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Title: The Earliest Arithmetics in English

Editor: Robert Steele

Contributor: de Villa Dei Alexander
Robert Record
active 1230 Joannes de Sacro Bosco

Release date: June 1, 2008 [eBook #25664]
Most recently updated: January 31, 2009

Language: English

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*** START OF THE PROJECT GUTENBERG EBOOK THE EARLIEST
ARITHMETICS IN ENGLISH ***

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ȝ, ſ (yogh, long s)
ɳ, łł (n with curl, crossed l: see below)
φ (Greek phi, sometimes used in printed text for 0)
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to change your browser’s default font.
In The Crafte of Nombrynge, final n was sometimes written with an extra curl
as . It has been rendered as ɳ for visual effect; the character is not intended
to convey phonetic information. In the same selection, the numeral “0” was
sometimes printed as Greek φ (phi); this has been retained for the e-text.
Double l with a line is shown as łł. The first few occurrences of d (for
“pence”) were printed with a curl as . The letter is shown with the same d’
used in the remainder of the text.
The word “withdraw” or “withdraw” was inconsistently hyphenated; it was left
as printed, and line-end hyphens were retained. All brackets [ ] are in the
original.
The diagrams in “Accomptynge by Counters” may not line up perfectly in all
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Typographical errors are shown in the text with mouse-hover popups. Other
underlined words are cross-references to the Index of Technical Terms and
the Glossary.

Contents
(added by transcriber)

Introduction v
The Crafte of Nombrynge 3
The Art of Nombryng 33
Accomptynge by Counters 52
The arte of nombrynge by the hande 66
App. I. A Treatise on the Numeration of Algorism 70
App. II. Carmen de Algorismo 72
Index of Technical Terms 81
Glossary 83

The Earliest Arithmetics
in English

EDITED WITH INTRODUCTION

BY

ROBERT STEELE

LONDON:

PUBLISHED FOR THE EARLY ENGLISH TEXT SOCIETY

BY HUMPHREY MILFORD, OXFORD UNIVERSITY PRESS,

AMEN CORNER, E.C. 4.
1922.

INTRODUCTION
The number of English arithmetics before the sixteenth century is very
small. This is hardly to be wondered at, as no one requiring to use even the
simplest operations of the art up to the middle of the fifteenth century was
likely to be ignorant of Latin, in which language there were several treatises
in a considerable number of manuscripts, as shown by the quantity of them
still in existence. Until modern commerce was fairly well established, few
persons required more arithmetic than addition and subtraction, and even in
the thirteenth century, scientific treatises addressed to advanced students
contemplated the likelihood of their not being able to do simple division.
On the other hand, the study of astronomy necessitated, from its earliest
days as a science, considerable skill and accuracy in computation, not only
in the calculation of astronomical tables but in their use, 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. (2653) f. 146-154 (15th c.) inc. “Of angrym ther be
IX figures in numbray . . .” A mere unfinished fragment, only getting
as far as Duplation.
(2) Camb. Univ. LI. IV. 14 (III.) f. 121-142 (15th c.) inc. “Al maner of
thyngis that prosedeth ffro the frist begynnyng . . .”
(3) Fragmentary passages or diagrams in Sloane 213 f. 120-3
(a fourteenth-century counting board), Egerton 2852 f. 5-13, Harl.
218 f. 147 and

(4) The two MSS. here printed; Eg. 2622 f. 136 and Ashmole 396 f. 48.
All of these, as the language shows, are of the fifteenth century.
The Crafte of Nombrynge is one of a large number of scientific treatises,
mostly in Latin, bound up together as Egerton MS. 2622 in the British
Museum Library. It measures 7” × 5”, 29-30 lines to the page, in a rough
hand. The English is N.E. Midland in dialect. It is a translation and
amplification of one of the numerous glosses on the de algorismo of
Alexander de Villa Dei (c. 1220), such as that of Thomas of Newmarket
contained in the British Museum MS. Reg. 12, E. 1. A fragment of another
translation of the same gloss was printed by Halliwell in his Rara
Mathematica (1835) p. 29. 1 It corresponds, as far as p. 71, l. 2, roughly to
p. 3 of our version, and from thence to the end p. 2, ll. 16-40.
The Art of Nombryng is one of the treatises bound up in the Bodleian MS.
Ashmole 396. It measures 11½” × 17¾”, and is written with thirty-three
lines to the page in a fifteenth century hand. It is a translation, rather literal,
with amplifications of the de arte numerandi attributed to John of
Holywood (Sacrobosco) and the translator had obviously a poor MS. before
him. The de arte numerandi was printed in 1488, 1490 (s.n.), 1501, 1503,
1510, 1517, 1521, 1522, 1523, 1582, and by Halliwell separately and in his
two editions of Rara Mathematica, 1839 and 1841, and reprinted by Curze
in 1897.
Both these tracts are here printed for the first time, but the first having been
circulated in proof a number of years ago, in an endeavour to discover other
manuscripts or parts of manuscripts of it, Dr. David Eugene Smith,
misunderstanding the position, printed some pages in a curious transcript
with four facsimiles in the Archiv für die Geschichte der
Naturwissenschaften und der Technik, 1909, and invited the scientific world
to take up the “not unpleasant task” of editing it.
Accomptynge by Counters is reprinted from the 1543 edition of Robert
Record’s Arithmetic, printed by R. Wolfe. It has been reprinted within the
last few years by Mr. F. P. Barnard, in his work on Casting Counters. It is
the earliest English treatise we have on this variety of the Abacus (there are
Latin ones of the end of the fifteenth century), but there is little doubt in my
mind that this method of performing the simple operations of arithmetic is

much older than any of the pen methods. At the end of the treatise there
follows a note on merchants’ and auditors’ ways of setting down sums, and
lastly, a system of digital numeration which seems of great antiquity and
almost world-wide extension.
After the fragment 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 Halliwell in Rara Mathemathica, but I
have added a number of stanzas from various manuscripts, selecting various
readings on the principle that the verses were made to scan, aided by the
advice of my friend Mr. Vernon Rendall, who is not responsible for the few
doubtful lines I have conserved. This poem is at the base of all other
treatises on the subject in medieval times, but I am unable to indicate its
sources.

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

The Pre-Medieval Instruments Used in Calculation.
The following are known:—

(1) A flat polished surface or tablets, strewn with sand, on which figures
were inscribed with a stylus.
(2) A polished tablet divided longitudinally into nine columns (or more)
grouped in threes, with which counters were used, either plain or marked
with signs denoting the nine numerals, etc.
(3) Tablets or boxes containing nine grooves or wires, in or on which ran
beads.
(4) Tablets on which nine (or more) horizontal lines were marked, each
third being marked off.
The only Greek counting board we have is of the fourth class and was
discovered at Salamis. It was engraved on a block of marble, and measures
5 feet by 2½. Its chief part consists of eleven parallel lines, the 3rd, 6th, and
9th being marked with a cross. Another section consists of five parallel
lines, and there are three rows of arithmetical symbols. This board could
only have been used with counters (calculi), preferably unmarked, as in our
treatise of Accomptynge by Counters.

Classical Roman Methods of Calculation.
We have proof of two methods of calculation in ancient Rome, one by the
first method, in which the surface of sand was divided into columns by a
stylus or the hand. Counters (calculi, or lapilli), which were kept in boxes
(loculi), were used in calculation, as we learn from Horace’s schoolboys
(Sat. 1. vi. 74). For the sand see Persius I. 131, “Nec qui abaco numeros et
secto in pulvere metas scit risisse,” Apul. Apolog. 16 (pulvisculo), Mart.
Capella, lib. vii. 3, 4, etc. Cicero says of an expert calculator “eruditum
attigisse pulverem,” (de nat. Deorum, ii. 18). Tertullian calls a teacher of
arithmetic “primus numerorum arenarius” (de Pallio, in fine). The counters
were made of various materials, ivory principally, “Adeo nulla uncia nobis
est eboris, etc.” (Juv. XI. 131), sometimes of precious metals, “Pro calculis
albis et nigris aureos argenteosque habebat denarios” (Pet. Arb.
Satyricon, 33).

There are, however, still in existence four Roman counting boards of a kind
which does not appear to come into literature. A typical one is of the third
class. It consists of a number of transverse wires, broken at the middle. On
the left hand portion four beads are strung, on the right one (or two). The
left hand beads signify units, the right hand one five units. Thus any number
up to nine can be represented. This instrument is in all essentials the same
as the Swanpan or Abacus in use throughout the Far East. The Russian
stchota in use throughout Eastern Europe is simpler still. The method of
using this system is exactly the same as that of Accomptynge by Counters,
the right-hand five bead replacing the counter between the lines.

The Boethian Abacus.
Between classical times and the tenth century we have little or no guidance
as to the art of calculation. Boethius (fifth century), at the end of lib. II. of
his Geometria gives us a figure of an abacus of the second class with a set
of counters arranged within it. It has, however, been contended with great
probability that the whole passage is a tenth century interpolation. As no
rules are given for its use, the chief value of the figure is that it gives the
signs of the nine numbers, known as the Boethian “apices” or “notae” (from
whence our word “notation”). To these we shall return later on.

The Abacists.
It would seem probable that writers on the calendar like Bede (a.d. 721) and
Helpericus (a.d. 903) were able to perform simple calculations; though we
are unable to guess their methods, and for the most part they were
dependent on tables taken from Greek sources. We have no early medieval
treatises on arithmetic, till towards the end of the tenth century we find a
revival of the study of science, centring for us round the name of Gerbert,
who became Pope as 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 (1563) edition of his works, I. 159, in
a somewhat enlarged form. Another tenth century treatise is that of Abbo of
Fleury (c. 988), preserved in several manuscripts. Very few treatises on the

use of the Abacus can be certainly ascribed to the eleventh century, but from
the beginning of the twelfth century their numbers increase rapidly, to judge
by those that have been preserved.
The Abacists used a permanent board usually divided into twelve columns;
the columns were grouped in threes, each column being called an “arcus,”
and the value of a figure in it represented a tenth of what it would have in
the column to the left, as in our arithmetic of position. With this board
counters or jetons were used, either plain or, more probably, marked with
numerical signs, which with the early Abacists were the “apices,” though
counters from classical times were sometimes marked on one side with the
digital signs, on the other with Roman numerals. Two ivory discs of this
kind from the Hamilton collection may be seen at the British Museum.
Gerbert is said by Richer to have made for the purpose of computation a
thousand counters of horn; the usual number of a set of counters in the
sixteenth and seventeenth centuries was a hundred.
Treatises on the Abacus usually consist of chapters on Numeration
explaining the notation, and on the rules for Multiplication and Division.
Addition, as far as it required any rules, came naturally under
Multiplication, while Subtraction was involved in the process of Division.
These rules were all that were needed in Western Europe in centuries when
commerce hardly existed, and astronomy was unpractised, and even they
were only required in the preparation of the calendar and the assignments of
the royal exchequer. In England, for example, when the hide developed
from the normal holding of a household into the unit of taxation, the
calculation of the geldage in each shire required a sum in division; as we
know from the fact that one of the Abacists proposes the sum: “If 200
marks are levied on the county of Essex, which contains according to Hugh
of Bocland 2500 hides, how much does each hide pay?” 3 Exchequer
methods up to the sixteenth century were founded on the abacus, though
when we have details later on, a different and simpler form was used.
The great difficulty of the early Abacists, owing to the absence of a figure
representing zero, was to place their results and operations in the proper
columns of the abacus, especially when doing a division sum. The chief
differences noticeable in their works are in the methods for this rule.

Division was either done directly or by means of differences between the
divisor and the next higher multiple of ten to the divisor. Later Abacists
made a distinction between “iron” and “golden” methods of division. The
following are examples taken from a twelfth century treatise. In following
the operations it must be remembered that a figure asterisked represents a
counter taken from the board. A zero is obviously not needed, and the result
may be written down in words.

(a) Multiplication. 4600 × 23.
Thousands
H H
u u
U U
n T n T
n n
d e d e
i i
r n r n
t t
e s e s
s s
d d
s s
4 6 Multiplicand.
1 8 600 × 3.
1 2 4000 × 3.
1 2 600 × 20.
8 4000 × 20.
1 5 8 Total product.
2 3 Multiplier.

(b) Division: direct. 100,000 ÷ 20,023. Here each counter in turn is
a separate divisor.
H. T. U. H. T. U.
2 2 3 Divisors.
2 Place greatest divisor to right of dividend.
1 Dividend.
2 Remainder.
1
1 9 9 Another form of same.

8 Product of 1st Quotient and 20.
1 9 9 2 Remainder.
1 2 Product of 1st Quotient and 3.
1 9 9 8 Final remainder.
4 Quotient.

(c) Division by Differences. 900 ÷ 8. Here we divide by (10-2).
H. T. U.
2 Difference.
8 Divisor.
49 Dividend.
41 8 Product of difference by 1st Quotient (9).
2 Product of difference by 2nd Quotient (1).
41 Sum of 8 and 2.
2 Product of difference by 3rd Quotient (1).
4 Product of difference by 4th Quot. (2). Remainder.
2 4th Quotient.
1 3rd Quotient.
1 2nd Quotient.
9 1st Quotient.
1 1 2 Quotient. (Total of all four.)

Division. 7800 ÷ 166.
Thousands
H. T. U. H. T. U.
3 4 Differences (making 200 trial divisor).
1 6 6 Divisors.
47 8 Dividends.
1 Remainder of greatest dividend.
1 2 Product of 1st difference (4) by 1st Quotient (3).
9 Product of 2nd difference (3) by 1st Quotient (3).
42 8 2 New dividends.

3 4 Product of 1st and 2nd difference by 2nd Quotient (1).
41 1 6 New dividends.
2 Product of 1st difference by 3rd Quotient (5).
1 5 Product of 2nd difference by 3rd Quotient (5).
43 3 New dividends.
1 Remainder of greatest dividend.
3 4 Product of 1st and 2nd difference by 4th Quotient (1).
1 6 4 Remainder (less than divisor).
1 4th Quotient.
5 3rd Quotient.
1 2nd Quotient.
3 1st Quotient.
4 6 Quotient.

Division. 8000 ÷ 606.
Thousands
H. T. U. H. T. U.
9 Difference (making 700 trial divisor).
4 Difference.
6 6 Divisors.
48 Dividend.
1 Remainder of dividend.
9 4 Product of difference 1 and 2 with 1st Quotient (1).
41 9 4 New dividends.
3 Remainder of greatest dividend.
9 4 Product of difference 1 and 2 with 2nd Quotient (1).
41 3 3 4 New dividends.
3 Remainder of greatest dividend.
9 4 Product of difference 1 and 2 with 3rd Quotient (1).
7 2 8 New dividends.
6 6 Product of divisors by 4th Quotient (1).
1 2 2 Remainder.
1 4th Quotient.

1 3rd Quotient.
1 2nd Quotient.
1 1st Quotient.
1 3 Quotient.

The chief Abacists are Gerbert (tenth century), Abbo, and Hermannus
Contractus (1054), who are credited with the revival of the art, Bernelinus,
Gerland, and Radulphus of Laon (twelfth century). We know as English
Abacists, Robert, bishop of Hereford, 1095, “abacum et lunarem compotum
et celestium cursum astrorum rimatus,” Turchillus Compotista (Thurkil),
and through him of Guilielmus R. . . . “the best of living computers,”
Gislebert, and Simonus de Rotellis (Simon of the Rolls). They flourished
most probably in the first quarter of the twelfth century, as Thurkil’s treatise
deals also with fractions. Walcher of Durham, Thomas of York, and Samson
of Worcester are also known as Abacists.
Finally, the term Abacists came to be applied to computers by manual
arithmetic. A MS. Algorithm of the thirteenth century (Sl. 3281, f. 6, b),
contains the following passage: “Est et alius modus secundum operatores
sive practicos, quorum unus appellatur Abacus; et modus ejus est in
computando per digitos et junctura manuum, et iste utitur ultra Alpes.”
In a composite treatise containing tracts written a.d. 1157 and 1208, on the
calendar, the abacus, the manual calendar and the manual abacus, we have a
number of the methods preserved. As an example we give the rule for
multiplication (Claud. A. IV., f. 54 vo). “Si numerus multiplicat alium
numerum auferatur differentia majoris a minore, et per residuum
multiplicetur articulus, et una differentia per aliam, et summa proveniet.”
Example, 8 × 7. The difference of 8 is 2, of 7 is 3, the next article being 10;
7 - 2 is 5. 5 × 10 = 50; 2 × 3 = 6. 50 + 6 = 56 answer. The rule will hold in
such cases as 17 × 15 where the article next higher is the same for both, i.e.,
20; but in such a case as 17 × 9 the difference for each number must be
taken from the higher article, i.e., the difference of 9 will be 11.

The Algorists.

Algorism (augrim, augrym, algram, agram, algorithm), owes its name to the
accident that the first arithmetical treatise translated from the Arabic
happened to be one written by Al-Khowarazmi in the early ninth century,
“de numeris Indorum,” beginning in its Latin form “Dixit Algorismi. . . .”
The translation, of which only one MS. is known, was made about 1120 by
Adelard of Bath, who also wrote on the Abacus and translated with a
commentary Euclid from the Arabic. It is probable that another version was
made by Gerard of Cremona (1114-1187); the number of important works
that were not translated more than once from the Arabic decreases every
year with our knowledge of medieval texts. A few lines of this translation,
as copied by Halliwell, are given on p. 72, note 2. Another translation still
seems to have been made by Johannes Hispalensis.
Algorism is distinguished from Abacist computation by recognising seven
rules, Addition, Subtraction, Duplation, Mediation, Multiplication,
Division, and Extraction of Roots, to which were afterwards added
Numeration and Progression. It is further distinguished by the use of the
zero, which enabled the computer to dispense with the columns of the
Abacus. It obviously employs a board with fine sand or wax, and later, as a
substitute, paper or parchment; slate and pencil were also used in the
fourteenth century, how much earlier is unknown. 5 Algorism quickly ousted
the Abacus methods for all intricate calculations, being simpler and more
easily checked: in fact, the astronomical revival of the twelfth and thirteenth
centuries would have been impossible without its aid.
The number of Latin Algorisms still in manuscript is comparatively large,
but we are here only concerned with two—an Algorism in prose attributed
to Sacrobosco (John of Holywood) in the colophon of a Paris manuscript,
though this attribution is no longer regarded as conclusive, and another in
verse, most probably by Alexander de Villedieu (Villa Dei). Alexander, who
died in 1240, was teaching in Paris in 1209. His verse treatise on the
Calendar is dated 1200, and it is to that period that his Algorism may be
attributed; Sacrobosco died in 1256 and quotes the verse Algorism. Several
commentaries on Alexander’s verse treatise were composed, from one of
which our first tractate was translated, and the text itself was from time to
time enlarged, sections on proofs and on mental arithmetic being added. We
have no indication of the source on which Alexander drew; it was most

likely one of the translations of Al-Khowarasmi, but he has also the
Abacists in mind, as shewn by preserving the use of differences in
multiplication. His treatise, first printed by Halliwell-Phillipps in his Rara
Mathematica, is adapted for use on a board covered with sand, a method
almost universal in the thirteenth century, as some passages in the algorism
of that period already quoted show: “Est et alius modus qui utitur apud
Indos, et doctor hujusmodi ipsos erat quidem nomine Algus. Et modus suus
erat in computando per quasdam figuras scribendo in pulvere. . . .” “Si
voluerimus depingere in pulvere predictos digitos secundum consuetudinem
algorismi . . .” “et sciendum est quod in nullo loco minutorum sive
secundorum . . . in pulvere debent scribi plusquam sexaginta.”

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

Digital Arithmetic.
The method of indicating numbers by means of the fingers is of
considerable age. The British Museum possesses two ivory counters marked
on one side by carelessly scratched Roman numerals IIIV and VIIII, and on
the other by carefully engraved digital signs for 8 and 9. Sixteen seems to
have been the number of a complete set. These counters were either used in
games or for the counting board, and the Museum ones, coming from the
Hamilton collection, are undoubtedly not later than the first century.
Frohner has published in the Zeitschrift des Münchener Alterthumsvereins a
set, almost complete, of them with a Byzantine treatise; a Latin treatise is
printed among Bede’s works. The use of this method is universal through

the East, and a variety of it is found among many of the native races in
Africa. In medieval Europe it was almost restricted to Italy and the
Mediterranean basin, and in the treatise already quoted (Sloane 3281) it is
even called the Abacus, perhaps a memory of Fibonacci’s work.
Methods of calculation by means of these signs undoubtedly have existed,
but they were too involved and liable to error to be much used.

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

The Counting Board.

The treatise on pp. 52-65 is the only one in English known on the subject. It
describes a method of calculation which, with slight modifications, is
current in Russia, China, and Japan, to-day, though it went out of use in
Western Europe by the seventeenth century. In Germany the method is
called “Algorithmus Linealis,” and there are several editions of a tract
under this name (with a diagram of the counting board), printed at Leipsic
at the end of the fifteenth century and the beginning of the sixteenth. They
give the nine rules, but “Capitulum de radicum extractione ad algoritmum
integrorum reservato, cujus species per ciffrales figuras ostenduntur ubi ad
plenum de hac tractabitur.” The invention of the art is there attributed to
Appulegius the philosopher.
The advantage of the counting board, whether permanent or constructed by
chalking parallel lines on a table, as shown in some sixteenth-century
woodcuts, is that only five counters are needed to indicate the number nine,
counters on the lines representing units, and those in the spaces above
representing five times those on the line below. The Russian abacus, the
“tchatui” or “stchota” has ten beads on the line; the Chinese and Japanese
“Swanpan” economises by dividing the line into two parts, the beads on one
side representing five times the value of those on the other. The “Swanpan”
has usually many more lines than the “stchota,” allowing for more extended
calculations, see Tylor, Anthropology (1892), p. 314.
Record’s treatise also mentions another method of counter notation (p. 64)
“merchants’ casting” and “auditors’ casting.” These were adapted for the
usual English method of reckoning numbers up to 200 by scores. This
method seems to have been used in the Exchequer. A counting board for
merchants’ use is printed by Halliwell in Rara Mathematica (p. 72) from
Sloane MS. 213, and two others are figured in Egerton 2622 f. 82 and f. 83.
The latter is said to be “novus modus computandi secundum inventionem
Magistri Thome Thorleby,” and is in principle, the same as the “Swanpan.”
The Exchequer table is described in the Dialogus de Scaccario (Oxford,
1902), p. 38.

1. Halliwell printed the two sides of his leaf in the wrong order. This
and some obvious errors of transcription—‘ferye’ for ‘ferthe,’ ‘lest’ for

‘left,’ etc., have not been corrected in the reprint on pp. 70-71.
2. For Egyptian use see Herodotus, ii. 36, Plato, de Legibus, VII.
3. See on this Dr. Poole, The Exchequer in the Twelfth Century, Chap.
III., and Haskins, Eng. Hist. Review, 27, 101. The hidage of Essex in
1130 was 2364 hides.
4. These figures are removed at the next step.
5. Slates are mentioned by Chaucer, and soon after (1410) Prosdocimo
de Beldamandi speaks of the use of a “lapis” for making notes on by
calculators.

Egerton 2622.

leaf 136 a.

HEc algorismus ars presens dicitur; in qua
Talibus indorum fruimur bis quinque figuris.
This boke is called þe boke of algorym, or Augrym after lewder A derivati
Algorism.
vse. And þis boke tretys þe Craft of Nombryng, þe quych crafte
is called also Algorym. Ther was a kyng of Inde, þe quich heyth
Algor, & he made þis craft. And after his name he called hit
algorym; or els anoþer cause is quy it is called Algorym, for þe Another
derivation
latyn word of hit s. Algorismus comes of Algos, grece, quid est word.
ars, latine, craft oɳ englis, and rides, quid est numerus, latine,
A nombur oɳ englys, inde dicitur Algorismus per addicionem
huius sillabe mus & subtraccionem d & e, quasi ars numerandi.
¶ fforthermore ȝe most vndirstonde þat in þis craft ben vsid teen
figurys, as here bene writen for ensampul, φ 9 8 7 6 5 4 3 2 1.
¶ Expone þe too versus afore: this present craft ys called
Algorismus, in þe quych we vse teen signys of Inde. Questio.
¶ Why teɳ fyguris of Inde? Solucio. for as I haue sayd afore þai
were fonde fyrst in Inde of a kynge of þat Cuntre, þat was called
Algor.

on and Numeration.
versus
[in margin
¶ Prima significat unum; duo vero secunda:
¶ Tercia significat tria; sic procede sinistre.
¶ Donec ad extremam venias, que cifra vocatur.

¶ Capitulum primum de significacione figurarum.

In þis verse is notifide þe significacion of þese figuris. And þus Expositi
versus.
expone the verse. Þe first signifiyth one, þe secunde leaf 136 b.
signi*fiyth tweyne, þe thryd signifiyth thre, & the fourte The mean
place of th
signifiyth 4. ¶ And so forthe towarde þe lyft syde of þe tabul or figures.
of þe boke þat þe figures bene writene in, til þat þou come to the
last figure, þat is called a cifre. ¶ Questio. In quych syde sittes þe
first figure? Solucio, forsothe loke quich figure is first in þe ryȝt
side of þe bok or of þe tabul, & þat same is þe first figure, for
þou schal write bakeward, as here, 3. 2. 6. 4. 1. 2. 5. The figure Which fig
read first.
of 5. was first write, & he is þe first, for he sittes oɳ þe riȝt syde.
And the figure of 3 is last. ¶ Neuer-þe-les wen he says ¶ Prima
significat vnum &c., þat is to say, þe first betokenes one, þe
secunde. 2. & fore-þer-more, he vndirstondes noȝt of þe first
figure of euery rew. ¶ But he vndirstondes þe first figure þat is
in þe nombur of þe forsayd teen figuris, þe quych is one of þese.
1. And þe secunde 2. & so forth.
versus
[in margin
¶ Quelibet illarum si primo limite ponas,
¶ Simpliciter se significat: si vero secundo,
Se decies: sursum procedas multiplicando.
¶ Namque figura sequens quamuis signat decies plus.
¶ Ipsa locata loco quam significat pertinente.
¶ Expone þis verse þus. Euery of þese figuris bitokens hym selfe Expositio
margin].
& no more, yf he stonde in þe first place of þe rewele / this
worde Simpliciter in þat verse it is no more to say but þat, & no
more. ¶ If it stonde in the secunde place of þe rewle, he betokens An explan
of the prin
tene tymes hym selfe, as þis figure 2 here 20 tokens ten tyme of notatio
hym selfe, leaf 137 a. *þat is twenty, for he hym selfe betokenes
tweyne, & ten tymes twene is twenty. And for he stondis oɳ þe
lyft side & in þe secunde place, he betokens ten tyme hym selfe.
And so go forth. ¶ ffor euery figure, & he stonde aftur a-noþer
toward the lyft side, he schal betokene ten tymes as mich more
as he schul betoken & he stode in þe place þere þat þe figure a-
fore hym stondes. loo an ensampulle. 9. 6. 3. 4. Þe figure of 4. An examp

þat hase þis schape . betokens bot hymselfe, for he stondes

units,

in þe first place. The figure of 3. þat hase þis schape .
betokens ten tymes more þen he schuld & he stde þere þat þe
figure of 4. stondes, þat is thretty. The figure of 6, þat hase þis tens,

schape , betokens ten tymes more þan he schuld & he stode

þere as þe figure of . stondes, for þere he schuld tokyne bot
sexty, & now he betokens ten tymes more, þat is sex hundryth. hundreds,

The figure of 9. þat hase þis schape . betokens ten tymes
more þane he schuld & he stode in þe place þere þe figure of sex
stondes, for þen he schuld betoken to 9. hundryth, and in þe
place þere he stondes now he betokens 9. þousande. Al þe hole thousands
nombur is 9 thousande sex hundryth & foure & thretty.
¶ fforthermore, when þou schalt rede a nombur of figure, þou How to re
number.
schalt begyne at þe last figure in the lyft side, & rede so forth to
þe riȝt side as here 9. 6. 3. 4. Thou schal begyn to rede at þe
figure of 9. & rede forth þus. 9. leaf 137 b. *thousand sex hundryth
thritty & foure. But when þou schalle write, þou schalt be-gynne
to write at þe ryȝt side.
¶ Nil cifra significat sed dat signare sequenti.
Expone þis verse. A cifre tokens noȝt, bot he makes þe figure to The mean
use of the
betoken þat comes aftur hym more þan he schuld & he were
away, as þus 1φ. here þe figure of one tokens ten, & yf þe cifre
were away 1 & no figure by-fore hym he schuld token bot one,
for þan he schuld stonde in þe first place. ¶ And þe cifre tokens
nothyng hym selfe. for al þe nombur of þe ylke too figures is bot
ten. ¶ Questio. Why says he þat a cifre makys a figure to signifye
(tyf) more &c. ¶ I speke for þis worde significatyf, ffor sothe it
may happe aftur a cifre schuld come a-noþur cifre, as þus 2φφ.
And ȝet þe secunde cifre shuld token neuer þe more excep he
schuld kepe þe order of þe place. and a cifre is no figure
significatyf.
¶ Quam precedentes plus ultima significabit /

Expone þis verse þus. Þe last figure schal token more þan alle þe The last fi
means mo
oþer afore, thouȝt þere were a hundryth thousant figures afore, as all the oth
þus, 16798. Þe last figure þat is 1. betokens ten thousant. And since it is
alle þe oþer figures ben bot betokene bot sex thousant seuyne highest va
hundryth nynty & 8. ¶ And ten thousant is more þen alle þat
nombur, ergo þe last figure tokens more þan all þe nombur
afore.

hree Kinds of Numbers
leaf 138 a.
* ¶ Post predicta scias breuiter quod tres numerorum
Distincte species sunt; nam quidam digiti sunt;
Articuli quidam; quidam quoque compositi sunt.

m
¶ Capitulum 2 de triplice divisione numerorum.
¶ The auctor of þis tretis departys þis worde a nombur into 3
partes. Some nombur is called digitus latine, a digit in englys. Digits.
Somme nombur is called articulus latine. An Articul in englys. Articles.
Some nombur is called a composyt in englys. ¶ Expone þis Composit
verse. know þou aftur þe forsayd rewles þat I sayd afore, þat
þere ben thre spices of nombur. Oone is a digit, Anoþer is an
Articul, & þe toþer a Composyt. versus.

, Articles, and Composites.
¶ Sunt digiti numeri qui citra denarium sunt.
¶ Here he telles qwat is a digit, Expone versus sic. Nomburs What are
digitus bene alle nomburs þat ben with-inne ten, as nyne, 8. 7. 6.
5. 4. 3. 2. 1.
¶ Articupli decupli degitorum; compositi sunt
Illi qui constant ex articulis degitisque.
¶ Here he telles what is a composyt and what is ane articul.
Expone sic versus. ¶ Articulis ben 2 alle þat may be deuidyt into What are
articles.
nomburs of ten & nothynge leue ouer, as twenty, thretty, fourty,
a hundryth, a thousand, & such oþer, ffor twenty may be

departyt in-to 2 nomburs of ten, fforty in to foure nomburs of
ten, & so forth.
leaf 138 b. *Compositys beɳ nomburs þat bene componyt of a What num
are compo
digyt & of an articulle as fouretene, fyftene, sextene, & such
oþer. ffortene is componyd of foure þat is a digit & of ten þat is
an articulle. ffiftene is componyd of 5 & ten, & so of all oþer,
what þat þai ben. Short-lych euery nombur þat be-gynnes with a
digit & endyth in a articulle is a composyt, as fortene
bygennynge by foure þat is a digit, & endes in ten.
¶ Ergo, proposito numero tibi scribere, primo
Respicias quid sit numerus; si digitus sit
Primo scribe loco digitum, si compositus sit
Primo scribe loco digitum post articulum; sic.
¶ here he telles how þou schalt wyrch whan þou schalt write a How to w
number,
nombur. Expone versum sic, & fac iuxta exponentis sentenciam;
whan þou hast a nombur to write, loke fyrst what maner nombur
it ys þat þou schalt write, whether it be a digit or a composit or
an Articul. ¶ If he be a digit, write a digit, as yf it be seuen, write if it is a di
seuen & write þat digit in þe first place toward þe ryght side. If it if it is a
composite
be a composyt, write þe digit of þe composit in þe first place &
write þe articul of þat digit in þe secunde place next toward þe
lyft side. As yf þou schal write sex & twenty. write þe digit of þe
nombur in þe first place þat is sex, and write þe articul next aftur
þat is twenty, as þus 26. But whan þou schalt sowne or speke leaf How to re
139 a. *or rede an Composyt þou schalt first sowne þe articul &
aftur þe digit, as þou seyst by þe comyne speche, Sex & twenty
& nouȝt twenty & sex. versus.
¶ Articulus si sit, in primo limite cifram,
Articulum vero reliquis inscribe figuris.
¶ Here he tells how þou schal write when þe nombre þat þou How to w
Articles:
hase to write is an Articul. Expone versus sic & fac secundum
sentenciam. Ife þe nombur þat þou hast write be an Articul, write
first a cifre & aftur þe cifer write an Articulle þus. 2φ. tens,
fforthermore þou schalt vndirstonde yf þou haue an Articul, loke
how mych he is, yf he be with-ynne an hundryth, þou schalt
write bot one cifre, afore, as here .9φ. If þe articulle be by hym- hundreds,

silfe & be an hundrid euene, þen schal þou write .1. & 2 cifers
afore, þat he may stonde in þe thryd place, for euery figure in þe
thryd place schal token a hundrid tymes hym selfe. If þe articul thousands
be a thousant or thousandes 3 and he stonde by hym selfe, write
afore 3 cifers & so forþ of al oþer.
¶ Quolibet in numero, si par sit prima figura,
Par erit & totum, quicquid sibi continuatur;
Impar si fuerit, totum tunc fiet et impar.
¶ Here he teches a generalle rewle þat yf þe first figure in þe To tell an
number
rewle of figures token a nombur þat is euene al þat nombur of
figurys in þat rewle schal be euene, as here þou may see 6. 7. 3.
5. 4. Computa & proba. ¶ If þe first leaf 139 b. *figure token an or an odd.
nombur þat is ode, alle þat nombur in þat rewle schalle be ode,
as here 5 6 7 8 6 7. Computa & proba. versus.
¶ Septem sunt partes, non plures, istius artis;
¶ Addere, subtrahere, duplare, dimidiare,
Sextaque diuidere, sed quinta multiplicare;
Radicem extrahere pars septima dicitur esse.

even Rules of Arithmetic.
¶ Here telles þat þer beɳ .7. spices or partes of þis craft. The first The seven
is called addicioñ, þe secunde is called subtraccioñ. The thryd is
called duplacioñ. The 4. is called dimydicioñ. The 5. is called
multiplicacioñ. The 6 is called diuisioñ. The 7. is called
extraccioñ of þe Rote. What all þese spices bene hit schalle be
tolde singillatim in here caputule.
¶ Subtrahis aut addis a dextris vel mediabis:
Thou schal be-gynne in þe ryght side of þe boke or of a tabul. Add, subt
halve, from
loke were þou wul be-gynne to write latyn or englys in a boke, & to left.
þat schalle be called þe lyft side of the boke, þat þou writest
toward þat side schal be called þe ryght side of þe boke. Versus.
A leua dupla, diuide, multiplica.
Here he telles þe in quych side of þe boke or of þe tabul þou
schalle be-gyne to wyrch duplacioñ, diuisioñ, and
multiplicacioñ. Thou schal begyne to worch in þe lyft side of þe

boke or of þe tabul, but yn what wyse þou schal wyrch in hym Multiply o
divide fro
dicetur singillatim in sequentibus capitulis et de vtilitate to right.
cuiuslibet artis & sic Completur leaf 140. *prohemium &
sequitur tractatus & primo de arte addicionis que prima ars
est in ordine.

raft of Addition.

Addere si numero numerum vis, ordine tali
Incipe; scribe duas primo series numerorum
Primam sub prima recte ponendo figuram,
Et sic de reliquis facias, si sint tibi plures.
¶ Here by-gynnes þe craft of Addicioñ. In þis craft þou most Four thing
be known
knowe foure thynges. ¶ Fyrst þou most know what is addicioñ.
Next þou most know how mony rewles of figurys þou most
haue. ¶ Next þou most know how mony diuers casys happes in
þis craft of addicioñ. ¶ And next qwat is þe profet of þis craft. what it is;
¶ As for þe first þou most know þat addicioñ is a castyng to-
gedur of twoo nomburys in-to one nombre. As yf I aske qwat is
twene & thre. Þou wyl cast þese twene nombres to-gedur & say
þat it is fyue. ¶ As for þe secunde þou most know þat þou schalle how many
of figures
haue tweyne rewes of figures, one vndur a-nother, as here þou
mayst se. ¶ As for þe thryd þou most know þat there ben 1234
foure diuerse cases. As for þe forthe þou most know þat þe 2168.
how many
profet of þis craft is to telle what is þe hole nombur þat comes of what is its
diuerse nomburis. Now as to þe texte of oure verse, he teches
there how þou schal worch in þis craft. ¶ He says yf þou wilt cast
one nombur to anoþer nombur, þou most by-gynne on þis wyse. How to se
the sum.
¶ ffyrst write leaf 140 b. *two rewes of figuris & nombris so þat
þou write þe first figure of þe hyer nombur euene vndir the first
figure of þe nether nombur, And þe secunde of þe nether 123
nombur euene vndir þe secunde of þe hyer, & so forthe of 234.

euery figure of both þe rewes as þou mayst se.

ases of the Craft of Addition.
¶ Inde duas adde primas hac condicione:
Si digitus crescat ex addicione priorum;

Primo scribe loco digitum, quicunque sit ille.
¶ Here he teches what þou schalt do when þou hast write too
rewes of figuris on vnder an-oþer, as I sayd be-fore. ¶ He says Add the fi
figures;
þou schalt take þe first figure of þe heyer nombre & þe fyrst
figure of þe neþer nombre, & cast hem to-geder vp-on þis
condicioɳ. Thou schal loke qweþer þe nomber þat comys þere-of
be a digit or no. ¶ If he be a digit þou schalt do away þe first rub out th
figure;
figure of þe hyer nombre, and write þere in his stede þat he stode
Inne þe digit, þat comes of þe ylke 2 figures, & so wrich forth oɳ write the r
in its plac
oþer figures yf þere be ony moo, til þou come to þe ende toward
þe lyft side. And lede þe nether figure stonde still euer-more til
þou haue ydo. ffor þere-by þou schal wyte wheþer þou hast done
wel or no, as I schal tell þe afterward in þe ende of þis Chapter.
¶ And loke allgate leaf 141 a. þat þou be-gynne to worch in þis
Craft of Addi*cioɳ in þe ryȝt side, here is an ensampul of þis Here is an
example.
case. Caste 2 to foure & þat wel be sex, do away 4. & write 1234
in þe same place þe figure of sex. ¶ And lete þe figure of 2 2142.
in þe nether rewe stonde stil. When þou hast do so, cast 3 & 4 to-
gedur and þat wel be seuen þat is a digit. Do away þe 3, & set
þere seueɳ, and lete þe neþer figure stonde stille, & so worch
forth bakward til þou hast ydo all to-geder.
Et si compositus, in limite scribe sequente
Articulum, primo digitum; quia sic iubet ordo.
¶ Here is þe secunde case þat may happe in þis craft. And þe
case is þis, yf of þe casting of 2 nomburis to-geder, as of þe Suppose i
Composit
figure of þe hyer rewe & of þe figure of þe neþer rewe come a down the
Composyt, how schalt þou worch. Þus þou schalt worch. Thou and carry
shalt do away þe figure of þe hyer nomber þat was cast to þe tens.
figure of þe neþer nomber. ¶ And write þere þe digit of þe
Composyt. And set þe articul of þe composit next after þe digit
in þe same rewe, yf þere be no mo figures after. But yf þere be
mo figuris after þat digit. And þere he schall be rekend for hym
selfe. And when þou schalt adde þat ylke figure þat berys þe
articulle ouer his hed to þe figure vnder hym, þou schalt cast þat
articul to þe figure þat hase hym ouer his hed, & þere þat Articul
schal tokeɳ hym selfe. lo an Ensampull leaf 141 b. *of all. Cast 326 Here is an
example.
6 to 6, & þere-of wil arise twelue. do away þe hyer 6 & write 216.

þere 2, þat is þe digit of þis composit. And þen write þe articulle
þat is ten ouer þe figuris hed of twene as þus. Now cast þe 1
articulle þat standus vpon þe figuris of twene hed to þe same 322
216.
figure, & reken þat articul bot for one, and þan þere wil arise
thre. Þan cast þat thre to þe neþer figure, þat is one, & þat wul be
foure. do away þe figure of 3, and write þere a figure of foure.
and lete þe neþer figure stonde stil, & þan worch forth. vnde
versus.
¶ Articulus si sit, in primo limite cifram,
¶ Articulum vero reliquis inscribe figuris,
Vel per se scribas si nulla figura sequatur.
¶ Here he puttes þe thryde case of þe craft of Addicioɳ. & þe
case is þis. yf of Addiciouɳ of 2 figuris a-ryse an Articulle, how Suppose i
Article, se
schal þou do. thou most do away þe heer figure þat was addid to a cipher a
þe neþer, & write þere a cifre, and sett þe articuls on þe figuris carry the t
hede, yf þat þere come ony after. And wyrch þan as I haue tolde
þe in þe secunde case. An ensampull. Cast 5 to 5, þat wylle be 25.
ten. now do away þe hyer 5, & write þere a cifer. And sette 15

ten vpon þe figuris hed of 2. And reken it but for on þus. lo an Here is an
example.
Ensampulle And leaf 142 a. *þan worch forth. But yf þere 1
come no figure after þe cifre, write þe articul next hym in þe 2φ
15
same rewe as here cast 5 to 5, and it wel be ten. do away 5
5. þat is þe hier 5. and write þere a cifre, & write after 5
hym þe articul as þus And þan þou hast done. 1φ
5
¶ Si tibi cifra superueniens occurrerit, illam
Dele superpositam; fac illic scribe figuram,
Postea procedas reliquas addendo figuras.
¶ Here he puttes þe fourt case, & it is þis, þat yf þere come a What to d
you have
cifer in þe hier rewe, how þou schal do. þus þou schalt do. do cipher in t
away þe cifer, & sett þere þe digit þat comes of þe addicioun as row.
þus In þis ensampul ben alle þe foure cases. Cast 3 to 1φφ84.
17743
foure, þat wol be seueɳ. do away 4. & write þere seueɳ;
An examp
þan cast 4 to þe figure of 8. þat wel be 12. do away 8, & sett þere all the
2. þat is a digit, and sette þe articul of þe composit, þat is ten, difficultie
vpon þe cifers hed, & reken it for hym selfe þat is on. þan cast
one to a cifer, & hit wulle be but on, for noȝt & on makes but

one. þan cast 7. þat stondes vnder þat on to hym, & þat wel be 8.
do away þe cifer & þat 1. & sette þere 8. þan go forthermore.
cast þe oþer 7 to þe cifer þat stondes ouer hym. þat wul be bot
seuen, for þe cifer betokens noȝt. do away þe cifer & sette þere
seueɳ, leaf 142 b. *& þen go forþermore & cast 1 to 1, & þat wel
be 2. do away þe hier 1, & sette þere 2. þan hast þou do. And yf
þou haue wel ydo þis nomber þat is sett here-after wel be þe
nomber þat schalle aryse of alle þe addicioɳ as here 27827.
¶ Sequitur alia species.

raft of Subtraction.
A numero numerum si sit tibi demere cura
Scribe figurarum series, vt in addicione.
¶ This is þe Chapter of subtraccioɳ, in the quych þou most know Four thing
know abo
foure nessessary thynges. the first what is subtraccioɳ. þe subtractio
secunde is how mony nombers þou most haue to subtraccioɳ, the
thryd is how mony maners of cases þere may happe in þis craft
of subtraccioɳ. The fourte is qwat is þe profet of þis craft. ¶ As
for þe first, þou most know þat subtraccioɳ is drawynge of one the first;
nowmber oute of anoþer nomber. As for þe secunde, þou most the second
knowe þat þou most haue two rewes of figuris one vnder anoþer,
as þou addyst in addicioɳ. As for þe thryd, þou moyst know þat the third;
foure maner of diuerse casis mai happe in þis craft. ¶ As for þe the fourth
fourt, þou most know þat þe profet of þis craft is whenne þou
hasse taken þe lasse nomber out of þe more to telle what þere
leues ouer þat. & þou most be-gynne to wyrch in þis craft in þe
ryght side of þe boke, as þou diddyst in addicioɳ. Versus.
¶ Maiori numero numerum suppone minorem,
¶ Siue pari numero supponatur numerus par.
leaf 143 a. * ¶ Here he telles þat þe hier nomber most be more þen Put the gr
number ab
þe neþer, or els eueɳ as mych. but he may not be lasse. And þe the less.
case is þis, þou schalt drawe þe neþer nomber out of þe hyer, &
þou mayst not do þat yf þe hier nomber were lasse þan þat. ffor
þou mayst not draw sex out of 2. But þou mast draw 2 out of sex.
And þou maiste draw twene out of twene, for þou schal leue noȝt
of þe hier twene vnde versus.

ases of the Craft of Subtraction.
¶ Postea si possis a prima subtrahe primam
Scribens quod remanet.
Here is þe first case put of subtraccioɳ, & he says þou schalt The first c
subtractio
begynne in þe ryght side, & draw þe first figure of þe neþer rewe
out of þe first figure of þe hier rewe. qwether þe hier figure be
more þen þe neþer, or eueɳ as mych. And þat is notified in þe
vers when he says “Si possis.” Whan þou has þus ydo, do away
þe hiest figure & sett þere þat leues of þe subtraccioɳ, lo an Here is an
example.
Ensampulle draw 2 out of 4. þan leues 2. do away 4 & write 234
þere 2, & latte þe neþer figure stonde stille, & so go for-by 122
oþer figuris till þou come to þe ende, þan hast þou do.
¶ Cifram si nil remanebit.
¶ Here he puttes þe secunde case, & hit is þis. yf it happe þat Put a ciph
nothing re
qwen þou hast draw on neþer figure out of a hier, & þere leue
noȝt after þe subtraccioɳ, þus leaf 143 b. *þou schalt do. þou
schalle do away þe hier figure & write þere a cifer, as lo an Here is an
example.
Ensampull Take foure out of foure þan leus noȝt. þerefore do 24
away þe hier 4 & set þere a cifer, þan take 2 out of 2, þan 24
leues noȝt. do away þe hier 2, & set þere a cifer, and so
worch whare so euer þis happe.
Sed si non possis a prima demere primam
Precedens vnum de limite deme sequente,
Quod demptum pro denario reputabis ab illo
Subtrahe totalem numerum quem proposuisti
Quo facto scribe super quicquid remanebit.
Here he puttes þe thryd case, þe quych is þis. yf it happe þat þe Suppose y
cannot tak
neþer figure be more þen þe hier figure þat he schalle be draw lower figu
out of. how schalle þou do. þus þou schalle do. þou schalle borro from the t
.1. oute of þe next figure þat comes after in þe same rewe, for þis borrow te
case may neuer happ but yf þere come figures after. þan þou
schalt sett þat on ouer þe hier figures hed, of the quych þou
woldist y-draw oute þe neyþer figure yf þou haddyst y-myȝt.
Whane þou hase þus ydo þou schalle rekene þat .1. for ten. ¶. take the lo
number fr
And out of þat ten þou schal draw þe neyþermost figure, And ten;

alle þat leues þou schalle adde to þe figure on whos hed þat .1. add the an
to the top
stode. And þen þou schalle do away alle þat, & sett þere alle that number.
arisys of the addicioɳ of þe ylke 2 figuris. And yf yt leaf 144 a.
*happe þat þe figure of þe quych þou schalt borro on be hym self
but 1. If þou schalt þat one & sett it vppoɳ þe oþer figuris hed,
and sett in þat 1. place a cifer, yf þere come mony figures after. Example.
lo an Ensampul. take 4 out of 2. it wyl not be, þerfore 2122
borro one of þe next figure, þat is 2. and sett þat ouer þe 1134
hed of þe fyrst 2. & rekene it for ten. and þere þe secunde
stondes write 1. for þou tokest on out of hym. þan take þe neþer
figure, þat is 4, out of ten. And þen leues 6. cast to 6 þe figure of
þat 2 þat stode vnder þe hedde of 1. þat was borwed & rekened
for ten, and þat wylle be 8. do away þat 6 & þat 2, & sette þere
8, & lette þe neþer figure stonde stille. Whanne þou hast do þus,
go to þe next figure þat is now bot 1. but first yt was 2, & þere-
of was borred 1. þan take out of þat þe figure vnder hym, þat is How to ‘P
back’ the
3. hit wel not be. þer-fore borowe of the next figure, þe quych is borrowed
bot 1. Also take & sett hym ouer þe hede of þe figure þat þou
woldest haue y-draw oute of þe nether figure, þe quych was 3. &
þou myȝt not, & rekene þat borwed 1 for ten & sett in þe same
place, of þe quych place þou tokest hym of, a cifer, for he was
bot 1. Whanne þou hast þus ydo, take out of þat 1. þat is rekent
for ten, þe neþer figure of 3. And þere leues 7. leaf 144 b. *cast þe
ylke 7 to þe figure þat had þe ylke ten vpon his hed, þe quych
figure was 1, & þat wol be 8. þan do away þat 1 and þat 7, &
write þere 8. & þan wyrch forth in oþer figuris til þou come to
þe ende, & þan þou hast þe do. Versus.
¶ Facque nonenarios de cifris, cum remeabis
¶ Occurrant si forte cifre; dum dempseris vnum
¶ Postea procedas reliquas demendo figuras.
¶ Here he puttes þe fourte case, þe quych is þis, yf it happe þat A very ha
is put.
þe neþer figure, þe quych þou schalt draw out of þe hier figure
be more pan þe hier figur ouer hym, & þe next figure of two or
of thre or of foure, or how mony þere be by cifers, how wold þou
do. Þou wost wel þou most nede borow, & þou mayst not borow
of þe cifers, for þai haue noȝt þat þai may lene or spare. Ergo 4
how woldest þou do. Certayɳ þus most þou do, þou most borow

on of þe next figure significatyf in þat rewe, for þis case may not
happe, but yf þere come figures significatyf after the cifers.
Whan þou hast borowede þat 1 of the next figure significatyf,
sett þat on ouer þe hede of þat figure of þe quych þou wold haue
draw þe neþer figure out yf þou hadest myȝt, & reken it for ten
as þou diddest in þe oþer case here-a-fore. Whaɳ þou hast þus y-
do loke how mony cifers þere were bye-twene þat figure
significatyf, & þe figure of þe quych þou woldest haue y-draw
the leaf 145 a. *neþer figure, and of euery of þe ylke cifers make a
figure of 9. lo an Ensampulle after. Take 4 out of 2. it wel 40002 Here is an
10004 example.
not be. borow 1 out of be next figure significatyf, þe
quych is 4, & þen leues 3. do away þat figure of 4 & write
þere 3. & sett þat 1 vppon þe figure of 2 hede, & þan take 4 out
of ten, & þan þere leues 6. Cast 6 to the figure of 2, þat wol be 8.
do away þat 6 & write þere 8. Whan þou hast þus y-do make of
euery 0 betweyn 3 & 8 a figure of 9, & þan worch forth in
goddes name. & yf þou hast wel y-do þou 5 schalt haue þis 39998 Sic.
nomber 10004

o prove the Subtraction.
¶ Si subtraccio sit bene facta probare valebis
Quas subtraxisti primas addendo figuras.
¶ Here he teches þe Craft how þou schalt know, whan þou hast How to pr
subtractio
subtrayd, wheþer þou hast wel ydo or no. And þe Craft is þis,
ryght as þou subtrayd þe neþer figures fro þe hier figures, ryȝt so
adde þe same neþer figures to þe hier figures. And yf þou haue
well y-wroth a-fore þou schalt haue þe hier nombre þe same þou
haddest or þou be-gan to worch. as for þis I bade þou schulde
kepe þe neþer figures stylle. lo an leaf 145 b. *Ensampulle of alle Here is an
example.
þe 4 cases togedre. worche welle þis case And yf þou 40003468
worch welle whan þou hast alle subtrayd þe þat hier 20004664
nombre here, þis schalle be þe nombre here foloyng
whan þou hast subtrayd. And þou schalt know þus. 39998804
adde þe neþer rowe of þe same nombre to þe hier rewe 20004664
as þus, cast 4 to 4. þat wol be 8. do away þe 4 & write Our autho
makes a s
þere 8. by þe first case of addicioɳ. þan cast 6 to 0 þat wol be 6. (3 for 1).
do away þe 0, & write þere 6. þan cast 6 to 8, þat wel be 14. do

away 8 & write þere a figure of 4, þat is þe digit, and write a
figure of 1. þat schall be-token ten. þat is þe articul vpon þe hed
of 8 next after, þan reken þat 1. for 1. & cast it to 8. þat schal be
9. cast to þat 9 þe neþer figure vnder þat þe quych is 4, & þat
schalle be 13. do away þat 9 & sett þere 3, & sett a figure of 1.
þat schall be 10 vpon þe next figuris hede þe quych is 9. by þe
secunde case þat þou hadest in addicioɳ. þan cast 1 to 9. & þat
wol be 10. do away þe 9. & þat 1. And write þere a cifer. and
write þe articulle þat is 1. betokenynge 10. vpon þe hede of þe
next figure toward þe lyft side, þe quych leaf 146 a. *is 9, & so do
forth tyl þou come to þe last 9. take þe figure of þat 1. þe quych He works
proof thro
þou schalt fynde ouer þe hed of 9. & sett it ouer þe next figures
hede þat schal be 3. ¶ Also do away þe 9. & set þere a cifer, &
þen cast þat 1 þat stondes vpon þe hede of 3 to þe same 3, & þat
schalle make 4, þen caste to þe ylke 4 the figure in þe neyþer
rewe, þe quych is 2, and þat schalle be 6. And þen 60003468 and brings
result.
schal þou haue an Ensampulle aȝeyɳ, loke & se, & but 20004664
þou haue þis same þou hase myse-wroȝt.

raft of Duplation.

Sequitur de duplacione

Si vis duplare numerum, sic incipe primo
Scribe figurarum seriem quamcunque velis tu.
¶ This is the Chapture of duplacioɳ, in þe quych craft þou most Four thing
be known
haue & know 4 thinges. ¶ Þe first þat þou most know is what is Duplation
duplacioɳ. þe secunde is how mony rewes of figures þou most
haue to þis craft. ¶ þe thryde is how many cases may 6 happe in
þis craft. ¶ þe fourte is what is þe profet of þe craft. ¶ As for þe Here they
first. duplacioɳ is a doublynge of a nombre. ¶ As for þe secunde
þou most leaf 146 b. *haue on nombre or on rewe of figures, the
quych called numerus duplandus. As for þe thrid þou most know
þat 3 diuerse cases may hap in þis craft. As for þe fourte. qwat is
þe profet of þis craft, & þat is to know what a-risyȝt of a nombre
I-doublyde. ¶ fforþer-more, þou most know & take gode hede in Mind whe
begin.
quych side þou schalle be-gyn in þis craft, or ellis þou mayst

spyl alle þi laber þere aboute. certeyn þou schalt begyɳ in the
lyft side in þis Craft. thenke wel ouer þis verse. ¶ 7A leua dupla,
diuide, multiplica. 7
The sentens of þes verses afore, as þou may see if þou take hede. Remembe
rules.
As þe text of þis verse, þat is to say, ¶ Si vis duplare. þis is þe
sentence. ¶ If þou wel double a nombre þus þou most be-gynɳ.
Write a rewe of figures of what nombre þou welt. versus.
Postea procedas primam duplando figuram
Inde quod excrescit scribas vbi iusserit ordo
Iuxta precepta tibi que dantur in addicione.
¶ Here he telles how þou schalt worch in þis Craft. he says, fyrst, How to w
sum.
whan þou hast writen þe nombre þou schalt be-gyn at þe first
figure in the lyft side, & doubulle þat figure, & þe nombre þat
comes þere-of þou schalt write as þou diddyst in addicioɳ, as ¶ I
schal telle þe in þe case. versus.

ases of the Craft of Duplation.
leaf 147 a.

* ¶ Nam si sit digitus in primo limite scribas.
¶ Here is þe first case of þis craft, þe quych is þis. yf of If the answ
digit,
duplacioɳ of a figure arise a digit. what schal þou do. þus þou
schal do. do away þe figure þat was doublede, & sett þere þe write it in
place of th
diget þat comes of þe duplacioɳ, as þus. 23. double 2, & þat wel figure.
be 4. do away þe figure of 2 & sett þere a figure of 4, & so
worch forth tille þou come to þe ende. versus.
¶ Articulus si sit, in primo limite cifram,
¶ Articulum vero reliquis inscribe figuris;
¶ Vel per se scribas, si nulla figura sequatur.
¶ Here is þe secunde case, þe quych is þis yf þere come an If it is an a
articulle of þe duplacioɳ of a figure þou schalt do ryȝt as þou
diddyst in addicioɳ, þat is to wete þat þou schalt do away þe
figure þat is doublet & sett þere a cifer, & write þe articulle ouer put a ciph
the place,
þe next figuris hede, yf þere be any after-warde toward þe lyft ‘carry’ the
side as þus. 25. begyn at the lyft side, and doubulle 2. þat wel be
4. do away þat 2 & sett þere 4. þan doubul 5. þat wel be 10. do

away 5, & sett þere a 0, & sett 1 vpon þe next figuris hede þe
quych is 4. & þen draw downe 1 to 4 & þat wolle be 5, & þen do
away þat 4 & þat 1, & sett þere 5. for þat 1 schal be rekened in
þe drawynge togedre for 1. wen leaf 147 b. *þou hast ydon þou
schalt haue þis nombre 50. yf þere come no figure after þe If there is
figure to ‘
figure þat is addit, of þe quych addicioɳ comes an articulle, þou them to, w
schalt do away þe figure þat is dowblet & sett þere a 0. & write them dow
þe articul next by in þe same rewe toward þe lyft syde as þus,
523. double 5 þat woll be ten. do away þe figure 5 & set þere a
cifer, & sett þe articul next after in þe same rewe toward þe lyft
side, & þou schalt haue þis nombre 1023. þen go forth & double
þe oþer nombers þe quych is lyȝt y-nowȝt to do. versus.
¶ Compositus si sit, in limite scribe sequente
Articulum, primo digitum; quia sic iubet ordo:
Et sic de reliquis faciens, si sint tibi plures.
¶ Here he puttes þe Thryd case, þe quych is þis, yf of duplacioɳ If it is a
Composit
of a figure come a Composit. þou schalt do away þe figure þat is
doublet & set þere a digit of þe Composit, & sett þe articulle write dow
digit, and
ouer þe next figures hede, & after draw hym downe with þe the tens.
figure ouer whos hede he stondes, & make þere-of an nombre as
þou hast done afore, & yf þere come no figure after þat digit þat
þou hast y-write, þan set þe articulle next after hym in þe same
rewe as þus, 67: double 6 þat wel be 12, do away 6 & write þere
þe digit leaf 148 a. *of 12, þe quych is 2, and set þe articulle next Here is an
example.
after toward þe lyft side in þe same rewe, for þere comes no
figure after. þan dowble þat oþer figure, þe quych is 7, þat wel
be 14. the quych is a Composit. þen do away 7 þat þou doublet
& sett þe þe diget of hym, the quych is 4, sett þe articulle ouer þe
next figures hed, þe quych is 2, & þen draw to hym þat on, &
make on nombre þe quych schalle be 3. And þen yf þou haue wel
y-do þou schalle haue þis nombre of þe duplacioɳ, 134. versus.
¶ Si super extremam nota sit monadem dat eidem
Quod tibi contingat si primo dimidiabis.
¶ Here he says, yf ouer þe fyrst figure in þe ryȝt side be such a How to do
w the mark f
merke as is here made, , þou schalle fyrst doubulle þe figure, half.
the quych stondes vnder þat merke, & þen þou schalt doubul þat

merke þe quych stondes for haluendel on. for too haluedels
makes on, & so þat wol be on. cast þat on to þat duplacioɳ of þe
figure ouer whos hed stode þat merke, & write it in þe same
w
place þere þat þe figure þe quych was doublet stode, as þus 23 .
double 3, þat wol be 6; doubul þat halue on, & þat wol be on.
cast on to 6, þat wel be 7. do away 6 & þat 1, & sett þere 7. þan
hase þou do. as for þat figure, þan go leaf 148 b. *to þe oþer figure
& worch forth. & þou schall neuer haue such a merk but ouer þe This can o
stand over
hed of þe furst figure in þe ryght side. And ȝet it schal not happe first figure
but yf it were y-halued a-fore, þus þou schalt vnderstonde þe
w
verse. ¶ Si super extremam &c. Et nota, talis figura significans
medietatem, unitatis veniat, i.e. contingat uel fiat super
extremam, i.e. super primam figuram in extremo sic versus
dextram ars dat: i.e. reddit monadem. i.e. vnitatem eidem. i.e.
eidem note & declinatur hec monos, dis, di, dem, &c. ¶ Quod
ergo totum hoc dabis monadem note continget. i.e. eveniet tibi si
dimidiasti, i.e. accipisti uel subtulisti medietatem alicuius unius,
in cuius principio sint figura numerum denotans imparem primo
i.e. principiis.

raft of Mediation.

¶ Sequitur de mediacione.

Incipe sic, si vis aliquem numerum mediare:
Scribe figurarum seriem solam, velut ante.
¶ In þis Chapter is taȝt þe Craft of mediaciouɳ, in þe quych craft The four t
to be know
þou most know 4 thynges. ffurst what is mediacioɳ. the secunde mediation
how mony rewes of figures þou most haue in þe wyrchynge of
þis craft. þe thryde how mony diuerse cases may happ in þis
craft. 8 ¶ As for þe furst, þou schalt vndurstonde þat mediacioɳ is the first
a takyng out of halfe a nomber out of a holle nomber, leaf 149 a.
*as yf þou wolde take 3 out of 6. ¶ As for þe secunde, þou schalt the second
know þat þou most haue one rewe of figures, & no moo, as þou
hayst in þe craft of duplacioɳ. ¶ As for the thryd, þou most the third;
vnderstonde þat 5 cases may happe in þis craft. ¶ As for þe the fourth

fourte, þou schalle know þat the profet of þis craft is when þou
hast take away þe haluendel of a nombre to telle qwat þere
schalle leue. ¶ Incipe sic, &c. The sentence of þis verse is þis. yf
þou wold medye, þat is to say, take halfe out of þe holle, or halfe
out of halfe, þou most begynne þus. Write one rewe of figures of Begin thu
what nombre þou wolte, as þou dyddyst be-fore in þe Craft of
duplacioɳ. versus.
¶ Postea procedas medians, si prima figura
Si par aut impar videas.
¶ Here he says, when þou hast write a rewe of figures, þou schalt See if the
number is
take hede wheþer þe first figure be eueɳ or odde in nombre, & or odd.
vnderstonde þat he spekes of þe first figure in þe ryȝt side. And
in the ryght side þou schalle begynne in þis Craft.
¶ Quia si fuerit par,
Dimidiabis eam, scribens quicquid remanebit:
¶ Here is the first case of þis craft, þe quych is þis, yf þe first If it is eve
halve it, a
figure be euen. þou schal take away fro þe figure euen halfe, & write the a
do away þat figure and set þere þat leues ouer, as þus, 4. take leaf in its plac
149 b. *halfe out of 4, & þan þere leues 2. do away 4 & sett þere
2. þis is lyght y-nowȝt. versus.

Mediation of an Odd Number.
¶ Impar si fuerit vnum demas mediare
Quod non presumas, sed quod superest mediabis
Inde super tractum fac demptum quod notat vnum.
Here is þe secunde case of þis craft, the quych is þis. yf þe first If it is odd
the even n
figure betokene a nombre þat is odde, the quych odde schal not less than i
be mediete, þen þou schalt medye þat nombre þat leues, when
the odde of þe same nombre is take away, & write þat þat leues
as þou diddest in þe first case of þis craft. Whaɳ þou hayst write
w Then writ
þat. for þat þat leues, write such a merke as is here vpon his sign for on
hede, þe quych merke schal betokeɳ halfe of þe odde þat was over it.
take away. lo an Ensampull. 245. the first figure here is Here is an
example.
betokenynge odde nombre, þe quych is 5, for 5 is odde; þere-
fore do away þat þat is odde, þe quych is 1. þen leues 4. þen

medye 4 & þen leues 2. do away 4. & sette þere 2, & make such
w
a merke upon his hede, þat is to say ouer his hede of 2 as þus.
w
242. And þen worch forth in þe oþer figures tyll þou come to
þe ende. by þe furst case as þou schalt vnderstonde þat þou Put the ma
only over
schalt leaf 150 a. *neuer make such a merk but ouer þe first figure first figure
hed in þe riȝt side. Wheþer þe other figures þat comyɳ after hym
be eueɳ or odde. versus.

ases of the Craft of Mediation.
¶ Si monos, dele; sit tibi cifra post nota supra.
¶ Here is þe thryde case, þe quych yf the first figure be a figure If the first
is one put
of 1. þou schalt do away þat 1 & set þere a cifer, & a merke ouer cipher.
þe cifer as þus, 241. do away 1, & sett þere a cifer with a merke
w
ouer his hede, & þen hast þou ydo for þat 0. as þus 0 þen worch
forth in þe oþer figurys till þou come to þe ende, for it is lyght as
dyche water. vnde versus.
¶ Postea procedas hac condicione secunda:
Impar si fuerit hinc vnum deme priori,
Inscribens quinque, nam denos significabit
Monos predictam.
¶ Here he puttes þe fourte case, þe quych is þis. yf it happeɳ the What to d
any other
secunde figure betoken odde nombre, þou schal do away on of is odd.
þat odde nombre, þe quych is significatiue by þat figure 1. þe
quych 1 schall be rekende for 10. Whan þou hast take away þat 1
out of þe nombre þat is signifiede by þat figure, þou schalt medie
þat þat leues ouer, & do away þat figure þat is medied, & sette in
his styde halfe of þat nombre. ¶ Whan þou hase so done, þou Write a fig
five over t
schalt write leaf 150 b. *a figure of 5 ouer þe next figures hede by- next lowe
fore toward þe ryȝt side, for þat 1, þe quych made odd nombre, number’s
schall stonde for ten, & 5 is halfe of 10; so þou most write 5 for
his haluendelle. lo an Ensampulle, 4678. begyɳ in þe ryȝt side as Example.
þou most nedes. medie 8. þen þou schalt leue 4. do away þat 8 &
sette þere 4. þen out of 7. take away 1. þe quych makes odde, &
sett 5. vpon þe next figures hede afore toward þe ryȝt side, þe
quych is now 4. but afore it was 8. for þat 1 schal be rekenet for

10, of þe quych 10, 5 is halfe, as þou knowest wel. Whan þou
hast þus ydo, medye þat þe quych leues after þe takyinge away
of þat þat is odde, þe quych leuynge schalle be 3; do away 6 5
& sette þere 3, & þou schalt haue such a nombre after go 4634.

forth to þe next figure, & medy þat, & worch forth, for it is lyȝt
ynovȝt to þe certayɳ.
¶ Si vero secunda dat vnum.
Illa deleta, scribatur cifra; priori
¶ Tradendo quinque pro denario mediato;
Nec cifra scribatur, nisi deinde figura sequatur:
Postea procedas reliquas mediando figuras
Vt supra docui, si sint tibi mille figure.
¶ Here he puttes þe 5 case, þe quych is leaf 151 a. *þis: yf þe If the seco
figure is o
secunde figure be of 1, as þis is here 12, þou schalt do away þat a cipher, a
1 & sett þere a cifer. & sett 5 ouer þe next figure hede afore write five
toward þe riȝt side, as þou diddyst afore; & þat 5 schal be haldel the next fi
of þat 1, þe quych 1 is rekent for 10. lo an Ensampulle, 214.
medye 4. þat schalle be 2. do away 4 & sett þere 2. þen go forth
to þe next figure. þe quych is bot 1. do away þat 1. & sett þere a
cifer. & set 5 vpon þe figures hed afore, þe quych is nowe 2, &
þen þou schalt haue þis nombre þen worch forth to þe nex 5
figure. And also it is no maystery yf þere come no figure 202,

after þat on is medyet, þou schalt write no 0. ne nowȝt ellis, but
set 5 ouer þe next figure afore toward þe ryȝt, as þus 14. medie 4 How to ha
fourteen.
then leues 2, do away 4 & sett þere 2. þen medie 1. þe quich is
rekende for ten, þe haluendel þere-of wel be 5. sett þat 5 vpon þe
hede of þat figure, þe quych is now 2, & do away þat 1, & þou 5
schalt haue þis nombre yf þou worch wel, vnde versus. 2,

o prove the Mediation.
¶ Si mediacio sit bene facta probare valebis
¶ Duplando numerum quem primo dimediasti
¶ Here he telles þe how þou schalt know wheþer þou hase wel How to pr
your medi
ydo or no. doubul leaf 151 b. *þe nombre þe quych þou hase
mediet, and yf þou haue wel y-medyt after þe dupleacioɳ, þou
schalt haue þe same nombre þat þou haddyst in þe tabulle or þou

began to medye, as þus. ¶ The furst ensampulle was þis. 4. þe First exam
quych I-mediet was laft 2, þe whych 2 was write in þe place þat
4 was write afore. Now doubulle þat 2, & þou schal haue 4, as
þou hadyst afore. þe secunde Ensampulle was þis, 245. When The secon
þou haddyst mediet alle þis nombre, yf þou haue wel ydo þou
w
schalt haue of þat mediacioɳ þis nombre, 122 . Now doubulle
þis nombre, & begyn in þe lyft side; doubulle 1, þat schal be 2.
do away þat 1 & sett þere 2. þen doubulle þat oþer 2 & sett þere
4, þen doubulle þat oþer 2, & þat wel be 4. þen doubul þat merke
þat stondes for halue on. & þat schalle be 1. Cast þat on to 4, & it
schalle be 5. do away þat 2 & þat merke, & sette þere 5, & þen
þou schal haue þis nombre 245. & þis wos þe same nombur þat
þou haddyst or þou began to medye, as þou mayst se yf þou take
hede. The nombre þe quych þou haddist for an Ensampul in þe 3 The third
example.
case of mediacioɳ to be mediet was þis 241. whan þou haddist
medied alle þis nombur truly leaf 152 a. *by euery figure, þou
w
schall haue be þat mediacioɳ þis nombur 120 . Now dowbul þis
nombur, & begyn in þe lyft side, as I tolde þe in þe Craft of
duplacioɳ. þus doubulle þe figure of 1, þat wel be 2. do away þat
1 & sett þere 2, þen doubul þe next figure afore, the quych is 2,
& þat wel be 4; do away 2 & set þere 4. þen doubul þe cifer, &
þat wel be noȝt, for a 0 is noȝt. And twyes noȝt is but noȝt.
þerefore doubul the merke aboue þe cifers hede, þe quych
betokenes þe haluendel of 1, & þat schal be 1. do away þe cifer
& þe merke, & sett þere 1, & þen þou schalt haue þis nombur
241. And þis same nombur þou haddyst afore or þou began to
medy, & yf þou take gode hede. ¶ The next ensampul þat had in The fourth
example.
þe 4 case of mediacioɳ was þis 4678. Whan þou hast truly
ymedit alle þis nombur fro þe begynnynge to þe endynge, þou
schalt haue of þe mediacioɳ þis nombur Now doubul this 5
nombur & begyn in þe lyft side, & doubulle 2 þat schal be 2334.

4. do away 2 and sette þere 4; þen doubule 3, þat wol be 6; do
away 3 & sett þere 6, þen doubul þat oþer 3, & þat wel be 6; do
away 3 & set þere leaf 152 b. *6, þen doubul þe 4, þat welle be 8;
þen doubul 5. þe quych stondes ouer þe hed of 4, & þat wol be
10; cast 10 to 8, & þat schal be 18; do away 4 & þat 5, & sett
þere 8, & sett that 1, þe quych is an articul of þe Composit þe

quych is 18, ouer þe next figures hed toward þe lyft side, þe
quych is 6. drav þat 1 to 6, þe quych 1 in þe dravyng schal be
rekente bot for 1, & þat 1 & þat 6 togedur wel be 7. do away þat
6 & þat 1. the quych stondes ouer his hede, & sett ther 7, & þen
þou schalt haue þis nombur 4678. And þis same nombur þou
hadyst or þou began to medye, as þou mayst see in þe secunde
Ensampul þat þou had in þe 4 case of mediacioɳ, þat was þis: The fifth
example.
when þou had mediet truly alle the nombur, a principio usque ad
finem. þou schalt haue of þat mediacioɳ þis nombur Now 5
doubul 1. þat wel be 2. do away 1 & sett þere 2. þen doubul 102.
0. þat will be noȝt. þerefore take þe 5, þe quych stondes ouer þe
next figures hed, & doubul it, & þat wol be 10. do away þe 0 þat
stondes betwene þe two figuris, & sette þere in his stid 1, for þat
1 now schal stonde in þe secunde place, where he schal betoken
10; þen doubul 2, þat wol be 4. do away 2 & sett þere 4. & leaf 153
a. *þou schal haue þus nombur 214. þis is þe same numbur þat
þou hadyst or þou began to medye, as þou may see. And so do
euer more, yf þou wil knowe wheþer þou hase wel ymedyt or
no. ¶. doubulle þe numbur þat comes after þe mediaciouɳ, & þou
schal haue þe same nombur þat þou hadyst or þou began to
medye, yf þou haue welle ydo. or els doute þe noȝt, but yf þou
haue þe same, þou hase faylide in þi Craft.

raft of Multiplication.

Sequitur de multiplicatione.

te down a Multiplication Sum.

S i tu per numerum numerum vis multiplicare
Scribe duas quascunque velis series numerorum
Ordo servetur vt vltima multiplicandi
Ponatur super anteriorem multiplicantis
A leua relique sint scripte multiplicantes.
¶ Here be-gynnes þe Chaptre of multiplicatioɳ, in þe quych þou Four thing
known of
most know 4 thynges. ¶ Ffirst, qwat is multiplicacioɳ. The Multiplica
secunde, how mony cases may hap in multiplicacioɳ. The thryde,

how mony rewes of figures þere most be. ¶ The 4. what is þe
profet of þis craft. ¶ As for þe first, þou schal vnderstonde þat the first:
multiplicacioɳ is a bryngynge to-geder of 2 thynges in on
nombur, þe quych on nombur contynes so mony tymes on, howe
leaf 153 b. *mony tymes þere ben vnytees in þe nowmbre of þat 2,
as twyes 4 is 8. now here ben þe 2 nombers, of þe quych too
nowmbres on is betokened be an aduerbe, þe quych is þe worde
twyes, & þis worde thryes, & þis worde foure sythes, 9 & so furth
of such other lyke wordes. ¶ And tweyn nombres schal be
tokenyde be a nowne, as þis worde foure showys þes tweyɳ
nombres y-broth in-to on hole nombur, þat is 8, for twyes 4 is 8,
as þou wost wel. ¶ And þes nombre 8 conteynes as oft tymes 4
as þere ben vnites in þat other nombre, þe quych is 2, for in 2
ben 2 vnites, & so oft tymes 4 ben in 8, as þou wottys wel. ¶ ffor the second
þe secunde, þou most know þat þou most haue too rewes of
figures. ¶ As for þe thryde, þou most know þat 8 maner of the third:
diuerse case may happe in þis craft. The profet of þis Craft is to the fourth
telle when a nombre is multiplyed be a noþer, qwat commys
þere of. ¶ fforthermore, as to þe sentence of oure verse, yf þou
wel multiply a nombur be a-noþer nombur, þou schalt write leaf
154 a. *a rewe of figures of what nomburs so euer þou welt, & þat The
multiplica
schal be called Numerus multiplicandus, Anglice, þe nombur the
quych to be multiplied. þen þou schalt write a-nother rewe of
figures, by þe quych þou schalt multiplie the nombre þat is to be
multiplied, of þe quych nombur þe furst figure schal be write
vnder þe last figure of þe nombur, þe quych is to be multiplied. How to se
the sum.
And so write forthe toward þe lyft side, as here you may se, And
þis one nombur schalle be called numerus multiplicans. 67324
Anglice, þe nombur multipliynge, for he schalle 1234
multiply þe hyer nounbur, as þus one tyme 6. And so
forth, as I schal telle the afterwarde. And þou schal begyn in þe
lyft side. ¶ ffor-þere-more þou schalt vndurstonde þat þere is two Two sorts
Multiplica
manurs of multiplicacioɳ; one ys of þe wyrchynge of þe boke mentally,
only in þe mynde of a mon. fyrst he teches of þe fyrst maner of
duplacioɳ, þe quych is be wyrchynge of tabuls. Afterwarde he and on pa
wol teche on þe secunde maner. vnde versus.

ltiply one Digit by another.

In digitum cures digitum si ducere maior
leaf 154 b.
* Per quantum distat a denis respice debes
¶ Namque suo decuplo totiens delere minorem
Sitque tibi numerus veniens exinde patebit.
¶ Here he teches a rewle, how þou schalt fynde þe nounbre þat How to m
two digits
comes by þe multiplicacioɳ of a digit be anoþer. loke how mony
[vny]tes ben. bytwene þe more digit and 10. And reken ten for
on vnite. And so oft do away þe lasse nounbre out of his owne Subtract th
greater fro
decuple, þat is to say, fro þat nounbre þat is ten tymes so mych is
þe nounbre þat comes of þe multiplicacioɳ. As yf þou wol
multiply 2 be 4. loke how mony vnitees ben by-twene þe quych
is þe more nounbre, & be-twene ten. Certen þere wel be vj
vnitees by-twene 4 & ten. yf þou reken þere with þe ten þe vnite,
as þou may se. so mony tymes take 2. out of his decuple, þe take the le
many time
quych is 20. for 20 is þe decuple of 2, 10 is þe decuple of 1, 30 is ten times
þe decuple of 3, 40 is þe decuple of 4, And þe oþer digetes til
þou come to ten; & whan þou hast y-take so mony tymes 2 out Example.
of twenty, þe quych is sex tymes, þou schal leue 8 as þou wost
wel, for 6 times 2 is twelue. take [1]2 out of twenty, & þere schal
leue 8. bot yf bothe þe digettes leaf 155 a. *ben y-lyech mych as
here. 222 or too tymes twenty, þen it is no fors quych of hem
tweyn þou take out of here decuple. als mony tymes as þat is fro Better use
table, thou
10. but neuer-þe-lesse, yf þou haue hast to worch, þou schalt
haue here a tabul of figures, where-by þou schalt se a-nonɳ ryght
what is þe nounbre þat comes of þe multiplicacioɳ of 2 digittes.
þus þou schalt worch in þis figure.
1
2 4
3 6 9
4 8 12 16
5 10 15 20 25
6 12 18 24 30 36
7 14 21 28 35 42 49
8 16 24 32 40 48 56 64
9 18 27 36 45 54 63 72 81

1 2 3 4 5 6 7 8 9

yf þe figure, þe quych schalle be multiplied, be euene as mych as How to us
þe diget be, þe quych þat oþer figure schal be multiplied, as two
tymes twayɳ, or thre tymes 3. or sych other. loke qwere þat The way t
the
figure sittes in þe lyft side of þe triangle, & loke qwere þe diget Multiplica
sittes in þe neþer most rewe of þe triangle. & go fro hym table.
vpwarde in þe same rewe, þe quych rewe gose vpwarde til þou
come agaynes þe oþer digette þat sittes in þe lyft side of þe
triangle. And þat nounbre, þe quych þou leaf 155 b. fyn*des þere is
þe nounbre þat comes of the multiplicacioɳ of þe 2 digittes, as yf
þou wold wete qwat is 2 tymes 2. loke quere sittes 2 in þe lyft
side in þe first rewe, he sittes next 1 in þe lyft side al on hye, as
þou may se; þe[n] loke qwere sittes 2 in þe lowyst rewe of þe
triangle, & go fro hym vpwarde in þe same rewe tylle þou come
a-ȝenenes 2 in þe hyer place, & þer þou schalt fynd ywrite 4, &
þat is þe nounbre þat comes of þe multiplicacioɳ of two tymes
tweyn is 4, as þow wotest welle. yf þe diget. the quych is
multiplied, be more þan þe oþer, þou schalt loke qwere þe more
diget sittes in þe lowest rewe of þe triangle, & go vpwarde in þe
same rewe tyl 10 þou come a-nendes þe lasse diget in the lyft side.
And þere þou schalt fynde þe nombre þat comes of þe
multiplicacioɳ; but þou schalt vnderstonde þat þis rewle, þe
quych is in þis verse. ¶ In digitum cures, &c., noþer þis triangle
schalle not serue, bot to fynde þe nounbres þat comes of the
multiplicacioɳ þat comes of 2 articuls or composites, þe nedes no
craft but yf þou wolt multiply in þi mynde. And leaf 156 a. *þere-to
þou schalt haue a craft afterwarde, for þou schall wyrch with
digettes in þe tables, as þou schalt know afterwarde. versus.

ltiply one Composite by another.
¶ Postea procedas postremam multiplicando
[Recte multiplicans per cunctas inferiores]
Condicionem tamen tali quod multiplicantes
Scribas in capite quicquid processerit inde
Sed postquam fuit hec multiplicate figure
Anteriorentur serei multiplicantis
Et sic multiplica velut isti multiplicasti

Qui sequitur numerum scriptum quiscunque figuris.
¶ Here he teches how þou schalt wyrch in þis craft. þou schalt How to m
one numb
multiplye þe last figure of þe nombre, and quen þou hast so ydo another.
þou schalt draw alle þe figures of þe neþer nounbre more taward
þe ryȝt side, so qwen þou hast multiplyed þe last figure of þe
heyer nounbre by alle þe neþer figures. And sette þe nounbir þat Multiply t
‘last’ figur
comes þer-of ouer þe last figure of þe neþer nounbre, & þen þou the higher
schalt sette al þe oþer figures of þe neþer nounbre more nere to ‘first’ of th
þe ryȝt side. ¶ And whan þou hast multiplied þat figure þat schal lower num
be multiplied þe next after hym by al þe neþer figures. And
worch as þou dyddyst afore til leaf 156 b. *þou come to þe ende.
And þou schalt vnderstonde þat euery figure of þe hier nounbre
schal be multiplied be alle þe figures of the neþer nounbre, yf þe
hier nounbre be any figure þen one. lo an Ensampul here Set the an
over the fi
folowynge. þou schalt begyne to multiplye in þe lyft 2465. the lower:
side. Multiply 2 be 2, and twyes 2 is 4. set 4 ouer þe hed 232
of þat 2, þen multiplie þe same hier 2 by 3 of þe nether then multi
second of
nounbre, as thryes 2 þat schal be 6. set 6 ouer þe hed of 3, þan lower, and
multiplie þe same hier 2 by þat 2 þe quych stondes vnder hym,
þat wol be 4; do away þe hier 2 & sette þere 4. ¶ Now þou most Then ante
lower num
antery þe nether nounbre, þat is to say, þou most sett þe neþer
nounbre more towarde þe ryȝt side, as þus. Take þe neþer 2
toward þe ryȝt side, & sette it eueɳ vnder þe 4 of þe hyer
nounbre, & antery alle þe figures þat comes after þat 2, as þus;
sette 2 vnder þe 4. þen sett þe figure of 3 þere þat þe figure of 2
stode, þe quych is now vndur þat 4 in þe hier nounbre; þen sett
þe oþer figure of 2, þe quych is þe last figure toward þe lyft side
of þe neþer nomber þere þe figure of 3 stode. þen þou schalt as thus.
haue such a nombre. leaf 157 a. * ¶ Now multiply 4, þe 464465
quych comes next after 6, by þe last 2 of þe neþer 232
nounbur toward þe lyft side. as 2 tymes 4, þat wel be 8.
sette þat 8 ouer þe figure the quych stondes ouer þe hede of þat
2, þe quych is þe last figure of þe neþer nounbre; þan multiplie
þat same 4 by 3, þat comes in þe neþer rewe, þat wol be 12. sette
þe digit of þe composyt ouer þe figure þe quych stondes ouer þe
hed of þat 3, & sette þe articule of þis composit ouer al þe
figures þat stondes ouer þe neþer 2 hede. þen multiplie þe same Now mult
the last bu
4 by þe 2 in þe ryȝt side in þe neþer nounbur, þat wol be 8. do

away 4. & sette þere 8. Euer more qwen þou multiplies þe hier of the high
figure by þat figure þe quych stondes vnder hym, þou schalt do
away þat hier figure, & sett þer þat nounbre þe quych comes of
multiplicacioɳ of ylke digittes. Whan þou hast done as I haue as thus.
byde þe, þou schalt haue suych an order of figure as is here, þen
take and antery þi neþer figures. And sett þe fyrst 1
figure of þe neþer figures vndre be figure of 6. ¶ And 82
11
4648[65]
draw al þe oþer figures of þe same rewe to hym-warde, 232.
leaf 157 b. *as þou diddyst afore. þen multiplye 6 be 2, &
sett þat þe quych comes ouer þere-of ouer al þe oþer figures
hedes þat stondes ouer þat 2. þen multiply 6 be 3, & sett alle þat
comes þere-of vpon alle þe figures hedes þat standes ouer þat 3;
þan multiplye 6 be 2, þe quych stondes vnder þat 6, þen do away
6 & write þere þe digitt of þe composit þat schal come þereof, &
sette þe articull ouer alle þe figures þat stondes ouer þe hede of
þat 3 as here, þen antery þi figures as þou diddyst afore, 11
and multipli 5 be 2, þat wol be 10; sett þe 0 ouer all þe 121
828
figures þat stonden ouer þat 2, & sett þat 1. ouer the next 464825
figures hedes, alle on hye towarde þe lyft side. þen 232
multiplye 5 be 3. þat wol be 15, write 5 ouer þe figures Antery the
hedes þat stonden ouer þat 3, & sett þat 1 ouer þe next figures figures ag
and multip
hedes toward þe lyft side. þen multiplye 5 be 2, þat wol be 10. do five:
away þat 5 & sett þere a 0, & sett þat 1 ouer þe figures hedes þat
stonden ouer 3. And þen leaf 158 a. þou schalt haue such a nounbre
as here stondes aftur.* ¶ Now draw alle þese figures 11
downe togeder as þus, 6.8.1. & 1 draw to-gedur; þat 1101
1215
wolle be 16, do away alle þese figures saue 6. lat hym 82820
stonde, for þow þou take hym away þou most write þer 4648
þe same aȝene. þerefore late hym stonde, & sett 1 ouer 232
þe figure hede of 4 toward þe lyft side; þen draw on to 4,
þat wolle be 5. do away þat 4 & þat 1, & sette þere 5. þen draw Then add
figures ab
4221 & 1 togedur, þat wol be 10. do away alle þat, & write þere line:
þat 4 & þat 0, & sett þat 1 ouer þe next figures hede toward þe
lyft side, þe quych is 6. þen draw þat 6 & þat 1 togedur, & þat
wolle be 7; do away 6 & sett þere 7, þen draw 8810 & 1, & þat
wel be 18; do away alle þe figures þat stondes ouer þe hede of
þat 8, & lette 8 stonde stil, & write þat 1 ouer þe next figuris
hede, þe quych is a 0. þen do away þat 0, & sett þere 1, þe quych

stondes ouer þe 0. hede. þen draw 2, 5, & 1 togedur, þat wolle be
8. þen do away alle þat, & write þere 8. ¶ And þen þou schalt and you w
have the a
haue þis nounbre, 571880.

ases of this Craft.
leaf 158 b.

* ¶ Sed cum multiplicabis, primo sic est operandum,
Si dabit articulum tibi multiplicacio solum;
Proposita cifra summam transferre memento.
¶ Here he puttes þe fyrst case of þis craft, þe quych is þis: yf þere What to d
first
come an articulle of þe multiplicacioɳ ysette before the articulle multiplica
in þe lyft side as þus multiplye 5 by 2, þat wol be 10; sette 51 results in
ouer þe hede of þat 2 a 0, & sett þat on, þat is þe articul, in 23. article.
þe lyft side, þat is next hym, þen þou schalt haue þis
nounbre ¶ And þen worch forth as þou diddist afore. And 1051.
þou schalt vnderstonde þat þou schalt write no 0. but 23
whan þat place where þou schal write þat 0 has no figure
afore hym noþer after. versus.
¶ Si autem digitus excreuerit articulusque.
Articulus 12 supraposito digito salit vltra.
¶ Here is þe secunde case, þe quych is þis: yf hit happe þat þere What to d
result is a
come a composyt, þou schalt write þe digitte ouer þe hede of þe composite
neþer figure by þe quych þou multipliest þe hier figure; and sett number.
þe articulle next hym toward þe lyft side, as þou diddyst afore, as
þus Multiply 8 by 8, þat wol be 64. Write þe 4 ouer 8, þat 83.
is to say, ouer þe hede of þe neþer 8; & set 6, þe quych leaf 83
159 a. *is an articul, next after. And þen þou schalt haue
such a nounbre as is here, And þen worch forth. 6483 13,
83
¶ Si digitus tamen ponas ipsum super ipsam.
¶ Here is þe thryde case, þe quych is þis: yf hit happe þat of þi What if it
digit.
multiplicaciouɳ come a digit, þou schalt write þe digit ouer þe
hede of þe neþer figure, by the quych þou multipliest þe hiere
figure, for þis nedes no Ensampul.
¶ Subdita multiplica non hanc que [incidit] illi

Delet eam penitus scribens quod prouenit inde.
¶ Here is þe 4 case, þe quych is: yf hit be happe þat þe neþer The fourth
of the craf
figure schal multiplye þat figure, þe quych stondes ouer þat
figures hede, þou schal do away þe hier figure & sett þere þat
þat comys of þat multiplicacioɳ. As yf þere come of þat
multiplicacioɳ an articuls þou schalt write þere þe hier figure
stode a 0. ¶ And write þe articuls in þe lyft side, yf þat hit be a
digit write þere a digit. yf þat hit be a composit, write þe digit of
þe composit. And þe articul in þe lyft side. al þis is lyȝt y-nowȝt,
þere-fore þer nedes no Ensampul.
¶ Sed si multiplicat aliam ponas super ipsam
Adiunges numerum quem prebet ductus earum.
leaf 159 b.

¶ Here is þe 5 case, þe quych is þis: yf *þe neþer figure schul The fifth c
the craft.
multiplie þe hier, and þat hier figure is not recte ouer his hede.
And þat neþer figure hase oþer figures, or on figure ouer his
hede by multiplicacioɳ, þat hase be afore, þou schalt write þat
nounbre, þe quych comes of þat, ouer alle þe ylke figures hedes,
as þus here: Multiply 2 by 2, þat wol be 4; set 4 ouer þe 236
hede of þat 2. þen multiplies þe hier 2 by þe neþer 3, þat
14 234
wol be 6. set ouer his hede 6, multiplie þe hier 2 by þe
neþer 4, þat wol be 8. do away þe hier 2, þe quych stondes ouer
þe hede of þe figure of 4, and set þere 8. And þou schalt haue þis
nounbre here And antery þi figures, þat is to say, set þi 46836
neþer 4 vnder þe hier 3, and set þi 2 other figures nere 234
hym, so þat þe neþer 2 stonde vndur þe hier 6, þe quych 6
stondes in þe lyft side. And þat 3 þat stondes vndur 8, as þus
aftur ȝe may se, Now worch forthermore, And multiplye 46836
þat hier 3 by 2, þat wol be 6, set þat 6 þe quych stondes 234
ouer þe hede of þat 2, And þen worch as I taȝt þe afore.
leaf 160 a.

* ¶ Si supraposita cifra debet multiplicare
Prorsus eam deles & ibi scribi cifra debet.
¶ Here is þe 6 case, þe quych is þis: yf hit happe þat þe figure by The sixth
the craft.
þe quych þou schal multiplye þe hier figure, þe quych stondes
ryght ouer hym by a 0, þou schalt do away þat figure, þe quych

ouer þat cifre hede. ¶ And write þere þat nounbre þat comes of
þe multiplicacioɳ as þus, 23. do away 2 and sett þere a 0. vnde
versus.
¶ Si cifra multiplicat aliam positam super ipsam
Sitque locus supra vacuus super hanc cifram fiet.
¶ Here is þe 7 case, þe quych is þis: yf a 0 schal multiply a The seven
of the craf
figure, þe quych stondes not recte ouer hym, And ouer þat 0
stonde no thyng, þou schalt write ouer þat 0 anoþer 0 as þus:
multiplye 2 be a 0, it wol be nothynge. write þere a 0 ouer 24
þe hede of þe neþer 0, And þen worch forth til þou come to 03
þe ende.
¶ Si supra 15 fuerit cifra semper est pretereunda.
¶ Here is þe 8 case, þe quych is þis: yf þere be a 0 or mony cifers The eighth
of the craf
in þe hier rewe, þou schalt not multiplie hem, bot let hem stonde.
And antery þe figures beneþe to þe next figure sygnificatyf as
þus: Ouer-lepe alle þese cifers & sett þat leaf 160 b. *neþer 00032.
2 þat stondes toward þe ryght side, and sett hym vndur þe 22
3, and sett þe oþer nether 2 nere hym, so þat he stonde
vndur þe thrydde 0, þe quych stondes next 3. And þan worch.
vnde versus.
¶ Si dubites, an sit bene multiplicacio facta,
Diuide totalem numerum per multiplicantem.
¶ Here he teches how þou schalt know wheþer þou hase wel I-do How to pr
multiplica
or no. And he says þat þou schalt deuide alle þe nounbre þat
comes of þe multiplicacioɳ by þe neþer figures. And þen þou
schalt haue þe same nounbur þat þou hadyst in þe begynnynge.
but ȝet þou hast not þe craft of dyuisioɳ, but þou schalt haue hit
afterwarde.
¶ Per numerum si vis numerum quoque multiplicare
¶ Tantum per normas subtiles absque figuris
Has normas poteris per versus scire sequentes.
¶ Here he teches þe to multiplie be þowȝt figures in þi mynde. Mental
multiplica
And þe sentence of þis verse is þis: yf þou wel multiplie on
nounbre by anoþer in þi mynde, þou schal haue þereto rewles in
þe verses þat schal come after.

¶ Si tu per digitum digitum vis multiplicare
Regula precedens dat qualiter est operandum.
¶ Here he teches a rewle as þou hast afore to multiplie a digit be Digit by d
easy.
anoþer, as yf þou wolde wete qwat is sex tymes 6. þou leaf 161 a.
*schalt wete by þe rewle þat I taȝt þe before, yf þou haue mynde
þerof.
¶ Articulum si per reliquum reliquum vis multiplicare
In proprium digitum debet vterque resolui.
¶ Articulus digitos post se multiplicantes
Ex digitus quociens retenerit multiplicari
Articuli faciunt tot centum multiplicati.
¶ Here he teches þe furst rewle, þe quych is þis: yf þou wel The first c
the craft.
multiplie an articul be anoþer, so þat both þe articuls bene with-
Inne an hundreth, þus þou schalt do. take þe digit of bothe the Article by
article;
articuls, for euery articul hase a digit, þen multiplye þat on digit
by þat oþer, and loke how mony vnytes ben in þe nounbre þat
comes of þe multiplicacioɳ of þe 2 digittes, & so mony
hundrythes ben in þe nounbre þat schal come of þe
multiplicacioɳ of þe ylke 2 articuls as þus. yf þou wold wete an examp
qwat is ten tymes ten. take þe digit of ten, þe quych is 1; take þe
digit of þat oþer ten, þe quych is on. ¶ Also multiplie 1 be 1, as
on tyme on þat is but 1. In on is but on vnite as þou wost welle,
þerefore ten tymes ten is but a hundryth. ¶ Also yf þou wold another ex
wete what is twenty tymes 30. take þe digit of twenty, þat is 2; &
take þe digitt of thrytty, þat is 3. multiplie 3 be 2, þat is 6. Now
in 6 ben 6 vnites, ¶ And so mony hundrythes ben in 20 tymes
30*, leaf 161 b. þerefore 20 tymes 30 is 6 hundryth eueɳ. loke &
se. ¶ But yf it be so þat one articul be with-Inne an hundryth, or
by-twene an hundryth and a thowsande, so þat it be not a
þowsande fully. þen loke how mony vnytes ben in þe nounbur
þat comys of þe multiplicacioɳ 16And so mony tymes 16 of 2
digittes of ylke articuls, so mony thowsant ben in þe nounbre, the
qwych comes of þe multiplicacioɳ. And so mony tymes ten
thowsand schal be in þe nounbre þat comes of þe multiplicacion
of 2 articuls, as yf þou wold wete qwat is 4 hundryth tymes [two
hundryth]. Multiply 4 be 2, 17 þat wol be 8. in 8 ben 8 vnites. ¶
And so mony tymes ten thousand be in 4 hundryth tymes [2] 17

hundryth, þat is 80 thousand. Take Mental
o work subtly without Figures. hede, I schall telle þe a generalle
multiplica

rewle whan þou hast 2 articuls, And Another
example.
þou wold wete qwat comes of þe multiplicacioɳ of hem 2.
multiplie þe digit of þat on articuls, and kepe þat nounbre, þen
loke how mony cifers schuld go before þat on articuls, and he
were write. Als mony cifers schuld go before þat other, & he
were write of cifers. And haue alle þe ylke cifers togedur in þi
mynde, leaf 162 a. *a-rowe ychoɳ aftur other, and in þe last plase
set þe nounbre þat comes of þe multiplicacioɳ of þe 2 digittes.
And loke in þi mynde in what place he stondes, where in þe
secunde, or in þe thryd, or in þe 4, or where ellis, and loke qwat
þe figures by-token in þat place; & so mych is þe nounbre þat Another
example.
comes of þe 2 articuls y-multiplied to-gedur as þus: yf þou wold
wete what is 20 thousant tymes 3 þowsande. multiply þe digit of
þat articulle þe quych is 2 by þe digitte of þat oþer articul þe
quych is 3, þat wol be 6. þen loke how mony cifers schal go to
20 thousant as hit schuld be write in a tabul. certainly 4 cifers
schuld go to 20 þowsant. ffor þis figure 2 in þe fyrst place
betokenes twene. ¶ In þe secunde place hit betokenes twenty. Notation.
¶ In þe 3. place hit betokenes 2 hundryth. .¶. In þe 4 place 2
thousant. ¶ In þe 5 place hit betokenes twenty þousant. þerefore
he most haue 4 cifers a-fore hym þat he may stonde in þe 5
place. kepe þese 4 cifers in thy mynde, þen loke how mony
cifers goɳ to 3 thousant. Certayn to 3 thousante leaf 162 b. *goɳ 3
cifers afore. Now cast ylke 4 cifers þat schuld go to twenty
thousant, And thes 3 cifers þat schuld go afore 3 thousant, &
sette hem in rewe ychoɳ after oþer in þi mynde, as þai schuld
stonde in a tabulle. And þen schal þou haue 7 cifers; þen sett þat
6 þe quych comes of þe multiplicacioɳ of þe 2 digittes aftur þe
ylke cifers in þe 8 place as yf þat hit stode in a tabul. And loke
qwat a figure of 6 schuld betoken in þe 8 place. yf hit were in a
tabul & so mych it is. & yf þat figure of 6 stonde in þe fyrst
place he schuld betoken but 6. ¶ In þe 2 place he schuld betoken
sexty. ¶ In the 3 place he schuld betokeɳ sex hundryth. ¶ In þe 4 Notation a
place sex thousant. ¶ In þe 5 place sexty þowsant. ¶ In þe sext
place sex hundryth þowsant. ¶ In þe 7 place sex þowsant
thousantes. ¶ In þe 8 place sexty þowsant thousantes. þerfore sett

6 in octauo loco, And he schal betoken sexty þowsant
thousantes. And so mych is twenty þowsant tymes 3 thousant, Mental
multiplica
¶ And þis rewle is generalle for alle maner of articuls, Whethir
þai be hundryth or þowsant; but þou most know well þe craft of
þe wryrchynge in þe tabulle leaf 163 a. *or þou know to do þus in
þi mynde aftur þis rewle. Thou most þat þis rewle holdyþe note
but where þere ben 2 articuls and no mo of þe quych ayther of
hem hase but on figure significatyf. As twenty tymes 3 thousant
or 3 hundryth, and such oþur.
¶ Articulum digito si multiplicare oportet
Articuli digit[i sumi quo multiplicate]
Debemus reliquum quod multiplicatur ab illis
Per reliquo decuplum sic summam latere nequibit.
¶ Here he puttes þe thryde rewle, þe quych is þis. yf þou wel The third
the craft;
multiply in þi mynde, And þe Articul be a digitte, þou schalt loke
þat þe digitt be with-Inne an hundryth, þen þou schalt multiply
the digitt of þe Articulle by þe oþer digitte. And euery vnite in þe
nounbre þat schalle come þere-of schal betoken ten. As þus: yf an examp
þat þou wold wete qwat is twyes 40. multiplie þe digitte of 40,
þe quych is 4, by þe oþer diget, þe quych is 2. And þat wolle be
8. And in þe nombre of 8 ben 8 vnites, & euery of þe ylke vnites
schuld stonde for 10. þere-fore þere schal be 8 tymes 10, þat wol
be 4 score. And so mony is twyes 40. ¶ If þe articul be a
hundryth or be 2 hundryth And a þowsant, so þat hit be notte a
thousant, leaf 163 b. *worch as þou dyddyst afore, saue þou schalt
rekene euery vnite for a hundryth.
¶ In numerum mixtum digitum si ducere cures
Articulus mixti sumatur deinde resoluas
In digitum post fac respectu de digitis
Articulusque docet excrescens in diriuando
In digitum mixti post ducas multiplicantem
¶ De digitis vt norma 18[docet] de [hunc]
Multiplica simul et sic postea summa patebit.
Here he puttes þe 4 rewle, þe quych is þis: yf þou multipliy on The fourth
of the craf
composit be a digit as 6 tymes 24, 19þen take þe diget of þat
composit, & multiply þat digitt by þat oþer diget, and kepe þe

nombur þat comes þere-of. þen take þe digit of þat composit, &
multiply þat digit by anoþer diget, by þe quych þou hast
multiplyed þe diget of þe articul, and loke qwat comes þere-of. Composit
digit.
þen take þou þat nounbur, & cast hit to þat other nounbur þat þou
secheste as þus yf þou wel wete qwat comes of 6 tymes 4 &
twenty. multiply þat articulle of þe composit by þe digit, þe Mental
multiplica
quych is 6, as yn þe thryd rewle þou was tauȝt, And þat schal be
6 score. þen multiply þe diget of þe composit, leaf 164 a. *þe quych
is 4, and multiply þat by þat other diget, þe quych is 6, as þou
wast tauȝt in þe first rewle, yf þou haue mynde þerof, & þat wol
be 4 & twenty. cast all ylke nounburs to-gedir, & hit schal be
144. And so mych is 6 tymes 4 & twenty.

o multiply without Figures.
¶ Ductus in articulum numerus si compositus sit
Articulum purum comites articulum quoque
Mixti pro digitis post fiat [et articulus vt]
Norma iubet [retinendo quod extra dicta ab illis]
Articuli digitum post tu mixtum digitum duc
Regula de digitis nec precipit articulusque
Ex quibus excrescens summe tu iunge priori
Sic manifesta cito fiet tibi summa petita.
¶ Here he puttes þe 5 rewle, þe quych is þis: yf þou wel multiply The fifth c
the craft:
an Articul be a composit, multiplie þat Articul by þe articul of þe
composit, and worch as þou wos tauȝt in þe secunde rewle, of þe
quych rewle þe verse begynnes þus. ¶ Articulum si per Relicum Article by
Composit
vis multiplicare. þen multiply þe diget of þe composit by þat oþir
articul aftir þe doctrine of þe 3 rewle. take þerof gode hede,
I pray þe as þus. Yf þou wel wete what is 24 tymes ten. An examp
Multiplie ten by 20, þat wel be 2 hundryth. þen multiply þe diget
of þe 10, þe quych is 1, by þe diget of þe composit, þe quych is
4, & þat leaf 164 b. *wol be 4. þen reken euery vnite þat is in 4 for
10, & þat schal be 40. Cast 40 to 2 hundryth, & þat wol be 2
hundryth & 40. And so mych is 24 tymes ten.

o work without Figures.
¶ Compositum numerum mixto si[c] multiplicabis

Vndecies tredecim sic est ex hiis operandum
In reliquum primum demum duc post in eundem
Vnum post denum duc in tria deinde per vnum
Multiplicesque demum intra omnia multiplicata
In summa decies quam si fuerit tibi doces
Multiplicandorum de normis sufficiunt hec.
¶ Here he puttes þe 6 rewle, & þe last of alle multiplicacioɳ, þe The sixth
the craft:
quych is þis: yf þou wel multiplye a composit by a-noþer
composit, þou schalt do þus. multiplie þat on composit, qwych Composit
Composit
þou welt of the twene, by þe articul of þe toþer composit, as þou
were tauȝt in þe 5 rewle, þen multiplie þat same composit, þe
quych þou hast multiplied by þe oþer articul, by þe digit of þe
oþer composit, as þou was tauȝt in þe 4 rewle. As þus, yf þou Mental
multiplica
wold wete what is 11 tymes 13, as þou was tauȝt in þe 5 rewle, & An examp
þat schal be an hundryth & ten, afterwarde multiply þat same
composit þat þou hast multiplied, þe quych is a .11. And
multiplye hit be þe digit of þe oþer composit, þe quych is 3, for 3
is þe digit of 13, And þat wel be 30. þen take þe digit of þat
composit, þe quych composit þou multiplied by þe digit of þat
oþer composit, leaf 165 a. *þe quych is a 11. ¶ Also of the quych 11 of the sixt
of the craf
on is þe digit. multiplie þat digitt by þe digett of þat other
composit, þe quych diget is 3, as þou was tauȝt in þe first rewle
in þe begynnynge of þis craft. þe quych rewle begynnes “In
digitum cures.” And of alle þe multiplicacioɳ of þe 2 digitt
comys thre, for onys 3 is but 3. Now cast alle þese nounbers
togedur, the quych is þis, a hundryth & ten & 30 & 3. And al þat
wel be 143. Write 3 first in þe ryght side. And cast 10 to 30, þat
wol be 40. set 40 next aftur towarde þe lyft side, And set aftur a
hundryth as here an Ensampulle, 143.
(Cetera desunt.)

1. In MS, ‘awiy.’
2. ‘ben’ repeated in MS.
3. In MS. ‘thausandes.’
4. Perhaps “So.”

5. ‘hali’ marked for erasure in MS.
6. ‘moy’ in MS.
7. ‘Subtrahas aut addis a dextris vel mediabis’ added on
margin of MS.
8. After ‘craft’ insert ‘the .4. what is þe profet of þis craft.’
9. After ‘sythes’ insert ‘& þis wordes fyue sithe & sex
sythes.’
10. ‘t’l’ marked for erasure before ‘tyl’ in MS.
11. Here ‘of þe same rew’ is marked for erasure in MS.
12. ‘sed’ deleted in MS.
13. 6883 in MS.
14. ‘þen’ overwritten on ‘þat’ marked for erasure.
15. ‘Supra’ inserted in MS. in place of ‘cifra’ marked for
erasure.
16–16. Marked for erasure in MS.
17. 4 in MS.
18. docet. decet MS.
19. ‘4 times 4’ in MS.

[Ashmole MS. 396, fol. 48.]

B
oys seying in the begynnyng of his Arsemetrike:—Alle
Fol. 48. thynges that bene fro the first begynnyng of
thynges have procedede, and come forthe, And by resoun
of nombre ben formede; And in wise as they bene, So

owethe they to be knowene; wherfor in vniuersalle
knowlechyng of thynges the Art of nombrynge is best, and
most operatyfe.

T
herfore sithen the science of the whiche at this tyme we The name
art.
intendene to write of standithe alle and about nombre:
ffirst we most se, what is the propre name therofe, and fro
whens the name come: Afterwarde what is nombre, And how
manye spices of nombre ther ben. The name is clepede
Algorisme, hade out of Algore, other of Algos, in grewe, That is Derivation
Algorism.
clepide in englisshe art other craft, And of Rithmus that is
callede nombre. So algorisme is clepede the art of nombryng, Another.
other it is had ofe en or in, and gogos that is introduccioun, and
Rithmus nombre, that is to say Interduccioun of nombre. And Another.
thirdly it is hade of the name of a kyng that is clepede Algo and
Rythmus; So callede Algorismus. Sothely .2. manere of nombres Kinds of
numbers.
ben notifiede; Formalle, 1 as nombre is vnitees gadrede to-gedres;
Materialle, 2 as nombre is a colleccioun of vnitees. Other nombre
is a multitude hade out of vnitees, vnitee is that thynge wher-by
euery thynge is callede oone, other o thynge. Of nombres, that
one is clepede digitalle, that othere Article, Another a nombre
componede oþer myxt. Another digitalle is a nombre with-in
.10.; Article is þat nombre that may be dyvydede in .10. parties
egally, And that there leve no residue; Componede or medlede is
that nombre that is come of a digite and of an article. And
vndrestande wele that alle nombres betwix .2. articles next is a
nombre componede. Of this art bene .9. spices, that is forto sey, The 9 rule
the Art.
numeracioun, addicioun, Subtraccioun, Mediacioun,
Duplacioun, Multipliacioun, Dyvysioun, Progressioun, And of
Rootes the extraccioun, and that may be hade in .2. maners, that
is to sey in nombres quadrat, and in cubices: Amonge the
whiche, ffirst of Numeracioun, and afterwarde of þe oþers by
ordure, y entende to write.

er I. Numeration.
Fol. 48 b.

*For-sothe numeracioun is of euery numbre by competent
figures an artificialle representacioun.
Sothly figure, difference, places, and lynes supposen o thyng Figures,
difference
other the same, But they ben sette here for dyuers resons. ffigure places, an
is clepede for protraccioun of figuracioun; Difference is callede limits.
for therby is shewede euery figure, how it hathe difference fro
the figures before them: place by cause of space, where-in me
writethe: lynees, for that is ordeynede for the presentacioun of
euery figure. And vnderstonde that ther ben .9. lymytes of The 9 figu
figures that representen the .9. digites that ben these. 0. 9. 8. 7. 6.
5. 4. 3. 2. 1. The .10. is clepede theta, or a cercle, other a cifre, The ciphe
other a figure of nought for nought it signyfiethe. Nathelesse she
holdyng that place givethe others for to signyfie; for withe-out
cifre or cifres a pure article may not be writte. And sithen that by The nume
these .9. figures significatifes Ioynede with cifre or with cifres
alle nombres ben and may be representede, It was, nether is, no
nede to fynde any more figures. And note wele that euery digite of digits,
shalle be writte with oo figure allone to it aproprede. And alle of articles
articles by a cifre, ffor euery article is namede for oone of the
digitis as .10. of 1.. 20. of. 2. and so of the others, &c. And alle
nombres digitalle owen to be sette in the first difference: Alle
articles in the seconde. Also alle nombres fro .10. til an .100.
[which] is excludede, with .2. figures mvst be writte; And yf it be
an article, by a cifre first put, and the figure y-writte towarde the
lift honde, that signifiethe the digit of the whiche the article is
namede; And yf it be a nombre componede, ffirst write the digit of compos
that is a part of that componede, and write to the lift side the
article as it is seide be-fore. Alle nombre that is fro an hundrede
tille a thousande exclusede, owithe to be writ by .3. figures; and
alle nombre that is fro a thousande til .x. Mł. mvst be writ by .4.
figures; And so forthe. And vnderstonde wele that euery figure The value
position.
sette in the first place signyfiethe his digit; In the seconde place
.10. tymes his digit; In the .3. place an hundrede so moche; In the
.4. place a thousande so moche; In the .5. place .x. thousande so
moche; In the .6. place an hundrede thousande so moche; In the
.7. place a thousande thousande. And so infynytly mvltiplying by
Fol. 49. *these .3. 10, 100, 1000. And vnderstande wele that
competently me may sette vpon figure in the place of a

thousande, a prike to shewe how many thousande the last figure
shalle represent. We writene in this art to the lift side-warde, as Numbers
written fro
arabiene writene, that weren fynders of this science, othere for right to le
this resoun, that for to kepe a custumable ordre in redyng, Sette
we alle-wey the more nombre before.

er II. Addition.

A
ddicioun is of nombre other of nombres vnto nombre or to Definition
nombres aggregacioun, that me may see that that is come
therof as excressent. In addicioun, 2. ordres of figures and
.2. nombres ben necessary, that is to sey, a nombre to be addede
and the nombre wherto the addicioun sholde be made to. The
nombre to be addede is that þat sholde be addede therto, and
shalle be vnderwriten; the nombre vnto the whiche addicioun
shalle be made to is that nombre that resceyuethe the addicion of
þat other, and shalle be writen above; and it is convenient that How the
numbers s
the lesse nombre be vnderwrit, and the more addede, than the be written
contrary. But whether it happe one other other, the same comythe
of, Therfor, yf þow wilt adde nombre to nombre, write the
nombre wherto the addicioun shalle be made in the omest ordre
by his differences, so that the first of the lower ordre be vndre
the first of the omyst ordre, and so of others. That done, adde the The metho
working.
first of the lower ordre to the first of the omyst ordre. And of
suche addicioun, other þere growith therof a digit, An article,
other a composede. If it be digitus, In the place of the omyst Begin at t
right.
shalt thow write the digit excrescyng, as thus:—
If the article; in the place of the omyst The resultant 2
put a-way by a cifre writte, and the To whom it shal be addede 1
digit transferrede, of þe whiche the The nombre to be addede 1
article toke his name, towarde the lift
side, and be it addede to the next The Sum
digit,
figure folowyng, yf ther be any figure folowyng; or no, and yf it
be not, leve it [in the] voide, as thus:—
or an artic

Resultans 2 7 8 2 7 The resultant 10
Cui debet addi 1 0 0 8 4 To whom it shalle be addede 7
Numerus addendus 1 7 7 4 3 The nombre to be addede 3

And yf it happe that the figure folowyng wherto the addicioun
shalle be made by [the cifre of] an article, it sette a-side;
In his place write the Fol. 49 b. *[digit
The resultant 17
of the] Article as thus:—
To whom it shalle be addede 10
The nombre to be addede 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
The resultant 10
write þe article towarde þe lift honde
To whom it shalle be addede 9
as bifore, and thus:—
The nombre to be addede 1

And yf 3 [therefrom grow a] nombre componed, 4 [in the place of or a comp
the nombre] put a-way 5
[let] the digit [be] 6 writ þat is part of
The resultant 12
þat composide, and þan put to þe lift
side the article as before, and þus:— To whom it shalle be addede 8
The nombre to be addede 4

This done, adde the seconde to the seconde, and write above The transl
note.
oþer as before. Note wele þat in addicions and in alle spices
folowyng, whan he seithe one the other shalle be writen aboue,
and me most vse euer figure, as that euery figure were sette by
halfe, and by hym-selfe.

er III. Subtraction.

S
ubtraccioun is of .2. proposede nombres, the fyndyng of the Definition
Subtractio
excesse of the more to the lasse: Other subtraccioun is
ablacioun of o nombre fro a-nother, that me may see a
some left. The lasse of the more, or even of even, may be
withdraw; The more fro the lesse may neuer be. And sothly that How it ma
done.
nombre is more that hathe more figures, So that the last be

signyficatifes: And yf ther ben as many in that one as in that
other, me most deme it by the last, other by the next last. More- What is re
ouer in with-drawyng .2. nombres ben necessary; A nombre to
be withdraw, And a nombre that me shalle with-draw of. The
nombre to be with-draw shalle be writ in the lower ordre by his
differences; The nombre fro the whiche me shalle withe-draw in Write the
number ab
the omyst ordre, so that the first be vnder the first, the seconde
vnder the seconde, And so of alle others. Withe-draw therfor the Subtract th
figure if
first of the lowere ordre fro the first of the ordre above his hede, possible.
and that wolle be other more or lesse, oþer egalle.
yf it be egalle or even the figure sette
The remanent 20
beside, put in his place a cifre. And
Wherof me shalle withdraw 22
yf it be more put away þerfro als
many of vnitees the lower figure The nombre to be withdraw 2

conteynethe, and writ the residue as
thus The remanent 2 2
Fol. 50. Wherof me shalle with-draw 2 8
Þe nombre to be withdraw 6

And yf it be lesse, by- If it is not
*Remanens 2 2 1 8 2 9 9 9 8 possible ‘b
cause the more may not ten,’
A quo sit subtraccio 8 7 2 4 3 0 0 0 4
be with-draw ther-fro,
Numerus subtrahendus 6 5 7 [6] . . . . 6
borow an vnyte of the
next figure that is
worthe 10. Of that .10. and of the figure that ye wolde have with-
draw fro be-fore to-gedre Ioynede, and then
subtract.
with-draw þe figure be-nethe, and
put the residue in the place of the The remanent 1 8
figure put a-side as þus:— Wherof me shalle with-draw 2 4

And yf the figure wherof me shal The nombre to be with-draw 0 6
borow the vnyte be one, put it a- If the seco
side, and write a cifre in the place þerof, lest the figures figure is o
folowing faile of thaire nombre, and þan worche as it shewith in
this figure here:—
And yf the vnyte wherof me shal borow be a cifre, go ferther to
the figure signyficatife, and ther borow one, and retournyng

bake, in the place of euery cifre
The remanent 3 0 98
þat ye passide ouer, sette figures
of .9. as here it is specifiede:— Wherof me shal with-draw 3 1 2
The nombre to be with-draw . . 3
And whan me comethe to the
nombre wherof me intendithe, If the seco
there remaynethe alle-wayes figure is a

.10. ffor þe whiche .10. &c. The remenaunt 2 9 9 9 9
The reson why þat for euery Wherof me shalle with-draw 3 0 0 0 3
cifre left behynde me setteth The nombre to be with-draw 4
figures ther of .9. this it is:—
If fro the .3. place me A justifica
borowede an vnyte, that vnyte by respect of the figure that he the rule gi

came fro representith an .C., In the place of that cifre [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 the .4., .5., .6., place, or ony other so vpwarde. This
done, withdraw the seconde of the lower ordre fro the figure
above his hede of þe omyst ordre, and wirche as before. And Why it is
to work fr
note wele that in addicion or in subtraccioun me may wele fro right to le
the lift side begynne and ryn to the right side, But it wol be more
profitabler to be do, as it is taught. And yf thow wilt prove yf How to pr
subtractio
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 wele; and and additi
in like wise in addicioun, whan thow hast addede alle thy
figures, withdraw them that thow first Fol. 50 b. *addest, and the
same wolle retourne. The subtraccioun is none other but a
prouffe of the addicioun, and the contrarye in like wise.

er IV. Mediation.

M
ediacioun is the fyndyng of the halfyng of euery Definition
mediation
nombre, that it may be seyne what and how moche is
euery halfe. In halfyng ay oo order of figures and oo
nombre is necessary, that is to sey the nombre to be halfede.
Therfor yf thow wilt half any nombre, write that nombre by his
differences, and begynne at the right, that is to sey, fro the first Where to
figure to the right side, so that it be signyficatife other represent

vnyte or eny other digitalle nombre. If it be vnyte write in his
place a cifre for the figures folowyng, [lest they signify less], and If the first
is unity.
write that vnyte without in the table, other resolue it in .60.
mynvtes and sette a-side half of tho minutes so, and reserve the
remenaunt without in the table, as thus .30.; other sette without
thus .dī: that kepethe none ordre of place, Nathelesse it hathe
signyficacioun. And yf the other figure signyfie any other digital
nombre fro vnyte forthe, oþer the nombre is ode or evene.
If it be even, write this half in this wise:—
Halfede 2 2
And if it be odde, Take the next even vndre to be halfede 4 4
hym conteynede, and put his half in the place
of that odde, and of þe vnyte that
halfede 2 3 [di]
remaynethe to be halfede do thus:—
To be halfede 4 7
This done, the seconde is to be halfede, yf
it be a cifre put it be-side, and yf it be significatife, other it is What to d
is not unit
even or ode: If it be even, write in the place of þe nombres Then halv
wipede out the halfe; yf it be ode, take the next even vnder it second fig
contenythe, and in the place of the Impar sette a-side put half of
the even: The 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
Halfede
that one, and adde that other to the next figure
to be halfede
precedent as here:—
And yf þe addicioun sholde be made to a cifre, sette it a-side, If it is odd
to the figu
and write in his place .5. before.
And vnder this fourme me
shalle write and worche, tille doublede 2 6 8 9 0 10 17 4
the totalle nombre be halfede. to be doublede 1 3 4 4 5 5 8 7

er V. Duplation.

D
uplicacioun is agregacion of nombre [to itself] þat me Definition
Duplation
may se the nombre growen. In doublynge ay is but one
ordre of 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. Fol. 51. *In the other .3. before we
begynne alle way fro the right side and fro the lasse nombre, In Where to

this spice and in alle other folowyng we wolle begynne fro the
lift side, ffor and me bigon the double fro the first, omwhile me
myght double oo thynge twyes. And how be it that me myght Why.
double fro the right, that wolde be harder in techyng and in
workyng. Therfor yf thow wolt double any nombre, write that
nombre by his differences, and double the last. And of that
doublyng other growithe a nombre digital, article, or componede.
[If it be a digit, write it in the place of the first digit.]
If it be article, write in his place a cifre and
double 10
transferre the article towarde the lift, as thus:—
to be doublede 5
And yf the nombre be componede,
write a digital that is part of his composicioun, and sette the What to d
the result.
article to the lift hande, as thus:—
doublede 16
That done, me most double the last save one,
and what growethe þerof me most worche as to be doublede 8

before. And yf a cifre be, touche it not. But yf
any nombre shalle be addede to the cifre,
in þe place of þe figure wipede out me most
doublede 6 0 6
write the nombre to be addede, as thus:—
to be doublede 3 0 3
In the same wise me shalle wirche of alle
others. And this probacioun: How to pr
your answ
If thow truly double the halfis, and truly
half the doubles, the same nombre and Doublede 6 1 8
figure shalle mete, suche as thow labourede to be doublede 3 0 9
vpone first, And of the contrarie.

er VI. Multiplication.

M
ultiplicacioun of nombre by hym-self other by a-nother, Definition
Multiplica
with proposide .2. nombres, [is] the fyndyng of the
thirde, That so oft conteynethe that other, as ther ben
vnytes in the oþer. In multiplicacioun .2. nombres pryncipally
ben necessary, that is to sey, the nombre multiplying and the
nombre to be multipliede, as here;—twies fyve. [The number Multiplier
multiplying] is designede aduerbially. The nombre to be Multiplica
multipliede resceyvethe a nominalle appellacioun, as twies .5. 5.

is the nombre multipliede, and twies is the nombre to be
multipliede.
Resultans 9 1 0 1 3 2 6 6 8 0 0 8
Multiplicandus . . 5 . . 4 . 3 4 0 0 4
Multiplicans . 2 2 . 3 3 2 2 2 . . .

Also me may thervpone to assigne the. 3. nombre, the whiche is Product.
Fol. 51 b. *clepede product or provenient, of takyng out of one fro
another: as twyes .5 is .10., 5. the nombre to be multipliede, and
.2. the multipliant, and. 10. as before is come therof. And
vnderstonde wele, that of the multipliant may be made the
nombre to be multipliede, and of the contrarie, remaynyng euer
the same some, and herofe comethe the comen speche, that
seithe all nombre is convertede by Multiplying in hym-selfe.
And ther ben .6 rules of Multiplicacioun; There are
ases of Multiplication. ffirst, yf a digit multiplie a digit, considre
of
Multiplica
how many
of vnytees ben betwix the digit 1 2 3 4 5 6 7 8 9 10
by multiplying and his .10. 2 4 6 8 10 1010 14 16 18 20
bethe to-gedre accomptede, 3 6 9 12 15 18 21 24 27 30
and so oft with-draw the digit 4 8 12 16 20 24 28 32 36 40
multiplying, vnder the article 5 10 15 20 25 30 35 40 45 50
of his denominacioun. 6 12 18 24 30 36 42 48 56 60
Example of grace. If thow wolt 7 14 21 28 35 42 49 56 63 70
wete how moche is .4. tymes 8 16 24 32 40 48 56 64 72 80
.8., 11se how many vnytees ben
9 18 27 36 45 54 63 72 81 90
betwix .8. 12 and .10. to-geder
10 20 30 40 50 60 70 80 90 100
rekenede, and it shewith that
.2.: withdraw ther-for the (1) Digit b
quaternary, of the article of his denominacion twies, of .40., And digit.
ther remaynethe .32., that is, to some of alle the multiplicacioun. See the ta
above.
Wher-vpon for more evidence and declaracion the seide table is
made. Whan a digit multipliethe an article, thow most bryng the (2) Digit b
article.
digit into þe digit, of þe whiche the article [has] 13 his name, and
euery vnyte shalle stonde for .10., and euery article an .100. (3) Comp
by digit.
Whan the digit multipliethe a nombre componede, þou most
bryng the digit into aiþer part of the nombre componede, so þat

digit be had into digit by the first rule, into an article by þe
seconde rule; and afterwarde Ioyne the produccioun, and þere
wol be the some totalle.
Resultans 1 2 6 7 3 6 1 2 0 1 2 0 8
Multiplicandus 2 3 2 6 4
Multiplicans 6 3 2 3 2 0 3 0 2

Whan an article multipliethe an article, the digit wherof he is (4) Article
article.
namede is to be brought Into the digit wherof the oþer is
namede, and euery vnyte wol be worthe Fol. 52. *an .100., and
euery article. a .1000. Whan an article multipliethe a nombre (5) Comp
by article.
componede, thow most bryng the digit of the article into aither
part of the nombre componede; and Ioyne the produccioun, and
euery article wol be worthe .100., and euery vnyte .10., and so
wolle the some be opene. Whan a nombre componede (6) Comp
by compo
multipliethe a nombre componede, euery part of the nombre
multiplying is to be hade into euery part of the nombre to be
multipliede, and so shalle the digit 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. Therfor yf thow wilt any nombre
by hym-self other by any other multiplie, write the nombre to be
multipliede in the ouer ordre by his differences, The nombre How to se
your numb
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 wolle grow a digit, an
article, other a nombre componede. If the resu
digit,
If it be a digit, even above the figure
multiplying is hede write his digit that The resultant 6
come of, as it apperethe here:— To be multipliede 3

And yf an article had be writ ouer the Þe nombre multipliyng 2
figure multiplying his hede, put a cifre an article,
þer and transferre the article towarde the lift hande, as thus:—
And yf a nombre componede be writ The resultant 1 0
ouer the figure multyplying is hede, to be multipliede 5
write the digit in the nombre þe nombre multipliyng 2

componede is place, and sette the article to the lift hande, as or a comp
thus:—
This done, me most bryng the last save Multiply n
Resultant 1 2
the last bu
to be multipliede 4 one of the multipliyng into the last of and so on
the nombre multipliyng 3 þe nombre to be multipliede, and se
what comythe therof 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 multipliede, wherof growithe oþer a digit, an
article, Fol. 52 b. *other a nombre componede.
If it be a digit, In the place of the
Resultant 6 6
ouerer, sette a-side, as here:
to be multipliede 3
If an article happe, there put a cifre in the nombre multipliyng 2 2
his place, and put hym to the lift hande,
as here:
The resultant 1 1 0
If it be a nombre componede, in the to be multipliede 5
place of the ouerer sette a-side, write þe nombre multiplying 2 2
a digit that 14 is a part of the
componede, and sette on the left
honde the article, as here: The resultant 1 315 2
to be multipliede 4
That done, sette forwarde the figures
þe nombre multipliant 3 3
of the nombre multiplying by oo
difference, so that the first of the Then ante
multipliant be vnder the last save one of the nombre to be multiplier
place.
multipliede, the other by o place sette forwarde. Than me shalle
brynge the last of the multipliant in hym to be multipliede, vnder
the whiche is the first multipliant. And than wolle growe oþer a Work as b
digit, an article, or a componede nombre. If it be a digit, adde
hym even above his hede; If it be an article, transferre hym to the
lift side; And if it be a nombre componede, adde a digit to the
figure above his hede, and sette to the lift hande the article. And
alle-wayes euery figure of the nombre multipliant is to be
brought to the last save one nombre to be multipliede, til me
come to the first of the multipliant, where me shalle wirche as it
is seide before of the first, and afterwarde to put forwarde the
figures by o difference and one tille they alle be multipliede. And How to de
ciphers.

yf it happe that the first figure of þe multipliant be a cifre, and
boue it is sette the figure signyficatife, write a cifre in the place
of the figure sette a-side, as thus, etc.:
The resultant 1 2 0
to be multipliede 6
the multipliant 2 0

And yf a cifre happe in the lower order be-twix the first and the How to de
ciphers.
last, and even above be sette the figure signyficatif,
leve it vntouchede, as here:—
The resultant 2 2 6 4 4
And yf the space above sette be To be multipliede 2 2 2
voide, in that place write thow a cifre. The multipliant 1 0 2
And yf the cifre happe betwix þe first
and the last to be multipliede, me
most sette forwarde the ordre of the figures by thaire differences,
for oft of duccioun of figures in cifres nought is the resultant, as
here,
Fol. 53. *wherof it is evident and open,
Resultant 8 0 0 8
yf that the first figure of the nombre
be to be multipliede be a cifre, vndir it to be multipliede 4 0 0 4
the multipliant 2 . . .
shalle be none sette as here:—
Resultant 3 2 016Vnder[stand] Leave roo
between th
To be multipliede 8 0
also that in multiplicacioun, divisioun, rows of fi
The multipliant 4
and of rootis the extraccioun,
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 ouer-hippede and sette out of mynde.

er VII. Division.

F
or to dyvyde oo nombre by a-nother, it is of .2. nombres Definition
division.
proposede, It is forto depart the moder nombre into as
many partis as ben of vnytees in the lasse nombre. And
note wele that in makynge of dyvysioun ther ben .3. nombres
necessary: that is to sey, the nombre to be dyvydede; the nombre Dividend,
Divisor,
dyvydyng and the nombre exeant, other how oft, or quocient. Ay Quotient.

shalle the nombre that is to be dyvydede be more, other at the
lest evene with the nombre the dyvysere, yf the nombre shalle be
made by hole nombres. Therfor yf thow wolt any nombre How to se
your Sum
dyvyde, write the nombre to be dyvydede in þe ouerer bordure
by his differences, the dyvisere in the lower ordure by his
differences, so that the last of the dyviser be vnder the last of the
nombre to be dyvyde, the next last vnder the next last, and so of
the others, yf it may competently be done; as here:— An examp

The residue 2 7
The quotient 5
To be dyvydede 3 4 2
The dyvyser 6 3

Residuum 8 2 7 2 6 Examples
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

And ther ben .2. causes whan the last figure may not be sette When the
the diviso
vnder the last, other that the last of the lower nombre may not be not be set
with-draw of the last of the ouerer nombre for it is lasse than the the last of
lower, other how be it, that it myght be with-draw as for hym- dividend.
self fro the ouerer the remenaunt may not so oft of them above,
other yf þe last of the lower be even to the figure above his hede,
and þe next last oþer the figure be-fore þat be more þan the
2
figure above sette. Fol. 53 . *These so ordeynede, me most wirche
from the last figure of þe nombre of the dyvyser, and se how oft
it may be with-draw of and fro the figure aboue his hede, namly How to be
so that the remenaunt may be take of so oft, and to se the residue
as here:—
And note wele that me may not withe-draw An examp
The residue 2 6
The quocient 9 more than .9. tymes nether lasse than ones.
To be dyvydede 3 3 2 Therfor se how oft þe figures of the lower
The dyvyser 3 4
ordre may be with-draw fro the figures of
the ouerer, and the nombre that shewith þe
quocient most be writ ouer the hede of þat
figure, vnder the whiche the first figure is, of the dyviser; And

by that figure me most withe-draw alle oþer figures of the lower Where to
quotiente
ordir and that of the figures aboue thaire hedis. This so done, me
most sette forwarde þe figures of the diuiser by o difference
towardes the right honde and worche as before; and thus:— Examples

Residuum . 1 2
quociens 6 5 4 2 0 0 4
Diuidendus 3 5 5 1 2 2 8 8 6 3 7 0 4
Diuisor 5 4 3 4 4 2 3

The quocient 6 5 4
To be dyvydede 3 5 5 1 2 2
The dyvyser 5 4 3

And yf it happe after þe settyng forwarde of the figures þat þe A special
last of the divisor may not so oft be withdraw of the figure above
his hede, above þat figure vnder the whiche the first of the
diuiser is writ me most sette a cifre in ordre of the nombre
quocient, and sette the figures forwarde as be-fore be o
difference alone, and so me shalle do in alle nombres to be
dyvidede, for where the dyviser may not be with-draw me most
sette there a cifre, and sette forwarde the figures; as here:—
And me shalle not cesse fro suche Another
The residue 1 2
example.
The quocient 2 0 0 4 settyng of figures forwarde, nether
To be dyvydede 8 8 6 3 7 0 4 of settynge of þe quocient into the
The dyvyser 4 4 2 3
dyviser, neþer of subtraccioun of
the dyvyser, tille the first of the
dyvyser be with-draw fro þe first
to be dividede. The whiche done, or ought, 17 oþer nought shalle
remayne: and yf it be ought, 17 kepe it in the tables, And euer vny
it to þe diviser. And yf þou wilt wete how many vnytees of þe
3 What the
divisioun Fol. 53 . *wol growe to the nombre of the divisere, the
quotient s
nombre quocient wol shewe it: and whan suche divisioun is
made, and þou lust prove yf thow have wele done or no, How to pr
your divis
Multiplie the quocient by the diviser, And the same figures wolle
come ayene that thow haddest bifore and none other. And yf
ought be residue, than with addicioun therof shalle come the
same figures: And so multiplicacioun provithe divisioun, and

dyvisioun multiplicacioun: as thus, yf multiplicacioun be made, or multipl
divide it by the multipliant, and the nombre quocient wol shewe
the nombre that was to be multipliede, etc.

er VIII. Progression.

P
rogressioun is of nombre after egalle excesse fro oone or Definition
Progressio
tweyne take agregacioun. of progressioun one is naturelle
or contynuelle, þat oþer broken and discontynuelle. Natural
Progressio
Naturelle it is, whan me begynnethe with one, and kepethe
ordure ouerlepyng one; as .1. 2. 3. 4. 5. 6., etc., so þat the
nombre folowynge passithe the other be-fore in one. Broken it is, Broken
Progressio
whan me lepithe fro o nombre tille another, and kepithe not the
contynuel ordire; as 1. 3. 5. 7. 9, etc. Ay me may begynne with
.2., as þus; .2. 4. 6. 8., etc., and the nombre folowyng passethe
the others by-fore by .2. And note wele, that naturelle
progressioun ay begynnethe with one, and Intercise or broken
progressioun, omwhile begynnythe with one, omwhile with
twayne. Of progressioun naturell .2. rules ther be yove, of the
whiche the first is this; whan the progressioun naturelle endithe The 1st ru
Natural
in even nombre, by the half therof multiplie þe next totalle Progressio
ouerere nombre; Example of grace: .1. 2. 3. 4. Multiplie .5. by
.2. and so .10. comethe of, that is the totalle nombre þerof. The The secon
seconde rule is suche, whan the progressioun naturelle endithe in
nombre ode. Take the more porcioun of the oddes, and multiplie
therby the totalle nombre. Example of grace 1. 2. 3. 4. 5.,
multiplie .5. by .3, and thryes .5. shalle be resultant. so the
nombre totalle is .15. Of progresioun intercise, ther ben also .2. 18 The first r
Broken
rules; and þe first is þis: Whan the Intercise progression endithe Progressio
in even nombre by half therof multiplie the next nombre to þat
halfe as .2. 18 4. 6. Multiplie .4. by .3. so þat is thryes .4., and .12.
the nombre of alle the progressioun, wolle folow. The seconde The secon
rule is this: whan the progressioun interscise endithe in ode, take
4
þe more porcioun of alle þe nombre, Fol. 53 . *and multiplie by
hym-selfe; as .1. 3. 5. Multiplie .3. by hym-selfe, and þe some of
alle wolle be .9., etc.

er IX. Extraction of Roots.

H
ere folowithe the extraccioun of rotis, and first in nombre The pream
the extrac
quadrates. Wherfor me shalle se what is a nombre roots.
quadrat, and what is the rote of a nombre quadrat, and
what it is to draw out the rote of a nombre. And before other note
this divisioun: Of nombres one is lyneal, anoþer superficialle, Linear,
superficia
anoþer quadrat, anoþer cubike or hoole. lyneal is that þat is solid num
considrede after the processe, havynge no respect to the
direccioun of nombre in nombre, As a lyne hathe but one
dymensioun that is to sey after the lengthe. Nombre superficial is Superficia
numbers.
þat comethe of ledynge of oo nombre into a-nother, wherfor it is
callede superficial, for it hathe .2. nombres notyng or mesurynge
hym, as a superficialle thynge hathe .2. dimensions, þat is to sey
lengthe and brede. And for bycause a nombre may be hade in a- Square nu
nother by .2. maners, þat is to sey other in hym-selfe, oþer in
anoþer, Vnderstonde yf it be had in hym-self, It is a quadrat. ffor
dyvisioun write by vnytes, hathe .4. sides even as a quadrangille.
and yf the nombre be hade in a-noþer, the nombre is superficiel
and not quadrat, as .2. hade in .3. makethe .6. that is þe first
nombre superficielle; wherfor it is open þat alle nombre quadrat
is superficiel, and not conuertide. The rote of a nombre quadrat The root o
square nu
is þat nombre that is had of hym-self, as twies .2. makithe 4. and
.4. is the first nombre quadrat, and 2. is his rote. 9. 8. 7. 6. 5. 4. 3.
2. 1. / The rote of the more quadrat .3. 1. 4. 2. 6. The most Notes of s
examples
nombre quadrat 9. 8. 7. 5. 9. 3. 4. 7. 6. / the remenent ouer the square roo
quadrat .6. 0. 8. 4. 5. / The first caas of nombre quadrat .5. 4. 7. interpolate
5. 6. The rote .2. 3. 4. The seconde caas .3. 8. 4. 5. The rote .6. 2.
The thirde caas .2. 8. 1. 9. The rote .5. 3. The .4. caas .3. 2. 1.
The rote .1. 7. / The 5. caas .9. 1. 2. 0. 4. / The rote 3. 0. 2. The Solid num
solide nombre or cubike is þat þat comytħe of double ledyng of
nombre in nombre; And it is clepede a solide body that hathe Three
dimension
þer-in .3 [dimensions] þat is to sey, lengthe, brede, and solids.
thiknesse. so þat nombre hathe .3. nombres to be brought forthe
in hym. But nombre may be hade twies in nombre, for other it is
hade in hym-selfe, oþer in a-noþer. If a nombre be hade twies in Cubic num
hym-self, oþer ones in his quadrat, þat is the same, þat a cubike
Fol. 54. *is, And is the same that is solide. And yf a nombre twies
be hade in a-noþer, the nombre is clepede solide and not cubike,
as twies .3. and þat .2. makithe .12. Wherfor it is opyne that alle

cubike nombre is solide, and not conuertide. Cubike is þat All cubics
solid num
nombre þat comythe of ledynge of hym-selfe twyes, or ones in
his quadrat. And here-by it is open that o nombre is the roote of a
quadrat and of a cubike. Natheles the same nombre is not
quadrat and cubike. Opyne it is also that alle nombres may be a No numbe
be both lin
rote to a quadrat and cubike, but not alle nombre quadrat or and solid.
cubike. Therfor sithen þe ledynge of vnyte in hym-self ones or
twies nought comethe but vnytes, Seithe Boice in Arsemetrike, Unity is n
number.
that vnyte potencially is al nombre, and none in act. And
vndirstonde wele also that betwix euery .2. quadrates ther is a
meene proporcionalle, That is openede thus; lede the rote of o Examples
square roo
quadrat into the rote of the oþer quadrat, and þan wolle þe meene
shew.
Residuum 0 4 0 0
Quadrande 4 3 5 6 3 0 2 9 1 7 4 2 4 1 9 3 6
Duplum 1 2 1 0 2 6 [8] 19
Subduplum 6 6 5 5 1 3 2 4 4

Also betwix the next .2. cubikis, me may fynde a double meene, A note on
proportion
that is to sey a more meene and a lesse. The more meene 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 of
the lesse.

er X. Extraction of Square Root.

T
o 20 draw a rote of the nombre quadrat it is What-euer
nombre be proposede to fynde his rote and to se yf it be
quadrat. And yf it be not quadrat the rote of the most To find a
root.
quadrat fynde out, vnder 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 Begin wit
last odd p
begynne worche vnder the last save one. And yf it be ode with
the last; and forto sey it shortly, al-weyes fro the last ode me
shalle begynne. Therfor vnder the last in an od place sette, me Find the n
square roo
most fynde a digit, the whiche lade in hym-selfe it puttithe away that numb
that, þat is ouer his hede, oþer as neighe as me may: suche a subtract,

digit founde and withdraw 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 me double it,
most fynde a-noþer digit vnder the next figure bifore the
doublede, the whiche Fol. 54 b. *brought in double settethe a-way and set the
double on
alle that is ouer his hede as to rewarde of the doublede: Than right.
brought into hym-self settithe all away in respect of hym-self,
Other do it as nye as it may be do: other me may with-draw the Find the s
figure by
digit 21[last] founde, and lede hym in double or double hym, and division.
after in hym-selfe; Than Ioyne to-geder the produccione of them
bothe, So that the first figure of the last product be addede before Multiply t
double by
the first of the first productes, the seconde of the first, etc. and so second fig
forthe, subtrahe fro the totalle nombre in respect of þe digit. and add af
the square
The residue 5 4 3 2 second fig
and subtra
To be quadrede 4 1 2 0 9 1 5 1 3 9 9 0 0 5 4 3 2
Examples
The double 4 0 2 4 6 0 0
The vnder double 2 0 3 1 2 3 [3] [0] [0] 0

And if it hap þat no digit may be founde, Than sette a cifre vndre
a cifre, and cesse not tille thow fynde a digit; and whan thow
hast founde it to double it, neþer to sette the doublede forwarde
nether the vnder doublede, Till thow fynde vndre the first figure Special ca
a digit, the whiche lade in alle double, settyng away alle that is
ouer hym in respect of the doublede: Than lede hym into hym-
selfe, and put a-way alle in regarde of hym, other as nyghe as
thow maist. That done, other ought or nought wolle be the The residu
residue. If nought, than it shewithe that a nombre componede
was the quadrat, and his rote a digit last founde with vndere-
double other vndirdoubles, so that it be sette be-fore: And yf
ought 22 remayne, that shewith that the nombre proposede was
not quadrat, 23 but a digit [last found with the subduple or
subduples is]
This table
1 2 3 4 5 6 7 8 9 constructe
use in cub
2 8 12 16 20 24 28 32 36
sums, givi
3 18 27 36 45 54 63 72 81
value of a
4 32 48 64 80 96 11224 128 144

5 50 75 100 125 150 175 200 225
6 72 108 144 180 216 252 288 324
7 98 147 196 245 294 343 393 441
8 128 192 256 320 384 448 512 576
9 168 243 324 405 486 567 648 72925

The rote of the most quadrat conteynede vndre the nombre
proposede. Therfor yf thow wilt prove yf thow have wele do or How to pr
square roo
no, Multiplie the digit last founde with the vnder-double oþer without or
vnder-doublis, and thow shalt fynde the same figures that thow remainder
haddest before; And so that nought be the Fol. 55. *residue. And yf
thow have any residue, than with the addicioun þerof that is
reseruede with-out in thy table, thow shalt fynde thi first figures
as thow haddest them before, etc.

er XI. Extraction of Cube Root.

H
eere folowithe the extraccioun of rotis in cubike nombres; Definition
cubic num
wher-for me most se what is a nombre cubike, and what is and a cube
his roote, And what is the extraccioun of a rote. A nombre
cubike it is, as it is before declarede, that comethe of ledyng of
any nombre twies in hym-selfe, other ones in his quadrat. The
rote of a nombre cubike is the nombre that is twies hade in hym-
selfe, or ones in his quadrat. Wher-thurghe it is open, that euery
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
þe nombre proposede yf it be a cubike; And yf it be not, than
thow most make extraccioun of his rote of the most cubike vndre
the nombre proposide his rote founde. Therfor proposede some
nombre, whos cubical rote þou woldest draw out; First thow Mark off t
places in t
most compt the figures by fourthes, that is to sey in the place of
thousandes; And vnder the last thousande place, thow most Find the f
digit;
fynde a digit, the whiche lade in hym-self cubikly puttithe a-way
that þat is ouer his hede as in respect of hym, other as nyghe as
thow maist. That done, thow most trebille the digit, and that treble it an
place it un
triplat is to be put vnder the .3. next figure towarde the right next but o
honde, And the vnder-trebille vnder the trebille; Than me most multiply b
fynde a digit vndre the next figure bifore the triplat, the whiche digit.
with his vnder-trebille had into a trebille, afterwarde other

vnder[trebille] 26 had in his produccioun, puttethe a-way alle that Then find
second dig
is ouer it in regarde of 27 [the triplat. Then lade in hymself
puttithe away that þat is over his hede as in respect of hym, other
as nyghe as thou maist:] That done, thow most trebille the digit Multiply t
triplate an
ayene, and the triplat is to be sette vnder the next .3. figure as second dig
before, And the vnder-trebille vnder the trebille: and than most twice by t
thow sette forwarde the first triplat with his vndre-trebille by .2. digit.
differences. And than most thow fynde a digit vnder the next
figure before the triplat, the whiche withe his vnder-triplat had in
his triplat afterwarde, other vnder-treblis lad in product Fol. 55 b. Subtract.
*It sittethe a-way ałł that is ouer his hede in respect of the triplat
than had in hym-self cubikly, 28 or as nyghe as ye may.
Examples
Residuum 5 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 6 0 1 8 4
Subtriplum 2 0 [3] 6 7 2 2

Nother me shalle not cesse of the fyndynge of that digit, neither Continue
process til
of his triplacioun, neþer of the triplat-is 29anterioracioun, that is first figure
to sey, settyng forwarde by .2. differences, Ne therof the vndre- reached.
triple to be put vndre the triple, Nether of the multiplicacioun
þerof, Neither of the subtraccioun, tille it come to the first figure,
vnder the whiche is a digitalle nombre to be founde, the whiche
withe his vndre-treblis most be hade in tribles, After-warde
without vnder-treblis to be hade into produccioun, settyng away
alle that is ouer the hede of the triplat nombre, After had into
hymselfe cubikly, and sette alle-way that is ouer hym. Examples
Also note wele that the
To be cubicede 1 7 2 8 3 2 7 6 8
produccion comynge of the
The triple 3 2 9
ledyng of a digite founde 30
me may adde to, and also The vnder triple 1 2 [3] 3 3

with-draw fro of the totalle
nombre sette above that digit so founde. 31 That done ought or The residu
nought most be the residue. If it be nought, It is open that the
nombre proposede was a cubike nombre, And his rote a digit
founde last with the vnder-triples: If the rote therof wex bade in
hym-selfe, and afterwarde product they shalle make the first

figures. And yf ought be in residue, kepe that without in the
table; and it is opene that the nombre was not a cubike. but a
digit last founde with the vndirtriplis is rote of the most cubike
vndre the nombre proposede conteynede, the whiche rote yf it be
hade in hym-selfe, And afterwarde in a product of that shalle Special ca
growe the most cubike vndre the nombre proposede conteynede,
And yf that be addede to a cubike the residue reseruede in the
table, wolle make the same figures that ye hade first. Fol. 56. *And Special ca
yf no digit after the anterioracioun 32 may not be founde, than put
there a cifre vndre a cifre vndir the thirde figure, And put
forwarde þe figures. Note also wele that yf in the nombre
proposede ther ben no place of thowsandes, me most begynne
vnder the first figure in the extraccioun of the rote. some vsen
forto distingue the nombre by threes, and ay begynne forto
wirche vndre the first of the last ternary other uncomplete
nombre, the whiche maner of operacioun accordethe with that
before.
Examples
The residue 0 1 1
The cubicandus 8 0 0 0 0 0 0 8 2 4 2 4 1 9
The triple 33 0 0 6
The vndertriple [2] 0 0 2 6 2

And this at this tyme suffisethe in extraccioun of nombres
quadrat or cubikes etc.

of Numbers, &c.
A table of
1 2 3 4 5 numbers;
one. x. an. hundrede / a thowsande / x. thowsande / probably f
the Abacu
6 7
An hundrede thowsande / A thowsande tymes a thowsande /

x. thousande tymes a thousande / An hundrede thousande tymes
a thousande A thousande thousande tymes a thousande / this is

the x place etc.

[Ende.]

1. MS. Materialle.
2. MS. Formalle.
3. ‘the’ in MS.
4. ‘be’ in MS.
5. ‘and’ in MS.
6. ‘is’ in MS.
7. 6 in MS.
8. 0 in MS.
9. 2 in MS.
10. sic.
11. ‘And’ inserted in MS.
12. ‘4 the’ inserted in MS.
13. ‘to’ in MS.
14. ‘that’ repeated in MS.
15. ‘1’ in MS.
16. Blank in MS.
17. ‘nought’ in MS.
18. 3 written for 2 in MS.
19. 7 in MS.
20. runs on in MS.
21. ‘so’ in MS.
22. ‘nought’ in MS.
23. MS. adds here: ‘wher-vpone se the table in the next side
of the next leefe.’
24. 110 in MS.
25. 0 in MS.
26. double in MS.
27. ‘it hym-selfe’ in MS.
28. MS. adds here: ‘it settethe a-way alle his respect.’
29. ‘aucterioracioun’ in MS.

30. MS. adds here: ’with an vndre-triple / other of an vndre-
triple in a triple or triplat is And after-warde with out vndre-
triple other vndre-triplis in the product and ayene that
product that comethe of the ledynge of a digit founde in
hym-selfe cubicalle’ /
31. MS. adds here: ‘as ther had be a divisioun made as it is
openede before.’
32. MS. anteriocacioun.
33. 4 in MS.

The original text was printed as a single continuous paragraph,
with no break between speakers; all examples were shown inline.
It has been broken up for this e-text.

116 b.

* ¶ The seconde dialoge of accomptynge by counters.
Mayster.

N
owe that you haue learned the commen kyndes of
Arithmetyke with the penne, you shall se the same art in
counters: whiche feate doth not only serue for them that
can not write and rede, but also for them that can do bothe, but
haue not at some tymes theyr penne or tables redye with them.
This sorte is in two fourmes commenly. The one by lynes, and
t
the other without lynes: in that y hath lynes, the lynes do stande
t
for the order of places: and in y that hath no lynes, there must be
sette in theyr stede so many counters as shall nede, for eche lyne
one, and they shall supplye the stede of the lynes.
S. By examples I shuld better perceaue your meanynge.
M. For example of the 117 a ly*nes:
Lo here you se .vi. lynes whiche stande 1 0 0 0 0 0
1 0 0 0 0
for syxe places so that the nethermost X 1 0 0 0
e 1 0 0
standeth for y fyrst place, and the next 1 0
1
aboue it, for the second: and so vpward
tyll you come to the hyghest, which is the syxte lyne, and
standeth for the syxte place. Now what is the valewe of euery Numeratio
place or lyne, 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 ones, and euery counter set in that lyne
betokeneth but one: and the seconde lyne is the place of 10, for
euery counter there, standeth for 10. The thyrd lyne the place of
hundredes: the fourth of thousandes: and so forth.
S. Syr I do perceaue that the same order is here of lynes, as was
in the other figures 117 b. *by places, so that you shall not nede
longer to stande about Numeration, excepte there be any other
difference.
M. Yf you do vnderstande it, then how wyll you set 1543?
S. Thus, as I suppose. X 1
5
e 4
M. You haue set y places truely, but your figures be 3
not mete for this vse: for the metest figure in this
behalfe, is the figure of a counter round, as you se here, where I
haue expressed that same summe.
S. So that you haue not one figure for 2, nor 3, × o
nor 4, and so forth, but as many digettes as you
haue, you set in the lowest lyne: and for euery
°
o o o o
10 you set one in the second line: and so of o o o

other. But I know not by what reason you set that one counter for
500 betwene two lynes.
M. you shall remember this, that when so euer you nede to set
downe 5, 50, or 500, or 5000, or so forth any other nomber,
whose numerator 118 a. *is 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, 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 sorte, 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
nombers aboue 5. And this is true bothe of digettes and articles.
And for example I wyll set downe this summe 287965,
which summe yf you marke well, you nede none X
other examples for to lerne the numeration of 118 o o

b. *this forme. But this shal you marke, that as °o
o o
you dyd in the other kynde of arithmetike, set a
X o o°
pricke in the places of thousandes, in this worke
you shall sette a starre, as you se here. o o° o o

o°

on on the Counting Board.
S. Then I perceave numeration, but I praye you, howe shall I do Addition.
in this arte to adde two summes or more together?
M. The easyest way in this arte is, to adde but 2 summes at ones
together: how be it you may adde more, as I wyll tell you anone.
Therfore when you wyll adde two summes, 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 set downe the other
summe, so that that lyne may be betwene them, as yf you wolde
adde 2659 to 8342, you must set your summes as you se here.
And then yf you lyst, you 119 a. *may
o° o o
adde the one to the other in the same X o o
place, or els you may adde them both
o o o o°
together in a newe place: which waye,
bycause it is moste playnest, I wyll o o o o °
showe you fyrst. Therfore wyl I o o o° o o o
begynne at the vnites, whiche in the
e
fyrst summe is but 2, and in y second summe 9, that maketh 11,
those do I take vp, and for them I set 11 in the new roume, thus,
e
Then do I take vp all y articles vnder a
hundred, which in the fyrst summe are X o° o o o o

40, and in the second summe 50, that o o o o°
maketh 90: or you may saye better, that
in the fyrste summe there are 4 articles
o o o o ° o
o
of 10, and in the seconde summe 5,
which make 9, but then take hede that you sette them in theyr 119
b. *ryght lynes as you se here.
Where I haue taken awaye 40
o° o o
from the fyrste summe, and 50 X o o
e
from y second, and in theyr stede

I haue set 90 in the thyrde,
t
o o o °
o

whiche I haue set playnely y you °o o o o
o
myght well perceaue it: how be it o
seynge that 90 with the 10 that
e
was in y thyrd roume all redy, doth make 100, I myghte better
for those 6 counters set 1 in the thyrde lyne, thus:
For it is all one summe as you may se, but it is beste,
X
o neuer to set 5 counters in any line, for that may be done
o
with 1 counter in a hygher place.
S. I iudge that good reason, for many are vnnedefull,
where one wyll serue.
M. Well, then 120 a. *wyll I adde forth of hundredes: I fynde 3 in
the fyrste summe, and 6 in the seconde, whiche make 900, them
do I take vp and set in the thyrd roume where is one hundred all
redy, to whiche I put 900, and it wyll be 1000, therfore I set one
counter in the fourth lyne for them all, as you se here.
e
Then adde I y thousandes together, whiche
in the fyrst summe are 8000, and in y
e X °o o
o o o o

second 2000, that maketh 10000: them do I o
take vp from those two places, and for
them I set one counter in the fyfte lyne, and then appereth as
you se,
o to be 11001, for so many doth amount of the addition of
X o
8342 to 2659.
o 120 b. *S. Syr, this I do perceave: but how shall I set one
summe to an other, not 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 I begynne with the smalest, which in the fyrst summe is
5, that do I take vp, and wold put to the other 5 in the seconde
summe, sauynge that two counters can not be set in a voyd place
of 5, but for them bothe I must set 1 in the seconde lyne, which is
the place of 10, therfore I take vp the 5 of the fyrst summe, and

the 5 of the seconde, and for them I set
1 in the second lyne, 121 a. *as you se o °
here.
Then do I lyke wayes take vp the 4
X o o o °
o o o o o o
counters of the fyrste summe and o o o o o o o
seconde lyne (which make 40) and adde ° o °
them to the 4 counters of the same lyne,
in the second summe, and it maketh 80,
But as I sayde I maye not conueniently o°

set aboue 4 counters in one lyne, X o o o °
therfore to those 4 that I toke vp in the o o o o o o
o o o o o o o o
fyrst summe, I take one also of the o
seconde summe, and then haue I taken
vp 50, for whiche 5 counters I sette downe one in the space ouer
e
y second lyne, as here doth appere.
t
121 b *and then is there 80, as well w those
e
4 counters, as yf I had set downe y other
o°
4 also. Now do I take the 200 in the fyrste X o o o °
summe, and adde them to the 400 in the o o o o o o

seconde summe, and it maketh 600, o° o o

therfore I take vp the 2 counters in the o

fyrste summe, and 3 of them in the seconde summe, and for them
e
5 I set 1 in y space aboue, thus.
e e
Then I take y 3000 in y fyrste summe, vnto
whiche there are none in the second summe o°

agreynge, therfore I do onely remoue those X o o o °
3 counters from the fyrste summe into the
o°
seconde, as here doth appere.
122 a *And so you see the hole o° o o

o°
summe, that amounteth of the o

addytion of 65436 with 3245
X o° o o
to be 6868[1]. And yf you haue marked these two
o ° examples well, you nede no farther enstruction in
Addition of 2 only summes: but yf you haue more
o° o o
o
then two summes to adde, you may adde them
thus.

Fyrst adde two of them, and
e
then adde the thyrde, and y X o o o o o o o°
fourth, or more yf there be so o° o o o° o o o
many: as yf I wolde adde
2679 with 4286 and 1391. o° o o°o o o°

Fyrste I adde the two fyrste o° o o o o° °
summes thus. 122 b. *And
then I adde the thyrde thereto thus.
And so of more yf you haue
them. X o o° o o o°

o o o o° o o o o o o

o°o o o °o °
o ° o°

action on the Counting Board.
S. Nowe I thynke beste that you passe forth to Subtraction,
except there be any wayes to examyn this maner of Addition,
then I thynke that were good to be knowen nexte.
M. There is the same profe here that is in the other Addition by Subtractio
the penne, I meane Subtraction, for that onely is a sure waye: but
consyderynge that Subtraction must be fyrste knowen, I wyl
fyrste teache you the arte of Subtraction, and that by this
example: I wolde subtracte 2892 out of 8746. These summes
must I set downe as I dyd in Addition: but here it is best 116 a (sic).
*to set the lesser nomber fyrste, thus.
Then shall I begynne to subtracte the
o o° o
greatest nombres fyrste (contrary to the X o o
t
vse of the penne) y is the thousandes in o° o o o° o
this example: therfore I fynd amongest
o° o o o o o o o
the thousandes 2, for which I withdrawe
so many from the seconde summe o o o°
(where are 8) and so remayneth there 6,
as this example showeth.
Then do I lyke wayes with the
hundredes, of whiche in the fyrste o °
summe 116 b. *I fynde 8, and is the

o° o o o° o
seconde summe but 7, out of whiche I +
can not take 8, therfore thus muste I do:
o° o o o o o o o
I muste loke how moche my summe
dyffereth from 10, whiche I fynde here o o o°
to be 2, then must I bate for my summe
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 from the fyrste summe I take that 800, and from the
second summe where are 6000, I take vp one thousande, and leue
t
5000; but then set I downe the 200 unto the 700 y are there all
redye, and make them 900 thus.
Then come I to the articles of tennes
where in the fyrste summe I fynde 90, + °
117 a. *and in the seconde summe but
o° o o o
only 40: Now consyderyng that 90 can
not be bated from 40, I loke how moche o o° o o o o o o
t
y 90 doth dyffer from the next summe o o o°
aboue it, that is 100 (or elles whiche is
all to one effecte, I loke how moch 9 doth dyffer from 10) and I
fynd it to be 1, then in the stede of that 90, I do take from the
second summe 100: but consyderynge that it is 10 to moche, I set
e
downe 1 in y nexte lyne beneth for it, as you se here.
e
Sauynge that here I haue set one counter in y
e
space in stede of 5 in y nexte lyne. And thus
+ °
haue I subtracted all saue two, which I must o° o o

bate from the 6 in the second summe, and °
there wyll remayne 4, thus.
t o o o°
So y yf I subtracte 2892 from 8746, the
remayner wyll be 5854, 117 b. *And that this is
truely wrought, you maye proue by Addition: for = °
yf you adde to this remayner the same summe that
o° o o
you dyd subtracte, then wyll the formar summe
8746 amount agayne. °
o o o o
S. That wyll I proue: and fyrst I set the summe that
was subtracted, which was 2892, and then the remayner 5854,

thus.
e
Then do I adde fyrst y 2 to 4, whiche
maketh 6, so take I vp 5 of those || o o °
counters, and in theyr stede I sette 1 in o°o o o o °o
the space, as here appereth.
118 a. *Then do I adde the 90 nexte o o° o o °
o o o o o o
aboue to the 50, and it maketh 140,
therfore I take vp those 6 counters, and
for them I sette 1 to the hundredes in y
e || o o °
e
thyrde lyne, and 4 in y second lyne, o° o o o° o o
thus. o° o o o °
Then do I come to the hundredes, of
o°
whiche I fynde 8 in the fyrst summe, and
e
9 in y second, that maketh 1700, therfore
I take vp those 9 counters, and in theyr
|| o o °
stede I sette 1 in the .iiii. lyne, and 1 in o°o o o° o o o
the space nexte beneth, and 2 in the o o o o

thyrde lyne, as you se here. °o
Then is there lefte in the fyrste summe
but only 2000, whiche I shall take vp from
e || o o o°
thence, and set 118 b. *in the same lyne in y
e t o° o
second summe, to y one y is there all redy: o o o o
and then wyll the hole summe appere (as
e
o°
you may wel se) to be 8746, which was y
fyrst grosse summe, and therfore I do perceaue, that I hadde well
subtracted before.
And thus you may se how Subtraction maye be
o o° o
tryed by Addition. X

t
S. I perceaue the same order here w counters, y I
t
o o °
lerned before in figures. o o o o

M. Then let me se howe can you trye Addition by
°o
Subtraction.
S. Fyrste I wyl set forth this example of Addition where I haue
added 2189 to 4988, and the hole summe appereth to be 7177,

119 a. *Nowe to trye whether that
summe be well added or no, || o o o o o o °o
o
I wyll subtract one of the fyrst
two summes from the thyrd, and
o o°o o o o
o° o o o° o o o° o
e
yf I haue well done y remayner
wyll be lyke that other summe. o° o o o o° o o o° o
As for example: I wyll subtracte
the fyrste summe from the thyrde, whiche I set thus in theyr
order.
Then do I subtract 2000 of the fyrste
e
summe from y second summe, and then || o o °o
o

remayneth there 5000 thus. o o

e
Then in the thyrd lyne, I subtract y 100 of
o° o o o° o
the fyrste summe, from the second summe, o° o o o o° o
e
where is onely 100 also, and then in y
thyrde lyne resteth nothyng. Then in the X °
second lyne with his space ouer hym, o o

I fynde 80, which I shuld subtract 119 b. o° o o o° o
*from the other summe, then seyng there
o° o o o o° o
are but only 70 I must take it out of some
hygher summe, which is here only 5000, therfore I take vp 5000,
and seyng that it is to moch by 4920, I sette downe so many in
the seconde roume, whiche with the 70 beynge there all redy do
make 4990, & then the summes doth stande thus.
Yet remayneth there in the fyrst summe
9, to be bated from the second summe, | | o o o o

where in that place of vnities dothe o° o o o

o° o o o
appere only 7, then I muste bate a
hygher summe, that is to saye 10, but
o° o o o o° o
seynge that 10 is more then 9 (which I
shulde abate) by 1, therfore shall I take
vp one counter from the seconde lyne, and set downe the same in
the fyrst 120 a. *or lowest lyne, as you se here.
And so haue I ended this worke, and the summe
e e || o o o o
appereth to be y same, whiche was y seconde
°o o o
o

summe of my addition, and therfore I perceaue, o°o o
I haue wel done.
o° o o
M. To stande longer about this, it is but folye:
excepte that this you maye also vnderstande, that many do
begynne to subtracte with counters, not at the hyghest summe, as
I haue taught you, but at the nethermoste, as they do vse to adde:
and when the summe to be abatyd, in any lyne appeareth greater
then the other, then do they borowe one of the next hygher
roume, as for example: yf they shuld abate 1846 from 2378, they
e
set y summes thus.
120 b. *And fyrste they take 6 whiche is in
the lower lyne, and his space from 8 in || o o o

e
the same roumes, in y second summe,
°o o
o o o o

and yet there remayneth 2 counters in the o o o o o° o

o° o° o o
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 3 of the
second summe can not be, therfore do they bate it from a hygher
roume, that is, from 1000, and bycause that 1000 is to moch 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 1000
in the fourth lyne of the fyrst summe, whiche yf I withdrawe
e
from the seconde summe, then doth all y figures stande in this
order.
So that (as you se) it differeth not greatly whether
you begynne subtraction at the hygher lynes, or at ||

121 a. *the lower. How be it, as some menne lyke the °
o o o
one waye beste, so some lyke the other: therfore o o
you now knowyng bothe, may vse whiche you lyst.

plication by Counters.
But nowe touchynge Multiplication: you shall set your nombers Multiplica
in two roumes, as you dyd in those two other kyndes, but so that
the multiplier be set in the fyrste roume. Then shall you begyn
e
with the hyghest nombers of y seconde roume, and multiply
them fyrst after this sort. Take that ouermost lyne in your fyrst

workynge, as yf it were the lowest lyne, setting on it some
mouable marke, as you lyste, and loke how many counters be in
hym, take them vp, and for them set downe the hole multyplyer,
so many tymes as you toke vp counters, reckenyng, I saye that
lyne for the vnites: and when you haue so done with the hygheest
nomber then come to the nexte lyne beneth, and do euen so with
e
it, and so with y next, tyll you haue done all. And yf there be
e
any nomber in a space, then for it 121 b. *shall you take y
multiplyer 5 tymes, and then must you recken that lyne for the
vnites whiche is nexte beneth that space: or els 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 chaunce your multyplyer be an odde nomber,
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 recken for the lyne of vnities, or 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 example: I wold multiply 1542 by 365, therfore I set
e
y nombers thus.
122 a. *Then fyrste I begynne at the 1000
e e || o
in y hyghest roume, as yf it were y fyrst
place, & I take it vp, settynge downe for it
o o o °
so often (that is ones) the multyplyer, °
o o o o o

which is 365, thus, as you se here: ° o o
where for the one counter
taken vp from the fourth o o o
lyne, I haue sette downe
e °
o
other 6, whiche make y
summe of the multyplyer,
X °
reckenynge that fourth lyne, o o o °
as yf it were the fyrste: °
o o o o o
whiche thyng I haue marked

by the hand set at the
e
° o o

begynnyng of y same,
S. I perceaue this well: for in dede, this summe that you haue set
downe is 365000, for so moche doth amount 122 b. *of 1000,
multiplyed by 365.
M. Well then to go forth, in the nexte space I fynde one counter
which I 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 182, and here I do styll let that fourth
e
place stand, as yf it were y fyrst:
as in this fourme o o o o
you se, where I o ° o°o o
haue set this
multiplycation with
|| ° o° o

e
y other: but for the
o o o °
ease of your o° o o o o

vnderstandynge,
I haue set a lytell
° o o

lyne betwene them: now shulde they both in one summe stand
thus.
123 a. *Howe be it an o o o o o
other fourme to o o o o

multyplye suche || o° o

counters in space is o o o °
this: Fyrst to remoue
o° o o o o
the fynger to the lyne
e
nexte benethe y space, ° o o

e
and then to take vp y
e
counter, and to set downe y multiplyer .v. tymes, as here you se.
o o o

o ° o o o o o o o o o o o o o o o

° °
o ° °
o °
o °
o

X o o o ° ° ° ° °

°
o o o o o

° o o

Which summes yf you do adde together into one summe, you
e t e
shal perceaue that it wyll be y same y appeareth of y other
working before, so that 123 b. *bothe sortes are to one entent, but
as the other is much shorter, so this is playner to reason, for
suche as haue had small exercyse in this arte. Not withstandynge
you maye adde them in your mynde before you sette them
downe, as in this example, you myghte haue sayde 5 tymes 300
is 1500, and 5 tymes 60 is 300, also 5 tymes 5 is 25, whiche all
put together do make 1825, which you maye at one tyme set
downe yf you lyste. But nowe to go forth, I must remoue the
hand to the nexte counters, whiche are in the second lyne, and
there must I take vp those 4 counters, settynge downe for them
my multiplyer 4 tymes, whiche thynge other I maye do at 4
tymes seuerally, or elles I may gather that hole summe in my
mynde fyrste, and then set it downe: as to saye 4 tymes 300 is
t
1200: 4 tymes 60 are 240: and 4 tymes 5 make 20: y is in all
t
1460, y shall I set downe also: as here you se.
124 a. *whiche yf I
ioyne in one summe °
o o o o o
with the formar
X o° o o o o o
nombers, it wyll
appeare thus. o o o ° o°
Then to ende this
o°
multiplycation, I
remoue the fynger to ° o o

the lowest lyne, where are onely 2,
° them do I take vp, and in theyr stede
o° do I set downe twyse 365, that is 730,
o o for which I set 124 b. *one in the space
o o o o
aboue the thyrd lyne for 500, and 2
°
o more in the thyrd lyne with that one
° o o that is there all redye, and the reste in

theyr order, and so haue I ended the hole summe thus.
Wherby you se, that 1542 (which is the
nomber of yeares syth Ch[r]ystes °
incarnation) beyng multyplyed by 365 o°
(which is the nomber of dayes in one o o

yeare) dothe amounte vnto 562830, which o o o °o o
o
e
declareth y nomber of daies sith Chrystes o° o o o

incarnation vnto the ende of 1542 1 yeares.
(besyde 385 dayes and 12 houres for lepe °
yeares).
S. Now wyll I proue by an other example, as this: 40 labourers
e
(after 6 d. y day for eche man) haue wrought 28 dayes, I wold
125 a. *know what theyr wages doth amount vnto: In this case
muste I worke doublely: fyrst I must multyplye the nomber of the
e e
labourers by y wages of a man for one day, so wyll y charge of
one daye amount: then secondarely shall I multyply that charge
of one daye, by the hole nomber of dayes, and so wyll the hole
summe appeare: fyrst therefore I shall set the summes thus.
t
Where in the fyrste space is the multyplyer (y is
o o o o
one dayes wages for one man) and in the second
space is set the nomber of the worke men to be
o°
multyplyed: then saye I, 6 tymes 4 (reckenynge
that second lyne as the lyne of vnites) maketh 24, for whiche
summe I shulde set 2 counters in the thyrde lyne, and 4 in the
seconde, therfore do I set 2 in the thyrde lyne, and let the 4 stand
styll in the seconde lyne, thus.* 125 b.
So apwereth the hole dayes wages to be 240d’.
that is 20 s. Then do I multiply agayn the same o o
summe by the nomber of dayes and fyrste I sette o o o o

the nombers, thus.
Then bycause there are counters in dyuers
o o lynes, I shall begynne with the hyghest, and
o o o o o o
take them vp, settynge for them the
o° o o multyplyer so many tymes, as I toke vp
t e
counters, y is twyse, then wyll y summe
stande thus.

e
Then come I to y seconde lyne, and take vp
those 4 counters, settynge for them the °
multiplyer foure tymes, so wyll the hole summe o°
appeare thus.* 126 a. o o o o
So is the hole wages of 40
o°
workemen, for 28 dayes (after 6d’. eche daye for a
man) 6720d’. that is 560 s. or 28 l’i.
o°o
o o

on on the Counting Board.
M. Now if you wold proue Multiplycation, the surest way is by Diuision.
e
Dyuision: therfore wyll I ouer passe it tyll I haue taught you y
arte of Diuision, whiche you shall worke thus. Fyrste sette downe
the Diuisor for feare of forgettynge, and then set the nomber that
e
shalbe deuided, at y ryghte syde, so farre from the diuisor, that
the quotient may be set betwene them: as for example: Yf 225
shepe cost 45 l’i. what dyd euery shepe cost? To knowe this,
I shulde diuide the hole summe, that is 45 l’i. by 225, but that can
not be, therfore must I fyrste reduce that 45 l’i. into a lesser
denomination, as into shyllynges: then I multiply 45 by 20, and it
is 900, that summe shall I diuide by the nomber of 126 b. *shepe,
whiche is 225, these two nombers therfore I sette thus.
Then begynne I at the hyghest lyne of the
diuident, and seke how often I may haue
the diuisor therin, and that maye I do 4 o o o° o o o

tymes, then say I, 4 tymes 2 are 8, whyche o o
yf I take from 9, there resteth but 1, thus °
And bycause I founde the diuisor 4 tymes
in the diuidente, I haue set (as you se) 4 in o o o
the myddle roume, which 127 a. *is the place o o
of the quotient: but now must I take the reste ° oooo
of the diuisor as often out of the remayner:
therfore come I to the seconde lyne of the diuisor, sayeng 2 foure
tymes make 8, take 8 from 10, and there resteth 2, thus.
Then come I to the lowest nomber,
which is 5, and multyply it 4 tymes, so | | o o
o o o o

is it 20, that take I from 20, and there ° oooo
remayneth nothynge, so that I se my
quotient to be 4, whiche are in valewe shyllynges, for so was the
diuident: and therby I knowe, that yf 225 shepe dyd coste 45 l’i.
euery shepe coste 4 s.
S. This can I do, as you shall perceaue by this example: Yf 160
sowldyars do spende euery moneth 68 l’i. what spendeth eche
man? Fyrst 127 b. *bycause I can not diuide the 68 by 160,
therfore I wyll turne the poundes into pennes by multiplicacion,
so shall there be 16320 d’. Nowe muste I diuide this summe by
the nomber of sowldyars, therfore I set them in order, thus.
Then begyn I at the hyghest place of the o
diuidente, sekynge my diuisor there, || o°
whiche I fynde ones, Therfore set I 1 in the o o o o

o°
nether lyne. o o

M. Not in the 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.* 128 a.
Then seke I agayne in the reste, how often I
may fynde my diuisor, and I se that in the ||
o o o o o
300 I myghte fynde 100 thre tymes, but then o° o o
the 60 wyll not be so often founde in 20,
therfore I take 2 for my quotient: then take I
100 twyse from 300, and there resteth 100, out of whiche with
the 20 (that maketh 120) I may take 60 also twyse, and then
standeth the nombers thus,
128 b. *where I haue sette the quotient 2 in the
lowest lyne: So is euery sowldyars portion ||
o o
102 d’. that is 8 s. 6 d’.
o°
M. But yet bycause you shall perceaue iustly o o

the reason of Diuision, it shall be good that you do set your
diuisor styll agaynst those nombres from whiche you do take it:
e
as by this example I wyll declare. Yf y purchace of 200 acres of
ground dyd coste 290 l’i. what dyd one acre coste? Fyrst wyl I

turne the poundes into pennes, so wyll there be 69600 d’· Then
in settynge downe these nombers I shall do thus.
Fyrst set the diuident on the ryghte
hande as it oughte, and then 129 a. *the o o o°
diuisor on the lefte hande agaynst those
X o° o o o
nombers, from which I entende to take
hym fyrst as here you se, wher I haue o °
set the diuisor two lynes hygher then is
theyr owne place.
S. This is lyke the order of diuision by the penne.
e
M. Truth you say, and nowe must I set y 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 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 from
the diuident, and farther I do set the diuisor one line lower, as
yow se here.
129 b. *And then seke I how often
the diuisor wyll be taken from the || o o o° o o o
nomber agaynste it, whiche wyll be
4 tymes and 1 remaynynge. o o o °o
S. But what yf it chaunce that when
the diuisor is so remoued, it can not be ones taken out of the
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 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, I fynde the diuisor 4 tymes in the diuidente, and 1
remaynynge, for 4 tymes 2 make 8, which I take from 9, and
there resteth 1, as this figure sheweth:
and in the myddle space for the
quotient I set 4 in the seconde lyne, || o o o

whiche is in this worke the place of
e
o o o °
o
o o o o
vnities.* 130 a. Then remoue I y diuisor
to the next lower line, and seke how
often I may haue it in the dyuident, which I may do here 8 tymes
iust, and nothynge remayne, as in this fourme,
where you may se that the hole quotient
is 348 d’, that is 29 s. wherby I knowe ||
o o o o o
that so moche coste the purchace of one o o o o

aker. o° o o

S. Now resteth the profes of Multiplycation, and also of Diuision.
M. Ther best profes are eche 130 b. *one by the other, for
Multyplication is proued by Diuision, and Diuision by
Multiplycation, as in the worke by the penne you learned.
t
S. Yf that be 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 I
suppose.
M. So wyll I do as touchynge hole nomber, and as for broken
nomber, I wyll not trouble your wytte with it, tyll you haue
t
practised this so well, y you be full perfecte, so that you nede not
to doubte in any poynte that I haue taught you, and thenne maye
e
I boldly enstructe you in y arte of fractions or broken nomber,
wherin I 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, procedynge by no grounded reason, but
onely by a receaued 131 a. *fourme, and that dyuersly of dyuers
men: for marchauntes vse one fourme, and auditors an other:

ants’ Casting Counters.
But fyrste for marchauntes fourme marke this example here, Merchants
casting.
in which I haue expressed this summe 198 l’i. 2 19 s.
11 d’. So that you maye se that the lowest lyne o oooo
serueth for pennes, the next aboue for shyllynges, the o ooo o
thyrde for poundes, and the fourth for scores of o

o oooo
poundes. And farther you maye se, that the space o
betwene pennes and shyllynges may receaue but one ooooo

counter (as all other spaces lyke wayes do) and that one standeth
in that place for 6 d’. Lyke wayes betwene the shyllynges and the
poundes, one counter standeth for 10 s. And betwene the poundes
and 20 l’i. one counter standeth for 10 poundes. But besyde those
you maye see at the left syde of shyllynges, that one counter
standeth alone, and betokeneth 5 s. 131 b. *So agaynste the
poundes, that one counter standeth for 5 l’i. And agaynst the 20
poundes, the one counter standeth for 5 score poundes, that is
100 l’i. so that euery syde counter is 5 tymes so moch as one of
them agaynst whiche he standeth. Now for the accompt of Auditors’
casting.
auditors take this example.
e
where I haue expressed y same summe o o o o o o
198 l’i. 19 s. 11 d’. But here you se the ooo ooo ooo ooo
o o oo
e
pennes stande toward y ryght hande, and
the other encreasynge orderly towarde the lefte hande. Agayne
you maye se, that auditours wyll make 2 lynes (yea and more) for
pennes, shyllynges, and all other valewes, yf theyr 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 132 a. *the lefte corner of the rowe it standeth for
e
10, of y same row. But now yf you wold adde other subtracte
e t
after any of both those sortes, yf you marke y order of y other
feate which I taught you, you may easely do the same here
without moch teachynge: for in Addition you must fyrst set
downe one summe and to the same set the other orderly, and lyke
maner yf you haue many: but in Subtraction you must sette
downe fyrst the greatest summe, and from it must you abate that
other euery denomination from his dewe place.
S. I do not doubte but with a lytell practise I shall attayne these
bothe: but how shall I multiply and diuide after these fourmes?
M. You can not duely do none of both by these sortes, therfore in
suche case, you must resort to your other artes.

S. Syr, yet I se not by these sortes how to expresse hundreddes,
yf they excede one hundred, nother yet thousandes.
M. They that vse such accomptes that it excede 200 132 b. *in one
summe, they sette no 5 at the lefte hande of the scores of
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
20000, whiche all I haue expressed in this example,
which is 97869 l’i. 12 s. 9 d’ ob. q. for I had not told
oooo
you before where, nother how you shuld set downe o
farthynges, which (as you se here) must be set in a o o o
o ooo
voyde space sydelynge beneth the pennes: for q one ooo
o oooo
counter: for ob. 2 counters: for ob. q. 3 counters: and o
more there can not be, for 4 farthynges 133 a. *do oo
o
make 1 d’. which must be set in his dewe place. ooo
oo
o

ors’ Casting Counters.
e
And yf you desyre y same summe after audytors maner, lo here
it is.
oo o o o o
ooo oo ooo ooo ooo oo ooo
o o oo
o

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 inuented dyuers and sundry
wayes 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 commoditie of it ryghte worthy
to be well marked. This feate hath ben vsed aboue 2000 yeares at
the leaste, and yet 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 conceaued in the 133 b. *mynde. And fyrst to begynne, yf
you wyll expresse 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.

¶ Here foloweth the table
of the arte of the
hande

e 1
134 b. *In which as you may se 1 is expressed by y lyttle fynger
e
of y lefte hande closely and harde croked.
3
2 is declared by lyke bowynge of the weddynge fynger (whiche 2
is the nexte to the lyttell fynger) together with the lytell fynger.
3 is signified by the myddle fynger bowed in lyke maner, with 3
those other two.
4 is declared by the bowyng of the myddle fynger and the rynge 4
fynger, or weddynge fynger, with the other all stretched forth.
5 is represented by the myddle fynger onely bowed. 5

And 6 by the weddynge fynger only crooked: and this you may 6
marke in these a certayne order. But now 7, 8, and 9, are
expressed with 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 fynger, as is 1, 7
saue that for 1 the fynger is clasped in, harde and 135 a. *rounde,
but for to expresse 7, you shall bowe the myddle ioynte of the
lytell fynger only, and holde the other ioyntes streyght.
S. Yf you wyll geue me leue to expresse it after my rude maner,
thus I vnderstand your meanyng: that 1 is expressed by
crookynge 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 8
lyttell fynger and the rynge fynger.
And yf you bowe lyke wayes with them the myddle fynger, then 9
doth it betoken 9.
Now to expresse 10, you shall bowe your fore fynger rounde, and 10
set the ende of it on the hyghest ioynte of the thombe.
And for to expresse 20, you must set your fyngers streyght, and 20
the ende of your thombe to the partition of the 135 b. *fore moste
and myddle fynger.

e 30
30 is represented by the ioynynge together of y headdes of the
foremost fynger and the thombe.
40 is declared by settynge of the thombe crossewayes on the 40
foremost fynger.
50 is signified by ryght stretchyng forth of the fyngers ioyntly, 50
and applyenge of 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
with the fore fynger.
70 is expressed by the bowynge of the foremost fynger, and 70
settynge the ende of the thombe between the 2 foremost or
hyghest ioyntes of it.
80 is expressed by settynge of the foremost fynger crossewayes 80
on the thombe, so that 80 dyffereth thus from 40, that for 80 the
forefynger is set crosse on the thombe, and for 40 the thombe is
e
set crosse ouer y forefinger.
136 a. *90 is signified, by bendynge the fore fynger, and settyng 90
e
the ende of it in the innermost ioynte of y thombe, that is euen at
the foote of it. And thus are all the nombers ended vnder 100.
S. In dede these be all the nombers from 1 to 10, and then all the
tenthes within 100, but this teacyed me not how to expresse 11, 11, 12, 13
21, 22, 23
12, 13, etc. 21, 22, 23, etc. and such lyke.
M. You can lytell vnderstande, yf you can not do that without
teachynge: what is 11? is it not 10 and 1? then expresse 10 as you
were taught, and 1 also, and that is 11: and for 12 expresse 10
and 2: for 23 set 20 and 3: and so for 68 you muste make 60 and
there to 8: and so of all other sortes.
But now yf you wolde represente 100 other any nomber aboue it, 100
you muste do that with the ryghte hande, after this maner.
You must expresse 100 in the ryght hand, with the lytell fynger
so bowed as you dyd expresse 1 in the left hand.

136 b. *And as you expressed 2 in the lefte hande, the same 200
fasshyon in the ryght hande doth declare 200.
The fourme of 3 in the ryght hand standeth for 300. 300

The fourme of 4, for 400. 400

Lykewayes the fourme of 5, for 500. 500

The fourme of 6, for 600. And to be shorte: loke how you did 600
expresse single vnities and tenthes in the lefte hande, so must
you expresse vnities and tenthes of hundredes, in the ryghte
hande.
S. I vnderstande you thus: that yf I wold represent 900, I must so 900
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 10, so in 1000
my ryght hande must I expresse 1000.
And so the fourme of euery tenthe in the lefte hande serueth to
e 4000
expresse lyke nomber of thousandes, so y fourme of 40 standeth
for 4000.
The fourme of 80 for 8000. 8000
137 a. 9000

*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 be it,
with dyuers fyngers you maye expresse
9999, and all at one tyme, and that lac
keth but 1 of 10000. So that vnder
10000 you may by your fyngers ex-
presse any summe. And this shal suf-
fyce for Numeration on the fyngers.
And as for Addition, Subtraction,
Multiplication, and Diuision (which
yet were neuer taught by any man as
farre as I do knowe) I wyll enstruct
you after the treatyse of fractions.
And now for this tyme fare well,

and loke that you cease not to
practyse that you haue lear
ned. S. Syr, with moste
harty mynde I thanke
you, bothe for your
good learnyng, and
also your good
counsel, which
(god wyllyng) I truste to folow.
Finis.

1. 1342 in original.
2. 168 in original.
3. Bracket ([) denotes new paragraph in original.
For this e-text, the brackets have been omitted in favor of
restoring the paragraph breaks. Numbers 200 and up were
printed as separate paragraphs and are unchanged.
Sidenote 4 was missing and has been supplied by the
transcriber; the pairs 5, 6 and 9, 10 (originally on one line)
have been separated.

APPENDIX I.

[From a MS. of the 14th Century.]

To alle suche even nombrys the most have cifrys as to ten.
twenty. thirtty. an hundred. an thousand and suche other. but ye
schal vnderstonde that a cifre tokeneth nothinge but he maketh
other the more significatyf that comith after hym. Also ye schal
vnderstonde that in nombrys composyt 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. the
sifre 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 born
oute of his first place where he schuld yf he stode ther tokene but
one. And there he stondith nowe in the ferye place he tokeneth a
thousand as by this rewle. In the first place he tokeneth but
hymself. In the secunde place he tokeneth ten times hymself. In
the thridde place he tokeneth an hundred tymes himself. In the
ferye he tokeneth a thousand tymes himself. In the fyftye place
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 tymes hymself, &c. And ye
schal vnderstond that this worde nombre is partyd into thre
partyes. Somme is callyd nombre of digitys for alle ben digitys
that ben withine ten as ix, viii, vii, vi, v, iv, iii, ii, i. Articules ben
alle thei that mow be devyded into nombrys of ten as xx, xxx, xl,
and suche other. Composittys be alle nombrys that ben
componyd of a digyt and of an articule as fourtene fyftene
thrittene and suche other. Fourtene is componyd of four that is a
digyt 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. In the firste place he tokeneth but
himself that is to say he tokeneth but that and no more. If that he
stonde in the secunde place he tokeneth ten tymes 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 tymes himself and
ten 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 figure of foure that hath this schape 4 tokeneth but himself

for he stondeth in the first place. The figure of thre that hath this
schape 3 tokeneth ten tyme himself for he stondeth in the
secunde place and that is thritti. The figure of sexe that hath this
schape 6 tokeneth ten tyme more than he schuld and he stode in
the place yer the figure of thre stondeth for ther he schuld tokene
but 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 schulde and he stode in the place ther the figure of
6 stondeth inne for thanne he schuld tokene but nyne hundryd.
And in the place that he stondeth inne nowe he tokeneth nine
thousand. Alle the hole nombre of these foure figurys. Nine
thousand sexe hundrid and foure and thritti.

APPENDIX II.

[From a B.M. MS., 8 C. iv., with additions from 12 E. 1 & Eg. 2622.]

Hec algorismus ars presens dicitur 1; in qua
Talibus Indorum 2 fruimur his quinque figuris.
0. 9. 8. 7. 6. 5. 4. 3. 2. 1.
Prima significat unum: duo vero secunda:
4
Tercia significat tria: sic procede sinistre
Donec ad extremam venies, qua cifra vocatur;
3[Que nil significat; dat significare sequenti.]

Quelibet illarum si primo limite ponas,
8
Simpliciter se significat: si vero secundo,
Se decies: sursum procedas multiplicando. 4
[Namque figura sequens quevis signat decies plus,
12
Ipsa locata loco quam significet pereunte:
Nam precedentes plus ultima significabit.]

5
Post predicta scias quod tres breuiter numerorum
Distincte species sunt; nam quidam digiti sunt;
16
Articuli quidam; quidam quoque compositi sunt.
[Sunt digiti numeri qui citra denarium sunt;
Articuli decupli degitorum; compositi sunt
Illi qui constant ex articulis digitisque.]
20
Ergo, proposito numero tibi scribere, primo
Respicias quis sit numerus; quia si digitus sit,
5
[Una figura satis sibi; sed si compositus sit,]
Primo scribe loco digitum post articulum fac
24
Articulus si sit, cifram post articulum sit,
[Articulum vero reliquenti in scribe figure.]
Quolibet in numero, si par sit prima figura,
Par erit et totum, quicquid sibi continetur;
28
Impar si fuerit, totum sibi fiet et impar.
Septem 6 sunt partes, non plures, istius artis;
Addere, subtrahere, duplare, dimidiare;
Sexta est diuidere, set quinta est multiplicare;
32
Radicem extrahere pars septima dicitur esse.
Subtrahis aut addis a dextris vel mediabis;
A leua dupla, diuide, multiplicaque;
Extrahe radicem semper sub parte sinistra.
36

Addere si numero numerum vis, ordine tali Addition.
Incipe; scribe duas primo series numerorum
Prima sub prima recte ponendo figuram,
Et sic de reliquis facias, si sint tibi plures.
40
Inde duas adde primas hac condicione;
Si digitus crescat ex addicione priorum,
Primo scribe loco digitum, quicunque sit ille;
Si sit compositus, in limite scribe sequenti
44

Articulum, primo digitum; quia sic iubet ordo.
Articulus si sit, in primo limite cifram,
Articulum vero reliquis inscribe figuris;
Vel per se scribas si nulla figura sequatur.
48
Si tibi cifra superueniens occurrerit, illam
Deme suppositam; post illic scribe figuram:
Postea procedas reliquas addendo figuras.
A numero numerum si sit tibi demere cura, Subtraction
52
Scribe figurarum series, vt in addicione;
Maiori numero numerum suppone minorem,
Siue pari numero supponatur numerus par.
Postea si possis a prima subtrahe primam,
56
Scribens quod remanet, cifram si nil remanebit.
Set si non possis a prima demere primam;
Procedens, vnum de limite deme sequenti;
Et demptum pro denario reputabis ab illo,
60
Subtrahe totaliter numerum quem proposuisti.
Quo facto, scribe supra quicquit remanebit,
Facque novenarios de cifris, cum remanebis,
Occurrant si forte cifre, dum demseris vnum;
64
Postea procedas reliquas demendo figuras.
7
[Si subtracio sit bene facta probare valebis, Proof.
Quas subtraxisti primas addendo figuras.
Nam, subtractio si bene sit, primas retinebis,
68
Et subtractio facta tibi probat additionem.]
Si vis duplare numerum, sic incipe; solam Duplation.
Scribe figurarum seriem, quamcumque voles que
Postea procedas primam duplando figuram;
72
Inde quod excrescet, scribens, vbi iusserit ordo,
Juxta precepta que dantur in addicione.

Nam si sit digitus, in primo limite scribe;
Articulus si sit, in primo limite cifram,
76
Articulum vero reliquis inscribe figuris;
Vel per se scribas, si nulla figura sequatur:
Compositus si sit, in limite scribe sequenti
Articulum primo, digitum; quia sic jubet ordo:
80
Et sic de reliquis facias, si sint tibi plures.
8
[Si super extremam nota sit, monadem dat eidem,
Quod tibi contingit, si primo dimidiabis.]
Incipe sic, si vis aliquem numerum mediare: Mediation.
84
Scribe figurarum seriem solam, velud ante;
Postea procedens medias, et prima figura
Si par aut impar videas; quia si fuerit par,
Dimidiabis eam, scribens quicquit remanebit;
88
Impar si fuerit, vnum demas, mediare,
Nonne presumas, sed quod superest mediabis;
Inde super tractum, fac demptum quod notat unum;
Si monos, dele; sit ibi cifra post nota supra.
92
Postea procedas hac condicione secunda: 9
Impar 10 si fuerit hic vnum deme priori,
Inscribens quinque, nam denos significabit
Monos prædictam: si vero secunda dat vnam,
96
Illa deleta, scribatur cifra; priori
Tradendo quinque pro denario mediato;
Nec cifra scribatur, nisi inde figura sequatur:
Postea procedas reliquas mediando figuras,
100
Quin supra docui, si sint tibi mille figure.
11
[Si mediatio sit bene facta probare valebis,
Duplando numerum quem primo dimidiasti.]
Si tu per numerum numerum vis multiplicare, Multiplicat
n.
104

Scribe duas, quascunque volis, series numerorum;
Ordo tamen seruetur vt vltima multiplicandi
Ponatur super anteriorem multiplicantis;
12
[A leua relique sint scripte multiplicantes.]
108
In digitum cures digitum si ducere, major
Per quantes distat a denis respice, debes
Namque suo decuplo tociens delere minorem;
Sicque tibi numerus veniens exinde patebit.
112
Postea procedas postremam multiplicando,
Juste multiplicans per cunctas inferiores,
Condicione tamen tali; quod multiplicantis
Scribas in capite, quicquid processerit inde;
116
Set postquam fuerit hec multiplicata, figure
Anteriorentur seriei multiplicantis;
Et sic multiplica, velut istam multiplicasti,
Qui sequitur numerum scriptum quicunque figuris.
120
Set cum multiplicas, primo sic est operandum,
Si dabit articulum tibi multiplicacio solum;
Proposita cifra, summam transferre memento.
Sin autem digitus excrescerit articulusque,
124
Articulus supraposito digito salit ultra;
Si digitus tamen, ponas illum super ipsam,
Subdita multiplicans hanc que super incidit illi
Delet eam penitus, scribens quod provenit inde;
128
Sed si multiplices illam posite super ipsam,
Adiungens numerum quem prebet ductus earum;
Si supraimpositam cifra debet multiplicare,
Prorsus eam delet, scribi que loco cifra debet,
132
12
[Si cifra multiplicat aliam positam super ipsam,
Sitque locus supra vacuus super hanc cifra fiet;]
Si supra fuerit cifra semper pretereunda est;

Si dubites, an sit bene multiplicando secunda,
136
Diuide totalem numerum per multiplicantem,
Et reddet numerus emergens inde priorem.
13
[Per numerum si vis numerum quoque multiplicare Mental
Multiplica
Tantum per normas subtiles absque figuris
140
Has normas poteris per versus scire sequentes.
Si tu per digitum digitum quilibet multiplicabis
Regula precedens dat qualiter est operandum
Articulum si per reliquum vis multiplicare
144
In proprium digitum debebit uterque resolvi
Articulus digitos post per se multiplicantes
Ex digitis quociens teneret multiplicatum
Articuli faciunt tot centum multiplicati.
148
Articulum digito si multiplicamus oportet
Articulum digitum sumi quo multiplicare
Debemus reliquum quod multiplicaris ab illis
Per reliquo decuplum sic omne latere nequibit
152
In numerum mixtum digitum si ducere cures
Articulus mixti sumatur deinde resolvas
In digitum post hec fac ita de digitis nec
Articulusque docet excrescens in detinendo
156
In digitum mixti post ducas multiplicantem
De digitis ut norma docet sit juncta secundo
Multiplica summam et postea summa patebit
Junctus in articulum purum articulumque
160
[Articulum purum comittes articulum que]
14

Mixti pro digitis post fiat et articulus vt
Norma jubet retinendo quod egreditur ab illis
Articuli digitum post in digitum mixti duc
164
Regula de digitis ut percipit articulusque

Ex quibus excrescens summe tu junge priori
Sic manifesta cito fiet tibi summa petita.
Compositum numerum mixto sic multiplicabis
168
Vndecies tredecem sic est ex hiis operandum
In reliquum primum demum duc post in eundem
Unum post deinde duc in tercia deinde per unum
Multiplices tercia demum tunc omnia multiplicata
172
In summa duces quam que fuerit te dices
Hic ut hic mixtus intentus est operandum
Multiplicandorum de normis sufficiunt hec.]
Si vis dividere numerum, sic incipe primo; Division.
176
Scribe duas, quascunque voles, series numerorum;
Majori numero numerum suppone minorem,
15
[Nam docet ut major teneat bis terve minorem;]
Et sub supprima supprimam pone figuram,
180
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;
184
Ut totiens demas demendas a remanente,
Que serie recte ponentur in anteriori,
Unica si, tantum sit ibi decet operari;
Set si non possis a prima demere primam,
188
Procedas, et eam numero suppone sequenti;
Hanc uno retrahendo gradu quo comites retrahantur,
Et, quotiens poteris, ab eadem deme priorem,
Ut totiens demas demendas a remanenti,
192
Nec plus quam novies quicquam tibi demere debes,
Nascitur hinc numerus quociens supraque sequentem
Hunc primo scribas, retrahas exinde figuras,
Dum fuerit major supra positus inferiori,

196
Et rursum fiat divisio more priori;
Et numerum quotiens supra scribas pereunti,
Si fiat saliens retrahendo, cifra locetur,
Et pereat numero quotiens, proponas eidem
200
Cifram, ne numerum pereat vis, dum locus illic
Restat, et expletis divisio non valet ultra:
Dum fuerit numerus numerorum inferiore seorsum
Illum servabis; hinc multiplicando probabis,
204 Proof.
Si bene fecisti, divisor multiplicetur
Per numerum quotiens; cum multiplicaveris, adde
Totali summæ, quod servatum fuit ante,
Reddeturque tibi numerus quem proposuisti;
208
Et si nil remanet, hunc multiplicando reddet,
Cum ducis numerum per se, qui provenit inde Square
Numbers.
Sit tibi quadratus, ductus radix erit hujus,
Nec numeros omnes quadratos dicere debes,
212
Est autem omnis numerus radix alicujus.
Quando voles numeri radicem querere, scribi
Debet; inde notes si sit locus ulterius impar,
Estque figura loco talis scribenda sub illo,
216
Que, per se dicta, numerum tibi destruat illum,
Vel quantum poterit ex inde delebis eandem;
Vel retrahendo duples retrahens duplando sub ista
Que primo sequitur, duplicatur per duplacationem,
220
Post per se minuens pro posse quod est minuendum.
16
Post his propones digitum, qui, more priori
Per precedentes, post per se multiplicatus,
Destruat in quantum poterit numerum remanentem,
224
Et sic procedens retrahens duplando figuram,
Preponendo novam donec totum peragatur,

Subdupla propriis servare docetque duplatis;
Si det compositum numerum duplacio, debet
228
Inscribi digitus a parte dextra parte propinqua,
Articulusque loco quo non duplicata resessit;
Si dabit articulum, sit cifra loco pereunte
Articulusque locum tenet unum, de duplicata resessit;
232
Si donet digitum, sub prima pone sequente,
Si supraposita fuerit duplicata figura
Major proponi debet tantummodo cifra,
Has retrahens solito propones more figuram,
236
Usque sub extrema ita fac retrahendo figuras,
Si totum deles numerum quem proposuisti,
Quadratus fuerit, de dupla quod duplicasti,
Sicque tibi radix illius certa patebit,
240
Si de duplatis fit juncta supprima figura;
Radicem per se multiplices habeasque
Primo propositum, bene te fecisse probasti;
Non est quadratus, si quis restat, sed habentur
244
Radix quadrati qui stat major sub eadem;
Vel quicquid remanet tabula servare memento;
Hoc casu radix per se quoque multiplicetur,
Vel sic quadratus sub primo major habetur,
248
Hinc addas remanens, et prius debes haberi;
Si locus extremus fuerit par, scribe figuram
Sub pereunte loco per quam debes operari,
Que quantum poterit supprimas destruat ambas,
252
Vel penitus legem teneas operando priorem,
Si suppositum digitus suo fine repertus,
Omnino delet illic scribi cifra debet,
A leva si qua sit ei sociata figura;
256

Si cifre remanent in fine pares decet harum
Radices, numero mediam proponere partem,
Tali quesita radix patet arte reperta.
Per numerum recte si nosti multiplicare
260
Ejus quadratum, numerus qui pervenit inde
Dicetur cubicus; primus radix erit ejus;
Nec numeros omnes cubicatos dicere debes,
Est autem omnis numerus radix alicujus;
264 Cube Roo
Si curas cubici radicem quærere, primo
Inscriptum numerum distinguere per loca debes;
Que tibi mille notant a mille notante suprema
Initiam, summa operandi parte sinistra,
268
Illic sub scribas digitum, qui multiplicatus
In semet cubice suprapositum sibi perdat,
Et si quid fuerit adjunctum parte sinistra
Si non omnino, quantum poteris minuendo,
272
Hinc triplans retrahe saltum, faciendo sub illa
Que manet a digito deleto terna, figuram
Illi propones quo sub triplo asocietur,
Ut cum subtriplo per eam tripla multiplicatur;
276
Hinc per eam solam productum multiplicabis,
Postea totalem numerum, qui provenit inde
A suprapositis respectu tolle triplate
Addita supprimo cubice tunc multiplicetur,
280
Respectu cujus, numerus qui progredietur
Ex cubito ductu, supra omnes adimetur;
Tunc ipsam delens triples saltum faciendo,
Semper sub ternas, retrahens alias triplicatas
284
Ex hinc triplatis aliam propone figuram,
Que per triplatas ducatur more priori;
Primo sub triplis sibi junctis, postea per se,

In numerum ducta, productum de triplicatis:
288
Utque prius dixi numerus qui provenit inde
A suprapositis has respiciendo trahatur,
Huic cubice ductum sub primo multiplicabis,
Respectumque sui, removebis de remanenti,
292
Et sic procedas retrahendo triplando figuram.
Et proponendo nonam, donec totum peragatur,
Subtripla sub propriis servare decet triplicatis;
Si nil in fine remanet, numerus datus ante
296
Est cubicus; cubicam radicem sub tripla prebent,
Cum digito juncto quem supprimo posuisti,
Hec cubice ducta, numerum reddant tibi primum.
Si quid erit remanens non est cubicus, sed habetur
300
Major sub primo qui stat radix cubicam,
Servari debet quicquid radice remansit,
Extracto numero, decet hec addi cubicato.
Quo facto, numerus reddi debet tibi primus.
304
Nam debes per se radicem multiplicare
Ex hinc in numerum duces, qui provenit inde
Sub primo cubicus major sic invenietur;
Illi jungatur remanens, et primus habetur,
308
Si per triplatum numerum nequeas operari;
Cifram propones, nil vero per hanc operare
Set retrahens illam cum saltu deinde triplata,
Propones illi digitum sub lege priori,
312
Cumque cifram retrahas saliendo, non triplicabis,
Namque nihil cifre triplacio dicitur esse;
At tu cum cifram protraxeris aut triplicata,
Hanc cum subtriplo semper servare memento:
316
Si det compositum, digiti triplacio debet

Illius scribi, digitus saliendo sub ipsam;
Digito deleto, que terna dicitur esse;
Jungitur articulus cum triplata pereunte,
320
Set facit hunc scribi per se triplacio prima,
Que si det digitum per se scribi facit illum;
Consumpto numero, si sole fuit tibi cifre
Triplato, propone cifram saltum faciendo,
324
Cumque cifram retrahe triplam, scribendo figuram,
Preponas cifre, sic procedens operare,
Si tres vel duo serie in sint, pone sub yma,
A dextris digitum servando prius documentum.
328
Si sit continua progressio terminus nuper
Per majus medium totalem multiplicato;
Si par, per medium tunc multiplicato sequentem.
Set si continua non sit progressio finis:
332
Impar, tunc majus medium si multiplicabis,
333
Si par per medium sibi multiplicato propinquum.

1. “Hec præsens ars dicitur algorismus ab Algore rege ejus
inventore, vel dicitur ab algos quod est ars, et rodos quod
est numerus; quæ est ars numerorum vel numerandi, ad
quam artem bene sciendum inveniebantur apud Indos bis
quinque (id est decem) figuræ.” —Comment. Thomæ de
Novo-Mercatu. MS. Bib. Reg. Mus. Brit. 12 E. 1.
2. “Hæ necessariæ figuræ sunt Indorum characteros.” MS.
de numeratione. Bib. Sloan. Mus. Brit. 513, fol. 58. “Cum
vidissem Yndos constituisse ix literas in universo numero
suo propter dispositionem suam quam posuerunt, volui
patefacere de opere quod sit per eas aliquidque esset levius
discentibus, si Deus voluerit. Si autem Indi hoc voluerunt et
intentio illorum nihil novem literis fuit, causa que mihi
potuit. Deus direxit me ad hoc. Si vero alia dicam preter

eam quam ego exposui, hoc fecerunt per hoc quod ego
exposui, eadem tam certissime et absque ulla dubitatione
poterit inveniri. Levitasque patebit aspicientibus et
discentibus.” MS. U.L.C., Ii. vi. 5, f. 102.
3. From Eg. 2622.
4. 8 C. iv. inserts Nullum cipa significat: dat significare
sequenti.
5. From 12 E. 1.
6.
En argorisme devon prendre
Vii especes . . . .
Adision subtracion
Doubloison mediacion
Monteploie et division
Et de radix eustracion
A chez vii especes savoir
Doit chascun en memoire avoir
Letres qui figures sont dites
Et qui excellens sont ecrites.—MS. Seld. Arch. B. 26.
7. From 12 E. 1.
8. From 12 E. 1.
9. 8 C. iv. inserts Atque figura prior nuper fuerit mediando.
10. I.e. figura secundo loco posita.
11. So 12 E. 1; 8 C. iv. inserts—
Si super extremam nota sit monades dat eidem
Quod contingat cum primo dimiabis
Atque figura prior nuper fuerit mediando.
12. 12 E. 1 inserts.
13. 12 E. 1 inserts to l. 174.
14. 12 E. 1 omits, Eg. 2622 inserts.
15. 12 E. 1 inserts.
16. 8 C. iv. inserts—
Hinc illam dele duplans sub ei psalliendo
Que sequitur retrahens quicquid fuerit duplicatum.

INDEX OF TECHNICAL TERMS 1
algorisme, 33/12; algorym, augrym, 3/3; the art of computing, using the so-called Arabic numerals.
The word in its various forms is derived from the Arabic al-Khowarazmi (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
ἀριθμός (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, 5/31; articuls, 9/36, 29/7,8; a number divisible by ten without remainder.
cast, 8/12; to add one number to another.
‘Addition is a casting together of two numbers into one number,’ 8/10.
cifre, 4/1; the name of the figure 0. The word is derived from the Arabic sifr = empty, nothing. Hence
zero.
A cipher is the symbol of the absence of number or of zero quantity. It may be used alone or in
conjunction with 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, 5/35; with reference to numbers, one compounded of a multiple of
ten and a digit.
conuertide = conversely, 46/29, 47/9.
cubicede, 50/13; to be c., to have its cube root found.
cubike nombre, 47/8; a number formed by multiplying a given number twice by itself, e.g. 27 =
3 × 3 × 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; the 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, 47/18.
lyneal nombre, 46/14; 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 themselves and unity, but it is not clear that this limitation is intended.
mediacioun, 16/36, 38/16; dividing by two (see 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.
produccioun, multiplication, 50/11.
quadrat nombre, 46/12; a number formed by multiplying a given number by itself, e.g. 9 = 3 × 3,
a square.
rote, 7/25; roote, 47/11; root. The roots of squares and cubes are the numbers from which the squares
and cubes are derived by multiplication 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 3, 6, 5, 4. 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, 46/37; a number which is the product of three other numbers, e.g. 66 = 11 × 2 × 3.
superficial nombre, 46/18; a number which is the product of two other numbers, e.g. 6 = 2 × 3.
ternary, consisting of three digits, 51/7.
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.

1. This Index has 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.

GLOSSARY

Words whose first appearance is earlier than the page cited in the Glossary
are identified in supplementary notes, and both occurrences are marked in the

main text.

ablacioun, taking away, 36/21
addyst, haddest, 10/37
agregacioun, addition, 45/22. (First example in N.E.D., 1547.)
a-ȝenenes, against, 23/10
allgate, always, 8/39
als, as, 22/24
and, if, 29/8;
&, 4/27;
& yf, 20/7
a-nendes, towards, 23/15
aproprede, appropriated, 34/27
apwereth, appears, 61/8
a-risyȝt, arises, 14/24
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. s. borowe, 12/20;
pp. borwed, 12/15;
borred, 12/19
boue, above, 42/34
caputule, chapter, 7/26
certayn, assuredly, 18/34
clepede, called, 47/7
competently, conveniently, 35/8
compt, count, 47/29
contynes, contains, 21/12;
pp. contenythe, 38/39
craft, art, 3/4
distingue, divide, 51/5
egalle, equal, 45/21
excep, except, 5/16
exclusede, excluded, 34/37
excressent, resulting, 35/16
exeant, resulting, 43/26

expone, expound, 3/23
ferye = ferþe, fourth, 70/12
figure = figures, 5/1
for-by, past, 12/11
fors; no f., no matter, 22/24
forseth, matters, 53/30
forye = forþe, forth, 71/8
fyftye = fyftþe, fifth, 70/16
grewe, Greek, 33/13
haluendel, half, 16/16;
haldel, 19/4;
pl. haluedels, 16/16
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;
holle, 17/1;
hoole, of three dimensions, 46/15
holdyþe, holds good, 30/5
how be it that, although, 44/4
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, remains, 11/19; First used in 10/40
leus, 11/28;
pp. laft, left, 19/24
lewder, more ignorant, 3/3
lust, desirest to, 45/13
lyȝt, easy, 15/31
lymytes, limits, 34/18;
lynes, 34/12;
lynees, 34/17;
Lat. limes, pl. limites.
maystery, achievement;

no m., no achievement, i.e. easy, 19/10
me, indef. pron. one, 42/1 First used in 34/16
mo, more, 9/16
moder = more (Lat. majorem), 43/22
most, must, 30/3 First used in 3/12
multipliede, to be m. = multiplying, 40/9
mynvtes, the sixty parts into which a unit is divided, 38/25
myse-wroȝt, mis-wrought, 14/11
nether, nor, 34/25
nex, next, 19/9
noȝt, nought, 5/7 First used in 4/8
note, not, 30/5
oo, one, 42/20; o, 42/21 First used in 34/27 (oo); 33/22 (o)
omest, uppermost, higher, 35/26;
omyst, 35/28
omwhile, sometimes, 45/31 First used in 39/17
on, one, 8/29
opyne, plain, 47/8
or, before, 13/25
or = þe oþer, the other, 28/34
ordure, order, 34/9;
row, 43/1 Word form is “order”
other, or, 33/13, 43/26; Note also “one other other” in 35/24
other . . . or, either . . . or, 38/37 First used in 37/5
ouerer, upper, 42/15
ouer-hippede, passed over, 43/19
recte, directly, 27/20 First used in 26/31
remayner, remainder, 56/28
representithe, represented, 39/14
resteth, remains, 63/29 First used in 57/29
rewarde, regard, 48/6
rew, row, 4/8
rewle, row, 4/20, 7/12;
rewele, 4/18;
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
‘significatyf’ in l. 14. But the ‘Solucio,’ which contained the word, has been omitted.
sithen, since, 33/8

some, sum, result, 40/17, 32 First used in 36/21
sowne, pronounce, 6/29
singillatim, singly, 7/25
spices, species, kinds, 34/4 First used in 5/34
spyl, waste, 14/26
styde, stead, 18/20
subtrahe, subtract, 48/12;
pp. subtrayd, 13/21
sythes, times, 21/16
taȝt, taught, 16/36
take, pp. taken;
t. fro, starting from, 45/22
taward, toward, 23/34
thouȝt, though, 5/20
trebille, multiply by three, 49/26
twene, two, 8/11 First used in 4/23
þow, though, 25/15
þowȝt, thought;
be þ., mentally, 28/4
þus = þis, this, 20/33
vny, unite, 45/10
wel, wilt, 14/31
wete, wit, 15/16;
wyte, know, 8/38;
pr. 2 s. wost, 12/38
wex, become, 50/18
where, whether, 29/12
wher-thurghe, whence, 49/15
worch, work, 8/19; First used in 7/35
wrich, 8/35;
wyrch, 6/19;
imp. s. worch, 15/9; First used in 9/6
pp. y-wroth, 13/24
write, written, 29/19; First used in 4/5
y-write, 16/1
wryrchynge = wyrchynge, working, 30/4
t
w , with, 55/8
y-broth, brought, 21/18
ychon, each one, 29/10

ydo, done, added, 9/6 First used in 8/37
ylke, same, 5/12
y-lyech, alike, 22/23
y-myȝt, been able, 12/2
y-nowȝt, enough, 15/31;
ynovȝt, 18/34
yove, given, 45/33
t
y , that, 52/8
y-write, v. write.
y-wroth, v. worch.

MARGINAL NOTES
Headnotes have been moved to the beginning of the appropriate paragraph.
Headnotes were omitted from the two Appendixes, as sidenotes give the same
information.
Line Numbers are cited in the Index and Glossary. They have been omitted from
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Instead, the Index and Glossary are linked directly to each word.
Numbered Notes:
Numbered sidenotes show page or leaf numbers from the original MSS. In the
e-text, sidenote numbers have been replaced with simple asterisks.
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numbered sequentially within each title.
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possible to their original format and location.

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