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(Extra Scats, Jto. cxviii. 1922 (for 1916).
Price 15s.
•4r**"W
EDITED WITH INTRODUCTION
BY
ROBERT STEELE
LONDON: PUBLISHED FOE THE EARLY ENGLISH TEXT SOCIETY
BY HUMPHREY MILFORD, OXFORD UNIVERSITY PRESS,
AMEN CORNER, E.C. 4. 1922.
\\v
(gjlra Strits, cxvin.
in Cheat Britain bv Richard Clav & Sons, Limitkd, bung \y, suffolk.
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 contem- plated 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 compu- tation, not only in the calculation of astronomical tables but in their use, a knowledge 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. (2053) f. 146-154 (15th c.) inc. "Of augrym ther be IX figures in numbray ..." A mere un- finished fragment, only getting as far as Duplation.
(2) Can.b. Univ. LI. IV. 14 (III.) f. 121-142 (15th c.) inc. "Al maner of thyngis that prosedeth ffro the f rist begynnyng . . ."
(3) Fragmentary passages or diagrams in Sloane 213 f. 120-3 (a fourteenth-century counting board), Egerton 2852 f. 5-13, Karl. 218 f. 117 and
(4) Th.- two MSS. here printed; Eg. 2622 f. 130 and Ashmole 396 f. 48. All of these, as the language shows, are of the fifteenth century.
The Ckafte of Nombrynge is one of a large number of scientific treatises, mostly in Latin, bound up together as Egerton MS. 2G22 in the British Museum Library. It measures 7" X 5", 29-30 lines to the page, in a rough hand. The English is X.K. Midland in dialect. It is a translation and amplification of one of the numerous glosses on the de algorismo of Alexander de Villa Lei (c. 1220), such as that of
vi Introduction
Thomas of Newmarket contained in the British Museum MS. Reg. 12, E. 1. A fragment of another translation of the same gloss was printed by Ilalliwell in his Ram Mathematica (1835) p. 29.* It corresponds, as far as p. 71, 1. 2, roughly to p. 3 of our version, and from thence to the end p. 2, 11. 10-40.
The Art of Nombbyng is cue of the treatises bound up in the Bodleian MS. Ashinole 396. It measures 1H-" x 17f" 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 nu- n)< ranch attributed to John of I [olywood (Sacrobosco) and the translator had obviously a poor MS. before him. The de arte numerandi was printed in 1488, 1490 («.«'.), 1501, 1503, 1510, 1517, 1521, 1522, 1523, 1582, and by Ilalliwell 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 fiir die Geschichte der Naturwissensch often und der Technilc, 1909, and invited the scientific world to take up the "not unpleasant task" of editing it.
AcCOMPTYNGE r.v 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 earliesl 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 doubl in my mind that this method of performing the simple operations of arithmetic is much older than ;m\ 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, a system of digital numeration which seems of great antiquity ami almost world-wide extension.
Alter the fragmenl already referred to, I print 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 Ilalliwell in Rara Mathematica, but I have added a number of stanzas from
• II dliwell printed the two sides of his leaf in the wrong order. This ami seme obvious errors of transcription 'ferye' for 'fertile,' ' 1> st ' for 'left.' etc., Ii i\ e not been correi ted in the reprint on pp. 70 71.
Introduction vii
various manuscripts, selecting various readings on the principle that the verses were made to scan, aided by the advice of my friend Mr. Vernon Kendall, 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 Loethius 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 Koinans 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,* were unable to devise methods for its use.
The Phe Medieval Instruments Used in Calculation. The following are known : —
(1) A Hat 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 oif.
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, Gth, and 9th being marked with a cross. Another section consists of live parallel lines, and there are three
Fur Egyptian use see Herodotus, ii. 3*5 , Plate, '/- Legibus, VII.
\iii Introduction
rows of arithmetical symbols. This hoard 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 tin' hand. Counters {calculi, ovlapilli), which were kept in boxes {loctdi), were used in calculation, as we Lain from Horace's schoolboys (Sat. 1. vi. 74). For the sand see Persius I. 131, "Nee qui abaco numeros et secto in pulvero metas scit risisse," Apul. Apolog. 10 (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 fun). The counters were madf of various materials, ivory principally, " Adeo nulla uncia nobis est eboris, etc." (Juv. XL 131), sometimes of precious metals, " Pro calculis albis et nigris aureos argenteosque habebat denarios" (Pet. Arb. Satyricon, 3.1).
There are, however, still in existence four Roman counting boards of a kind which does not appear to come into literature. A typical one is of the third class. Ifc consists of a number of transverse wires, broken at (he middle. On the left hand portion four heads are strung, on the right one (or two). The left hand heads signify units, the light h;ind one five units. Thus any number up to nine can be represented'. This instrument is in all essentials the same as tin' Swanpan or Abacus in use Ihroughoul 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 live 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 (liftli century), at tl nd of lib. II. of his Geometria gives us a figure of an abacus
.if the second class with a set of counters arranged within it. It has, however, been contended with greal pi bability that the whole pa age 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
Introduction ix
nine numbers, known as the Boethian "apices" or "notae" (from whence our word "notation"). To these we shall return later on.
The Abacists.
It would seem probable that writers on the calendar like Bede (a.d. 721) and Ilelpericus (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 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 the works of Bede in the Basle (15G3) edition of his works, I. 159, in a somewhat enlarged form. Another tenth century treatise is that of Abbo of Fleury (c. 98K), 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 rides 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
x Introduction
of the calendar and the assignments of the royal exchequer. In England, for example, when tin- 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]"* Exchequer methods up to the sixteenth century were founded on the abacus, though when we have details later on, a different and simpler form was n^vi\.
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 arc 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 he remembered that a figure asterisked represents a counter taken from the board. A zero is obviously not needed, and the result may he written down in words.
[a) Multiplication. 4600 x 23.
|
Thousands |
- |
|||||
|
' s |
||||||
|
c |
tr. a V |
a 3 pq |
£ |
1 |
||
|
4 |
• i |
Multiplicand. |
||||
|
1 |
8 |
600 .:. |
||||
|
1 |
4000 '.. |
|||||
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1 |
2 |
600 20. |
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|
8 |
I |
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1 |
5 |
8 |
Total product |
|||
|
•J |
a |
Multiplier. |
' S«m .,n tin.. Dr. Poole, Tfa ExcJieqver in tht Twelfth Century, Chap. 111., and Haskins, Eng. Hist, Review, 27, 101. The hidage of Essex in 1130 way 2364 hi les
Introduction
XI
(b) Division: direct. 100,000 -r 20,023. Here each counter in turn is a separate divisor.
|
Thousands |
|||||||
|
a. |
T. |
u. |
II. |
1 |
u. |
||
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2 |
2 |
3 |
Divisors. |
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2 |
Place greatest divisor to right o |
' dividend |
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1 |
2 |
1 9 |
Dividend. Remainder. |
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1 |
9 |
Another form of same. |
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8 |
Product of 1st Quotient and 20. |
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1 |
9 |
9 |
■1 |
Remainder. |
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1 2 |
Product of 1st Quotient and 3. |
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1 |
9 |
9 |
S |
Final remainder. |
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4 |
Quotient. |
(c) Division i;v Differences. 900 — 8. Here we divide by (10-2).
Difference. Divisor. Dividend.
Product of difference by 1st Quotient (it). Produci of different by 2nd Quotient (1). Sum of 8 and 2.
Product of difference by 3rd Quotient (1). Produci of difference by 1th Quot. (2). Remainder. 1th Quotient. 3rd Quotient. 2nd Quotient. 1st Quotient,
Quotient. Total of all four.) • I Ue i figures ir< i emo\ ed al tlie next stuje
|
II. |
T. |
U. |
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•_' |
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*1 |
8 ■1 |
8 |
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*1 |
2 |
1 |
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1 1 9 |
•J |
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1 |
1 |
■■ |
Xll
Introduction
Division. 7800 -f- 166.
|
Thousands |
|||||
|
II |
T. |
r. |
ii. |
T. |
U. |
|
3 |
•1 |
||||
|
i |
(i |
G |
|||
|
*r |
8 |
||||
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1 |
1 9 |
2 |
|||
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*■> |
8 3 |
2 1 |
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|
*i |
1 1 |
6 2 5 |
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*3 1 |
3 3 |
4 |
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1 |
6 1 3 |
1 I 5 |
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|
1 |
6 |
Differences (making 200 trial divisor).
Divisors.
Dividends.
Remainder ofgreatesl dividend.
Product of 1st difference (4) by 1st Quotient (3).
Product of 2nd difference (3) by 1st Quotient (3).
New dividends.
Product of 1 t and 2nd difference by 2nd Quotient (1).
New di\ idends.
Product oflsl difference l>\ 3rd Quotieni (5).
Product of 2nd difference by 3rd Quotient 5),
New dividends.
Remainder of greatest dividend.
Producf of 1st and 2nd difference by 1th Quotient (1).
Remainder (less than divisor).
4th Quotient.
3rd Quotient.
2nd Quotient.
1st Quotient.
Quotient.
• 1 1, removed at the m \t step.
Introdtiction
Xlll
Division. S000 H- 606.
|
Thousands i |
|||||
|
!: |
T. |
U. *8 1 *1 *1 |
H. 6 9 9 3 3 3 7 »■• 1 |
T. 9 4 4 9 3 9 2 2 1 1 |
0". 1 6 4 4 4 8 6 2 1 1 1 3 |
Difference (making 700 trial divisor).
Difference.
Divisors.
Dividend.
Remainder of dividend.
Product of difference 1 and 2 with 1st Quotient (1).
New dividends.
Remainder of greatest dividend.
Product of difference 1 and 2 with 2nd Quotient (1).
New dividends.
Remainder of greatest dividend.
Product of difference 1 and 2 with 3rd Quotient (1).
New dividends.
Product of divisors by 4th Quotient (1).
Remainder.
Hli Quotient.
3rd Quotient.
2nd Quotient.
1st Quotient.
Quotient.
These figures are removed (it the next step.
The chief Abacists arc Gerbert (tenth century), Abbo, and Eer- mannus Contractus (1051), who arc credited with the revival of the art, ISernelinus, Gerland, and Badulphus of Laon (twelfth century). We know as English Abacists, Hubert, bishop of Hereford, 1095, " abacum et lunarem compotum et celestium cursum astrorum rimatus," Turchillus Compotista (Thurkil), and through him of Guilielmus B. . . . "the best of living computers," Gislebert, and Simonus de Rotellis (Simon of the Rolls). They flourished most probably in the
xiv IntrodiLction
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 Ahacists.
Finally, the term Ahacists came to he applied to computers by manual arithmetic. A MS. Algorithm of the thirteenth century (SI. 3281, f. 6, h), contains the following passage: "Est et alius modus secundum operatores sive practicos, quorum unus appellator Abacus; et modus ejus est in computando per digitos et junctura manuum, et isle 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. 1 V ., f. 5 1 vo). "Si numcrus multiplicat alium numerum auferatur differentia majoris a minore, et per residuum multiplicetur articulus, et una differentia per aliam, et summa proveniet." Example, 8 x 7. The difference of 8 is 2, of 7 is 3, the next article being 10; 7-2 is 5. 5 x 10 = 50; 2 X 3 = 6. 50 + G = 5G answer. The rule will hold in such cases as 17 X 15 where the article next higher is the same for both, i.e., 20 ; but in such a case as 17 x 9 the difference for each number must be taken from the higher article, i.e., the difference of 0 will be 11.
The Alqoeists.
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 A I Khowarazmi in the early ninth century, " de numeris Endorum," beginning in its Latin form '• l>i\it Algorismi. . . ." The translation, of which only one ISIS, 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 (1111-1187); the number of importanl works that were not translated more than once from the Arabic decreases every year with our know- ledge of medieval texts. A few lines of this translation, as copied by Ealliwell, are given on p. 72. note 2. Another translation still seems to have been made by Johannes Hispalensis.
Algorism is distinguished from Abacist compulation by recognis- ing seven rules, Addition, Subtraction, Duplation, Mediation, Multi- plication, Division, and Extraction of Roots, to which were afterwards
Introduction xv
added Numeration and Progression. It is farther distinguished by the use of the zero, which enabled the computer to dispense with the columns of tbe 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 un- known.* 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.
The number of Latin Algorisms still in manuscript is compara- tively large, but we are here only concerned with tAvo — ah Algorism in prose attributed to Sacrobosco (John of Holy wood) 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 ; Sacro- bosco died in 1256 and quotes the verse Algorism. Several com- mentaries 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 Kara Mathematica, is adapted for use on a board covered with sand, a method almost universal in the thirteenth century, as some passages in the algorism of that period already quoted show : " Est et alius modus qui utitur apud Indos, et doctor hujusmodi ipsos "erat quidem nomine Algus. Et modus suus erat in computando per quasdam figuras scribendo in pulvere. . . ." " Si voluerimus depingere in pulvere predictos digitos secundum consuetiulinem 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
ft Slates are mentioned by Chancer, and soon after (1410) Prosdocimo do Beldamandi speaks of the use of a " lapis " for making notes on by calculators.
xvi Introduction
rather in the range of its problems and the methods of attack than
in mere methods of calculation, which arc of its period. Its sole interest as regards the presenl work is that Leonard] makes use of the digital signs described in Record's treatise on The arte of nombryngi by the Imnd 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 1 1 1 V and Villi, and on the other by carefully engraved digital signs for 8 and 9. Sixteen seems to have been the number of a complete set. These counters were either used in games or for the counting board, and the Museum ones, coming from the Hamilton collection, are undoubtedly not later than the first century. Frohner has published in the Zeitschrift des Miinchener Alterthumsvereins a set, almost complete, of them with a Byzantine treatise ; a Latin treatise is printed among Bede's works. The use of this method is universal through the Last, 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 treat already quoted (Sloane 32S1) it is even called the Abacus, perhaps a memory of Fibonacci's work.
Methods ofcalculation 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;
l! may now be regarded as proved by Bubnoy that our present numerals are derived from Greek sources through the so-called Boethian "apices," which arc first found in late tenth century manuscripts. Thai they were not derived directly from the Arabic seems certain Prom 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 Basing- stoke in the thirteenth century, and is figured by Matthew Paris (V. 285); but this form had no success. The date' of t lie intro- duction of the zero has been hotly debated, but it seems otmous that the twelfth century Latin translators from the Arabic were
Introduction xvii
perfectly •well acquainted with the system they met in their Arahic 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
c c c
Mij. lviii., the second Mij. lxi., the third Mij. 63, the fourth 1264, and the fifth 126G. Another example is given in a set of astronomical tables for 1269 in a manuscript of Eoger 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 " Algorithmic 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 extractiono ad algoritmum integrorum reser- vato, cujus species per ciffralea figuras ostenduntur ubi ad plenum de hac tractabitur." The invention of the art is there attributed to Api>ulegius 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" econo- mises by dividing the line into two parts, the beads on one side representing five times the value of those on the other. The "Swan- pan" has usually many more lines than the "stchota," allowing for more extended calculations, see Tylor, Anthropology (1892), p. 314.
xviii Introduction
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 Ex- chequer. A counting board for merchants' use is printed by Ilalliwell 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.
CIjc Earliest Hritljmctics in ^nglisij.
NOMBRTNGE.
£{k Oirnftc of ilombrnngc.
H
Egerton 2622.
Ec algorism//- ars presens elicit///-; in qua • ieafi36a.
Talib//- inclor///;/ fruimwr bis qui//q//' figuris. This boke is called be boke of algorym, or Augryra after lewder a derivation
. . . *1 of Algorism.
4 vse. Ana bis boke tretys be Craft oi Nombryng, be quych crafte is called also Algorym. Ther was a kyng of Inde, be quich heyth Algor, & lie made bis craft. And aft/'/' his name he called hit algorym ; or els anober cause is quy it is called Algorym, for be
8 latyn word of hit s. Algorismws comes of AlgoSj .^rece, quid est Another
.. c . .. i-i -7 i- t derivation of
ars, latme, cratt on) euglis, ami rides, quid est Humerus, latino, Atheword. nombwr on englys, inde dicitux Algorisniws per addicaonew hnius sillabe xa.us & subtracc&'oneni d A' e, qwasi ars numerandi. IT fforther-
12 more y most vndi'rstonde b'/t in bis craft ben vsid teen figurys, as here bene writen for ensampul, 098765432 1. IF Expone be too versus afore: this present craft ys called Algorismws, in be quych we vse teen signys of Inde. Questio. H Why ten) fyguris
16 of Inde] Solucio. for as I haue sayd afore bai were fonde fyrsl in Inde of a kynge of bat Cuntre, b//t was called Algor.
H Prima sigm/Zcat xmum ; duo vero sc////da: venusim
H Tercia sigtiijicat tria; sic procede sinistre.
20 IT Doiv c ad extremal venias, que cifra voca////-.
H Cap/7/////m primum de significac/o//e figurar ////'. exPo»«;o
x . . . . . verius.
In bis verse is notinde be signincacton of bese Bguris. And bus expone the vers*'. J;e firsl signifiyth one, be secuwde signi2fiyth *ieafi366. 21 tweyne, be thryd signifiyth thre, & the fourte signifiyth I. * And ^f^cTof81 so forthe towarde be Iyfl syde of be tabid or of be boke bat be the fi§uies- figures bene writene in, til bat bou come to the last figure, bat is
4 Notation and Numeration.
called a cifre. 1 Questio. In quych syde sittes be first figure? Sohiczo, forsothe loke quich figure is first in be ry$t side of be bok or of be tabul, & pat .same is be first figure, for bou sclial write Which flBuro bakeward, as here, 3. 2. 6. 1. 1. 2. 5. The figure of 5. was first 4
is read first. .
write, & he is be first, for he sittes on) be rijt syde. And the figure of 3 is last. 11 Neuer-be-les wen he say- 1 IV/ma sigu^/icat vnu/// Are., bat is to say, pa first betokenes one, pa secuwde. 2. & fore-J>er-niore, lie vndirstondes no^t of be first figure of euery rew. 8 II But he vndi'rstondes be first figure b"t i.s in be nonibur of pe forsayd teen figuris, pa quych is one of bese. 1. And pa secunde 2. & so forth. venutVn 1F Quelib/ 7 illar///// si pr////o limite ponas, 12
11 Simpliciter se significat: si vero see////do,
Se decies : sursu/// y//ocedas m/dtiplicando. IT Na///q//c figura seque//s q/?'n//uis signat decies plus. IT Ipsa locata loco quam sign///Vat p'/////ente. 16
Expositio [in IT Expone bis verse bus. Euery of bese figuris bitokens hym selfe & no more, yf he stoude in be first place of pa rewele this Anexpiana- wordc Simpliciter in bat verse it is no more to say hut bat, &
linn ot the ' i
principles of no more. IT If it stonde in the secmide place of be rewle, he 20
nutation. _ '
betokens ten/.' tymes hym selfe, as bis figure 2 here 20 tokens
i leaf is7 a. ten tyme hym selfe, lbat is twenty, for he hym selfe betokenes
tweywe, & ten tymes twene is twenty. And for he stondis on)
be lyft side & in be secuwde place, he betokens ten tyme hywi 21
selfe. And so ge forth. • ffor euery figwe, & he stonde aft///'
a-nob'/- toward the lyft side, he schal betokene ten tymes as mich
more as he schul betoken & he stode in be place pert bat be
An example : figure a-f< ire hym stondes. loo an ensampulle. 9. 6. 3. I. J3e 28
units, figure of 1. pat hase bis schape \. betokens bol liymselfe, for he
stondes in be first place. The figure of •">. bal hase bis schape
ten?, }. betokens ten tymes mor< ben he schuld & he stode pen pat
pa figure <>f I. stondes, bat is thretty. The figur< of 6, b"t hase 32 bis schape 6, betokens ten tymes nior< ban he schuld & he stode pere as be figure of }. stondes, for pert he schuld tokynt bol hundreds, sexty, & now he betokens ten tymes more, bat is sex hundryth.
The figurt of 9. bal hase bis schape 9. betokens ten tymes more 36 bane he schuld & he stode in be place peri be fig?<r< of sex stondes, for ben he schuld betoken to 9. hundryth, and in be place peri he thousands, stondes now he betokens 9. bousande. Al be hole nombur is !>
thousande sex hundryth & fourt & thretty. H fforthermore, when 40
The Tin'" Kinds of Numbers. 5
bou schalt rede a nombzir of figure, bou sclialt begyne at Be last How to read
' ' r . OJ r the number.
figure in the lyft side, & rede so forth to be ri^t side as here 9. G. 3. 4. Thou schal begyn to rede at be figure of 9. & rede forth i bus. 9. Hhousand sex hun'dryth thritty & fuiire. But when bou Uearm&. schalle write, pun schalt be-gynne to write at be ry^t side. 1f Nil cifra significat &ed dat signage sequenti. Expone bis v< rse. A cifre tokens 1103 1, hot he makes be figure The meaning 8 to betoken bat comes aftwr hym more ban he schuld A: he were the cipher. away, as pus \cf>. here pe figure of one tokens ten, & yf pe cifre were away'2 & no figure by-fore hym he schuld token hot one, for ban he schwZd stonde in be first place. If And pe cifre tokens 12 nothyng hym selfe. for al pe nombur of pe ylke too 'figures is bot ten. IF Questio. Why says he fat a cifre makys a figure to signifye (tyf) more &e. H I speke for Jns worde significatyf, ffor sothe it may happe aft?//' a cifre schuld come a-nobwr cifre, as bus 2<p(p. And 16 }et pe secunde cifre shuld token neuer pe more excep he schuld kepe pe order of pe place, and a cifre is no figure significatyf. IF Quam precedentes plus ultima significabit / Expone bis verse bus. be last figure schal token more ban alle peiast
1 ' ' ° ' figure means
20 be ober afore, thoust bere were a hundrvth thousant figures afore, [pore than ail
i I ' > > - ° ' the others,
as bus,- 16798. be last Rgure bat is 1. betokens ten thousant. And *h';^,;;/ alle be ober figures ben bot betokene bot sex thousant seuyne value- hzmdryth nynty & 8. H And ten thousant is more ben alle bat 24 noinli///', ergo be last figure tokens more ban all be nombwr afore.
31I Post p/> dicta scias breuit'r q^oo" tres num< >roium 3 ieafi38«.
Distincte species sunt ; nam quidam digiti sunt ; Articuli quidam ; quidam quoque compositi sunt. 28 IF Capit/////m 2m tie t/Vplice divisione nnmerorum.
IF The auctor of bis tretis departys bis worde a nombur into 3 partes. Some nombur is called digitus latine, a digit in englys. Digits. SoTTime nombur is called articulus latine. An Articul in englys. Articles. 32 Some nombur is called a composyl in englys. 1F Expone bis verse. Composites, know Jjou aftur be forsayd rewles pnt I sayd afore, fat bere ben thre spices of nombur. Oone is a digit, Anober is an Articul, & be tober a Composyt. versus. 36 IF Sunt digiti num< / i qui citra denariu/// s//nt.
IF Here he telles qwat is a digit, Expone versus sir. Nomburs what are digitus bene alle nomburs bat hen wit/j-inne ten, as nyne, 8. 7. 6. 5. 4. 3. 2. 1.
- In MS ' awiy."
6 Digits, Articles, and Composites.
If Articupli clecupli degito////// ; compositi zicaX Illi qui constant ex articulis degitisque. 11 Heir he telles what is a composyl and what is ane articul. What aw Expone sic versus. H Articulis ben1 all- bat may be deuidyt in- 4 to nombwrs of ten & nothynge leue oner, as twenty, thretty, fourtj , a hundryth, a thousand, & such oper, ffior twenty may be departyt in-to 2 nombzws of ten, fforty in to foure nombwrs of ten, & so forth. ! ieafi386, 2Compositys beiO nomlw//'s jjat bene coniponyt of a digyt & of an 8 niunbers articulle as fouretene, fyftene, sextene, & such o]>er. (fortene is posites?; cowponyd of foure bat is a digit & of ten J>at is an articulle. ffiftene is componyd of 5 & ten, & so of all o\>er, whal J>at }>ai ben. Short-lycb euery nomhwr pat be-gynnes witli a digit & endyth in a 12 articulle is a< com posy t, as fortene bygennynge by four< J>at is a digit, & endes in ten.
II Ergo, pyoposito nume7'0 tibi scribere, p/7mo
Respicias quid sit Humerus; si digitus sit 1G
P/v'mo scribe loco digitus, si compositus sit Py/mo scribe loco digitus post articulii/// ; sic. How to write IF here he telles how Jjou schalt wyrch whan pmi schalt write a
a number, , . , „ „ . , ,. .
nombwr. Expone vei'swm sic, & rac mxta expone?ins sentenciam ; 20 whan Jv'ii hast a nombzir to write, loke fyrsl what nianer nombi»' it ys put pou schalt write, whether it be a digit or a composit or an e n is a Articul. H If he be a digit, write a digit, as vf it be seuen, write
digit; . ' '
seuen & write pal digit in pe first place toward be ryghl side. H it 24 if it is a be a composyt, write be digit of be composit in be first place &
composite. . .. ,
write be articul of bat digit in be secunde plan' next toward be 1 v ft side. As yf bnii schal write sex & twenty, write }v digit of be nombwr in be first place bat is sex, and write be articul next after 28 but is twenty, as bus 26. But whan b<<u schalt sowno or speke s leaf 139 a. :i<>r rede an Composyt bou schalt lirst sowne be articul & aft///- be How to read <jigit, as j,,m Seyst- by be comyne speche, Sex & twenty & uoujt
t wenty & sex. versus. 32
II ArticuL/s si sit, in p/7mo limite cifram, Articulu/// /■- ro reliq///s insc/ /be figures.
How to write H Here he tells how JJOU schal write when Jie nonibre p<d b.<u
hase to write is an Articul. Expone versus sic & fac secundum. 36 sentenciam. [fe be nombw?' pa\ bou hast write be an Articul. write ien», first a cit'i'e & after be cifer write an Articulle bus. --/.. fforther
more bou schalt vndzVstonde yf bou haue an Articul. loke how 1 • ben ' repeated in MS.
The Seven Rules of Arithmetic. 7
mych he is, yf he be w/t//-ynno an hundryth, bou schalt write bol one cif re, afore, as liere .9^. It' be articulle be by hym-silfe & be hundreds, an hundrid euene, ben schal bou write. 1. & 2 cifers afore, bat he 4 may stonde in be fchryd place, for euery figure in be tliryd place schal token a hundrid tymes hvm selfe. If be articul be a thousant thensands,
' O.C.
or thousandes1 and he stonde by lay??? selfe, write afore 3 cifers & > for]) of al ober. 8 IT Quolib< -/ in numcro, si par sit pr/ma figura,
Par erit & to/"m, quicquid sibi coni&miatur ; Impar si fu- /it, totu/// tu//c fiet e£ impar. f[ Here lie teches a gcneralle rewle bat vf be first fi^wre in be To ten "»
° . even "umber
12 rewle of figwres token a noinbwr bat is euene al \>a\ nonibMr of tigurys in bat rewle schal be euene, as here bou may see 6. 7. 3. 5. 4. Coniputa ov pn/ba. * If be iirst -lii:><ve token an nombwr bat is ode, *ieafi396. alle bat nonibw in bat rewle schalle be ode, as her< 5 6 7 8 6 7. "l;1,1°ai1- 16 Computa & proba. versus.
H Septet su//t partes, no// plwres, istius artis ; 11 Adders, subtroheiv, duplare, dimidiare, Sextaq^' diuidere, sed qui//ta mtdtiplicare ; 20 Radices extrahere pws septi///a dicitur esse.
IF Here telles bat ber ben) .7. spices or partes of bis craft. The Theseven first is called addicion, be secunde is called subtraccion. The tliryd is called duplacion. The 4. is called dimydicion. The 5. is called 24 uwdtiplicacioii. The G is called diuision. The 7. is called extraccion of be Rote. What all bese spices bene hit schalle be tolde singilla- tim in here caput'ule.
II Subt/^his aut addis a dext/7s vel mediabis: 28 Thou schal be-gynne in be rv-ht side of be boke or of n tabul. Add, sub-
. Hact, or
loke were \»>n wnl be-gynne to write latyn or englys in a boke, & halve, from
r OJ Jo. . rjght t0 |e,t
jwt schalle 1).' called be lyft side of the boke, bat bou writes! toward
bat side schal be called be ryght side of be boke. Versus. 32 A leua dupla, diuide, m/dtiplica.
Here he tidies be in quych side of be boke or of be t ibid bou
schalle be-gyn* to wyrch duplacion, diuision, and rnwltiplicacioii.
Thou schal begyne to worch in be lyft side of be boke or of be Multiply or 36 tabul, but yn what wyse bou schal wyrch in hym dicetur singil- urt to right.
latim in seque/>tib//v c&jiitulis et de vtilitate cmuslibet bxUs &
sic Completur 3prohemiw?H & sequit?«r tractates & p/7mo de arte 3 leaf ho
addictonis que prmia ars est in ordine.
1 III MS. ' ili:;, 1- in ll .'
The Craft of Addition.
Four tilings must be known :
what il is;
how many rows of
figures ;
how many cases ;
what is its result.
How to set down the sum.
Add the first figures ;
mil out the top figure ;
write the result in its phi 18
Alddeiv si xixxmerQ nuimru/// vis, ordine tali J Incipe; scribe duas p?'*mo series Rumeroium P/ /ma/// sub p/7ma recte pone//do figuraw, Et sic de reliq///s facias, si sint tibi plures. 4
U Here by-gynnes be craft of Addicion. In bis craft bmi most knowe foure thynges. fl Fyrst pint most know what is addicion. Next b/m most know how mony rewles of figurys boii most haue. 1i Xcxt b<m most know how mony diuers casys happes in bis craft 8 of addicion. n1 And next qwat is be profet of bis craft. II A- for he first bou most know bat addicion is a castyng to-gedw of twob nomburys in-to om? nombre. As yf I aske qwat is twene & thre. Jjou wyl cast bese twene nombres to-gedw & say b<>t it is fyne. 12 * As for be secunde bou most know jwt boii schalle haue tweyne rewes of figures, one vndur a-notKr, as her/- b/<u mayst se. 1034 11 As for be thryd bou most know pot there beu foure diuerse -1,;s- cases. As for be forthe bou must know \>at be profet of bis craft is 16 to telle what is be hole nombwr bat comes of diuerse nomburis. Now as to be texts of oure verse, he teches there how bou schal worch in bis craft, H He says yf b<<u wilt cast one nombwr to anober nombur, bou most by-gynne on bis wyse. H ffyrst write 20 Hwo rewes of figuris & nombris so bat b<m write be first figure of be liver nombur euene vnd/V the first figure of be nether nonibwr, And be secunde of be nether nombur euene vnd/V be secunde of be hyer, ,V s.i forthe of euery figure of both be rewesas bou mayst se i-:'« 24 f Inde duas adde p/ /mas hac condic/one :
Si digitus crescat ex addL ne prior/;/// ;
P/ /mo scribe loco digitus, quicu//q//' sit ille. •li Here he teches what bou schalt lo when b'"i hast write too 28 rewes of figuris on ruder an-ober, as I sayd be-fore. If He says b<<u schalt take be first figure of be heyer nombr< & be fyrst figur< of be ueber nombre, & cast hem to-geder ■ 1 bis condicion). Thou schal hike qweber be noniber bat comys bere-of be a digit or no. 32 f If he be a digit bou schalt do away be first figure of be hyei nombre, and write berein his stede bal he kode [nne be digit, bat e.iUMs of be ylke 2 figs/res, A: so wrich forth on) ober figures yf ber< be ony moo, til bou come to be ende toward be lyft side. And 36 lede be aether Bgwe stonde still euer-more til bou haue ydo. ffor berc-by bou schal wyte wheber bou hast don< v*e\ or no, as I schal tell be afterward in be ende of bis Chapter. U And loke allgate J at bnii be gynne to worch in bis Craft of Addi'2cioh) in be ryjt side, 40
The Cases of the Craft of Addition. 9
here is an ensampul of bis case 1234 Caste 2 to foure & bat wel be Here is an
* 2142 example.
sex, do away 4. A' write in be ~ " same place be figwre of sex.
U And lete be figure of 2 in be nether rewe stonde stil. When
■1 bou hast do so, cast 3 & i to-gedw?1 and bat wel he seuen bat is
a digit. l>n away be 3, & set bere seuen), and lete be neber figure
stonde stille, & so worch forth bakward til bou hast ydo all to-geder.
Et si composite, in limite scribe seque?*te
8 Articulum, p//mo digitum; quia sic iubet ordo.
IT Here is be secunde case put may happe in bis craft. And be case is bis, yf of be easting of 2 nomburis to-ged< /■, as of be figure of suppose it is
r > J r o o > f o a Composite,
be hyer rewe & of be figure of be neber rewe come a Composyt, how set down 12 schalt bou worch. bus bou sclialt worch. Thou shalt do away be jnd carry
i ' i •> i the tens.
figwre of be hyer noinber bat was east to be figure of be neb'/' nomber. 11 And write bere be 'igit of be Composyt. And set be articul of be coniposit next aft'//1 be digit in be same rewe, yf here
16 be no mo figures aft'/'. But yf bere lie mo figuris after bat digit. And bere be schall be rekend for hym selfe. And when bou schalt adde b'd ylke figure b/d berys be articulle ouer his bed to be figure vnder bym, bou schalt cast bat articul to be figure bot hase hym ouer
20 his bed. & here bat Articul schal token) hym selfe. lo an Ensam- Here is an pull xof- all 326. Cast G to G, & bere-of wil arise twelue. do away , lt..lful 6 be hyer G & write bere 2, bat is be digit of bis coniposit.
And be// write be articulle bat is ten oue?' be figuris bed of twene
2 \ as b«s * Now cast be articulle bat standus vpon be figwis of twene 216. bed to be same fgwre, & reken bat articul bot for one, and ban bere wil arise thre. pan cast bat thre to be neb/-/' figure, bat is one, & bat wul be foure. do away be figure of 3, and write
28 bere a figure of foure. and lete be neber figwre stonde stil. & ban worch forth, vnde versus.
IF Articulus si sit, in p/7mo limite cifram, If Ai'ticuhi//' verq 'iquis inscribe figuris,
32 Vel p< ■/• se scribao si nulla figura sequat///'.
11 Here he puttes be thryde case of be craft of Addicion). & be case is bis. yf of Ad^'cioun) of 2 figuris a-ryse an Articulle, bow suppose u is
an Article,
schal bou do. thou most do away be beer figure b"t was addid to setdowna
' - ' ° ' cipher and
30 be neb//' & write bere a cifre, and sett be articuls on be figuris •»«■* th«
' ' ' r o tens.
hede, yf bot b/ re come ony after. And wyrch ban as I haue tolde
be in be secunde case. An ensampull 25. Cast 5 to 5, bat wylle
be ten. now do away be hyer 5, & '' write bere a cifer. And
40 sette ten vpon be figuris hed of 2. And reken it but for on bus. lo
10 Tht Craft of Subtraction.
Here is exmnpU
an Ensampnlle IT ■ -^"'1 lJ,an worch forth. But vf K/v come no figure after be ' 7-, | cifre, write be articulnext hym in be same rewe as here j 5 . cast 5 to 5, and it we] be ten. do away 5. |>at is j?e
hier 5. _ and write ben a cifre, & write afte?' hym be articul as i
bus i,', j. And Jjan b/m hast done. 5
IT Si tibi cifra sup// ueniens occurrerit, ilia///
Dele sup /posita/// ; fac illic scribe figura///, 8
Postea procedas reliquas addendo figuras. wuattodo U Here he puttes be fourt case, & it is bis. bat yf here come a
when you ' Y i i i
have a cipher cifer ill be hier lvwc, how h/ii M'lial (In. hi b0U SChalt do. do
in the top ' ' .
row. away be cifer, & sett b<??\s be digit bat comes of be addicioun as bus 12
Ln example 1 </.</> M. Ill bis ciisail 1 1 till In-n alb be foure Cases. Cast 3 to foill'e, ofall the 17743 Y
difficulties. p(,t wol be seiien). do away 1. & write be?-e seueri); ban cast
I to be figure of 8. \a\ wel be 12. do away 8, & .-''it bere 2. bat is a digit, and sette be articul of be composit, bat is ten, vpon be cifers 16 lied, & reken it, for hym selfe bat is on. ban cast one to a cifer, & hit wulb be but on, for 1103) & on makes but one. ban east 7. bat stond es vml'7' bat on to hym, & bat wel be 8. do away be cifer & bat 1. & sette bere 8. ban go forthermore. cast Jv ober 7 to be cifer 20 bat stondes ouer hym. bat wul be bot seuen, for be cifer betokens
»ieafi42o. uojt. do away be cifer & sette |' ■/•■ ;euen), 2& ben go forberrnore & c.i t I to 1, & bat wel be 2. do away be hier 1, & sette bere 2. ban hast bou do. And yf bmi haue wel ydo bis nomber bat is sett 24 her* alt-/- wel be be nomber bat schalle aryse of all be addicioh) as her* 27827. IT Sequifrtr alia spec/es.
Viiu///'/0 wamernm si sit tibi denier- cura Scribe figurarw»i series, vt in addicione. 28
Fourthi II Tins is be Chapter of subtraccion), in the quych bou most
to know ' ' .
tsub- know foure uessessarv thyncres. the first what is subtraccion). be
traction ... i
secunde is how mony uombers bou must haue to subtraccion), the
thryd is how niony maner; of 1 b< ■■ may happe in bis craft of 32
Libtraccion). Tie- fourte is qwat is be p?*ofel of |i- craft. If As for i; be first, bou most know bat subtraccion is drawyngi of one thcBecondj nowmber oute of anobej nomber. A.s for be secunde, bou most
known bat bou mo t hauo two rewes of figuris on< vnder anober, 36 the third] bou addyst in addioion). As for be thryd, bou nioyst know bat
fourc maner of diuerse casis mai happe in bis craft. If A.s for be thefourth. fourt, bou most know bat be prof el of bis craft is whenne bou ha
taken be 1 1 ■ nomber out ot be mor< to telle what bere leues ouer I"
The Cases of th Croft oj Subtraction. 11
b'/t. & J5ou mosl be-gynne to wyrch in pis craft in be ryght side of be boke, as bou diddyst in addiciou). Versus.
IT Maiori numero nmaei-uju suppone minorem, 1 H Siue pari nuwero supponat"/- mini' / us par.
XU Here he telles bat be hier nomber most be more ben be neber, ■ leaf 143 a. or els eueri) as mych. but be may noi be lasse. And be case is greater bis, bou schalt drawe be neber nomber out of be byer, & bou mayst above the 8 not do b'/t yf be hier nomber were lasso ban bat. ffor bou mayst not es draw sex out of 2. But bou mast draw 2 out of sex. And bou maiste draw twene out of twene, for bou schal leue no}t of be hier twene vnde versus. 12 H Postea si possis a prima subtrr/he pim&m
Scribens quod remanet. Here is be first case put of subtraccion), & be says bou schalt The first ease " begynne in be ryght side, & draw be first figwe of be nebe?1 rewe Hon. 16 out of be first figure of be hier rewe. qwether be bier Bgwre be more ben be nebe/', or eueu) as mych. And bat is notified in be vers when lie says " Si possis." Whan bou has bus ydo, do away be hiest figwre & sett bere tat leues of be subtraccion), lo an Ensampulle Here is an 20 23F"- draw 2 out of 4. ban leues 2. do away 1 & write bere 2, & I l22 latte be nebe?' figur< sfo rcde stille, & so go for-by ober figuris till b'/u come to be ende, ban Last bou do.
11 Cifram si nil remanebit. 24 IT Here he putt's be secuude case, & hit is bis. yf it happe bat Put a cipher
i i i • „ , . if nothing
qwen pou hast draw on neber tigim out oi n hier, & bere leue no2t remains, after be subfa'accion), bus 2bou schalt do. bou scballe do away be hier - leaf i48 6, figwre & write pere a cifer, as lo an Ensampull V,y. Take foure Here i« an 28 out of foure ban Lais iM,>i. berefor< do away ; 24 ; be hier 1 & set pere a cifer, ban take 2 out of 2. ban leues nojt. do away b hier 2. & set ber< a cifer, and so worch whare so euer bis happe. Sed si wan possis a p/ /ma denize p/ /ma//' 32 P/- cedens vnu/// de limite deme seque//te,
Quod demptu//' pro denario reputabis ah illo Subt/v<he to/(dem numemw qiiam proposnisti Quo facto sc/7be super quicquid remanebit. 36 Here he puttes be thryd ca e, be quych is bis. yf it happe bat suppose you
1 r . , ' r cannol take
be neber ngttre be mor< ben be hierfigwe bat he scballe be draw out the lower , , ? > r or figure from
of. how schalle bou do. bus bou schalle do. bou scballe borro .1. the top one,
iii > borrow ten ;
oute of be next figwre bal conn after in be same rewe, for bis case In may neuer happ but yf pere come figures after, ban bou schall < it
12 The Cases of the Craft of Subtraction.
bat on ouer be hier figures lied, of the quych b<«u woldist y-draw onto be neyber figure yf bmi haddyst y-mv^t. Whane bou hase
take the bus. ydo bon scliull'- rekene bat .1. for ten. H. And out of bat ten
lower number
fromtenj b<m sclial draw be neybermost figure, And alle b«t leues bou schalle d add the adde to be figure on vvhos bed bat . 1. stode. And ben bf/ii schalle
niiswer to ' ' ' r
thetop d<> away alle bat, & sett bere alle that arisys of tin- addicion) of be
number. .
iieafuia. ylke - figuris. And yf yt 'liappe bat be figure of be quych J>ou
schalt bonxi on be hym self but 1. If bou schalt bat one & sett it 8 vppon) be ober figuris hed, and sett in bat 1. place a cifer, yf ber<
Kxampie. come mony figures after, lo an Ensampul. |~ 2122- j- take 1 out of 2. it wyl not be, berfore borro one of Jv next L__J figure, bat is 2. and sett bat one/' be hed of pe fyrst 2. & rekene it for ten. and bere be 12 seeunde stondes write 1. for bou tokest on out of 1 iy /// . ban take be nej>er figure, bat is 4, out of ten. And ben leues 6. cast to G be figure of bat 2 bat stode vnder be hedde of 1. bat was borwed & rekened for ten, and bat wylle be 8. do away brtt 0 & bat 2, & 16 sette bere 8, & lette be neber figure stonde stille. Whanne bou hast
how to do bus, go to be next figwe b<d is now but 1. but first yt was 2. &
'Pay luck' Y ' ° r ' J
the borrowed bere-of was borred 1. ban take out of bat be figure vnder hym, b</t
is 3. hit wel not be. ber-fore borowe of the next figttre, be quych is 20 bot 1. Also take & sett hym ouer be hede of be figu?'e bat bou woldest bane y-draw oute of be nether figure, be quych was 3. & bou myjt not, & rekene bat borwed 1 for ten & sett in be same place, of be quych place bop tokest hym of, a cifer, for he was boi 1. 24 Whanne bou hast Jras ydo, take oul of bat 1. bai is rekent for ten,
»ieaii446. }v neb< /• figure of •">. And bei'i leiies 7. 2cast be ylke 7 to }v figwre bat bad be ylke ten vpon his bed, be quych figure was 1, & bat wol lie 8. ban do away bat 1 and bat 7. & write ben 8. & ban wyrch 28 forth in obe/* figuris til bou come to be ende, & ban bou hast be do.
Versus.
11 Facque nonenarios de cifris, c\\m remeabis
■ Occ///-rant si forte cifre; dum demps- /is vnum 32
U Postea p/vcedas reliquas deme//do figuras.
A. very l»ard " "'r' '"' puttes J>e foiirte ease, be quych LS bis, yf it liappe bat
cuseisput. |)(, llt.j1r/. figWj.e, be quych bou schalt draw out of be hier figure be
mon ban be hier figur ouer hym, & }v next figure of two or of 3(i thre or of foure, or how monj ber< be by cifers, how wold b<>\\ do. )?ou wost. wel bou must mile borow, & bou mays! nol borow of be cifers, for bai haue 003! bat bai may lene or spar-. Ergo3 bo\*
Pel baps " So."
How to 'prove the Sttbtraction. 13
woldest jj'Hi do. Certayn) pus most pou do, bou most borow on of pe next figure significatyf in pat rewe, for pis case may not happe, but yf pere come figures significatyf niter the cifers. "Whan pou 4 hast borowede pat 1 of the next figure significatyf, sett pot on oner pe hede of pot figwe of pe quych pou wold haue draw pe nep' r figure out yf Jjou hadest niy^t, & reken it for ten as pou diddest in J>e oper case here-a-fore. Whan) pou hast pus y-do loke how 8 niony cifers pere were bye-twene pat figure significatyf, & pe figwe of pe quych pou woldest haue y-draw the * neper figure, and of euery i leaf 145 a. of pe ylke cifers make a figure of 9. lo an Ensampulle after, f 40002 | Here is an Take 4 out of 2. it wel not be. borow 1 out of pe next figure i_10004j
12 significatyf, pe quych is 4, & pen leues 3. do away pat figure of 4 & write pere 3. & sett pot 1 vppon pe figure of 2 hede, & pan take 4 out of ten, & pan pere leues G. Cast G to the figure of 2, pot avoI be 8. do away pat 6 & write pere 8. Whan poll hast pus y-do
16 make of euery 0 betweyn 3 & 8 a figure of 9, & pan worch forth in goddes name. & yf pou hast wel y-do bou2 schalt haue pis nomber
H Si subtracc/o sit bene facta p/ "bare valebis j~39998l Sic.
Quas s«btraxisti p/v'mas addendo figuras. [_10004 J
20 U Here he teches be Craft how pou schalt know, whan pou hast How to prove
. . :i subtraction
subtrayd, wheper pou hast wel ydo or no. And pe Craft is pis, sum. ryglit as pou subtrayd pe neper figures fro pe hier figures, ry^t so adde pe. same neper figures to pe hier figures. And yf pou haue
24 well y-wroth a-fore pou schalt haue pe hier nombre pe same pou haddest or puu be-gan to worch. as for pis I bade pou schulde kepe pe neper figures stylle. lo an 3Ensampulle of alle pe 4 cases Meafi45&. togedre. worche welle bis case ^0003468^. And yf bou worch welle Here is an
28 whan pou hast alle subtrayd ! 20004664 I pe pat bier nomine here, pis schalle be pe nombre here foloyng whan pou hast subtrayd f39998804~j. And pou schalt know pus. adde pe neper rewe of pe Our author I 20004664 I same noml)re to pe hier rewe as pus, cast 4 to 4. pat wol >>«« 3f<>r 1 •
32 be 8. do away pe 4 & write pere 8. by pe first case of addicion). pan cast G to 0 pat wol be G. do away pe 0, & write pere G. pan cast 6 to 8, pot wel be 11. do away 8 ec write pere a figure of 4, pat is pe digit, and write a figwre of 1. pat schall be-token ten. pat
3g is pe artieul vpon be hed of <s Dext afte7', pan reken pat 1. for 1. &
cast it to 8. pat schal be 9. cast to pat 9 pe neber figure vnder pat
pe quych is 1, & pat schall' be 13. do away bat 9 & sett f> re 3, &
sett a figure of 1. pot schall be 10 vpon pe next figur/.s hede pe
2 'hali ' marked for erasure in MS.
14 The Graft of Duplatu n.
quych is 9. by be secu^de case b«t bou hades! in addicion). bun cast
I in 9. & }?at wo] be 1". '1" awaj be 9. & J1. it 1. And write ber< a
cifer. and write be articulL bat is I . betokenyngi LO. vpon pehedeof
i leal it'-' be next figure toward be 1 \ It side, be quych 4s 9, iV so do forth hi 1
Reworks bou come to bf last 9. lake be \\<<uv> of bat 1. be qnych bou schalt
his i t ' r T , ,
through, fynde oue»' be hed of 9. & sett ii ouer be nexl figures hede bat schal be .'3. •: Also do away be 9. & set be;*< a cifer, & ben cast bat 1 bat stondes vpon be hede of 3 to be same ■"», & pal schalle make 8 I, ben caste to be ylke 1 the figure in be ueybe?" rewe, be quych is
andbringa 2, and bat sch, ill- be 6. And pen schal bou haue an Ensanipulle ajeyn), loke & se, & but bou haue bis same bou hase my-v wrojt. I 60003168 i Sequit^/' de duplac/one 12
I 2000-l(3t34 !
Hi vis duplaiv numru/e, sic i//cipe primo y^ Scribe figurarum series quamcunque velis tu. Four tilings 11 Tliis is the Chapture of duplacion), in be quych crafi bou mosl
knowiiin haue & know I thinges. *,' jje first bat bou most know is what is 1G
Dnplalion. . i • 1 •
duplacion). pe secuwde is bow mony rewes o| ngwes b"U mosl haue to bis craft. U be thryde is how many cases may2 happe in
Here they bis craft. *,\ be fourte is what is be prof el of be craft. • As for be
first, duplacion) is a doublyng of a uombre. * As for be secuwde 20
si«afi466. bou most 'haue on nombre or on rewe of figures, the quych called ivivwus duplandws. As for pe thrid b"ti most know bat •"> diuerse ease.- may hap in bis craft. As for be fourte. qwat is be profet of bis craft, & bal is to know what u-risy;l of a nombre [-doublyde. 24
Mind where ' fforber-inore, bou mosl know »V take gode hede in qnych side b<m schalh be-gyn-in bis craft, or ellis bou maysl spyl all* bj'laber pere aboute. certcyn bou schalt begyh) in the 1\ ft side in bis Craft, thenke wel ouer bis verse. IT lA leua dupla, diuide, niwltiplica.4 28 The sentens of pes verses afore, as bou may see if pou take hede.
,!""1 ,1,1 As be texl of bis verse, bal is to say, 1 Si vis duplare. bis is be
your n ' » ' • err
sentence. 8 If bou wel double a nombre bus bou mosl be-gynn).
Write a rewe of figures of what nombr< pou well, versus. 32
Postea p/ -cedas pn'maw duplando figura//' Inde qwod excrescit scribas vbi iusserit ordo Iuxta pr< cepta tibi que dant"/- in addic/one. Ho towork ' Her- he telles how pon schall worcli in bis Craft, he says, 36
f\ rat, whan pou hast writen be nombre bou schall be-gyn al be first
- 'moy ' iu MS. 4 Subtrakaa ai t addis a dextns ve\ mediabis' added on margin of MS.
.i sum.
Tht Cases oj th Graft of Ditplation. 15
Bgur< in tin- lyft sidi-, iV" doubulli bal figwre, & be nombre bat comes bere-of b"U schalt write as bou diddyst in addicion), as 1 I schal telle be in be case. v< rsus. 4 *H Nam si sit digitus in prinio limite scribas. ' ieai U7<*.
H Here is be first case of bis craft, be quych is bis. yf of dupla- ifthj
' > ' * x " > -j isa digit,
cioii) of a figure arise a digit, what schal bou do. bus pou schal
do. do away be figure bat was doublede, & sett here be diget bat write it in
' ' ° r !ii Hie place of
8 comes of be duplacion), as bus. 23. double 2, & bat wel be 4. do tiietop
' l ' ' _ figure.
away be figure of 2 & sett bere a figure of -4, & so worch forth tille Jvm come to be ende. versus.
H Articub/-' si sit, in pn'mo limite cifram, 12 *I Articulu;// vero reliquis inscribe figuris ;
II Vel p<r se scribas, si nulla figura sequat*// . U Here is be secunde case, ba quych is b's yf ber< come an if it is an
r „ iii article,
articulle of b'- duplacion) of a figure bou schalt do ryjt as bou 16 diddyst in addicion), bat is to wete bat b"ii schalt do away be
figure bat is doublet & sett bere a cifer, & write be articulle ouer b-e pui acipher next figuris hede, vf ber<> be any after-warde toward be lyft side as and 'carry*
° ' ' the tens.
bus. 25. begyn at the lyft side, and doubulle 2. bat wel be 4. do 20 away bat 2 iV' sett bere 4. ban doubul 5. bat wel be 10. do away 5,
& sett bere a 0, & sett 1 vpon be next figum hede be quych is 4.
& ben draw downe 1 to 4 & bat wolle be 5, & ben do away bat 4
& bit 1, & sett bere 5. for bat 1 schal be rekened in be drawynge to- 24 gedre for 1. wen 2bou hast ydon bou schalt baue bis nombre 50. 2ieafU76.
yf bere come no figure after be figwre bat is addit, of be quych if there is
. . . no figure i o
addicion) comes an articulle, bou schalt do away be figure bat is 'carry' them dowblet & sett pen a 0. & write be articul next by in be same "'em down. 28 rewe toward be lyft syde as bu<. 523. double 5 bat woll be ten. do away be figure 5 & set be?*e a cifer, & sett be articul next after in be same rewe toward be lyft side, & bou schall haue bis nombre 1023. ben go forth & double be ober nombers be-quych is ly^t y- 32 nowjt to do. versus.
11 Compositus si sit, in limite scr/be seq/'cute Articulu//', p/v'mo digitus ; quia sic iubet ordo : Et sic de reliquis facie/>s, si suit tibi plures. 30 H Here he puttes be Thryd ease, be quych is bis, yf of dupla- mtisa
\ c n r* •• i i t, i ,. Composite,
cion) ot a figure come a < omposit. pou schalt do away be ngwre b"t is doublet & set bere a digit of be Composit, & sett be articulle ouer writ. be next figures hede, & after draw hyra downe. with be figure ouer and 'carry ■ 40 whus hede lie stondes, & make pere-of an nombre as bou hast done
Here is an example.
lft The Craft oj Mediation.
afore, & yf \>ert conic no figwre after pat digit pat pow hasl y-write,
paw set pe articulle next, after hym in pe same rewe as pus, t "» T : double
' ieafU8a. o' pat wel be 12, do away 6 & write pei'e pe digit !of 12, pe quyi h
is 2, and set jje articulle nexl after toward pe lyft .side in pe same 4 rewe, for pere comes no figure after. pan dowble pat oper figure, pe quych is 7, fat wel be 14. the quych is a Composit. fen do away 7 pat pou doublet & sett pe pe diget of hym, the quych is I, sett pe articulle ouer pe next figures bod, pe quych is 2, & pen draw to hym 8 pat on, & make on nombre pe quych schalle be 3. And pen yf p<m haue wel y-do pou schalle haue pis nombre of pe duplacion), 134. versus. U Si super ext/<ma//> nota sit monade//' dat eid- >m
Quod tibi '"^tingat si p/v'mo dimidiabis. 12
Howto fl Here be says, yf ouer pa fyrst figitre in pe ryjt side be such a
mark t..i- merke as is li«'i ' made, ", pou schalle fyrst doubul]" pe figure, the
one-half.
quych stondes vnder p<d merke, & pen pou schalt doubul pat merke f"' quych stondes for haluendel on. for too haluedels makes on, & 16 so pat wol be on. east p"t on to pat duplacion) of pe figure ouer whos hed stode Jjat merke, & write it in pe same place J>ere fiat pe figure pe quych was doublet stode, as Jms 23w. double •">, pat wol be 6 ; doubul pat halue on, & pat wol be on. cast on to 6, pat wel be 20 7. do away 6 & pat 1, & set! pert 7. pan base poll do. as for pal »ieafi*86. figure, pan go -to pe oper figure & worch forth. & pou schall ueuer Tins can only j)aue sllt.], ;l merk but oil'/' be hed of be furst figure in be ryght
stand over ' ' ° ' • °
neure?' s^e- And Je* '' acna' 11"t happe but yf it were y-halued a-fore, pus 24 pou schalt vnde7'stonde pe verse. 11 Si super extrernaw &c. Et nota, talis figwra " significans medietate?H, unitatrs veniat, i.e. con- tingal uel fiat super extreinam, i.e. super prtmaw figuraw? in extrenio sic versus dextram ars da! : i.e. reddil monade///. i.e. vnitatew eidem. 28 i.e. eidem note & declinatfwr hec monos, d-is, di, dem,&c. ■ Quod ergo to////// hoc dabis monad em note conting* t. i.e. eveniet tibi si dimi- diasti, i.e. accipisti uel subtulisti medietatem alicuius unius, in cuius principio sint figura num< rum denotaus imparew primo i.e. principiis, 32 11 Sequit'o- de mediacione.
ncipe sic. si vis alique// nuwerum mediar< : Sc/-/be figurar///// seriem sola///, velut an/ . Thefour "' In pis Chapter is ta^l pe Crafl of mediacioun), in pe quych 36
crafl bou mosl know 1 thynges. Hurst what is mediacion). the
mediation! ; ' , , , . , ,
secunde how inony rewes oi figures pou most haue m po wyrcnyng< of bis craft. p>- thryde how mony diuerse cases may happ in pis theSni craft.1 1 Ai for pe furst, pou schall vndurstonde pat mediacion) isa 4'j ; After ' crafl ' insert ' the .4. what is pe profet of |»is craft.'
J
The Mediation of an Odd Number. 17
takyng out of halfe a nonibcr out of a holle nomber, 1as yf b"U Ueafuoa. wolde bake 3 out of 6. 51 As for be secunde, Jjou schalt know b«t the econd; Jvu most haue one rewe of figures, & no moo, as bou hayst in be 4 craft of duplaciou). r As for the thryd, bou most vnderstonde J?at the third; 5 cases may happe in bis craft. U As for be fourte, Jjou schalle the fourth, know pat the profet of pis craft is when Jjou hast take away be haluendel of a nombre to telle qwat bere schalle leue. ^ Incipe 8 sic, &c. The sentence of bis verse is bis. yf Jjou wold medye, bat is to say, take lialfe out of be- holle, or halfe out of halfe, Jjou most begynne bws. "Write one rewe of figures of what nombre jjou wolte, Begin thus. as bou dyddyst be-fore in b1' Craft of duplacion). versus. 1- II Postea p/ncedas medians, si p/v'ma figura
Si par aut i//'par videas. If Here he says, when }vm hast write a rewe of figures, bou schalt take hede wheber be first figure be euen) or odde in nombre, see if the
-.. . number is
lb & vnderstonde bot he spekes of be first figure in be ry3t side. And evenorodd. in the ryght side bou schalle begynne in bis Craft.
IT Quia si fucrit par, Dimidiab/s earn, scribes quicq>'/d remanebit: 20 1i Here is the first case of bis craft, be quych is bis, yf be first [fit is even,
■ * l'li-ci halve it, and
figure he eiien. bou schal take away fro be figure euen halfe, oc dowritethe
answer m
away bat figure and set bere bat leues ouer, as bus, 4. take 2 halfe its place. out of 4, A; ban bere leues 2. do away 4 ec sett bere 2. bis is lyght 2 1 y-nowjt. versus.
H Impar si fuerit vnu//< demas mediaiv
Quod no// prestunas, Bed quod sup' rest mediabis Inde sup- r tractu//* fac demptii/// quod no/ot vntttn. 28 Here is be secunde case of bis craft, the quych is bis. yf be u it is odd,
first figure betokene a nombre bat is odde, the quych odde schal not even number
ii less tlia" ll- be mediete, ben bou schalt medye bat nombre bat leues, when the
odde of be same nombre is take away, & write bat bat leues as bou
32 diddest in be first case of bis craft. Whan) b"ii hayst write bat. for
b'd bat leues, write such a mcrke as is here " vpon his hede, be quych Then write
merke schal betoken) halfe of be odde bat was take away, lo an one-half over
Ensampull. 245. the first figure her< is betokenynge odde nombre,
36 be quych is 5, for 5 is odde ; b< re fore do away bat bot is odde, be Herein an
C X 111 1 1 1 1 1 1 • .
quych is 1, ben leues 4. ben medye 1 & ben leues '_'. do away I. &
settc bere 2, & make such a merke " upon his hede, bat is to say
ouer his hede of 2 as bus. 212."' And ben worch forth in be ober
40 figures tyll b<m come to be ende. by be furst case as bou schalt
KOMBRYNGE. C
18 The Cast of the Croft of Mediation.
iieafisoa. vnderstonde bat bou schall J.neue/" make such a merk but ouer be markboniy ^rs^ figure hed in pe ri$l side. Wheber J»e other figures bat comyii) figure." r8< after hym be run,' or odde. ve?-sus.
H Si monos, dele ; sit UU cifra post no/V< supra. 4
if the first ^[ Here is be thryde case, be quych vf the first fieiire be a figure
figure is one ' ' ' ' J ° D
put a cipher. 0f ]_ j,,m schalt do away J>at 1 & set bere a cifer, & a merke on*0/- be cifer as bus, 241. do away 1, & sett bere a cifer with a merke oner his hede, & ben hast b<m ydo for bat 0. as bus 0W ben worch forth 8 iu pe ober figwys till b"U come to pe ende, for it is lyght as dyche water, vn.de vc?*sus.
IT Postea p/<<cedas hac condic/one secu//da:
Imp-// si fu< /it hinc vnu/// deme p/v'ori, 12
Inscribens quinque, nam denos significant Monos p/ ■' d/'7am. Whattodo H Here he puttes be fourte case, pr quych is bis. yf it happen)
figure is odd. the secunde figure betoken odde nombre, bou schal do away on of 16 bat odde nombre, pe quych is significatiue by bat figure 1. be quych 1 schall be rekende for 1". Whan bou hast take away pat 1 out of be nombre p«t is signifiede by bat figure, bou schalt medie bat bat leues ouer, & do away pal figure bat is medied, & sette in his styde 20 halfe of bat nombre. ' Whan bou hase so dime, bou schall write * leaf u,n6. -a figure of 5 one/1 be next figures hede by-fore toward be ryjt side, figure of five f"r Pa^ 1, be quych made odd nombre, schall stonde for ten, & 5 is iowermnn- halfe of 10; so J>ou inosl write 5 for his halucndelle. 1" an Kn- 24
sampulle, 1678. begyn) in be ryjl side as bou most nedes. medie S. Example. ben bou schall leue 1. do away bat 8 >V sette bere 1. ben out of 7. take away 1. pe quych makes odde, & sett 5. vpon be next figures hede afore toward be rj ;i ide, J1*1 quycji is now 1. but afore it was 28 8. for pat 1 schal be rekenel Eor 10, of be quych 10, 5 is halfe, as bou knowest wel. Whan bou hast bus ydo, medye bat be quych leues after pe takyinge away of pat pit is odde, be quych leuynge schalle be •"> ; '1" away 6 A sette be?'e .">, & bou schalt haue such a 32 nombre 4634. after go forth to p<i next figwre, & medy bat, & worch forth, for it is lyjl vimv;t to be certayn).
• Si V' o S' cwwda dat vnu>/>. Ilia deleta, sc//bat///- cifra; pr/ori 36
IT Tradendo quinque pro denario mediato ;
Nee cifra sc//batur. nisi dei//de fig>//a seq>/at>//- :
Postea pr "cedas reliqt^s mediando figuras
Vt sup/v docui, si sint tibi mille figure. -JO
How to prove the Mediation. 19
IF Here he puttes pe 5 case, pe quyeh is ^is : yf pe secunde Ucafisia. figure be of 1, as pis is here 12, pou schalt do away fiat 1 & sett figm-euCon^ \>ere a cifer. & sett 5 ouer J?e next figure hede afore toward pe rijt andwritefive
4 side, as pou diddyst afore j & pat 5 schal ho haldel of pat 1, pe figi,rC.'e quych 1 is rekent fur 10. lo an Ensampulle, 211. medye 4. pat schalle he 2. do away -1 & sett pere 2. pe>j go forth to pe next figure, pe quych is hot 1. do away pat 1. & sett pere a cifer. & set
8 5 vpon pe figures hed afore, pe quych is nowe 2, & pen pou schalt
haue pis no/nhre 202, pen worch forth to pe nex figure. And also it is no maystery yf pere come no figure after pat on is medyet, pou schalt write no 0. no now^t elUs, but set 5 oner pe next figure afore 12 toward pe ryst, as pus 14. ruedie 4 then leues 2, do away 4 & sett How to halve
.. •!• in l-iii fourteen.
yere 2. pen medie 1. pe qtticn is rekende for ten, pe halueftdel pere- <>f wel he 5. sett pot 5 vpon pe hede of p«t figure, pe quych is now 2, & do away pat 1, & pou schalt haue pis nombre yf pou
1 6 worch wel, 2 . xn.de versus.
11 Si mediacio sit bene f^-ta probare valeb/x IF Duplando numerum que/// p/7mo di///ediasti 1F Here he telles pe how pou schalt know whoper pou base wel How to prove
iiio i i t in your media-
20 ydo or no. doubul -pe nombre pe quych pou base mediet, and yf tion.
pou haue wel y-medyt after pe dupleacion), pou schalt haue pe neafl°16- same nombre pat pou haddyst in pe tahulle or pou began to medye, as pus. II The furst ensampulle was pis. 4. pe quych I-mediet was First
24 laft 2, pe whych 2 was write in pe place pat 4 was write afore. Now douhulle pat 2, & pou schal banc 4, as pou hadyst afore, pe secunde Ensampulle was pis, 245. When pou haddyst mediet alle The second, pis nombre, yf pou haue wel ydo pou schalt haue of p«t mediacion)
28 pis nombre, 1 22". Now douhulle pis nombre, & begyn in pe lyft side; douhulle 1, pat schal be 2. do away pat 1 & sett pere 2. pen doubulle pat ope)' 2 & s stt J" r< 4, pen douhulle pal ober 2, & pat wel be 4. yen doubul pat merke pat stondes for liable on. & pat schalle
32 be 1. Cast pat on to 4, & it schalle be 5. do away pat 2 & pat merke, & sette pere 5, & pen pou schal haue pis nombre 245. & pis wos pe same nombur pat pou haddyst or pou began to medye, as pou mayst se yf pou take hede. Tie' nombre pe quych pou haddist
3G for an Ensampul in pe 3 ease of mediacion) to be mediet was pis 'f lie third 241. whan pou haddist medied alle pig nombur truly 3by eue?y »ieafi52«. figure, pou 'schall bane be pot mediacion) pis nombur 120w. Now dowbul pis nomh///\, A' begyn in pe lyft, .side, as I tolde pe in |>e
40 Craft of duplacion). pus doubulle pe figwre of 1, pat wel be 2. do
20 Tin Craft of Multiplication.
away bat 1 & sett bere 2, ben doubul be next figur< afore, the quych is 2, & bat wel be 4 ; do away 2 & set ber< I. ben doubul be cifer, & bat wel be 11031, for a 0 is imjt. .And twyes no^t is but no$t. berefore doubul the merke aboue be cifers hede, be quych be- -1 tokenes be haluewdel of 1, & bat schal be 1. do away be cifei & |>e merke, & sett Ipere 1, & ben bou schalt haue bis nombur 241.
And bis same nombur Jvai haddyst afore or bou began to 1 ly, &
The fourth yf b-/u take sode liede. H Tbe next ensampul bat bad in be 4 ease 8
example. ' '
of mediacion) was bis 4G78. AVlian bou bast truly ymedit alle bis nombur fro be begynnynge to be endynge, bou schalt haue of be
mediacion) bis nombur 2334. Now doubul this nombur & begyn in belyft side, & doubulle 2 bat schal be 1. do away 2 and sette bere 12 4 ; ben doubule 3, b^t wol be 6 ; do away 3 & sett \>pre G, ben 1 leaf 1525. doubul bat o}w 3, & bat wel be G ; do away 3 & set \ere 1G, ben doubul be 4, bat welle be 8; ben doubul 5. be quych stondes oue?" be lied of 4, & bat wol be 10; east 10 to 8, & J>«t schal be 18; do 16 away 4 & bat 5, & set! pere 8, & sett that 1, be quych is an articul of be Composit J?e quych is 18, ouer be next figures bed toward be lyft side, be qnych is G. drav bat 1 to 6, be quych 1 in be dravyng schal be rekente bot for 1, & bat 1 & bat G togedur wel be 7. do 20 away bat G & bat 1. the quych stondes oiu /• his hede, & setl ther 7, & ben bim schalt haue bis nombur 1678. And bis same nombur bou hadyst or Jjou began to medye, as bou mayst see in be secunde ThenWi Ensampul bat bou had in be 4 case of mediacion), bat was bis : when 24
example. ' ' > > ' • *
bou had niediet truly alle the nombur, a pn'ncipio usque ad finem.
5 bou schalt haue of bat mediacion) bis nombur 102. Now doubul
1. bat wel be 2. do away 1 & sett here 2. ben doubul 0. bat will be
nojt. berefore take be 5, be quych stondes oue>" be nexi figures 28
hed, & doubul it, & bat wol be 1<». do away be 0 bat stondes
betwene be two figwr/s, A sette bere in his stid 1, for bal 1 now
schal stonde in be secunde place, where he schal betoken 10; ben
ieafi5Sa. doubul 2, bat wol lie I. do away 2 & set! bere 1. A' -bou schal haue 32
bus nombur 214. bis is be sane' nuwibur bat bou hadyst or b<<u
began to medye, as bou may see. And so do eue?" mor< . yf bou wil
knowe wheber bou base wel ymedyi or no. '. doubull< be nuwbur
J1, it conies aft'/' be mediaciouii), & bon schal haue be same nombur 36
bal Jjou hadyst or Jjou began to medye, yf bou haue welle ydo. or
els doute be no?t, but yf Jjou haue be same, bou base faylide in b/
Craft. „.,,.,.. ,„
Sequitur de multiphcatione. 40
To write down a Multiplication Sum. 21
Si tu per num'vui/) namerum vis ni/dtiplicar- . !
►O Scribe duas qua&cunque velis series muneroium Ordo servetur vt vltima ni«ltiplicandi 4 Ponat'// sup-/ ant>/iorem multiplicantts
A leua reliq'/e sint scripte ni/dtiplicantes. If Here be-gynnes pe Chapt?'e of multiplication), in be quych Four things p<>\\ most know -1 thynges. If Ffirst, qwat is multiplicacion). The of Haitlpiica- 8 secunde, how mony cases may hap in multiplicacion). The thryde, how mony rewes of figures fere most he. IF The 4. what is pe profet of pis craft. 1F As for pe first, pou schal vnderstonde ]>at the firsts multiplication) is a bryngynge to-geder of 2 thynges in on nombur,
12 pa quych on nombur coutynes so mony tymes on, howe hnony iieafi53&. tymes pere ben vnytees in pe nowmbre of }>at 2, as twyes 4 is 8. now here ben pe 2 nombe?'s, of Jje quych too nowmbres on is betokened be an aduerbc, pe quych is pe worde twyes, iV; his worde
1G thryes, & f»is worde foure sythes,2 & so furth of such other lyke wordes. H And tweyn nombres schal he tokenyde he a nowne, as fis worde foure showys j>es tweyn) nombres y-broth in-to on hole nombur, Jiat is 8, for twyes 4 is 8, as pou wost wel. IT And J>es
20 nombre 8 conteynes as "ft tymes 4 as pere ben vnites in pat other nombre, be quych is 2, for in 2 ben 2 vnites, & so oft tymes 4 ben in 8, as J>ou wottys wel. H ffor J>e secuude, pm\ most know bat Jv.m the second: most haue t«.«» rewes of figures. II As for pe thryde, Jjou most know the third:
24 pat 8 mane/1 of diuerse case may happe in J?is craft. The p/v/fet of
j>is Craft is to telle when a nomb?'e is multiplyed be a nojje?*, qwat the fourth, cowimys p^re of. r\ fforthermore, as to pe sentence of oure verse, yf Jj^u wel multiply a nombur be a-nofe?1 nombur, Jwu schalt write
28 3n rewe of figures of what noniburs so euer pow welt, & jjat schal be 3 leafista. called Numerus multiplicands, Anglice, pe nonibur the quych to The muitipH- be multiplied, fen Jjou schall write a-nother rewe of figures, by pe quych pou schalt rnultiplie the nombre pat is to be multiplied, of pe
32 quych nombur p<- furst figure schal be write vnder pe last figure of pe nombur, pe quych is to be multiplied. And so write forthe toward pe lyft side, as here you may se, j~ 67324~l And bis one Howtoset nombur schalh be called nu??ierus m/dti- ! \ plicans. Aug'- sum!'
36 Nee, pe nombur multipliynge, for he schalle multiply pe hyer noun- bur, as Jjus one tyme 6. And so forth, as I schal telle the afterwarde. And Jjou schal begyn in pe lyft side. U ffor-Jjere-niore )>ou schalt vndurstonde pat pere is two manurs of multiplicacion); one ys of Muitipiiea-
40 be wyrchynge of pe boke only in pe mynde of a mon. fyrst he mentally, - A iter 'sythes' insert '& |)is wordes fyne sithe & sex sythes.'
22 The Craft of Multiplication.
and on paper, teches of be fyrst man'/' of duplacion), be 'quych is be wyrchynge of tabids. Afterwarde he wol beche on be secunde maner. vnde versus.
In digitus cures digitus si ducere ma/or 4
i ieafX5*6. aP//' qua//tu/// distat a denis respice debes
IT Namq//' suo decuplo totiens deler- mi//ore//> Sitq//' tibi numerus veniens exinde patebit. Howto H Here he teches a rewle, how pnw schalt fynde J1'' nounbre bat 8
digits. conies by be multiplication) of a digit be anober. loke how mony
[vny]tes ben. bytwene be more digit and 10. And reken ten for on vnite. And so oft do away be lasse nounbre out of his owne subtract tiie decuple, bal is to say, fro Jjat nounbre bat is ten tymes so mych is 12 Tun; be nounbre pat comes of be multiplication). As yf bou wol multiply
2 be 4. loke how mony vnitees ben by-twene be quych is be more nounbre, & be-twene ten. Certen bere wel be vj vnitees by-twene 4 take the less & tin. yf b<m ivkni pere with be ten be vnite, as bou may se. so 16 tiraesfrom mony tymes take 2. out of his decuple, be quych is 20. for 20 is be itself. decuple of 2, 10 is be decuple of 1 , 30 is be decuple of 3, t0, is be
decuple of 4, And be ober digetes til bou come to ten ; & whan b<>n Example. hast y-take so mony tymes 2 out of twenty, be quych is sex tymes, 20 bou schal leue 8 as bou wost wel, for G times 2 is twelue. take 1 1 12 ont of twenty, & pere schal leue 8. hot yf bothe be digettes »ieafi55o. '-'hen y-lyech mych as here, l'l'2 or too tymes twenty, ben it is no
fors quych. of hem tweyn bou take out of here decuple, als mony 24 Better use tymes as pat is fro 10. hut neuer-be-lesse, yf bou haue has! to though. worch, bou schalt haue her< a tabu! of figures, wher< by bou schall se a nonn) ryght what is be nounbre bat comes of be multiplication) of 2 digittes. bus bou schalt worch in bis BgMre. 28
1|
|
2| |
1 |
||
|
3| |
6 |
9 |
|
|
8 |
12 16 |
||
|
5 |
10 |
15 20|25 |
1 |
|
6 |
12 1 |
18 24|30 |
|36 |
|
7,1! |
2] |28 35 12 19 |
|
|
8 L6 |
24 32 10 i- 56 Q4 |
i |
|
9 18 |
27 36 !.'• 54 |
81 1 |
|
1| 2| |
3 l 5 6 , 3 |
9 |
Howtouseit. yf be figure, jv quych schalL be multiplied, be euen< as mych as be 29 diget be, be quych bat ober figure schal be uiultiplied, as two tymes twa\ n1, <>r thre tymes 3. or sj ch other, loke qwer< bat figure sittes in
To multiply one Digit by another. 23
be lyft side of be Wangle, & loke qwere be diget sittes in be neper The way to most rewe of be triangle. cV go fro hvm vpwarde in be same rewe, tipiication
table.
be quycn rewe gose vpwarde til bou come agaynes be oJ>er digette bat
4 sittes in be lyft side of be tro'angle. And bat nounbre, be quych bou
fyn1des bere is be nounbre bat comes of the multiplication) of be 2 ' leafisst.
digittes, as yf bun wold wete qwat is 2 tymes 2. loke quere sittes
2 in be lyft side in be first rewe, he sittes next 1 in be lyft side al
8 on hye, as b<m may se ; be[w] loke qwere sittes 2 in be lowyst rewe
of be troangle, & go fro hym vpwarde in be same rewe tylle bun
come a-^enenes 2 in be hyer place, & ber bou schalt fynd ywrite ^,
& bat is be nounbre bat comes of be multiplication) of two tymes
12 tweyn is 4, as bow wotest welle. yf be diget. the quych is mwlti-
plied, he more ban be ob^/1, bon schalt loke qwere be more diget
sittes in be lowest rewe of be triangle, & go vpwarde in be same
rewe tyl2 bou come a-nendes be lasse diget in the lyft side. And
16 fere bou schalt fynde be nombre bat comes of be multiplication);
hut bou schalt vnderstonde bat bis rewle, be quych is in bis verse.
IT In digitum cures, &c, nober bis tn'angle schalle not serue, hot to
fynde be nounhres Jvvt comes of the multiplication) bat comes of 2
20 articuls or composites, be nedes no craft but yf bou wolt multiply
in bi mynde. And -bere-to bou schalt haue a craft aftenvarde, for 3 icafi56«. bou schall wyrch with digettes in be tables, as bou schalt know afterwarde. versus. 24 H Postea p/</cedas postrema/// multiplicand©
[Recte multiplicans per cu^ctas i^feriores] Condic/onem tamen tali quod multiplicand* Scribas in capite quicq>//d p/<<cesserit inde 28 Sed postq«c</;^ fuit hec m^ltiplicate figure
Anteriorent///' serei m>dtiplica?<t/>- Et sic m/dtiplica velut isti ni/dtiplicasti Qui sequit/o- RumenLm sc//ptu//< quiscu/<q«< figur/s. 32 " Here he teches how b<m schalt wyrch in bis craft, bou schalt n°wto
7-iiii,- ri 1 •> . multiply one
niwmplye be last ngure <>t be nombre, and quen b-u hast so vdo bou number by
* another.
schalt draw alle be figures of be neb'/- nounbre more taward be rot side, so qwen bou hast umltiplyed be last figure of be heyer nounbre 36 by alle be neber figures. And sette be nounbir bat conies fer-of oner Multiply the be last figure of be nefer nounbre, & ben bou schalt sette al be ojje?* oftbe l.fgher figures of be nej>er nounbre hum-'/ nere to be ryjt side. «: And whan oftiwiowM bou hast multiplied bat figure bat schal be multiplied be next after 2 't'l' marked for erasure before ' tyl' in JIS.
Tlit-n antery the lower
24 The Craft of Multiplication.
hym by al be neper figures. And worch as bou dyddyst afore til i leaf 1566. 1boa come to be ende. And bou sclialt vnderstonde bat euery set thean- ficmre of be hicr nounbre schal be multiplied be alle be figures of the
swer over the or i r o
first of the neber nounbre, yf be hier nounbre be any figure ben one. lo an 4 Ensampul here folowynge. j" 2465 j. bou schalt begyne to lmdtiplye
in be lyft side. Multiply L2!2. ! 2 he 2, and twyes 2 is 4. set 4
then mniti- ouer be bed of brtt 2, ben nmltiplie be same hier 2 by 3 of be nether of the lower, nounbre, as thryes 2 pat schal he G. set G ouer be hed of 3, ban s mwltiplie be same hier 2 by bat 2 be quych stondes vnder hym, bat wol be 4 ; do away be hier 2 & sette be?*e 4. II Now bou most antery be nether nounbre, bat is to say, b'<u most sett be neb'/1 nounbre more towarde be ry$t side, as bus. Take be neb'-/- 2 toward 12 be ryjt side, & sette it eueu) Viuh/- be 1 of be liver nounbre, & antery alle be figures bat comes after bat 2, as bus; sette 2 vnder be 4. ben sett be figure of 3 bere bat be figure of 2 stode, be quych is now vndur \>nt 4 ill be hier nounbre ; ben sett be ober figure of 2, 16 be quych is be last figwe toward be lyft side of be neb'/1 noudw bere as thus. be figure of 3 stode. ben b<m schalt liaue such a nombre ' 4G44t35 "
* kaiir,7<(. -11 Now multiply 1, be <piych comes next after G, by be last 1 232 !
2 of be neber nounbur toward be lyft side, as 2 tymes 4, bat wel be 20 8. sette bat 8 ouer be figure the quych stondes ou< r }v hede of bat 2, be quych is be last figure of be nebe?" nounbre; ban multiplie bat same -1 by 3, bat comes in be nebe?" re we, bat wol be 12. sette be digit of be composyt ouer be figure be quych stondes one/' be hed of 24 bat 3, & sette J>e articule of bis co//qiosit ouer al be figures bat Nowmulti- stondes ouer be neber 2 hede. ben nmltiplie be same 4 by be 2 in h\si but one be n^t side in b.e neber nounbur, bat wol be 8. do away 4. & sette
bere 8. Euer more qwen b'<u multiplies be bier figure by bat figur< 28 be quych stondes vnder hym, bou schalt do away bat hier figure, & sett ber bat nounbre be quych comes of imdtiplicacion) of ylke digittes. Whan bou hast done as 1 haue byde be, b"ii schalt haue astini . suych an order of figure as is here, i~ >2 j. ben take and antery 32
bi neber figures. And sett be fyrst I 4*>48[65] j figure of be neber
figures :: vndre be figure of G. ' ' U And draw al be
*leafi57&. ober figure-, of be suae lvwe to hyni-warde, 'as b>ai diddyst albre.
ben mttltiplye G be 2, & sett bat be quych comes oue?" bm'-of 36 OUer al J)e ober figures hi'des bat stondes ouer bat 2. ben m/dti- ply 6 be •">, & sett alle bat comes ber< of vpon alle be figures hedes bat standes oner bal ."> : ba« multiplye 6 be 2, be quych J Here 'of I • ame rew ' is marked for erasure in MS.
To multiply one Composite hy another. 25
stondes vnder fat G, fen do away G & write fere fe digitt of fe composit fat schal come fereof, & sette fe articull ouer alle jje figures fat stondes out/- fie hede of fat 3 as heir, fen j n j 4 antery fi figures as fou diddyst afore, and niwltipli 5| J2* j Antery the
| 828 | figures again,
be 2, fat wol be 10; -sett fe 0 ouer all fe figures fat | 464825 | *|,d multiply
stonden oner fat 2, & sett fat 1. ouer the next figures | 232 1
hedes, alle on hye towarde fe lyft side, fen midtiplye 5 he 3. fat 8 wol he 15, write 5 ouer fe figures hedes fat stonden ouer fat 3, & sett fat 1 ouer fe next figures hedes toward fe lyft side, fen umltiplye 5 he 2, fat wol he 10. do away fat 5 & sett fere a 0, & sett fat 1 ouer fe figures hedes fat stonden ouer 3. And feu
12 fou schalt haue such a nounhre as here stondes aftur. 1f 11 j Ueafisso. 11 Now draw alle fese figures downe togeder as fus, G.8. 1. | j*01 i & 1 draw to-gedur; fat wolle he 10, do away alle fese j 82820 I figures saue G. lat hyni stonde, for fow fou take hym i 464Li;) I
16 away fou most wiite fer fe same a^ene. ferefore late ' — — '
hym stonde, & sett 1 ouer fe figure hede of 4 toward fe lyft side ; Then add nil
t lit* figures
fen draw on to 4, fat wolle he 5. do away fat 4 A; fat 1, it sette above the fere 5. fen draw 4221 it 1 togedwr, fat wol be 10. do away alle
20 fat, <t write fere fat 4 & fat 0, & sett fat 1 ouer fe next figures hede toward fe lyft side, fe quych is G. fen draw fat G it fat 1 togedur, <t fat wolle he 7 ; do away G & sett fere 7, fen draw 8810 & 1, A: fat wel he 18; do away alle fe figures fat stondes oner fe
24 hede of fat 8, & lette 8 stonde stil, it write fat 1 ouer fe next figttm hede, fe quych is a 0. fen do away fat 0, & sett fere 1, fe quych stondes ouer fe 0. hede. fen draw 2, 5, it 1 togedwr, fat wolle he 8. fen do away alle fat, it write fere 8. 11 And fen fou and you will
28 schalt haue fis nouuhre, 571880. answer.
-1f Sed cu/// m/dtiplicabis, pr/mo sic est operandi!///, " leanssa.
Si dabit articulu//^ tibi m^ltiplicacio solu//> ; P/'oposita cifra su///ma//> tronsferre memento.
32 11 Here he puttes fe fyrst case of fis craft, fe quych is f is : What to do
yf fere come an articulle of be nmltiplicacion) ysette before the multipiica- . ,. . , , „ .. , , tion results
articulle in fe Jylt side as fus | 51 ;. multiplye 5 by 2, fat wol he in an article.
10; sette ouer f e hede of fat 2 ! 23 ! a 0, it sett fat on, fat is fe 3G articul, in fe lyft side, fat is next hym, fen fou schalt haue
fis nounhre fio5l~l U And fen worch forth as fou diddist afore.
Ami fou j 23 ! schalt vnderstonde fat fou schalt write no 0.
hut whan fat place where foil schal write fat 0 has no figure afore 40 hym nofer after, versus.
20 The Cm/I of Multiplication.
IT Si aut< vm digitus excreuerit articvlusque. Articul//.-.1 svprajiosito dig-ito salit vltra. what toa.. IT II en- is be secunde case, be quych is bis: yf hit happe but
if the result r , ,. . , . , / ,
is a composite bere come a comnosyt, buu sehalt write be digitte ouer be hede 01 be 4 number. r f J > r ,.,.?.„ i
neper figure by be quych bou multipliest be hier figure; and sett be articulle next liym toward be lyft side, as bou diddyst afore, as pus j~~83~j. Multiply 8 by 8, bat wol be 64. Write be 4 ouer 8, bat is
2 ieafi59o. [Jjp_ j to say, ouer be liede of be neb'/' S; & set G, be quych 2is an 8
articul, next after. And ben bou sehalt haue such a nounbre as is
here, j^(MS3:1~j, And ben worch forth.
183 J
11 Si digitus tarae?2 ponas ipx//m super ip.vam. 12
whatifit 11 Here is be thryde ease, be quych is jus : yf hit happe bat of bi
mwltiplicacioun) come a digit, buu sehalt write be digit ouer be hede of be neber figure, by the quych bmi multipliest be hieri figure, for bis nedes no Ensampul. 16
•i Subdita m^/ltiplica non hanc que [incidit] illi Delet earn penitws scribens quod pruuenit inde. ihefourth * Here is be 1 case, be quych is: yf hit be bappe bat be neper
ca t the ' "
w-aft. fig\ire schal multiplye bat figure, be quych stondes ouer pat hgures £\)
hede, bou schal do away be hier figure & sett bere bat pat comys of b//t multiplication). As yf pere come of bat niztltiplicaciori) an articuls buu schalt write bere be hier figure stode a 0. H And write be articuls in be lyft side, yf bat hit be a digit write pere a 24 digit, yf pat hit 'be a composit, write be digii of be composit. And be articul in be lyft side, al bis is lyjt y-nowjt, pere-fore per nedes no Ensampul.
11 Sed si m//ltiplicat alia/// ponas sup- r ipsam 28
Adiu//ges nunv/U/// que/// p/>bet duct//* ear/////. * leaf 1596. «1 Here is pe 5 case, pe quych is pis : yf 4be neber figure schul Jfthecraft!86 nw/ltinlie be hier, and bat hier figure is not recto otier his hede.
And pal neber figun base oper figures, or on figure ouer his hede by 32 multiplication), bal hase be afore, pou sehalt write pat nounbre, be quych comes of pat, ouer all* pe ylko figures hedes, as pus here : j 2361 Multiply 2 1>\ 2, pal wol be I ; sel I ouer pe hede of pal 2. j 234 1 ben6 multiplies be hier 2 by pe nepe?- .">. bat vvol be G. sel 36 ouer his hede 6, multiplie pe hier 2 by pe ne]>er 1, pal wol be 8. do away pe hier 2, pe quych stondes ouer pe hede <>f be figure of 4,
1 'sed' deleted in MS. 6883 in Ms.
5 '|>eu' overwritten on ' bat' marked for erasure.
The Cases of this Craft. 27
and set pere 8. And fou schalt haue fis nounbre here f4683G~i. And
antery f i figures, fat is to say, set pi neper 4 vnder f e [_?_ ! hier •'!,
and set fi 2 other figures nere liym, so fat J>e nefer 2 stonde vndwr 4 fe hier 6, fe quych 6 stondes in \>e lyft side. And pat 3 fat stondes vndur 8, as fus aftur je may se, !~46836~j Now worch forthermore, And nmltiplye fat hier 3 by 2, | 234 ! fat wol he G, set f'd G fe quych stondes ouer fe hede of fat 2, And fen worch as I tajt fe 8 afore.
JH Si suprnposita cifra debet m/dtiplicare > ieai woo.
Prorsus ea/// deles & ibi scribi cifra debet. 11 Here is fe G case, fe quych is fis : yf hit happe fat fe figure ruesixtiicase 12 by fe quych fou schal rawltiplye fe hier figure, fe quych stondes ryght ouer hym by a 0, fou schalt do away fat figure, fe quych oner fat cifre hede. 11 And write fere fat nounhre fat conies of fe niwltiplicaciori) as fus, 23. do away 2 and sett fere a 0. vntZe ] 6 versus.
11 Si cifra m/dtiplicat alia/// posita/// sup/ ir ip.-am Sitq//'' locus supra vacuus super banc cifra/// fiet. H Here is fe 7 case, fe quych is fis: yf a 0 schal mzdtiply a The seventh 20 figure, fe quych stondes not recte ouer hym. And ouer fat 0 craft, stonde no thyng, fou schalt write oner fat 0 anofer 0 as fus : j~ 24~j
multiplye 2 be a 0, it wol be nothynge. write fere a 0 ouer be 1 I
hede of fe neber 0, And fen worch forth til fou come to fe ende. 24 H Si sup/a- fuerit cifra semper est p/ete/eunda.
H Here is fe 8 case, fe quych is fis : yf fere be a 0 or niony Theeighth cifers in fe hier rewe, fou schalt not mwltiplie hem, hot let hem craft, stonde. And antery fe figures benefe to fe next figure sygnificatyf 28 as f us : roo032~l Ouer-lepe alle fese cifers & sett fat 3nefer 2 fat Meafi6o&.
stondes !_^ ' toward fe ryght side, and sett hym vndwr fe 3,
and sett fe ofer nether 2 nere hym, so fat he stonde vndwr fe thrydde 0, f<- quych stondes next 3. And fan worch. vnde versus. 32 H Si dubites, an sit bene m/dtiplicac/o facta,
Diuicle totalem wimernm p'v multiplicante///. H Here he teches how fou schalt know whefer fou base wel I- Kowtoprova
aii lii-iii l''1' multipli-
du or no. And he says fat fou schalt delude alle fe nounbre fat cation. 36 comes of fe nmltiplicacion) by fe nefer figures. And fen fou schalt haue fe same nounbur fat fou hadyst in fe begynnynge. hut }et fou hast not fe craft of dyuision), hut fou schalt haue hit after- wards.
- ' Supra ' inserted in MS. in place of ' cifra ' marked for erasure.
28 The Craft of Multiplication.
U Per numerum si vis nwnemm q>/oq^" m>/ltiplicare U Tcottxm. per normas subtiles absq<7< figuris Has normas pot>ris p< r versus sciiv sequentes. Mental imiiii- 1i Here he teches be to mwltiplie be bowat figures in bi niynde. 4
plication. r ..,',, r
And be sentence of bis verse is bis : yf pon wel mzutrpke on nounbre by anober in bi mynde, bou schal haue bereto rewles in be verses bat schal come after.
U Si tu per digitus digitus vis mwltiplicare 8
Re<//ifa p/vcedens dat quality est operandi!.///. Digit by digit H Here he teches a rewle as bou hast afore to mwltiplie a digit
is eaej'. ' l
' leaf leia. ^c anober, as yf bou wolde wete qwat is sex tymes C>. bon lschalt
wete by be rewle bat I ta$t be before, yf bou haue mynde berof. 12 U Articulii/// si per reliquu/// reliquu/// vis multiplicare
In propriu/// digitii/// debet vterque resolui. H Articul//* digitos post se m//ltiplicantes Ex digitus quociens retenerit multijAicaxi 1 G
Artieuli faciu//t tot centu/// m//ltiplieati. Thefiretcase H Here he; teches be furst rewle, be quych is bis : vf bon \wl
of the craft. . . r ' * * .
mwltiplie an articul be anober, so bat both be articuls bene v/itJi- Inne an hundreth, bus Jwu schalt do. take be digit of bothe the 20 Article by articuls, for euery articul base a digit, ben nmltiplye bat on digit by
article; , r . l J ' ° J
bat ober, and loke how niony vnytes hen in be nounbre bat comes of be umltiplicacion) of be 2 digittes, & so mony bundrytbes ben in be nounbre bat schal come of be mwltiplicacion) of be ylke 2 articuls _ I
anexampie: as bus. yf bou wold wete qwat is ten tymes ten. take be digit of ten, be quych is 1 ; take be digit <>f bat ober ten, be quych is on. H Also mwltipfie 1 be 1, as on tyme on bat is but 1. In on is bul on vnite as bou wost welle, berefort ten tymes ten is but a hun- 2s
anotherex- di'yth. 1! Also yf bou wold wete what is twenty tymes 30. take be digit of twenty, bat is 2: >V take be digitt of tbrytty, bat is 3. mwltiplie 3 be 2, bat is 6. N"ow in t! ben 6 vnites, ' And so mony
Motion, hundrythes hen in 20 tymes 302, berefore 20 tymes .".() is 6 hun- 32 dryth euen). loke »V se. • Bui yf it be so bat on< articul be wiih- Inne an hundryth, or by-twene an hundryth and a thowsande, so bat it be not a bowsande fully, ben loke how mony vnytes ben in he nounbur bat comys of be mwltiplicaciori) 8And so monj tymes3 36 of "_' digittes of ylke articuls, so mony thowsant hen in be nounbre, the qwycb (Mines of }v umltiplicacion). And so monj tymes ten thowsand schal be In be nounbre bal comes of be multiplication of y— 3 M irked for erasure in .VS.
// w to work subtly without Pigun >. 29
2 articuls, as yf bou wold wete qwat is 4 hundryth tymes [two hundryth]. Multiply 4 be 2,1 bat wol he 8. in 8 ben 8 vnites. IT And so mony tymes ten thousand be in 4 hundryth tymes Mental mum-
J «' plication.
i [2]1 hundryth, b"t is 80 thousand. Take hede, I schall telle be a generalle rewle whan bou bast 2 articuls, And bou wold wete qwat Another ex-
o / > A ample.
comes of be mMltiplicacion) of hem 2. nmltiplie be digit of bat on articuls, and kepe bat nounhre, ben loke how mony cifers schuld go 8 before bat on articuls, and be were write. Als mony cifers schuld go before bat other, & he were write of cifers. And haue alle be ylke cifers togedur in bi mynde, 2a-rowe ychon) aftur other, and <ieafi62a. in be last plase set be nounbre bat comes of be mwltiplicacion) of be 12 2 digittes. And loke in bi mynde in what place he stondes, where in be secundc, or in be thryd, or in bo 4, or where ellis, and Like qwat be figures by-token in bat place ; & so mych is be nounbre bat comes of be 2 articuls y-mwltiplied to-gedwr as bus: vf bou wold Another ex-
r J 1 ° . . ample.
16 wete what is 20 thousant tymes 3 bowsande. multiply be digit of bal articulle be quych is 2 by be digitte of bat ober articul be quych is 3, bat wol be G. ben loke how mony cifers schal go to 20 thousant as hit schuld be write in a tabul. certainly 1 cifers schuld go to
20 20 bowsant. ffor bis figure 2 in be fyrst place betokenes twene.
U In be secunde place hit betokenes twenty. r In be 3. place bit Notation. betokenes 2 hundryth. .r. In be 4 place 2 thousant. 1i In be 5 place lu't betokenes twenty bousant. b'/refore he most bane 4 cifers
24 a-fore hym bat he may stowde in be 5 place, kepe bese 4 cifers in ' thy mynde, ben loke bow mony cifers gon) to 3 thousant. Certayn to 3 thousante 3gori) ."> cifers afore. Now cast ylke 4 cifers bat 3 ieafiG2&, schuld go to twenty thousant, And thes 3 cifers bat schuld go
28 afore 3 thousant, & settc hem in rewe ychon) aft''/' obey in bi mynde, as bai schuld stonde in a tabulle. And ben schal bou bane 7 cifers; ben sett bat G be quych comes of be nittltiplicaciori) of be 2 digittes aftwi be ylke cifers in be 8 place as yf bat hit stode in a
32 tabul. And loke qwat a figure of G schuld betoken in be 8 place, yf hit were in a tabul & so mych it is. A; yf bat figure of 6 stonde in be fyrst place he schuld betoken but 6. U In be 2 place he schuld betoken sexty. IT In the •"> place he schuld betoken) sex hundryth.
3G If In be 4 place sex thousant. H In be 5 place sexty bowsant. Notation H In be sext place sex hundryth bowsant. 1F In be 7 place sex bowsant thousantes. H In be 8 place sexty bowsant thousantes. berfore sett G in octauo loco, And he schal betoken sexty bowsant
1 4 in .MS.
30 The Graft oj Multiplication.
Mentaimui- thousantes. And so mych is twenty bowsant tymes •"> fcliousant, H And J^is rewle is generalle for alle man'/- of articuls, Whethir bai be hnndryth or bowsant ; but bou most know well be craft of be i leafiesa. wryrchynge in be tabulle lor bou know to do bus in bi mynde 4. aftur bis rewle. Thou most bat Jris rewle holdybe note but wher< y> n ben 2 articuls and no mo of be quych aytlier of hem lmse but on figure significatyf. As twenty tymes 3 thousant or 3 hundryth, and such opnr. <S
H Articulum digito si m//ltiplicaie oportet Articuli digit[i sumi quo multiplicate] Debem//* rsliquu/// quod m/dtiplicat///- ab Wis Per reliq//o decuplu/// sic summ&m latere neq"/b/t. 12
The third H Here he puttes be thryde rewle, be quych is bis. yf bou wel
craft i multiply in bi mynde, And be Articul be a digitte, bou schalt loke
bat be digitt be \v«t/i-Inne an hundryth, ben bun schalt multiply the digitt of be Articulle by be ober digitte. And euery vnite in }>e 16 nounbre bat schalle come J?ere-of schal betoken ten. As bus: yf an example, bat bou wold wete qwat is twyes to. m?dtiplie J?e digitte of -10, be quych is 4, by be ober diget, be quych is 2. And bat wolle be 8. And in be nombre of 8 ben -s vnites, & euery of pe ylke vnites 20 schidd stonde for 10. pen fore ber< schal be 8 tymes 1", J>a1 wol lie 1 score. And so mony is twyes 10. * If be articul be a hun- dryth or ho 2 hundryth And a bowsant, so bat hit be notte a -1 ieafi6S6. thousant, 2worcb as poit dyddyst afore, saue bon schali rekene euery 24 vnite for a hundryth.
11 In nu///'/U/// mixtu//' digitus si ducer- cures Articul"- mixti sum&tur deinde resoluas In digitu/// post fac respectu de digitis Articul/ sq//< docet excrescens in diriuawdo In digitu/// mixti post ducas m?iltiplicarctew H De digitis vt norma1 (docet] de [hunc] Multiplica simwl et sic postea summa patebit. 32
The fourth Here he puttes be I rewle, be quycb is Jus : yf bun niMltipliy
CA96 of t llC
crafi on composit be a digit as <"> tjrmes 2 1. lben take |>e digcl of psA com-
posit, & multiply bed digitt by bai ober diget, and kepe be nombur bai comes pert of. ben take j)e digil of bat composii, & nu/ltiply bat 36 digit by anober diget, by }>e quych bou hast mwltiplyed be diget of
Composite be art icul , and loke qwal comes beri of. ben take bou bat uounbur,
by digil ' , , ' ' . '
\- cast hit to bat other nounbur bai pou secheste as pus yl pou wet :; docet. decel MS. ' ' I times 1 ' in MS .
How to multiply without Figures. 31
wete qwat conies of 6 tymes 4 & twenty, multiply bat articxillc of Mental mui-
t ■ /. i tiptication.
pe composit by pe digit, be quych is 6, as yn be thryd rewle b"ii ■was tau$t, And bat schal be C> score, ben multiply be diget of be 4 composit, Jpe quych is 4, and multiply bat by pat other diget, be iieafi6ia. quycli is 6, as bou wast tau^t in be first rewle, yf pou haue mynde berof, & pat wol be 4 & twenty, cast all ylke nounburs to-gecKr, & hit schal lie 144. And so mych is 6 tymes 4 & twenty. 8 11 "Ductus in articulu/// nttmerws si compo&itus sit
Articuluw. puru/// comites articulu/// quoque Mixti pro digit/* post fiat [et articulus vt] Norma iubet [retinendo quod extra dicta ab illis] 12 Articuli digitus post tu mixtii/^ digitus due
"ELegula de digitis nee p/< cipit articulwsqwe Ex quib>/>' exc/v scens su///me tu iunge pn'ori Sic ma/ifesta cito fiet tibi su///ma petita. 16 If Here he puttes be 5 rewle, be quych is bis: yf bou vvel The fifth case
,, . i , .. , , .. , . , . . , of the craft :
multiply an Articul be a com posit, niwltipne bat Articnl by be articul of pe composit, and worch as bou wos tai^t in be secunde rewle, of be quych rewle pe verse begynnes pus. U Articulu//? si Article by 20 pe;1 Relicuw vis mwltiplicare. ben multiply pe diget of pe composit omposi by pat op/V articul afl/V pe doctrine of pe 3 rewle. take Jvrof gode hedc, I pray pe as pus. Yf pou wel wete what is 24 tymes ten. Multiplie ten by 20, pat wel be 2 hundryth. pen multiply pe diget Anexampie. 24 of pe 10, pe quych is 1, by pe diget of pe composit, pe quych is 4,
& pat 2wol be 4. pen reken euery vnite pat is in 4 for 10, & pat Meafiei*. schal be 40. Cast 40 to 2 hundryth, & pat wol be 2 hundryth & 40. And so mych is 24 tymes ten. 28 « Compositu/// nunvrii/// mixto si[c] m>dtiplicabis
Vndecies tredeci/// sic est ex hiis op'/andum In reliquu/// jiHmum demu/// due post in eund^// Vnu/// post den//m due in tn'a dei//de per vnu/// 32 Multiplices'/?" dem//m intra omnia, m//ltiplicata
In su/;/ma decies q//om si fuerit tibi doces Multiplicandoiv//// de normis sufficiunt hec. H Here he puttes pe 6 rewle, & be last of alle multiplicacion), The sixth case 36 pe quych is pis: yf J>ou wel mwltiplye a romposit by a-noper com- posit, bou schaltdo bus. rmdtiplie bat on composit, qwych bou welt composite by
. ,, • i j. Composite.
ot the twene, by be articul oJ pe toper composit, as pou were tat^t in
pe 5 rewle, J-en mwltiplie pal same composit, pe quych bou hast
40 multiplied by pe oper articul, by be digit of be ober composit, as
'■'■>- How to work wit In ) u I Figures.
Mental mui- h<m was taint in be I rewle. As bus, yf bull wold wete what is 11
tiplication. r ? * Y " '
An example tymes 13, as bou was taujt in be 5 rewle, & bat sehal he an hun- dryfch & ten, aftenvarde multiply bat same composit pat bou hast multiplied, be quych is a .11. And nwltiplye hit be be digit of be 4 oper composit, be quych is 3, for 3 is be digit of 13, And pat wel he 30. ben take be digit of bat composit, be quych composit bou iieafi65«. multiplied by be digit of bat oper composit, Jpe quych is a 11. of the sixth 1[ Also of be quych 11 on is be digit, nmltiplie bat digitt by be 8 craft. digctt of pat other composit, be quych diget is 3, as bou was tau^t in
be first rewle m be begynnynge of pis craft, be quych rewle begynnes "In digitus? cures." And of alle be nraltiplicacion) of be 2 digitt comys thre, for onys 3 is hut 3. Now cast alle pese nounhers 12 toged?^-, the quych is pis, a hundiyth & ten & 30 & 3. And al bat wel be 143. AVrite 3 first in be ryglit side. And cast 10 to 30, bat wol be 40. set 40 next after towardc be lyft side, And set aftur a bundryth as here an Ensampulle, 143. 16
(Cetera desunt.)
33
^Ijc %xt jof |lom.brnng.
A TRANSLATION OF
Sotjn of Iftotofoooti's ©r Hvtr Numcrantu.
B
T
[Ashmole MS. 396, /o/. 48.]
oys seying in the begynnyng of his Arsemetrikc : —AIL' thynges that bem fro the first begynnyng of thynges Ful- 4S- have p/v^ceded', and come forth'', And by reso/m of nombre ben formed' ; And in wise as they bene, So oweth'' they to be knowen- ; wherfor in vniu'/sall'' knowlechyng of thynges the Art of nombrynge is best, and most operatyfe.
herfore sithen the science of the whiche at this lyme we intendene to write of standithe alle and about nombre : The name of
. the art.
Hirst we most se, what is the propro name therofe, and fro whena the name come: Afterward',' what is nombre, And how manye spices of nombre ther ben. The name is clepede Algorisme,
12 hade out of Algore, other of Algos, in grewe, That is clepide in Derivation of englisshe art other craft, And of Rithmws that is callede nombre. So algorisme is cfepede the art of nombryng, other it is had ofe en or in, and gogos that is iutroducciown, and Rithimts nombre, thai is Another.
16 to say Interducciown of nombre. And thirdly it is hade of the name of a kyng thai is clepede Algo and Rythmws; So callede Algorisms. Sothely .2. manere of nombres hen notifiede; Formalle,1 as nombre is vnitees gadrede to-gedres ; Materially2 as Another.
20 nombre is a collecciowri of vnitees. other nombre is a multitude had'-' out of vnitees, vnitee is that thynge wher-by enery thynge is called'? oone, other o thynge. of nombres, that one is clepede digitalle, that othere Article, Another a nombre componede obe?'
24 myxt. Another digitalle is a nombre w/t//-in .10.: Article is batKm<38<>f
' numbers,
nombre that may be dyvydede in .10. parties egally, And that there
1 MS. Materially « MS. Formalle.
NOMBRYNGE. n
3 f ( 'ha/pU r 1. Numeration.
leve no residue; Componede or medlede is thai nombre that is come of a digite and of an article. And vndrestande wele that alle nombres betwix .2. articles nexl is a nonibr< componede. Of this The n mi, s art bene .9. spices, thai is forto sey, nume/'aciottn, addicioun, Sub- 4 tracciozm, Mediawozm, Duplaciomi, IVlultipIiacio?m, 1 )yvysiown, Pro- gression!), And of Rootes the extracciown, ami that may be hade in .2. manors, that is to sey in nombres quadrat, and in cubices: Amonge the whiche, ffirst of Nume?"aciown, and afterwarde of J>c s ojjers l»y ordure, y entende to write.
> Poi. 48i ^or-soth? numeracioun is of eivry numbre by competent figures an artificialle rep/vsentacio/m.
Figures, /* ^J otlily figure, difference, places, and lynes supposen o tbyng 12
piaces,"mid ,^^^ other the same, Bui they ben sette here for dyuers resons. f^^y fiigure is clepede for protracciown of figuracioun : 1 >ifference is railed'' tor therby is shewede eue?'y figure, how it hath* difference fro the figures before them: place by cause of space, where-in me L6 writethe : lynees, for that is ordeyned* for the presentaciown of
The o figures, euery figure. And vnderstonde (hat ther ben .9. lymytes of figures that representen the .'.>. digites that 1 »< • 1 1 tin'-', it. 9. 8. 7. 6.
Thecipher. 5. 4. 3. 2. 1. The .10. is clepede theta, or a cercle, other a cifre, 20 other a figure of nought for nought it signyfiethe. Nathelesse she holdyng that place giveth< others for to signyfie ; for with* oul cifre or cifres a pure article may not be writte. And sithen that by
rhenumera- these .'.'. figures significatifes [oynede with cifre or with cifres alle 21 nombres ben and may be repn sentedi . It was, nether is, no nede to
ofdigits, fynde any more figures. And note wele that euery digite shall* be writte with oo figure allone to it aproprede. And alle articles by
of articles, a cifre, ffor euery article is namede for oone of the digitis as .10. of 28 1.. 20. of. 2. and so of the others. &c. And all- nombres digitalle
nwell to lie Sette ill the fil'Si dillerelice : All articles ill the seeulide.
Also all- nombres fro . I<». til an .100. [which] is excluded* , with .2. figures inv i l>e writte; And yf it he an article, by a cifre first put, 32 and the figure y-writte toward' the lift honde, thai signifiethe the digit of the whichc fche article is named' ; And yf it be a nombre ofcompo- componede, ffirst write the digil that is a part of that componede,
and write to the lift side the article as it is seid* he fore. All- 36 nombre that is fro an hundred* tille a thousand* exclusede, owithe to lie writ by .3. figures ; ami alle nombre that is fro a thousand''
Chapter II Addition. 35
til .x. Mt. mvsl be writ by A. figures; And so forth e. And vnder- stond* wele that euery figure sette in the first place signyfiethe his lvalue
J ° J ° ^ due to posi-
digit ; In tin' seconde place .1". tynies his digit; In the .3. place an tiol>- 4 hundrede so moche ; In the .4. place a thousands so moche ; In the .5. place .x. thousande so nioche; In the .6. place an hundrede thousands so moche ; In the .7. placea thousande thousande. And so infynytly mvltiplying by 'these .3. 10, 100, 1000. And vnder- > Foi.49. 8 stande wele that competently me may sette vpon figure in the place of a thousand/', a prike to shewe how many thousand'' the last figure shade represent. We writene in this art to the lift side-warde, as Numbers are
■ written from
arabiene writene, that weren fynders of this science, othere for this right to left.
12 resomi, that for to kepe a custumable ordre in redyng, Sette we alle-wey the more nombre before.
ddiciottn is of nombre other of nombres vnto nombre or to nombres aggregation, that me may see that that is come Definition.
10 1 \ therof as excressent. In addicio/m, 2. ordres of figures and .2. nombres hen necessary, that is to sey, a nombre to he addede and the nombre wherto the addiczoun sholde he made to. The nombre to he addede is that bat sholde be added/' therto, and shade
20 he vndervvriten ; the nombre vnto the whiche addiciomi shalle be made to is that nombre that resceyuethe the addicion of fat other, and shall/' he writen above; and it is convenient that the lesse Ho«- the
ii i • iii iiii numbers
nombre be vndenvrit, and the more addede, than the contrary, should be
written.
24 But whether it happe one other other, the same comythe of, Therfor, yf bow wilt adde nombre to nombre, write the nombre wherto the addiciown shalle be made in the omest ordre by his differences, so that the first of the lower ordre he Andre the first
28 of the omyst ordre, and so of others. That done, adde the first of The method the lower ordre to the first of the omyst ordre. And of suche addition, other J?ere gvo\vit7i therof a digit, An article, other a composed/'. If it he digitas, In the place of the omyst shalt thow Betfn at the
32 write the <li- i t excrescyng, as thus : — ■
A
The resultant I 2 If the article; in the place of the The sum is
To whom it slml be addedfl | 1 l)|llV,, put a.way },v a 0jfre writte, *'
The »omi,i-p. to be added/- | l and t]l(. (]igi| transferrede, of be 36 which/' the article toke his name, toward/' the lift side, and be it added/' to the next figure folowyng, yf fcher be any figure folowyng ; or no, and yf it he not, leve it [in the] voide, as thus : —
36
Chapter Til. Subtraction.
or an article.
The resultant
in
To whom it shall? he added':
The noinbre to be addede
|
Resultans |
|2|7|8|2|7 |
|
Cui debet addi |
| 1 I 0 j 0 | 8 | 4 |
|
Numcriw addendws |
|1|7|7|4J8 |
• Fol. 49 A.
or a compo- site.
The trans- lator's note.
Definition of Subtraction,
How it may be done,
What is re- quired, i
And yf it happe that the figure folowyng wherto tlio addiciown shalle be made by [the cifre of] an article, it sette a-side ; In his place write the ] [digit of the] Article, The resultant j 1 7
as thus : —
Te whom it shalh be addede | 10
The nombre to be added'-
I 7
And yf it happe that a figure of .9. by the figure that me mvst adde [one] to, In the place of that 9. put a cifre and write be article toward'' be lift honde as bifore, and thus : —
|
The resultant |
1 io |
|
To whom it shalle be addede |
1 9 |
|
The nombre to be addede |
I 1 |
And yf2 [therefrom grow a] nombre componed,3 [in the place of the nombre] put a- way4 [let] the digit [be]5 writ bat is part of bat composide, and ban put to be lift, T]|l, ,vsu|(;m, side the article as before, and
bus : —
I 12
To whom it shall be added* | 8 12
The nombre to be addede
I 4
This done, adde the scconde to the second'', and write above ober as before. Note wele bat in addic/ons and in alb- spices folowyng, whan he seithe one the other shall' be writen aboue, and me most \q vse eiUT figure, as that v\v ry figure were sette by half', and by hym-selfe.
Subtraction is of .2. proposede nombres, the fyndyng of the excesse of the more to the lasse ; Other subtraccioMn is ablatio Mil of o nombre fro a nother, that me may see a some left. The lasse of the more, or even of even, may be w/t/'draw ; The more fro the lesse may neuej' be. And sothly that nombre is more thai hath' more figures, So that the last be lignyficatifes : And yf ther ben as many in that one as in thai other, me most deme it by the last, other by the next last. More-oue?1 in w/t//- drawyng .'_'. nombres hen necessary: A nombre to be w/t/«lraw, And a nombre that me shalh w/t//-draw of. The nombre to be w/tA-draw shall- be writ in the lower ordre by his differences j The
20
24
28
3 'the' in MS.
■ ■ be ' in MS. 5 'is' in MS.
4 'and" in .MS.
The remanent
20
Wherof me shalle withdraw | 22
The nombre to be withdraw
Chapter III. Subtraction.
nombre fro the whiche mo shalle withe-draw in the omyst ordre, so that the first be vnder the first, the seconde vnder thg second?, And so of allc others. Withe-draw therfor the first of the lowere 4 ordre fro the first of the ordre above his hede, and that wolle be other more or lesse, ober egall?. yf it be egalle or even the figure sctte beside, put in his place a 8 cifre. And yf it be more put away be?"fro als many of vnitees the lower figure eonteynethe, and writ the residue as thus
12 _ And yf it he
lesse, by-cause the more may not be wiih-
1G draw ther-fro, borow an vnyte of the next figure that is worthe 10.
Of that .10. and of the figure that ye wolde have w/tA-draw fro
be-fore to-gedre Ioynede, wtt/i-draw be figure be-nethe, and put the
residue in the place of the figure
20 put a-side as bws :—
37
Write tlie greater num- ber above.
Subtract the first figure if possible.
|
The remanent | 2 2 |
|
Wherof me shalle w/tA-draw | 2 | 8 |
|
f>e nombre to be withdraw | 6 |
|
Remaneiis | 2 | 2 | 1 | 8 | 2 | 9 | 9 | 9 | 8 |
|
A quo sit subtraccio | 8 | 7 | 2 | 4 | 3 | 0 | 0 [ 0 | 4 |
|
Numerus subtrahends | 6 | 5 | 2 |[6] | . | . | . | . | 6 |
1 Fol. 50.
If it is not possible ' borrow ten,
ami then sub- tract.
The remanent
1 I 8
And yf the figure wherof me shal borow the vnyte be one,
Wherof me shalle witA-draw | 2 | 4
The nombre to be wtt/i-draw 10 16
If t lie second figure is one.
put it aside, and write a cifre in the place bm>f, lest the figures 24 folowing fade of thaire nombre, and ban worche as it shew/tA in
this figure here : —
And yf the vnyte wherof me
shal borow be a cifre, go 28 ferther to the figure signy-
ficatife, and ther borow one, and retowmyng bake, in the place of
euer)' cifre fat ye passide oue?*, sette figures of .'.». as here it is
specified*' : — 32 And whan me comethe
|
The remanent | 3 | 0 | 93 |
|
Wherof me shal witA-draw | 3 | 1 | 2 |
|
The nombre to be witA-draw . | . | 3 |
If the secon.l figure is a cipher.
|
The remenaunt |
1-2 | 9 |
!' |
9|9 |
|
Wherof me shalle n ith draw |
|3|0 |
" |
ii | 3 |
|
The nombre to be \\ itA-draw |
1 1 |
14 |
to the nombre wherof me intendithe, there re maynethe alle-wayes .10. ffor J'e whiche .10. &c. The reson why 3G bat for euery cifre leff behynde me setteth figures ther of .9. this it is : — If fro the ..'!. place me borowede an vnyte, that vnyte by respect of the figure that he came fro repj'esentith an .C, In the
A jil^titi. :i- tiim of I he rule given.
2 6 in MS.
3 Oin .MS.
38 Chapter IV. Mediation.
place of that cifrc [passed over] is left .9., [which is worth ninety],
and yit it remaynethe as .10., And the same resone wolde be yf
me hade borowede an vnyte fro tlie .4., .5., .G., place, or ony
other so vpwarde. This done, withdraw the seconde of tlie lower 4
ordre fro the figure above his hede of be omyst ordre, and wirche
why it is as before. And note wele that in addicion or in subtracciown me
workfrom m;iv wele fro the lift side begynne and ryn to the right side, l.ut it right to left. • J»' . " . ° '
wol be more profitabler to be do, as it is taught. And yf thow 8
How to prove wilt p?'ove yf thow have do wele or no, The figures that thow hast
withdraw, adde them ayene to the omyst figures, and they wolle
accorde with the first that thow haddest yf thow have labored
nnd addition, wele ; and in like wise iii addiciozin, whan thow hast added-' w\\e 12
iFoi.561. thy figures, w/t//draw them that thow first 'addest, and the same
wolle reto///-ne. The subtraccio/m is none other but a prouffe of the
addiciown, and the contrarye in like wise.
Definition of ~M M~ediacio*m is the fyndyng of the halfyng of euery nombre, 16
that it may he seyne what and how moche is euery half'.
M
In halfyng ay oo order of figures and oo nombre is neces- sary, that is to sey the nonibre to be halfede. Therfor yf thow wilt half any nombre, write that nombre by his differences, and 20 whereto besrvnne at the right, that- is to sey, fro the first 'figure to the right
begin. OJ .
side, so that it he siguyncatife other represent vnyte or eny other
digitalle nombre. If it he vnyte write m his place a cifre for tlie
[fthefn-st figures folowyng, [lest they signify less], and write that vnyte 24
unity. w/t//oiit in the table, other resolue it in .00. mynvtes and sette a-
side half of tho minutes so, and reserve the remenawnt w-'t//ont in
the table, as thus ..'Hi. ; other sette wzt/touf thus .,/, : that kepethe
none ordre of place, Nathelesse it bathe signyficaciown. And yf -,s
the other figure signyfie any other digital nombre fro vnyte forthe,
wimt to do ober the nombre is ode or evene. If it he
if it is m,t . . . Halfedi 2 12
unity. even, write tins halt in this wise : —
. to bi halfedi i I
And it It lie Odde, lake tlie lie\l eVell Vlldl'e
32
liym conteynede, and put his half in the place of that odde. and of be vnyte that remaynethe to he halfed
do thus : —
Then halve This done, the SCCOnde is to he halted- . \ f
the Bei'iind
balfed<
To be halfed.
[di] 36
tignre. it be a cifre put it be side, and yf it he significatifi . other it is even
or od( : It' it he even, write in the place of be nombres wiped-- out the half- : yf it he od . take the next even vnder it wwtenythe, and in the place of the lmpar sette a-side put half of the even: The 40
Chapter J". Duplation.
89
vnyte that remaynethe to be halfede, respect hade to them before,
is worthe .10. Dyvide that .10. in .2., 5. is, and sette a-side that x™ u odd>
J ' add "• t" the
one, and adde that other to the next figur( 4 precedent as here : — And yf fe addicioam sholde ho made to a cifre,
Halfed<
to be halfede
sette it a-side, and write in his place .5. And vnder this fozmne me shalle write ami worch< . 8 till- tin' total le nombre be halfede.
|
1 doubled*; | 2 | 6 |
8 | 9 | 0 |
10| 17 | 4 |
|
to be doublede | 1 | 3 |
4|4|5 |
5| 8|7 |
D
uplicaciotm is asr< p acion of nombre [to itself] bat me raav se Definition of
i i t -i " i i ' Duplation.
the nombre growen. In doublynge ay is but one ordre of 12 | J figures necessarie. And me most be-gynne with the lift side, other of the more figure, And after the nombre of the more figure representithe. 1In the ether .3. before we begynne alle way ' r-oi.si. fro the right side and fro the lasse nombre, In this spice and in alle whereto
° L begin.
16 other folpwyng we wolle begynne fro the lift side, ffor and me bigon the double fro the first, omwhiie me myght double oo thyn'ge twyes. And how he it that me myght double fro the right, that why. wolde be harder in techyng and in workyng. Therfor yf thow
20 wolt double any nombre, write that nombre by his differences, and double the last. And of that doublyrcg other growithe a nombre digital, article, or componede. [If it be a digit, write it in the place of the first digit.] If it he article, write' in his place a cifre
24 and transferre the article toward' the lift, as thus : — And yf the nombre be componede, write a
double
to be doubled*
Hi
n ith the 1 1 suit.
doubled* to I"' donblei
16
digital that is part of his coinposiciomi, and sette the article to the 28 lift hmd', as thus:—
That done, me most double the last save one,
and what growethe peroi me most worche as
before. And yf a cifre be, touche it not. But yf any nombre
32 dial]- be added- to the cifre, in be place of be figure wipede out
me most write the nombre to be addede, •
i doubled* | 6 | 0 6|
to be doubled* 3 | 0 j 3
as thus : —
In the same wise me shalle wirche of 36 alle others. And this probacio?m : If thow truly-double the halfis
and truly half the doubles, the same
nombre and figure shalle mete, m hi n
thow labow>-ed< rpone first, And of the 40 contrarie.
Ifow to provi j our answer.
led* to be doubled*
1 ^
40
Definition of Multiplica- tion.
Multiplier. Multiplicand.
M'
Chapter VI. Multiplication.
ultiplicaciown of nonibre by hym-self other by a-nother, wiUi proposide .2. nombres, [is] tlie fyndyng of the thirde, That
so nft uteynethe that other, as tlier ben vnytes in the
In multiplicackmn .2. nombres pryncipally ben necessary, 4 that is to sey, the norabre multiplying and the nombre to be multiplied^, as here ; — twies fyve. [The number multiplying] is designede adue^bially. The nombre to be multiplied^ resceyvethe a no»ii??alle appellaciotm, as twies .5. 5. is the nombre multiplied^, 8 and twies is the nombre to be multipliede.
of
|
j Reaultans | * | 1 | 0 | |
1 |3 | 2 |
6 | 6 | 8 |
0 | 0 | 8 |
|
| Multiplicand's | . | . | 5 | |
. | . | 4 |
• |3|4 |
0 | 0 | 4 |
|
J Multiplicans | . | 2 | 2 1 |
. | 3 | 3 |
- : - 1 - |
. | . | . |
Product.
* Fol. 51 6.
There are G rules .if Mul- tiplication.
I Digit by digit.
Also me may thervpone to assigne the. 3. nombre, the whiche is 2clepede product or p?*ovenient, of takyng out of one fro another: as twyes .5 is .10., 5. the nombre to be multiplied^, and .1*. the 12 multipliant, and. 10. as before is come therof. And vnderstonde welf, that of the multipliant may be made the nombre to be mul- tipliede, and of the con- trarie, remaynyng eue?1 the same some, and her- ofe comethe the comen speche, that seithe all
nombre is convertede by 6 j_l2 | 18 | 24 j 30 | 36 ! 12 ; 48 | 56 j 60 Multiplying in hym selfe. And thcr ben .6 rules of Multiplicacioftn ; ftlrst, yf a digit multiplie a -
|
1 | 2 |
3 |
1 |
5| 6| 7 |
8 |
9| 10 |
|
2| 4 |
6 |
8 |
in i<>:; 1 1 |
16 |
18 ! 20 |
|
3 | 6 |
9 |
12 |
15 | 18 | 21 |
24 |
| 27 | 30 |
|
4 | 8 |
12 |
16 |
| 20 | 24 j 28 |
32 |
1 36 | 40 |
|
5 | 10 |
15 |
20 |
25 | 30 | 35 |
40 |
| 45 | 50 |
|
6 | 12 | 7 11 |
18 |
24 |
30 ' 36 1 42 |
L8 |
56 | 60 |
|
21 |
28 |
35 12 in |
56 |
||
|
8 | 1(5 | |
24 |
32 |
40 | 48 | 56 |
64 |
|
|
9 | 18 |
27 |
36 |
15 54 63 |
72 |
81 J 90 |
|
in | 20 | |
30 |
■10 |
60 1 ro |
80 |
90 | 100 ': |
See Hi. table ubove.
■1 Digit l.y :n tide.
digit, considre how many of vnytees ben betwix the digit by multi- plying and his .10. bethc to-gedre accomptede, and so ofl with -draw the digit multiplying, vnder the article of his denomiwaciozm. Example of grace. If thow woll wete how mocht is .1. tymes .8., 28 *se how many vnytees ben betwix .8.5 and .10. to-geder rekenede, and it shrw/iA that .'_'.: withdraw ther-for the quaternary, of the article of his denomination twies, of .1".. And ther remaynethe" .32., thai is, to some of all< the multiplicaciowu. Wher-vpon for 32 more evidence and declaration the seid< table is made. Whan a digit mulliplieth* an article, thow most bryng the di.it into \>e digit, of \>e which* the article [has]0 his name, and euery vnyte
MS.
• 1 ih
" SIC.
in-' rted in Ms.
Ami ' insei ted in MS.
' 'to' in MS.
The Cases of Multiplication.
shall*- stondfi for .10., and euery article an .100. Whan the digit niultipliethe a nombre componede, Jjou most bryng the digit into aiber part of the nombre componede, so pat digit be had into digit 4 by the first rule, into an article by be seconds rule; and afterwarde Ioyne the producciown, and pere vvol be the some totalle.
41
:; Composite by digit.
|
Resultans |
| 1 | 2 | 6 |
7 | 3 ! 6 |
1 |
2(0' |
1 j 2 | 0 | 8 |
|
Multiplicands |
1 1 |2 |
| 3 | 2 |
|6 1 |
MM |
|
|
Multiplicans |
1 1 6 | 3 |
2|3| |
2 | 0 | |
1 3 | 0 | 2 |
"Whan an article multipliethe an article, the digit wherof he is namede is to be brought Into the digit wherof the oper is nainede, 8 and euery vnyte wol he worthe xan .100., and euery article, a .1000. Whan an article multipliethe a nombre componede, thow most bryng the digit of the article into aitlier part of the nombre componede; and Ioyne the p/v/duecio//n, and euery article wol be
12 worthe .100., and euery vnyte .10., and so wolle the some be opene. Whan a nombre componede multipliethe a nombre com- ponede, euery part of the nombre multiplying is to be hade into euery part of the nombre to be multiplied^', and so shalle the digit
16 be hade twies, onys in the digit, that other in the article. The article also twies, ones in the digit, that other in the article. Ther- for yf thow wilt any nombre by hym-self other by any other multiplie, write tlie nombre to be multipliede in the ouer ordre by
20 Ids differences, The nombre multiplying in the lower ordre by his differences, so that the first of the lower ordre be vnder the last of the ouer ordre. This done, of the multiplying, the last is to be hade into the last of the nombre to be multipliede. Wherof than
2 1 wolle grow a digit, ari article, other a nombre componede. If it be a digit, even above the figure multiplying is hede write his digit that come of, as it apperethe here : — The resultant HT
I Article by article.
5 Composite
by article.
(6 Composite by composite.
How to set down your numbers.
If the result is a ilk'it.
To be multiplied!
pc nombre multipliyng | 2 And yf an article had be writ oue?" the figure multiplying his hede, 2S put a cifre per and transferre the article towarde the lift hande, as
thus : —
in arti.le,
|
The resultant |
|1 |0 |
|
to 1"- mull iplii dfl |
1 1 5 |
|
be nombre multipliyng |
1 |2 |
And yf a nombre componede be writ ouer the figure multyplying is hede, write the digit in the nombre componede is place, and sette
32 the article to the lift hand'-, as thus : — •
or a compo- site.
42
Chapter VI. Multiplication.
|
The resultant Co be multiplied^ |
1 |2 1 M |
|
the nombre multiplying |
I 13 |
|
Resultant |
l« |
6 |
|
|
to be multiplied^ |
1 |
3 |
|
|
the nombre nwdti |
>liyng |
! 2 |
2 |
|
The resultant | 1 | 1 | 0 |
|
tn be multiplied*! | 5 |
|
be nombre multiplying 2 i: |
Multiply next _ TlllS doilO, 1110 lllOSt blVim the last
by the last The resultant i 1 ] 2 ' ,. .J .?
but one, and save one 0f the iiiultn>hyii£ mto
the last of be nombre to be multi- plied'-, and se what comythe tberof as before, and so do with alle, tille me come to the first of the nombre multiplying, that must be brought into the last of the nombre to be multiplied'', wherof growithe obe?" a digit, an article, i Foi.526. * other a nombre componede. If it be a digit, In the place of the ouerer, sette a-side, as here :
If an article happe, there put a
cifre in his place, and put hym to
the lift hande, as here :
11' it be a nombre componede, in
the place of the ouerer sette a-side, write a digit that9 is a part of
the componede, and sette on the
left honde the article, as here :
Then nnter.y That done, sette forwards the the multiplier
one place. figures of the nombre multiplying
by oo difference, so that the first of the multiplianl be vnder the last save one of the nombre to be multipliede, the other by o place sette forwarde. Than me shall brynge the last of the multipliant in hym to be multipliede, vnder the whiche is the first multipliant. Ami than wolle growe obe?" a digit, an article, or a componedi nombre. If it he a digit, ad<le hym even above his hede ; Lf it be an article, transferre hym to the lift side; And if it Le a nombre componede, adde a digit to the figure above his hede, and sette to the lift hand'- the article. And all- waves euery figure of the nombre multipliant is to be brought to the last save one nombre to be multiplied^', til me come to the first of the multipliant, where me halh win-he as ii is seide before of the first, and afterward* to put forward* the figures by o difference and one tille they alle be multiplied'. And yf it happe that the firsl figure of pe multi- pliant be a cifre, and boue it is sette the figure signyficatife, write a cifrc in the place of the figur* sette a side, as thus, etc. :
|
The resultant |
| 1 | 3S| 2 |
|
to be multipliede |
1 1' |4 |
|
be nombre multipliant |
1 1 3 | 3 |
L2
1G
20
Work aa I.. fore.
How to ileal wuh ciphers,
|
The resultanl |
1 |
|2 |
0 |
|
to 1"' multipliede |
|6 |
||
|
i lie multiplianl |
|2 |
11 |
'•' ■ thai ' repi ited in .MS.
-1 • J ' iri MS
Chapter VII. Division.
43
|
The resultant |
[ 2 | 2 | 6 | 4 | 4 |
|
To lie multiplied*' |
1 | | 2 | 2 | 2 |
|
The multipliant |
|1|0|2| | | |
|
Resultant | 8 | 0 | 0 | 8 | |
|
to be multiplied*' | 4 j 0 | 0 | 4 | |
|
the nwltiplianl 2 | . | . | . | |
|
j Resultant |
3 |
2 | 01 |
|
|
l To be mwlt: |
pliede |
8 ! 0 |
|
|
1 The nmltip |
!iaut |
1 |
4 1 |
And yf a cifre happe in the lower order be-t\vix the first and the
last, and even above be sette the figure signyficatif, leve it vn-Howtode
' ° * with ciplu
fcouchede, as here : — 4 And yf the ?pace above sette be
void'', in that place write thow
a cifre. And yf the cifre happe
betwix be first and the last to be rmdtipliede, me most sette 8 forwards the ordre of the figures by thaire differences, for oft of
ducciomi of figures in cifres nought is the resultant, as here, 1\vherof
it is evident and open, yf that
the first figure of the nombre be 12 to be multipliede be a cifre, vndir
it shall? be none sette as here : —
Vnder Tstandl also tliat in multiplica- J-eave room
L J i between the
ciozm, divisio/m, and of rootis the ex- rowa.°l
figures.
10 ,-tt,, rrr-^r. — tracciown, competently me may leve
a mydel space betwix .2. ordres of figures, that me may write there what is come of addyng other withe-drawyng, lest any thynge sholde be ouerdiippede and sette'
20 out of mynde.
For to dyvyde oo nombre by a-nother, it is of .2. nombres pro- Definition or ' ' . J l division.
. posed*', It is forto depart the moder nombre into as many partis as ben of vnytees in the lasse nombre. And note 24 wele that in makynge of dyvysiown ther ben .3. nombres necessary :
that is to sey, the nombre to be dyvydede; the nombre dyvydyng Dividend, and the nombre exeant, other how oft, or quocient. Ay shalle the Quotient. nombre that is to be dyvydede be more, other at the lest evene with 28 tbe nombre the dyvysere, yf the nombre shalle be made by hole
nombres. Therfor yf thow wolt any nombre dyvyde, write tbe?owtoset
* ^ "' J down your
nombre to be dyvydede in be ouerer bordure by Ids differences, the Sum* dyvisere in the lower ordure by bis differences, so thai the last of 32 the dyviser be vnder the lasl of the nombre to be dyvyde, the next last vnder the next last, and sunt' the others, yf it may <■< mpetently be done ; as here : —
|
The residue |
1 1 2 |
7 |
|
The quotienl |
1 1 |
5 |
|
To be dyvydede |
|3|4 |
2 |
|
The d\ \ \ ser |
I |« |
3 |
An example.
1 Blank in Ms.
44
Chapter VII. Division,
Examples.
When the last of the divisor must not be set below the last of the dividend.
i Fol. 5:P.
How to begin.
An example.
|
Residuum | 8 || | | 2 | 7 || | 2 j 6 |
|
Quociens \ | 2 | 1 2 | 2 || | | 5 || | | 9 |
|
Diuidendus 6 8 0 6 | 6 || 3 | 4 | 2 || 3 | 3 | 2 |
|
Diuiser | 3 | 2 | || 3 | || | 6 | 3 || | 3 | 4 |
Where t>> set the quotiente
And thcr ben .2. causes whan the last figure may not be sette vnder the last, otlier that the last of the lower nombre may not be with' draw of the last of the ouerer nombre for it is lasse than the lower, other how be it, that it myght be wtt/i-draw as for iiyin self fro 4 the ouerer the remenaunt may not so oft of them above, other yf be last of the lower be even to the figure above his hede, and be next last oJ>er the figure be-fore brtt be more ban the figure above sette. ] These so ordeynede, me most wirche from the last figure of 8 be nombre of the dyvyser, and se how oft it may be wtt/i-draw of and fro the figure aboue his hede, namly so that the remenaunt may be take of so oft, and to se the residue as here : —
And note wele thai me may not withe- 12 draw more than .9. tymes nether lasse than ones. Therfor se how oft be figures of the lower ordre may be wiili- draw fro the figures of the ouerer, and the nombre that shew/t// be 1G qjfocient most be writ owr the hede of bat figure, vnder the whichc the first figure is, of the dyviser ; And by that figure me most withe- draw alle ober figures of the lower ordir and that of the figures aboue thaire hedis. This so don-, me most sette. forward* |?e figures 20 of the diuiser by o difference towards the right hondt and worche as he fore ; and Jams : —
|
'Die residue | 2 | G |
|
The quoeienl | 9 To be dyvydede | 3 | 3 | 2 The dyvyser | | 3 | 4 |
Examples.
|
Residuum |
1 1 1 1 1 1 1! 1 i 1 1 • 1 1 ■-' |
|
quociens |
i 1 , 6 | 5 | 4 1 1 1 2 1 0 | 0 | 4 |
|
Diuidendus |
3 | 5 | 5 1 1 | 2 | 2 8|8|6|3|7|0|4 |
|
Diuisor |
| | 5 1 3 1 1 2 : 3 | | |
|
The quocient |
| | | |G 5 1 |
|
To 1" dj vydedi |
| 3 | 5 | 5 | 1 -J 'J |
|
The dy \ J i i |
:, 1 :; |
a special
And yf ii happ< after be setlyng forwardc of the figures bat }v last of the divisor may not so oft be \\/t//draw of the figure above '-'I his hede, above bat figure vnder the whichi the first of the diuiser is writ uk' most sette a cifre in ordre of the uombre quocient, and sette the figures forward* as be-fore be o difference alone, and so me shalle do in all* nombres to be d\ vidi d< . for where the dyviser may 28
Chapter VIII. Progression. 45
e w«t7t-draw me mosl sette there a cifre, and sette forwarde the figures ; as here :—
And Die shalle not CeSSe fro Anollier ex- ample.
such? settyng of figures for-
|
, The residue |
1 1 1 |
I ! |
in |
2 |
|
Tlie quocient |
1 1 1 |
|2| |
,, n |
» |
|
To be dyvydede |
| 8 ! 8 | |
« I s.i |
7 1 0 |
i |
|
The dyvyser |
1 1 |
2 | 3 |
i 1 |
J_ ward'', nether of settynge of be quocient into the dyviser, neb'/- of subt?'acciown of the dyvyser, tille the first of the dyvyser 8 be wit7i-draw fro be first to he divided''. The whiche don?, or ought,1 ober nought shall? remayne : and yf it be ought,1 kepe it in the tables, And euer vny it to be diviser. And yf bou wilt wete how many vnytees of be divisio»n 2wol growe to the nonibre of the *Foi.33». 12 diviser?, the nombre quocient wol shewe it: and whan suche quotient divisiown is made, ami b<»u lust prove yf thow have wele done or no, Multiplie the quocient by the diviser, And the same figures How to prove wolle come ayene that thow haddest bifore and none other. And ' 16 yf ought be residue, than with addiciown therof shalle come the same figures : And so mul'tiplicacioMn provithe divisiown, and dyvi- siown niultiplicaciown : as thus, yf rnultiplicaciown be made, divide it ormuitipUca- by the multipliant, and the nombre quocient wol shewe the nombre 20 that was to be multiplied'3, etc.
rogressiozra is of nombre after egalle excesse fro oone or tweyne Definition o.
. . Progression.
take agregacioun. oi progression one is naturelle or con-
P
tynuelle, bat ober broken aud discontynuelle. Naturelle it
24 is, whan me begynnethe with one, and kepethe ordure ouerlepyng Natural Pro- one ; as .1. 2. 3. 4. 5. C, etc., so bat the nombre folowynge passithe the other be-fore in one. Broken it is, whan me lepithe fro o nombre tille another, and kepithe not the contynuel ordire; as 1. 3. Broken Pro-
28 5. 7. 9, etc. Ay me may begynne with .2., as bus ; .2. 4. G. 8., etc., and the nombre folowyng passethe the others by-fore by .2. And note wele, that naturelle progressioioi ay begynnethe with one, and Intercise or broken progression, omwhile begynnythe w/th one,
32 omwhile with twayue. Of progression naturell .2. rule.- ther be
yove, of the which) tin in t is this] whan the progression naturelle The 1st role endithe in even nombre, by the half therof multiplie be next totalb. Progression. ou?/-er? nombre : Example "f grace : .1. 2. •">. L. Multiplie .-">. by .2.
36 ami so .10. comethe of, that is the totalb nombre berof. The seconde
rule is such'1, whan the progression naturelle endith in nombre The second ode. Take the more portion of the oddes, and multiplie therby
i() the totalle nombre. Example of grace 1. 2. 3. 4. 5., multiplie
1 ' uoudit ' iii M.S.
46 Chapter IX. Extraction of Boots.
.5. by .^>, and fchryes ."». shall* be resultant, so the nombre totalle riie flret rule is .15. Of progresiown inte?'cise, ther ben also .2. ] rules: and be I'rogreseion. first is bis : Whan the Lntercisc progression enditn* ra even nombre
by half therof multiplie the next noinbre to Jjat halfe as .2.1 t. G. 4
Multiplie . I. by .•">. so J>ai is thryes .1.. and .12. the nombre of alle Thesecona the progression, wolle folow. The second* rule is this: whan the
progression interseise endithe in ode, take be more porciown of alle *Foi.5s*. Jje nombre, 2 and multiplie by hym-selfe; as .1. 3. 5. Multiplie .3. 8
by hym-selfe, and be some of alle wolle be .'.'.. etc. The preamble ~B ■""ere folowithe the ext .racci* ,/m of rotis, and first in nombre ii fr.M.tij, I — I (|;/^dra.t''.<. Wherfor me shall* se what is a nombre quadrat,
J_JL and what is the rote of a nombre quadrat, and what it 12
is to draw out the rote of a nombre. And before other note Linear, this divisiown : Of nombres one is lyneal, ano)>er superficialle,
superficial, ill i • i
andsoiid anober quadrat, anob'r cubike or hoole. lyneal is that bat is con- numbers. ' .
sidrede after the processe, havynge no respect to the direccio?«n lb
of nombre in nombre, As a lyne^atbe but one dyraensiown that is to sey after the lengthe. Nombre superficial is b<d comethe of ledynge of oo nombre into a nother, wherfor it is callede super- superficial ficial, for it, hath' .2. nombres notyng or niesuryng* hym, as a 20
numbers. ,, , , , , .-. , • • , , • 1 ,1 1
superficialle thynge hathe .2, dimensions, Jvt, is to sey lengthe and brede. And for bycause a nombre may be hade in a-nother by .2. mane?*s, b"t is to sey other in hym-selfe, oJ?er in anojer, Vnder-
squarenum- stonde yf it he liad in hym -self. It is a quadrat, ffor dyvisiottn 24 write by vnytes, -hathe .1. sides even as a quadrangille. and yf the nombre be hade in a-nober, the nombre is superficiel and not, quadrat, as .2.4jade in .3. raakethe .6. thai is be firsi nombre sup ficiellej wlierfor it is open bat alle nombre quadrat is superficiel, 28
riwrootofa and not co?iuertide. The rote of a nombre quadrat is ba1 nombre
I-'''" that is had of hym-self, as twies .2. makithe t. and .1. is the first
nombre quadrat, and 2. is his rote. 9. 8. 7. 6. 5. I. 3. 2. 1. The
Notesofsomi rote of the more quadrat .■">. 1. I. 2. 6. The most nombre quadrat 32
exampli ol
squareroots [) g 7. 5 9. •">. 4. 7. •'». the lviilellelit OU* r the quadrat .<>. (.). O.
hereinterp
Lated. |. ;,. The first caas of nombre quadrat .5. I. 7. o. • >. Lhe rote .2.
3.4. The second* caas .3. 8. I. 5. The rote .6. 2. The thirde caas .2. 8. I. 9. The rote .5. 3. The .4. caas .3. 2. 1. The rote 36
soUdnum. .1. 7. The 5. caas .9. I. 2. 0. I. The rote 3. 0. 2. The solide nombre or cubik* is bat bat comythe of double ledyng of nombre in nomhiv ; And ii is clepede a solide body that hath* per-ia .3
1 3 written for '1 in MS.
bers.
Ghapter A'. Extraction of Squaw Root. 47
[dimensions] but is to sey, lengthe, brede, and thiknesse. so bat Three di-
L " ' inensions oi
nombre liatli' ,3. nonibres to be broughl forthe in hym. l!ut solids. nombre may be hade twies in nombre, for other it is hade in hym- 4 self'/, ober in a-nobe;\ II' a nombre be hade twies in hyin-self, ob< r
. . ' . ' Fol. 54.
ones in his quadrat, b"t is tin* same, bat a cubike lis, And is the cubic uum- same that is solide. And yf a nombre twies be had' in a-nober, the nombre is clepede solide and not cubike, as twies .3. and bat .2. 8 makithe .12. Wherfor it is opyne that alle cubike nombre is solid'', Aiicubics
an- solid
and not co?iuertide. Cubike is bat nombre. bat comythe of ledyngi numbers, of hym-selfe twyes, or ones in his quadrat. And here-by it is open that o nombre is the route of a quadrat and of a cubike. Natheles 12 the same nombre is not qwadrat and cubike. Opyne it is also that No number
1 l " may I e both
alle nombres may be a rote to a qwadrat and cubike, but not alk lin.?'ir a,ld nombre quadrat or cubike. Therfor sithen be ledynge of vnyte in hym-self ones or twies nought comethe but vnytes, Seithe Boice in 16 Arsemetrike, that vnyte potencially is al nombre, and none in act. unity is not
^ L J a number.
And vndirstonde wele also that betwix euery .2. quadrates ther is a
Examples of square roots.
meene proporcionalle, That is opened'' thus; lede the rote of o quadrat into the rote of the ob' •/• quadrat, and ban wolle be meene 20 shew. Also betwix the next .2. cubikis, me may fynde a double a note on
. i mean propor-
meene, that is to sey a more meene and a lesse. The more meene tionais. thus, as to brynge the rote of the lesse into a quadrat of the more. The lesse thus, If the rote of the more be brought Into the quadrat
2 1 of the lesse.
•"TI^o draw a rote of the nombre quadrat it is What-euer nombre be
1 proposed* to fynde his rote and to so yf it be quadrat. And Tofinda yf it be not quadrat the rote of the most quadrat fynde out, vnder
28 the nombre proposede. Therfor yf thow wilt the rote of any quadrat nombre draw out, write the nombre by his differences, and compt the nombre of the figures, and wete yf it be ode or even. And yf it be even, than most thow begynne worche vnder the last save one. Begin with
.32 And yf it be ode w/tA the last ; and forto sey it shortly, al-weyes place.
fro the last ode me shalle begynne. Therfor vnder the last in an
od place sette, me most fynd« a digit, the whiche lad< in hym-selfe
it puttithe away that, bat is ouer his hede, ober as neighe as me
- 7 in MS. :t runs on in MS.
|
Residuum |
1 1 1 o i |
1 1 |4 |
1 |
! o |
1 1 o | |
|
|
Quadrande |
|4|3|5|6 |
3 | 0 | 2 | 9 |
i ! |
' ! 4 |
2 | 4 |
1 | 9 | 3 | 6 |
|
Duplum |
|1|2| | |
1|0| | |
2| |
|6 |
| |
|[8] | 2 |
|
Subduplu»i |
1 |6| |6 |
|5| j 5 |
1 1 |
18 |
- |
M| |4 | |
48
Chapter X. Extraction of Square Root.
Kind the nearest square root Ol thai num- ber, subtract,
double it,
I Fol. 546. and set the double one to the ri(,'lit. Find the second figure by division. Multiply the double by the second figure, and add after it the square of the second figure, and subtract.
Examples.
may: suche a digit founde and w/t//draw fro his ouerer, me most double that digit and sette the double vnder the next figure towarde the right honde, and his vnder double vnder hym. That done, than uiemosl fyred< a nober digil vnder the next figure bifore the doublede, 4 the whiche 1brought in double settethe away alle that is oner his hede as to reward'' of the doublede : Than brought into hym-self settithe all away in respect of hym-self, Other do it as nye as it may be d<>: other me may vvz't/j-draw the digit 2[last] founde, and 8 lede hym in double or double hym, and after in hym-self''; Than [oyne to-geder the pj'oduccione of them bothe, So that the Brsl figure of the last pj'oduct be addede before the first of the first productes, the seconds of the first, etc. and so forthe, subtrahe fro the totalle 12 nombre in respect of be digit. And if it hap fat no digit may be
|
The residue |
MM 1 1 1 1 II 1 1 ' 1 5 | 4 | 3 | 2 |
|
To lie qnadredfl Tbe double |
|4|1|2|0|9||1|5|1|3|9||9|0|0|5|4|3|2 |
|
| 4 | 0 | | || | 2 | | 4 | | 6 I | 0 | | | 0 |
|
|
The vnder double |
2| I 0 | |3||1| |2| |3||[3]| | CO] 1 ICO] | |0 |
founde, Than sette a cifre vndre a eifre, and cesse uot tille thow fynde a digit; and whan thow hast founde it to double it, neb'/' to
special cases, sette the doubled' forward! nether the vnder doublede, Till thow \<] fynde vndre the firs! figure a digit, the which' lad< in all- double, settyng away all- that is ou< r hym in respect of the doublede : Than lede hym into hym selfe, and put a-way all' in regarde of hym, other
i he residue, as nyghe as thow maist. That done, other oughl or nought wolle 20 be the residue. It' nought, than it shewithe thai a nombre cdm- ponede was the quadrat, and his rote a digil last founde with vndere-double other vndirdoubles, so that it be sette be-fore: And yf ought3 rcmayii' . that . -hew/i/' that the nombre proposede was not 24 quadrat,4 but a digit [last found with the subduple or subduplea
11,1- tabic is
consti iici. d for use in cube root sum . gh ing the value ol ab.a
|
1 |
1 2 |
1 3 | 4 | 5| 6 | |
7| 3| 9 |
|
2 |
8 |
12 | 16 | 20 | 24 | |
28 | 32 | 36 |
|
3 4 |
18 32 |
27 36 , 15 | 54 | |
63 | ri\ 81 |
|
| 48 | (54 | 80 | 96 | |
i L2 128 114 |
||
|
5 |
50 |
| 75 | 100 | 125 | 150 | |
175 | 200 | 225 |
|
'; |
n |
L08 ,MI 180 216 ! |
252 | 288 |
|
i |
98 |
1 17 | L96 | 245 | 294 j |
111 |
|
9 |
L28 168 |
L92 | 256 320 ] 384 243 [ 324 | 105 1 486 | |
148 512 576 567 | CIS | 72!tf |
'-' ' so' iii MS. :1 ' nougbl ' in MS.
1 M>. adds here : ' wher-vponc -■ the table in the aexl side of the next leefc' 110 in MS. 6 0 in MS.
Chapter XL Extraction of Cuh Hoot. 40
is] The rote of the most quadrat conteynede vndre the nombre proposede. Therfor vf thow wilt prove yf thow have wele <lo or How to prove
tlie square
no, Multiplic tlie digit last iounde w/t/' the vnder-double ober vnder- i<»>t without
' r ° ' or with a
4 doublis, ami thow shalt fynde the same figures that thow haddest remainder.
before; And so that nought be the 'residue. And yf thow have iFol.55. any residue, than with the addicioztt) berof that is rese?-uede with-out in thy table, thow shalt fynde tin first figures as thow haddest them 8 before, etc.
Heere folowithe the extraceio«n of rotis in cubike nonibres ; Definition ' of a cubic wher-for me most se what is a nomine cubike, and what number and a cube root, is his route, And what is the extracciown of a rote. A
12 nombre cubike it is, as it is before declarede, that comethe of ledyng "f any nombre twies in hyni-selfe, other ones in his quadrat. The rote of a nombre cubike is the nombre that is twies had'/ in hym-selfe, or ones in his quadrat. Wher-thnrghe it is open, that
16 eiury nombre quadrat or cubike have the same rote, as it is seide before. And forto draw out the rote of a cubike, It is first to fynde be nombre p?-oposede yf it be a cubike; Am! yf it be not, than thow most make extraeciown of his rote of the most cubike
20 vndre tlie nombre proposide his role founde. Therfor p?'oposede some nomine, whos cubical rote boil woldest draw out; First thow Mark off
, . . the places in
most compl the figures by fourtnes, that is to sey in the place of threes, thousandes; And vnder the last thousand'' place, thow most fynde Findthe first
21 a digit, the whiche lad' in hym-self cubikly puttithe a-way that bat is one/- his hede as in respect of hym, other as nyghe as thow
maist. That done, thow most trebille the digit, and that triplat treble it and
n ii-iii I)'il<'e it under
is to be put vnder the .3. next figure towarde the right lionae, tlie next but
' pi <"'e> am' |n|1'"
to And the vnder-trebille vnder the trebille; Than me most fymh' a tipiybythe
. . . . 'li-il-
digit vndre the next figure bifore the triplat, tlie whiche With his Then find the
. i mi r i i ri i "ii no sec°nd digit.
vnder-trebille had into a trebille, aftenvarde other vnderLtrebuleJ * had in his produccfcwn, puttethc a-way alle that is ouer it in 32 regarde of3 [the triplat. Then lade in hymself puttithe away that bat is over his hede as in respect of hym, other as nyghe as thou maist:] That done, thow must trebille the digit ayene, and the Multiply the
... , "rst triplate
triplat is to be sette vnder the next .3. figure as before, And and the sec-
0 . ond digit,
ob the vnder-trebille vnder the trebille: and than most thow sette twice by this
digit.
forward'' the first triplat with his vndre-trebille by .2. differences.
And than most thow fynde a digit vnder the next figure before the triplat, the which' withe his vnder-t?'iplat had in his triplat after-
2 double in MS. :; 'it hym-selfe5 in MS.
KOMBKYNGE. E
Cliapfir XL Extraction of Cube Boot.
Subtract. warde, other vnder treblis lad in product ] It sitteth* a-way aH that is ouer his hede in respect of the triplat than had in hym-self cubikly,2 or as nyghe as ye may.
Examples
Continue i his process till the fust figure is readied.
|
Residuum | | | | | | | 5 II | | | | | 4 || | 1 | 0 | 1 | 9 | |
|
Cubicandus |8|3|6|5|4|3|2||3|0|0|7|6|7||1|1|6|6|7| |
|
Triplum 1 | | 6 | 0 | | | 1! | | | 1 | 8 | | | | 4 | | |
|
Subtriplim |2 | 1 | 0| | |[8]|| | | 6| | | 7 II | 2 |. | | 2 j |
Kxnmples.
Nother me sballe not cesse of tlic fyndynge of that digit, neither of 4 his fcriplaciown, nejrer of the triplat-is 3anteriorac?own, that is to sey, settyng forwarde by .•_'. differences, Ne therof the vudro-triple to be put vndre the triple, Nether of the multiplicackmn berof, Neither of the subtracciozm, tills it conic to the first figure, vnder the 8 whiche is a digitally nomine to be found'', the whiche withe bis vndre-treblis most be hade in tribles, After-warde wt'fc/ioul vnder- treblis to he had' into producciozm, settyng away alle that is ouer the hede of the triplat nombre, After had into hymselfe enhikly, 12 and sette alle-way that is ouer livni.
|
| To be cubicede | 1 | 7 | 2 | 8 || 3 | 2 | 7 | 6 |-8 |
|
The triple | | | 3 | 2 || | | | 9 | |
|
1 The vnder triple | | | 1 | 2 || |[3]| | 3 | 3 |
5pi ciali mi
Also note vvele that the producc/on coni- ynge of the ledyng of a digite founde4 me may adde to, and also with draw fro of the totalle nombre sette above that digit so founde.5 That done ought or non-lit most be the residue. If it he nought, ft is open that the nombre pj-oposed* was a cubik< 16 nombre, And his vote a digit founde last with the vnder-triples ; If the rote therof w'ex bade in hym-selfe, and afterwarde product they sballe make the first figwres. And yf ought be in residue, kepe that wiViOut in the table ; and it is open* that the nomine was not 20 a cubike. bid a digit last founde with the vndirtriplis is rote of the in. .,-1 cubike vndre the nombre proposed* conteynede, the . which* rote yf it be hade in hym selfe, And afterward* in a product of that shalle growe the most cubike vndre the nombre proposed* 24 couteynede, And yf that be added- to a cubiki the residue reserued* in the table, woll* make the same figures that ye had* first. 6And
- MS. adds here: 'it sctteth* a-way all* his respect.'
:; ' ancterioracioun ' i" MS.
1 MS. adds here: 'with an viidre-triple / other of an vndre-triple in a triple or triplal is And after-warde with oul mire triple other vndre-triplis in the p?-oducl and ayene tlial product thai cometh* of the ledynga of a digit foundi in hym self) cubicalli ' /
■ MS. adds here i 'as thei had be a divisiouu made as it ia opened*: before.
/" '• of Numbers, &c.
51
yf no digit after the anterioraciown1 may not be found'', than put there a cifre vndre a cifre vndir the thirde figure, And put forwards special case, be figures. Note also wele that yf in the nomine proposede ther i ben ii" place of thowsandes, me most begynne vnder the first figure in the extracciown of the rote, some vsen forto distingue the nornbre by threes, and ay begynne forto wirche vndre the first of
Examples.
|
The residue |
1 1 1 1 1 1 1 0 || | | | | | 1 | 1 |
|
The cubicanchts |
|8|0|0|0|0|0|0]!8|2|4|2|4|1|9 |
|
The triple |
I I I 3 1 0 | 0 | | | |6| | | | |
|
The vndert/iple |
|[2]| | | 0 | 0 | | 2| | |6|2| | |
the last ternary other unco7»plete nombre, the whiche maner of 8 operaciozm accordethe with that before. And this at this tyme suffisethe in extracciown of nomhres quadrat or cuhikes etc.
12 3 4 5 6 A table of
one. x. an. hundred* a thowsande / x. thowsande / An hundred': pl'^'iy' 7
thowsande / A thowsande tymes a thowsand* x. thousand'5 tyines 12 a thousand'; / An hundred* thousande tymes a thousande A thou- sande thousande tymes a thousande this is the x place etc.
[Ende.] 1 Ms. anteriocaciotm. - 4 in MS.
from the
Abacus.
52
^ccomptnnqc bn counters.
1 "«»• ^[ Tlic secondc dialoge of accomptynge by counters.
Mayster. "Tk"T"Owe that you haue learned the conmien kyndes of Arithme- \\\ tyke with the penne, you shall se the same art in counters : 1 ^ whiche feate doth not only'seruc for them that can not write 4 and rede, hut also for them that can do bothe,but haue not at sonic tymes theyr penne or tables redye with them. This sovtc is in two fourmes commenly. The one by lynes, and the other without lynes : in that y* hath lynes, the lynes do stande for the order of places : 8 and in y* that hath no lynes, there must he sette in theyr stede so many counters as shall nede, for echo lyne one, and they shall supplye the stede of the lynes. 8. By examples I shuld better 3 117-7 perceaue your meanyuge. M. For example of the ly2ncs : Lo here 12 you se .vi. lynes whiche stande for syxe places so — im«»»
J J l 1-0 0-0-0
that the nethermost standeth for y" fyrst place, and l*.j .$?<,'" ~
the next ahoue it, for the second : and so vpward tyll — >
you come to the hyghest, which is the syxte lyne, and standeth for 16 the syxte place. Now whal is the valewe of euery place or lyne, Numeration, you may perceaue by the figures whiche I haue set on them, which is accordynge as you learned before in the Numeration of figures by the penne: for the fyrste place is the place of vnities or one,-, and 20 euery counter set, in thai lyne betokeneth hut one : and the second e • lyne is the place of 10, for euery counter there, standeth for 10. The thyrd lyne tin' place of hundredes: the Fourth of thousandes : and so forth. S. Syr 1 do perceaue that the same order is here of 21 3 iiT'-. lynes, as was in the other figures 8by places, so that you shall DOl nede longer to stande about Numeration, excepte there he any other difference. M. W you do vndersta?ide it, then how wyll you set 15431 S. Thus, as I suppose. __x_, — M. You haue set yc 28 places truely. hut your figures he — \ not mete for this vse :
Addition on the Counting Board. 53
for the inetest figure in this behalfe, is the figure of a counter round, as you se here, where I haue expressed that same _^_# _ZIZ= suinme. S. So that you haue not one figure for 2, — #~#~#~# —
4 nor 3, nor 4, and so forth, but as many digettes as you haue, you set in the lowest lyne : and for euery 10 you set one in the second line : and so of other. But I know not by what reason you set that one counter for 500 betwene two lynes. M. you shall re- 8 member this, that when so euer you nede to set downe 5, 50, or 500, or 5000, or so forth any other nomber, whose numerator lis » nsa. 5, you shall set one counter for it, in the next space aboue the lyne that it hath his denomination of, as in this example of that 500,
12 bycause the numerator is 5, it must be set in a voyd space: and bycause the denominator is hundred, I knowe that his place is the voyde space next aboue hundredes, that is to say, aboue the thyrd lyne. And farther you shall marke, that in all workynge by this
1G suite, yf you shall sette downe any summe betwene 4 and 10, for the fyrste parte of that nomber you shall set downe 5, & then so many counters more, as there reste nowbers aboue 5. And this is true bothe of digettes and articles. And for example I wyll set
20 downe this summe 287965, ~* *-*# which sumine yf you
marke well, you nede none - *~j~l»i*m other examples for to lerne the numeration of ^i_ 2this forme. But this mis*.
shal you marke, that as you dyd in the other kynde of arithmetike,
24 set a pricke in the places of thousa?Rles, in this worke you shall sette a starre, as you se here. S. Then I perceave numeration, but I praye you, howe shall I do in this arte to adde two summes or Addition. more together'? M. The easyest way in this arte is, to adde but 2
28 summes at ones together : how be it you may adde more, as I wyll tell you anone. Therfore when you wyll adde two suwimes, you shall fyrst set downe one of them, it forseth not whiche, and then by it drawe a lyne crosse the other lynes. And afterward
32 set downe the other su?nme, so that that lyne may be betwene them,
as yf you wolde adde 2G5'J to 8342, you must set your sumnies as you se
here. And then yf you lyst, you 3may adde the one to the other 36 in the same place, or els you may adde them both together in a newe place : which waye, bycause it is moste playnest, 1 wyll showe you fyrst. Therfore wyl I begynne at the vnites, whiche in the fyrst summe is but 2, and in y' second surame 9, that maketh 11, 40 those do 1 take vp, and fur them 1 set 11 in the new roume, thus.
54
Addition hy Counters.
1 119 b.
a 120 a.
0-0-0-
_#^#-I- Then do T take vp all ye articles vnder
— *-j-g — a hundred, which in the fyrst sumnie
^^
are 40, and in the second suinme 50, that maketh 90 : or you may saye better, that in the fyrste suinme there are 1 articles of 10, and in the seconde suinme 5, which make 9, but then take hede that you sette them in theyr xryght -*-#*j lynes as you se here. Where I -
haue taken awaye 40 from the fyrste summe, and 50 from ye 8 second, and in theyr stede I haue set 90 in the thyrde, whiche I haue set playnely y* you myght well perceaue it : how be it seynge that 90 with the 10 that was in yp thyrd roume all redy,doth make 100, 1 myghte better for tliose G counters set 1 in the thyrde 12 lyne, thus ; _^ ~Z For it is all one summe as you may se, but it is beste, — neuer to set 5 counters iu any line, for that
may be done with 1 counter in a hygher place. S. I iudge thai good reaso?i, for many are vnnedefull, where one wyll serue. 16 M. Well, then -wyll 1 adde forth of hundredes : I fynde 3 in the fyrste summe, and G in the seconde, whiche make 900, them <.h> I take vp and set in the thyrd roume where is one hundred all redy, to whiche 1 put 900, and it wyll be 1000, therfore I set one 20 counter in the fourth lyne for them all, as you se here. Then adde I ye
thou-
-0*0 -0 \-0-0-
sandes together, whiche in the fyrst summe are 8000, an>/ in \p second "2000, that maketh 10000: them do 1 take vp from those -I two places, and for them I set one counter in the fyfte lyne. and then appereth as z^ljzz : you se, to be 11001, for so many doth
3 1206. amount of the m addition of 8342 to 2659. >. Syr,
this 1 do pereeave : but how shall 1 sel one sumnie to an other, not 28 chaungynge them to a thyrde place ! M. Marke well how I do it : I wyll adde together 65436, and 3245, whiche fyrste I set downe thus. Then do 1 begynne with the smalest, which in the fyrst summe is . thai do 1 take vp, and wold put to the other 5 in the seconde summe, sauynge that two counters can nol be set in a voyd place of 5, but for them bothe I musl sei I in the seconde lyne, which is the place of L0, therfore 1 take vp the 5 of 36 the fyrsi Bumme, and the 5 of the seconde, and for them I set 1
* 121 a. in the second lyne, las you se here. Then do I lyke wayes, take vp the 1 counters of the fyrste summe and — -# " "— - 40
J«t
0000
32
|
-Jk |
|
|
*0 |
|
|
— 0-0 0 0-0-0- |
0-0-0-0 - |
Subtraction on the Counting Board. 55
second e lyne (which make 40) and adde them to the 4 counters of the same lyne, in the second summe, and it maketh 80, But as 1 sayde 1 maye not conueniently set aboue 4 counters in one lyne,
4 therfore to those 4 that I toke vp in the fyrst summe, I take one also of the seconde summe, and then haue I taken vp 50, for whiche 5 counters I sette downe one in the space oner yc second lyne, as here doth, appere. 1 — 9 1 and then is there 80, 1121 6
8 as well w* those ~x"#"#~*~l~# ^» # » 4 counters, as yf I
had set downe ye I • — other 4 also. Xow
do I take the 200 in the fyrste su??znie, and adde them to the 400 in the seconde summe, and it maketh GOO, therfore I take vp the 2 12 counters in the fyrste summe, and 3 of them in the seconde summe, and for them 5 I set 1 in ye space aboue, ———- thus. Then I take yp 3000 in ye fyrste Zx1^1 gumrne, vnto whiche there are none in the
3=
16 second summe agreynge, therfore I do onely remoue those 3 counters from the fyrste summe into the seconde, as here doth appere.
—» — -And so you see the hole sivmme, that amourcteth l2 i--«
•*•-•— of the addytiow of 654.30 with 3245 to be 6868[1].
0 And yf you haue marked these two examples well,
-x-
20 =
you nede no farther enstructioM in Addition of 2 only summes : but yf you haue more then two summes to adde, you may adde them thus. Fyrst adde two of them, and then adde the thyrde,
24 and ye fourth, or more yf there be so many : as yf I wolde adde 2679 with 4286 and 1391. Fyrste I adde the two fyrste summes thus. -x_#a# I_tf-#-#-#_ir^>' 3And then I adde the
m~TV^ i*B ~m t-*I — thyrde thereto thus.
28 And so of more yf you haue x 9 T~^*~
them. S. Xowe I thynke ===g#»=#IgIt— &
3 122 b.
m -0 »•
beste that you passe forth to Subtraction, except there be any wayes to examyn this maner of Addition, then I thynke that were
32 good to be knowen nexte. M. There is the same profe here that is
in the other Addition by the penne, I meane Subtraction, for that Subtraction, onely is a sure waye : but consyderynge that Subtraction must be fyrste knowen, I wyl fyrste teache you the arte of Subtraction, and
36 that by this example : I wolde subtracte 2892 out of 8746. These summes must 1 set downe as 1 dyd in Addition: but here it is best 'to set the lesser nomber fyrste, Xl 0^ T~#I»*» uea'sic)
thus. Then shall I begynne to sub- — •*»-»-#-t1#a#-#-# —
40 tracte the greatest nombres fyrste (contrary to the vse of the penne)
G Subtraction by Counters.
v1 is the bhousandes in this example : tlierfore I fynd amongost the thousandes 2, for which I withdrawe so many ivom the seconde sunime (where arc 8) and so remayneth there G, as this example
showeth. -+—m T-gS • ^ien ^° ^ ^ce wayes with 1
the hun- -*1*1*1*1) ij r>— dredes, of whiche in the
i ii66. fyrste sunime ll fynde 8, and is the seconde summe but 7, out of whiche I can not take 8, therfore thus muste 1 do : I muste loke how moche my summe dyffereth from 10, whiche 1 fynde here to 8 be 2, tlien must I bate for my sumrne of 800, one thousande, and set downe the excesse of hundredes, that is to saye 2, for so moche 100[0] is more then I shuld take vp. Therfore fro??! the fyrsto su??zme I take that 800, and from the second sumuie where are 1 2 G000, I take vp one thousande, and leue 5000; but then set I downe the 200 unto the 700 y* are there all redye, and make them 900 thus. ~| T 9 Then come I to the articles
of tt'/mes — #-#*#-»-J— 0gm-e-0 — where in tlie fyrste summe 16
- iir.i. 1 fynde 90, -and in the seconde sumnie but only 40: Now con- syderyng that 90 can not be bated from 10, I loke how moche yl 90 doth dyffer from the next summe aboue it, that is 100 (or elles whiche is all to one effecte, 1 loke how moch 9 doth dyffer 20 from 10) and I fynd it to be 1, then in the stede of that 90, 1 do take from the second summe 100: but consyderynge that it is 10 to moche, 1 set downe 1 in y' nexte lyne beneth for it, as you se here. Sauynge that lure , T » 1 hane sel one 24
counter in ye space in stede — g m I ~~3t OI 5 in y' nexte
lyne. And thus haue 1 subtracted all saue two, which I must bate from the 6 in the second summe, and there wyll remayne 1, thus.
■ So y* yf 1 subtracte 2892 fro??? 8746, the re- 28
-* — mayncr wyll be 5854, sAnd that this is truely
wrought, yon inaye proue by Addition: for yf you adde to this remayner the same su??inic that yon dyd subtracte, then wyll the formar su???mc 8746 amount agayne. S. That wyll 1 proue: and 32 fyrst 1 set the su???me that was subtracted, which was 2892, and the// the i < -i i i.i \ uer 585 1, thus, 'then do 1 adde fyrst \' 2 to i, whiche
-i—m-0 0 0 —
maketh 6, so take 1 vp 5 of those counters, and in theyr stede 1 3G
sette 1 in the spaco, as liere appereth. -ii-#i
■'Then do 1 adde the 90 nexte aboue to *
the 50, and it maketh I I", tlierfore 1 take vp those 6 counters, and
for them I sette 1 to the hundredes in ye thyrde lyne, and 1 in y' 40
*^i£i
Subtraction by Counters. 57
second lyne, thus, -n-g»»- |~g Then do I come to
the hundredes, of ===- *~* |'«|'» whiche 1 fynde 8 in
the fyrst summe, and 9 in y" second, that malteth 1700, therfore I
■1 take vp those 9 counters, and in theyr stede 1 sette 1 in tlie .iiii.
lyne, and 1 in the space nexte beneth, and 2 in the thyrde lyne,
as you se here, _,|_#_#_ili» Then is there lefte in the
•*•
Egg
f\ rste summe |-ga»*-» — hut only 2000, whiche I
8 shall take vp from thence, ami sit lin the same lyne in ye second i ns&. su?nnie, to y" one y* is there all recly : and then wyll the hole sumine appere (as you may wel se) to he >s7 16, which was y" fyrst grosse summe, and therfore
12 1 do perceaue, that I hadde well subtracted before. And thus you may se how Subtraction maye be tryed by Addition. S. I perceaue the same order here w* counters, y* I lerned before in figures. M. Then let me se howe can you trye Addition by
16 .Subtraction. S. Fyrste I wyl set forth this example of Addition where I haue added 2189 to 4988, and the hole summe appereth to he 7177. -w-0-0 — J tmtrirtrYif9 — -Xuwe to trye 2 U9 (I.
0- ■ — . ■ -000—0 0 I 00
whether that ~z~%»%i%Z^Azm^\^ZAZ&%ZZZ summe be well
20 added or no, I wyll subtract one of the fyrst two siwinies from the thynl, and yf I haue well done ye remayner wyll be lyke that other summe. As for example : I wyll subtracte the fyrste summe from the thyrde, whiche I set thus -1 \-0-m ^0^0
24 in theyr order. Then do I subtract 2000 =§Sg g~i~T~g«S== of the fyrst. • .-ummc frow \ ' second su??ime, and then remayneth there 5000 thus. x 1 • Then in the thynl lyne,
I subtract ye 100 ~~i$i'i~m~T~m^=~ of the fyrste summe,
28 fro/// the second summe, where is onely 100 also, and then in y1' thyrde lyne resteth nothyng. Then in the second lyne with his space om-r hym, 1 fynde 80, which I shuld subtract 3from the 3 1194. other su?nme, then seyng there are bul only 70 I must take it out
32 of some hygher summe, which is here only 5000, therfore 1 take vp 5000, and seyng that it is to inoch by 4920, 1 sette downe so many in the secondc roume, whiche with the 70 beyngo there all redy do make 1990, & then the summes x t t »0 B 0
36 doth .-tuple thus. Yet remayneth there -,. _ _ f *%* * * —
0*0 0-0 *-—0*0
in the fyrst summe 9, to be hated from the second summe, where
in that place of vnities dothe appere only 7, then 1 nmste bate a
hygher summe, that is to saye L0, but seynge that L0 is more then
40 9 (which 1 shulde abate) by 1, therfore shall 1 take vp one counter
from the seconde lyne, awl set downe the same- in the fyrst *or '- 120 «.
58 Multiplication hy Counters.
lowest lyne, as you se here. Z^zi_###_#_# And so haue I
ended this worke, and the — *£*%% Biimme appereth
to be ye .same, whiche was yc seeonde sumnie of my addition, and therfore I perceaue, I haue wel done. M. To stande longer about 4 this, it is Lnt folye : excepte that this yon maye also vnderstande, that many do begynne to subtracte with counters, not at the hyghest summe, as I haue taughl you, but at the nethermoste, as they do vse to adde : and when the sumnie to be abatyd, in any 8 lyne appeareth greater then the other, then do they borowe one of the next hygher roume, as for example: yf they shuld abate
18 1G from 2378, they set ye summes thus. -u_#g !-»-»
1 1201.- lAnd fyrste they take 6 whiche is in the z=ziJ»-*-~ *\Z$%-J—- I2
lion
lower lyne, and his space from 8 in the same ronmes, in y' second sumnie, and yet there remayneth 2 counters in the lowest lyne. Then in the second lyne must 4 be subtracte from 7, and so remayneth there 3. Then 8 in the thyrde lyne and his space, from 16 3 of the second sumnie can not he, therfore do they bate it from a hygher roume, that is, from 1000, and bycause that 1000 is to luoeh by 200, therfore must I sette downe 200 in the thyrde lyne, after I haue taken vp 1000 from the fourth lyne ; then is there yet 20 1000 in the fourth lyne of the fyrst sumnie, whiche yf 1 withdrawe from the seeonde summe, then dotli all ye figures stande in this order. ~7TT ~~ So that (as you se) it differeth not greatly whether
— ' ~b~b # — you ^e§ynne subtraction at the hygher lynes, or 24
*i2ia. at '-'the lower. How be it, as some menne lyke the one wave beste, Muitipiica- so some lyke tlie other: therfore you now knowyng bothe, may vse whiche you lyst. lint nowe touchynge Multiplication : you shall set your nombers in two roumes, as yon dyd in those two other 28 kyndes, but so thai the multiplier he set in the fyrste roume. Then shall you begyn with the hyghest no?nbers of y'' seeonde roume, and multiply them fyrst after this sort. Take that ouer- niosl lyne in your fyrsi workynge, as yf it were the lowest lyne, 32 setting mi it some mouable marke, as you Iyste, and loke how- many counters he in hym, take them vp, and for them set downe the hole multyplyer, so many tymes as you toke vp counters, reckenyng, 1 saye that lyne for the vnites: and when you haue so 36 done with the hvgheest no?ftber then come to the nexte lyne beneth, and do euen so with it. and so with ye next, tyll you haue done all. And yf there lie any uomber in a space, then for it
*vi\i. 3shall you take ye multiplyer 5 tymes, and then must you recken 40 that lyne for the vnites whiche is nexte beneth that space : or els
Multiplication hy Con/din.
59
after a shorter way, you shall take only halfe the multyplyer, but then shall you take the lyne nexte aboue that space, for the lyne of vnites : but in suche workynge, yf chau?zce your multyplyer be an
4 odde noniber, so that you can not take the halfe of it iustly, then muste you take the greater halfe, and set downe that, as if that it were the iuste halfe, and farther you shall set one counter in the space beneth that line, which you reckon for the lyne of vnities, or
8 els only remoue forward the same that is to be multyplyed. S. Yf you set forth an example hereto I thynke I shal perceaue you. M. Take this exa?3iple : I wold multiply 1542 by 3G5, therfore 1 set yp numbers thus. _, ,_ \-*m 1Then fyrste I be-
12 gynne at the 1000 in ~=0£z^z\z+-»»-»:=z y" hyghest rounie, as yf it were yp fyrst place, & I take it vp, settynge downe for it so often (that is ones) the multyplyer, which is 3G5, thus, as
you se here :
u*=±
where for the one
4^> vp from the
haue sette downe
16 counter taken -x- fourth lyne, I HI other G, whiche make yc siu/nne of the multyplyer, reckenynge that fourth lyne, as yf it were the fyrste : whiche thyng I haue marked
20 by the baud set at the begynnyng of ye same, S. I perceaue this well : fur in dede, this sumnie that you haue set downe is 3G5000, for so moche doth amount 2of 1000, multiplyed by 3G5. M. Well then to go forth, in the nexte space I fynde one counter which I
24 remoue forward but take not vp, but do (as in such case I must) set downe the greater halfe of my multiplier (seyng it is an odde nomber) which is 1S2, and here I do styll let that fourth place- stand, as yf it were y"
28 fyrst: as in this fourme you so, where I haue set
%eL
m »-
P*£T
±&i
•£*>
this multiplycatiu// with yp other : but fur the ease of your vnder-
sta/zdynge, 1 haue set a ly till lyne betwene them : now shulde they
32 both in one suwinie stand thus.
3 Howe be it an other fourme ~"~j to multyplye suche counters
::^
<£0
3 123 a.
in space is this: Fyrst to remoue the fynger to the lyne nexte
3G benethe y" space, and then to take vp y€ counter, arid to set downe
ye multiplyer .v. tymes, as here you se. Which suwmes yf you do
^==fi£i=
::••
•»• •
_#
|
v^ |
|
adde together into one suwzme, you dial perceaue that it wyll be y
CO
Multiplication by Counters,
l 121) I,.
2 Vila.
«&
* 185 a.
same y* appeareth of ye other working before, so that 1botlie sortes are to one entent, but as the other is much shorter, so this is playner to reason, for suclic as hauc had small exercyse in this arte. Not withstandynge you maye adde them in your mynde before you 4 sette them dovvne, as in this example, yen myghte hauc sayde 5 tymes 300 is 1500, and 5 tymes GO is 300, also 5 tymes 5 is 25, whiche all put together do make 1825, which yon maye at one tyme sot downc yf you lyste. But nowe to go forth, I must 8 remoue the hand to the iiexte counters, whiche are in the second lyne, and there must 1 take vp those 4 counters, settynge downe for them my multiplyer 4 tymes, whiche thynge other 1 maye do at 4 tymes seuerally, or elles I may gather that hole summe in my 12 mynde fyrste, and then set it downe: as to -aye 4 tymes 3(>0 is 1200: 4 tymes GO are 240: and 4 tymes 5 make 20: y* is in all 1460, vl shall I set downe also : as here you se. 2 whiche yf I ioyne *® — -* — .'-#-#-
in one summe with the forruar nombers, it wyll appeare thus. _( • Then to eiule this inultiplycatiou, 1 re-
Ufzlz
i&±±
%*-
1G
Ut±
^m- -»-•-
moue the fynger to the lowest, lyne, 20 where are onelj 2, them do 1 take vp, and in thcyr stede do I set downc twyse 3G5, that is 730, for which 1 set sone in the space abone the thyrd lyne for 500, and 2 more in the thyrd lyne with that one that is there all redye, and 24 the reste in theyr order, and so banc 1 ended the hole summe thus.
, . » Wherby you se, that 1542 (which is
K» the nomber of yeares syth Ch[r]ystes
incarnation) beyng multyplyed by 3G5 28
*£L
which is the nomber of dayes in one yeare) dothe amounts vnto 5G2830, which declareth ye nomber of daies sith Chrystes incarna- tion vnto the ende of 15424 yeares. (besyde 385 dayes and 12 houres for lepe yeares). S. Now wyll 1 proue by an other example, 32 as this: 40 labourers (alter 6d. \' day for eche man) haue wrought 28 dayes, I wold J know what theyr wages doth amount vnto: In this case muste I worke doublely : fyrsl I must multyplye the nomber of the labourers by \' wages of a man for one day, so wyll 3G \' charge of one daye amount: then secondarely shall I multyply thai charge of one daye, by the hole nomber of dayes, and so wyll the hole summe appeare : l\ rsl therefore I shall set the sumines thus. 4 1342 in original.
Division on the Counting Board. 61
T Where in the fyiste space is the multyplyer
— -^\-0-0-0-0 — (y* is one dayes wages for one man) and in the second space is set the nomber of the worke men to be multy- 4 plyed : the?? save I, G tymes 1 (reckenynge that second lyne as the lyne of vnites) raaketh 24, for whiche summe I shulde set 2 counters in the thyrcle lyne, and 4 in the seconde, therfore do I set 2 in the thyrde lyne, and let the 4 stand styll in the seconde
So apwereth the hole dayes wages > 125 6. •z*E*Ez. that is 20s. Then do I multiply agayn the same summe by the nomber of dayes and fyrste I sette
lyne, thus.1
to be 2-t0d. =z
the nombers, thus.
IZZZZ Tliera by cause there
-0-0-0-0 — dyuers lynes, I shall
12 are counters in - — Z*V~0
begynne with the hyghest, and take them vp, settynge for them the multyplyer so many tymes, as] I fcoke vp counters, ye is twyse, then wyll ye su??ime stande thus. ZZZIZZ g Then come
10 1 to ye seconde lyne, and take \-0-0 0-0 — vp those 4
cou/ders, settynge for them the multiplyer foure tymes, so wyll the
hole summe appeare thus.- \~iP 80 is the hole wages
000
of 40 workeme/;, for 28 \-0-m dayes (after 6d(. eche
20 daye for a man) G720d'. that is 5G0s. or 28 l'i. M. Now if you
wold proue Multiplycatum, the surest way is by Dyuision : therfore Diuision. wyll 1 oner passe it tyll 1 haue taught you ye arte of Diuision, whiche you shall worke thus. Fyrste sette downe the Diuisor for
24 fcare of forgettynge, and then set the nomber that shalhe deuided, at ye ryghte syde, so fane from the diuisor, that the quotient may he set hetwene them: as for example: Yf 225 shepe cost 45 l'i. what dyd euery shepe cost? To knowe this, I shulde diuide Hie
28 hole summe, that is 45 l'i. by 225, but that can not be, therfore must I fyrste reduce that 4.") l'i. into a lessor denomination, as into 6hyllynges: then I multiply 15 by 20, ami it is 900, that summe shall I diuide by the nowiber of -sliepe, whiche is 225, these s 12r,,,.
32 two nombers therfore I sette thus. t — r
Then begynne I at the hyghest lyne zE**?i:i J—
0-0-0-
of the diuident, and seke how often T may haue the diuisor therin,
and that maye I do 4 tymes, then say I, 1 tymes 2 are 8, whyche yf
36 I take from 9, there resteth hut 1, thus —
And bycause 1 founde the diuisor I — 0J0
4==
1 -00-0-0
tymes in the diuidente, I haue set (as you se) 4 in the myddle roume, which 'is the place of the quotient: hut now must t take tma. 40 the reste of the diuisor as often out, of the remayner : therfore come
62
Division by Counters.
' 127 6.
I to the seconde lyne of the diuisor, sayeag 2 foure fcymes make 8, take 8 from 10, and there resteth 2, | , t
**» f
thus. Then come I to the lowest
nomber, which is 5, and multyply it i tyines, so is it 20, that take I from 20, and there remayneth nothynge, so that I se my quotient to be i, whiche are in valewe shyllynges, for so was the diuident; and therby I knowe, that yf 225 shepe dyd coste -15 l'i. euery shepe coste 1 s. S. Tliis can 1 do, as you shall perceaue by this example : 8 Yf 160 sowldyars do spende euery monetli 68 l'i. what spendeth eehe man ? Fyrst 1byca"use I can not diuide the 68 by 160,therfore 1 wyll turne the poundes into pennes by multiplicacion, so shall there be 16320d'. Nowe mush' 1 diuide this summe by the 12
nomber of sowldyars, therfore I set them _lt .
in order, thus. Then begyn 1 at the — •*—
u
rt+
hyghest place of the diuidente, sekynge my diuisor there, whiche I fynde ones, Therfore set I 1 in the nether lyne. M. Not in the 16 nether line of the hole summe, but in the nether lyne of that worke, whiche is the thyrde lyne. S. So standeth it with reason. M. Then thus do they stande.2 | x T T Then seke
1 agayne in the reste, how - ** — \-»~m — - often I may 20 fynde my diuisor, and I se that in the 300 1 myghte fynde 100 thre tymes, but then the 60 wyll not be so often founde in 20, therfore I fake 2 for my quotient: then take I 100 twyse from 300, and there resteth 100, out of whiche with the 20 (that maketh 24 120) I may take GO also twyse, and then standeth the nombers thus, n T~ T~ swhere I haue sette the quotient 2 in the — #*— f [ lowest lyne: So is euery sowldyars portion
102d'. that is 8s. 6d\ M. But ye1 bycause you shall perceaue 28 iustly the reason of Diuision, if shall he good that you do set your diuisor styll agaynst those nombres from whiche yon do take it: as by this example 1 wyll declare. Yf y" purchace of 200 of ground dyd coste 290 l'i. what dyd one acre coste 1 Fyrst 32 wy] I turne the poundes into pennes, so wyll there he 69600 d'- Then in settynge downe these nombers 1 shall do thus. Fyrsi set the diuident on the ryghte hande as it oughte, and then
—2 'the diuisor on (lie lefte hande agaynsl 36
***#* » z those nombers, fro/// which 1 entende
" '■ to take hym fyrst as here you se,
wher 1 haue set the diuisor two lynes hygher the// is theyr owne place. S. This is lyke the order of diuision by the penne. 40
Division by Court 63
M. Truth you say, and nowe must I set ye quotient of this worke in the thyrde lyne, for that is the lyne of vnities in respecte to the diuisor in this worke. Then I seke howe often the diuisor
4 maye be founde in the diuident, and that I fynde 3 tymes, then set I 3 in the thyrde lyne for the quotient, and take awaye that 60000 fro?//, the diuident, and farther I do set the diuisor one line lower, as yow se here. H \-0-0-
8 'And then seke I how often the
0-0-0
p^z^i
diuisor wyll be taken from the nomber agaynste it, whiche wyll be 4 tymes and 1 remaynynge. S. But what yf it chaunce that when the diuisor is so remoued, it can not be ones taken out of the
12 diuident agaynste it ? M. Then must the diuisor be set in an other line lower. S. So was it in diuision by the penne, and therfore was there a cypher set in the quotient : but howe shall that be noted here ? M. Here nedeth no token, for the lynes do
16 represente the places : onely loke that you set your quotient in that place which standeth for vnities in respecte of the diuisor : but now to returne to the example, 1 fynde the diuisor -1 tymes in the diuidente, and 1 remaynynge, for -1 tymes 2 make 8, which I take
20 from 9, and there resteth 1, as this figure sheweth : and in the myddle space for the quotient I set 1 in the seconde lyne, whiche is in this worke the place of vnities.- H)_#_#_ Then remoue I ve diuisor to the next
-0-0 -0-0— \
1 129 6.
24 lower line, and seke how often I may haue it in the dyuident, which 1 may do here 8 tymes iust, and nothynge remayne, as
in this fourme, ~\x r i where you may se that
the hole quoti- zzzzzz:izj#jijz*zi~ ent is 348 d', that is
28 29 s. wherhy I knowe that so moche coste the purchace of one aker. S. Now resteth the profes of Multiply cation, and also of Diuision. M. Ther best profes are eche 3one by the other, for: *i3o&. Multyplication is proued by Diuision, and Diuision by Multiplyca-
32 lion, as in the worke by the prune you learned. S. Yf that lie all, you shall not nede to repete agayne that, y* was sufficyently taughte all redye : and excepte you wyll teache me any other feate, here maye you make an ende of this arte J suppose. .1/. So
36 wyll I do as touchynge hole nomber, and as for broken nomber, 1 wyll not trouble your wytte with it, tyll you haue practised tin's so well, y* you be full perfecte, so that you nede not to doubte in any poynte that I haue taught you, ami thenne maye I boldly
40 enstructe you in y" arte of fractions or broken nomber, wherin I
64 Merchants Casting Counters.
wyll also showe you the reasons of all that you haue nowe learned.
But yet before I make an ende, I wyll showe you the order of
commen castyng, wher in are bothe pennes, shyllynges, and poundes,
iisia. procedynge by no grounded reason, but onely by a receaued 4
Merchants' 'fourine, and that dyuersly of dyuers men: for marchauntes vse
casting. . J J
one fourme, and auditors an other : Iiut fyrste for niarchauntes fourme marke this example here, • ##* # # in which I haue expressed this summe 1981'i.'2 19s. • lid'. So that 8
you maye se that the lowest #*# 000 lyne serueth for
pe/mes, the next aboue for shyllynges, the thyrde for poundes, and the fourth for scores of pouwdes. Ami farther you maye se, that the space betwene pennes and shyllynges may receaue but one 12 counter (as all other spaces lyke wayes do) and that one standi th in that place for G d'. Lyke wayes betwene the shyllynges >i)t<! the poumles, one eoimter standeth fur 10 s. And betwene the poundes and 20 l'i. one counter standeth for 10 poumles. But 16 besyde those }rou maye see at the left syde of shyllynges, that one arsis, counter standeth alone, and betokeneth 5s. 3So agaynste the poundes, that one coulter standeth for 5 l'i. And agaynsl the 20 poundes, the one counter standeth for 5 score pouwdes, that is 20 Auditors' 100 l'i. so that euery syde counter is ■"> tymes so moch as one of them agaynst whiche he standeth. Now for the accompt of auditors take this example. 00000 0 where 1 haue
expressed yr same m 000 summe 1981'i. 24
19s. lid'. But here you se the ]ie//nes slaude toward yr ryght hande, and the other encreasynge orderly towarde the lofte hande. Agayne you maye se, that auditours wyll make 2 lynes (yea and nmre) h»r pennes, shyllynges, nn<l all other valewes, yf theyr 28 summes extende therto. Also you se, that they set one counter at the ryght ende of eche rowe, whiche so set there standeth for 5 of • ]:::„. that roume : and on 'the lefte corner of the rowe it stawdeth for
10, of ye same row. But now yf you wold addo other subtiacte 32 after any of both those sortes, yf you marke yr order of y* other feate which 1 taught you. you may easely do the same here without nioeh teachynge : for in A.dditio?J you must fyrst set. downe one summe and to the same set- the other orderly, and lyke maner yf 36 you haue many: but in Subtraction you must sette downe fyrst the greatest SUUinie, and from it inn I you abate that other euery denomination from his drwv place. S. I do not doubte but with a
- 16* in ori'-inal.
Aitditors Casting Counter*. 6i
ly tell practise I shall attayne these bothe : but how shall 1 multiply and diuide after these fourmes 1 M. You can not duely do none of both by these sortes, therfore in suche case, you must resort to t your other artes. S. Syr, yet I se uot by these sortes how to expresse buredreddes, yf they excede one hundred, nother yet thousandes. M. They that vse such accomptes that it excede 200 Mn one summe, they sette no 5 at the lefte hande of the scores of ' 1326. 8 poundes, but they set all the hundredes in an other farther rowe and 500 at the lefte hand therof, and the thousandes they set in a farther rowe yet, and at the lefte syde therof they sette the 5000, and in the space ouer they sette the 10000, and in a hygher rowe
12 20000, whiche all I haue expressed in this example, which is 978691'i. 12s. 9d' oh. q. for 1 had not told you before where, nother how you shuld set downe farthynges, which • • • •
(as you se here) must be set in a voyde space # m » m
16 sydelynge beneth the pernios : for q one counter: • • • » •
for ob. 2 counters : for ob. q. 3 counters : and m %
1 000
more there can not be, for 4 farthynges 2do make J 0 - 133 a.
1 d\ which must be set in his dewe place. And yf you desyre
20 yp same summe after audytors manor, lo here it is.
i * * * 9 0 *
0 But in this thyng, you shall take this for suffycyent, and the reste
you shall obserue as you maye se by the working of eche sorte : for
the dyuers wittes of men haue iuuented dyuers and sundry waves
24 almost vnnumerable. But one feate I shall teache you, whiche not only for the straungenes and secretnes is moche pleasaunt, but also for the good cowmoditie of it ryghte worthy to be well marked. This feate hath hen vsed ahoue 2000 yeans at the leaste, and yet
28 was it neuer comenly knowen, especyally in Englysshe it was neuer taughte yet. This is the arte of nombrynge on the hand, with diuers gestures of the fyngers, expressynge any summe con- ceaued in the 3mynde. And fyrst to begynne, yf you wyll expresse :i isst.
32 any summe vnder 100, you shall expresse it with your lefte hande : and from 100 vnto 10000, you shall expresse it with your ryght hande, as here orderly by this table folowynge you may perceaue.
^f Here foloweth the table
of the arte of the
hande
NOMBRYNGE. F
6G
%\t wck of nombvpgc bn tbc lgmbt.
*14
tooo
JOOO
-4-00 0
i isi6. i 1 In which as you may se 1 is expressed by y' lyttle fynger of \"
2 lefte hande closely and harde croked. *[2 is declared by lyke bow-
ynge of the weddyuge fynger (whiche is the iiexte to the lyttell
s fynger) together with the lytell fynger. [3 is signified by the |
myddle fynger bowed in lyke maner, with those otheT two. [-1 is
declared by the bowyng of the myddle fynger and the rynge
Bracket ( [ ) denotes new paragraph in original.
Digital Signs of Numbers. 07
fynger, or weddynge fynger, with the other all stretched forth. [5 is represented bj the myddle fynger onely bowed. [And 6 by 5,6 the weddynge fynger only crooked: and this you may marke in 4 these a certayne order. But now 7. 8, and 9, are expressed witJi the bowynge of the .same fyngers as are 1, 2, and 3, but after an other fourme. [For 7 is declared by the bowynge of the lytell 7 fynger, as is 1, saue that for 1 the fynger is clasped in, harde and 8 1rounde, but for to expresse 7, you shall bowe the myddle ioynte i ,:;:,„. of the lytell fynger only, and holde the other ioyntes streyght. S. Yf you wyll geue me Ieue to expresse it after my rude maner, thus I vnderstand your meanyng : that 1 is expressed by crookynge
12 in the lyttell fynger lyke the head of a bysshoppes bagle : and 7 is declared by the same fynger bowed lyke a gybbet. M. So I perceaue, you vnderstande it. [Then to expresse 8, you shall bowe - after the same maner both the lyttell fynger and the rynge fynger.
16 [And yf you bowe lyke wayes with them the myddle fynger, then doth it betoken 'J. [JSow to expresse 10, you .shall bowe your 9, 10 fore fynger rounde, and set the ende of it on the hyghest ioynte of the thombe. [And for to expresse 20, you must set your fyngers 20
20 streyght, and the ende of your thombe to the partitio?* of the 2fore - 135&. moste and myddle fynger. [30 is represented by the ioynynge 30 together of ye headdes of the foremost fynger and the thombe. [40 is declared by settynge of the thombe crossewayes on the fore- to
24 most fynger. [50 is signified by ryght stretchyng forth of the 50 fyngers ioyntly, and applyengeof the thombes ende to the partition of the myddle fynger and the rynge fynger, or weddynge fynger. [60 is formed by bendynge of the thombe croked and crossynge it 60
28 with the fore fynger. [70 is expressed by the bowynge of the 70 foremosl fynger, and settynge the ende of the thombe between the 2 foremost or hyghest ioyntes of it. [SO is expressed by settynge so of the foremost fynger crossewayes on the thombe, so that <s0
32 dyffereth thus fro?« 10, that for 80 the forefynger is set crosse on the thombe, and for 10 the thombe is set crosse oner y1' forefinger. 3[90 is signified, by bendynge the fore fynger,and settyng the ende 90 a i.w,,. of it in the innermost ioynte of y" I limn be, that is euen at the foote
3G of it. And thus are ill the no??ibers ended vnder 100. 8. In dede these be all the nombers Ivom 1 to l'», and then all the tenthes within 100, bul this teacyed me nol how to expresse 11, u 12, 13, eta. 21, 22, 23, etc. and such lyke. M. You can lytell 12, is, 21, 22,
40 vnderstande. yf yon can not do that without teachynge: whal is
68 Digital Numeration.
11 ? is it not lo and 1 i then expresse 10 as you were taught, and 1 also, ami that is 1 1 : and for 12 expresse 10 and 2 : for 23 set 20 and 3 : and so for 68 you muste make GO and there to 8 : and so ioo of all other sortes. [But now yf you wolde represent* 100 other 4 any nomber aboue it, you muste do that with the ryghte hande, after this maner. [You must expresse 100 in the ryght hand, with the ly tell fynger so bowed as you dyd expresse 1 in the left hand. 8
1 1366. ^And as you expressed 2 in the lefte hande, the same fasshyon
■-;11'1 in the ryght hande doth declare 200. 300 The fonrme of ."> in the ryght hand standeth for 300. too The fourme of 4, for 400. 12
•' Lykewayes the fourme of 5, for 500.
ooo The fourme of G, for GOO. And to he shorte : loke how you did expresse single vnities and tenthes in the lefte hande, so must you expresse vnities and tenthes of hundredes, in the ryghte hande. 1G boo S. I vnderstande you thus: (hat yf 1 wold represent 900, 1 must so fourme the fyngers of my ryghte hande, as I shuld do in my left hand to expresse 9, And as in my lefte hand I expressed iooo 10, so in my ryght hande must I expresse 1000. 20
And so the fourme of euery tenthe in the lefte hande serueth to expresse lyke no??iber of thousawdes, so y' fourme of 40 standeth *ooo for 4000.
- ' The fourme of 80 for 8000. -'
-And the fourme of 90 (whiche is the greatest) for 9000, and aboue that I can not expresse any nomber. M. No not with one fynger: how he it, 28
with dyuers fyngers you maye expresse 9999, and all at one tyme, and that lac keth hut 1 of 10000. So that vnder 10000 you may by your fyngers ex- 32
presse any summe. And this shal suf- fice for Numeration on the fyng( 1 3. And as for Addition, Subtraction, Multiplication, and Diuision (which 36
yet were neuer taughl by any man as farre as 1 do knowe) I wyll enstruct you after the treatyse of fractions. And now for this tyme fare well. 40
137 a
Digital Numeration. 09
and loke thai \ ou cease not to
practyse that you haue Lear
ned. S. Syr, with moste
harty niyiulc I thanke
you, bothe for your
■• 1 learnyng, and
also your good
course], which
(god wyllyng) I truste to folow.
Finis.
70
APPENDIX I.
% &xmtm on tin flttmcrntion of Rigorism.
[From a MS. of tfo l't>h Century.']
To alio suche oven nombrys the most have cifrys as to ten. twenty. Ihirtty. an hundred, an thousand and suche other, but ye schal vnderstonde that a cifre tokeneth nothinge lmt ho maketh other the more significatyf thai comith after hym. Also ye schal vnderstonde thai in nombrys composyl and in alle other nombrys that ben of diverse figurys ye schal begynne in the ritht syde and to rekene backwarde and so he schal be wryte as thus — 1000. 1hc sif ii' in the ritht side was first wryte and yit he tokeneth nothinge to the secunde no the thridde but thei maken that figure of 1 the more signyficatyf that comith after hem by as moche as he bora oute of his first place where he schuld yf he stode ther tokene lmt one. Ainl there he stondith nowe in the ferye place he tokeneth 12 a thousand as by tins rewle. In the firsl place he tokeneth hut hymself. In the secunde place he tokeneth ten times hymself. In tin1 thridde place he tokeneth an hundred tymes himself. In the ferye he tokeneth a thousand tymes himself. In the fyftye place 16 he tokeneth ten thousand tymes himself. In the sexte place he tokeneth an hundred thousand tymes hymself. In the seveth place he tokeneth ten hundred thousand lymes hymself, &c. And ye schal vnderstond thai this worde nombre is partyd into thre 20 partyes. Somme is callyd nombre of digitys fop alle hen digitys thai hen withine ten as ix, viii. vii, vi, v. iv, iii. ii. i. Al'ticules hen alle thei thai mow 1"' devyded into nombrys of ten as xx, xxx, xl, and suche other. Composittys be alio nombrys thai hen com- 24 ponyd of a digyl and of an articule as fourtene fyftene thrittene and suche other. Fourtene is componyd of four thai is a digyl
Numeration. 71
and of ten that is an articule. Fyftene is componyd of fyve that
is a digyt and of ten that is an articule and so of others
But as to this rewle. Jn the firste place lie tokeneth hut himself 4 that is to say he tokeneth hut that and no more. If that he stonde in the seeumle place he tokeneth ten tyines himself as this figure 2 here 21. this is oon and twenty. This figure 2 stondith in the secunde place and therfor he tokeneth ten tyraes himself and ten 8 tymes 2 is twenty and so forye of every figure and he stonde after another toward the lest syde he schal tokene ten tymes as moche more as he schuld token and he stode in that place ther that the figure afore him stondeth : lo an example as thus 9634. This
12 figure of foure that hath this schape 4 tokeneth hut himself for he stondeth in the first place. The figure of thre that hath this schape 3 tokeneth ten tyine himself for he stondeth in the secunde place and that is thritti. The figure of sexe that hath this schape G
1G tokeneth ten tyme more than he schuld and he stode in the place yer the figure of thre stondeth for ther he schuld tokene hut sexty. And now he tokeneth ten tymes that is sexe hundrid. The figure of nyne that hath this schape 9 tokeneth ten tymes more than he
20 schulde and he stode in the place ther the figure of G stondeth inne for thanne he schuld tokene hut nyne hundryd. And in the place that he stondeth inne m>we he tokeneth nine thousand. Alle the hole nombre of these foure figurys. Nine thousand sexe hundrid
24 and foure and thritti.
APPENDIX IT.
Carmen tic ^(prisma.
[From a B.M. MS., 8 C. iv., with additions from 12 E. 1 & Eg. 2622.] Hec algorismus ars presens dicitur1 ; in qua Talibus Indorum2 fruimur l>is quinque figuris. 0. 9. 8. 7. C. 5. 4. 3. 2. 1. Prima significat unum : duo vero secunda : 4
Tercia significat tria: sic procede sinistra Donee ad extremam venies, qua cifra vocatur; 8[Que nil significat; dat significare sequenti.] Quelibet illarum si primo liruite ponas, 8
Simpliciter se significat : si vero seenndo, Se decies : sursuni procedas tnultiplicando.4 [Namque figura sequens quevis signat decies plus, Ipsa locata loco quam signilicet percunte : 12
Nam precedentes plus \iltima significabit.]
5 Post predicta scias quod tres breuiter numerorum
Distincte species sunt; nam quidam digiti sunt;
Articuli quidam ; quidam quoque compositi -nut. 10
[.Sunt digiti numeri qui citra denarium sunt ;
Articuli decupli degrtoruro ; compositi sunt
Illi tpii constant ex articulis digitisque.]
Ergo, proposito numero tilii scribere, primo 20
Respicias quis sit numerus ; quia si digitus sit,
5 [Una figura satis sibi ; sed si compositus sit,]
I'rimo scribe Lien d'i^itum post articulum fac
Articulus si sit, cifram post articulum sit, 24
[Articulum vero reliquenti in scribe figure.]
1 " Hoc prsesens ars dicitur algorismus ab Algore rege ejus inventore, vel dicitur ab algos quod est ars, et rodos quod est nuraerus; quse est ars numer- orum vel numerandi, ad quam artem bene sciendum inveniebantur apud Imtos Ms qninque i^iil est decern) ligurse. " — Comment. Tkotna de Novo-Meratiu. Ms. Bib. Reg. Mus. Brit. 12 E. 1.
- "Il;i' necessarise figura sunt rndorum characteros. " .'/.v. de numcra- tione. Bib. Sloan. .Mus. Brit. 513, fol. 58. "Cum ridissem Yndos constituisse i\ litems in mi i verso numero sun propter dispositionem suam quam posuerunt, volui patefacere de opere quod sit per eas aliquidque esset levius discentibus, si Deus voluerit. Si autem Indi 1km' voluerunl et intentio illorum nihil novem literis fuit, causa que mibi potuit. Dens direxit me ad hoc. Si \rero alia dicam preter earn quam ego exposui, hoc fecerunl per hoc quod ego exposui, eadem tarn certissime et absque ulla dubitatione poterit inveniri. Leritasque patebil aspicientibus et discentibus." MS. 1'. I;.''., li. vi. 5, f. 102.
'•' From Eg. 2622.
* 8 0. iv. inserts Nullum cipa significat : dat significare sequenti.
« From 12 E. 1.
Addition, Subtraction. 73
Quolibet in numero, si par sit prima figura, Par erit el totum, quicquid sibi continetur; Era par si fuerit, totum sibi fiet et inrpar. 28
Septem.1 sunt partes, non plures, istius artis;
Addere, subtrahere, duplare, dimidiare ;
Soxta est diuidere, set quinta est rnultiplicare ;
Radicem extrabere pars septhna dicitur esse. .".•_'
Subtrabis aut addis a dextris vel mediabis ;
A leua dupla, diuide5 multiplicaque :
Extrahe radicem semper sub parte sinistra.
Addere si numero numerum vis, ordine tali 36 Addition,
Incipc ; scribe duas primo series numerorum
Prima sub prima recte poneudo figurani,
Et sic de reliquis facias, si shit tibi plures.
Tndc duas adde primas hac condicione; 40
Si digitus crescat ex addicione priorum,
Primo scribe loco digitum, quicunque sit ille ;
Si sit compositus, in limite scribe sequent:
Articulum, primo digitum; quia sic iubet ordo. 44
Articulus si sit, in primo limite cifram,
Articulum vein reliquis inscribe figuris;
Vel per se scribas si nulla figura sequatur.
Si tibi cifra superueniens occurrerit, illam 48
I >eme suppositam ; post illic scribe figuram :
Postea procedas reliquas addendo figuras.
A numero numerum si sit tibi demere cura, subtraction.
Scribe figurarum scries, vt in addicione; 52
Maiori numero numerum suppone minorem,
Sine pari numero supponatur numerus par.
1'.. tea si possis a prima subtrahe primam,
Scribens quod remanet, cifram si nil remanebit. 56
Set si non possis a prima demere primam ;
Procedens, vnuin tie Limite deme sequent! ;
1 En argorisme devon prendre Vii especes .... Adi ion ubtracion I loubloison mediacion Monteploie ot division El de radix enst racion A chez \ ii especes savoir Doil chai eun i n memoire avoir Letres qui figures sunt dites Ml quiexcellens lonl ecrites. MS, Scld. Arch, B. 26,
74 Duplation, Mediation.
Et demptum pro denario reputabis ab illo,
Subtrahe totaliter numerum quem proposuisti. 60
Quo facto, scribe supra quicquit remanebit, Facque novcnarios de cifris, cum rernanebis, Occurrant si forte cifre, dum demseris viuim ; Postea procedas reliquas demendo figuras. 6-t
Proof. ] [Si subtracio sit bene facta probare valebis,
Quas subtraxisti primas addendo iiguras. Nam, subtractio si bene sit, primas retinebis, Et subtractio facta tibi probat additionem.] 68
Dupiation. Si vis duplare numerum, sic iucipe; solam
Scribe figurarum seriem, quamcamque voles que
Postea procedas primam duplando figuram ;
Inde quod excrescet, scribeus, vbi iusserit ordo, 72
Juxta precepta que dantur in addicione.
Nam si sit digitus, in prinio limite scribe ;
Articulus si sit, in primo limite cifram,
Articulum vero reliquis inscribe iiguris ; 76
\Y1 per se scribas, si nulla figura sequatur :
Compositus si sit, in limite scribe sequent]
Articulum primo, digitum ; quia sic jubet ordo :
Et sic de reliquis facias, si .sint tibi plures. 80
1 1 Si super extremam nota sit, monadem dat (idem,
Quod tibi contingit, si primo dimidiabis. ] Mediation. Tncipc sic, si vis aliquem numerum mediare :
Scribe figurarum seriem solam, velud ante ; 84
Postea procedens medias, el prima figura
Si par aut impar videas ; quia si fuerit par,
Dimidiabis earn, scribens quicquit remanebil ;
Impar si fuerit, vnum demas, mediare, S8
Nonne presumas, sed quod superesl mediabis ;
Inde super tractum, fac demptum quod notat mniiii :
Si monos, dele ; sit ilii cifra post nota supra.
Postea procedas hac condicione secunda : - 92
[mpar8 si fuerit hie vnum deme priori,
[nscribens quinque, nam denos significabil
Monos prsedictam : si vero secunda dai vnam,
Ilia deleta, scribatur cifra ; priori 96
1 From 12 E. 1.
2 8 0. iv. inserts A.tque figura prior auper merit mediando. ' /. e. 6gura secundo loco posita.
Multiplication. 7c
Tradendo quinque pro denario rnediato;
N cifra scribatuTj nisi hide figura sequatur :
Postea procdeas reliquas mediando figuras,
Quin supra doeui, si sint iibi rnille figure. 100
1 [Si mediatio sit bene facta probare valebis, Duplaudo numerum quein primo dimidiasti.]
Si tu per numerum numerum vis multiplicare, tion.
Scribe duas, quascunque volis, series numerorum ; 104
Ordo tamen seruetur vt vltima multiplicandi Ponatur super anteriorem multiplicantis ;
2 [A leua relique sint scripte multiplicantes.]
In digitum euros digitum si ducere, major 108
Per quantes distat a denis respice, debes
Namque suo decuplo tociens delere minorem ;
Sicque til ji numerus veniens exinde patebit.
Postea procedas postremam multiplicando, 112
Juste multiplicans per cunctas inferiores,
Condicione tamen tali ; quod multiplicantis
Scribas in capite, qtiicquid processerit hide;
Set postquam fuerit hec multiplicata, figure 116
Anteriorontur seriei multiplicautis ;
El sic multiplica, velut istam multiplicasti,
Qui sequitur numerum scriptum quicunque figuris.
Set cum niultiplicas, primo sic est operandum, 120
Si dabit articulum tibi multiplicacio solum ;
Proposita cifra, summam transferre memento.
sin autem digitus excrescent articulusque,
Articulus supraposito digito salit ultra; 124
Si digitus tamen, ponas ilium super ipsam,
Subdita multiplicans banc que super Lncidit illi
Delel earn penitus, scribens quod provenit inde;
Sed -i multiplices illam posite super ipsam, 128
Adiungens numerum quern prebet ductus earum;
Si supraimpositam cifra debet multiplicare,
Prorsus earn delet, scribi que loco cifra debet,
2 [Si cifra multiplica! aliam positam super ipsam, 132
Sitque locus supra vacuus super hanc cifra fiet;]
1 So 12 E. 1 ; 8 0. iv. inserts- Si gnpei extremam aota -it monades dat eidem Quod contingat cum primo dimiabis Atijue figura prior ouper fuerit mediando.
- 12 K. l inserts.
76 Multiplication Without Figures,
Si supra fuerit cifra semper pretereunda estj
Si dubitee, an sit bene multiplicando secunda,
Diuide totalem numerum per multiplicantem, 13G
Et reddet numerus emergens inde priorem. Mental * [Per numerum si vis numerum quoque multiplicare ^uHiphca- rp.ul^um per normas subtiles absque figuris
Has norm as poteris per versus scire sequentes. HO
Si tu per digitum digitum quilibet multiplicabis
Regula precedens dat qualiter est operandum
Articulum si per reliquum vis multiplicare
In proprium digitum debebit uterque resolvi 1 14
Articulus digitos post per se multiplicantes
Ex digitis quociens ten ere t multiplicatum
Articuli faciunt tot centum multiplicati.
Articulum digito si multiplicamus oportet 148
Articulum digitum sumi quo multiplicare
Debemus reliquum quod multiplicaris ab illis
Per reliquo decuplum sic omne latere nequibit
In numerum mix turn digitum si ducere cures L52
Articulus mixti sumatur deinde resolvas
In digitum post hec fac ita de digitis nee
Articulusque docet excrescens in detinendo
In digitum mixti post ducas multiplicantem 156
De digitis ut norma docet sit juncta secunda
Multiplica summam et postea summa patebil
Junctus in articulum purum articulumque
2 [Articulum purum comittes articulum que] 1G0
Mixti pre digitis post fiat et articulus vt
Norma jubet retinendo quod egreditur ab illis
Articuli digitum post iu digitum mixti due
Regula tie digitis ut percipit articulusque 1G4
Ex quibus excrescens summe tu junge priori
Sic manifesta cito lid tibi summa petita.
Compositum numerum mixto sic multiplicabis
Vhdecies tredecem sic est ex hiis operandum 168
In reliquum primum demum due post in eundem
Tiium post deinde due in tercia deinde per unum
Multiplices tercia demum tunc omnia multiplicata
In summa duces quam que fueril te dices 172
1 12 E. 1 ins, its to 1. 174. a 12 E. 1 omits, Eg. 2G22 inserts.
1 1 ,- rion, Square Numbers.
77
Hie ut hie mixtus intentus est operanduni Multiplicandorura de norniis sufficiunt hec.]- Si vis dividere numerum, sic incipe primo ; Scribe duas, quascunque vol 3, eries numeroruni; Majori numero numerum suppone miuorem, 1[X:mi dicct ut major ton at bis terve minorem;] Et sub supprima supprimam pone figuram, Sic reliquis reliquas a dextra parte locabis ; Postea de prima primam sub parte sinistra Subtrahe, si possis, quociens potes adminus istud, Scribens quod remanet sub tali conditione; Ut totiens demas demendas a remanente, Que scrie recte ponentur in anteriori, I nica si, tantum sit ibi decet operari : Set si dou ] ossis a prima demere primam, Procedas, et earn numero suppone sequenti ; Hanc uno retrahendo gradu quo comites retrahantur, Et, qUotiens poteris, ab eadem deme priorem, Ut totiens demas demendas a remanenti, Nee plus ipuun novies quicquam tibi demere deb 3, Nascitur liinc numerus quociens supraque sequentem Hunc primo scribas, retrabas exinde figuras, Dum fuerit major supra positus inferiori, Et rarsum Bat divisio mure priori ; Et numerum quotiens supra scribas pereunti, Si fiat saliens retrahendo, cifra locetur, Et pereat numero quotiens, proponas eidem C if ram, ne numerum pereat vis, dum locus illic I.' tat, et expletis divisio non valet ultra: Dum fuerit numerus numerorum inferiore seorsum Ilium servabis; hinc multiplicand© probabis, Si bene fecisti, divisor multiplicetur Per numerum quotiens; cum multiplicaveris, adde Totali sumuue, quod servatum fuit ante, Reddeturque tibi numerus quern proposuisti ; Et si nil remanet, hunc multiplicando reddet, Cum ducis numerum per se, qui provenit inde Sit tibi quadratus, ductus radix erit hujus, Nee numeros omnes quadratos dicere debes, Est autem oinnis numerus radix alicujus. 1 12 E. 1 inserts.
176
LSI!
184
188
192
IDG
•JIM I
JU1 Proof.
208
Siiuare Numbers.
212
78 Sq uare Boot.
Quando voles numeri radicem querere, scribi
Debet; hide notes si sit locus ulterius impar,
Estque figura loco talis scribenda sub ill",
Que, per se dicta, numerum tibi destruat ilium, 216
Vel quantum potent ex inde delebis eandem;
Vel retrahendo duples retrahens duplando snl> isla
Que prime sequitur, duplicatur per duplacationem,
Post per se minuens pro posse quod est minuenduni, ' 220
1 Post liis propones digitum, qui, more priori
Per precedentes, post per se multiplicatus,
Destruat in quantum poterit numerum remanentem,
Et sic procedens retraliens duplando figuram, 22 1
Preponendo novam donee totum peragatur,
Subdupla propriis servare docetque duplatis ;
Si det compositum numerum duplacio, debel
[nscribi digitus a parte dextra parte propinqua, -L's
Articulusque loco quo non duplicata resessit;
Si dabit articulum, sit cifra loco pereunte
ArtiEulusque locum tenet unum, de duplicata rcsessil ;
Si donet digitum, sub prima pone sequente, 232
Si supraposita fuerit duplicata figura
Major proponi debet tantummodo cifra,
Has retrahens solito propones more figuram,
Usque sub extrema ita fac retrabendo figuras, 236
Si totum deles numerum quern proposuisti,
Quadratus fuerit, de dupla quod duplicasti,
Sicque tibi radixUlius certa patebit,
Si de duplatis fitjuncta supprima figura; 240
Radicem per se multiplices habeasque
Primo propositum, bene te fecisse probasti ;
Non est quadratus, si quis restat, sed habentur
Radix quadrati qui stat major sub eadein ; J 1 1
Vel quicquitl remanel tabula servare memento ;
Hoc casu radix per se quoque midtiplicetur,
\ el sir quadratus sub primo major habetur,
lliii" addas remanens, el prius debes haberi ; 248
Si locus extremus fueril par, scribe figuram
Sub pereunte loco per quam debes operari,
Que quantum poteril supprimas destruat amb
1 S ('. iv. inserts—
linn ill.iin dele duplans sub ei psallii ado
Que sequitur retraliens quicquid fuerit duplicatum.
Cube Boot. 79
Vel penitus legem teneas operando priorem, 252
Si stippositum digitus suo fine repcrtus,
Omnino delet illie scribi cifra debet,
A leva si qua sit ei sociata figura ;
Si cifre remanent in fine pares decefc harum 256
Radices, numero mediam proponere partem,
Tali quesita radix patet arte reperta.
I'll niimerum recte >i nosti multiplicare
Ejus quadratum, numerus qui pervenit inde 2G0
Dicetur cubicus; primus radix erit ejus;
Nee numeros omnes cubicatos dicere debes,
Est autem omnis numerus radix alicujus;
Si curas cubici radicem quaerere, primo 264 cube Root.
Enscriptum numerum distinguere per loca debes ;
Que tibi mille notant a mille notaute suprema
Initiam, sum ma operandi parte sinistra,
Illic sub scribas digitum, qui multiplicatus 268
In semet cubicc suprapositum sibi perdat,
Et si quid fuerit adjunctum parte sinistra
Si non omnino, quantum poteris minuendo,
Hinc triplans retrahe saltum, faciendo sub ilia 272
Que manet a digito deleto tenia, figuram
Illi propones que sub triplo asocietur,
Et cum subtriplo per earn tripla multiplicatur ;
Him' per earn solam productum multiplicabis, 276
Postca totaleni numerum, qui provenit inde
A suprapositis respectu tulle triplate
Addita supprimo cubice tune niultiplicetur,
Respectu cujus, numerus qui progredietur 280
Ex cubito ductu, supra omnes adimetur ;
Tunc ipsam delens triples saltum faciendo,
Semper sub ternas, retrahens alias triplicatas
Ex hinc triplatis aliam propone figuram, 284
Que per triplatas ducatur more priori ;
Primo sub triplis sibi junctis, postea per se,
In numerum ducta, productum de triplicatis :
Utque prius dixi numerus qui provenil inde 288
A suprapositis has respiciendo trahatur,
Huic cubice ductum sub primo multiplicabis,
EcspectunKjuc sui, removebis de remanenti,
Et sic procedas retrahendo triplando figuram. 292
80 Ouhe Boot
Et proponendo nonam, donee totum peragatur,
Subtripla sub propriis servare decet triplicatis ;
Si nil in fine remanet, nnmerus datus ante
Est cubicus; cubicam radicem sub tripla prebent, 29G
Cum digito juncto quern supprimo posuisti,
Hec cubice ducta, numerum reddant tibi priumm.
Si quid erit remanens non est cubicus, sed habetur
Major sub primo qui skit radix cubicam, 300
Servari debet quicquid radice remansit,
Extraeto numero, decet hec addi cubicato.
Quo facto, nnmerus reddi debet tilii primui .
Nam debes per se radicem niultiplicare 304
Ex hinc in numerum duces, qui provenit inde
Sub primo cubicus major sie invenietur;
I Hi jungatur remanens, et primus habetur,
Si per fcriplatum numerum nequeas operari ; 3ns
Cifram propones, nil vero per bam' operare
Set retrahens illam cum saltu deinde triplata,
Propones illi digitum sub lege priori,
Cumque cifram retrahas saliendo, non triplicabis, 312
Namque nihil cifre triplacio dicitur esse ;
At tu cum cifram protraxeris aut triplicata,
llanc cum subtriplo semper servare memento :
Si det compositum, digiti triplacio debet 316
Illius scribi, digitus saliendo sub ipsam ;
Digito delete, que tenia dicitur esse ;
Jungitur articulus cum triplata pereunte,
Set tacit hunc scribi per se triplacio prima, 320
Que si det digitum per se scribi facit ilium ;
Consumpto numero, si sole fuit tibi cifre
Triplato, propone cifram saltum faciendo,
Cumque cifram retrahe triplam, scribendo figuram, 324
Preponas cifre, sic procedens operare,
Si tres vel duo serie in sint, pone sub yma,
A dextris digitum servando prius documentum.
si sit continua progressio terminus nuper 328
Per majus me Hum totalem multiplicato;
Si par, per medium tune multiplicato sequentem.
Set si continua non sit progressio finis :
[nipar, tune majus medium si rnultiplicabis, 332
Si par per medium sibi multiplicato propinquum. 333
INDEX OF TECHNICAL TERMS1
algorisme, 33/i2; algorym, augrym, •'! 3 ; the art of computing, using the so-called Arabic numerals.
The word in its various forms is derived from the Arabic nl- Khawarazmi (i.e. the native of Khwarazm (Khiva)). This was the surname of Ja'far Mohammad ben Musa, who wrote a treatise early in the 9th century (see p. xiv).
The form algorithm is also found, being suggested by a supposed derivation from the Greek api8/j.6s (number).
antery, 24 11 ; to move figures to the right of the position in which they are first written. This operation is performed repeatedly upon the multiplier in multiplication, and upon certain figures which arise in the process of root extraction.
anterioracioun, 50 5 ; the operation of moving figures to the right.
article, 34 23 ; articul, 031 ; artiCUls, 0 36, 29, 7, 8 ; a number divisible by ten without remainder.
Cast, 812 : to add one number to another.
'Addition is a cutting together of two numbers into one number,' 8/10.
Cifre, 4/i ; the name of the figure 0. The word is derived from the Arabic sifr= empty, nothing. Hence :<-r<>.
A cipher is the symbol of the absence of number or of zero quantity. It may be used alone or in conjunction witli digits or other ciphers, and in the latter case, according to the position which it occupies relative to the other figures, indicates the absence of units. or tens, or hundreds, etc. The great superiority of the Arabic to all other systems of notation resides in the employment of this symbol. When the cipher is not used, the place value of digits has to be indicated by writing them in assigned rows or columns. Ciphers, however, may be interpolated amongst the significant figures used, and as they sufficiently indicate the positions of the empty rows or columns, the latter need not be indicated in any other way. The practical performance of calculations is thus enormously facilitated (see p. xvi).
componede, 33/24; composyt, 035; with reference to numbers, one compounded of a multiple of ten and a digit.
conuertide = conversely. 41; 29, 47/9.
CUbicede, 50, 13 ; to be C, to have its cube root found.
1 This Index lias been kindly prepared by Professor J. B. Dale, of King's College, University of London, and the best thanks of the Society are due to him for his valuable contribution.
NOMBRYNGE. 81 °
82 Index of Technical Terms.
Cllbike nombre, 47/8 ; a number formed by multiplying a given number twice by itself, e. ;/. 27 = 3 x 3 x 3. Now called simply a cube.
decuple, 22/12 ; the product of a number by ten. Tenfold.
departys = divides, 5/29.
digit, 5/30; digitalle, 33/24; a number less than ten, represented by one of the nine Arabic numerals.
dimydicion, 7/23 ; the operation of dividing a number by two. Halving.
duccioun, multiplication, 43/9.
duplacion, 7/23, 14/15 ; tne operation of multiplying a number by two. Doubling.
i-mediet — halved, 19/23.
intercise — broken, 46/2 ; intercise Progression is the name given to either of the Progressions 1, 3, 5, 7, etc. ; 2, 4, 6, 8, etc., in which the common difference is 2.
lede into, multiply by, 47/l8.
lyneal nombre, 4*> '14 ; a number such as that which expresses the measure of the length of a line, and therefore is not necessarily the product of two or more numbers (vide Superficial, Solid). This appears to be the meaning of the phrase as used in The Art of Nombryng. It is possible that the numbers so designated are the prime numbers, that is, numbers not divisible by any other number except- them- selves and unity, but it is not clear that this limitation is intended.
mediacioun, 16 36, 38/i6 ; dividing by two (set' also dimydicion).
medlede nombre, 34 1 ; a number formed of a multiple of ten and a digit (vide componede, composyt).
medye, 17/8, to halve ; mediete, halved, 17 30 ; ymedit, 20/9.
naturelle progressioun, 45/22 ; the series of numbers 1, 2, 3, etc.
prodllCCiOUn, multiplication, aO/n.
quadrat nombre, 16 12 ; a number formed by multiplying a given number by itself, e.g. '.» — 3 x ;», a square.
rote, 7 25 ; rootej"47'n ; root. The roots of squares and cubes are the numbers from which the squares and cubes are derived by multi- plication into themselves.
significatyf, significant, 5/14. The significant figures of a number are, strictly speaking, those other than zero, e.g. in 3 6 5 0 4 0 0, the
significant figures are .">, 6, 5, -I. Modern usage, however, regards all figures between the two extreme significant figures as significant,
even when some are zero. Thus, in the above example, 3 6 5 0 4 are considered significant.
solide nombre, -4 « > 37 ; a number which is the producl of three other
numbers, • . g. 66 = 11 x 2 x 3. superficial nombre, 46 iS ; a number which is the product of two other
numbers, e. g. 6 = 2 x 3. ternary, consisting of three digits, 51 . vnder double, a digit which has been doubled, 48 3. vnder-trebille, a digit which has been trebled, 49/28 ; vnder-triplat,
49/39-
W, a symbol used to denote half a unit, 17/33.
GLOSSARY
ablacioun, taking away, 36/21
addyst, haddest, IO/37
agregacioun, addition, 45/22. (First
example in X.E.D., 15 17.) a-3enen.es, against, 23/io allgate, always, 8/39 als, as, 22/24
and, if, 29/8 ; &, 4/27 ; & yf, 2O/7 a-nendes, towards, 23/15 aproprede, appropriated, 34/27 apwereth, appears, 61/8 arisy;t, arises, H/24 a-rowe, in a row, 29/ 10 arsemetrike, arithmetic, 33/ 1 ayene, again, 45/15
bagle, crozier, 67/ 12
bordure = ordure, row, 43/30
borro, inf. borrow, 11 /'38 ; imp. ?.
borowe, 12/20; pp. borwed, 12/15 ;
borred, 12/19 boue, above, 42/34
ferye = ferj>e, fourth, 70/ 12 figure == figures, 5/i for-by, past, H/21 fors; no f, no matter, 22/24 forseth, matters, 53/30 forye = forbe, forth, 71/8 fyftye = fjftbe, fifth, 70/i6
grewe, Greek, 33/13
haluendel, half, I6/16; haldel, 19/4 ;
pL haluedels, I6/16 hayst, hast, 17/3, 32 hast, haste, 22/25 heer, higher, 9/35 here, their, 7/26 here-a-fore, heretofore, 13/7 heyth, was called, 3/5 hole, whole, 4/39 ; nolle, 17/i ; hoole,
of three dimensions, 46/i 5 holdy!>e, holds good, 30/5 how be it that, although, 44/4
caputule, chapter, 7/26 certayn, assuredly, I8/34 clepede, called, 47/7 competently, conveniently, 35/S ODmpt, count, 47/29 contynes, contains, 21/i2 ; pp.
tenythe, 38/39 craft, art, 3/4
distingue, divide, 51/5
egalle, equal, 45/21 excep, except, 5/i6 exclusede, excluded, 34/37 excressent, resulting, 35/ 16 exeant, resulting, 43/26 expone, expound, 3/23
lede = lete, let, 8/37
lene, lend, 12/39
lest, least, 43/27
lest = left, 71/9
leue, leave, 6/5 ; pr. 3 s. leues, re- mains, II/19 ; leus, H/28 ; pp. laft, left, 19/24
lewder, more ignorant, 3/3
lust, desirest to, 45/ 13
ly3t, easy, 15 31
lymytes, limits, 34/iS ; lynes, 34/i2 ; lynees, 34/i7 ; Lat. limes, pi. limit, s.
maystery, achievement ; no m., no
achievement, i.e. easy, 19/io me, indef. pron. one, 42/ 1 mo, more, 9/ 16
83
84
Glossary.
moder = more (Lat. majorem), 43/22
most, must, 30/3
multipliede, to be m. = multiplying,
40/9 mynvtes, the sixty parts into which a,
unit is divided, 38/25 myse-wroU, mis-wrought, 14/u
nether, nor, 3 1/25 nex, next, 19/9 no;t nought, 5/7 note, not, 30/5
00, one, 42/20-; 0, 42/21
omest, uppermost, higher, 35/26 ;
omyst, 35/28 omwhile, sometimes, 45/31 on, oue, 8/29 opyne. plain, 47/8 or, before, 13/25 or = ))e o}>cr, the other, 2S/34 ordure, order, 34/9 ; row, 43/ 1 other, or, 33/i3, 43/26 ; other . . .
or, either . . . or, 38/37 ouerer, upper, 42/15 ouer-hippede, passed over, 43/ig
recte, directly, 27/2o remayner, remainder, 56/28 reprosentithe, represented, 39/14 resteth, remains, 63/29 rewarde, regard, 48/6 rew, row, 4/8
rewle, row, 4/20, 7/i2; rewele, 4/iS; rewles, rules, 5/33
s. = scilicet, 3/8
sentens, meaning, 14/29
signifye(tyf), 5/13. The last three letters are added above the line, evidently because of the word 'sig- nificatyf in 1. 14. But the «So- lucio,' which contained tlie word, has been omitted.
sithen, since, 33/8
some, sum, result, 40/ 17, 32
sowne, pronounce, 6/29
Bingillatim, singly, 7/25
spices, species, kinds, 34/4
spyl, waste, H/26
styde, stead, I8/20
subtrahe, subtract, 48/i2 ; pp,
trayd, 13/21 sythes, times, 21/ 16
sub-
tan, taught, I6/36
take, pp. taken ; t. fro, starting from,
45/22 taward, toward, 23/34 thou^t, though, 5/20 trebille, multiply by three, 49/26 twene, two, 8/ 11 ]>OW, though, 25/15
|'Ow;t, thought; be ]>., mentally, 28/4 ])us = pis, this, 2O/33
vny, unite, 45/io
wel, wilt, H/31
wete, wit, 15/i6; wyte, know, 8/38;
pr. 2 s. wost, 12/38 wex, become, 50/i8 where, whether, 29/ 12 wher thurghe, whence, 4 9/ 15 worch, work, 8/19; wrica, 8/35;
wyrch, *'»/i9; iimp. s. worch, 15/9 ;
pp. y-wroth, 13/24 write, written, 29/19; y-write, I6/1 wryrchynge = wyrchynge, working,
30/4 wt, with, 55/8
y-broth, brought, 21/ iS ychon, each one, 29/ 10 ydo, done, added, 9/6 ylke, same, 5/12 ydyech, alike, 22/23 y-my;t, been able, 12/2 y-now;t, enough, 15,31 yove, given, 45/33 y'. that, 52/8 y-write, v. write, y-wroth, v. worch.
ynov^t, I8/34
(ftaiilfl (ftitfllish jfest £orictii
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L. Charlemagne Romances :— 12. Huon of Burdeux, by Lord Berners, ed. S. L. Lee, B.A. Part IV. 5*. ,,
I.I. Torrent of Portyngale, from the unique MS. in the Chetham Library, ed. E. Adam, Ph.D. 10*. ,,
LIL Bullein's Dialogue against the Feuer Pestilence, 1578 (ed. 1, 1584). Ed. M. A A. 11. Bullen. 10s. 18S8
LIII. Vicary's Anatomie of the Body of Man, 1548, id. I'i77, ed. 1''. ,1. A 1 rn\ Furnivall. Parti. 15*. ,.
I. IV. Caxton's Englishing of Alain Chartier's Curial, I ! Dr P. ■ L Furnivall * Prof. P. Meyer. 5s. ,,
The Extra Series of the " Early English Text Society." 7
LV. Barbour's Bruce, ed. Rev. Prof. Skeat, Litt.D., LL.D. Part IV. 5s. 1SS9
LVI. Early English Pronunciation, by A. J. Ellis, Esq.j F.R.S. Pt. V., the present English Dialects. 25*. ,,
LVII. Caxton's Eneydos, a.d. 1490, coll. with its French, ed. M. T. Culley, M.A. & Dr. F.J. Furnivall. 13s. 1S90
LVIII. Caxton's Blanchardyn & Eglantine, c. 14S9, extracts from ed. 1595, <fc French, ed. Dr. L. Kellner. 17s.
LIX. Guy of Warwick, 2 texts (Auchinleck and Cains MSS.), Tart 111., ed. Prof. J. Zupitza, Ph.D. 15s. 1S91
LX. Lydgate's Temple of Glass, re-edited from the MSS. by Dr. J. Schick. 15*.
LXI. Hoccleve's Minor Poems. I , from the Phillipps and Durham MSS., ed. F. J. Furnivall, Ph.D. 15s. 1892
LXII. The Chester Plays, re-edited from the MSS. by the late Dr. Hermann Deimling. Parti. 15*.
LXI1I. Thoma3 a Kempis's De Imitatione Christi, englisht ab. 1440, & 1502, ed. Prof. J. K. Ingram, l.'.s. 1893
LX1V. Caxton's Godfrey of Boloyne, or Last Siege of Jerusalem, 1481, ed. Dr. Mary N. Colvin. 15s.
LXV. Sir Bevis of Ham ton, ed. Prof. E. Kolbing, Ph.D. Part III. 15s. 18?'4
LXV1 Lydgate's and Burgh's Secrees of Philisoffres (' Governance of Kings and Princes '), ab. 1445 — 50.
ed. B. Steele, B.A. 15*. LXVII. The Three Kings' Sons, a Romance, ab. 1500, Part I., the Text, ed. Dr. Furnivall. 10s. 1895
LXVIII. Melusine, the prose Romance, ah. 1500, Part I, the Text, ed. A. K. Donald. 20s. „
LXIX. Lydgate's Assembly of the Gods, ed. Prof. Oscar L. Triggs, M.A., Ph.D. 15s. 1891
LXX. The Digby Plays, edited by Dr. F. J. Furnivall. 15s.
LXXI. The Towneley Plays, ed. Geo. England and A. W. Pollard, M.A. 15s. 1897
LXXII. Hoccleve's Eegcment of Princes, 1411-12, and 14 Poems, edited by Dr. F. J. Furnivall. 15s. LXXIII. Hoccleve's Minor Poems, II., from the Ashburnham MS., ed. I. Gollancz, M.A. [At Press. l.XXIV. Secreta Secretorum, 3 prose Englishings, one by Jas Yonge, 1428, ed. R. Steele, B.A.
Part I. ■_"■. 1898
LXXV. Speculum Guidonis de Warwyk. edite 1 by Mis G. L. Morrill, M.A., Ph.D. 10s. ,,
LXXVI. George Ashby's Poems, &c, ed. Miss Mary Bateson. 15s. 1S'J9
LXXVII. Lydgate's DeGuilleville's Pilgrimage of the Life of Man, 1426, ed. Dr. F. J. Furnivall. Part I. 10s ,, LXXY1II. The Life and Death of Mary Magdalene, by T. Robinson, c. 1020, ed. Dr. II. O. Sommer. 5s. LXXIX. Caxton's Dialogues, English and French, c. 1483, ed. Henry Bradley, MA. 10s. 1900
LXXX. Lydgate's Two Kightingale Poems, ed. Dr. Otto Claiming. 5s. ,,
LXXXI. Gower's Confessio Amantis, edited 1>. G. C. Macaulay, M.A. Vol.1. 15s. „
LXXXII. Gower's Confessio Amantis, edited by G. C. Macaulay, M.A. Vol. II. 1901
LXXXIII. Lydgate's DeGuilleville's Pilgrimage of the Life of Man, 1420, ed. Dr. F.J. I'm uivall. Pt. II. 10 LXXXIV. Lydgate's Reason and Sensuality, edited by Dr. E. Sieper. Part I. 5s. ,,
LXXXV. Alexander Scott's Poems, 15CS, from the unique Edinburgh MS., ed. A. K. Donald, B.A. 10*. LXXXVI. William of Shoreham's Poems, re ed. from the unique .MS. by Dr. M. Konrath. Part I. 10*. ,, LXXXV1I. Two Coventry Corpus-Christi Plays, re-edited by Hardin Craig, M.A. 10s.
LXXXVI1I. Le Morte Arthur, re-edited from the Harleian MS. 2252 by Prof. Bruce, Ph.D. 15s. 1903
1.XXX1X. Lydgate's Reason and Sensuality, edited by Dr. E. Sieper. Part II. 15*. ,,
XC. English Fragments from Latin Medieval Sei vice-Books, ed. by Ily. Littlehales. 5s.
XCI. The Macro Plays, from Mr. Cuniey's unique MS, ed. Dr. Furnivall and A. W. Pollard, M.A. 10* 1904 XCII. Lydgate's DeGuileville's Pilgrimage of the Life of Man, Part III, cd. Miss Locock. 10s. XCIII. Lovelich's Romance of Merlin, from the unique MS., ed. Dr. E A. Kock. Part I. 10s XCIV. Respublica, a Play on Social England, a.d. 1553, ed. L. A. Magnus, LL.B. 12s. XCV. Lovelich's History of the Holy Grail, Pt. V. : The Legend of the Holy Grail, by Dorothy Kempe. 6*. ,, XCVI. Mirk's Peatial, edited from the MSS. by Dr. Erbe. Part I. 12*. „
XCVIl. Lydgate's Troy Book, edited from the best MSS. by Dr. Ily. Bergen. Tart I, Books [and I I. 1 XCVHI. Skelton's Magnyfycence, edited by Dr. R. I.. Ramsay, with an Introduction. Is. • I. „
XCIX. The Romance of Emare, re-edited from the MS. by Miss Edith Rickert, Ph.D. 7 . C. The Harrowing of Hell, and The Gospel of Nicodemus, reed, by Prof. Huline, M.A., Ph.D. 15*. CI. Songs, Carols, &c, from Richard Hill's Balliol MS., edited by It. Roman Dyboski. 1."..,-. „
CII. Promptorium Parvulorum, the 1st English-Latin Dictionary, ed. Rev. A. L. Mayhew, M.A. 21s. 1003 CHI. Lydgate's Troy Book, edited from the bet MSS. by Dr. Ily. Bergen. Part II, Book III. 10*. CIV. The Non-Cycle Mystery Plays, re-edited by O. Waterhouse, M.A. 15*.
CV. The Tale of Beryn, with the Pardoner and Tapster, ed. Dr. I". .). Furnivall and W. G. Stone. 15*. C VI. Lydgate's Troy Book, edited from the best MSS. by Dr. Hy. Bergen. Pari III. 15s. 1910
CVIL Lydgate's Minor Poems, edited by Dr. II. N. MacCracken. Put I, Religious Poems. 1 ,,
CVIII. Lydgate's Siege of Thebes, re-edited from the MSS. by Prof. Dr. A. Erdmann. Pt. J, The Text. 15«. 1911 CIX Partonope, re-edited from its 3 MSS. by Dr. A. T. Bodtker. The Texts. 15s. CX. Caxton's Mirrour of the World, with all the woodcuts, ed. by O. II. Pi ioi ,*M. A., Litt.D. 15*. CXI. Caxton's History of Jason, the Text, Part I, ed. by John Munro. 15s.
CXII. Lovelich's Romance of Merlin, ed. from the un tque US. by Prof. E. A. Kock, PhD. 15*. 1013
CX1II. Poems by Sir John Salusbury, Robert Chester, and others, from Christ Church MS. 184 &c • d
by Prof. Carleton Brown, Ph.D. 15s. CM V. The Gild of St. Mary, Lichfield, ed. by the late Dr. F. J. Furnivall. 15s. 1914
CXV. Th; Chester Plays. Part II, re-edited by Dr. Matthews. 15s.
|'X\ I. The Paukne Epistles, ed. Miss M. J. Powell. 15s. 1016
CXVII. Bp. Fisher's English Works, Pt. II, ed. by the Rev. Ronald IJ ayne. 15«.
CXVIII. The Craft of Nombrynge, ed. by R. Bteele, B.A. 15*. 1916
CXIX. The Owl and Nightingale, 2 Texts parallel, ed. by the late G. F. II. 3yk< s and .1. H, G.Grattan.
15*. [At Pr* . CXX. Ludus Co entriae, ed. by Miss K. S. Block, M.A. 30». [Nearly Ready. 1917
ORIGINAL SERIES.
Forthcoming issues will be chosen from the following :—
Harmony of Life of Christ, from Pepj'sian Library, ed. by Miss Margery Goates. [Ready, The Alliterative Siege of Jerusalem, edited by the late Prof. E. Kolbing, Ph.D. [At Press. A Stanzaic Exposition of the Feasts of the Church and the Life of Christ based on the Ltgenda Aurea, ed. from
the MS9. Hail. 3900, llarl. 2200, and Addit. 3S066, by Miss F. A. Foster, Ph.D. [At Press. The Earliest English Apocalypse with a Commentary, edited by Dr. Anna C. Paues. [At Press. Trevisa's Dialogus inter Militem et Clericum, Sermon by FitzRalph, and Begynyng* of the World, edited fiom
the MSS. by Aaron J. Perry, M.A. A Critical Edition of the Old English Heptiteuch (MS. Iiodl. Misc. i09), ed. by S. J. Crawford, M.A., B.Litt. The Sege or Battsll of Troy, ed. by Miss Baruicle. The Pepysian Southern Passion, ed. by Mrs. Carloton Brown. Three Old English Prose Tracts, from MS. Cott. Vitell. A. XV. ; ed. by Dr. S. I. Rypins.
EXTRA SERIES.
The Publications due up to 1920 iu ill probably be chosen from : —
The Owl and Nightingale, 2 Texts parallel, ed. by the late G. F. II. Sykes and J. H. G. Grattan. [At Press.
The "Coventry'' Plays, ed. by Miss Block, Royal Holloway College.
Lydgate's Minor Poems, ed. Dr. II. N. MacCracken. Part II, Secular Poems. [At Press.
Lydgate's Troy Book, ed. by Dr. Hy. Bergen. Part IV, Introduction, Notes, &c. [At Press.
Lydgate's Siege of Thebes, re-edited from the MSS. by Prof. Dr. A. Erdmann. Part II, Notes, Ac.
Seoreta Seoretorum : three prose Englisliings, ab. 1440, ed. R. Steele, b. A. Part II. [At Press.
The Book of the Foundation of St. Bartholomew's Hospital, London, MS. ab. 142J, ed. Sir Norman Moore. |£. t.
Piers Plowman, the A Text, re-edited from the MSS. by Prof. R. W. Chambers, M.A., D.Lit., and J. H. G.
Grattan, M.A. [At Press. King Alisaunder, two parallel texts, ed. from Lincoln's Inn MS. 150 and Laud. Misc. (322 by L. F. Powell, Esq. Caxton's Paris and Vienne, ed. by O. II. Prior, Litt.D. Interludium de Clerico et Puella and Dux Moraud, ed. Prof. J. M. Manly.
Other texts ore in preparation. August 1922.
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