotista (Thurkil), and through him of Guilielmus R. . . . the best of livingcomputers, Gislebert, and Simonus de Rotellis (Simon of the Rolls). They flourishedmost probably in the first quarter of the twelfth century, as Thurkils treatise deals alsowith fractions. Walcher of Durham, Thomas of York, and Samson of Worcester are alsoknown as Abacists.

Finally, the term Abacists came to be applied to computers by manual arithmetic. A MS.Algorithm of the thirteenth century (Sl. 3281, f. 6, b), contains the following passage:

Est et alius modus secundum operatores sive practicos, quorum unus appellaturAbacus; et modus ejus est in computando per digitos et junctura manuum, et iste utiturultra Alpes.

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 themethods 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 summaproveniet. 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 acase as 17 9 the difference for each number must be taken from the higher article,i.e., the difference of 9 will be 11.

THEALGORISTS.

Algorism (augrim, augrym, algram, agram, algorithm), owes its name to the accidentthat the first arithmetical treatise translated from the Arabic happened to be one writtenby Al-Khowarazmi in the early ninth century, de numeris Indorum, beginning in itsLatin form Dixit Algorismi. . . . The translation, of which only one MS. is known, wasmade about 1120 by Adelard of Bath, who also wrote on the Abacus and translated witha commentary Euclid from the Arabic. It is probable that another version was made byGerard of Cremona (1114-1187); the number of important works that were nottranslated more than once from the Arabic decreases every year with our knowledge ofmedieval 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, towhich were afterwards added Numeration and Progression. It is further distinguished bythe use of the zero, which enabled the computer to dispense with the columns of the


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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, howmuch earlier is unknown. 5Algorism quickly ousted the Abacus methods for all intricatecalculations, being simpler and more easily checked: in fact, the astronomical revival ofthe twelfth and thirteenth centuries would have been impossible without its aid.

The number of Latin Algorisms still in manuscript is comparatively large, but we are hereonly concerned with twoan Algorism in prose attributed to Sacrobosco (John ofHolywood) 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 versetreatise on the Calendar is dated 1200, and it is to that period that his Algorism may beattributed; Sacrobosco died in 1256 and quotes the verse Algorism. Severalcommentaries on Alexanders verse treatise were composed, from one of which our firsttractate was translated, and the text itself was from time to time enlarged, sections onproofs and on mental arithmetic being added. We have no indication of the source onwhich Alexander drew; it was most likely one of the translations of Al-Khowarasmi, buthe has also the Abacists in mind, as shewn by preserving the use of differences inmultiplication. His treatise, first printed by Halliwell-Phillipps in his Rara Mathematica, isadapted 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 quotedshow: Est et alius modus qui utitur apud Indos, et doctor hujusmodi ipsos erat quidemnomine Algus. Et modus suus erat in computando per quasdam figuras scribendo inpulvere. . . . Si voluerimus depingere in pulvere predictos digitos secundumconsuetudinem algorismi . . . et sciendum est quod in nullo loco minutorum sivesecundorum . . . in pulvere debent scribi plusquam sexaginta.

MODERNARITHMETIC.

Modern Arithmetic begins with Leonardi Fibonaccis treatise de Abaco, written in 1202

and re-written in 1228. It is modern rather in the range of its problems and themethods of attack than in mere methods of calculation, which are of its period. Its soleinterest as regards the present work is that Leonardi makes use of the digital signsdescribed in Records treatise on The arte of nombrynge by the handin mentalarithmetic, calling it modus Indorum. Leonardo also introduces the method of proof by

casting out the nines.

DIGITALARITHMETIC.

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 carelesslyscratched Roman numerals IIIV and VIIII, and on the other by carefully engraveddigital 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 Museumones, coming from the Hamilton collection, are undoubtedly not later than the firstcentury. Frohner has published in the Zeitschrift des Mnchener Alterthumsvereinsa set,almost complete, of them with a Byzantine treatise; a Latin treatise is printed amongBedes works. The use of this method is universal through the East, and a variety of it isfound among many of the native races in Africa. In medieval Europe it was almostrestricted to Italy and the Mediterranean basin, and in the treatise already quoted(Sloane 3281) it is even called the Abacus, perhaps a memory of Fibonaccis work.


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

THEUSEOFARABIC FIGURES.

It may now be regarded as proved by Bubnov that our present numerals are derivedfrom Greek sources through the so-called Boethian apices, which are first found in latetenth century manuscripts. That they were not derived directly from the Arabic seemscertain from the different shapes of some of the numerals, especially the 0, whichstands for 5 in Arabic. Another Greek form existed, which was introduced into Europe byJohn 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 hotlydebated, but it seems obvious that the twelfth century Latin translators from the Arabicwere perfectly well acquainted with the system they met in their Arabic text, while theearliest astronomical tables of the thirteenth century I have seen use numbers ofEuropean and not Arabic origin. The fact that Latin writers had a convenient way ofwriting hundreds and thousands without any cyphers probably delayed the general useof 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 thirteenthcentury and a very interesting set of family notes concerning births in a British Museummanuscript, Harl. 4350 shows their extension. The first is dated Mijc. lviii., the secondMijc. lxi., the third Mijc. 63, the fourth 1264, and the fifth 1266. Another example isgiven in a set of astronomical tables for 1269 in a manuscript of Roger Bacons works,where the scribe began to write MCC6. and crossed out the figures, substituting the

Arabic form.

THECOUNTINGBOARD.

The treatise on pp. 52-65 is the only one in English known on the subject. It describesa method of calculation which, with slight modifications, is current in Russia, China, andJapan, 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 editionsof a tract under this name (with a diagram of the counting board), printed at Leipsic atthe end of the fifteenth century and the beginning of the sixteenth. They give the ninerules, but Capitulum de radicum extractione ad algoritmum integrorum reservato, cujusspecies per ciffrales figuras ostenduntur ubi ad plenum de hac tractabitur. Theinvention of the art is there attributed to Appulegius the philosopher.

The advantage of the counting board, whether permanent or constructed by chalkingparallel 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 representingunits, 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 Chineseand Japanese Swanpan economises by dividing the line into two parts, the beads onone side representing five times the value of those on the other. The Swanpan hasusually many more lines than the stchota, allowing for more extended calculations, seeTylor,Anthropology(1892), p. 314.

Records treatise also mentions another method of counter notation (p. 64) merchantscasting and auditors casting. These were adapted for the usual English method ofreckoning numbers up to 200 by scores. This method seems to have been used in the

x


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A derivation ofAlgorism.

Anotherderivation of thword.

Exchequer. A counting board for merchants use is printed by Halliwell in RaraMathematica(p. 72) from Sloane MS. 213, and two others are figured in Egerton 2622f. 82 and f. 83. The latter is said to be novus modus computandi secunduminventionem 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.

1.Halliwell printed the two sides of his leaf in the wrong order. This and some obviouserrors of transcriptionferye for ferthe, lest for left, etc., have not been corrected inthe reprint on pp. 70-71.

2.For Egyptian use see Herodotus, ii. 36, Plato, de Legibus, VII.

3.See on this Dr. Poole, The Exchequer in the Twelfth Century, Chap. III., and Haskins,Eng. Hist. Review, 27, 101. The hidage of Essex in 1130 was 2364 hides.

4.These figures are removed at the next step.

5.Slates are mentioned by Chaucer, and soon after (1410) Prosdocimo de Beldamandispeaks of the use of a lapis for making notes on by calculators.

The Earliest Arithmetics in English.

Egerton2622.

HEc algorismusars presens dicitur; in quaTalibusindorumfruimurbis quinquefiguris.

This boke is called e boke of algorym, or Augrym afterlewdervse.And is boke tretys e Craftof Nombryng, e quych crafte is calledalso Algorym. Ther was a kyng of Inde, e quich heythAlgor, & hemade is craft. And afterhis name he called hit algorym; or elsanoercause is quy it is called Algorym, for e latyn word of hit s.

Algorismuscomesof Algos, grece, quidestars, latine, craft o englis,and rides, quidestnumerus, latine, A nomburo englys, inde dicitur

Algorismusperaddicionemhuiussillabe mus& subtraccionem d & e,quasi ars numerandi. fforthermoree mostvndirstonde at in is

leaf 136 a.


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Expositioversus.

The meaningand place of thefigures.

Which figure isread first.

Expositio[inmargin].

An explanationof the principlesof notation.

An example:

units,

tens,

versus[in margin].

versus

[in margin].

craft ben vsid teen figurys, as here benewriten for ensampul, 9 8 76 5 4 3 2 1. Exponee too versus afore: this present craft ys called

Algorismus, in e quych we vse teen signys of Inde. Questio. Whyte fyguris of Inde? Solucio. for as I haue sayd afore ai werefondefyrst in Inde of a kyngeof at Cuntre, at was called Algor.

Notation and Numeration.

Prima significat unum; duo vero secunda: Tercia significat tria; sic procede sinistre. Donec ad extremamvenias, que cifra vocatur.

Capitulum primum de significacione figurarum.

In is verse is notifide e significacion of ese figuris. And us exponethe verse. e first signifiyth one, e secunde signi*fiyth tweyne, ethryd signifiyth thre, & the fourte signifiyth 4. And so forthe towardee lyft syde of e tabul or of e boke at e figures benewritenein,til at ou come to the last figure, at is called a cifre. Questio. In

quych syde sittes e first figure? Solucio, forsothe loke quich figure isfirst in e ryt side of e bok or of e tabul, & at same is e firstfigure, for ou schal write bakeward, as here, 3. 2. 6. 4. 1. 2. 5. Thefigure of 5. was first write, & he is e first, for he sittes o e ritsyde. And the figure of 3 is last. Neuer-e-les wen he says Primasignificat vnum&c., at is to say, e first betokenes one, e secunde.2. & fore-er-more, he vndirstondes notof e first figure of eueryrew. But he vndirstondes e first figure at is in e nomburof eforsayd teen figuris, e quych is oneof ese. 1. And e secunde 2. &so forth.

Quelibetillarumsi pr imo limite ponas, Simpliciterse significat: si vero secundo,

Se decies: sursumprocedas multiplicando. Namquefigura sequens quamuis signat decies plus. Ipsa locata loco quam significat pertinente.

Expone is verse us. Euery of ese figuris bitokens hym selfe & nomore, yf he stonde in e first place of e rewele/ this wordeSimpliciterin at verse it is no more to say but at, & no more. If itstonde in the secunde place of e rewle, he betokens tenetymes hymselfe, as is figure2 here 20 tokens ten tyme hym selfe, *at istwenty, for he hym selfe betokenes tweyne, & ten tymes tweneistwenty. And for he stondis o e lyft side & in e secunde place, hebetokens ten tyme hymselfe. And so go forth. ffor euery figure, &he stonde aftura-noertoward the lyft side, he schal betokenetentymes as mich moreas he schul betoken & he stode in e place ereat e figurea-forehym stondes. loo an ensampulle. 9. 6. 3. 4. efigureof 4. at hase is schape . betokens bot hymselfe, for hestondes in e first place. The figureof 3. at hase is schape .betokens ten tymes moreen he schuld &he stde ereat e figureof 4. stondes, at is thretty. The figureof 6, at hase is schape ,

leaf 136 b.

leaf 137 a.


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hundreds,

thousands.

How to read thenumber.

The meaningand use of thecipher.

The last figuremeans morethan all theothers, since it of the highest

value.

Digits.

Articles.

Composites.

What are digits

betokens ten tymes morean he schuld & he stode ereas e figureof . stondes, for erehe schuld tokynebot sexty, & now he betokensten tymes more, at is sex hundryth. The figureof 9. at hase isschape . betokens ten tymes moreanehe schuld & he stode in eplace eree figureof sex stondes, for en he schuld betoken to 9.hundryth, and in e place erehe stondes now he betokens 9.ousande. Al e holenomburis 9 thousande sex hundryth & foure&thretty. fforthermore, when ou schalt rede a nomburof figure, ou

schalt begyneat e last figurein the lyft side, & rede so forth to erit side as here9. 6. 3. 4. Thou schal begyn to rede at e figureof 9.& rede forth us. 9. *thousand sex hundryth thritty & foure. But whenou schallewrite, ou schalt be-gynne to write at e ryt side.

Nil cifra significat seddat signaresequenti.

Expone is verse. A cifre tokens not,bot he makes e figuretobetoken at comes afturhym morean he schuld & he wereaway, asus 1. heree figureof onetokens ten, & yf e cifre wereaway 1&no figureby-forehym he schuld token bot one, for an he schuldstonde in e first place. And e cifre tokens nothyng hym selfe. for

al e nomburof e ylketoo figures is bot ten. Questio. Why says heat a cifre makys a figureto signifye (tyf)more&c. I speke for isworde significatyf, ffor sothe it may happe aftura cifre schuld come a-nourcifre, as us 2. And et e secunde cifre shuld token neuere moreexcephe schuld kepe e orderof e place. and a cifre is nofiguresignificatyf.

Quam precedentes plus ultima significabit /

Expone is verse us. e last figureschal token morean allee oerafore, thouterewerea hundryth thousant figures afore, as us,16798. e last figureat is 1. betokens ten thousant. And allee oer

figures ben bot betokenebot sex thousant seuynehundryth nynty &8. And ten thousant is moreen alleat nombur, ergo e last figuretokens morean all e nomburafore.

The Three Kinds of Numbers

* Post predicta scias breuiterquodtres numerorumDistincte species sunt; nam quidam digiti sunt;

Articuli quidam; quidam quoquecompositi sunt.

Capitulum 2mde triplice divisione numerorum.

The auctor of is tretis departysis worde a nomburinto 3 partes.Some nomburis called digituslatine, a digitin englys. Somme nomburis called articuluslatine. AnArticulin englys. Some nomburis called acomposytin englys. Expone is verse. know ou afture forsaydrewlesat I sayd afore, at ereben thre spicesof nombur. Ooneis adigit, Anoeris an Articul, & e toera Composyt. versus.

Digits, Articles, and Composites.

Sunt digiti numeri qui citradenariumsunt.

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What arearticles.

What numbersare composites

How to write anumber,

if it is a digit;

if it is acomposite.

How to read it.

How to writeArticles:

tens,

hundreds,

thousands, &c.

Herehe telles qwat is a digit, Expone versussic. Nomburs digitusbeneallenomburs at ben with-inne ten, as nyne, 8. 7. 6. 5. 4. 3.2. 1.

Articupli decupli degitorum; compositi suntIlli qui constant ex articulis degitisque.

Herehe telles what is a composyt and what is anearticul. Exponesic versus. Articulis ben 2alleat may be deuidyt into nombursoften & nothyngeleueouer, as twenty, thretty, fourty, a hundryth,a thousand, & such oer, ffor twenty may be departyt in-to 2nomburs of ten, fforty in to fourenomburs of ten, & so forth.

*Compositys be nomburs at bene componyt of a digyt & of anarticulleas fouretene, fyftene, sextene, & such oer. ffortene iscomponyd of foureat is a digit & of ten at is an articulle. ffiftene iscomponyd of 5 & ten, & so of all oer, what at ai ben. Short-lycheuery nomburat be-gynnes witha digit & endyth in a articulleis acomposyt, as fortene bygennyngeby foureat is a digit, & endes inten.

Ergo, proposito numero tibi scribere, primoRespicias quid sit numerus; si digitus sitPrimo scribe loco digitum, si compositus sitPrimo scribe loco digitumpost articulum; sic.

here he telles how ou schalt wyrchwhan ou schalt write anombur. Expone versumsic, & fac iuxta exponentis sentenciam; whanou hast a nomburto write, loke fyrst what manernomburit ys atou schalt write, whether it be a digit or a composit or an Articul. Ifhe be a digit, write a digit, as yf it be seuen, write seuen & write atdigit in e first place toward e ryght side. If it be a composyt, writee digit of e composit in e first place & write e articul of at digit

in e secunde place next toward e lyft side. As yf ou schal write sex& twenty. write e digit of e nomburin e first place at is sex, andwrite e articul next afturat is twenty, as us 26. But whan ouschalt sowne or speke *or rede an Composyt ou schalt first sowne earticul & afture digit, as ou seyst by e comynespeche, Sex &twenty & nout twenty & sex. versus.

Articulussi sit, in primo limite cifram,Articulum veroreliquis inscribe figuris.

Here he tells how ou schal write when e nombre at ou hase towrite is an Articul. Expone versus sic & fac secundum sentenciam. Ifee nomburat ou hast write be an Articul, write first a cifre & afture cifer write an Articulleus. 2. fforthermoreou schalt vndirstondeyf ou haue an Articul, loke how mych he is, yf he be with-ynne anhundryth, ou schalt write bot onecifre, afore, as here.9. If earticullebe by hym-silfe & be an hundrid euene, en schal ou write.1. & 2 cifers afore, at he may stonde in e thryd place, for eueryfigurein e thryd place schal token a hundrid tymes hym selfe. If earticul be a thousant or thousandes3and he stonde by hymselfe,write afore3 cifers & so for of al oer.

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of figures;

12342168.

how many casewhat is its resu

How to set dowthe sum.

123234.

Add the firstfigures;

rub out the topfigure;

write the resultin its place.

Here is anexample.

12342142.

Suppose it is aComposite, setdown the digit,and carry thetens.

ou schallehaue tweyne rewes of figures, onevndur a-nother, ashereou mayst se. As for e thryd ou most know at thereben foure diuerse cases. As for e forthe ou most know at eprofet of is craft is to telle what is e hole nomburat comes ofdiuerse nomburis. Now as to e texte of oure verse, he teches therehow ou schal worchin is craft. He says yf ou wilt cast onenomburto anoernombur, ou most by-gynne on is wyse. ffyrstwrite *two rewes of figuris & nombris so at ou write e first figure

of e hyer nombureuenevndirthe first figureof e nether nombur,And e secunde of e nether nombureuenevndire secunde ofe hyer, & so forthe of euery figure of both e rewes as oumayst se.

The Cases of the Craft of Addition.

Inde duas adde primas hac condicione:Si digitus crescat ex addicione priorum;Primo scribe loco digitum, quicunquesit ille.

Here he teches what ou schalt do when ou hast write too rewes

of figuris onvnder an-oer, as I sayd be-fore. He says ou schalttake e first figure of e heyer nombre& e fyrst figureof e neernombre, & cast hem to-gedervp-on is condicio. Thou schal lokeqweere nomberat comys ere-of be a digit or no. If he be adigit ou schalt do away e first figure of e hyer nombre, and writeerein his stede at he stode Inne e digit, at comes of e ylke 2figures, & so wrichforth o oerfigures yf erebe ony moo, til oucome to e ende toward e lyft side. And lede e nether figurestondestill euer-moretil ou haue ydo. ffor ere-by ou schal wytewheerou hast donewel or no, as I schal tell e afterward in e ende of isChapter. And loke allgateat ou be-gynne to worch in is Craft of

Addi*cio in e ryt side, here is an ensampul of is case. Caste2 to foure& at wel be sex, do away 4. & write in e sameplace e figure of sex. And lete e figure of 2 in e nether rewestonde stil. When ou hast do so, cast 3 & 4 to-gedurand at wel beseuen at is a digit. Do away e 3, & set ereseue, and lete eneerfigurestonde stille, & so worchforth bakward til ou hast ydoall to-geder.

Et si compositus, in limite scribe sequenteArticulum, primo digitum; quiasic iubet ordo.

Here is e secunde case at may happe in is craft. And e case isis, yf of e casting of 2 nomburis to-geder, as of e figure of e hyerrewe & of e figure of e neerrewe come a Composyt, how schaltouworch. usou schalt worch. Thou shalt do away e figure of ehyer nomberat was cast to e figure of e neernomber. Andwrite eree digit of e Composyt. And set e articul of e compositnext aftere digit in e same rewe, yf erebe no mofigures after.But yf erebe mo figuris afterat digit. And ere he schall be rekendfor hym selfe. And when ou schalt adde at ylke figure at berys earticulleouerhis hed to e figurevnderhym, ou schalt cast atarticul to e figure at hase hym ouerhis hed, & ereat Articul schal

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Here is anexample.

326216.

1322216.

Suppose it is anArticle, set dowa cipher andcarry the tens.

25.

15 Here is anexample.1

2155

5

What to do wheyou have acipher in the torow.184.

17743An example ofall thedifficulties.

Four things to

1 5

toke hym selfe. lo an Ensampull *of all. Cast 6 to 6, & ere-ofwil arise twelue. do away e hyer 6 & write ere2, at is e digitof is composit. And enwrite e articulleat is ten ouere figurished of twene as us. Now cast e articulleat standus vpon efiguris of twene hed to e same figure, & reken at articul bot forone, and an erewil arise thre. an cast at thre to e neerfigure, at is one, & at wul be foure. do away e figure of 3, andwrite erea figure of foure. and lete e neerfigure stonde stil, &

an worch forth. vndeversus. Articulus si sit, in primo limite cifram, Articulumvero reliquis inscribe figuris,Vel perse scribas si nulla figura sequatur.

Herehe puttes e thryde case of e craft of Addicio. & e case isis. yf of Addiciou of 2 figuris a-ryse an Articulle, how schal ou do.thou most do away e heerfigure at was addid to e neer, & writeerea cifre, and sett e articulson e figuris hede, yf at erecomeony after. And wyrch an as I haue tolde e in e secunde case. Anensampull. Cast 5 to 5, at wylle be ten. now do away e hyer 5,

& write erea cifer. And sette ten vpon e figuris hed of 2. Andreken it but for on us. lo an EnsampulleAnd *an worch forth.But yf erecome no figure aftere cifre, write e articul nexthym in e same rewe as here cast 5 to 5, and it wel be ten.do away 5. at is e hier 5. and write erea cifre, & writeafterhym e articul as us And an ou hast done.

Si tibi cifra superueniens occurrerit, illamDele superpositam; fac illic scribe figuram,Postea procedas reliquas addendo figuras.

Herehe puttese fourt case, & it is is, at yf erecome a cifer ine hier rewe, how ou schal do. us ou schalt do. do away e cifer,& sett eree digit at comes of e addiciounas us In isensampul ben allee fourecases. Cast 3 to foure, at wol beseue. do away 4. & write ereseue; an cast 4 to e figureof 8.at wel be 12. do away 8, & sett ere2. at is a digit, and sette earticul of e composit, at is ten, vpon e cifers hed, & reken it forhym selfe at is on. an cast oneto a cifer, & hit wullebe but on, fornot & on makes but one. an cast 7. at stondes vnderat on tohym, & at wel be 8. do away e cifer & at 1. & sette ere8. an goforthermore. cast e oer7 to e cifer at stondes ouerhym. at wul

be bot seuen, for e cifer betokens not. do away e cifer & sette ereseue, *& en go forermore& cast 1 to 1, & at wel be 2. do awaye hier 1, & sette ere2. an hast ou do. And yf ou haue wel ydois nomber at is sett here-afterwel be e nomber at schallearyseof allee addicio as here27827. Sequitur alia species.

The Craft of Subtraction.

A numero numerumsi sit tibi demerecuraScribe figurarumseries, vt in addicione.

This is e Chapterof subtraccio, in the quych ou most know foure

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know aboutsubtraction:

the first;

the second;

the third;

the fourth.

Put the greaternumber abovethe less.

The first case osubtraction.

Here is anexample.234

122

Put a cipher ifnothing remains

Here is an

example.2424

Suppose youcannot take the

nessessary thynges. the first what is subtraccio. e secunde is howmony nombers ou most haue to subtraccio, the thryd is how monymaners of cases eremay happe in is craft of subtraccio. The fourteis qwat is e profet of is craft. As for e first, ou most know atsubtraccio is drawyngeof onenowmberoute of anoernomber. Asfor e secunde, ou most knowe at ou most haue two rewes offiguris onevnderanoer, as ou addystin addicio.As for e thryd,ou moyst know at fouremanerof diuerse casis mai happe in is

craft. As for e fourt, ou most know at e profet of is craft iswhenne ou hasse taken e lasse nomber out of e moreto tellewhat ereleuesouerat. & ou most be-gynne to wyrch in iscraftin e ryght side of e boke, as ou diddyst in addicio. Versus.

Maiori numero numerumsuppone minorem, Siue pari numero supponaturnumerus par.

* Herehe telles at e hier nomber most be moreen e neer, orels eue as mych. but he may not be lasse. And e case is is, ouschalt drawe e neernomber out of e hyer, & ou mayst not do atyf e hier nomber werelasse an at. ffor ou mayst not draw sex

out of 2. But ou mast draw 2 out of sex. And ou maiste draw tweneout of twene, for ou schal leue not of e hier twene vndeversus.

The Cases of the Craft of Subtraction.

Postea si possis a prima subtrahe primamScribens quod remanet.

Hereis e first case put of subtraccio, & he says ou schalt begynnein e ryght side, & draw e first figureof e neerrewe out of e firstfigureof e hier rewe. qwether e hier figure be moreen e neer,or eue as mych. And at is notified in e vers when he says Si

possis. Whan ou has us ydo, do away e hiest figure & sett ereat leuesof e subtraccio, lo an Ensampulledraw 2 out of 4.an leues 2. do away 4 & write ere2, & latte e neerfigurestonde stille, & so go for-byoerfiguris till ou come to eende, an hast ou do.

Cifram si nil remanebit.

Herehe puttese secunde case, & hit is is. yf it happe at qwenou hast draw on neerfigureout of a hier, & ere leue not afteresubtraccio, us *ou schalt do. ou schalledo away e hier figure &write erea cifer, as lo an Ensampull Take foureout of fourean

leusnot. ereforedo away e hier 4 & set erea cifer, an take2 out of 2, an leues not. do away e hier 2, & set erea cifer,and so worch whareso eueris happe.

Sed si nonpossis a prima demere primamPrecedens vnumde limite deme sequente,Quod demptumprodenario reputabis ab illoSubtrahe totalem numerumquemproposuistiQuo facto scribe super quicquid remanebit.

Herehe puttes e thryd case, e quych is is. yf it happe at e

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lower figure frothe top one,borrow ten;

take the lowernumber fromten;add the answer

to the topnumber.

Example.

21221134

How to Payback theborrowed ten.

A very hard casis put.

neerfigure be moreen e hier figure at he schallebe draw out of.how schalleou do. us ou schalledo. ou schalleborro.1. oute ofe next figure at comes afterin e same rewe, for is case mayneuerhapp but yf erecome figures after. an ou schalt sett at onouere hier figureshed, of the quych ou woldist y-draw oute eneyerfigure yf ou haddyst y-myt. Whane ou hase us ydo ouschallerekene at .1. for ten. . And out of at ten ou schal draw eneyermost figure, And alleat leues ou schalleadde to e figureon

whos hed at .1. stode. And en ou schalledo away alleat, & settereallethat arisys of the addicio of e ylke 2 figuris. And yf yt*happe at e figure of e quych ou schalt borro on be hym self but1. If ou schalt at one& sett it vppo e oerfigurished, and settin at 1. place a cifer, yf erecome mony figuresafter. lo anEnsampul. take 4 out of 2. it wyl not be, erforeborro oneofe next figure, at is 2. and sett at ouere hed of e fyrst 2.& rekene it for ten. and ere e secunde stondes write 1. forou tokest on out of hym. an take e neerfigure, at is 4, out often. And en leues 6. cast to 6 e figure of at 2 at stode vnderehedde of 1. at was borwed& rekened for ten, and at wylle be 8. do

away at 6 & at 2, & sette ere8, & lette e neerfigure stondestille. Whanne ou hast do us, go to e next figure at is now bot 1.but first yt was 2, & ere-of was borred1. an take out of at efigure vnderhym, at is 3. hit wel not be. er-foreboroweof the nextfigure, e quych is bot 1. Also take & sett hym ouere hede of efigureat ou woldest haue y-draw oute of e nether figure, e quychwas 3. & ou myt not, & rekene at borwed 1 for ten & sett in esame place, of e quych place ou tokest hymof, a cifer, for he wasbot 1. Whanne ou hast usydo, take out of at 1. at is rekent forten, e neerfigure of 3. And ereleues 7. *cast e ylke 7 to efigure at had e ylke ten vpon his hed, e quych figure was 1, & at

wol be 8. an do away at 1 and at 7, & write ere8. & an wyrchforth in oerfiguris til ou come to e ende, & an ou hast e do.

Versus.

Facque nonenarios de cifris, cumremeabis Occurrant si forte cifre; dum dempseris vnum Postea procedas reliquas demendo figuras.

Herehe puttese fourte case, e quych is is, yf it happe at eneerfigure, e quych ou schalt draw out of e hier figure be morepan e hier figur ouerhym, & e next figure of two or of thre or offoure, or how mony erebe by cifers, how wold ou do. ou wostwel

ou most nede borow, & ou mayst not borow of e cifers, for aihaue not at ai may leneor spare. Ergo 4how woldest ou do.Certay us most ou do, ou most borow on of e next figuresignificatyf in at rewe, for is case may not happe, but yf erecomefigures significatyf afterthe cifers. Whan ou hast borowede at 1 ofthe next figure significatyf, sett at on ouere hede of at figure ofe quych ou wold haue draw e neerfigure out yf ou hadestmyt, & reken it for ten as oudiddest ine oercase here-a-fore.Wha ou hast us y-do loke how mony cifers erewerebye-tweneat figuresignificatyf, & e figure of e quych ou woldest haue y-

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Here is anexample.40002

10004

Sic.

How to prove asubtraction sum

Here is anexample.

4000346820004664

3999880420004664

Our authormakes a slip he(3 for 1).

He works hisproof through,

and brings out result.

39998

10004

6000346820004664

draw the *neerfigure, and of euery of e ylke cifers make a figureof 9. lo an Ensampulleafter. Take 4 out of 2. it wel not be.borow 1 out of be next figure significatyf, e quych is 4, & enleues 3. do away at figureof 4 & write ere3. & sett at 1vppon e figure of 2 hede, & an take 4 out of ten, & an ere leues6. Cast 6 to the figure of 2, at wol be 8. do away at 6 & write ere8. Whan ou hast us y-do make of euery 0 betweyn 3 & 8 a figure of9, & an worch forth in goddes name. & yf ou hast wel y-do

ou5

schalt haue is nomber

How to prove the Subtraction.

Si subtraccio sit benefacta probarevalebisQuas subtraxisti primas addendo figuras.

Herehe teches e Craft how ou schalt know, whan ou hastsubtrayd, wheerou hast wel ydo or no. And e Craft is is, ryght asou subtrayde neerfigures fro e hier figures, ryt so adde esame neerfigures to e hier figures. And yf ou haue well y-wrotha-foreou schalt haue e hier nombre e same ou haddest orou be-

gan to worch. as for is I bade ou schulde kepe e neerfiguresstylle. lo an *Ensampulleof allee 4 cases togedre. worche welleiscase And yf ou worch wellewhan ou hast allesubtrayde at hier nombrehere, is schallebe e nombre herefoloyng whan ou hast subtrayd. And ou schaltknow us. adde e neerrowe of e samenombre to e hier rewe as us, cast 4 to 4. atwol be 8. do away e 4 & write ere8. by e first case of addicio.an cast 6 to 0 at wol be 6. do away e 0, & write ere 6. an cast6 to 8, at wel be 14. do away 8 & write erea figure of 4, at is edigit, and write a figure of 1. at schall be-token ten. at is e articul

vpon e hed of 8 next after, an reken at 1. for 1. & cast it to 8. atschal be 9. cast to at 9 e neerfigure vnderat e quych is 4, &at schallebe 13. do away at 9 & sett ere 3, & sett a figure of 1.at schall be 10 vpon e next figurishede e quych is 9. by esecunde case at ou hadest in addicio. an cast 1 to 9. & at wolbe 10. do away e 9. & at 1. And write erea cifer. and write earticulleat is 1. betokenynge10. vpon e hede of e next figuretoward e lyft side, e quych *is 9, & so do forth tyl ou come to elast 9. take e figureof at 1. e quych ou schalt fynde ouere hedof 9. & sett it ouere next figures hede at schal be 3. Also doaway e 9. & set erea cifer, & en cast at 1 at stondes vpon e

hede of 3 to e same 3, & at schallemake 4, en caste to e ylke 4the figurein e neyerrewe, e quych is 2, and at schallebe 6. Anden schal ou haue an Ensampulleaey, loke & se, & butou haue is same ou hase myse-wrot.

The Craft of Duplation.

Sequiturde duplacione

Si vis duplarenumerum, sic incipe primo

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Four things musbe known inDuplation.

Here they are.

Mind where youbegin.

Remember yourules.

How to work asum.

If the answer isa digit,write it in theplace of the topfigure.

If it is an article

put a cipher inthe place, and

carry the tens.

Scribe figurarumseriemquamcunquevelistu.

This is the Chaptureof duplacio, in e quych craft ou most haue& know 4 thinges. e first at ou most know is what is duplacio.e secunde is how mony rewes of figures ou most haue to is craft. e thryde is how many cases may6happe in is craft. e fourte iswhat is e profet of e craft. As for e first. duplacio is adoublyngeof a nombre. As for e secunde ou most *haue onnombre or on rewe of figures, the quych called numerusduplandus.

As for e thrid ou most know at 3 diuerse cases may hap in iscraft. As for e fourte. qwat is e profet of is craft, & at is to knowwhat a-risytof a nombre I-doublyde. fforer-more, ou most know& take gode hede in quych side ou schallebe-gyn in is craft, or ellisou mayst spylalleilaberere aboute. certeyn ou schalt begy inthe lyft side in is Craft. thenke wel oueris verse. 7A leua dupla,diuide, multiplica. 7

The sentensof es verses afore, as ou may see if ou take hede. Ase text of is verse, at is to say, Si vis duplare. is is e sentence. If ou weldouble a nombre us ou most be-gyn. Write a rewe of

figures of what nombreou welt. versus.Postea procedas primamduplando figuramInde quod excrescit scribas vbi iusserit ordoIuxta precepta tibi que danturin addicione.

Herehe telles how ou schalt worch in is Craft. he says, fyrst,whan ou hast writen e nombre ou schalt be-gyn at e first figurein the lyft side, & doubulleat figure, & e nombre at comes ere-ofou schalt write as ou diddyst in addicio, as I schal telle e in ecase. versus.

The Cases of the Craft of Duplation.* Nam si sit digitus in primo limite scribas.

Hereis e first case of is craft, e quych is is. yf of duplacio of afigurearise a digit. what schal ou do. us ou schal do. do away efigure at was doublede, & sett eree diget at comes of eduplacio, as us. 23. double 2, & at wel be 4. do away e figureof2 & sett erea figureof 4, & so worchforth tilleou come to eende. versus.

Articulussi sit, in primo limite cifram, Articulumvero reliquis inscribe figuris;

Vel perse scribas, si nulla figura sequatur.

Here is e secunde case, e quych is is yf erecome an articulleofe duplacio of a figure ou schalt do ryt as ou diddyst in addicio,at is to weteat ou schalt do away e figureat is doublet & setterea cifer, & write e articulleouere next figurishede, yf erebeany after-warde toward e lyft side as us. 25. begyn at the lyft side,and doubulle2. at wel be 4. do away at 2 & sett ere 4. an doubul5. at wel be 10. do away 5, & sett erea 0, & sett 1 vpon e nextfigurishede e quych is 4. & en draw downe 1 to 4 & at wollebe 5,& en do away at 4 & at 1, & sett ere5. for at 1 schal be

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The four thingsto be known in

mediation:

the first

the second;

the third;

the fourth.

Begin thus.

See if thenumber is evenor odd.

If it is even,halve it, andwrite the answein its place.

If it is odd, halvthe even numbeless than it.

Then write thesign for one-haover it.Here is anexample.

The Craft of Mediation.

Sequiturde mediacione.

Incipe sic, si vis aliquemnumerummediare:Scribe figurarumseriem solam, velut ante.

In is Chapter is tate Craft of mediaciou, in e quych craft ou

most know 4 thynges. ffurst what is mediacio. the secunde howmony rewes of figuresou most haue in e wyrchyngeof is craft. ethryde how mony diuerse cases may happ in is craft.8 As for efurst, ou schalt vndurstonde at mediacio is a takyng out of halfe anomber out of a hollenomber, *as yf ou wolde take 3 out of 6. Asfor e secunde, ou schalt know at ou most haue onerewe offigures, & no moo, as ou haystin e craft of duplacio. As for thethryd, ou most vnderstonde at 5 cases may happe in is craft. Asfor e fourte, ou schalleknow at the profet of is craft is when ouhast take away e haluendel of a nombreto telle qwat ereschalleleue. Incipe sic, &c. The sentence of is verse is is. yf ou wold

medye, at is to say, take halfe out of e holle, or halfe out of halfe,ou most begynne us. Write onerewe of figuresof what nombre ouwolte, as ou dyddyst be-forein e Craft of duplacio. versus.

Postea procedas medians, si prima figuraSi par aut impar videas.

Herehe says, when ou hast write a rewe of figures, ou schalttake hede wheere first figurebe eue or odde in nombre, &vnderstonde at he spekes of e first figure in e ryt side. And intheryght side ou schallebegynne in is Craft.

Quia si fuerit par,Dimidiabiseam, scribens quicquid remanebit:

Hereis the first case of is craft, e quych is is, yf e first figurebe euen. ou schal take away fro e figureeuen halfe, & do away atfigure and set ereat leues ouer, as us, 4. take *halfe out of 4, &an ereleues 2. do away 4 & sett ere2. is is lyght y-nowt.versus.

The Mediation of an Odd Number.

Impar si fuerit vnumdemas mediareQuod nonpresumas, sedquod superest mediabisInde supertractumfac demptumquod notat vnum.

Hereis e secunde case of is craft, the quych is is. yf e first figurebetokenea nombre at is odde, the quych odde schal not be mediete,en ou schalt medye at nombre at leues, when the odde of esame nombreis take away, & write at at leues as ou diddest in efirst case of is craft. Wha ou hayst write at. for at at leues,write such a merke as is herewvpon his hede, e quych merke schalbetoke halfe of e odde at was take away. lo an Ensampull. 245.the first figurehereis betokenyngeodde nombre, e quych is 5, for 5

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Put the markonly over thefirst figure.

If the first figuris one put acipher.

What to do ifany other figureis odd.

Write a figure ofive over the

next lowernumbers head.

Example.

54634.

If the secondfigure is one, pa cipher, andwrite five over

is odde; ere-foredo away at at is odde, e quych is 1. en leues4. en medye 4 & en leues 2. do away 4. & sette ere2, & makesuch a merke wupon his hede, at is to say ouerhis hede of 2 as us.242.wAnd en worch forth in e oerfigures tyll ou come to eende. by e furst case as ou schalt vnderstonde at ou schalt*neuermake such a merk but ouere first figure hed in e rit side.Wheere other figures at comy afterhym be eue or odde.versus.

The Cases of the Craft of Mediation.

Si monos, dele; sit t ibicifra post notasupra.

Here is e thryde case, e quych yf the first figurebe a figureof 1.ou schalt do away at 1 & set erea cifer, & a merke ouere ciferas us, 241. do away 1, & sett erea cifer witha merke ouerhishede, & en hast ou ydo for at 0. as us 0wen worch forth in eoer figurys till ou come to e ende, for it is lyght as dyche water.vndeversus.

Postea procedas hac condicione secunda:

Imparsi fuerit hinc vnumdeme priori,Inscribens quinque, nam denos significabitMonos predictam.

Herehe puttese fourte case, e quych is is. yf it happe thesecunde figurebetoken odde nombre, ou schal do away on of atodde nombre, e quych is significatiue by at figure 1. e quych 1schall be rekende for 10. Whan ou hast take away at 1 out of enombre at is signifiede by at figure, ou schalt medie at at leuesouer, & do away at figureat is medied, & sette in his stydehalfe ofat nombre. Whan ou hase so done, ou schalt write *a figure of 5

ouere next figureshede by-foretoward e ryt side, for at 1, equych made odd nombre, schall stonde for ten, & 5 is halfe of 10; soou most write 5 for his haluendelle. lo an Ensampulle, 4678. begy ine ryt side as ou most nedes. medie 8. en ou schalt leue 4. doaway at 8 & sette ere4. en out of 7. take away 1. e quych makesodde, & sett 5. vpon e next figureshede aforetoward e ryt side,e quych is now 4. but aforeit was 8. for at 1 schal be rekenet for10, of e quych 10, 5 is halfe, as ou knowest wel. Whan ou hastus ydo, medye at e quych leues aftere takyingeaway of at atis odde, e quych leuyngeschallebe 3; do away 6 & sette ere3, & ou schalt haue such a nombre aftergo forth to e next

figure, & medy at, & worch forth, for it is lyt ynovtto e certay. Si vero secunda dat vnum.

Illa deleta, scribaturcifra; priori Tradendo quinque pro denario mediato;

Nec cifra scribatur, nisi deinde figura sequatur:Postea procedas reliquas mediando figuras

Vt supradocui, si sint tibi mille figure.

Herehe puttese 5 case, e quych is *is: yf e secunde figurebeof 1, as is is here 12, ou schalt do away at 1 & sett erea cifer. &

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the next figure.

5202,

How to halvefourteen.

52,

How to proveyour mediation.

First example.

The second.

The thirdexample.

The fourthexample.

52334.

sett 5 ouere next figure hede aforetoward e rit side, as oudiddyst afore; & at 5 schal be haldel of at 1, e quych 1 is rekentfor 10. lo an Ensampulle, 214. medye 4. at schallebe 2. do away 4& sett ere2. engo forth to e next figure. e quych is bot 1. doaway at 1. & sett erea cifer. & set 5 vpon e figureshed afore, equych is nowe 2, & en ou schalt haue is nombreen worchforth to e nex figure. And also it is no maysteryyf erecome nofigureafter at on is medyet, ou schalt write no 0. ne nowt ellis, but

set 5 ouere next figure aforetoward e ryt, as us 14. medie 4then leues 2, do away 4 & sett ere2. en medie 1. e quich isrekende for ten, e haluendel ere-of wel be 5. sett at 5 vpon ehede of at figure, e quych is now 2, & do away at 1, & ouschalt haue is nombre yf ou worch wel, vndeversus.

How to prove the Mediation.

Si mediacio sit benefacta probarevalebis Duplando numerumquemprimo dimediasti

Herehe telles e how ou schalt know wheerou hase wel ydo or

no. doubul *e nombre e quych ou hase mediet, and yf ou hauewel y-medyt after e dupleacio, ou schalt haue e same nombre atou haddyst in e tabulleor ou began to medye, as us. The furstensampullewas is. 4. e quych I-medietwas laft2, e whych 2 waswrite in e place at 4 was write afore. Now doubulleat 2, & ouschal haue 4, as ou hadyst afore. e secunde Ensampullewas is,245. When ou haddyst mediet alleis nombre, yf ou haue wel ydoou schalt haue of at mediacio is nombre, 122w. Now doubulleisnombre, & begyn in e lyft side; doubulle1, at schal be 2. do awayat 1 & sett ere2. en doubulleat oer2 & sett ere4, endoubulleat oer2, & at wel be 4. endoubul at merke at

stondes for halue on. & at schallebe 1. Cast at on to 4, & it schallebe 5. do away at 2 & at merke, & sette ere5, & en ou schalhaue is nombre 245. & is wos e same nombur at ou haddyst orou began to medye, as ou mayst se yf ou take hede. The nombree quych ou haddist for an Ensampul in e 3 case of mediacio to bemediet was is 241. whan ou haddist medied alleis nombur truly*by euery figure, ou schall haue be at mediacio is nombur 120w.Now dowbul is nombur, & begyn in e lyft side, as I tolde e in eCraft of duplacio. us doubullee figure of 1, at wel be 2. do awayat 1 & sett ere2, en doubul e next figureafore, the quych is 2, &at wel be 4; do away 2 & set ere4. en doubul e cifer, & at wel

be not, for a 0 is not. And twyes not is but not. ereforedoubulthe merke aboue e cifers hede, e quych betokenes e haluendel of1, & at schal be 1. do away e cifer & e merke, & sett ere1, &en ou schalt haue is nombur 241. And is same nombur ouhaddyst afore or ou began to medy, & yfou take gode hede. Thenext ensampul at had in e 4 case of mediacio was is 4678. Whanou hast truly ymeditalleis nombur fro e begynnyngeto eendynge, ou schalt haue of e mediacio is nombur Nowdoubul this nombur & begyn in e lyft side, & doubulle2 atschal be 4. do away 2 and sette ere 4; en doubule3, at wol be 6;

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the second:

the third:

the fourth.

The multiplicand

How to set dowthe sum.

673241234

Two sorts ofMultiplication:

mentally,

and on paper.

How to multiplytwo digits.

Subtract thegreater from te

take the less somany times fromten times itself.

Example.

Better use thistable, though.

nombre8 conteynes as oft tymes 4 as ereben vnites in at othernombre, e quych is 2, for in 2 ben 2 vnites, & so oft tymes 4 ben in8, as ou wottys wel. ffor e secunde, ou most know at ou mosthaue too rewes of figures. As for e thryde, ou most know at 8manerof diuerse case may happe in is craft. The profet of is Craftis to telle when a nombreis multiplyed be a noer, qwat commysereof. fforthermore, as to e sentence of oureverse, yf ou welmultiply a nombur be a-noernombur, ou schalt write *a rewe of

figures of what nomburs so euerou welt, & at schal be calledNumerusmultiplicandus, Anglice, e nomburthe quych to bemultiplied. en ou schalt write a-nother rewe of figures, by e quychou schalt multiplie the nombre at is to be multiplied, of e quychnombure furst figure schal be write vndere last figureof enombur, e quych is to be multiplied. And so write forthe toward elyft side, as hereyou may se, And is onenomburschallebe called numerus multiplicans. Anglice, e nomburmultipliynge, for he schallemultiply e hyer nounbur, asus onetyme 6. And so forth, as I schal telle the afterwarde. And ouschal begyn in e lyft side. ffor-ere-more ou schalt vndurstonde

at ereis two manurs of multiplicacio; one ys of e wyrchyngeofe boke only in e mynde of a mon. fyrst he teches of e fyrst manerof duplacio, e quych is be wyrchyngeof tabuls. Afterwarde he wolteche on e secunde maner. vndeversus.

To multiply one Digit by another.

In digitumcures digitumsi ducere maior* Perquantumdistat a denis respice debes Namquesuo decuplo totiens delereminorem

Sitquetibi numerus veniens exinde patebit.

Herehe teches a rewle, how ou schalt fynde e nounbre atcomes by e multiplicacio of a digit be anoer. loke how mony[vny]tes ben. bytwene e moredigit and 10. And reken ten for onvnite. And so oft do away e lasse nounbre out of his owne decuple,at is to say, fro at nounbre at is ten tymes so mych is e nounbreat comes of e multiplicacio. As yf ou wol multiply 2 be 4. lokehow mony vnitees ben by-twene e quych is e morenounbre, & be-twene ten. Certen erewel be vj vnitees by-twene 4 & ten. yf oureken erewithe ten e vnite, as ou may se. so mony tymes take2. out of his decuple, e quych is 20. for 20 is e decuple of 2, 10 ise decuple of 1, 30 is e decuple of 3, 40 is e decuple of 4, And e

oerdigetes til ou come to ten; & whan ou hast y-take so monytymes 2 out of twenty, e quych is sex tymes, ou schal leue 8 as ouwost wel, for 6 times 2 is twelue. take [1]2 out of twenty, & ereschalleue 8. bot yf bothe e digettes*ben y-lyechmych as here. 222 ortoo tymes twenty, en it is no forsquych of hem tweyn ou take outof here decuple. alsmony tymes as at is fro 10. but neuer-e-lesse,yf ou haue hastto worch, ou schalt haue herea tabul of figures,where-by ou schalt se a-non ryght what is e nounbre at comes ofe multiplicacio of 2 digittes. us ou schalt worch in is figure.

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How to use it.

The way to usethe Multiplicatiotable.

How to multiplyone number byanother.

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 e figure, e quych schallebe multiplied, be eueneas mych as ediget be, e quych at oerfigureschal be multiplied, as two tymestway, or thre tymes 3. or sych other. loke qwereat figure sittes ine lyft side of e triangle, & loke qweree diget sittes in e neermost rewe of e triangle. & go fro hym vpwarde in e same rewe, equych rewe gose vpwarde til ou come agaynes e oerdigette at

sittes in e lyft side of e triangle. And at nounbre, e quych oufyn*des ereis e nounbre at comes of the multiplicacio of e 2digittes, as yf ou wold wete qwat is 2 tymes 2. loke queresittes 2 ine lyft side ine first rewe, he sittes next 1 in e lyft side al on hye,as ou may se; e[n] loke qweresittes 2 in e lowyst rewe of etriangle, & go fro hym vpwarde in e same rewe tylleou come a-enenes2 in e hyer place, & er ou schalt fynd ywrite 4, & at is enounbre at comes of e multiplicacio of two tymes tweyn is 4, asow wotest welle. yf e diget. the quych is multiplied, be morean eoer, ou schalt loke qweree morediget sittes in e lowest rewe ofe triangle, & go vpwarde in e same rewe tyl10ou come a-nendes

e lasse diget in the lyft side. And ereou schalt fynde e nombreat comes of e multiplicacio; but ou schalt vnderstonde at isrewle, e quych is in is verse. In digitumcures, &c., noeristriangle schallenot serue, bot to fynde e nounbres at comes of themultiplicacio at comes of 2 articuls or composites, e nedes no craftbut yf ou wolt multiply in i mynde. And *ere-to ou schalt haue acraft afterwarde, for ou schall wyrch withdigettes in e tables, asou schalt know afterwarde. versus.

To multiply one Composite by another.

Postea procedas postremammultiplicando[Recte multiplicans per cunctas inferiores]Condicionem tamen tali quodmultiplicantesScribas in capite quicquid processerit indeSed postquamfuit hec multiplicate figureAnteriorenturserei multiplicantisEt sic multiplica velut isti multiplicastiQui sequiturnumerumscriptumquiscunquefiguris.

Herehe teches how ou schalt wyrch in is craft. ou schaltmultiplye e last figureof e nombre, and quen ou hast so ydo ou

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Multiply the lasfigure of thehigher by the

first of thelower number.

Set the answerover the first ofthe lower: 2465.

232

then multiply thsecond of thelower, and soon.

Then antery thelower number:

as thus.464465 232

Now multiply bythe last but oneof the higher:

as thus.

1 824648[65] 232.

11 121 828464825 232

schalt draw allee figures of e neernounbre moretawarde rytside, so qwenou hast multiplyed e last figureof e heyer nounbreby allee neerfigures. And sette e nounbir at comes er-of ouere last figureof e neernounbre, & en ou schalt sette al e oerfigures of e neernounbremorenereto e ryt side. And whanou hast multiplied at figureat schal be multiplied e next afterhym by al e neerfigures. And worch as ou dyddyst aforetil *oucome to e ende. And ou schalt vnderstonde at euery figureof e

hier nounbreschal be multiplied be allee figures of the neernounbre, yf e hier nounbrebe any figureen one. lo an Ensampulherefolowynge. ou schalt begyne to multiplye in e lyftside. Multiply 2 be 2, and twyes 2 is 4. set 4 ouere hed ofat 2, en multiplie e same hier 2 by 3 of e nethernounbre, as thryes 2 at schal be 6. set 6 ouere hed of 3, anmultiplie e same hier 2 by at 2 e quych stondes vnderhym, atwol be 4; do away e hier 2 & sette ere4. Now ou most anterye nether nounbre, at is to say, ou most sett e neernounbremore towarde e ryt side, as us. Take e neer2 toward e rytside, & sette it eue vndere 4 of e hyer nounbre, & antery allee

figures at comes afterat 2, as us; sette 2 vndere 4. en sett efigureof 3 ereat e figure of 2 stode, e quych is now vndur at 4in e hier nounbre; en sett e oer figureof 2, e quych is e lastfigure toward e lyft side of e neernombereree figureof 3stode. en ou schalt haue such a nombre. * Now multiply4, e quych comes next after6, by e last 2 of e neernounbur toward e lyft side. as 2 tymes 4, at wel be 8. setteat 8 ouere figure the quych stondes ouere hede of at 2, equych is e last figureof e neernounbre; an multiplie at same 4by 3, at comes in e neerrewe, at wol be 12. sette e digit of ecomposyt ouere figure e quych stondes ouere hed of at 3, &

sette e articule of is composit oueral e figures at stondes ouere neer2 hede. en multiplie e same 4 by e 2 in e ryt side in eneernounbur, at wol be 8. do away 4. & sette ere8. Euermoreqwen ou multiplies e hier figureby at figuree quych stondesvnderhym, ou schalt do away at hier figure, & sett er at nounbree quych comes of multiplicacio of ylke digittes. Whan ou hast doneas I haue byde e, ou schalt haue suych an orderof figureas ishere, en take and antery i neerfigures. And sett efyrst figure of e neerfigures 11vndre be figureof 6. And draw al e oerfigures of e same rewe to hym-warde, *as ou diddyst afore. en multiplye 6 be 2, & settat e quych comes ouerere-of oueral e oerfigures hedes atstondes ouerat 2. en multiply 6 be 3, & sett alleat comes ere-ofvpon allee figures hedes at standes ouerat 3; anmultiplye 6 be2, e quych stondes vnderat 6, en do away 6 & write eree digittof e composit at schal come ereof, & sette e articull oueralleefigures at stondes ouere hede of at 3 as here, en anteryi figures as ou diddyst afore, and multipli 5 be 2, at wolbe 10; sett e 0 ouerall e figures at stonden ouerat 2, &sett at 1. ouerthe next figures hedes, alleon hye towardee lyft side. en multiplye 5 be 3. at wol be 15, write 5 ouer

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The fourth caseof the craft.

The fifth case othe craft.

236234

46836234

46836

234

The sixth case othe craft.

The seventh casof the craft.

2403

The eighth case

e neerfigure, by the quych ou multipliest e hierefigure, for isnedes no Ensampul.

Subdita multiplica non hanc que [incidit] illiDelet eampenitusscribens quod prouenit inde.

Hereis e 4 case, e quych is: yf hit be happe at e neerfigureschal multiplye at figure, e quych stondes ouerat figures hede,ou schal do away e hier figure& sett ere at at comys of atmultiplicacio. As yf ere come of at multiplicacio an articuls ouschalt write ere e hier figurestode a 0. And write e articuls in elyft side, yf at hit be a digit write ere a digit. yf at hit be acomposit, write e digit of e composit. And e articul in e lyft side.al is is lyt y-nowt, ere-foreer nedes no Ensampul.

Sedsi multiplicat aliamponas superipsamAdiunges numerumquemprebet ductusearum.

Hereis e 5 case, e quych is is: yf *e neerfigureschulmultiplie e hier, and at hier figureis not recteouerhis hede. Andat neerfigurehase oerfigures, or on figure ouerhis hede by

multiplicacio, at hase be afore, ou schalt write at nounbre, equych comes of at, ouerallee ylke figures hedes, as us here:Multiply 2 by 2, at wol be 4; set 4 ouere hede of at 2.en 14multiplies e hier 2 by e neer3, at wol be 6. setouerhis hede 6, multiplie e hier 2 by e neer4, at wol be8. do away e hier 2, e quych stondes ouere hede of e figureof 4, and set ere 8. And ou schalt haue is nounbrehere

And antery i figures, at is to say, set i neer4 vnderehier 3, and set i 2 other figures nerehym, so at e neer2stonde vndure hier 6, e quych 6 stondes in e lyft side. And at 3at stondes vndur 8, as us aftur e may se, Now worch

forthermore, And multiplye at hier 3 by 2, at wol be 6, setat 6 e quych stondes ouere hede of at 2, And en worchas I tat e afore.

* Si supraposita cifra debet multiplicareProrsus eamdeles & ibi scribi cifra debet.

Hereis e 6 case, e quych is is: yf hit happe at e figureby equych ou schal multiplye e hier figure, e quych stondes ryght ouerhym by a 0, ou schalt do away at figure, e quych ouerat cifrehede. And write ereat nounbre at comes of e multiplicacio asus, 23. do away 2 and sett ere a 0. vndeversus.

Si cifra multiplicat aliampositamsuperipsamSitquelocus supravacuussuperhanc ciframfiet.

Hereis e 7 case, e quych is is: yf a 0 schal multiply a figure, equych stondes not recteouerhym, And ouerat 0 stonde no thyng,ou schalt write ouerat 0 anoer0 as us: multiplye 2 be a 0,it wol be nothynge. write ere a 0 ouere hede of e neer0,

And en worch forth til ou come to e ende.

Si supra15fuerit cifra semperestpretereunda.

Hereis e 8 case, e quych is is: yf erebe a 0 or mony cifers in

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of the craft.

00032.22

How to provethemultiplication.

Mentalmultiplication.

Digit by digit iseasy.

The first case othe craft.

Article by article

an example:

anotherexample:

e hier rewe, ou schalt not multiplie hem, bot let hem stonde. Andantery e figures benee to e next figuresygnificatyf as us: Ouer-lepe alleese cifers & sett at *neer2 at stondes towarde ryght side, and sett hym vndure 3, and sett e oernether 2 nere hym, so at he stonde vndure thrydde 0, equych stondes next 3. And an worch. vndeversus.

Si dubites, an sit benemultiplicacio facta,Diuide totalem numerumpermultiplicantem.

Herehe teches how ou schalt know wheerou hase wel I-do orno. And he says at ou schalt deuide allee nounbre at comes ofe multiplicacio by e neerfigures. And en ou schalt haue esame nounbur at ou hadyst in e begynnynge. but et ou hast note craft of dyuisio, but ou schalt haue hit afterwarde.

Pernumerumsi vis numerumquoquemultiplicare Tantum pernormas subtiles absquefiguris

Has normas poteris perversus sciresequentes.

Herehe teches e to multiplie be owtfigures in i mynde. And e

sentence of is verse is is: yf ouwel multiplie on nounbre by anoerin i mynde, ou schal haue ereto rewles in e verses at schalcome after.

Si tu perdigitumdigitumvis multiplicareRegulaprecedens dat qualiterest operandum.

Herehe teches a rewle as ou hast aforeto multiplie a digit beanoer, as yf ou wolde wete qwat is sex tymes 6. ou *schalt weteby e rewle at I tat e before, yf ou haue mynde erof.

Articulumsi perreliquumreliquumvis multiplicare

In propriumdigitumdebet vterqueresolui. Articulusdigitos post se multiplicantesEx digitusquociens retenerit multiplicariArticuli faciunt tot centummultiplicati.

Herehe teches e furst rewle, e quych is is: yf ou wel multipliean articul be anoer, so at both e articuls bene with-Inne anhundreth, us ou schalt do. take e digit of bothe the articuls, foreuery articul hase a digit, en multiplye at on digit by at oer, andloke how mony vnytes ben in e nounbre at comes of emultiplicacio of e 2 digittes, & so mony hundrythes ben in enounbreat schal come of e multiplicacio of e ylke 2 articuls as

us. yf ou wold wete qwat is ten tymes ten. take e digit of ten, equych is 1; take e digit of at oerten, e quych is on. Alsomultiplie 1 be 1, as on tyme on at is but 1. In on is but on vnite asou wost welle, ereforeten tymes ten is but a hundryth. Also yfou wold wete what is twenty tymes 30. take e digit of twenty, at is2; & take e digitt of thrytty, at is 3. multiplie 3 be 2, at is 6. Nowin 6 ben 6 vnites, And so mony hundrythes ben in 20 tymes 30*,erefore20 tymes 30 is 6 hundryth eue. loke & se. But yf it be soat onearticul be with-Inne an hundryth, orby-twene an hundrythand a thowsande, so at it be not a owsande fully. en loke how

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How to work subtly without Figures.Mentalmultiplication.

Anotherexample.

Anotherexample.

Notation.

Notation again.


mony vnytes ben in e nounbur at comys of e multiplicacio 16Andso mony tymes 16of 2 digittesof ylke articuls, so mony thowsant benin e nounbre, the qwych comes of e multiplicacio. And so monytymes ten thowsand schal be in e nounbre at comes of emultiplicacion of 2 articuls, as yf ou wold wete qwat is 4 hundrythtymes [two hundryth]. Multiply 4 be 2, 17at wol be 8. in 8 ben 8

vnites. And so mony tymes ten thousandbe in 4 hundryth tymes [2] 17hundryth, at is

80 thousand. Take hede, I schall telle e agenerallerewle whan ou hast 2 articuls, And ou wold wete qwatcomes of e multiplicacio of hem 2. multiplie e digit of at onarticuls, and kepe at nounbre, en loke how mony cifers schuld gobeforeat on articuls, andhe werewrite. Als mony cifers schuld gobeforeat other, & he werewrite of cifers. And haue allee ylkecifers togedurin i mynde, *a-roweychoaftur other, and in e lastplase set e nounbre at comes of e multiplicacio of e 2 digittes.

And loke in i mynde in what place he stondes, wherein e secunde,or in e thryd, or in e 4, or whereellis, and loke qwat e figures by-token in at place; & so mych is e nounbre at comes of e 2 articuls

y-multiplied to-geduras us: yf ou wold wete what is 20 thousanttymes 3 owsande. multiply e digit of at articullee quych is 2 bye digitte of at oerarticul e quych is 3, at wol be 6. en lokehow mony cifers schal go to 20 thousant as hit schuld be write in atabul. certainly 4 cifers schuld go to 20 owsant. ffor is figure 2 in efyrst place betokenes twene. In e secunde place hit betokenestwenty. In e 3. place hit betokenes 2 hundryth. .. In e 4 place 2thousant. In e 5 place hit betokenes twenty ousant. ereforehemost haue 4 cifers a-forehym at he may stonde in e 5 place. kepeese 4 cifers in thy mynde, en loke how mony cifers go to 3thousant. Certayn to 3 thousante *go 3 cifers afore. Now cast ylke 4

cifers at schuld go to twenty thousant, And thes 3 cifers at schuldgo afore3 thousant, & sette hem in rewe ycho afteroerin imynde, as ai schuld stonde in a tabulle. And en schal ou haue 7cifers; en sett at 6 e quych comes of e multiplicacio of e 2digittesaftur e ylke cifers in e 8 place as yf at hit stode in a tabul.

And loke qwat a figureof 6 schuld betoken in e 8 place. yf hit werein a tabul & so mych it is. & yf at figure of 6 stonde in e fyrst placehe schuld betoken but 6. In e 2 place he schuld betoken sexty. Inthe 3 place he schuld betoke sex hundryth. In e 4 place sexthousant. In e 5 place sexty owsant. In e sext place sexhundryth owsant. In e 7 place sex owsant thousantes. In e 8place sexty owsant thousantes. erforesett 6 in octauo loco, And heschal betoken sexty owsant thousantes. And so mych is twentyowsant tymes 3 thousant, And is rewle is generallefor allemanerof articuls, Whethir ai be hundryth or owsant; but ou mostknowwell e craft of e wryrchyngein e tabulle*or ou know to do us ini mynde aftur is rewle. Thou most at is rewle holdyenotebutwhereereben 2 articuls and no mo of e quych ayther of hem hasebut on figuresignificatyf. As twenty tymes 3 thousant or 3 hundryth,and such our.

leaf 162 a.

leaf 162 b.

leaf 163 a.


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The third case othe craft;

an example.

The fourth caseof the craft:

Composite by

digit.


The fifth case othe craft:

Articulum digito si multiplicare oportetArticuli digit[i sumi quo multiplicate]Debemusreliquumquod multiplicaturab illisPerreliquo decuplumsic summamlaterenequibit.

Herehe puttes e thryde rewle, e quych is is. yf ou wel multiplyin i mynde, And e Articul be a digitte, ou schalt loke at e digittbe with-Inne an hundryth, en ou schalt multiply the digitt of e

Articulle by e oer digitte. And euery vnite in e nounbre at schalle

come ere-of schal betoken ten. As us: yf at ou wold wete qwat istwyes 40. multiplie e digitteof 40, e quych is 4, by e oerdiget,e quych is 2. And at wolle be 8. And in e nombre of 8 ben 8vnites, & euery of e ylke vnites schuld stonde for 10. ere-fore ereschal be 8 tymes 10, at wol be 4 score. And so mony is twyes 40. If e articul be a hundryth or be 2 hundryth And a owsant, so athit be notte a thousant, *worch as oudyddyst afore, saue ou schaltrekene euery vnite for a hundryth.

In numerummixtumdigitumsi ducerecuresArticulusmixti sumaturdeinde resoluas

In digitumpost fac respectu de digitisArticulusquedocet excrescens in diriuandoIn digitummixti post ducas multiplicantem

De digitis vt norma 18[docet] de [hunc]Multiplica simul et sic postea summa patebit.

Here he puttes e 4 rewle, e quych is is: yf ou multipliy oncomposit be a digit as 6 tymes 24, 19en take e diget of atcomposit, & multiply at digitt by at oerdiget, and kepe e nomburat comes ere-of. en take e digit of at composit, & multiply atdigit by anoerdiget, by e quych ou hast multiplyed e diget of earticul, and loke qwat comes ere-of. en take ou at nounbur, &cast hit to at other nounbur at ou secheste as us yf ou wel weteqwat comes of 6 tymes 4 & twenty. multiply at articulleof ecomposit by e digit, e quych is 6, as yn e thryd rewle ou wastaut, And at schal be 6 score. en multiply e diget of e composit,*e quych is 4, and multiply at by at other diget, e quych is 6, asou wast taut in e first rewle, yf ou haue mynde erof, & at wolbe 4 & twenty. cast all ylke nounburs to-gedir, & hit schal be 144. Andso mych is 6 tymes 4 & twenty.

How to multiply without Figures.

Ductusin articulumnumerussi compositussitArticulumpurumcomites articulumquoqueMixti pro dig

The Earliest Arithmetics in English

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