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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
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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.

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underlined words are cross-references to the Index of Technical Terms and the Glossary.

The Earliest Arithmetics in English / Early English Text Society. / Extra Series, No. CXVIII. / 1922 (for 1916).

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
vi

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
vii

various manuscripts, selecting various readings on the principle that
the verses were made to scan, aided by the advice of my friend Mr.
Vernon 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
viii

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
ix

nine numbers, known as the Boethian “apices” or “notae” (from whence our
word “notation”). To these we shall return later on.

The Abacists.

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

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 u n d r e d s	T e n s	U n i t s	H u n d r e d s	T e n s	U n i t 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.)

xii

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.

xiii

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
xiv

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
xv

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
xvi

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
xvii

perfectly well acquainted with the system they met in their Arabic text,
while the earliest astronomical tables of the thirteenth century I have
seen use numbers of European and not Arabic origin. The fact that Latin
writers had a convenient way of writing hundreds and thousands without
any cyphers probably delayed the general use of the Arabic notation. Dr.
Hill has published a very complete survey of the various forms of
numerals in Europe. They began to be common at the middle of the
thirteenth century and a very interesting set of family notes concerning
births in a British Museum manuscript, Harl. 4350 shows their extension.
The first is dated Mij^c. lviii.,
the second Mij^c. lxi., the third Mij^c. 63, the
fourth 1264, and the fifth 1266. Another example is given in a set of
astronomical tables for 1269 in a manuscript of Roger Bacon’s works,
where the scribe began to write MCC6. and crossed out the figures,
substituting the “Arabic” form.

The Counting Board.

The treatise on pp. 52-65 is the only one in English known on the
subject. It describes a method of calculation which, with slight
modifications, is current in Russia, China, and Japan, to-day, though it
went out of use in Western Europe by the seventeenth century. In Germany
the method is called “Algorithmus Linealis,” and there are several
editions of a tract under this name (with a diagram of the counting
board), printed at Leipsic at the end of the fifteenth century and the
beginning of the sixteenth. They give the nine rules, but “Capitulum de
radicum extractione ad algoritmum integrorum reservato, cujus species
per ciffrales figuras ostenduntur ubi ad plenum de hac tractabitur.” The
invention of the art is there attributed to Appulegius the
philosopher.

The advantage of the counting board, whether permanent or constructed
by chalking parallel lines on a table, as shown in some
sixteenth-century woodcuts, is that only five counters are needed to
indicate the number nine, counters on the lines representing units, and
those in the spaces above representing five times those on the line
below. The Russian abacus, the “tchatui” or “stchota” has ten beads on
the line; the Chinese and Japanese “Swanpan” economises by dividing the
line into two parts, the beads on one side representing five times the
value of those on the other. The “Swanpan” has usually many more lines
than the “stchota,” allowing for more extended calculations, see Tylor,
Anthropology (1892), p. 314.

xviii

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.

The Earliest Arithmetics in English.

Egerton 2622.

leaf 136 a.

HEc algorismus ars
presens dicitur; in qua

Talibus indorum fruimur bis
quinque figuris.

A derivation of Algorism.
This boke is called þe boke of algorym, or Augrym after
lewder 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;
Another derivation of the word.
or els anoþer cause is quy it is called Algorym, for þe latyn
word of hit s. Algorismus comes of Algos, grece,
quid est 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.

Notation 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.

Expositio versus.
In þis verse is notifide þe significacion of þese figuris.
And þus expone the verse.
The meaning and place of the figures.
Þe first signifiyth one, þe secunde
leaf 136 b.
signi*fiyth tweyne, þe thryd signifiyth thre, & the fourte
signifiyth 4. ¶ And so forthe towarde þe lyft syde of þe tabul or
of þe boke þat þe figures bene writene in, til þat
þou come to the last figure, þat is
4

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.
Which figure is read first.
The figure 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.

Expositio [in margin].
¶ Expone þis verse þus. Euery of þese figuris
bitokens hym selfe & 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.
An explanation of the principles of
notation.
¶ If it stonde in the secunde place of þe rewle, he betokens
tene tymes hym selfe, as þis figure 2 here 20 tokens ten
tyme 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.
An example:
loo an ensampulle. 9. 6. 3. 4. Þe figure of
4. þat hase þis schape . betokens bot hymselfe, for he stondes in þe first
place.
units,
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.
tens,
The figure of 6, þat hase þis 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.
thousands.
Al þe hole nombur is 9 thousande sex hundryth &
foure & thretty. ¶ fforthermore, when
5

þou schalt rede a nombur of figure,
How to read the number.
þou 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.

The meaning and use of the cipher.
Expone þis verse. A cifre tokens noȝt, bot he makes þe
figure to 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
away1 & 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 /

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

The Three 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.

¶ Capitulum 2^m 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.
Composites.
¶ Expone þis 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.

Digits, Articles, and Composites.

¶ Sunt digiti numeri qui citra denarium
sunt.

What are digits.
¶ Here he telles qwat is a digit, Expone versus
sic. Nomburs 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.
What are articles.
¶ Articulis ben2
alle þat may be deuidyt into 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.
What numbers are composites.
*Compositys beɳ nomburs þat bene componyt of a 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.

How to write a number,
¶ here he telles how þou schalt wyrch whan þou schalt
write a 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 it is a digit;
¶ If he be a digit, write a digit, as yf it be seuen, write seuen
& write þat digit in þe first place toward þe ryght side.
if it is a composite.
If it be a composyt, write þe digit of þe composit in þe first place
& write þe articul of þat digit in þe secunde place next toward þe
lyft side. As yf 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.
How to read it.
But whan þou schalt sowne or speke
leaf 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.

How to write Articles:
¶ Here he tells how þou schal write when þe nombre þat
þou 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
7

mych he is, yf he be with-ynne an hundryth, þou
schalt write bot one cifre, afore, as here .9φ.
hundreds,
If þe articulle be by hym-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.
thousands, &c.
If þe articul be a thousant or thousandes3
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.

To tell an even number
¶ Here he teches a generalle rewle þat yf þe
first figure in þe 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.
or an odd.
¶ If þe first
leaf 139 b.
*figure token an 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.

The Seven Rules of Arithmetic.

The seven rules.
¶ Here telles þat þer beɳ .7. spices or
partes of þis craft. The first is called addicioñ, þe secunde is
called subtraccioñ. The thryd is called duplacioñ.
The 4. is called dimydicioñ. The 5. is called
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:

Add, subtract, or halve, from right to
left.
Thou schal be-gynne in þe ryght side of þe boke or of a tabul. loke
were þou wul be-gynne to write latyn or englys in a boke,
& 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.
Multiply or divide from left to right.
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 dicetur
singillatim in sequentibus
capitulis et de vtilitate cuiuslibet artis
& sic Completur
leaf 140.
*prohemium & sequitur tractatus &
primo de arte addicionis que prima ars est in
ordine.

The Craft 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.

Four things must be known:
¶ Here by-gynnes þe craft of Addicioñ. In þis craft þou
most 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.
how many rows of figures;
¶ As for þe secunde þou most know þat þou schalle haue
tweyne rewes of figures, one vndur a-nother, as here
þou mayst se.

1234
2168.
how many cases;
¶ As for þe thryd þou most know þat there ben foure
diuerse cases.
what is its result.
As for þe forthe þou most know þat þe profet of þis
craft is to telle what is þe hole nombur þat comes of
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 set down 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,

123
234.
And þe secunde of þe nether nombur euene vndir þe
secunde of þe hyer, & so forthe of euery figure of
both þe rewes as þou mayst se.

The Cases 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.
Add the first figures;
¶ He says 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.
rub out the top figure;
¶ If he be a digit þou schalt do away þe first 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
write the result in its place.
wrich forth oɳ 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,
9

Here is an example.
here is an ensampul of þis case.

1234
2142.
Caste 2 to foure & þat wel be sex, do away 4. & write in
þe same place þe figure of sex. ¶ And lete þe figure
of 2 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,
Suppose it is a Composite, set down the digit,
and carry the tens.
yf of þe casting of 2 nomburis to-geder, as of þe figure
of þe hyer rewe & of þe figure of þe neþer rewe come a
Composyt, how schalt þou worch. Þus þou schalt
worch. Thou shalt do away þe figure of þe hyer nomber þat
was cast to þe 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.
Here is an example.
lo an Ensampull
leaf 141 b.
*of all.

326
216.
Cast 6 to 6, & þere-of wil arise twelue. do away þe hyer 6
& write 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.

1
322
216.
Now cast þe articulle þat standus vpon þe figuris
of twene hed to þe same 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.
Suppose it is an Article, set down a cipher and
carry the tens.
yf of Addiciouɳ of 2 figuris a-ryse an Articulle, how schal
þou do. thou most do away þe heer figure þat was
addid to þe neþer, & write þere a cifre, and sett þe
articuls on þe figuris hede, yf þat
þere come ony after. And wyrch þan as I haue tolde þe in
þe secunde case. An ensampull.

25.
15
Cast 5 to 5, þat wylle be ten. now do away þe hyer 5, & write
þere a cifer. And sette ten vpon þe figuris hed of 2. And reken
it but for on þus. lo
Here is an example.
10

an Ensampulle

1
2φ
15
And
leaf 142 a.
*þan worch forth. But yf þere come no figure after þe
cifre, write þe articul next hym in þe same rewe as here

5
5
cast 5 to 5, and it wel be ten. do away 5. þat is þe hier 5. and write
þere a cifre, & write after hym þe articul as þus

1φ
5
And þan þou hast done.

¶ Si tibi cifra superueniens occurrerit, illam

Dele superpositam; fac illic scribe figuram,

Postea procedas reliquas addendo figuras.

What to do when you have a cipher in the top
row.
¶ Here he puttes þe fourt case, & it is þis, þat
yf þere come a cifer in þe hier rewe, how þou schal do.
þus þou schalt do. do away þe cifer, & sett þere þe
digit þat comes of þe addicioun as þus

1φφ84.
17743
An example of all the difficulties.
In þis ensampul ben alle þe foure cases. Cast 3 to foure,
þat wol be seueɳ. do away 4. & write þere seueɳ; þan
cast 4 to þe figure of 8. þat wel be 12. do away 8, &
sett þere 2. þat is a digit, and sette þe articul of þe composit,
þat is ten, 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.

The Craft of Subtraction.

A numero numerum si sit tibi demere
cura

Scribe figurarum series, vt in addicione.

Four things to know about
subtraction:
¶ This is þe Chapter of subtraccioɳ, in the quych þou most
know foure nessessary thynges. the first what is subtraccioɳ. þe
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
the first;
þe first, þou most know þat subtraccioɳ is drawynge
of one nowmber oute of anoþer nomber.
the second;
As for þe secunde, þou most knowe þat þou most haue two rewes of
figuris one vnder anoþer, as þou addyst
in addicioɳ.
the third;
As for þe thryd, þou moyst know þat foure
maner of diuerse casis mai happe in þis craft.
the fourth.
¶ As for þe 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
11

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
Put the greater number above the less.
þe hier nomber most be more þen þe neþer, or els eueɳ as
mych. but he may not be lasse. And þe 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.

The Cases of the Craft of Subtraction.

¶ Postea si possis a prima subtrahe primam

Scribens quod remanet.

The first case of subtraction.
Here is þe first case put of subtraccioɳ, & he says þou
schalt 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,
Here is an example.
lo an Ensampulle

234
122
draw 2 out of 4. þan leues 2. do away 4 & write þere 2, &
latte þe neþer figure stonde stille, & so go
for-by oþer figuris till þou come to þe
ende, þan hast þou do.

¶ Cifram si nil remanebit.

Put a cipher if nothing remains.
¶ Here he puttes þe secunde case, & hit is þis.
yf it happe þat 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
Here is an example.
lo an Ensampull

24
24
Take foure out of foure þan leus noȝt.
þerefore do away þe hier 4 & set þere a cifer,
þan take 2 out of 2, þan 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.

Suppose you cannot take the lower figure
from the top one, borrow ten;
Here he puttes þe thryd case, þe quych is þis. yf it happe þat þe
neþer figure be more þen þe hier figure þat
he schalle be draw out of. how schalle þou do. þus
þou schalle do. þou schalle borro .1. oute of þe
next figure þat comes after in þe same rewe, for þis case
may neuer happ but yf þere come figures after. þan
þou schalt sett
12

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 lower number from ten;
¶. And out of þat ten þou schal draw þe neyþermost figure,
And alle þat leues þou schalle
add the answer to the top number.
adde to þe figure on whos hed þat .1. stode. And þen þou
schalle do away alle þat, & sett þere
alle that 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.

2122
1134
take 4 out of 2. it wyl not be, þerfore borro one of þe
next figure, þat is 2. and sett þat ouer þe 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.
How to ‘Pay back’ the borrowed ten.
þan take out of þat þe figure vnder hym, þat
is 3. hit wel not be. þer-fore borowe of the next
figure, þe quych is 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.

A very hard case is put.
¶ Here he puttes þe fourte case, þe quych is þis, yf it
happe þat 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. Ergo4
how
13

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.
Here is an example.
lo an Ensampulle after.

40002
10004
Take 4 out of 2. it wel 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.
Sic.

39998
10004
& yf þou hast wel y-do þou5
schalt haue þis nomber

How to prove the Subtraction.

¶ Si subtraccio sit bene facta
probare valebis

Quas subtraxisti primas addendo figuras.

How to prove a subtraction sum.
¶ Here he teches þe Craft how þou schalt know, whan
þou hast 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.
Here is an example.
lo an
leaf 145 b.
*Ensampulle of alle þe 4 cases togedre. worche
welle þis case

40003468
20004664
And yf þou worch welle whan þou hast alle subtrayd þe
þat hier nombre here, þis schalle be þe
nombre here foloyng whan þou hast subtrayd.

39998804
20004664
Our author makes a slip here (3
for 1).
And þou schalt know þus. adde þe neþer rowe of þe same
nombre to þe hier rewe as þus, cast 4 to 4. þat wol be 8. do away þe 4
& write þere 8. by þe first case of addicioɳ. þan cast 6 to 0
þat wol be 6. 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
14

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.
He works his proof through,
take þe figure of þat 1. þe quych 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 brings out a result.

60003468
20004664
And þen schal þou haue an Ensampulle aȝeyɳ, loke & se,
& but þou haue þis same þou hase myse-wroȝt.

The Craft of Duplation.

Sequitur de duplacione

Si vis duplare
numerum, sic incipe primo

Scribe figurarum seriem
quamcunque velis tu.

Four things must be known in
Duplation.
¶ This is the Chapture of duplacioɳ,
in þe quych craft þou most haue & know 4 thinges.
¶ Þe first þat þou most know is what is duplacioɳ. þe
secunde is how mony rewes of figures þou most haue
to þis craft. ¶ þe thryde is how many cases may6
happe in þis craft. ¶ þe fourte is what is þe profet of þe
craft.
Here they are.
¶ As for þe 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.
Mind where you begin.
¶ fforþer-more, þou most know & take gode
hede in 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.
Remember your 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.

How to work a sum.
¶ Here he telles how þou schalt worch in þis Craft. he
says, fyrst, whan þou hast writen þe nombre þou schalt
be-gyn at þe first
15

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.

The Cases of the Craft of Duplation.

leaf 147 a.

* ¶ Nam si sit digitus in primo limite
scribas.

If the answer is a digit,
¶ Here is þe first case of þis craft, þe quych is þis. yf of
duplacioɳ of a figure arise a digit. what schal þou do.
þus þou schal do.
write it in the place of the top figure.
do away þe figure þat was doublede, & sett þere þe
diget þat comes of þe duplacioɳ, as þus. 23. double 2, & þat
wel 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.

If it is an article,
¶ Here is þe secunde case, þe quych is þis yf þere come an
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 &
put a cipher in the place, and ‘carry’ the
tens.
sett þere a cifer, & write þe articulle ouer þe
next figuris hede, yf þere be any after-warde
toward þe lyft 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.
If there is no figure to ‘carry’ them to, write
them down.
yf þere come no figure after þe figure
þat is addit, of þe quych addicioɳ comes an articulle,
þou schalt do away þe figure þat is dowblet &
sett þere a 0. & write 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.

If it is a Composite,
¶ Here he puttes þe Thryd case, þe quych is þis, yf of
duplacioɳ of a figure come a Composit. þou schalt do away
þe figure þat is doublet & set þere a digit of
þe Composit,
write down the digit, and ‘carry’ the
tens.
& sett þe articulle ouer þe next figures hede, &
after draw hym downe with þe figure
ouer whos hede he stondes, & make þere-of an nombre as
þou hast done
16

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,
Here is an example.
and set þe articulle next 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.

How to double the mark for one-half.
¶ Here he says, yf ouer þe fyrst figure in þe
ryȝt side be such a merke as is here made, ^w,
þou schalle fyrst doubulle þe figure, 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 place þere
þat þe figure þe quych was doublet stode, as þus 23^w.
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.
This can only stand over the first
figure.
& þou schall neuer haue such a merk but ouer þe
hed of þe furst figure in þe ryght side. And ȝet it schal not happe but
yf it were y-halued a-fore, þus þou schalt
vnderstonde þe verse. ¶ Si super
extremam &c. Et nota, talis figura ^w
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.

The Craft of Mediation.

¶ Sequitur de mediacione.

Incipe sic, si vis aliquem
numerum mediare:

Scribe figurarum seriem solam, velut
ante.

The four things to be known in
mediation:
¶ In þis Chapter is taȝt þe Craft of mediaciouɳ, in þe quych craft þou most
know 4 thynges. ffurst what is mediacioɳ. the secunde 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
the first
¶ As for þe furst, þou schalt vndurstonde þat mediacioɳ is a
17

takyng out of halfe a nomber out of a holle nomber,
leaf 149 a.
*as yf þou
the second;
wolde take 3 out of 6. ¶ As for þe secunde, þou schalt know
þat þou most haue one rewe of figures, & no
moo, as þou hayst in þe
the third;
craft of duplacioɳ. ¶ As for the thryd, þou most vnderstonde
þat
the fourth.
5 cases may happe in þis craft. ¶ As for þe 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.
Begin thus.
Write one rewe of figures of 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 even or odd.
take hede wheþer þe first figure be eueɳ or odde in
nombre, & 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:

If it is even, halve it, and write the
answer in its place.
¶ Here is the first case of þis craft, þe quych is þis, yf
þe first figure be euen. þou schal take away fro þe figure
euen halfe, & do away þat figure and set þere þat
leues ouer, as þus, 4. take
leaf 149 b.
*halfe out of 4, & þan þere leues 2. do away 4 & sett
þere 2. þis is lyght y-nowȝt. versus.

The 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.

If it is odd, halve the even number less
than it.
Here is þe secunde case of þis craft, the quych is þis. yf þe
first figure betokene a nombre þat is odde, the quych odde
schal not 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 þat. for þat þat leues,
Then write the sign for one-half over
it.
write such a merke as is here ^w vpon his hede, þe
quych merke schal betokeɳ halfe of þe odde þat was take away.
Here is an example.
lo an Ensampull. 245. the first figure here is
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 a merke ^w upon his hede, þat
is to say ouer his hede of 2 as þus. 242.^w And þen
worch forth in þe oþer figures tyll þou come to þe ende.
by þe furst case as þou schalt
18

vnderstonde þat
Put the mark only over the first figure.
þou schalt
leaf 150 a.
*neuer make such a merk but ouer þe first figure
hed in þe riȝt side. Wheþer þe other figures þat comyɳ
after hym be eueɳ or odde. versus.

The Cases of the Craft of Mediation.

¶ Si monos, dele; sit tibi cifra post nota supra.

If the first figure is one put a
cipher.
¶ Here is þe thryde case, þe quych yf the first figure be a
figure of 1. þou schalt do away þat 1 & set
þere a cifer, & a merke ouer þe cifer as þus, 241. do
away 1, & sett þere a cifer with a merke
ouer his hede, & þen hast þou ydo for þat 0. as þus
0^w 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.

What to do if any other figure is
odd.
¶ Here he puttes þe fourte case, þe quych is þis. yf
it happeɳ the secunde figure betoken odde nombre, þou schal do
away on of 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.
Write a figure of five over the next lower
number’s head.
¶ Whan þou hase so done, þou schalt write
leaf 150 b.
*a figure of 5 ouer þe next figures hede by-fore
toward þe ryȝt side, for þat 1, þe quych made odd nombre, schall
stonde for ten, & 5 is halfe of 10; so þou most write 5 for
his haluendelle.
Example.
lo an Ensampulle, 4678. begyɳ in þe ryȝt side as 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;

5
4634.
do away 6 & sette þere 3, & þou schalt haue such a nombre
after go 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.

19

¶ Here he puttes þe 5 case, þe quych is
leaf 151 a.
*þis:
If the second figure is one, put a cipher, and
write five over the next figure.
yf þe secunde figure be of 1, as þis is here 12, þou schalt do
away þat 1 & sett þere a cifer. & sett 5 ouer þe
next figure hede afore toward þe riȝt side, as þou diddyst
afore; & þat 5 schal be haldel 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

5
202,
þen worch forth to þe nex figure. And also it is no maystery yf þere come no
figure 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.
How to halve fourteen.
medie 4 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,

5
2,
& do away þat 1, & þou schalt haue þis nombre yf
þou worch wel, vnde versus.

How to prove the Mediation.

¶ Si mediacio sit bene facta probare
valebis

¶ Duplando numerum quem primo
dimediasti

How to prove your mediation.
¶ Here he telles þe how þou schalt know wheþer þou hase
wel 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.
First example.
¶ The furst ensampulle was þis. 4. þe 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.
The second.
þe secunde Ensampulle was þis, 245. When þou haddyst
mediet alle þis nombre, yf þou haue wel ydo þou schalt
haue of þat mediacioɳ þis nombre, 122^w. 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 third example.
The nombre þe quych þou haddist for an Ensampul in þe 3 case of
mediacioɳ to be mediet was þis 241. whan þou haddist medied
alle þis nombur truly
leaf 152 a.
*by euery figure, þou schall haue be þat mediacioɳ
þis nombur 120^w. 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
20

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 fourth example.
¶ The next ensampul þat had in 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

5
2334.
Now doubul this nombur & begyn in þe lyft side, &
doubulle 2 þat schal be 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

5
102.
Now doubul 1. þat wel be 2. do away 1 & sett þere 2. þen
doubul 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.

The Craft of Multiplication.

Sequitur de multiplicatione.

To write 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.

Four things to be known of
Multiplication:
¶ Here be-gynnes þe Chaptre of multiplicatioɳ,
in þe quych þou most know 4 thynges. ¶ Ffirst, qwat is
multiplicacioɳ. The 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.
the first:
¶ As for þe first, þou schal vnderstonde þat
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.
the second:
¶ ffor þe secunde, þou most know þat þou most
haue too rewes of figures.
the third:
¶ As for þe thryde, þou most know þat 8 maner
of diuerse case may happe in þis craft.
the fourth.
The profet of þis Craft is to 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,
The multiplicand.
& þat 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 set down the sum.
And so write forthe toward þe lyft side, as here you may se,

67324
1234
And þis one nombur schalle be called numerus
multiplicans. Anglice, þe nombur
multipliynge, for he schalle 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.
Two sorts of Multiplication: mentally,
¶ ffor-þere-more þou schalt vndurstonde þat þere is
two manurs of multiplicacioɳ; one ys of þe
wyrchynge of þe boke only in þe mynde of a mon. fyrst he
22

teches of þe fyrst maner of duplacioɳ, þe quych is be
wyrchynge of tabuls.
and on paper.
Afterwarde he wol teche on þe secunde maner. vnde
versus.

To multiply 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.

How to multiply two digits.
¶ Here he teches a rewle, how þou schalt fynde þe
nounbre þat 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.
Subtract the greater from ten;
And so oft do away þe lasse nounbre out of his owne 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.
take the less so many times from ten times
itself.
so mony tymes take 2. out of his decuple, þe quych is 20. for 20 is þe
decuple of 2, 10 is þe decuple of 1, 30 is þe decuple of 3, 40 is þe
decuple of 4, And þe oþer digetes til þou come to ten;
& whan þou
Example.
hast y-take so mony tymes 2 out 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
Better use this table, though.
tymes as þat is fro 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

How to use it.
yf þe figure, þe quych schalle be multiplied, be
euene as mych as þe diget be, þe quych þat oþer
figure schal be multiplied, as two tymes twayɳ, or thre
tymes 3. or sych other.
The way to use the Multiplication table.
loke qwere þat figure sittes in
23

þe lyft side of þe triangle, & loke qwere þe diget
sittes in þe neþer most rewe of þe triangle. & go fro hym
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
tyl10
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.

To multiply 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.

How to multiply one number by
another.
¶ Here he teches how þou schalt wyrch in þis craft.
þou schalt multiplye þe last figure of þe nombre, and quen
þou hast so ydo 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.
Multiply the ‘last’ figure of the higher by the
‘first’ of the lower number.
And sette þe nounbir þat comes þer-of ouer þe last figure
of þe neþer nounbre, & þen þou schalt sette al þe
oþer figures of þe neþer nounbre more
nere to þe ryȝt side. ¶ And whan þou hast multiplied
þat figure þat schal be multiplied þe next after
24

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.
Set the answer over the first of the
lower:
lo an Ensampul here folowynge.

2465.
232
þou schalt begyne to multiplye in þe lyft side. Multiply 2
be 2, and twyes 2 is 4. set 4
then multiply the second of the lower, and so
on.
ouer þe hed of þat 2, þen multiplie þe same hier 2
by 3 of þe nether nounbre, as thryes 2 þat schal be 6. set 6 ouer
þe hed of 3, þan 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.
Then antery the lower number:
¶ Now þou most 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.
as thus.
þen þou schalt haue such a nombre.

464465
232
leaf 157 a.
* ¶ Now multiply 4, þe quych comes next after 6, by þe
last 2 of þe neþer 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.
Now multiply by the last but one of the
higher:
þen multiplie þe same 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
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.
as thus.
Whan þou hast done as I haue byde þe, þou schalt haue suych an
order of figure as is here,

1
82
4648[65]
232.
þen take and antery þi neþer figures. And sett þe fyrst
figure of þe neþer figures11
vndre be figure of 6. ¶ And draw al þe oþer figures
of þe same rewe to hym-warde,
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
25

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,

11
121
828
464825
232
Antery the figures again, and multiply by
five:
þen antery þi figures as þou diddyst afore, and
multipli 5 be 2, þat wol be 10; sett þe 0 ouer all þe
figures þat stonden ouer þat 2, & sett þat 1.
ouer the next figures hedes, alle on hye towarde þe lyft
side. þen multiplye 5 be 3. þat wol be 15, write 5 ouer þe
figures hedes þat stonden ouer þat 3, & sett þat 1
ouer þe next figures hedes toward þe lyft side. þen
multiplye 5 be 2, þat wol be 10. do 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.*

11
1101
1215
82820
4648
232
¶ Now draw alle þese figures downe togeder as þus, 6.8.1.
& 1 draw to-gedur; þat wolle be 16, do away alle þese figures
saue 6. lat hym stonde, for þow þou take hym away þou most write þer
þe same aȝene. þerefore late hym stonde, & sett 1
ouer þe figure hede of 4 toward þe lyft side; þen draw on
to 4, þat wolle be 5.
Then add all the figures above the line:
do away þat 4 & þat 1, & sette þere 5. þen draw 4221
& 1 togedur, þat wol be 10. do away alle þat, &
write þere 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 you will have the answer.
¶ And þen þou schalt haue þis nounbre, 571880.

The Cases of this Craft.

leaf 158 b.

* ¶ Sed cum multiplicabis, primo sic
est operandum,

Si dabit articulum tibi multiplicacio solum;

Proposita cifra summam transferre
memento.

What to do if the first multiplication
results in an article.
¶ Here he puttes þe fyrst case of þis craft, þe quych is
þis: yf þere come an articulle of þe multiplicacioɳ ysette
before the articulle in þe lyft side as þus

51
23.
multiplye 5 by 2, þat wol be 10; sette ouer þe hede of þat 2 a 0,
& sett þat on, þat is þe articul, in þe lyft side, þat is next hym,
þen þou schalt haue þis nounbre

1051.
23
¶ And þen worch forth as þou diddist afore. And þou schalt
vnderstonde þat þou schalt write no 0. but whan þat place
where þou schal write þat 0 has no figure afore hym noþer
after. versus.

¶ Si autem digitus excreuerit
articulusque.

Articulus12
supraposito digito salit vltra.

What to do if the result is a composite
number.
¶ Here is þe secunde case, þe quych is þis: yf hit happe þat
þere come a composyt, þou schalt write þe digitte ouer þe
hede of þe neþer figure by þe quych þou multipliest
þe hier figure; and sett þe articulle next hym toward þe lyft
side, as þou diddyst afore, as þus

83.
83
Multiply 8 by 8, þat wol be 64. Write þe 4 ouer 8, þat is to say,
ouer þe hede of þe neþer 8; & set 6, þe quych
leaf 159 a.
*is an articul, next after.
And þen þou schalt haue such a nounbre as is here,

648313,
83
And þen worch forth.

¶ Si digitus tamen ponas ipsum super
ipsam.

What if it be a digit.
¶ Here is þe thryde case, þe quych is þis: yf hit happe þat of þi
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.

The fourth case of the craft.
¶ Here is þe 4 case, þe quych is: yf hit be happe þat þe
neþer 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.

The fifth case of the craft.
¶ Here is þe 5 case, þe quych is þis: yf *þe neþer
figure schul 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:

236
234
Multiply 2 by 2, þat wol be 4; set 4 ouer þe hede of þat 2. þen14
multiplies þe hier 2 by þe neþer 3, þat 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,
27

and set þere 8. And þou schalt haue þis nounbre here

46836
234
And antery þi figures, þat is to say, set þi neþer 4
vnder þe hier 3, and set þi 2 other figures nere 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,

46836
234
Now worch forthermore, And multiplye þat hier 3 by 2, þat
wol be 6, set þat 6 þe quych stondes 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.

The sixth case of the craft.
¶ Here is þe 6 case, þe quych is þis: yf hit happe þat þe
figure by 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.

The seventh case of the craft.
¶ Here is þe 7 case, þe quych is þis: yf a 0 schal
multiply a 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:

24
03
multiplye 2 be a 0, it wol be nothynge. write þere a 0
ouer þe hede of þe neþer 0, And þen worch forth til þou
come to þe ende.

¶ Si supra15 fuerit cifra semper
est pretereunda.

The eighth case of the craft.
¶ Here is þe 8 case, þe quych is þis: yf þere be a 0
or mony cifers 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:

00032.
22
Ouer-lepe alle þese cifers & sett þat
leaf 160 b.
*neþer 2 þat stondes toward þe ryght side, and sett hym
vndur þe 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.

How to prove the multiplication.
¶ Here he teches how þou schalt know wheþer þou hase
wel I-do 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.

Mental multiplication.
¶ Here he teches þe to multiplie be þowȝt figures in
þi mynde. 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.

Digit by digit is easy.
¶ Here he teches a rewle as þou hast afore to
multiplie a digit be 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.

The first case of the craft.
¶ Here he teches þe furst rewle, þe quych is þis: yf þou wel
multiplie an articul be anoþer, so þat both þe articuls
bene with-Inne an hundreth, þus þou schalt do.
Article by article;
take þe digit of bothe the 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.
an example:
yf þou wold wete 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.
another example:
¶ Also yf þou wold 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 tymes16 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
29

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.

How to work subtly without Figures.
Mental multiplication.
¶ And so mony tymes ten thousand be in 4 hundryth tymes [2]17 hundryth, þat is 80 thousand.
Take hede, I schall telle þe a
Another example.
generalle rewle whan þou hast 2 articuls, And 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.
Notation.
¶ In þe secunde place hit betokenes twenty. ¶ 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.
Notation again.
¶ In þe 4 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
30

thousantes.
Mental multiplication.
And so mych is twenty þowsant tymes 3 thousant, ¶ 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.

The third case of the craft;
¶ Here he puttes þe thryde rewle, þe quych is þis. yf
þou wel 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:
an example.
yf 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.

The fourth case of the craft:
Here he puttes þe 4 rewle, þe quych is þis: yf þou multipliy on
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.
Composite by digit.
þen take þou þat nounbur, & cast hit to þat other nounbur þat
þou secheste as þus yf þou wel
31

wete qwat comes of 6 tymes 4 & twenty.
Mental multiplication.
multiply þat articulle of þe composit by þe digit, þe 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.

How to 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.

The fifth case of the craft:
¶ Here he puttes þe 5 rewle, þe quych is þis: yf þou
wel multiply 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.
Article by Composite.
¶ Articulum si per Relicum 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 example.
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.

How to 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.

The sixth case of the craft:
¶ Here he puttes þe 6 rewle, & þe last of alle
multiplicacioɳ, þe quych is þis: yf þou wel multiplye a
composit by a-noþer composit, þou schalt do þus.
Composite by Composite.
multiplie þat on composit, qwych 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,
Mental multiplication.
as
32

þou was tauȝt in þe 4 rewle.
An example
As þus, yf þou wold wete what is 11 tymes 13, as þou was tauȝt in
þe 5 rewle, & 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.
of the sixth case of the craft.
¶ Also of the quych 11 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.]

Boys 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.

Therfore sithen the science of
the whiche at this tyme we
The name of the 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,
Derivation of Algorism.
hade out of Algore, other of Algos, in grewe, That is
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.
Another.
And thirdly it is hade of the name of a kyng that is
clepede Algo and Rythmus; So callede
Algorismus.
Kinds of numbers.
Sothely .2. manere of nombres 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
34

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.
The 9 rules of the Art.
Of this art bene .9. spices, that is forto sey,
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.

Chapter I. Numeration.

Fol. 48 b.

*For-sothe numeracioun is of euery
numbre by competent figures an artificialle
representacioun.

Figures, differences, places, and
limits.
Sothly figure, difference, places, and lynes supposen o thyng other
the same, But they ben sette here for dyuers resons. ffigure is
clepede for protraccioun of figuracioun;
Difference is callede 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.
The 9 figures.
And vnderstonde that ther ben .9. lymytes of figures that
representen the .9. digites that ben these. 0. 9. 8. 7. 6.
5. 4. 3. 2. 1.
The cipher.
The .10. is clepede theta, or a cercle, other a cifre, 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.
The numeration
And sithen that by 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.
of digits,
And note wele that euery digite shalle be writte
with oo figure allone to it aproprede.
of articles,
And alle 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;
of composites.
And yf it be a nombre componede, ffirst write the digit 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
35

til .x. Mł. mvst be writ by .4. figures; And so forthe.
The value due to position.
And vnderstonde wele that euery figure 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.
Numbers are written from right to left.
We writene in this art to the lift side-warde, as
arabiene writene, that weren fynders of this science,
othere for this resoun, that for to kepe a custumable
ordre in redyng, Sette we alle-wey the more nombre
before.

Chapter II. Addition.

Definition.
Addicioun is of nombre other of
nombres vnto nombre or to 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;
How the numbers should be written.
and it is convenient that the lesse nombre be vnderwrit, and the more
addede, than the 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.
The method of working.
That done, adde the 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.
Begin at the right.
If it be digitus, In the place of the omyst shalt thow write the
digit excrescyng, as thus:—

The resultant

To whom it shal be addede

The nombre to be addede

The Sum is a digit,
If the article; in the place of the omyst put a-way by a cifre writte,
and the digit transferrede, of þe whiche the article toke
his name, towarde the lift side, and be it addede to the
next figure folowyng, yf ther be any figure folowyng; or no, and yf it
be not, leve it [in the] voide, as thus:—

36

or an article,

The resultant

To whom it shalle be addede

The nombre to be addede

Resultans	2	7	8	2	7
Cui debet addi	1	0	0	8	4
Numerus addendus	1	7	7	4	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;

The resultant

To whom it shalle be addede

The nombre to be addede

In his place write the
Fol. 49 b.
*[digit of the] Article as thus:—

And yf it happe that a figure of .9. by the figure that me mvst adde
[one] to,

The resultant

To whom it shalle be addede

The nombre to be addede

In the place of that 9. put a cifre and write þe article
towarde þe lift honde as bifore, and thus:—

or a composite.
And yf3
[therefrom grow a] nombre componed,4
[in the place of the nombre] put a-way5

The resultant

To whom it shalle be addede

The nombre to be addede

[let] the digit [be]6
writ þat is part of þat composide, and þan
put to þe lift side the article as before, and þus:—

The translator’s note.
This done, adde the seconde to the seconde, and write above
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.

Chapter III. Subtraction.

Definition of Subtraction.
Subtraccioun is of .2.
proposede nombres, the fyndyng of the 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.
How it may be done.
And sothly that 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.
What is required.
More-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;
Write the greater number above.
The
37

nombre fro the whiche me shalle withe-draw in the
omyst ordre, so that the first be vnder the first, the seconde
vnder the seconde, And so of alle others.
Subtract the first figure if possible.
Withe-draw therfor the first of the lowere ordre fro the
first of the ordre above his hede, and that wolle be other more
or lesse, oþer egalle.

The remanent

Wherof me shalle withdraw

The nombre to be withdraw

The remanent	2	2
Wherof me shalle with-draw	2	8
Þe nombre to be withdraw		6

yf it be egalle or even the figure sette beside, put in his place
a cifre. And yf it be more put away þerfro als many of vnitees
the lower figure conteynethe, and writ the residue as thus

Fol. 50.

*Remanens	2	2	1	8	2	9	9	9	8
A quo sit subtraccio	8	7	2	4	3	0	0	0	4
Numerus subtrahendus	6	5	7	[6]	.	.	.	.	6

If it is not possible ‘borrow ten,’
And yf it be lesse, by-cause the more may not be with-draw
ther-fro, borow an vnyte of the next figure that is worthe 10. Of
that .10. and of the figure that ye wolde have
with-draw fro
and then subtract.
be-fore to-gedre Ioynede,

The remanent	1	8
Wherof me shalle with-draw	2	4
The nombre to be with-draw	0	6

with-draw þe figure be-nethe, and put the residue in the
place of the figure put a-side as þus:—

If the second figure is one.
And yf the figure wherof me shal borow the vnyte be one, put it a-side,
and write a cifre in the place þerof, lest the figures folowing
faile of thaire nombre, and þan worche as it
shewith in this figure here:—

The remanent	3	0	98
Wherof me shal with-draw	3	1	2
The nombre to be with-draw	.	.	3

If the second figure is a cipher.
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 þat ye
passide ouer, sette figures of .9. as here it is
specifiede:—

The remenaunt	2	9	9	9	9
Wherof me shalle with-draw	3	0	0	0	3
The nombre to be with-draw					4

And whan me comethe to the nombre wherof me intendithe,
there remaynethe alle-wayes .10. ffor þe whiche
.10. &c.
A justification of the rule given.
The reson why þat for euery cifre left behynde me setteth figures
ther of .9. this it is:—If fro the .3. place me borowede an
vnyte, that vnyte by respect of the figure that he came fro
representith an .C., In the
38

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.
Why it is better to work from right to
left.
And note wele that in addicion or in subtraccioun me may
wele fro the lift side begynne and ryn to the right side, But it wol be
more profitabler to be do, as it is taught.
How to prove subtraction,
And yf thow wilt prove yf thow have do wele or no, The figures
that thow hast withdraw, adde them ayene to the omyst figures, and they
wolle accorde with the first that thow haddest yf thow
have labored wele;
and addition.
and 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.

Chapter IV. Mediation.

Definition of mediation.
Mediacioun is
the fyndyng of the halfyng of euery 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
Where to begin.
begynne at the right, that is to sey, fro the first 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
If the first figure is unity.
figures folowyng, [lest they signify less], and 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.

Halfede

to be halfede

halfede

[di]

To be halfede

What to do if it is not unity.
If it be even, write this half in this wise:—

And if it be odde, Take the next even vndre hym conteynede,
and put his half in the place of that odde, and of þe vnyte that
remaynethe to be halfede do thus:—

Then halve the second figure.
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 even
or ode: If it be even, write in the place of þe nombres
wipede out the halfe; yf it be ode, take the next
even vnder it contenythe, and in the place
of the Impar sette a-side put half of the even: The
39

vnyte that remaynethe to be halfede, respect hade
to them before, is worthe .10.

Halfede

to be halfede

If it is odd, add 5 to the figure
before.
Dyvide that .10. in .2., 5. is, and sette a-side that one, and adde that
other to the next figure precedent as here:—

And yf þe addicioun sholde be made to a cifre, sette it
a-side, and write in his place .5.

doublede	2	6	8	9	0	10	17	4
to be doublede	1	3	4	4	5	5	8	7

And vnder this fourme me shalle write and worche,
tille the totalle nombre be halfede.

Chapter V. Duplation.

Definition of Duplation.
Duplicacioun is agregacion
of nombre [to itself] þat me 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,
Where to begin.
In 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.
Why.
And how be it that me myght 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.]

double

to be doublede

What to do with the result.
If it be article, write in his place a cifre and transferre the article
towarde the lift, as thus:—

And yf the nombre be componede,

doublede

to be doublede

write a digital that is part of his composicioun, and sette the
article to the lift hande, as thus:—

That done, me most double the last save one, and what growethe
þerof me most worche as before. And yf a cifre be, touche
it not. But yf any nombre shalle be addede to the
cifre,

doublede	6	0	6
to be doublede	3	0	3

in þe place of þe figure wipede out me most write the nombre to
be addede, as thus:—

In the same wise me shalle wirche of alle
others.
How to prove your answer.
And this probacioun:

Doublede	6	1	8
to be doublede	3	0	9

If thow truly double the halfis, and truly half the doubles, the same
nombre and figure shalle mete, suche as thow
labourede vpone first, And of the contrarie.

Chapter VI. Multiplication.

Definition of Multiplication.
Multiplicacioun of nombre by
hym-self other by a-nother, 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.
Multiplier.
[The number multiplying] is designede aduerbially.
Multiplicand.
The nombre to be 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	.	.	.

Product.
Also me may thervpone to assigne the. 3. nombre, the
whiche is
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.

The Cases of Multiplication.

There are 6 rules of Multiplication.

1	2	3	4	5	6	7	8	9	10
2	4	6	8	10	1010	14	16	18	20
3	6	9	12	15	18	21	24	27	30
4	8	12	16	20	24	28	32	36	40
5	10	15	20	25	30	35	40	45	50
6	12	18	24	30	36	42	48	56	60
7	14	21	28	35	42	49	56	63	70
8	16	24	32	40	48	56	64	72	80
9	18	27	36	45	54	63	72	81	90
10	20	30	40	50	60	70	80	90	100

And ther ben .6 rules of Multiplicacioun;
(1) Digit by digit.
ffirst, yf a digit multiplie a digit, considre how many of
vnytees ben betwix the digit by multiplying and his .10. bethe
to-gedre accomptede, and so oft with-draw the digit
multiplying, vnder the article of his
denominacioun. Example of grace. If thow wolt wete
how moche is .4. tymes .8.,
11se how many vnytees ben betwix .8.12
and .10. to-geder rekenede, and it shewith that
.2.: withdraw ther-for the quaternary, of the article of his
denominacion twies, of .40., And ther remaynethe
.32., that is, to some of alle the multiplicacioun.
See the table above.
Wher-vpon for more evidence and declaracion the seide table is
made.
(2) Digit by article.
Whan a digit multipliethe an article, thow most bryng the digit
into þe digit, of þe whiche the article [has]13
his name, and euery vnyte
41

shalle stonde for .10., and euery article an .100.
(3) Composite 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

(4) Article by article.
Whan an article multipliethe an article, the digit wherof he is
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.
(5) Composite by article.
Whan an article multipliethe a nombre 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.
(6) Composite by composite.
Whan a nombre componede 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,
How to set down your numbers.
The nombre multiplying in the lower ordre by his differences, so that
the first of the lower ordre be vnder the last of the ouer ordre.
This done, of the multiplying, the last is to be hade into the
last of the nombre to be multipliede. Wherof than wolle grow a
digit, an article, other a nombre componede.
If the result is a digit,

The resultant

To be multipliede

Þe nombre multipliyng

If it be a digit, even above the figure multiplying is hede write his
digit that come of, as it apperethe here:—

an article,
And yf an article had be writ ouer the figure multiplying
his hede, put a cifre þer and transferre the article
towarde the lift hande, as thus:—

The resultant	1	0
to be multipliede		5
þe nombre multipliyng		2

or a composite.
And yf a nombre componede be writ ouer the figure
multyplying is hede, write the digit in the nombre componede is
place, and sette the article to the lift hande, as
thus:—

Resultant	1	2
to be multipliede		4
the nombre multipliyng		3

Multiply next by the last but one, and so
on.
This done, me most bryng the last save one of the multipliyng into
the last of 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.

Resultant	6	6
to be multipliede		3
the nombre multipliyng	2	2

The resultant	1	1	0
to be multipliede			5
þe nombre multiplying		2	2

The resultant	1	315	2
to be multipliede			4
þe nombre multipliant		3	3

If it be a digit, In the place of the ouerer, sette a-side, as
here:

If an article happe, there put a cifre in his place, and put hym to
the lift hande, as here:

If it be a nombre componede, in the place of the ouerer
sette a-side, write a digit that14
is a part of the componede, and sette on the left
honde the article, as here:

Then antery the multiplier one place.
That done, sette forwarde the figures of the nombre multiplying
by oo
difference, so that the first of the multipliant be vnder the last save
one of the nombre to be 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.
Work as before.
And than wolle growe oþer a 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.
How to deal with ciphers.
And 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

How to deal with ciphers.
And yf a cifre happe in the lower order be-twix the first and the last,
and even above be sette the figure signyficatif,

The resultant	2	2	6	4	4
To be multipliede			2	2	2
The multipliant	1	0	2

leve it vntouchede, as here:—

And yf the space above sette be voide, in that place write
thow a cifre. 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,

Resultant	8	0	0	8
to be multipliede	4	0	0	4
the multipliant	2	.	.	.

Fol. 53.
*wherof it is evident and open, yf that the first figure of the nombre
be to be multipliede be a cifre, vndir it shalle be none
sette as here:—

Resultant	3	2	016
To be multipliede		8	0
The multipliant		4

Leave room between the rows of
figures.
Vnder[stand] also that in multiplicacioun, divisioun, 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.

Chapter VII. Division.

Definition of division.
For to dyvyde oo nombre by a-nother, it
is of .2. nombres 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:
Dividend, Divisor, Quotient.
that is to sey, the nombre to be dyvydede; the nombre dyvydyng
and the nombre exeant, other how oft, or quocient. Ay
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.
How to set down your Sum.
Therfor yf thow wolt any nombre 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;
An example.
as here:—

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

44

Examples.

Residuum			8				2	7		2	6
Quociens		2	1	2	2			5			9
Diuidendus	6	8	0	6	6	3	4	2	3	3	2
Diuiser	3	2		3			6	3		3	4

When the last of the divisor must not be set
below the last of the dividend.
And ther ben .2. causes whan the last figure may not be sette vnder the
last, other that the last of the lower nombre may not be
with-draw of the last of the ouerer nombre for it
is lasse than the lower, other how be it, that it myght be
with-draw as for hym-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 figure above sette.
Fol. 53².
*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
How to begin.
and fro the figure aboue his hede, namly so that the remenaunt
may be take of so oft, and to se the residue as here:—

The residue		2	6
The quocient			9
To be dyvydede	3	3	2
The dyvyser		3	4

An example.
And note wele that me may not withe-draw more than .9. tymes
nether lasse than ones. Therfor se how oft þe figures of the lower 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;
Where to set the quotiente
And by that figure me most withe-draw alle oþer
figures of the lower 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

A special case.
And yf it happe after þe settyng forwarde of the
figures þat þe 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
45

not be with-draw me most sette there a cifre, and sette
forwarde the figures; as here:—

The residue						1	2
The quocient				2	0	0	4
To be dyvydede	8	8	6	3	7	0	4
The dyvyser	4	4	2	3

Another example.
And me shalle not cesse fro suche settyng of
figures forwarde, nether of settynge of þe quocient
into the 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 divisioun
Fol. 53³.
*wol growe to the nombre of the divisere,
What the quotient shows.
the nombre quocient wol shewe it: and whan suche divisioun
is made, and þou lust prove yf thow have wele done or
How to prove your division,
no, 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:
or multiplication.
as thus, yf multiplicacioun be made, divide it by the
multipliant, and the nombre quocient wol shewe the nombre that was to be
multipliede, etc.

Chapter VIII. Progression.

Definition of Progression.
Progressioun is of nombre after
egalle excesse fro oone or tweyne
take
agregacioun. of
progressioun one is naturelle or
contynuelle, þat oþer broken and
discontynuelle.
Natural Progression.
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 Progression.
Broken it is, 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;
The 1st rule for Natural Progression.
whan the progressioun naturelle endithe in
even nombre, by the half therof multiplie þe next totalle
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 second rule.
The 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
46

.5. by .3, and thryes .5. shalle be resultant. so the nombre
totalle is .15.
The first rule of Broken Progression.
Of progresioun intercise, ther ben
also .2.18
rules; and þe first is þis: Whan the Intercise progression
endithe 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 second rule.
The seconde rule is this: whan the progressioun
interscise endithe in ode, take þe more
porcioun of alle þe nombre,
Fol. 53⁴.
*and multiplie by hym-selfe; as .1. 3. 5. Multiplie .3. by
hym-selfe, and þe some of alle wolle be .9.,
etc.

Chapter IX. Extraction of Roots.

The preamble of the extraction of
roots.
Here folowithe the
extraccioun of rotis, and first in nombre
quadrates. Wherfor me shalle se what is a nombre
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:
Linear, superficial, and solid numbers.
Of nombres one is lyneal, anoþer superficialle, anoþer
quadrat, anoþer cubike or hoole. lyneal is that þat
is 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.
Superficial numbers.
Nombre superficial is 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.
Square numbers.
And for bycause a nombre may be hade in a-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 root of a square number.
The rote of a nombre quadrat 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.
Notes of some examples of square roots here
interpolated.
The most nombre quadrat 9. 8. 7. 5. 9. 3. 4. 7. 6. / the remenent
ouer the quadrat .6. 0. 8. 4. 5. / The first caas of nombre
quadrat .5. 4. 7. 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.
Solid numbers.
The solide nombre or cubike is þat
þat comytħe of double ledyng of nombre in nombre;
Three dimensions of solids.
And it is clepede a solide body that hathe
þer-in .3
47

[dimensions] þat is to sey, lengthe, brede, and 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.
Cubic numbers.
If a nombre be hade twies in 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.
All cubics are solid numbers.
Wherfor it is opyne that alle cubike nombre is
solide, and not conuertide. Cubike
is þat 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.
No number may be both linear and solid.
Opyne it is also that alle nombres may be a rote to a
quadrat and cubike, but not alle nombre quadrat or
cubike. Therfor sithen þe ledynge of vnyte in hym-self
ones or twies nought comethe but vnytes, Seithe Boice in
Arsemetrike,
Unity is not a 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,
Examples of square roots.
That is openede thus; lede the rote of o 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

A note on mean proportionals.
Also betwix the next .2. cubikis, me may fynde a double meene, 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.

Chapter X. Extraction of Square Root.

To20 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.
To find a square root.
And yf it be not quadrat the rote of the most 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
Begin with the last odd place.
it be even, than most thow 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,
Find the nearest square root of that number,
subtract,
me most fynde a digit, the whiche lade in
hym-selfe it puttithe away that, þat is ouer his
hede, oþer as neighe as me
48

may: suche a 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.
double it,
That done, than me most fynde a-noþer digit vnder
the next figure bifore the doublede,
and set the double one to the right.
the whiche
Fol. 54 b.
*brought in double settethe a-way alle that is ouer
his hede as to rewarde of the doublede: Than brought
into hym-self settithe all away in respect of hym-self,
Find the second figure by division.
Other do it as nye as it may be do: other me may with-draw
the digit
21[last] founde, and lede hym in double or
double hym, and after in hym-selfe;
Multiply the double by the second figure, and
add after it the square of the second figure, and subtract.
Than Ioyne to-geder the produccione of them bothe, So that
the first figure of the last product be addede before the
first of the first productes, the seconde of the
first, etc. and so forthe, subtrahe fro the
totalle nombre in respect of þe digit.

Examples.

The residue														5	4	3	2
To be quadrede	4	1	2	0	9	1	5	1	3	9	9	0	0	5	4	3	2
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,
Special cases.
Till thow fynde vndre the first figure 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.
The residue.
That done, other ought or nought wolle be the 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 ought22
remayne, that shewith that the nombre
proposede was not quadrat,23
but a digit [last found with the subduple or subduples
49

is]

This table is constructed for use in cube root
sums, giving the value of ab.²

1	2	3	4	5	6	7	8	9
2	8	12	16	20	24	28	32	36
3	18	27	36	45	54	63	72	81
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.
How to prove the square root without or with a
remainder.
Therfor yf thow wilt prove yf thow have wele do or no, Multiplie
the digit last founde with the vnder-double
oþer vnder-doublis, and thow shalt fynde the same figures that
thow 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.

Chapter XI. Extraction of Cube Root.

Definition of a cubic number and a cube
root.
Heere folowithe the
extraccioun of rotis in cubike nombres; wher-for me most
se what is a nombre cubike, and what is 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;
Mark off the places in threes.
First thow most compt the figures by fourthes, that is to sey in the
place of thousandes;
Find the first digit;
And vnder the last thousande place, thow most 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.
treble it and place it under the next but one,
and multiply by the digit.
That done, thow most trebille the digit, and
that triplat is to be put vnder the .3. next figure towarde the
right honde, And the vnder-trebille vnder the
trebille;
Then find the second digit.
Than me most fynde a digit vndre the next figure bifore the
triplat, the whiche with his vnder-trebille
had into a trebille, afterwarde other vnder[trebille]26
had in his produccioun, puttethe a-way alle
that is ouer it in regarde of27
[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:]
Multiply the first triplate and the second
digit, twice by this digit.
That done, thow most trebille the digit ayene, and the triplat is
to be sette vnder the next .3. figure as before, And the
vnder-trebille vnder the trebille: and than most thow
sette forwarde the first triplat with his
vndre-trebille by .2. differences. And than most thow fynde a
digit vnder the next figure before the triplat, the whiche
withe his vnder-triplat
had in his triplat afterwarde,
50

Subtract.
other vnder-treblis lad in product
Fol. 55 b.
*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

Continue this process till the first figure
is reached.
Nother me shalle not cesse of the fyndynge of that digit,
neither of his triplacioun, neþer of the triplat-is
29 anterioracioun, that is to
sey, settyng forwarde by .2. differences, Ne therof the
vndre-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,
Examples.
and sette alle-way that is ouer hym.

To be cubicede	1	7	2	8	3	2	7	6	8
The triple			3	2				9
The vnder triple			1	2		[3]		3	3

Also note wele that the produccion comynge of the
ledyng of a digite founde30
me may adde to, and also with-draw fro of the
totalle nombre sette above that digit so founde.31
The residue.
That done ought or 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,
Special cases.
And afterwarde in a product of that shalle
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.
Special case.
Fol. 56.
*And
51

yf no digit after the anterioracioun32
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.

Table of Numbers, &c.

A table of numbers; probably from the
Abacus.

1	2	3	4	5
one.	x.	an. hundrede /	a thowsande /	x. thowsande /

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.

Nowe 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 the other without lynes:
in that y^t hath lynes, the lynes do stande for the
order of places: and in y^t 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:

	1	0 0 0 0 0
	1	0 0 0 0
X	1	0 0 0
	1	0 0
	1	0
	1

Lo here you se .vi. lynes whiche stande for syxe places so that the
nethermost standeth for y^e fyrst place, and the next 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.
Numeration.
Now what is the valewe of euery 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?


X	1
	5
	4
	3

S. Thus, as I suppose.

M. You haue set y^e places truely, but your figures
be not mete for this vse:
53

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.


×	o
	°
	o o o o
	o o o

S. So that you haue not one figure for 2, nor 3, nor 4, and so
forth, but as many digettes as you haue, you set in the lowest lyne: and
for euery 10 you set one in the second line: and so of other. But I know
not by what reason you set that one counter for 500 betwene two
lynes.

M. you shall 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,

X
	o o
	o o°o
X	o o°
	o o°o o
	o°

which summe yf you marke well, you nede none other
examples for to lerne the numeration of
118 b.
*this forme. But this shal you marke, that as you dyd in the other kynde
of arithmetike, set a pricke in the places of thousandes, in this
worke you shall sette a starre, as you se here.

Addition on the Counting Board.

Addition.
S. Then I perceave numeration, but I praye you, howe shall I do
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.


X	o°o o	o o
	o o o	o°
	o o o o	°
	o o	o°o o o

And then yf you lyst, you
119 a.
*may adde the one to the other in the same place, or els you may adde
them both together in a newe place: which waye, bycause it is moste
playnest, I wyll showe you fyrst. Therfore wyl I begynne at the
vnites, whiche in the fyrst summe is but 2, and in
y^e second summe 9, that maketh 11, those do I take vp,
and for them I set 11 in the new roume, thus,


X	o°o o	o o
	o o o	o°
	o o o o	°	o
			o

Then do I take vp all y^e articles vnder a hundred, which in
the fyrst summe are 40, and in the second summe 50, that maketh
90: or you may saye better, that in the fyrste summe there are 4
articles 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.


X	o°o o	o o
	o o o	o°
			o°o o o o
			o

Where I haue taken awaye 40 from the fyrste summe, and 50
from y^e second, and in theyr stede I haue set 90 in the
thyrde, whiche I haue set playnely y^t you myght well perceaue
it: how be it seynge that 90 with the 10 that was in y^e thyrd
roume all redy, doth make 100, I myghte better for those 6
counters set 1 in the thyrde lyne, thus:


X
	o

	o

For it is all one summe as you may se, but it is beste, neuer to set 5
counters in any line, for that may be done 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.


X	o°o o	o o	o


			o

Then adde I y^e thousandes together, whiche in the fyrst
summe are 8000, and in y^e second 2000, that
maketh 10000: them do I take vp from those two places, and for
them I set one counter in the fyfte lyne, and then appereth as
you se,

	o
X	o


	o

to be 11001, for so many doth amount of the addition of 8342 to
2659.

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.


		o °
X	o o o	°
	o o	o o o o
	o o o o	o o o
	°	o °

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 here.


		o°
X	o o o	°
	o o	o o o o
	o o o o	o o o o
		o

Then do I lyke wayes take vp the 4 counters of the fyrste summe
and
55

seconde lyne (which make 40) and adde them to the 4 counters of the same
lyne, in the second summe, and it maketh 80, But as I sayde I
maye not conueniently set aboue 4 counters in one lyne, therfore
to those 4 that I toke vp in the fyrst summe, I take one
also of the seconde summe, and then haue I taken vp 50, for
whiche 5 counters I sette downe one in the space ouer y^e
second lyne, as here doth appere.


		o°
X	o o o	°
	o o	o o o o
		o°o o
		o

121 b
*and then is there 80, as well w^t those 4 counters, as yf I had set
downe y^e other 4 also. Now do I take the 200 in the fyrste
summe, and adde them to the 400 in the seconde summe, and it
maketh 600, therfore I take vp the 2 counters in the fyrste summe, and 3
of them in the seconde summe, and for them 5 I set 1 in y^e
space aboue, thus.


		o°
X	o o o	°
		o°
		o°o o
		o

Then I take y^e 3000 in y^e fyrste summe,
vnto whiche there are none in the second summe agreynge, therfore I do
onely remoue those 3 counters from the fyrste summe into the seconde, as
here doth appere.


	o°
X	o°o o
	o °
	o°o o
	o

122 a
*And so you see the hole summe, that amounteth of the
addytion of 65436 with 3245 to be 6868[1]. And yf you haue marked
these two examples well, you nede no farther enstruction
in Addition of 2 only summes: but yf you haue more then two summes to
adde, you may adde them thus.


X	o o	o o o o	o°
	o°	o o	o°o o o
	o°o	o°o o	o°
	o°o o o	o°	°

Fyrst adde two of them, and then adde the thyrde, and y^e
fourth, or more yf there be so many: as yf I wolde adde 2679 with 4286
and 1391. Fyrste I adde the two fyrste summes thus.
122 b.
*And then I adde the thyrde thereto thus.


X	o	o°	o o o°
	o o o	o°o o o	o o o
	o°o o o	°o	°
	o	°	o°

And so of more yf you haue them.

Subtraction 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
Subtraction.
in the other Addition by 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.


X	o o	o o°o
	o°o o	o°o
	o°o o o	o o o o
	o o	o°

Then shall I begynne to subtracte the greatest nombres fyrste (contrary
to the vse of the penne)
56

y^t is the thousandes in this example: therfore I fynd
amongest the thousandes 2, for which I withdrawe so many from the
seconde summe (where are 8) and so remayneth there 6, as this
example showeth.


		o°
+	o°o o	o°o
	o°o o o	o o o o
	o o	o°

Then do I lyke wayes with the hundredes, of whiche in the fyrste summe
116 b.
*I fynde 8, and is the seconde summe but 7, out of whiche I can not take
8, therfore thus muste I do: I muste loke how moche my summe
dyffereth from 10, whiche I fynde here 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 5000; but then set I downe the 200 unto the 700
y^t are there all redye, and make them 900 thus.


+		°
		o°o o o
	o o°o o	o o o o
	o o	o°

Then come I to the articles of tennes where in the fyrste
summe I fynde 90,
117 a.
*and in the seconde summe but only 40: Now consyderyng that 90
can not be bated from 40, I loke how moche y^t 90 doth
dyffer from the next summe 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 downe 1 in y^e nexte lyne beneth for it, as you se
here.


+		°
		o°o o
		°
	o o	o°

Sauynge that here I haue set one counter in y^e space in stede
of 5 in y^e nexte lyne. And thus haue I subtracted all saue
two, which I must bate from the 6 in the second summe, and there wyll
remayne 4, thus.


=	°
	o°o o
	°
	o o o o

So y^t 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 you dyd subtracte, then
wyll the formar summe 8746 amount agayne.

S. That wyll I proue: and fyrst I set the summe that
was subtracted, which was 2892, and then the remayner
5854, thus.


\|\|	o o	°
	o°o o	o o°o
	o o°o o	°
	o o	o o o o

Then do I adde fyrst y^e 2 to 4, whiche maketh 6, so take I vp
5 of those counters, and in theyr stede I sette 1 in the space, as here
appereth.


\|\|	o o	°
	o°o o	o°o o
	o°o o o	°
		o°

118 a.
*Then do I adde the 90 nexte 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 thyrde lyne, and 4 in y^e
57

second lyne, thus.


\|\|	o o	°
	o°o o	o°o o o
		o o o o
		°o

Then do I come to the hundredes, of whiche I fynde 8 in the fyrst summe,
and 9 in y^e second, that maketh 1700, therfore I take vp
those 9 counters, and in theyr stede I sette 1 in the .iiii. lyne, and 1
in the space nexte beneth, and 2 in the thyrde lyne, as you se here.


\|\|	o o	o°
		o°o
		o o o o
		o°

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


X	o o°o
	o o°
	o o o o
	°o

And thus you may se how Subtraction maye be tryed by Addition.

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

M. Then let me se howe can you trye Addition by
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,


\|\|	o o	o o o o	o°o
	o	o°o o o	o
	o°o o	o°o o	o°o
	o°o o o	o°o o	o°o

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


\|\|	o o	o°o
	o	o
	o°o o	o°o
	o°o o o	o°o

Then do I subtract 2000 of the fyrste summe from y^e
second summe, and then remayneth there 5000 thus.


X		°
	o	o
	o°o o	o°o
	o°o o o	o°o

Then in the thyrd lyne, I subtract y^e 100 of the fyrste
summe, from the second summe, where is onely 100 also, and
then in y^e thyrde lyne resteth nothyng. Then in the
second lyne with his space ouer hym, I fynde 80, which I shuld
subtract
119 b.
*from the other summe, then seyng there 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.


\|\|		o o o o
		o°o o o
		o°o o o
	o°o o o	o°o

Yet remayneth there in the fyrst summe 9, to be bated from the
second summe, where in that place of vnities dothe appere only 7, then I
muste bate a hygher summe, that is to saye 10, but 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
58

lowest lyne, as you se here.


\|\|	o o o o
	o°o o o
	o°o o
	o°o o

And so haue I ended this worke, and the summe appereth to
be y^e same, whiche was y^e seconde summe of my
addition, and therfore I perceaue, I haue wel done.

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 set y^e summes thus.


\|\|	o	o o
	o°o o	o o o
	o o o o	o°o
	o°	o°o o

120 b.
*And fyrste they take 6 whiche is in the lower lyne, and his space from
8 in the same roumes, in y^e second summe, and yet
there remayneth 2 counters in the lowest lyne. Then in the second lyne
must 4 be subtracte from 7, and so remayneth there 3. Then 8 in the
thyrde lyne and his space, from 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 from the seconde summe, then doth all y^e figures
stande in this order.


\|\|
	°
	o o o
	o o

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 one waye beste, so some
lyke the other: therfore you now knowyng bothe, may vse whiche you
lyst.

Multiplication by Counters.

Multiplication.
But nowe touchynge Multiplication: you shall set your
nombers 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
with the hyghest nombers of y^e 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 it, and so
with y^e next, tyll you haue done all. And yf there be any
nomber in a space, then for it
121 b.
*shall you take y^e multiplyer 5 tymes, and then must you
recken that lyne for the vnites whiche is nexte beneth that space: or
els
59

after a shorter way, you shall take only halfe the multyplyer, but then
shall you take the lyne nexte aboue that space, for the lyne of vnites:
but in suche workynge, yf 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 y^e nombers thus.


\|\|		o
	o o o	°
	o°	o o o o
	°	o o

122 a.
*Then fyrste I begynne at the 1000 in y^e hyghest roume, as yf
it were y^e fyrst place, & I take it vp, settynge downe
for it so often (that is ones) the multyplyer, which is 365, thus, as
you se here:


			o o o
			o°
X		°
	o o o		°
	o°	o o o o
	°	o o

where for the one counter taken vp from the fourth lyne, I haue
sette downe other 6, whiche make y^e summe of the
multyplyer, reckenynge that fourth lyne, as yf it were the fyrste:
whiche thyng I haue marked by the hand set at the begynnyng of
y^e 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 place stand, as yf it were y^e fyrst:

			o o o	o
			o°	o°o o
\|\|			°	o°o
	o o o			°
	o°	o o o o
	°	o o

as in this fourme you se, where I haue set this multiplycation
with y^e other: but for the ease of your
vnderstandynge, I haue set a lytell lyne betwene them: now
shulde they both in one summe stand thus.

			o o o o o
			o o o o
\|\|			o°o
	o o o		°
	o°	o o o o
	°	o o

123 a.
*Howe be it an other fourme to multyplye suche counters in
space is this: Fyrst to remoue the fynger to the lyne nexte benethe
y^e space, and then to take vp y^e
counter, and to set downe y^e 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°
°	°	°	°	°
	o°	o o o o
	°	o o

Which summes yf you do adde together into one summe, you
shal perceaue that it wyll be y^e
60

same y^t appeareth of y^e 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 1200: 4 tymes 60 are 240: and 4 tymes 5 make 20:
y^t is in all 1460, y^t shall I set downe also: as
here you se.

			°
			o o o o	o
X			o°o	o o o o
	o o o		°	o°
	o°
	°	o o

124 a.
*whiche yf I ioyne in one summe with the formar nombers, it wyll appeare
thus.

		°
		o°
		o o
o o o		o
o°
°	o o

Then to ende this multiplycation, I remoue the fynger to the lowest
lyne, where are onely 2, them do I take vp, and in theyr stede do I set
downe twyse 365, that is 730, for which I set
124 b.
*one in the space aboue the thyrd lyne for 500, and 2 more in the thyrd
lyne with that one that is there all redye, and the reste in theyr
order, and so haue I ended the hole summe thus.

		°
		o°
		o o
o o o		o°o o
o°		o o o
°

Wherby you se, that 1542 (which is the nomber of yeares syth Ch[r]ystes
incarnation) beyng multyplyed by 365 (which is the nomber of dayes in one
yeare) dothe amounte vnto 562830, which declareth y^e
nomber of daies sith Chrystes incarnation vnto the ende of
15421
yeares. (besyde 385 dayes and 12 houres for lepe yeares).

S. Now wyll I proue by an other example, as this: 40
labourers (after 6 d. y^e 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 labourers by
y^e wages of a man for one day, so wyll y^e 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.




	o o o o
o°

Where in the fyrste space is the multyplyer (y^t is one dayes
wages for one man) and in the second space is set the nomber of
the worke men to be 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.



	o o
	o o o o

So apwereth the hole dayes wages to be 240d’. that is
20 s. Then do I multiply agayn the same summe by the nomber
of dayes and fyrste I sette the nombers, thus.



	o o
o o	o o o o
o°o o

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


	°
	o°
	o o o o

Then come I to y^e seconde lyne, and take vp those 4
counters, settynge for them the multiplyer foure tymes, so wyll
the hole summe appeare thus.*
126 a.


	o°
	o°o
	o o

So is the hole wages of 40 workemen, for 28 dayes (after 6d’.
eche daye for a man) 6720d’. that is 560 s. or 28 l’i.

Division on the Counting Board.

Diuision.
M. Now if you wold proue Multiplycation, the surest way is
by Dyuision: therfore wyll I ouer passe it tyll I haue taught you
y^e arte of Diuision, whiche you shall worke thus. Fyrste
sette downe the Diuisor for feare of forgettynge, and then set the
nomber that shalbe deuided, at y^e 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.



o o		o°o o o
o o
°

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 tymes, then say I, 4
tymes 2 are 8, whyche yf I take from 9, there resteth but 1, thus



o o		o
o o
°	o o o o

And bycause I founde the diuisor 4 tymes in the diuidente, I haue
set (as you se) 4 in the myddle roume, which
127 a.
*is the place of the quotient: but now must I take the reste of the
diuisor as often out of the remayner: therfore come
62

I to the seconde lyne of the diuisor, sayeng 2 foure tymes make 8,
take 8 from 10, and there resteth 2, thus.


\|\|
	o o
	o o		o o
	°	o o o o

Then come I to the lowest nomber, which is 5, and multyply it 4 tymes,
so is it 20, that take I from 20, and there 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.

		o
\|\|		o°
	o	o o o
	o°	o o

Then begyn I at the hyghest place of the diuidente, sekynge my diuisor
there, whiche I fynde ones, Therfore set I 1 in the nether lyne.

M. Not in the 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.


\|\|
	o	o	o o o
	o°		o o

Then seke I agayne in the reste, how often I may fynde my diuisor, and I
se that in the 300 I myghte fynde 100 thre tymes, but then 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,


\|\|
	o	o
	o°
		o o

128 b.
*where I haue sette the quotient 2 in the lowest lyne: So is euery
sowldyars portion 102 d’. that is 8 s. 6 d’.

M. But yet bycause you shall perceaue iustly the reason of
Diuision, it shall be good that you do set your diuisor styll agaynst
those nombres from whiche you do take it: as by this example I
wyll declare. Yf y^e 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.


	o o	o°
X		o°o o o
		o °

Fyrst set the diuident on the ryghte hande as it oughte, and then
129 a.
*the diuisor on the lefte hande agaynst those nombers, from which
I entende to take hym fyrst as here you se, wher I haue set the diuisor
two lynes hygher then is theyr owne place.

S. This is lyke the order of diuision by the penne.
63

M. Truth you say, and nowe must I set y^e 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.


\|\|	o o		o°o o o
		o o o	°o

129 b.
*And then seke I how often the diuisor wyll be taken from the nomber
agaynste it, whiche wyll be 4 tymes and 1 remaynynge.

S. But what yf it chaunce that when the diuisor is so remoued,
it can not be ones taken out of the 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:


\|\|	o o		o
		o o o	o°
		o o o o

and in the myddle space for the quotient I set 4 in the seconde lyne,
whiche is in this worke the place of vnities.*
130 a.
Then remoue I y^e 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,


\|\|
	o o	o o o
		o o o o
		o°o o

where you may se that the hole quotient is 348 d’, that is
29 s. wherby I knowe that so moche coste the purchace of one
aker.

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.

S. Yf that be all, you shall not nede to repete agayne that,
y^t 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
practised this so well, y^t you be full perfecte, so that you
nede not to doubte in any poynte that I haue taught you, and thenne maye
I boldly enstructe you in y^e arte of fractions or broken
nomber, wherin I
64

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:

Merchants’ Casting Counters.

Merchants’ casting.
But fyrste for marchauntes fourme marke this example here,

o	o o o o
	o
o	o o o
	o
o	o o o o
	o
	o o o o o

in which I haue expressed this summe 198 l’i.2 19 s.
11 d’. So that you maye se that the lowest lyne serueth for
pennes, the next aboue for shyllynges, the thyrde for poundes,
and the fourth for scores of poundes. And farther you maye se,
that the space betwene pennes and shyllynges may receaue but one 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.
Auditors’ casting.
Now for the accompt of auditors take this example.

o	o o	o o	o
o o o	o o o	o o o	o o o
o		o	o o

where I haue expressed y^e same summe 198 l’i.
19 s. 11 d’. But here you se the pennes stande toward
y^e 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 10, of
y^e same row. But now yf you wold adde other subtracte after
any of both those sortes, yf you marke y^e order of
y^t 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
65

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,

	o o o o
	o
o	o o
o	o o o
	o o o
o	o o o o
	o
	o o
	o
	o o o
	o o
	o

which is 97869 l’i. 12 s. 9 d’ ob. q. for I had not told you
before where, nother how you shuld set downe farthynges, which (as you
se here) must be set in a voyde space sydelynge beneth the pennes: for q
one counter: for ob. 2 counters: for ob. q. 3 counters: and more
there can not be, for 4 farthynges
133 a.
*do make 1 d’. which must be set in his dewe place.

Auditors’ Casting Counters.

And yf you desyre y^e same summe after audytors maner, lo
here it is.

	o o	o		o	o	o
o o o	o o	o o o	o o o	o o o	o o	o o o
o				o			o o
							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

hand numbering as described in text

1
134 b.
*In which as you may se 1 is expressed by y^e lyttle fynger of
y^e lefte hande closely and harde croked.

2
32 is declared by lyke bowynge of the weddynge fynger
(whiche is the nexte to the lyttell fynger) together with the lytell
fynger.

3
3 is signified by the myddle fynger bowed in lyke maner, with those
other two.

4
4 is declared by the bowyng of the myddle fynger and the rynge
67

fynger, or weddynge fynger, with the other all stretched forth.

5
5 is represented by the myddle fynger onely bowed.

6
And 6 by the weddynge fynger only crooked: and this you may 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.

7
For 7 is declared by the bowynge of the lytell fynger, as is 1, 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.

8
Then to expresse 8, you shall bowe after the same maner both the lyttell
fynger and the rynge fynger.

9
And yf you bowe lyke wayes with them the myddle fynger, then doth it
betoken 9.

10
Now to expresse 10, you shall bowe your fore fynger rounde, and set the
ende of it on the hyghest ioynte of the thombe.

20
And for to expresse 20, you must set your fyngers streyght, and the ende
of your thombe to the partition of the
135 b.
*fore moste and myddle fynger.

30
30 is represented by the ioynynge together of y^e headdes of
the foremost fynger and the thombe.

40
40 is declared by settynge of the thombe crossewayes on the foremost
fynger.

50
50 is signified by ryght stretchyng forth of the fyngers ioyntly, and
applyenge of the thombes ende to the partition of the myddle fynger
and the rynge fynger, or weddynge fynger.

60
60 is formed by bendynge of the thombe croked and crossynge it with the
fore fynger.

70
70 is expressed by the bowynge of the foremost fynger, and settynge the
ende of the thombe between the 2 foremost or hyghest ioyntes of it.

80
80 is expressed by settynge of the foremost fynger crossewayes 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 set
crosse ouer y^e forefinger.

90
136 a.
*90 is signified, by bendynge the fore fynger, and settyng the ende of
it in the innermost ioynte of y^e 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,
11, 12, 13,
21, 22, 23
but this teacyed me not how to expresse 11, 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
68

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.

100
But now yf you wolde represente 100 other any nomber aboue it, you muste
do that with the ryghte hande, after this maner.

You must expresse 100 in the ryght hand, with the lytell fynger so
bowed as you dyd expresse 1 in the left hand.

200
136 b.
*And as you expressed 2 in the lefte hande, the same fasshyon in the
ryght hande doth declare 200.

300
The fourme of 3 in the ryght hand standeth for 300.

400
The fourme of 4, for 400.

500
Lykewayes the fourme of 5, for 500.

600
The fourme of 6, for 600. And to be shorte: loke how you did expresse
single vnities and tenthes in the lefte hande, so must you expresse
vnities and tenthes of hundredes, in the ryghte hande.

900
S. I vnderstande you thus: that yf I wold represent 900,
I must so fourme the fyngers of my ryghte hande, as I shuld do in
my left hand to expresse 9,
1000
And as in my lefte hand I expressed 10, so in my ryght hande must I
expresse 1000.

And so the fourme of euery tenthe in the lefte hande serueth to
expresse lyke nomber of thousandes,
4000
so y^e fourme of 40 standeth for 4000.

8000
The fourme of 80 for 8000.

9000
137 a.

*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,
69

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.

A Treatise on the Numeration of Algorism.

[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
71

and of ten that is an articule. Fyftene is componyd of fyve that is a
digyt and of ten that is an articule and so of others
. . . . . . But as to this rewle. 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 dicitur1; in qua

Talibus Indorum2 fruimur his quinque figuris.

0. 9. 8. 7. 6. 5. 4. 3. 2. 1.

Prima significat unum: duo vero secunda:

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,

Simpliciter se significat: si vero secundo,

Se decies: sursum procedas multiplicando.4

[Namque figura sequens quevis signat decies plus,

Ipsa locata loco quam significet pereunte:

Nam precedentes plus ultima significabit.]

5Post predicta scias quod tres breuiter numerorum

Distincte species sunt; nam quidam digiti sunt;

Articuli quidam; quidam quoque compositi sunt.

[Sunt digiti numeri qui citra denarium sunt;

Articuli decupli degitorum; compositi sunt

Illi qui constant ex articulis digitisque.]

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

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;

Impar si fuerit, totum sibi fiet et impar.

Septem6 sunt partes, non plures, istius artis;

Addere, subtrahere, duplare, dimidiare;

Sexta est diuidere, set quinta est multiplicare;

Radicem extrahere pars septima dicitur esse.

Subtrahis aut addis a dextris vel mediabis;

A leua dupla, diuide, multiplicaque;

Extrahe radicem semper sub parte sinistra.

Addition.
Addere si numero numerum vis, ordine tali

Incipe; scribe duas primo series numerorum

Prima sub prima recte ponendo figuram,

Et sic de reliquis facias, si sint tibi plures.

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

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.

Si tibi cifra superueniens occurrerit, illam

Deme suppositam; post illic scribe figuram:

Postea procedas reliquas addendo figuras.

Subtraction.
A numero numerum si sit tibi demere cura,

Scribe figurarum series, vt in addicione;

Maiori numero numerum suppone minorem,

Siue pari numero supponatur numerus par.

Postea si possis a prima subtrahe primam,

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,

Subtrahe totaliter numerum quem proposuisti.

Quo facto, scribe supra quicquit remanebit,

Facque novenarios de cifris, cum remanebis,

Occurrant si forte cifre, dum demseris vnum;

Postea procedas reliquas demendo figuras.

Proof.
7[Si subtracio sit bene facta probare valebis,

Quas subtraxisti primas addendo figuras.

Nam, subtractio si bene sit, primas retinebis,

Et subtractio facta tibi probat additionem.]

Duplation.
Si vis duplare numerum, sic incipe; solam

Scribe figurarum seriem, quamcumque voles que

Postea procedas primam duplando figuram;

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,

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:

Et sic de reliquis facias, si sint tibi plures.

8[Si super extremam nota sit, monadem dat eidem,

Quod tibi contingit, si primo dimidiabis.]

Mediation.
Incipe sic, si vis aliquem numerum mediare:

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;

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.

Postea procedas hac condicione secunda:9

Impar10 si fuerit hic vnum deme priori,

Inscribens quinque, nam denos significabit

Monos prædictam: si vero secunda dat vnam,

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.]

Multiplication.
Si tu per numerum numerum vis multiplicare,

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.

Mental Multiplication.

13[Per numerum si vis numerum quoque multiplicare

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

14[Articulum purum comittes articulum que]

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.]

Division.

Si vis dividere numerum, sic incipe primo;

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,

Proof.
204

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,

Square Numbers.

Cum ducis numerum per se, qui provenit inde

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.

16Post 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,

79

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;

Cube Root.
264

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.

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
84

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, 34/4First
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

w^t, 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

y^t, 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 the e-text except in the one verse selection
(App. II, Carmen de Algorismo). 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.

Footnotes give textual information such as variant readings. They
have been numbered sequentially within each title.

Sidenotes giving a running synopsis of the text have been kept
as close as possible to their original format and location.

Contents (added by transcriber)

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.

(a) Multiplication. 4600 × 23.

(b) Division: direct. 100,000 ÷ 20,023. Here each counter in turn is a separate divisor.

(c) Division by Differences. 900 ÷ 8. Here we divide by (10-2).

Division. 7800 ÷ 166.

Division. 8000 ÷ 606.

¶ Capitulum primum de significacione figurarum.

¶ Capitulum 2m de triplice divisione numerorum.

Sequitur de duplacione

¶ Sequitur de mediacione.

Sequitur de multiplicatione.

*For-sothe numeracioun is of euery numbre by competent figures an artificialle representacioun.

* ¶ The seconde dialoge of accomptynge by counters.

¶ Here foloweth the table of the arte of the hande

Contents
(added by transcriber)

The Earliest Arithmetics
in English

AMEN CORNER, E.C. 4.
1922.

(a) Multiplication. 4600
× 23.

(b) Division: direct.
100,000 ÷ 20,023. Here each counter in turn is a separate divisor.

(c) Division by Differences.
900 ÷ 8. Here we divide by (10-2).

¶ Capitulum primum de significacione
figurarum.

¶ Capitulum 2^m de triplice divisione
numerorum.

*For-sothe numeracioun is of euery
numbre by competent figures an artificialle
representacioun.

¶ Here foloweth the table
of the arte of the
hande