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[Transcriber's Note:

This text is intended for users whose text readers cannot use the "real" (unicode/utf-8) version of the file. Characters that could not be fully displayed have been "unpacked" and shown in brackets:

[gh] (yogh) n, [l~l] (n with curl, crossed l: see below) 0 (Greek phi: see below)

In The Crafte of Nombrynge, final "n" was sometimes written with an extra curl. In this Latin-1 text it is shown as n. In the same selection, the numeral "0" was sometimes printed as the Greek letter phi. It is shown here as 0 rather than the usual ph because the physical form is more significant than the sound of the letter. Double "l" with a line is shown as [l~l]. The first few occurrences of "d" (for "pence") were printed with a decorative curl. The letter is shown with the same "d" used in the remainder of the text.

The word "withdraw" or "w{i}t{h}draw" was inconsistently hyphenated; it was left as printed, and line-end hyphens were retained. Superscripts are shown with carets as ^e. Except for markers and similar, and the letters noted above, all brackets are in the original.

Individual letters were italicized to show expanded abbreviations; these are shown in br{ac}es. Other italicized words are shown conventionally with lines, boldface with marks. When a footnote called for added text, the addition is shown in the body text with [[double brackets]].

The original text contained at least five types of marginal note. Details are given at the end of the e-text, followed by a listing of typographical errors.]

* * * * * * * * * * * * * *

The Earliest Arithmetics in English

Early English Text Society.

Extra Series, No. CXVIII.

1922 (for 1916).



THE EARLIEST ARITHMETICS IN ENGLISH

Edited With Introduction

by

ROBERT STEELE

London: Published for the Early English Text Society By Humphrey Milford, Oxford University Press, Amen Corner, E.C. 4. 1922.



[Titles (list added by transcriber):

The Crafte of Nombrynge The Art of Nombryng Accomptynge by Counters The arte of nombrynge by the hande APP. I. A Treatise on the Numeration of Algorism APP. II. Carmen de Algorismo]



INTRODUCTION

The number of English arithmetics before the sixteenth century is very small. This is hardly to be wondered at, as no one requiring to use even the simplest operations of the art up to the middle of the fifteenth century was likely to be ignorant of Latin, in which 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 students contemplated the likelihood of their not being able to do simple division. On the other hand, the study of astronomy necessitated, from its earliest days as a science, considerable skill and accuracy in computation, not only in the calculation of astronomical tables but in their use, aknowledge of which latter was fairly common from 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 in numbray..." Amere 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 that prosedeth ffro the frist begynnyng..."

(3) Fragmentary passages or diagrams in Sloane 213 f.120-3 (afourteenth-century counting 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 CRAFTE OF NOMBRYNGE is one of a large number of scientific treatises, mostly in Latin, bound up together as Egerton MS. 2622 in the British Museum Library. It measures 7"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 algorismo of Alexander de Villa Dei (c. 1220), such as that of Thomas of Newmarket contained in the British Museum MS. Reg. 12, E.1. Afragment of another translation of the same gloss was printed by Halliwell in his Rara Mathematica (1835) p.29.[1*] It corresponds, as far as p.71, l.2, roughly to p.3 of our version, and from thence to the end p.2, ll.16-40.

[Footnote 1*: Halliwell printed the two sides of his leaf in the wrong order. This and some obvious errors of transcription— 'ferye' for 'ferthe,' 'lest' for 'left,' etc., have not been corrected in the reprint on pp.70-71.]

The ART OF NOMBRYNG is 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 a fifteenth century hand. It is a translation, rather literal, with amplifications of the de arte numerandi attributed to John of Holywood (Sacrobosco) and the translator had obviously a poor MS. before him. The de arte numerandi was printed in 1488, 1490 (s.n.), 1501, 1503, 1510, 1517, 1521, 1522, 1523, 1582, and by Halliwell separately and in his two editions of Rara Mathematica, 1839 and 1841, and reprinted by Curze in 1897.

Both these tracts are here printed for the first time, but the first having been circulated in proof a number of years ago, in an endeavour to discover other manuscripts or parts of manuscripts of it, Dr. David Eugene Smith, misunderstanding the position, printed some pages in a curious transcript with four facsimiles in the Archiv fr die Geschichte der Naturwissenschaften und der Technik, 1909, and invited the scientific world to take up the "not unpleasant task" of editingit.

ACCOMPTYNGE BY COUNTERS is reprinted from the 1543 edition of Robert Record's Arithmetic, printed by R.Wolfe. It has been reprinted within the last few years by Mr. F.P. Barnard, in his work on Casting Counters. It is the earliest English treatise we have on this variety of the Abacus (there are Latin ones of the end of the fifteenth century), but there is little doubt in my mind that this method of performing the simple operations of arithmetic is much older than any of the pen methods. At the end of the treatise there follows a note on merchants' and auditors' ways of setting down sums, and lastly, asystem of digital numeration which seems of great antiquity and almost world-wide extension.

After the fragment already referred to, Iprint as an appendix the 'Carmen de Algorismo' of Alexander de Villa Dei in an enlarged and corrected form. It was printed for the first time by Halliwell in Rara Mathemathica, but I have added a number of stanzas from various manuscripts, selecting various readings on the principle that the verses were made to scan, aided by the advice of my friend Mr. Vernon Rendall, who is not responsible for the few doubtful lines I have conserved. This poem is at the base of all other treatises on the subject in medieval times, but I am unable to indicate its sources.

THE SUBJECT MATTER.

Ancient and medieval writers observed a distinction between the Science and the Art of Arithmetic. The classical treatises on the subject, those of Euclid among the Greeks and Boethius among the Latins, are devoted to the Science of Arithmetic, but it is obvious that coeval with practical Astronomy the Art of Calculation must have existed and have made considerable progress. If early treatises on this art existed at all they must, almost of necessity, have been in Greek, which was the language of science for the Romans as long as Latin civilisation existed. But in their absence it is safe to say that no involved operations were or could have been carried out by means of the alphabetic notation of the Greeks and Romans. Specimen sums have indeed been constructed by moderns which show its possibility, but it is absurd to think that men of science, acquainted with Egyptian methods and in possession of the abacus,[2*] were unable to devise methods for its use.

[Footnote 2*: For Egyptian use see Herodotus, ii.36, Plato, de Legibus, VII.]

THE PRE-MEDIEVAL INSTRUMENTS USED IN CALCULATION.

The following are known:—

(1) A flat polished surface or tablets, strewn with sand, on which figures were inscribed with a stylus.

(2) A polished tablet divided longitudinally into nine columns (or more) grouped in threes, with which counters were used, either plain or marked with signs denoting the nine 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 being marked off.

The only Greek counting board we have is of the fourth class and was discovered at Salamis. 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 arithmetical symbols. This board could only have been used with counters (calculi), preferably unmarked, as in our treatise of Accomptynge by Counters.

CLASSICAL ROMAN METHODS OF CALCULATION.

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 Horace's schoolboys (Sat.1. vi. 74). For the sand see Persius I.131, "Nec qui abaco 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 attigisse pulverem," (de nat. Deorum, ii.18). Tertullian calls a teacher of arithmetic "primus numerorum 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 of precious 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 does not appear to come into literature. Atypical one is of the third class. It consists of a number 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 hand one five units. Thus any number up to nine can be represented. This instrument is in all essentials the same as the Swanpan or Abacus in use throughout the Far East. The Russian stchota in use throughout Eastern Europe is simpler still. The method of using this system is exactly the same as that of Accomptynge by Counters, the right-hand five bead replacing the counter between the lines.

THE BOETHIAN ABACUS.

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 Geometria gives us a figure of an abacus of the second class with a set of counters arranged within it. It has, however, been contended with great probability that the whole passage is a tenth century interpolation. As no rules are given for its use, the chief value of the figure is that 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 lateron.

THE ABACISTS.

It would seem probable that writers on the calendar like Bede (A.D. 721) and Helpericus (A.D. 903) were able to perform simple calculations; though we are unable to guess their methods, and for the most part they were dependent on tables taken from Greek sources. We have no early medieval treatises on arithmetic, till towards the end of the tenth century we find a revival of the study of science, centring for us round the name of Gerbert, who became Pope as SylvesterII. in 999. His treatise on the use of the Abacus was written (c.980) to a friend Constantine, and was first printed among the works of Bede in the Basle (1563) edition of his works, I.159, in a somewhat enlarged form. 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 certainly ascribed to the eleventh century, but from the beginning of the twelfth century their numbers increase rapidly, to judge by those that have been preserved.

The Abacists used a permanent board usually divided into twelve columns; the columns were grouped in threes, each column being called an "arcus," and the value of a figure in it represented a tenth of what it would have in the column to the left, as in our arithmetic 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 with the digital signs, on the other with Roman numerals. Two ivory discs of this kind from the 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 usual number 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 the notation, and on the rules for Multiplication and Division. Addition, as far as it required any rules, came naturally under Multiplication, while Subtraction was involved in the process of Division. These rules were all that were needed in Western Europe in centuries when commerce hardly existed, and astronomy was unpractised, and even they were only required in the preparation of the calendar and the assignments of the royal exchequer. In England, for example, when the hide developed from the normal holding 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 Abacists proposes the sum: "If 200 marks are levied on the county of Essex, which contains according to Hugh of Bocland 2500 hides, how much does each hide pay?"[3*] Exchequer methods up to the sixteenth century were founded on the abacus, though when we have details later on, adifferent and simpler form was used.

[Footnote 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.]

The great difficulty of the early Abacists, owing to the absence of a figure representing zero, 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 are in the methods for this rule. Division was either done directly or by means of differences between 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 are examples taken from a twelfth century treatise. In following the operations it must be remembered that a figure asterisked represents a counter taken from the board. Azero is obviously not needed, and the result may be written down in words.

(a) MULTIPLICATION. 4600 23.

+ -+ -+ Thousands + -+ -+ -+ -+ -+ -+ H T U H T U u e n u e n n n i n n i d s t d s t r s r s e e d d s s + -+ -+ -+ -+ -+ -+ 4 6 +Multiplicand.+ + -+ -+ -+ -+ -+ -+ 1 8 6003. 1 2 40003. 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 separate divisor.

+ -+ -+ Thousands + -+ -+ -+ -+ -+ -+ 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 and20. + -+ -+ -+ -+ -+ -+ 1 9 9 2 Remainder. 1 2 Product of 1st Quotient and3. + -+ -+ -+ -+ -+ -+ 1 9 9 8 +Final remainder.+ 4 Quotient. + -+ -+ -+ -+ -+ -+

(c) DIVISION BY DIFFERENCES. 900 8. Here we divide by (10-2).

+ -+ -+ -+ -+ -+ -+ H. T. U. + -+ -+ -+ -+ -+ -+ 2 Difference. 8 Divisor. + -+ -+ -+ -+ -+ -+ [4*]9 +Dividend.+ [4*]1 8 Product of difference by 1st Quotient (9). 2 Product of difference by 2nd Quotient (1). + -+ -+ -+ -+ -+ -+ [4*]1 Sum of 8 and2. 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.+) + -+ -+ -+ -+ -+ -+

[Footnote 4*: These figures are removed at the next step.]

DIVISION. 7800 166.

- -+ Thousands + - - H. T. U. H. T. U. - - + 3 4 Differences (making 200 trial divisor). 1 6 6 Divisors. + - - [4*]7 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). - - + [4*]2 8 2 New dividends. 3 4 Product of 1st and 2nd difference by 2nd Quotient (1). + - - [4*]1 1 6 New dividends. 2 Product of 1st difference by 3rd Quotient (5). 1 5 Product of 2nd difference by 3rd Quotient (5). - - + [4*]3 3 New dividends. 1 Remainder of greatest dividend. 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.+ + - -

[Footnote 4*: These figures are removed at the next step.]

DIVISION. 8000 606.

- -+ Thousands + - - - - - - H. T. U. H. T. U. - - - - - -+ 9 Difference (making 700 trial divisor). 4 Difference. 6 6 Divisors. + - - - - - - [4*]8 Dividend. 1 Remainder of dividend. 9 4 Product of difference 1 and 2 with 1st Quotient (1). - - - - - -+ [4*]1 9 4 New dividends. 3 Remainder of greatest dividend. 9 4 Product of difference 1 and 2 with 2nd Quotient (1). + - - - - - - [4*]1 3 3 4 New dividends. 3 Remainder of greatest dividend. 9 4 Product of difference 1 and 2 with 3rd Quotient (1). - - - - - -+ 7 2 8 New dividends. 6 6 Product of divisors by 4th Quotient (1). + - - - - - - 1 2 2 Remainder. 1 4th Quotient. 1 3rd Quotient. 1 2nd Quotient. 1 1st Quotient. - - - - - -+ 1 3 +Quotient.+ + - - - - - -

[Footnote 4*: These figures are removed at the next step.]

The chief Abacists are Gerbert (tenth century), Abbo, and Hermannus Contractus (1054), who are credited with the revival of the art, Bernelinus, Gerland, and Radulphus of Laon (twelfth century). We know as English Abacists, Robert, bishop of Hereford, 1095, "abacum et lunarem compotum et celestium cursum astrorum rimatus," Turchillus Compotista (Thurkil), and through him of Guilielmus R.... "the best of living computers," Gislebert, and Simonus de Rotellis (Simon of the Rolls). They flourished most probably in the first quarter of the twelfth century, as Thurkil's treatise deals also with fractions. Walcher of Durham, Thomas of York, and Samson of Worcester are also known as Abacists.

Finally, the term Abacists came to be applied to computers by manual arithmetic. AMS. Algorithm of the thirteenth century (Sl. 3281, f.6,b), contains the following passage: "Est et alius modus secundum operatores sive practicos, quorum unus appellatur Abacus; et modus ejus est in computando per digitos et junctura manuum, et iste utitur ultra 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 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, 87. The difference of 8 is 2, of 7 is 3, the next article being 10; 7-2 is5. 510 = 50; 23 = 6. 50+6 = 56 answer. The rule will hold in such cases as 1715 where the article next higher is the same for both, i.e., 20; but in such a case as 179 the difference for each number must be taken from the higher article, i.e., the difference of 9 will be11.

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. Afew lines of this translation, as copied by Halliwell, are given on p.72, note2. 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, amethod 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 and9. 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 Mnchener Alterthumsvereins a set, almost complete, of them with a Byzantine treatise; aLatin 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. Acounting 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 e boke of algorym, or Augrym aft{er} lewd{er} vse. And is boke tretys e Craft of Nombryng, e quych crafte is called also Algorym. Ther was a kyng of Inde, e quich heyth Algor, &he made is craft. And aft{er} his name he called hit algory{m}; or els ano{er} cause is quy it is called Algorym, for e latyn word of hit s. Algorism{us} com{es} of Algos, grece, q{uid} e{st} ars, latine, craft on englis, and rides, q{uid} e{st} {nu}me{rus}, latine, Anomb{ur} on 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} [gh]e most vnd{ir}stonde {a}t in is craft ben vsid teen figurys, as here ben{e} writen for ensampul, +0+9 8 7 6 5 4 3 2 1. Expone e too v{er}sus afor{e}: this p{re}sent craft ys called Algorism{us}, in e quych we vse teen signys of Inde. Questio. Why ten fyguris of Inde? Solucio. for as I haue sayd afore ai wer{e} fonde fyrst in Inde of a kyng{e} of at Cuntre, {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 is verse is notifide e significac{i}on of ese figur{is}. And us expone the verse. e first signifiyth on{e}, e secu{n}de [*leaf 136b] signi[*]fiyth tweyn{e}, e thryd signifiyth thre, &the fourte signifiyth4. And so forthe towarde e lyft syde of e tabul or of e boke {a}t e figures ben{e} writen{e} in, til at {o}u come to the last figure, {a}t is called a cifre. Questio. In quych syde sittes e first figur{e}? Soluc{io}, forsothe loke quich figure is first in e ry[gh]t side of e bok or of e tabul, &{a}t same is e first figur{e}, for {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 e first, for he sittes on e ri[gh]t syde. And the fig{ur}e of 3 is last. Neu{er}-e-les wen he says P{ri}ma sig{nifica}t vnu{m} &c., at is to say, e first betokenes on{e}, e secu{n}de. 2. & fore-{er}-mor{e}, he vnd{ir}stondes no[gh]t of e first fig{ur}e of eu{er}y rew. But he vnd{ir}stondes e first figure {a}t is in e nomb{ur} of e forsayd teen figuris, e quych is on{e} of {e}se. 1. And e 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 is v{er}se us. Eu{er}y of ese figuris bitokens hym selfe & no mor{e}, yf he stonde in e first place of e rewele / this worde Simplicit{er} in at verse it is no more to say but at, & no mor{e}. If it stonde in the secu{n}de place of e rewle, he betokens ten{e} tymes hym selfe, as is figur{e} 2 here 20 tokens ten tyme hym selfe, [*leaf 137a] at is twenty, for he hym selfe betokenes twey{ne}, &ten tymes twene is twenty. And for he stondis on e lyft side & in e 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-no{er} toward the lyft side, he schal betoken{e} ten tymes as mich mor{e} as he schul betoken & he stode in e place {ere} at e fig{ure} a-for{e} hym stondes. loo an ensampull{e}. 9. 6. 3. 4. e fig{ure} of 4. {a}t hase is schape 4. betokens bot hymselfe, for he stondes in e first place. The fig{ure} of 3. at hase is schape 3. betokens ten tymes mor{e} en he schuld & he stode {ere} {a}t e fig{ure} of 4. stondes, {a}t is thretty. The fig{ure} of 6, {a}t hase is schape 6, betokens ten tymes mor{e} an he schuld & he stode {ere} as e fig{ure} of 3. stondes, for {ere} he schuld tokyn{e} bot sexty, &now he betokens ten tymes mor{e}, at is sex hundryth. The fig{ure} of 9. {a}t hase is schape 9. betokens ten tymes mor{e} an{e} he schuld & he stode in e place {ere} e fig{ure} of sex stondes, for en he schuld betoken to 9. hundryth, and in e place {ere} he stondes now he betokens 9. ousande. Al e hole nomb{ur} is 9 thousande sex hundryth & four{e} & thretty. fforthermor{e}, when {o}u schalt rede a nomb{ur} of fig{ure}, {o}u schalt begyn{e} at e last fig{ure} in the lyft side, &rede so forth to e ri[gh]t side as her{e} 9.6. 3.4. Thou schal begyn to rede at e fig{ure} of 9. & rede forth us.9. [*leaf 137b] thousand sex hundryth thritty & foure. But when {o}u schall{e} write, {o}u schalt be-gynne to write at e ry[gh]t side.

Nil cifra sig{nifica}t s{ed} dat signa{re} sequenti.

[Sidenote: The meaning and use of the cipher.]

Expone is v{er}se. Acifre tokens no[gh]t, bot he makes e fig{ure} to betoken at comes aft{ur} hym mor{e} an he schuld & he wer{e} away, as us 1+0+. her{e} e fig{ure} of on{e} tokens ten, &yf e cifre wer{e} away[{1}] & no fig{ure} by-for{e} hym he schuld token bot on{e}, for an he sch{ul}d stonde in e first place. And e cifre tokens nothyng hym selfe. for al e nomb{ur} of e ylke too fig{ure}s is bot ten. Questio. Why says he at a cifre makys a fig{ure} to signifye (tyf) mor{e} &c. Ispeke for is worde significatyf, ffor sothe it may happe aft{ur} a cifre schuld come a-no{ur} cifre, as us 2+0+0+. And [gh]et e secunde cifre shuld token neu{er} e mor{e} excep he schuld kepe e ord{er} of e 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 is v{er}se us. e last figu{re} schal token mor{e} an all{e} e o{er} afor{e}, thou[gh]t {ere} wer{e} a hundryth thousant figures afor{e}, as us, 16798. e last fig{ure} at is 1. betokens ten thousant. And all{e} e o{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} en all{e} at nomb{ur}, {er}go e last figu{re} tokens mor{e} an all e 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 is tretis dep{ar}tys is worde a nomb{ur} into 3 p{ar}tes. Some nomb{ur} is called digit{us} latine, adigit 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 is v{er}se. know {o}u aft{ur} e forsayd rewles {a}t I sayd afore, at {ere} ben thre spices of nomb{ur}. Oon{e} is a digit, Ano{er} is an Articul, &e to{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 at 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} {a}t may be deuidyt into nomb{urs} of ten & nothyng{e} leue ou{er}, as twenty, thretty, fourty, ahundryth, athousand, &such o{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 ben nomb{ur}s at bene componyt of a digyt & of an articull{e} as fouretene, fyftene, sextene, &such o{er}. ffortene is co{m}ponyd of four{e} at is a digit & of ten at is an articull{e}. ffiftene is componyd of 5 & ten, &so of all o{er}, what at ai ben. Short-lych eu{er}y nomb{ur} at be-gynnes w{i}t{h} a digit & endyth in a articull{e} is a composyt, as fortene bygennyng{e} by four{e} at 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 {o}u schalt wyrch whan {o}u schalt write a nomb{ur}. Expone v{er}su{m} sic, &fac iuxta expon{ent}is sentencia{m}; whan {o}u hast a nomb{ur} to write, loke fyrst what man{er} nomb{ur} it ys {a}t {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 {a}t digit in e first place toward e ryght side. If it be a composyt, write e 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 {o}u schal write sex & twenty. write e digit of e nomb{ur} in e first place at is sex, and write e articul next aft{ur} at is twenty, as us 26. But whan {o}u schalt sowne or speke [*leaf 139a] or rede an Composyt ou schalt first sowne e articul & aft{ur} e digit, as {o}u seyst by e comyn{e} speche, Sex & twenty & nou[gh]t 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 {o}u schal write when e nombre {a}t {o}u hase to write is an Articul. Expone v{er}sus sic & fac s{ecundu}m sentenciam. Ife e nomb{ur} {a}t {o}u hast write be an Articul, write first a cifre & aft{ur} e cifer write an Articull{e} us. 2+0+. fforthermor{e} {o}u schalt vnd{ir}stonde yf {o}u haue an Articul, loke how mych he is, yf he be w{i}t{h}-ynne an hundryth, {o}u schalt write bot on{e} cifre, afore, as her{e} .9+0+. If e articull{e} be by hym-silfe & be an hundrid euen{e}, en schal {o}u write .1. & 2 cifers afor{e}, at he may stonde in e thryd place, for eu{er}y fig{ure} in e thryd place schal token a hundrid tymes hym selfe. If e articul be a thousant or thousandes[{3}] and he stonde by hy{m} selfe, write afor{e} 3 cifers & so for of al o{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 {a}t yf e first fig{ure} in e rewle of fig{ure}s token a nomb{ur} at is euen{e} al {a}t nomb{ur} of fig{ur}ys in at rewle schal be euen{e}, as her{e} {o}u may see 6. 7. 3. 5.4. Computa & p{ro}ba. If e first [*leaf 139b] fig{ur}e token an nomb{ur} at is ode, all{e} at nomb{ur} in at rewle schall{e} be ode, as her{e} 5 6 7 8 67. 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 {a}t {er} ben .7. spices or p{ar}tes of is craft. The first is called addicio, e secunde is called subtraccio. The thryd is called duplacio. The 4. is called dimydicio. The 5. is called m{u}ltiplicacio. The 6 is called diuisio. The 7. is called extraccio of e Rote. What all ese 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 e ryght side of e boke or of a tabul. loke wer{e} {o}u wul be-gynne to write latyn or englys in a boke, & {a}t schall{e} be called e lyft side of the boke, at {o}u writest toward {a}t side schal be called e ryght side of e boke. V{er}sus.

A leua dupla, diuide, m{u}ltiplica.

[Sidenote: Multiply or divide from left to right.]

Here he telles e in quych side of e boke or of e tabul {o}u schall{e} be-gyn{e} to wyrch duplacio, diuisio, and m{u}ltiplicacio. Thou schal begyn{e} to worch in e lyft side of e boke or of e tabul, but yn what wyse {o}u schal wyrch in hym dicetur singillatim in seque{n}tib{us} capi{tulis} et de vtilitate cui{us}li{bet} art{is} & sic Completur [*leaf 140.] p{ro}hemi{um} & sequit{ur} tractat{us} & p{ri}mo de arte addic{ion}is que p{ri}ma ars est in ordine.

[Headnote: The Craft of Addition.]

+Adder{e} si nu{mer}o num{e}ru{m} vis, ordine tali Incipe; scribe duas p{rim}o series nu{mer}or{um} P{ri}ma{m} sub p{ri}ma recte pone{n}do figura{m}, Et sic de reliq{ui}s facias, si sint tibi plures.

[Sidenote: Four things must be known: what it is; how many rows of figures; how many cases; what is its result. How to set down the sum.]

Her{e} by-gynnes e craft of Addicio. In is craft {o}u most knowe foure thyng{es}. Fyrst {ou} most know what is addicio. Next {o}u most know how mony rewles of figurys ou most haue. Next {o}u most know how mony diue{r}s casys happes in is craft of addicio. And next qwat is e p{ro}fet of is craft. As for e first ou most know at addicio is a castyng to-ged{ur} of twoo nomburys in-to on{e} nombr{e}. As yf I aske qwat is twene & thre. {o}u wyl cast ese twene nomb{re}s to-ged{ur} & say {a}t it is fyue. As for e secunde ou most know {a}t ou schall{e} haue tweyne rewes of figures, on{e} vndur a-nother, as her{e} {o}u maystse.

1234 2168.

As for e thryd ou most know {a}t ther{e} ben foure diu{er}se cases. As for e forthe {o}u most know {a}t e p{ro}fet of is craft is to telle what is e hole nomb{ur} {a}t comes of diu{er}se nomburis. Now as to e texte of oure verse, he teches ther{e} how {o}u schal worch in is craft. He says yf {o}u wilt cast on{e} nomb{ur} to ano{er} nomb{ur}, ou most by-gynne on is wyse. ffyrst write [*leaf 140b] two rewes of figuris & nombris so at {o}u write e first figur{e} of e hyer nomb{ur} euen{e} vnd{ir} the first fig{ure} of e nether nomb{ur}, And e secunde of e nether nomb{ur} euen{e} vnd{ir} e secunde of e hyer, & so forthe of eu{er}y fig{ur}e of both e rewes as {o}u maystse.

123 234.

[Headnote: The Cases of the Craft of Addition.]

Inde duas adde p{ri}mas hac condic{i}one: Si digitus crescat ex addic{i}one prior{um}; P{ri}mo scribe loco digitu{m}, quicu{n}q{ue} sit ille.

[Sidenote: Add the first figures; rub out the top figure; write the result in its place. Here is an example.]

Here he teches what {o}u schalt do when {o}u hast write too rewes of figuris on vnder an-o{er}, as I sayd be-for{e}. He says {o}u schalt take e first fig{ur}e of e heyer nomb{re} & e fyrst figur{e} of e ne{er} nombre, &cast hem to-ged{er} vp-on is condicion. Thou schal loke qwe{er} e nombe{r} at comys {ere}-of be a digit or no. If he be a digit {o}u schalt do away e first fig{ur}e of e hyer nomb{re}, and write {ere} in his stede at he stode Inne e digit, {a}t comes of e ylke 2 fig{ur}es, &so wrich forth on o{er} figures yf {ere} be ony moo, til {o}u come to e ende toward e lyft side. And lede e nether fig{ure} stonde still eu{er}-mor{e} til {o}u haue ydo. ffor {ere}-by {o}u schal wyte whe{er} {o}u hast don{e} wel or no, as I schal tell e aft{er}ward in e ende of is Chapt{er}. And loke allgate at ou be-gynne to worch in is Craft of [*leaf 141a] Addi[*]cion in e ry[gh]t side, here is an ensampul of is case.

1234 2142.

Caste 2 to four{e} & at wel be sex, do away 4. & write in e same place e fig{ur}e of sex. And lete e fig{ur}e of 2 in e nether rewe stonde stil. When {o}u hast do so, cast 3 & 4 to-ged{ur} and at wel be seuen {a}t is a digit. Do away e 3, &set {ere} seuen, and lete e ne{er} fig{ure} stonde still{e}, &so worch forth bakward til {o}u hast ydo all to-ged{er}.

Et si composit{us}, in limite scribe seque{n}te Articulum, p{ri}mo digitum; q{uia} sic iubet ordo.

[Sidenote: Suppose it is a Composite, set down the digit, and carry the tens. Here is an example.]

Here is e secunde case {a}t may happe in is craft. And e case is is, yf of e casting of 2 nomburis to-ged{er}, as of e fig{ur}e of e hyer rewe & of e figure of e ne{er} rewe come a Composyt, how schalt {ou} worch. {us} {o}u schalt worch. Thou shalt do away e fig{ur}e of e hyer nomb{er} at was cast to e figure of e ne{er} nomber. And write {ere} e digit of e Composyt. And set e articul of e composit next aft{er} e digit in e same rewe, yf {ere} be no mo fig{ur}es aft{er}. But yf {ere} be mo figuris aft{er} at digit. And ere he schall be rekend for hym selfe. And when {o}u schalt adde {a}t ylke figure {a}t berys e articull{e} ou{er} his hed to e figur{e} vnd{er} hym, {o}u schalt cast at articul to e figure {a}t hase hym ou{er} his hed, &{ere} at Articul schal token hym selfe. lo an Ensampull [*leaf 141b] of all.

326 216.

Cast 6 to 6, & {ere}-of wil arise twelue. do away e hyer 6 & write {ere} 2, {a}t is e digit of is composit. And e{n} write e articull{e} at is ten ou{er} e figuris hed of twene as {us}.

1 322 216.

Now cast e articull{e} {a}t standus vpon e fig{ur}is of twene hed to e same fig{ur}e, &reken at articul bot for on{e}, and an {ere} wil arise thre. an cast at thre to e ne{er} figure, at is on{e}, &at wul be four{e}. do away e fig{ur}e of 3, and write {ere} a fig{ur}e of foure. and lete e ne{er} fig{ur}e stonde stil, &an worch forth. vn{de} {ver}sus.

Articulus si sit, in p{ri}mo limite cifram, Articulu{m} v{er}o reliquis inscribe figuris, Vel p{er} se scribas si nulla figura sequat{ur}.

[Sidenote: Suppose it is an Article, set down a cipher and carry the tens. Here is an example.]

Her{e} he puttes e thryde case of e craft of Addicion. & e case is is. yf of Addicioun of 2 figuris a-ryse an Articull{e}, how schal {o}u do. thou most do away e heer fig{ur}e {a}t was addid to e ne{er}, &write {ere} a cifre, and sett e articuls on e figuris hede, yf {a}t {ere} come ony aft{er}. And wyrch an as I haue tolde e in e secunde case. An ensampull.

25. 15

Cast 5 to 5, at wylle be ten. now do away e hyer 5, &write {ere} a cifer. And sette ten vpon e figuris hed of 2. And reken it but for on us.] lo an Ensampull{e}

1 2+0+ 15

And [*leaf 142a] an worch forth. But yf {ere} come no figure aft{er} e cifre, write e articul next hym in e same rewe as here

- 5 5 -

cast 5 to 5, and it wel be ten. do away 5. at is e hier 5. and write {ere} a cifre, &write aft{er} hym e articul as us

1+0+ 5

And an {o}u hast done.

Si tibi cifra sup{er}ueniens occurrerit, illa{m} Dele sup{er}posita{m}; fac illic scribe figura{m}, Postea procedas reliquas addendo figuras.

[Sidenote: What to do when you have a cipher in the top row. An example of all the difficulties.]

Her{e} he putt{es} e fourt case, &it is is, at yf {ere} come a cifer in e hier rewe, how {o}u schal do. us {o}u schalt do. do away e cifer, &sett {ere} e digit {a}t comes of e addiciou{n} as us

1+0+0+84. 17743

In is ensampul ben all{e} e four{e} cases. Cast 3 to foure, {a}t wol be seuen. do away 4. & write {ere} seuen; an cast 4 to e figur{e} of 8. {a}t wel be 12. do away 8, &sett {ere} 2. at is a digit, and sette e articul of e composit, at is ten, vpon e cifers hed, &reken it for hym selfe at is on. an cast on{e} to a cifer, & hit wull{e} be but on, for no[gh]t & on makes but on{e}. an cast 7. {a}t stondes vnd{er} at on to hym, &at wel be 8. do away e cifer & at 1. & sette {ere} 8. an go forthermor{e}. cast e o{er} 7 to e cifer {a}t stondes ou{er} hy{m}. {a}t wul be bot seuen, for e cifer betokens no[gh]t. do away e cifer & sette {ere} seuen, [*leaf 142b] & en go for{er}mor{e} & cast 1 to 1, &at wel be 2. do away e hier 1, &sette {ere} 2. an hast {o}u do. And yf {o}u haue wel ydo is nomber at is sett her{e}-aft{er} wel be e nomber at schall{e} aryse of all{e} e addicion as her{e} 27827. Sequi{tu}r alia sp{eci}es.

[Headnote: The Craft of Subtraction.]

+A nu{mer}o num{er}u{m} si sit tibi demer{e} cura Scribe figurar{um} series, vt in addicione.

[Sidenote: Four things to know about subtraction: the first; the second; the third; the fourth.]

This is e Chapt{er} of subtraccion, in the quych ou most know foure nessessary thyng{es}. the first what is subtraccion. e secunde is how mony nombers ou most haue to subt{ra}ccion, the thryd is how mony maners of cases {ere} may happe in is craft of subtraccion. The fourte is qwat is e p{ro}fet of is craft. As for e first, {o}u most know {a}t subtraccion is drawyng{e} of on{e} nowmb{er} oute of ano{er} nomber. As for e secunde, ou most knowe {a}t ou most haue two rewes of figuris on{e} vnd{er} ano{er}, as {o}u addyst in addicion. As for e thryd, {o}u moyst know {a}t four{e} man{er} of diu{er}se casis mai happe in is craft. As for e fourt, ou most know {a}t e p{ro}fet of is craft is whenne {o}u hasse taken e lasse nomber out of e mor{e} to telle what {ere} leues ou{er} {a}t. & {o}u most be-gynne to wyrch in {is} craft in e ryght side of e boke, as {o}u diddyst in addicion. V{er}sus.

Maiori nu{mer}o num{er}u{m} suppone minorem, Siue pari nu{mer}o supponat{ur} num{er}us par.

[Sidenote: Put the greater number above the less.]

[*leaf 143a] Her{e} he telles at e hier nomber most be mor{e} en e ne{er}, or els euen as mych. but he may not be lasse. And e case is is, ou schalt drawe e ne{er} nomber out of e hyer, &ou mayst not do {a}t yf e hier nomber wer{e} lasse an at. ffor {o}u mayst not draw sex out of 2. But {o}u mast draw 2 out of sex. And ou maiste draw twene out of twene, for ou schal leue no[gh]t of e hier twene vn{de} v{er}sus.

[Headnote: The Cases of the Craft of Subtraction.]

Postea si possis a prima subt{ra}he p{ri}ma{m} Scribens quod remanet.

[Sidenote: The first case of subtraction. Here is an example.]

Her{e} is e first case put of subtraccion, &he says ou schalt begynne in e ryght side, &draw e first fig{ure} of e ne{er} rewe out of e first fig{ure} of e hier rewe. qwether e hier fig{ur}e be mor{e} en e ne{er}, or euen as mych. And at is notified in e vers when he says "Si possis." Whan {o}u has us ydo, do away e hiest fig{ur}e & sett {ere} at leues of e subtraccion, lo an Ensampull{e}

- 234 122 -

draw 2 out of 4. an leues 2. do away 4 & write {ere} 2, & latte e ne{er} figur{e} sto{n}de stille, &so go for-by o{er} figuris till {o}u come to e ende, an hast {o}udo.

Cifram si nil remanebit.

[Sidenote: Put a cipher if nothing remains. Here is an example.]

Her{e} he putt{es} e secunde case, &hit is is. yf it happe {a}t qwen {o}u hast draw on ne{er} fig{ure} out of a hier, &{er}e leue no[gh]t aft{er} e subt{ra}ccion, us [*leaf 143b] ou schalt do. {o}u schall{e} do away e hier fig{ur}e & write {ere} a cifer, as lo an Ensampull

24 24

Take four{e} out of four{e} an leus no[gh]t. {er}efor{e} do away e hier 4 & set {ere} a cifer, an take 2 out of 2, an leues no[gh]t. do away e hier 2, &set {ere} a cifer, and so worch whar{e} so eu{er} is happe.

Sed si no{n} possis a p{ri}ma dem{er}e p{ri}ma{m} P{re}cedens vnu{m} de limite deme seque{n}te, Quod demptu{m} p{ro} denario reputabis ab illo Subt{ra}he to{ta}lem num{er}u{m} qu{em} p{ro}posuisti Quo facto sc{ri}be super quicquid remaneb{i}t.

[Sidenote: Suppose you cannot take the lower figure from the top one, borrow ten; take the lower number from ten; add the answer to the top number. How to 'Pay back' the borrowed ten. Example.]

Her{e} he puttes e thryd case, e quych is is. yf it happe at e ne{er} fig{ur}e be mor{e} en e hier fig{ur}e at he schall{e} be draw out of. how schall{e} ou do. us {o}u schall{e} do. ou schall{e} borro .1. oute of e next fig{ur}e at comes aft{er} in e same rewe, for is case may neu{er} happ but yf {ere} come figures aft{er}. an {o}u schalt sett at on ou{er} e hier figur{es} hed, of the quych ou woldist y-draw oute e ney{er} fig{ur}e yf {o}u haddyst y-my[gh]t. Whane ou hase us ydo ou schall{e} rekene {a}t .1. for ten. . And out of at ten {o}u schal draw e neyermost fig{ur}e, And all{e} {a}t leues ou schall{e} adde to e figur{e} on whos hed at .1. stode. And en {o}u schall{e} do away all{e} at, &sett {ere} all{e} that arisys of the addicion of e ylke 2 fig{ur}is. And yf yt [*leaf 144a] happe at e fig{ur}e of e quych {o}u schalt borro on be hym self but 1. If {o}u schalt at on{e} & sett it vppon e o{er} figur{is} hed, and sett in {a}t 1. place a cifer, yf {ere} come mony figur{es} aft{er}. lo an Ensampul.

2122 1134

take 4 out of 2. it wyl not be, erfor{e} borro on{e} of e next figur{e}, {a}t is 2. and sett at ou{er} e hed of e fyrst 2. & rekene it for ten. and ere e secunde stondes write 1. for {o}u tokest on out of hy{m}. an take e ne{er} fig{ur}e, at is 4, out of ten. And en leues 6. cast to 6 e fig{ur}e of at 2 at stode vnd{er} e hedde of 1. at was borwed & rekened for ten, and at wylle be 8. do away {a}t 6 & at 2, & sette {ere} 8, &lette e ne{er} fig{ur}e stonde stille. Whanne {o}u hast do us, go to e next fig{ur}e {a}t is now bot 1. but first yt was 2, & {ere}-of was borred1. an take out of {a}t e fig{ur}e vnd{er} hym, {a}t is 3. hit wel not be. er-for{e} borowe of the next fig{ur}e, e quych is bot 1. Also take & sett hym ou{er} e hede of e fig{ure} at ou woldest haue y-draw oute of e nether figure, e quych was 3. & ou my[gh]t not, &rekene {a}t borwed 1 for ten & sett in e same place, of e quych place {o}u tokest hy{m} of, acifer, for he was bot 1. Whanne {o}u hast {us} ydo, take out of at 1. {a}t is rekent for ten, e ne{er} figure of 3. And {ere} leues7. [*leaf 144b] cast e ylke 7 to e fig{ur}e at had e ylke ten vpon his hed, e quych fig{ur}e was1, &at wol be8. an do away {a}t 1 and {a}t7, &write {ere} 8. & an wyrch forth in o{er} figuris til {o}u come to e ende, &an {o}u hast e do. V{er}sus.

Facque nonenarios de cifris, cu{m} remeabis Occ{ur}rant si forte cifre; dum demps{er}is vnum Postea p{ro}cedas reliquas deme{n}do figuras.

[Sidenote: Avery hard case is put. Here is an example.]

Her{e} he putt{es} e fourte case, e quych is is, yf it happe at e ne{er} fig{ur}e, e quych {o}u schalt draw out of e hier fig{ur}e be mor{e} pan e hier figur ou{er} hym, &e next fig{ur}e of two or of thre or of foure, or how mony {ere} be by cifers, how wold {o}u do. {o}u wost wel {o}u most nede borow, &{o}u mayst not borow of e cifers, for ai haue no[gh]t at ai may lene or spar{e}. Ergo[{4}] how woldest {o}u do. Certayn us most {o}u do, {o}u most borow on of e next figure significatyf in at rewe, for is case may not happe, but yf {ere} come figures significatyf aft{er} the cifers. Whan {o}u hast borowede {a}t 1 of the next figure significatyf, sett {a}t on ou{er} e hede of {a}t fig{ur}e of e quych {o}u wold haue draw e ne{er} figure out yf {o}u hadest my[gh]t, &reken it for ten as o{u} diddest i{n} e o{er} case her{e}-a-for{e}. Whan {o}u hast us y-do loke how mony cifers {ere} wer{e} bye-twene at figur{e} significatyf, &e fig{ur}e of e quych {o}u woldest haue y-draw the [*leaf 145a] ne{er} figure, and of eu{er}y of e ylke cifers make a figur{e} of 9. lo an Ensampull{e} after.

- 40002 10004 -

Take 4 out of 2. it wel not be. borow 1 out of be next figure significatyf, e quych is 4, &en leues 3. do away {a}t figur{e} of 4 & write {ere} 3. & sett {a}t 1 vppon e fig{ur}e of 2 hede, &an take 4 out of ten, &an ere leues 6. Cast 6 to the fig{ur}e of 2, {a}t wol be 8. do away at 6 & write {er}e 8. Whan {o}u hast us y-do make of eu{er}y 0 betweyn 3 & 8 a figure of 9, &an worch forth in goddes name. & yf {o}u hast wel y-do {o}u[{5}] schalt haue is nomb{er}

- 39998 Sic. 10004 -

[Headnote: How to prove the Subtraction.]

Si subt{ra}cc{i}o sit b{e}n{e} facta p{ro}bar{e} valebis Quas s{u}btraxisti p{ri}mas addendo figuras.

[Sidenote: How to prove a subtraction sum. Here is an example. He works his proof through, and brings out a result.]

Her{e} he teches e Craft how {o}u schalt know, whan {o}u hast subt{ra}yd, whe{er} ou hast wel ydo or no. And e Craft is is, ryght as {o}u subtrayd e ne{er} figures fro e hier figures, ry[gh]t so adde e same ne{er} figures to e hier figures. And yf {o}u haue well y-wroth a-for{e} ou schalt haue e hier nombre e same {o}u haddest or ou be-gan to worch. as for is I bade ou schulde kepe e ne{er} figures stylle. lo an [*leaf 145b] Ensampull{e} of all{e} e 4 cases toged{re}. worche well{e} is case

40003468 . 20004664

And yf ou worch well{e} whan ou hast all{e} subtrayd e {a}t hier nombr{e} her{e}, is schall{e} be e nombre here foloyng whan {o}u hast subtrayd.

39998804 . [Sidenote: Our author makes a slip here (3 for1).] 20004664

And ou schalt know {us}. adde e ne{er} rowe of e same nombre to e hier rewe as us, cast 4 to 4. at wol be 8. do away e 4 & write {ere} 8. by e first case of addicion. an cast 6 to 0 at wol be 6. do away e 0, &write ere 6. an cast 6 to 8, {a}t wel be 14. do away 8 & write {ere} a fig{ur}e of 4, at is e digit, and write a fig{ur}e of 1. {a}t schall be-token ten. {a}t is e articul vpon e hed of 8 next aft{er}, an reken {a}t 1. for 1. & cast it to 8. at schal be 9. cast to at 9 e ne{er} fig{ur}e vnd{er} at e quych is 4, &at schall{e} be 13. do away at 9 & sett {er}e 3, & sett a figure of 1. {a}t schall be 10 vpon e next figur{is} hede e quych is 9. by e secu{n}de case {a}t {o}u hadest in addicion. an cast 1 to 9. & at wol be 10. do away e 9. & at 1. And write {ere} a cifer. and write e articull{e} at is 1. betokenyng{e} 10. vpon e hede of e next figur{e} toward e lyft side, e quych [*leaf 146a] is 9, &so do forth tyl {o}u come to e last9. take e figur{e} of at1. e quych {o}u schalt fynde ou{er} e hed of 9. & sett it ou{er} e next figures hede at schal be3. Also do away e 9. & set {ere} a cifer, &en cast at 1 at stondes vpon e hede of 3 to e same 3, &{a}t schall{e} make4, en caste to e ylke 4 the figur{e} in e ney{er} rewe, e quych is2, and at schall{e} be 6. And en schal {o}u haue an Ensampull{e} a[gh]eyn, loke & se, &but {o}u haue is same {o}u hase myse-wro[gh]t.

60003468 20004664

Sequit{ur} de duplac{i}one

[Headnote: The Craft of Duplation.]

+Si vis duplar{e} num{er}u{m}, sic i{n}cipe p{rim}o Scribe fig{ur}ar{um} serie{m} q{ua}mcu{n}q{ue} vel{is} tu.

[Sidenote: Four things must be known in Duplation. Here they are. Mind where you begin. Remember your rules.]

This is the Chaptur{e} of duplacion, in e quych craft {o}u most haue & know 4 thing{es}. e first {a}t {o}u most know is what is duplacion. e secu{n}de is how mony rewes of fig{ur}es {o}u most haue to is craft. e thryde is how many cases may[{6}] happe in is craft. e fourte is what is e p{ro}fet of e craft. As for e first. duplacion is a doublyng{e} of a nombre. As for e secu{n}de {o}u most [*leaf 146b] haue on nombre or on rewe of figures, the quych called nu{merus} dupland{us}. As for e thrid {o}u most know at 3 diu{er}se cases may hap in is craft. As for e fourte. qwat is e p{ro}fet of is craft, &{a}t is to know what a-risy[gh]t of a nombre I-doublyde. ffor{er}-mor{e}, {o}u most know & take gode hede in quych side {o}u schall{e} be-gyn in is craft, or ellis {o}u mayst spyl all{e} {i} lab{er} {er}e aboute. c{er}teyn {o}u schalt begyn in the lyft side in is Craft. thenke wel ou{er} is verse. [{7}]A leua dupla, diuide, m{u}ltiplica.[{7}] [[Subt{ra}has a{u}t addis a dext{ri}s {ve}l medi{a}b{is}]] The sentens of es verses afor{e}, as {o}u may see if {o}u take hede. As e text of is verse, at is to say, Si vis duplare. is is e sentence. If {o}u wel double a nombre us {o}u most be-gynn. Write a rewe of figures of what nomb{re} ou welt. v{er}sus.

Postea p{ro}cedas p{ri}ma{m} duplando figura{m} Inde q{uo}d excrescit scribas vbi iusserit ordo Iuxta p{re}cepta tibi que dant{ur} in addic{i}one.

[Sidenote: How to work a sum.]

Her{e} he telles how {o}u schalt worch in is Craft. he says, fyrst, whan {o}u hast writen e nombre {o}u schalt be-gyn at e first figur{e} in the lyft side, &doubull{e} at fig{ur}e, &e nombre at comes {ere}-of {o}u schalt write as {o}u diddyst in addicion, as Ischal telle e in e case. v{er}sus.

[Headnote: The Cases of the Craft of Duplation.]

[*leaf 147a]

Nam si sit digitus in primo limite scribas.

[Sidenote: If the answer is a digit, write it in the place of the top figure.]

Her{e} is e first case of is craft, e quych is is. yf of duplacion of a figur{e} arise a digit. what schal {o}u do. us {o}u schal do. do away e fig{ur}e at was doublede, &sett {ere} e diget at comes of e duplacion, as us. 23. double 2, &{a}t wel be 4. do away e figur{e} of 2 & sett {ere} a figur{e} of 4, &so worch forth till{e} {o}u come to e ende. v{er}sus.

Articul{us} si sit, in p{ri}mo limite cifram, Articulu{m} v{er}o reliquis inscribe figuris; Vel p{er} se scribas, si nulla figura sequat{ur}.

[Sidenote: If it is an article, put a cipher in the place, and 'carry' the tens. If there is no figure to 'carry' them to, write them down.]

Here is e secunde case, e quych is is yf {ere} come an articull{e} of e duplacion of a fig{ur}e {o}u schalt do ry[gh]t as {o}u diddyst in addicion, at is to wete at {o}u schalt do away e figur{e} at is doublet & sett {ere} a cifer, &write e articull{e} ou{er} e next figur{is} hede, yf {ere} be any aft{er}-warde toward e lyft side as us. 25. begyn at the lyft side, and doubull{e} 2. at wel be 4. do away at 2 & sett ere 4. an doubul 5. at wel be 10. do away 5, & sett {ere} a 0, &sett 1 vpon e next figur{is} hede e quych is 4. & en draw downe 1 to 4 & at woll{e} be 5, &en do away {a}t 4 & at 1, &sett {ere} 5. for at 1 schal be rekened in e drawyng{e} toged{re} for 1. wen [*leaf 147b] ou hast ydon ou schalt haue is nomb{r}e 50. yf {ere} come no figur{e} aft{er} e fig{ur}e {a}t is addit, of e quych addicion comes an articull{e}, {o}u schalt do away e figur{e} {a}t is dowblet & sett {ere} a 0. & write e articul next by in e same rewe toward e lyft syde as us, 523. double 5 at woll be ten. do away e figur{e} 5 & set {ere} a cifer, &sett e articul next aft{er} in e same rewe toward e lyft side, &ou schalt haue is nombre 1023. en go forth & double e o{er} nombers e quych is ly[gh]t y-now[gh]t to do. v{er}sus.

Compositus si sit, in limite sc{ri}be seq{uen}te Articulu{m}, p{ri}mo digitu{m}; q{uia} sic iubet ordo: Et sic de reliq{ui}s facie{n}s, si sint tibi plures.

[Sidenote: If it is a Composite, write down the digit, and 'carry' the tens. Here is an example.]

Her{e} he putt{es} e Thryd case, e quych is is, yf of duplacion of a fig{ur}e come a Composit. {o}u schalt do away e fig{u}re {a}t is doublet & set {ere} a digit of e Composit, & sett e articull{e} ou{er} e next figures hede, &aft{er} draw hym downe w{i}t{h} e figur{e} ou{er} whos hede he stondes, &make {ere}-of an nombre as {o}u hast done afore, &yf {ere} come no fig{ur}e aft{er} at digit at {o}u hast y-write, a{n} set e articull{e} next aft{er} hym in e same rewe as us, 67: double 6 at wel be 12, do away 6 & write {ere} e digit [*leaf 148a] of 12, e quych is2, and set e articull{e} next aft{er} toward e lyft side in e same rewe, for {ere} comes no figur{e} aft{er}. an dowble at o{er} figur{e}, e quych is7, at wel be 14. the quych is a Composit. en do away 7 at {o}u doublet & sett e e diget of hy{m}, the quych is 4, sett e articull{e} ou{er} e next figur{es} hed, e quych is 2, &en draw to hym at on, &make on nombre e quych schall{e} be 3. And en yf {o}u haue wel y-do {o}u schall{e} haue is nombre of e duplacion, 134. v{er}sus.

Si super ext{re}ma{m} nota sit monade{m} dat eid{em} Quod t{ibi} {con}tingat si p{ri}mo dimidiabis.

[Sidenote: How to double the mark for one-half. This can only stand over the first figure.]

Her{e} he says, yf ou{er} e fyrst fig{ur}e in e ry[gh]t side be such a merke as is her{e} made, ^w, {o}u schall{e} fyrst doubull{e} e figur{e}, the quych stondes vnd{er} {a}t merke, &en ou schalt doubul at merke e quych stond{es} for haluendel on. for too haluedels makes on, & so {a}t wol be on. cast {a}t on to at duplacion of e figur{e} ou{er} whos hed stode at merke, &write it in e same place {ere} at e figur{e} e quych was doublet stode, as us 23^w. double 3, at wol be 6; doubul at halue on, &at wol be on. cast on to 6, {a}t wel be 7. do away 6 & at 1, &sett {ere} 7. an hase ou do. as for at figur{e}, an go [*leaf 148b] to e o{er} fig{ure} & worch forth. &{o}u schall neu{er} haue such a merk but ou{er} e hed of e furst figure in e ryght side. And [gh]et it schal not happe but yf it were y-halued a-for{e}, us {o}u schalt vnd{er}stonde e verse. Si sup{er} ext{re}ma{m} &c. Et nota, talis fig{ur}a ^w significans medietate{m}, unitat{is} veniat, {i.e.} contingat u{e}l fiat sup{er} ext{re}ma{m}, {i.e.} sup{er} p{ri}ma{m} figura{m} in ext{re}mo sic v{er}sus dextram ars dat: {i.e.} reddit monade{m}. {i.e.} vnitate{m} eide{m}. {i.e.} eidem note & declina{tur} hec monos, d{i}s, di, dem, &c. Quod {er}g{o} to{tum} ho{c} dabis monade{m} note {con}ting{et}. {i.e.} eveniet tibi si dimidiasti, {i.e.} accipisti u{e}l subtulisti medietatem alicuius unius, in cuius principio sint figura nu{mer}u{m} denotans i{m}pare{m} p{rim}o {i.e.} principiis.

[Headnote: The Craft of Mediation.]

Sequit{ur} de mediacione.

+Incipe sic, si vis alique{m} nu{me}ru{m} mediar{e}: Sc{ri}be figurar{um} seriem sola{m}, velut an{te}.

[Sidenote: The four things to be known in mediation: the first the second; the third; the fourth. Begin thus.]

In is Chapter is ta[gh]t e Craft of mediacioun, in e quych craft {o}u most know 4 thynges. ffurst what is mediacion. the secunde how mony rewes of figur{es} {o}u most haue in e wyrchyng{e} of is craft. e thryde how mony diu{er}se cases may happ in is craft.[{8}] [[the .4. what is e p{ro}fet of is craft.]] As for e furst, {o}u schalt vndurstonde at mediacion is a takyng out of halfe a nomber out of a holle nomber, [*leaf 149a] as yf {o}u wolde take 3 out of 6. As for e secunde, {o}u schalt know {a}t {o}u most haue on{e} rewe of figures, &no moo, as {o}u hayst in e craft of duplacion. As for the thryd, ou most vnd{er}stonde at 5 cases may happe in is craft. As for e fourte, ou schall{e} know at the p{ro}fet of is craft is when {o}u hast take away e haluendel of a nomb{re} to telle qwat er{e} schall{e} leue. Incipe sic, &c. The sentence of is verse is is. yf {o}u wold medye, at is to say, take halfe out of e holle, or halfe out of halfe, ou most begynne {us}. Write on{e} rewe of figur{es} of what nombre ou wolte, as {o}u dyddyst be-for{e} in e Craft of duplacion. v{er}sus.

Postea p{ro}cedas medians, si p{ri}ma figura Si par aut i{m}par videas.

[Sidenote: See if the number is even or odd.]

Her{e} he says, when {o}u hast write a rewe of figures, {o}u schalt take hede whe{er} e first figur{e} be euen or odde in nombre, & vnd{er}stonde {a}t he spekes of e first figure in e ry[gh]t side. And i{n} the ryght side {o}u schall{e} begynne in is Craft.

Quia si fu{er}it par, Dimidiab{is} eam, scribe{n}s quicq{ui}d remanebit:

[Sidenote: If it is even, halve it, and write the answer in its place.]

Her{e} is the first case of is craft, e quych is is, yf e first figur{e} be euen. ou schal take away fro e figur{e} euen halfe, &do away at fig{ur}e and set {ere} at leues ou{er}, as us, 4. take [*leaf 149b] halfe out of 4, &an {ere} leues 2. do away 4 & sett {ere} 2. is is lyght y-now[gh]t. v{er}sus.

[Headnote: The Mediation of an Odd Number.]

Impar si fu{er}it vnu{m} demas mediar{e} Quod no{n} p{re}sumas, s{ed} quod sup{er}est mediabis Inde sup{er} tractu{m} fac demptu{m} quod no{ta}t vnu{m}.

[Sidenote: If it is odd, halve the even number less than it. Here is an example. Then write the sign for one-half over it. Put the mark only over the first figure.]

Her{e} is e secunde case of is craft, the quych is is. yf e first figur{e} betoken{e} a nombre at is odde, the quych odde schal not be mediete, en {o}u schalt medye at nombre at leues, when the odde of e same nomb{re} is take away, &write at {a}t leues as {o}u diddest in e first case of is craft. Whan {o}u hayst write at. for {a}t at leues, write such a merke as is her{e} ^w vpon his hede, e quych merke schal betoken halfe of e odde at was take away. lo an Ensampull. 245. the first figur{e} her{e} is betokenyng{e} odde nombre, e quych is 5, for 5 is odde; {er}e-for{e} do away at {a}t is odde, e quych is 1. en leues 4. en medye 4 & en leues 2. do away 4. & sette {ere} 2, &make such a merke ^w upon his hede, at is to say ou{er} his hede of 2 as us. 242.^w And en worch forth in e o{er} figures tyll {o}u come to e ende. by e furst case as {o}u schalt vnd{er}stonde at {o}u schalt [*leaf 150a] neu{er} make such a merk but ou{er} e first fig{ur}e hed in e ri[gh]t side. Whe{er} e other fig{ur}es at comyn aft{er} hym be euen or odde. v{er}sus.

[Headnote: The Cases of the Craft of Mediation.]

Si monos, dele; sit t{ibi} cifra post no{ta} supra.

[Sidenote: If the first figure is one put a cipher.]

Here is e thryde case, e quych yf the first figur{e} be a figur{e} of1. {o}u schalt do away at 1 & set {ere} a cifer, &a merke ou{er} e cifer as us, 241. do away1, &sett {ere} a cifer w{i}t{h} a merke ou{er} his hede, &en hast {o}u ydo for at 0. as us 0^w en worch forth in e oer fig{ur}ys till {o}u come to e ende, for it is lyght as dyche water. vn{de} v{er}sus.

Postea p{ro}cedas hac condic{i}one secu{n}da: Imp{ar} si fu{er}it hinc vnu{m} deme p{ri}ori, Inscribens quinque, nam denos significabit Monos p{re}d{ict}am.

[Sidenote: What to do if any other figure is odd. Write a figure of five over the next lower number's head. Example.]

Her{e} he putt{es} e fourte case, e quych is is. yf it happen the secunde figur{e} betoken odde nombre, ou schal do away on of at odde nombre, e quych is significatiue by {a}t figure 1. e quych 1 schall be rekende for 10. Whan {o}u hast take away {a}t 1 out of e nombre {a}t is signifiede by at figur{e}, {o}u schalt medie {a}t at leues ou{er}, &do away at figur{e} at is medied, &sette in his styde halfe of {a}t nombre. Whan {o}u hase so done, {o}u schalt write [*leaf 150b] a figure of 5 ou{er} e next figur{es} hede by-for{e} toward e ry[gh]t side, for at 1, e quych made odd nombr{e}, schall stonde for ten, &5 is halfe of 10; so {o}u most write 5 for his haluendell{e}. lo an Ensampull{e}, 4678. begyn in e ry[gh]t side as {o}u most nedes. medie 8. en {o}u schalt leue 4. do away at 8 & sette {ere} 4. en out of 7. take away 1. e quych makes odde, &sett 5. vpon e next figur{es} hede afor{e} toward e ry[gh]t side, e quych is now 4. but afor{e} it was 8. for at 1 schal be rekenet for 10, of e quych 10, 5 is halfe, as ou knowest wel. Whan {o}u hast us ydo, medye {a}t e quych leues aft{er} e takying{e} away of at at is odde, e quych leuyng{e} schall{e} be 3; do away 6 & sette {er}e 3, &ou schalt haue such a nombre

5 4634.

aft{er} go forth to e next fig{ur}e, &medy at, & worch forth, for it is ly[gh]t ynov[gh]t to e c{er}tayn.

Si v{er}o s{e}c{un}da dat vnu{m}. Illa deleta, sc{ri}bat{ur} cifra; p{ri}ori Tradendo quinque pro denario mediato; Nec cifra sc{ri}batur, nisi dei{n}de fig{ur}a seq{u}at{ur}: Postea p{ro}cedas reliq{ua}s mediando figuras Vt sup{ra} docui, si sint tibi mille figure.

[Sidenote: If the second figure is one, put a cipher, and write five over the next figure. How to halve fourteen.]

Her{e} he putt{es} e 5 case, e quych is [*leaf 151a] is: yf e secunde figur{e} be of 1, as is is here 12, ou schalt do away at 1 & sett {ere} a cifer. & sett 5 ou{er} e next fig{ur}e hede afor{e} toward e ri[gh]t side, as ou diddyst afor{e}; & at 5 schal be haldel of at 1, e quych 1 is rekent for 10. lo an Ensampull{e}, 214. medye 4. {a}t schall{e} be 2. do away 4 & sett {ere} 2. e{n} go forth to e next figur{e}. e quych is bot 1. do away at 1. & sett {ere} a cifer. & set 5 vpon e figur{es} hed afor{e}, e quych is nowe 2, &en ou schalt haue is no{m}b{re}

5 202,

en worch forth to e nex fig{ur}e. And also it is no mayst{er}y yf {ere} come no figur{e} after at on is medyet, {o}u schalt write no 0. ne now[gh]t ellis, but set 5 ou{er} e next fig{ur}e afor{e} toward e ry[gh]t, as us 14. medie 4 then leues 2, do away 4 & sett {ere} 2. en medie 1. e q{ui}ch is rekende for ten, e halue{n}del {ere}-of wel be 5. sett {a}t 5 vpon e hede of {a}t figur{e}, e quych is now 2, &do away {a}t 1, &ou schalt haue is nombre yf {o}u worch wel,

5 2.

vn{de} v{er}sus.

[Headnote: How to prove the Mediation.]

Si mediacio sit b{e}n{e} f{ac}ta p{ro}bar{e} valeb{is} Duplando num{er}u{m} que{m} p{ri}mo di{m}ediasti

[Sidenote: How to prove your mediation. First example. The second. The third example. The fourth example. The fifth example.]

Her{e} he telles e how ou schalt know whe{er} ou hase wel ydo or no. doubul [*leaf 151b] e nombre e quych {o}u hase mediet, and yf {o}u haue wel y-medyt after e dupleacion, ou schalt haue e same nombre at {o}u haddyst in e tabull{e} or {o}u began to medye, as us. The furst ensampull{e} was is. 4. e quych I-mediet was laft2, e whych 2 was write in e place {a}t 4 was write afor{e}. Now doubull{e} at 2, &{o}u schal haue 4, as {o}u hadyst afor{e}. e secunde Ensampull{e} was is, 245. When {o}u haddyst mediet all{e} is nomb{re}, yf ou haue wel ydo ou schalt haue of {a}t mediacion is nombre, 122^w. Now doubull{e} is nombre, &begyn in e lyft side; doubull{e} 1, at schal be 2. do away at 1 & sett {ere} 2. en doubull{e} {a}t o{er} 2 & sett {ere} 4, en doubull{e} at o{er} 2, &at wel be 4. e{n} doubul at merke at stondes for halue on. & at schall{e} be 1. Cast at on to 4, &it schall{e} be 5. do away at 2 & at merke, &sette {ere} 5, &en {o}u schal haue is nombre 245. & is wos e same nombur {a}t {o}u haddyst or {o}u began to medye, as {o}u mayst se yf ou take hede. The nombre e quych ou haddist for an Ensampul in e 3 case of mediacion to be mediet was is 241. whan {o}u haddist medied all{e} is nombur truly [*leaf 152a] by eu{er}y figur{e}, ou schall haue be {a}t mediacion is nombur 120^w. Now dowbul is nomb{ur}, &begyn in e lyft side, as I tolde e in e Craft of duplacion. us doubull{e} e fig{ur}e of 1, at wel be 2. do away at 1 & sett {ere} 2, en doubul e next figur{e} afore, the quych is 2, &at wel be 4; do away 2 & set {ere} 4. en doubul e cifer, & at wel be no[gh]t, for a 0 is no[gh]t. And twyes no[gh]t is but no[gh]t. {ere}for{e} doubul the merke aboue e cifers hede, e quych betokenes e halue{n}del of 1, &at schal be 1. do away e cifer & e merke, &sett {ere} 1, &en {o}u schalt haue is nombur 241. And is same nombur {o}u haddyst afore or {o}u began to medy, & yf {o}u take gode hede. The next ensampul at had in e 4 case of mediacion was is 4678. Whan {o}u hast truly ymedit all{e} is nombur fro e begynnyng{e} to e endyng{e}, {o}u schalt haue of e mediacion is nombur

5 2334.

Now doubul this nombur & begyn in e lyft side, &doubull{e} 2 at schal be 4. do away 2 and sette ere 4; en doubul{e} 3, {a}t wol be 6; do away 3 & sett {ere} 6, en doubul at o{er} 3, &at wel be 6; do away 3 & set {ere} [*leaf 152b] 6, en doubul e 4, at welle be 8; en doubul 5. e quych stondes ou{er} e hed of 4, &at wol be 10; cast 10 to 8, &{a}t schal be 18; do away 4 & at 5, &sett {ere} 8, &sett that 1, e quych is an articul of e Composit e quych is 18, ou{er} e next figur{es} hed toward e lyft side, e quych is 6. drav {a}t 1 to 6, e quych 1 in e dravyng schal be rekente bot for 1, &{a}t 1 & {a}t 6 togedur wel be 7. do away at 6 & at 1. the quych stondes ou{er} his hede, &sett ther 7, & en ou schalt haue is nombur 4678. And is same nombur {o}u hadyst or {o}u began to medye, as {o}u mayst see in e secunde Ensampul at ou had in e 4 case of mediacion, at was is: when {o}u had mediet truly all{e} the nombur, ap{ri}ncipio usque ad fine{m}. {o}u schalt haue of at mediacion is nombur

5 102.

Now doubul 1. at wel be 2. do away 1 & sett {ere} 2. en doubul 0. {a}t will be no[gh]t. {ere}for{e} take e 5, e quych stondes ou{er} e next figur{es} hed, &doubul it, &at wol be 10. do away e 0 at stondes betwene e two fig{u}r{i}s, &sette {ere} in his stid 1, for {a}t 1 now schal stonde in e secunde place, wher{e} he schal betoken 10; en doubul 2, at wol be 4. do away 2 & sett ere 4. & [*leaf 153a] ou schal haue us nombur 214. is is e same nu{m}bur at {o}u hadyst or {o}u began to medye, as {o}u may see. And so do eu{er} mor{e}, yf {o}u wil knowe whe{er} ou hase wel ymedyt or no. .doubull{e} e nu{m}bur at comes aft{er} e mediacioun, &{o}u schal haue e same nombur {a}t {o}u hadyst or {o}u began to medye, yf {o}u haue welle ydo. or els doute e no[gh]t, but yf {o}u haue e same, {o}u hase faylide in {i} Craft.

Sequitur de multiplicatione.

[Headnote: The Craft of Multiplication.]

[Headnote: To write down a Multiplication Sum.]

+Si tu p{er} num{er}u{m} num{er}u{m} vis m{u}ltiplicar{e} Scribe duas q{ua}scu{nque} velis series nu{me}ror{um} Ordo s{er}vet{ur} vt vltima m{u}ltiplicandi Ponat{ur} sup{er} ant{er}iorem multiplicant{is} A leua reliq{u}e sint scripte m{u}ltiplicantes.

[Sidenote: Four things to be known of Multiplication: the first: the second: the third: the fourth. How to set down the sum. Two sorts of Multiplication: mentally, and on paper.]

Her{e} be-gynnes e Chapt{r}e of m{u}ltiplication, in e quych ou most know 4 thynges. Ffirst, qwat is m{u}ltiplicacion. The secunde, how mony cases may hap in multiplicacion. The thryde, how mony rewes of figur{es} {ere} most be. The 4. what is e p{ro}fet of is craft. As for e first, {o}u schal vnd{er}stonde at m{u}ltiplicacion is a bryngyng{e} to-ged{er} of 2 thyng{es} in on nombur, e quych on nombur {con}tynes so mony tymes on, howe [*leaf 153b] mony tymes {ere} ben vnytees in e nowmb{re} of at 2, as twyes 4 is 8. now her{e} ben e 2 nomb{er}s, of e quych too nowmbr{e}s on is betokened be an adu{er}be, e quych is e worde twyes, &is worde thryes, &is worde four{e} sythes,[{9}] [[& is wordes fyue sithe & sex sythes.]] &so furth of such other lyke wordes. And tweyn nombres schal be tokenyde be a nowne, as is worde four{e} showys es tweyn nombres y-broth in-to on hole nombur, at is 8, for twyes 4 is 8, as {o}u wost wel. And es nomb{re} 8 conteynes as oft tymes 4 as {ere} ben vnites in {a}t other nomb{re}, e quych is 2, for in 2 ben 2 vnites, &so oft tymes 4 ben in 8, as {o}u wottys wel. ffor e secu{n}de, {o}u most know at {o}u most haue too rewes of figures. As for e thryde, {o}u most know {a}t 8 man{er} of diu{er}se case may happe in is craft. The p{ro}fet of is Craft is to telle when a nomb{re} is m{u}ltiplyed be a no{er}, qwat co{m}mys {ere}of. fforthermor{e}, as to e sentence of our{e} verse, yf {o}u wel m{u}ltiply a nombur be a-no{er} nomb{ur}, ou schalt write [*leaf 154a] a rewe of figures of what nomb{ur}s so eu{er} {o}u welt, &at schal be called Num{erus} m{u}ltiplicand{us}, Anglice, e nomb{ur} the quych to be m{u}ltiplied. en {o}u schalt write a-nother rewe of figur{e}s, by e quych {o}u schalt m{u}ltiplie the nombre at is to be m{u}ltiplied, of e quych nomb{ur} e furst fig{ur}e schal be write vnd{er} e last figur{e} of e nomb{ur}, e quych is to be m{u}ltiplied. And so write forthe toward e lyft side, as her{e} you mayse,

67324 1234

And is on{e} nomb{ur} schall{e} be called nu{meru}s m{u}ltiplicans. An{gli}ce, e nomb{ur} m{u}ltipliyng{e}, for he schall{e} m{u}ltiply e hyer nounb{ur}, as us on{e} tyme 6. And so forth, as I schal telle the aft{er}warde. And ou schal begyn in e lyft side. ffor-{ere}-more ou schalt vndurstonde at {ere} is two man{ur}s of m{u}ltiplicacion; one ys of e wyrchyng{e} of e boke only in e mynde of a mon. fyrst he teches of e fyrst man{er} of duplacion, e quych is be wyrchyng{e} of tabuls. Aft{er}warde he wol teche on e secunde man{er}. vn{de} v{er}sus.

[Headnote: To multiply one Digit by another.]

In digitu{m} cures digitu{m} si duc{er}e ma{i}or [*leaf 154b.] P{er} qua{n}tu{m} distat a denis respice debes Namq{ue} suo decuplo totiens deler{e} mi{n}ore{m} Sitq{ue} tibi nu{meru}s veniens exinde patebit.

[Sidenote: How to multiply two digits. Subtract the greater from ten; take the less so many times from ten times itself. Example.]

Her{e} he teches a rewle, how {o}u schalt fynde e nounb{r}e at comes by e m{u}ltiplicacion of a digit be ano{er}. loke how mony [vny]tes ben. bytwene e mor{e} digit and 10. And reken ten for on vnite. And so oft do away e lasse nounbre out of his owne decuple, at is to say, fro at nounb{r}e at is ten tymes so mych is e nounb{re} {a}t comes of e m{u}ltiplicacion. As yf {o}u wol m{u}ltiply 2 be 4. loke how mony vnitees ben by-twene e quych is e mor{e} nounb{re}, &be-twene ten. C{er}ten {ere} wel be vj vnitees by-twene 4 & ten. yf {o}u reken {ere} w{i}t{h} e ten e vnite, as ou may se. so mony tymes take 2. out of his decuple, e quych is 20. for 20 is e decuple of 2, 10 is e decuple of 1, 30 is e decuple of 3, 40 is e decuple of 4, And e o{er} digetes til {o}u come to ten; & whan {o}u hast y-take so mony tymes 2 out of twenty, e quych is sex tymes, {o}u schal leue 8 as {o}u wost wel, for 6 times 2 is twelue. take [1]2 out of twenty, &{ere} schal leue 8. bot yf bothe e digett{es} [*leaf 155a] ben y-lyech mych as her{e}. 222 or too tymes twenty, en it is no fors quych of hem tweyn {o}u take out of here decuple. als mony tymes as {a}t is fro 10. but neu{er}-e-lesse, yf {o}u haue hast to worch, {o}u schalt haue her{e} a tabul of figures, wher{e}-by {o}u schalt se a-nonn ryght what is e nounbre {a}t comes of e multiplicacion of 2 digittes. us {o}u schalt worch in is fig{ur}e.

[Sidenote: Better use this table, though. How to use it. The way to use the Multiplication table.]

1 - 2 4 3 6 9 - 4 8 12 16 5 10 15 20 25 - 6 12 18 24 30 36 7 14 21 28 35 42 49 - 8 16 24 32 40 48 56 64 9 18 27 36 45 54 63 72 81 1 2 3 4 5 6 7 8 9

yf e fig{ur}e, e quych schall{e} be m{u}ltiplied, be euen{e} as mych as e diget be, e quych at o{er} figur{e} schal be m{u}ltiplied, as two tymes twayn, or thre tymes 3. or sych other. loke qwer{e} at fig{ur}e sittes in e lyft side of e t{ri}angle, &loke qwer{e} e diget sittes in e ne{er} most rewe of e triangle. & go fro hym vpwarde in e same rewe, e quych rewe gose vpwarde til {o}u come agaynes e o{er} digette at sittes in e lyft side of e t{ri}angle. And at nounbre, e quych ou [*leaf 155b] fyn[*]des {ere} is e nounbre at comes of the m{u}ltiplicacion of e 2 digittes, as yf ou wold wete qwat is 2 tymes 2. loke quer{e} sittes 2 in e lyft side i{n} e first rewe, he sittes next 1 in e lyft side al on hye, as {o}u may se; e[{n}] loke qwer{e} sittes 2 in e lowyst rewe of e t{ri}angle, &go fro hym vpwarde in e same rewe tyll{e} ou come a-[gh]enenes 2 in e hyer place, &er ou schalt fynd ywrite 4, & at is e nounb{r}e at comes of e multiplicacion of two tymes tweyn is 4, as ow wotest well{e}. yf e diget. the quych is m{u}ltiplied, be mor{e} an e o{er}, ou schalt loke qwer{e} e mor{e} diget sittes in e lowest rewe of e t{ri}angle, &go vpwarde in e same rewe tyl[{10}] {o}u come a-nendes e lasse diget in the lyft side. And {ere} {o}u schalt fynde e no{m}b{r}e at comes of e m{u}ltiplicacion; but {o}u schalt vnd{er}stonde at is rewle, e quych is in is v{er}se. In digitu{m} cures, &c., no{er} is t{ri}angle schall{e} not s{er}ue, bot to fynde e nounbres {a}t comes of the m{u}ltiplicacion at comes of 2 articuls or {com}posites, e nedes no craft but yf ou wolt m{u}ltiply in i mynde. And [*leaf 156a] ere-to ou schalt haue a craft aft{er}warde, for ou schall wyrch w{i}t{h} digettes in e tables, as ou schalt know aft{er}warde. v{er}sus.

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