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Title: The Earliest Arthmetics in English
Author: Anonymous
Editor: Robert Steele
Release Date: June 1, 2008 [EBook #25664]
Language: English
Character set encoding: UTF-8
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* * * * *
* * * *
* * * * *
The Earliest Arithmetics
in English
Early English Text Society.
Extra Series, No. CXVIII.
1922 (for 1916).
THE EARLIEST ARITHMETICS
IN ENGLISH
Edited With Introduction
by
ROBERT STEELE
London:
Published for the Early English Text Society
By Humphrey Milford, Oxford University Press,
Amen Corner, E.C. 4.
1922.
[Titles (list added by transcriber):
The Crafte of Nombrynge
The Art of Nombryng
Accomptynge by Counters
The arte of nombrynge by the hande
APP. I. A Treatise on the Numeration of Algorism
APP. II. Carmen de Algorismo]
INTRODUCTION
The number of English arithmetics before the sixteenth century is very
small. This is hardly to be wondered at, as no one requiring to use even
the simplest operations of the art up to the middle of the fifteenth
century was likely to be ignorant of Latin, in which language there were
several treatises in a considerable number of manuscripts, as shown by
the quantity of them still in existence. Until modern commerce was
fairly well established, few persons required more arithmetic than
addition and subtraction, and even in the thirteenth century, scientific
treatises addressed to advanced students contemplated the likelihood of
their not being able to do simple division. On the other hand, the study
of astronomy necessitated, from its earliest days as a science,
considerable skill and accuracy in computation, not only in the
calculation of astronomical tables but in their use, a knowledge of
which latter was fairly common from the thirteenth to the sixteenth
centuries.
The arithmetics in English known to me are:--
(1) Bodl. 790 G. VII. (2653) f. 146-154 (15th c.) _inc._ “Of angrym
ther be IX figures in numbray . . .” A mere unfinished fragment,
only getting as far as Duplation.
(2) Camb. Univ. LI. IV. 14 (III.) f. 121-142 (15th c.) _inc._
“Al maner of thyngis that prosedeth ffro the frist begynnyng . . .”
(3) Fragmentary passages or diagrams in Sloane 213 f. 120-3
(a fourteenth-century counting board), Egerton 2852 f. 5-13,
Harl. 218 f. 147 and
(4) The two MSS. here printed; Eg. 2622 f. 136 and Ashmole 396
f. 48. All of these, as the language shows, are of the fifteenth
century.
The CRAFTE OF NOMBRYNGE is one of a large number of scientific
treatises, mostly in Latin, bound up together as Egerton MS. 2622 in
the British Museum Library. It measures 7” × 5”, 29-30 lines to the
page, in a rough hand. The English is N.E. Midland in dialect. It is a
translation and amplification of one of the numerous glosses on the _de
algorismo_ of Alexander de Villa Dei (c. 1220), such as that of Thomas
of Newmarket contained in the British Museum MS. Reg. 12, E. 1.
A fragment of another translation of the same gloss was printed by
Halliwell in his _Rara Mathematica_ (1835) p. 29.[1*] It corresponds, as
far as p. 71, l. 2, roughly to p. 3 of our version, and from thence to
the end p. 2, ll. 16-40.
[Footnote 1*: Halliwell printed the two sides of his leaf in the
wrong order. This and some obvious errors of transcription--
‘ferye’ for ‘ferthe,’ ‘lest’ for ‘left,’ etc., have not been
corrected in the reprint on pp. 70-71.]
The ART OF NOMBRYNG is one of the treatises bound up in the Bodleian MS.
Ashmole 396. It measures 11½” × 17¾”, and is written with thirty-three
lines to the page in a fifteenth century hand. It is a translation,
rather literal, with amplifications of the _de arte numerandi_
attributed to John of Holywood (Sacrobosco) and the translator had
obviously a poor MS. before him. The _de arte numerandi_ was printed in
1488, 1490 (_s.n._), 1501, 1503, 1510, 1517, 1521, 1522, 1523, 1582, and
by Halliwell separately and in his two editions of _Rara Mathematica_,
1839 and 1841, and reprinted by Curze in 1897.
Both these tracts are here printed for the first time, but the first
having been circulated in proof a number of years ago, in an endeavour
to discover other manuscripts or parts of manuscripts of it, Dr. David
Eugene Smith, misunderstanding the position, printed some pages in a
curious transcript with four facsimiles in the _Archiv für die
Geschichte der Naturwissenschaften und der Technik_, 1909, and invited
the scientific world to take up the “not unpleasant task” of editing it.
ACCOMPTYNGE BY COUNTERS is reprinted from the 1543 edition of Robert
Record’s Arithmetic, printed by R. Wolfe. It has been reprinted within
the last few years by Mr. F. P. Barnard, in his work on Casting
Counters. It is the earliest English treatise we have on this variety of
the Abacus (there are Latin ones of the end of the fifteenth century),
but there is little doubt in my mind that this method of performing the
simple operations of arithmetic is much older than any of the pen
methods. At the end of the treatise there follows a note on merchants’
and auditors’ ways of setting down sums, and lastly, a system of digital
numeration which seems of great antiquity and almost world-wide
extension.
After the fragment already referred to, I print as an appendix the
‘Carmen de Algorismo’ of Alexander de Villa Dei in an enlarged and
corrected form. It was printed for the first time by Halliwell in
_Rara Mathemathica_, but I have added a number of stanzas from various
manuscripts, selecting various readings on the principle that the verses
were made to scan, aided by the advice of my friend Mr. Vernon Rendall,
who is not responsible for the few doubtful lines I have conserved. This
poem is at the base of all other treatises on the subject in medieval
times, but I am unable to indicate its sources.
THE SUBJECT MATTER.
Ancient and medieval writers observed a distinction between the Science
and the Art of Arithmetic. The classical treatises on the subject, those
of Euclid among the Greeks and Boethius among the Latins, are devoted to
the Science of Arithmetic, but it is obvious that coeval with practical
Astronomy the Art of Calculation must have existed and have made
considerable progress. If early treatises on this art existed at all
they must, almost of necessity, have been in Greek, which was the
language of science for the Romans as long as Latin civilisation
existed. But in their absence it is safe to say that no involved
operations were or could have been carried out by means of the
alphabetic notation of the Greeks and Romans. Specimen sums have indeed
been constructed by moderns which show its possibility, but it is absurd
to think that men of science, acquainted with Egyptian methods and in
possession of the abacus,[2*] were unable to devise methods for its use.
[Footnote 2*: For Egyptian use see Herodotus, ii. 36, Plato, _de
Legibus_, VII.]
THE PRE-MEDIEVAL INSTRUMENTS USED IN CALCULATION.
The following are known:--
(1) A flat polished surface or tablets, strewn with sand, on which
figures were inscribed with a stylus.
(2) A polished tablet divided longitudinally into nine columns (or more)
grouped in threes, with which counters were used, either plain or marked
with signs denoting the nine numerals, etc.
(3) Tablets or boxes containing nine grooves or wires, in or on which
ran beads.
(4) Tablets on which nine (or more) horizontal lines were marked, each
third being marked off.
The only Greek counting board we have is of the fourth class and was
discovered at Salamis. It was engraved on a block of marble, and
measures 5 feet by 2½. Its chief part consists of eleven parallel lines,
the 3rd, 6th, and 9th being marked with a cross. Another section
consists of five parallel lines, and there are three rows of
arithmetical symbols. This board could only have been used with counters
(_calculi_), preferably unmarked, as in our treatise of _Accomptynge by
Counters_.
CLASSICAL ROMAN METHODS OF CALCULATION.
We have proof of two methods of calculation in ancient Rome, one by the
first method, in which the surface of sand was divided into columns by a
stylus or the hand. Counters (_calculi_, or _lapilli_), which were kept
in boxes (_loculi_), were used in calculation, as we learn from Horace’s
schoolboys (Sat. 1. vi. 74). For the sand see Persius I. 131, “Nec qui
abaco numeros et secto in pulvere metas scit risisse,” Apul. Apolog. 16
(pulvisculo), Mart. Capella, lib. vii. 3, 4, etc. Cicero says of an
expert calculator “eruditum attigisse pulverem,” (de nat. Deorum,
ii. 18). Tertullian calls a teacher of arithmetic “primus numerorum
arenarius” (de Pallio, _in fine_). The counters were made of various
materials, ivory principally, “Adeo nulla uncia nobis est eboris, etc.”
(Juv. XI. 131), sometimes of precious metals, “Pro calculis albis et
nigris aureos argenteosque habebat denarios” (Pet. Arb. Satyricon, 33).
There are, however, still in existence four Roman counting boards of a
kind which does not appear to come into literature. A typical one is of
the third class. It consists of a number of transverse wires, broken at
the middle. On the left hand portion four beads are strung, on the right
one (or two). The left hand beads signify units, the right hand one five
units. Thus any number up to nine can be represented. This instrument is
in all essentials the same as the Swanpan or Abacus in use throughout
the Far East. The Russian stchota in use throughout Eastern Europe is
simpler still. The method of using this system is exactly the same as
that of _Accomptynge by Counters_, the right-hand five bead replacing
the counter between the lines.
THE BOETHIAN ABACUS.
Between classical times and the tenth century we have little or no
guidance as to the art of calculation. Boethius (fifth century), at the
end of lib. II. of his _Geometria_ gives us a figure of an abacus of the
second class with a set of counters arranged within it. It has, however,
been contended with great probability that the whole passage is a tenth
century interpolation. As no rules are given for its use, the chief
value of the figure is that it gives the signs of the nine numbers,
known as the Boethian “apices” or “notae” (from whence our word
“notation”). To these we shall return later on.
THE ABACISTS.
It would seem probable that writers on the calendar like Bede (A.D. 721)
and Helpericus (A.D. 903) were able to perform simple calculations;
though we are unable to guess their methods, and for the most part they
were dependent on tables taken from Greek sources. We have no early
medieval treatises on arithmetic, till towards the end of the tenth
century we find a revival of the study of science, centring for us round
the name of Gerbert, who became Pope as Sylvester II. in 999. His
treatise on the use of the Abacus was written (c. 980) to a friend
Constantine, and was first printed among the works of Bede in the Basle
(1563) edition of his works, I. 159, in a somewhat enlarged form.
Another tenth century treatise is that of Abbo of Fleury (c. 988),
preserved in several manuscripts. Very few treatises on the use of the
Abacus can be certainly ascribed to the eleventh century, but from the
beginning of the twelfth century their numbers increase rapidly, to
judge by those that have been preserved.
The Abacists used a permanent board usually divided into twelve columns;
the columns were grouped in threes, each column being called an “arcus,”
and the value of a figure in it represented a tenth of what it would
have in the column to the left, as in our arithmetic of position. With
this board counters or jetons were used, either plain or, more probably,
marked with numerical signs, which with the early Abacists were the
“apices,” though counters from classical times were sometimes marked on
one side with the digital signs, on the other with Roman numerals. Two
ivory discs of this kind from the Hamilton collection may be seen at the
British Museum. Gerbert is said by Richer to have made for the purpose
of computation a thousand counters of horn; the usual number of a set of
counters in the sixteenth and seventeenth centuries was a hundred.
Treatises on the Abacus usually consist of chapters on Numeration
explaining the notation, and on the rules for Multiplication and
Division. Addition, as far as it required any rules, came naturally
under Multiplication, while Subtraction was involved in the process of
Division. These rules were all that were needed in Western Europe in
centuries when commerce hardly existed, and astronomy was unpractised,
and even they were only required in the preparation of the calendar and
the assignments of the royal exchequer. In England, for example, when
the hide developed from the normal holding of a household into the unit
of taxation, the calculation of the geldage in each shire required a sum
in division; as we know from the fact that one of the Abacists proposes
the sum: “If 200 marks are levied on the county of Essex, which contains
according to Hugh of Bocland 2500 hides, how much does each hide
pay?”[3*] Exchequer methods up to the sixteenth century were founded on
the abacus, though when we have details later on, a different and
simpler form was used.
[Footnote 3*: See on this Dr. Poole, _The Exchequer in the Twelfth
Century_, Chap. III., and Haskins, _Eng. Hist. Review_, 27, 101.
The hidage of Essex in 1130 was 2364 hides.]
The great difficulty of the early Abacists, owing to the absence of a
figure representing zero, was to place their results and operations in
the proper columns of the abacus, especially when doing a division sum.
The chief differences noticeable in their works are in the methods for
this rule. Division was either done directly or by means of differences
between the divisor and the next higher multiple of ten to the divisor.
Later Abacists made a distinction between “iron” and “golden” methods of
division. The following are examples taken from a twelfth century
treatise. In following the operations it must be remembered that a
figure asterisked represents a counter taken from the board. A zero is
obviously not needed, and the result may be written down in words.
(_a_) MULTIPLICATION. 4600 × 23.
+-----------+-----------+
| Thousands | |
+---+---+---+---+---+---+
| H | T | U | H | T | U |
| u | e | n | u | e | n |
| n | n | i | n | n | i |
| d | s | t | d | s | t |
| r | | s | r | | s |
| e | | | e | | |
| d | | | d | | |
| s | | | s | | |
+---+---+---+---+---+---+
| | | 4 | 6 | | | +Multiplicand.+
+---+---+---+---+---+---+
| | | 1 | 8 | | | 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.
+-----------+-----------+
| Thousands | |
+---+---+---+---+---+---+
| H.| T.| U.| H.| T.| U.|
+---+---+---+---+---+---+
| | 2 | | | 2 | 3 | +Divisors.+
+---+---+---+---+---+---+
| | 2 | | | | | Place greatest divisor to right of dividend.
| 1 | | | | | | +Dividend.+
| | 2 | | | | | Remainder.
| | | | 1 | | |
| | 1 | 9 | 9 | | | Another form of same.
| | | | | 8 | | Product of 1st Quotient 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.
+---+---+---+-----+---+---+
| | | |[4*]9| | | +Dividend.+
| | | |[4*]1| 8 | | Product of difference by 1st Quotient (9).
| | | | | 2 | | Product of difference by 2nd Quotient (1).
+---+---+---+-----+---+---+
| | | |[4*]1| | | Sum of 8 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.+)
+---+---+---+-----+---+---+
[Footnote 4*: These figures are removed at the next step.]
DIVISION. 7800 ÷ 166.
+---------------+---------------+
| Thousands | |
+----+----+-----+-----+----+----+
| H. | T. | U. | H. | T. | U. |
+----+----+-----+-----+----+----+
| | | | | 3 | 4 | Differences (making 200 trial
| | | | | | | divisor).
| | | | 1 | 6 | 6 | Divisors.
+----+----+-----+-----+----+----+
| | |[4*]7| 8 | | | +Dividends.+
| | | 1 | | | | Remainder of greatest dividend.
| | | | 1 | 2 | | Product of 1st difference (4)
| | | | | | | by 1st Quotient (3).
| | | | 9 | | | Product of 2nd difference (3)
| | | | | | | by 1st Quotient (3).
+----+----+-----+-----+----+----+
| | |[4*]2| 8 | 2 | | New dividends.
| | | | 3 | 4 | | Product of 1st and 2nd difference
| | | | | | | by 2nd Quotient (1).
+----+----+-----+-----+----+----+
| | |[4*]1| 1 | 6 | | New dividends.
| | | | | 2 | | Product of 1st difference by
| | | | | | | 3rd Quotient (5).
| | | | 1 | 5 | | Product of 2nd difference by
| | | | | | | 3rd Quotient (5).
+----+----+-----+-----+----+----+
| | | |[4*]3| 3 | | New dividends.
| | | | 1 | | | Remainder of greatest dividend.
| | | | | 3 | 4 | Product of 1st and 2nd difference
| | | | | | | by 4th Quotient (1).
+----+----+-----+-----+----+----+
| | | | 1 | 6 | 4 | +Remainder+ (less than divisor).
| | | | | | 1 | 4th Quotient.
| | | | | | 5 | 3rd Quotient.
| | | | | 1 | | 2nd Quotient.
| | | | | 3 | | 1st Quotient.
+----+----+-----+-----+----+----+
| | | | | 4 | 6 | +Quotient.+
+----+----+-----+-----+----+----+
[Footnote 4*: These figures are removed at the next step.]
DIVISION. 8000 ÷ 606.
+-------------+-----------+
| Thousands | |
+---+---+-----+---+---+---+
| H.| T.| U. | H.| T.| U.|
+---+---+-----+---+---+---+
| | | | | 9 | | Difference (making 700 trial divisor).
| | | | | | 4 | Difference.
| | | | 6 | | 6 | Divisors.
+---+---+-----+---+---+---+
| | |[4*]8| | | | +Dividend.+
| | | 1 | | | | Remainder of dividend.
| | | | 9 | 4 | | Product of difference 1 and 2 with
| | | | | | | 1st Quotient (1).
+---+---+-----+---+---+---+
| | |[4*]1| 9 | 4 | | New dividends.
| | | | 3 | | | Remainder of greatest dividend.
| | | | | 9 | 4 | Product of difference 1 and 2 with 2nd
| | | | | | | Quotient (1).
+---+---+-----+---+---+---+
| | |[4*]1| 3 | 3 | 4 | New dividends.
| | | | 3 | | | Remainder of greatest dividend.
| | | | | 9 | 4 | Product of difference 1 and 2 with 3rd
| | | | | | | Quotient (1).
+---+---+-----+---+---+---+
| | | | 7 | 2 | 8 | New dividends.
| | | | 6 | | 6 | Product of divisors by 4th Quotient (1).
+---+---+-----+---+---+---+
| | | | 1 | 2 | 2 | +Remainder.+
| | | | | | 1 | 4th Quotient.
| | | | | | 1 | 3rd Quotient.
| | | | | | 1 | 2nd Quotient.
| | | | | 1 | | 1st Quotient.
+---+---+-----+---+---+---+
| | | | | 1 | 3 | +Quotient.+
+---+---+-----+---+---+---+
[Footnote 4*: These figures are removed at the next step.]
The chief Abacists are Gerbert (tenth century), Abbo, and Hermannus
Contractus (1054), who are credited with the revival of the art,
Bernelinus, Gerland, and Radulphus of Laon (twelfth century). We know as
English Abacists, Robert, bishop of Hereford, 1095, “abacum et lunarem
compotum et celestium cursum astrorum rimatus,” Turchillus Compotista
(Thurkil), and through him of Guilielmus R. . . . “the best of living
computers,” Gislebert, and Simonus de Rotellis (Simon of the Rolls).
They flourished most probably in the first quarter of the twelfth
century, as Thurkil’s treatise deals also with fractions. Walcher of
Durham, Thomas of York, and Samson of Worcester are also known as
Abacists.
Finally, the term Abacists came to be applied to computers by manual
arithmetic. A MS. Algorithm of the thirteenth century (Sl. 3281,
f. 6, b), contains the following passage: “Est et alius modus secundum
operatores sive practicos, quorum unus appellatur Abacus; et modus ejus
est in computando per digitos et junctura manuum, et iste utitur ultra
Alpes.”
In a composite treatise containing tracts written A.D. 1157 and 1208, on
the calendar, the abacus, the manual calendar and the manual abacus, we
have a number of the methods preserved. As an example we give the rule
for multiplication (Claud. A. IV., f. 54 vo). “Si numerus multiplicat
alium numerum auferatur differentia majoris a minore, et per residuum
multiplicetur articulus, et una differentia per aliam, et summa
proveniet.” Example, 8 × 7. The difference of 8 is 2, of 7 is 3, the
next article being 10; 7 - 2 is 5. 5 × 10 = 50; 2 × 3 = 6. 50 + 6 = 56
answer. The rule will hold in such cases as 17 × 15 where the article
next higher is the same for both, _i.e._, 20; but in such a case as
17 × 9 the difference for each number must be taken from the higher
article, _i.e._, the difference of 9 will be 11.
THE ALGORISTS.
Algorism (augrim, augrym, algram, agram, algorithm), owes its name to
the accident that the first arithmetical treatise translated from the
Arabic happened to be one written by Al-Khowarazmi in the early ninth
century, “de numeris Indorum,” beginning in its Latin form “Dixit
Algorismi. . . .” The translation, of which only one MS. is known, was
made about 1120 by Adelard of Bath, who also wrote on the Abacus and
translated with a commentary Euclid from the Arabic. It is probable that
another version was made by Gerard of Cremona (1114-1187); the number of
important works that were not translated more than once from the Arabic
decreases every year with our knowledge of medieval texts. A few lines
of this translation, as copied by Halliwell, are given on p. 72, note 2.
Another translation still seems to have been made by Johannes
Hispalensis.
Algorism is distinguished from Abacist computation by recognising seven
rules, Addition, Subtraction, Duplation, Mediation, Multiplication,
Division, and Extraction of Roots, to which were afterwards added
Numeration and Progression. It is further distinguished by the use of
the zero, which enabled the computer to dispense with the columns of the
Abacus. It obviously employs a board with fine sand or wax, and later,
as a substitute, paper or parchment; slate and pencil were also used in
the fourteenth century, how much earlier is unknown.[5*] Algorism
quickly ousted the Abacus methods for all intricate calculations, being
simpler and more easily checked: in fact, the astronomical revival of
the twelfth and thirteenth centuries would have been impossible without
its aid.
[Footnote 5*: Slates are mentioned by Chaucer, and soon after
(1410) Prosdocimo de Beldamandi speaks of the use of a “lapis”
for making notes on by calculators.]
The number of Latin Algorisms still in manuscript is comparatively
large, but we are here only concerned with two--an Algorism in prose
attributed to Sacrobosco (John of Holywood) in the colophon of a Paris
manuscript, though this attribution is no longer regarded as conclusive,
and another in verse, most probably by Alexander de Villedieu (Villa
Dei). Alexander, who died in 1240, was teaching in Paris in 1209. His
verse treatise on the Calendar is dated 1200, and it is to that period
that his Algorism may be attributed; Sacrobosco died in 1256 and quotes
the verse Algorism. Several commentaries on Alexander’s verse treatise
were composed, from one of which our first tractate was translated, and
the text itself was from time to time enlarged, sections on proofs and
on mental arithmetic being added. We have no indication of the source on
which Alexander drew; it was most likely one of the translations of
Al-Khowarasmi, but he has also the Abacists in mind, as shewn by
preserving the use of differences in multiplication. His treatise, first
printed by Halliwell-Phillipps in his _Rara Mathematica_, is adapted for
use on a board covered with sand, a method almost universal in the
thirteenth century, as some passages in the algorism of that period
already quoted show: “Est et alius modus qui utitur apud Indos, et
doctor hujusmodi ipsos erat quidem nomine Algus. Et modus suus erat in
computando per quasdam figuras scribendo in pulvere. . . .” “Si
voluerimus depingere in pulvere predictos digitos secundum consuetudinem
algorismi . . .” “et sciendum est quod in nullo loco minutorum sive
secundorum . . . in pulvere debent scribi plusquam sexaginta.”
MODERN ARITHMETIC.
Modern Arithmetic begins with Leonardi Fibonacci’s treatise “de Abaco,”
written in 1202 and re-written in 1228. It is modern rather in the range
of its problems and the methods of attack than in mere methods of
calculation, which are of its period. Its sole interest as regards the
present work is that Leonardi makes use of the digital signs described
in Record’s treatise on _The arte of nombrynge by the hand_ in mental
arithmetic, calling it “modus Indorum.” Leonardo also introduces the
method of proof by “casting out the nines.”
DIGITAL ARITHMETIC.
The method of indicating numbers by means of the fingers is of
considerable age. The British Museum possesses two ivory counters marked
on one side by carelessly scratched Roman numerals IIIV and VIIII, and
on the other by carefully engraved digital signs for 8 and 9. Sixteen
seems to have been the number of a complete set. These counters were
either used in games or for the counting board, and the Museum ones,
coming from the Hamilton collection, are undoubtedly not later than the
first century. Frohner has published in the _Zeitschrift des Münchener
Alterthumsvereins_ a set, almost complete, of them with a Byzantine
treatise; a Latin treatise is printed among Bede’s works. The use of
this method is universal through the East, and a variety of it is found
among many of the native races in Africa. In medieval Europe it was
almost restricted to Italy and the Mediterranean basin, and in the
treatise already quoted (Sloane 3281) it is even called the Abacus,
perhaps a memory of Fibonacci’s work.
Methods of calculation by means of these signs undoubtedly have existed,
but they were too involved and liable to error to be much used.
THE USE OF “ARABIC” FIGURES.
It may now be regarded as proved by Bubnov that our present numerals are
derived from Greek sources through the so-called Boethian “apices,”
which are first found in late tenth century manuscripts. That they were
not derived directly from the Arabic seems certain from the different
shapes of some of the numerals, especially the 0, which stands for 5 in
Arabic. Another Greek form existed, which was introduced into Europe by
John of Basingstoke in the thirteenth century, and is figured by Matthew
Paris (V. 285); but this form had no success. The date of the
introduction of the zero has been hotly debated, but it seems obvious
that the twelfth century Latin translators from the Arabic were
perfectly well acquainted with the system they met in their Arabic text,
while the earliest astronomical tables of the thirteenth century I have
seen use numbers of European and not Arabic origin. The fact that Latin
writers had a convenient way of writing hundreds and thousands without
any cyphers probably delayed the general use of the Arabic notation.
Dr. Hill has published a very complete survey of the various forms
of numerals in Europe. They began to be common at the middle of the
thirteenth century and a very interesting set of family notes concerning
births in a British Museum manuscript, Harl. 4350 shows their extension.
The first is dated Mij^c. lviii., the second Mij^c. lxi., the third
Mij^c. 63, the fourth 1264, and the fifth 1266. Another example is given
in a set of astronomical tables for 1269 in a manuscript of Roger
Bacon’s works, where the scribe began to write MCC6. and crossed out
the figures, substituting the “Arabic” form.
THE COUNTING BOARD.
The treatise on pp. 52-65 is the only one in English known on the
subject. It describes a method of calculation which, with slight
modifications, is current in Russia, China, and Japan, to-day, though it
went out of use in Western Europe by the seventeenth century. In Germany
the method is called “Algorithmus Linealis,” and there are several
editions of a tract under this name (with a diagram of the counting
board), printed at Leipsic at the end of the fifteenth century and the
beginning of the sixteenth. They give the nine rules, but “Capitulum de
radicum extractione ad algoritmum integrorum reservato, cujus species
per ciffrales figuras ostenduntur ubi ad plenum de hac tractabitur.” The
invention of the art is there attributed to Appulegius the philosopher.
The advantage of the counting board, whether permanent or constructed by
chalking parallel lines on a table, as shown in some sixteenth-century
woodcuts, is that only five counters are needed to indicate the number
nine, counters on the lines representing units, and those in the spaces
above representing five times those on the line below. The Russian
abacus, the “tchatui” or “stchota” has ten beads on the line; the
Chinese and Japanese “Swanpan” economises by dividing the line into two
parts, the beads on one side representing five times the value of those
on the other. The “Swanpan” has usually many more lines than the
“stchota,” allowing for more extended calculations, see Tylor,
_Anthropology_ (1892), p. 314.
Record’s treatise also mentions another method of counter notation
(p. 64) “merchants’ casting” and “auditors’ casting.” These were adapted
for the usual English method of reckoning numbers up to 200 by scores.
This method seems to have been used in the Exchequer. 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.
+The Earliest Arithmetics in English.+
+The Crafte of Nombrynge+
_Egerton 2622._
[*leaf 136a]
Hec algorism{us} ars p{re}sens dicit{ur}; in qua
Talib{us} indor{um} fruim{ur} bis qui{n}q{ue} figuris.
[Sidenote: A derivation of Algorism. Another derivation of the word.]
This boke is called þe boke of algorym, or Augrym aft{er} lewd{er} 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 aft{er} his name he called hit algory{m}; or els anoþ{er}
cause is quy it is called Algorym, for þe latyn word of hit s.
Algorism{us} com{es} of Algos, grece, q{uid} e{st} ars, latine, craft oɳ
englis, and rides, q{uid} e{st} {nu}me{rus}, latine, A nomb{ur} oɳ
englys, inde d{icitu}r Algorism{us} p{er} addic{i}one{m} hui{us} sillabe
m{us} & subtracc{i}onem d & e, q{ua}si ars num{er}andi. ¶ fforthermor{e}
ȝe most vnd{ir}stonde þ{a}t in þis craft ben vsid teen figurys, as here
ben{e} writen for ensampul, φ 9 8 7 6 5 4 3 2 1. ¶ Expone þe too
v{er}sus afor{e}: this p{re}sent craft ys called Algorism{us}, in þe
quych we vse teen signys of Inde. Questio. ¶ Why teɳ fyguris of Inde?
Solucio. for as I haue sayd afore þai wer{e} fonde fyrst in Inde of a
kyng{e} of þat Cuntre, þ{a}t was called Algor.
[Headnote: Notation and Numeration.]
[Sidenote: v{ersus} [in margin].]
¶ Prima sig{nifica}t unu{m}; duo ve{r}o s{e}c{un}da:
¶ Tercia sig{nifica}t tria; sic procede sinistre.
¶ Don{e}c ad extrema{m} venias, que cifra voca{tur}.
+¶ Cap{itulu}m primum de significac{i}o{n}e figurar{um}.+
[Sidenote: Expo{sitio} v{ersus}.]
[Sidenote: The meaning and place of the figures. Which figure is
read first.]
In þis verse is notifide þe significac{i}on of þese figur{is}. And þus
expone the verse. Þe first signifiyth on{e}, þe secu{n}de [*leaf 136b]
signi[*]fiyth tweyn{e}, þe thryd signifiyth thre, & the fourte
signifiyth 4. ¶ And so forthe towarde þe lyft syde of þe tabul or of þe
boke þ{a}t þe figures ben{e} writen{e} in, til þat þ{o}u come to the
last figure, þ{a}t is called a cifre. ¶ Questio. In quych syde sittes þe
first figur{e}? Soluc{io}, forsothe loke quich figure is first in þe
ryȝt side of þe bok or of þe tabul, & þ{a}t same is þe first figur{e},
for þ{o}u schal write bakeward, as here, 3. 2. 6. 4. 1. 2. 5. The
fig{ur}e of 5. was first write, & he is þe first, for he sittes oɳ þe
riȝt syde. And the fig{ur}e of 3 is last. ¶ Neu{er}-þe-les wen he says
¶ P{ri}ma sig{nifica}t vnu{m} &c., þat is to say, þe first betokenes
on{e}, þe secu{n}de. 2. & fore-þ{er}-mor{e}, he vnd{ir}stondes noȝt of
þe first fig{ur}e of eu{er}y rew. ¶ But he vnd{ir}stondes þe first
figure þ{a}t is in þe nomb{ur} of þe forsayd teen figuris, þe quych is
on{e} of þ{e}se. 1. And þe secu{n}de 2. & so forth.
[Sidenote: v{ersus} [in margin].]
¶ Quelib{et} illar{um} si pr{im}o limite ponas,
¶ Simplicite{r} se significat: si v{er}o se{cun}do,
Se decies: sursu{m} {pr}ocedas m{u}ltiplicando.
¶ Na{m}q{ue} figura seque{n}s q{uam}uis signat decies pl{us}.
¶ Ipsa locata loco quam sign{ific}at p{ertin}ente.
[Transcriber’s Note:
In the following section, numerals shown in +marks+ were printed in
a different font, possibly as facsimiles of the original MS form.]
[Sidenote: Expo{sitio} [in margin].]
[Sidenote: An explanation of the principles of notation. An example:
units, tens, hundreds, thousands. How to read the number.]
¶ Expone þis v{er}se þus. Eu{er}y of þese figuris bitokens hym selfe &
no mor{e}, yf he stonde in þe first place of þe rewele / this worde
Simplicit{er} in þat verse it is no more to say but þat, & no mor{e}.
¶ If it stonde in the secu{n}de place of þe rewle, he betokens ten{e}
tymes hym selfe, as þis figur{e} 2 here 20 tokens ten tyme hym selfe,
[*leaf 137a] þat is twenty, for he hym selfe betokenes twey{ne}, & ten
tymes twene is twenty. And for he stondis oɳ þe lyft side & in þe
secu{n}de place, he betokens ten tyme hy{m} selfe. And so go forth.
¶ ffor eu{er}y fig{ure}, & he stonde aft{ur} a-noþ{er} toward the lyft
side, he schal betoken{e} ten tymes as mich mor{e} as he schul betoken &
he stode in þe place þ{ere} þat þe fig{ure} a-for{e} hym stondes. loo an
ensampull{e}. 9. 6. 3. 4. Þe fig{ure} of 4. þ{a}t hase þis schape +4.+
betokens bot hymselfe, for he stondes in þe first place. The fig{ure} of
3. þat hase þis schape +3.+ betokens ten tymes mor{e} þen he schuld & he
stode þ{ere} þ{a}t þe fig{ure} of 4. stondes, þ{a}t is thretty. The
fig{ure} of 6, þ{a}t hase þis schape +6+, betokens ten tymes mor{e} þan
he schuld & he stode þ{ere} as þe fig{ure} of +3.+ stondes, for þ{ere}
he schuld tokyn{e} bot sexty, & now he betokens ten tymes mor{e}, þat is
sex hundryth. The fig{ure} of 9. þ{a}t hase þis schape +9.+ betokens ten
tymes mor{e} þan{e} he schuld & he stode in þe place þ{ere} þe fig{ure}
of sex stondes, for þen he schuld betoken to 9. hundryth, and in þe
place þ{ere} he stondes now he betokens 9. þousande. Al þe hole nomb{ur}
is 9 thousande sex hundryth & four{e} & thretty. ¶ fforthermor{e}, when
þ{o}u schalt rede a nomb{ur} of fig{ure}, þ{o}u schalt begyn{e} at þe
last fig{ure} in the lyft side, & rede so forth to þe riȝt side as
her{e} 9. 6. 3. 4. Thou schal begyn to rede at þe fig{ure} of 9. & rede
forth þus. 9. [*leaf 137b] thousand sex hundryth thritty & foure. But
when þ{o}u schall{e} write, þ{o}u schalt be-gynne to write at þe ryȝt
side.
¶ Nil cifra sig{nifica}t s{ed} dat signa{re} sequenti.
[Sidenote: The meaning and use of the cipher.]
Expone þis v{er}se. A cifre tokens noȝt, bot he makes þe fig{ure} to
betoken þat comes aft{ur} hym mor{e} þan he schuld & he wer{e} away, as
þus 1φ. her{e} þe fig{ure} of on{e} tokens ten, & yf þe cifre wer{e}
away[{1}] & no fig{ure} by-for{e} hym he schuld token bot on{e}, for
þan he sch{ul}d stonde in þe first place. ¶ And þe cifre tokens nothyng
hym selfe. for al þe nomb{ur} of þe ylke too fig{ure}s is bot ten.
¶ Questio. Why says he þat a cifre makys a fig{ure} to signifye (tyf)
mor{e} &c. ¶ I speke for þis worde significatyf, ffor sothe it may happe
aft{ur} a cifre schuld come a-noþ{ur} cifre, as þus 2φφ. And ȝet þe
secunde cifre shuld token neu{er} þe mor{e} excep he schuld kepe þe
ord{er} of þe place. and a cifre is no fig{ure} significatyf.
+¶ Q{ua}m p{re}cedentes plus ulti{m}a significabit+ /
[Sidenote: The last figure means more than all the others,
since it is of the highest value.]
Expone þis v{er}se þus. Þe last figu{re} schal token mor{e} þan all{e}
þe oþ{er} afor{e}, thouȝt þ{ere} wer{e} a hundryth thousant figures
afor{e}, as þus, 16798. Þe last fig{ure} þat is 1. betokens ten
thousant. And all{e} þe oþ{er} fig{ure}s b{e}n bot betoken{e} bot sex
thousant seuyn{e} h{u}ndryth nynty & 8. ¶ And ten thousant is mor{e} þen
all{e} þat nomb{ur}, {er}go þe last figu{re} tokens mor{e} þan all þe
nomb{ur} afor{e}.
[Headnote: The Three Kinds of Numbers]
[*leaf 138a]
¶ Post p{re}dicta scias breuit{er} q{uod} tres num{er}or{um}
Distincte species sunt; nam quidam digiti sunt;
Articuli quidam; quidam q{uoque} compositi sunt.
¶ Capit{ulu}m 2^m de t{ri}plice divisione nu{mer}or{um}.
[Sidenote: Digits. Articles. Composites.]
¶ The auctor of þis tretis dep{ar}tys þis worde a nomb{ur} into 3
p{ar}tes. Some nomb{ur} is called digit{us} latine, a digit in englys.
So{m}me nomb{ur} is called articul{us} latine. An Articul in englys.
Some nomb{ur} is called a composyt in englys. ¶ Expone þis v{er}se. know
þ{o}u aft{ur} þe forsayd rewles þ{a}t I sayd afore, þat þ{ere} ben thre
spices of nomb{ur}. Oon{e} is a digit, Anoþ{er} is an Articul, & þe
toþ{er} a Composyt. v{er}sus.
[Headnote: Digits, Articles, and Composites.]
¶ Sunt digiti num{er}i qui cit{ra} denariu{m} s{u}nt.
[Sidenote: What are digits.]
¶ Her{e} he telles qwat is a digit, Expone v{er}su{s} sic. Nomb{ur}s
digitus ben{e} all{e} nomb{ur}s þat ben w{i}t{h}-inne ten, as nyne,
8. 7. 6. 5. 4. 3. 2. 1.
¶ Articupli decupli degito{rum}; compositi s{u}nt
Illi qui constant ex articulis degitisq{ue}.
[Sidenote: What are articles.]
¶ Her{e} he telles what is a composyt and what is an{e} articul. Expone
sic v{er}sus. ¶ Articulis ben[{2}] all{e} þ{a}t may be deuidyt into
nomb{urs} of ten & nothyng{e} leue ou{er}, as twenty, thretty, fourty,
a hundryth, a thousand, & such oþ{er}, ffor twenty may be dep{ar}tyt
in-to 2 nomb{ur}s of ten, fforty in to four{e} nomb{ur}s of ten, & so
forth.
[Sidenote: What numbers are composites.]
[*leaf 138b] Compositys beɳ nomb{ur}s þat bene componyt of a digyt & of
an articull{e} as fouretene, fyftene, sextene, & such oþ{er}. ffortene
is co{m}ponyd of four{e} þat is a digit & of ten þat is an articull{e}.
ffiftene is componyd of 5 & ten, & so of all oþ{er}, what þat þai ben.
Short-lych eu{er}y nomb{ur} þat be-gynnes w{i}t{h} a digit & endyth in a
articull{e} is a composyt, as fortene bygennyng{e} by four{e} þat is a
digit, & endes in ten.
¶ Ergo, p{ro}posito nu{mer}o tibi scriber{e}, p{ri}mo
Respicias quid sit nu{merus}; si digitus sit
P{ri}mo scribe loco digitu{m}, si compositus sit
P{ri}mo scribe loco digitu{m} post articulu{m}; sic.
[Sidenote: How to write a number, if it is a digit; if it is a
composite. How to read it.]
¶ here he telles how þ{o}u schalt wyrch whan þ{o}u schalt write a
nomb{ur}. Expone v{er}su{m} sic, & fac iuxta expon{ent}is sentencia{m};
whan þ{o}u hast a nomb{ur} to write, loke fyrst what man{er} nomb{ur} it
ys þ{a}t þ{o}u schalt write, whether it be a digit or a composit or an
Articul. ¶ If he be a digit, write a digit, as yf it be seuen, write
seuen & write þ{a}t digit in þe first place toward þe ryght side. 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 þ{o}u schal write sex & twenty. write þe digit of þe nomb{ur} in þe
first place þat is sex, and write þe articul next aft{ur} þat is twenty,
as þus 26. But whan þ{o}u schalt sowne or speke [*leaf 139a] or rede an
Composyt þou schalt first sowne þe articul & aft{ur} þe digit, as þ{o}u
seyst by þe comyn{e} speche, Sex & twenty & nouȝt twenty & sex.
v{er}sus.
¶ Articul{us} si sit, in p{ri}mo limite cifram,
Articulu{m} {vero} reliq{ui}s insc{ri}be figur{is}.
[Sidenote: How to write Articles: tens, hundreds, thousands, &c.]
¶ Here he tells how þ{o}u schal write when þe nombre þ{a}t þ{o}u hase to
write is an Articul. Expone v{er}sus sic & fac s{ecundu}m sentenciam.
Ife þe nomb{ur} þ{a}t þ{o}u hast write be an Articul, write first a
cifre & aft{ur} þe cifer write an Articull{e} þus. 2φ. fforthermor{e}
þ{o}u schalt vnd{ir}stonde yf þ{o}u haue an Articul, loke how mych he
is, yf he be w{i}t{h}-ynne an hundryth, þ{o}u schalt write bot on{e}
cifre, afore, as her{e} .9φ. If þe articull{e} be by hym-silfe & be an
hundrid euen{e}, þen schal þ{o}u write .1. & 2 cifers afor{e}, þat he
may stonde in þe thryd place, for eu{er}y fig{ure} in þe thryd place
schal token a hundrid tymes hym selfe. If þe articul be a thousant or
thousandes[{3}] and he stonde by hy{m} selfe, write afor{e} 3 cifers &
so forþ of al oþ{er}.
¶ Quolib{et} in nu{mer}o, si par sit p{ri}ma figura,
Par erit & to{tu}m, quicquid sibi co{n}ti{nua}t{ur};
Imp{ar} si fu{er}it, totu{m} tu{n}c fiet {et} impar.
[Sidenote: To tell an even number or an odd.]
¶ Her{e} he teches a gen{er}all{e} rewle þ{a}t yf þe first fig{ure} in
þe rewle of fig{ure}s token a nomb{ur} þat is euen{e} al þ{a}t nomb{ur}
of fig{ur}ys in þat rewle schal be euen{e}, as her{e} þ{o}u may see 6.
7. 3. 5. 4. Computa & p{ro}ba. ¶ If þe first [*leaf 139b] fig{ur}e token
an nomb{ur} þat is ode, all{e} þat nomb{ur} in þat rewle schall{e} be
ode, as her{e} 5 6 7 8 6 7. Computa & p{ro}ba. v{er}sus.
¶ Septe{m} su{n}t partes, no{n} pl{u}res, istius artis;
¶ Adder{e}, subt{ra}her{e}, duplar{e}, dimidiar{e},
Sextaq{ue} diuider{e}, s{ed} qui{n}ta m{u}ltiplicar{e};
Radice{m} ext{ra}her{e} p{ar}s septi{m}a dicitur esse.
[Headnote: The Seven Rules of Arithmetic.]
[Sidenote: The seven rules.]
¶ Her{e} telles þ{a}t þ{er} beɳ .7. spices or p{ar}tes 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
m{u}ltiplicacioñ. The 6 is called diuisioñ. The 7. is called extraccioñ
of þe Rote. What all þese spices ben{e} hit schall{e} be tolde
singillati{m} in her{e} caputul{e}.
¶ Subt{ra}his aut addis a dext{ri}s vel mediabis:
[Sidenote: Add, subtract, or halve, from right to left.]
Thou schal be-gynne in þe ryght side of þe boke or of a tabul. loke
wer{e} þ{o}u wul be-gynne to write latyn or englys in a boke, & þ{a}t
schall{e} be called þe lyft side of the boke, þat þ{o}u writest toward
þ{a}t side schal be called þe ryght side of þe boke. V{er}sus.
A leua dupla, diuide, m{u}ltiplica.
[Sidenote: Multiply or divide from left to right.]
Here he telles þe in quych side of þe boke or of þe tabul þ{o}u
schall{e} be-gyn{e} to wyrch duplacioñ, diuisioñ, and m{u}ltiplicacioñ.
Thou schal begyn{e} to worch in þe lyft side of þe boke or of þe tabul,
but yn what wyse þ{o}u schal wyrch in hym +dicetur singillatim in
seque{n}tib{us} capi{tulis} et de vtilitate cui{us}li{bet} art{is} & sic
Completur [*leaf 140.] p{ro}hemi{um} & sequit{ur} tractat{us} & p{ri}mo
de arte addic{ion}is que p{ri}ma ars est in ordine.+
[Headnote: The Craft of Addition.]
++Adder{e} si nu{mer}o num{e}ru{m} vis, ordine tali
Incipe; scribe duas p{rim}o series nu{mer}or{um}
P{ri}ma{m} sub p{ri}ma recte pone{n}do figura{m},
Et sic de reliq{ui}s facias, si sint tibi plures.
[Sidenote: Four things must be known: what it is; how many rows of
figures; how many cases; what is its result. How to set down the sum.]
¶ Her{e} by-gynnes þe craft of Addicioñ. In þis craft þ{o}u most knowe
foure thyng{es}. ¶ Fyrst þ{ou} most know what is addicioñ. Next þ{o}u
most know how mony rewles of figurys þou most haue. ¶ Next þ{o}u most
know how mony diue{r}s casys happes in þis craft of addicioñ. ¶ And next
qwat is þe p{ro}fet of þis craft. ¶ As for þe first þou most know þat
addicioñ is a castyng to-ged{ur} of twoo nomburys in-to on{e} nombr{e}.
As yf I aske qwat is twene & thre. Þ{o}u wyl cast þese twene nomb{re}s
to-ged{ur} & say þ{a}t it is fyue. ¶ As for þe secunde þou most know
þ{a}t þou schall{e} haue tweyne rewes of figures, on{e} vndur a-nother,
as her{e} þ{o}u mayst se.
1234
2168.
¶ As for þe thryd þou most know þ{a}t ther{e} ben foure diu{er}se cases.
As for þe forthe þ{o}u most know þ{a}t þe p{ro}fet of þis craft is to
telle what is þe hole nomb{ur} þ{a}t comes of diu{er}se nomburis. Now as
to þe texte of oure verse, he teches ther{e} how þ{o}u schal worch in
þis craft. ¶ He says yf þ{o}u wilt cast on{e} nomb{ur} to anoþ{er}
nomb{ur}, þou most by-gynne on þis wyse. ¶ ffyrst write [*leaf 140b] two
rewes of figuris & nombris so þat þ{o}u write þe first figur{e} of þe
hyer nomb{ur} euen{e} vnd{ir} the first fig{ure} of þe nether nomb{ur},
And þe secunde of þe nether nomb{ur} euen{e} vnd{ir} þe secunde of þe
hyer, & so forthe of eu{er}y fig{ur}e of both þe rewes as þ{o}u
mayst se.
123
234.
[Headnote: The Cases of the Craft of Addition.]
¶ Inde duas adde p{ri}mas hac condic{i}one:
Si digitus crescat ex addic{i}one prior{um};
P{ri}mo scribe loco digitu{m}, quicu{n}q{ue} sit ille.
[Sidenote: Add the first figures; rub out the top figure;
write the result in its place. Here is an example.]
¶ Here he teches what þ{o}u schalt do when þ{o}u hast write too rewes of
figuris on vnder an-oþ{er}, as I sayd be-for{e}. ¶ He says þ{o}u schalt
take þe first fig{ur}e of þe heyer nomb{re} & þe fyrst figur{e} of þe
neþ{er} nombre, & cast hem to-ged{er} vp-on þis condicioɳ. Thou schal
loke qweþ{er} þe nombe{r} þat comys þ{ere}-of be a digit or no. ¶ If he
be a digit þ{o}u schalt do away þe first fig{ur}e of þe hyer nomb{re},
and write þ{ere} in his stede þat he stode Inne þe digit, þ{a}t comes of
þe ylke 2 fig{ur}es, & so wrich forth oɳ oþ{er} figures yf þ{ere} be ony
moo, til þ{o}u come to þe ende toward þe lyft side. And lede þe nether
fig{ure} stonde still eu{er}-mor{e} til þ{o}u haue ydo. ffor þ{ere}-by
þ{o}u schal wyte wheþ{er} þ{o}u hast don{e} wel or no, as I schal tell
þe aft{er}ward in þe ende of þis Chapt{er}. ¶ And loke allgate þat þou
be-gynne to worch in þis Craft of [*leaf 141a] Addi[*]cioɳ in þe ryȝt
side, here is an ensampul of þis case.
1234
2142.
Caste 2 to four{e} & þat wel be sex, do away 4. & write in þe same place
þe fig{ur}e of sex. ¶ And lete þe fig{ur}e of 2 in þe nether rewe stonde
stil. When þ{o}u hast do so, cast 3 & 4 to-ged{ur} and þat wel be seuen
þ{a}t is a digit. Do away þe 3, & set þ{ere} seueɳ, and lete þe neþ{er}
fig{ure} stonde still{e}, & so worch forth bakward til þ{o}u hast ydo
all to-ged{er}.
Et si composit{us}, in limite scribe seque{n}te
Articulum, p{ri}mo digitum; q{uia} sic iubet ordo.
[Sidenote: Suppose it is a Composite, set down the digit,
and carry the tens. Here is an example.]
¶ Here is þe secunde case þ{a}t may happe in þis craft. And þe case is
þis, yf of þe casting of 2 nomburis to-ged{er}, as of þe fig{ur}e of þe
hyer rewe & of þe figure of þe neþ{er} rewe come a Composyt, how schalt
þ{ou} worch. Þ{us} þ{o}u schalt worch. Thou shalt do away þe fig{ur}e of
þe hyer nomb{er} þat was cast to þe figure of þe neþ{er} nomber. ¶ And
write þ{ere} þe digit of þe Composyt. And set þe articul of þe composit
next aft{er} þe digit in þe same rewe, yf þ{ere} be no mo fig{ur}es
aft{er}. But yf þ{ere} be mo figuris aft{er} þat digit. And þere he
schall be rekend for hym selfe. And when þ{o}u schalt adde þ{a}t ylke
figure þ{a}t berys þe articull{e} ou{er} his hed to þe figur{e} vnd{er}
hym, þ{o}u schalt cast þat articul to þe figure þ{a}t hase hym ou{er}
his hed, & þ{ere} þat Articul schal tokeɳ hym selfe. lo an Ensampull
[*leaf 141b] of all.
326
216.
Cast 6 to 6, & þ{ere}-of wil arise twelue. do away þe hyer 6 & write
þ{ere} 2, þ{a}t is þe digit of þis composit. And þe{n} write þe
articull{e} þat is ten ou{er} þe figuris hed of twene as þ{us}.
1
322
216.
Now cast þe articull{e} þ{a}t standus vpon þe fig{ur}is of twene hed to
þe same fig{ur}e, & reken þat articul bot for on{e}, and þan þ{ere} wil
arise thre. Þan cast þat thre to þe neþ{er} figure, þat is on{e}, & þat
wul be four{e}. do away þe fig{ur}e of 3, and write þ{ere} a fig{ur}e of
foure. and lete þe neþ{er} fig{ur}e stonde stil, & þan worch forth.
vn{de} {ver}sus.
¶ Articulus si sit, in p{ri}mo limite cifram,
¶ Articulu{m} v{er}o reliquis inscribe figuris,
Vel p{er} se scribas si nulla figura sequat{ur}.
[Sidenote: Suppose it is an Article, set down a cipher and carry
the tens. Here is an example.]
¶ Her{e} he puttes þe thryde case of þe craft of Addicioɳ. & þe case is
þis. yf of Addiciouɳ of 2 figuris a-ryse an Articull{e}, how schal þ{o}u
do. thou most do away þe heer fig{ur}e þ{a}t was addid to þe neþ{er},
& write þ{ere} a cifre, and sett þe articuls on þe figuris hede, yf
þ{a}t þ{ere} come ony aft{er}. 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 þ{ere} a
cifer. And sette ten vpon þe figuris hed of 2. And reken it but for on
þus.] lo an Ensampull{e}
+----+
| 1 |
| 2φ |
| 15 |
+----+
And [*leaf 142a] þan worch forth. But yf þ{ere} come no figure aft{er}
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
þ{ere} a cifre, & write aft{er} hym þe articul as þus
+----+
| 1φ |
| 5 |
+----+
And þan þ{o}u hast done.
¶ Si tibi cifra sup{er}ueniens occurrerit, illa{m}
Dele sup{er}posita{m}; fac illic scribe figura{m},
Postea procedas reliquas addendo figuras.
[Sidenote: What to do when you have a cipher in the top row.
An example of all the difficulties.]
¶ Her{e} he putt{es} þe fourt case, & it is þis, þat yf þ{ere} come a
cifer in þe hier rewe, how þ{o}u schal do. þus þ{o}u schalt do. do away
þe cifer, & sett þ{ere} þe digit þ{a}t comes of þe addiciou{n} as þus
1φφ84.
17743
In þis ensampul ben all{e} þe four{e} cases. Cast 3 to foure, þ{a}t wol
be seueɳ. do away 4. & write þ{ere} seueɳ; þan cast 4 to þe figur{e} of
8. þ{a}t wel be 12. do away 8, & sett þ{ere} 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 on{e} to a cifer, & hit wull{e} be
but on, for noȝt & on makes but on{e}. þan cast 7. þ{a}t stondes vnd{er}
þat on to hym, & þat wel be 8. do away þe cifer & þat 1. & sette þ{ere}
8. þan go forthermor{e}. cast þe oþ{er} 7 to þe cifer þ{a}t stondes
ou{er} hy{m}. þ{a}t wul be bot seuen, for þe cifer betokens noȝt. do
away þe cifer & sette þ{ere} seueɳ, [*leaf 142b] & þen go forþ{er}mor{e}
& cast 1 to 1, & þat wel be 2. do away þe hier 1, & sette þ{ere} 2. þan
hast þ{o}u do. And yf þ{o}u haue wel ydo þis nomber þat is sett
her{e}-aft{er} wel be þe nomber þat schall{e} aryse of all{e} þe
addicioɳ as her{e} 27827. ¶ Sequi{tu}r alia sp{eci}es.
[Headnote: The Craft of Subtraction.]
++A nu{mer}o num{er}u{m} si sit tibi demer{e} cura
Scribe figurar{um} series, vt in addicione.
[Sidenote: Four things to know about subtraction: the first;
the second; the third; the fourth.]
¶ This is þe Chapt{er} of subtraccioɳ, in the quych þou most know foure
nessessary thyng{es}. the first what is subtraccioɳ. þe secunde is how
mony nombers þou most haue to subt{ra}ccioɳ, the thryd is how mony
maners of cases þ{ere} may happe in þis craft of subtraccioɳ. The fourte
is qwat is þe p{ro}fet of þis craft. ¶ As for þe first, þ{o}u most know
þ{a}t subtraccioɳ is drawyng{e} of on{e} nowmb{er} oute of anoþ{er}
nomber. As for þe secunde, þou most knowe þ{a}t þou most haue two rewes
of figuris on{e} vnd{er} anoþ{er}, as þ{o}u addyst in addicioɳ. As for
þe thryd, þ{o}u moyst know þ{a}t four{e} man{er} of diu{er}se casis mai
happe in þis craft. ¶ As for þe fourt, þou most know þ{a}t þe p{ro}fet
of þis craft is whenne þ{o}u hasse taken þe lasse nomber out of þe
mor{e} to telle what þ{ere} leues ou{er} þ{a}t. & þ{o}u most be-gynne to
wyrch in þ{is} craft in þe ryght side of þe boke, as þ{o}u diddyst in
addicioɳ. V{er}sus.
¶ Maiori nu{mer}o num{er}u{m} suppone minorem,
¶ Siue pari nu{mer}o supponat{ur} num{er}us par.
[Sidenote: Put the greater number above the less.]
[*leaf 143a] ¶ Her{e} he telles þat þe hier nomber most be mor{e} þen þe
neþ{er}, or els eueɳ as mych. but he may not be lasse. And þe case is
þis, þou schalt drawe þe neþ{er} nomber out of þe hyer, & þou mayst not
do þ{a}t yf þe hier nomber wer{e} lasse þan þat. ffor þ{o}u mayst not
draw sex out of 2. But þ{o}u mast draw 2 out of sex. And þou maiste draw
twene out of twene, for þou schal leue noȝt of þe hier twene vn{de}
v{er}sus.
[Headnote: The Cases of the Craft of Subtraction.]
¶ Postea si possis a prima subt{ra}he p{ri}ma{m}
Scribens quod remanet.
[Sidenote: The first case of subtraction. Here is an example.]
Her{e} is þe first case put of subtraccioɳ, & he says þou schalt begynne
in þe ryght side, & draw þe first fig{ure} of þe neþ{er} rewe out of þe
first fig{ure} of þe hier rewe. qwether þe hier fig{ur}e be mor{e} þen
þe neþ{er}, or eueɳ as mych. And þat is notified in þe vers when he says
“Si possis.” Whan þ{o}u has þus ydo, do away þe hiest fig{ur}e & sett
þ{ere} þat leues of þe subtraccioɳ, lo an Ensampull{e}
+-----+
| 234 |
| 122 |
+-----+
draw 2 out of 4. þan leues 2. do away 4 & write þ{ere} 2, & latte þe
neþ{er} figur{e} sto{n}de stille, & so go for-by oþ{er} figuris till
þ{o}u come to þe ende, þan hast þ{o}u do.
¶ Cifram si nil remanebit.
[Sidenote: Put a cipher if nothing remains. Here is an example.]
¶ Her{e} he putt{es} þe secunde case, & hit is þis. yf it happe þ{a}t
qwen þ{o}u hast draw on neþ{er} fig{ure} out of a hier, & þ{er}e leue
noȝt aft{er} þe subt{ra}ccioɳ, þus [*leaf 143b] þou schalt do. þ{o}u
schall{e} do away þe hier fig{ur}e & write þ{ere} a cifer, as lo an
Ensampull
+----+
| 24 |
| 24 |
+----+
Take four{e} out of four{e} þan leus noȝt. þ{er}efor{e} do away þe hier
4 & set þ{ere} a cifer, þan take 2 out of 2, þan leues noȝt. do away þe
hier 2, & set þ{ere} a cifer, and so worch whar{e} so eu{er} þis happe.
Sed si no{n} possis a p{ri}ma dem{er}e p{ri}ma{m}
P{re}cedens vnu{m} de limite deme seque{n}te,
Quod demptu{m} p{ro} denario reputabis ab illo
Subt{ra}he to{ta}lem num{er}u{m} qu{em} p{ro}posuisti
Quo facto sc{ri}be super quicquid remaneb{i}t.
[Sidenote: Suppose you cannot take the lower figure from the top one,
borrow ten; take the lower number from ten; add the answer to the top
number. How to ‘Pay back’ the borrowed ten. Example.]
Her{e} he puttes þe thryd case, þe quych is þis. yf it happe þat þe
neþ{er} fig{ur}e be mor{e} þen þe hier fig{ur}e þat he schall{e} be draw
out of. how schall{e} þou do. þus þ{o}u schall{e} do. þou schall{e}
borro .1. oute of þe next fig{ur}e þat comes aft{er} in þe same rewe,
for þis case may neu{er} happ but yf þ{ere} come figures aft{er}. þan
þ{o}u schalt sett þat on ou{er} þe hier figur{es} hed, of the quych þou
woldist y-draw oute þe neyþ{er} fig{ur}e yf þ{o}u haddyst y-myȝt. Whane
þou hase þus ydo þou schall{e} rekene þ{a}t .1. for ten. ¶. And out of
þat ten þ{o}u schal draw þe neyþermost fig{ur}e, And all{e} þ{a}t leues
þou schall{e} adde to þe figur{e} on whos hed þat .1. stode. And þen
þ{o}u schall{e} do away all{e} þat, & sett þ{ere} all{e} that arisys of
the addicioɳ of þe ylke 2 fig{ur}is. And yf yt [*leaf 144a] happe þat þe
fig{ur}e of þe quych þ{o}u schalt borro on be hym self but 1. If þ{o}u
schalt þat on{e} & sett it vppoɳ þe oþ{er} figur{is} hed, and sett in
þ{a}t 1. place a cifer, yf þ{ere} come mony figur{es} aft{er}. lo an
Ensampul.
+------+
| 2122 |
| 1134 |
+------+
take 4 out of 2. it wyl not be, þerfor{e} borro on{e} of þe next
figur{e}, þ{a}t is 2. and sett þat ou{er} þe hed of þe fyrst 2. & rekene
it for ten. and þere þe secunde stondes write 1. for þ{o}u tokest on out
of hy{m}. þan take þe neþ{er} fig{ur}e, þat is 4, out of ten. And þen
leues 6. cast to 6 þe fig{ur}e of þat 2 þat stode vnd{er} þe hedde of 1.
þat was borwed & rekened for ten, and þat wylle be 8. do away þ{a}t 6 &
þat 2, & sette þ{ere} 8, & lette þe neþ{er} fig{ur}e stonde stille.
Whanne þ{o}u hast do þus, go to þe next fig{ur}e þ{a}t is now bot 1. but
first yt was 2, & þ{ere}-of was borred 1. þan take out of þ{a}t þe
fig{ur}e vnd{er} hym, þ{a}t is 3. hit wel not be. þer-for{e} borowe of
the next fig{ur}e, þe quych is bot 1. Also take & sett hym ou{er} þe
hede of þe fig{ure} þat þou woldest haue y-draw oute of þe nether
figure, þe quych was 3. & þou myȝt not, & rekene þ{a}t borwed 1 for ten
& sett in þe same place, of þe quych place þ{o}u tokest hy{m} of,
a cifer, for he was bot 1. Whanne þ{o}u hast þ{us} ydo, take out of þat
1. þ{a}t is rekent for ten, þe neþ{er} figure of 3. And þ{ere} leues 7.
[*leaf 144b] cast þe ylke 7 to þe fig{ur}e þat had þe ylke ten vpon his
hed, þe quych fig{ur}e was 1, & þat wol be 8. þan do away þ{a}t 1 and
þ{a}t 7, & write þ{ere} 8. & þan wyrch forth in oþ{er} figuris til þ{o}u
come to þe ende, & þan þ{o}u hast þe do. V{er}sus.
¶ Facque nonenarios de cifris, cu{m} remeabis
¶ Occ{ur}rant si forte cifre; dum demps{er}is vnum
¶ Postea p{ro}cedas reliquas deme{n}do figuras.
[Sidenote: A very hard case is put. Here is an example.]
¶ Her{e} he putt{es} þe fourte case, þe quych is þis, yf it happe þat þe
neþ{er} fig{ur}e, þe quych þ{o}u schalt draw out of þe hier fig{ur}e be
mor{e} pan þe hier figur ou{er} hym, & þe next fig{ur}e of two or of
thre or of foure, or how mony þ{ere} be by cifers, how wold þ{o}u do.
Þ{o}u wost wel þ{o}u most nede borow, & þ{o}u mayst not borow of þe
cifers, for þai haue noȝt þat þai may lene or spar{e}. Ergo[{4}] how
woldest þ{o}u do. Certayɳ þus most þ{o}u do, þ{o}u most borow on of þe
next figure significatyf in þat rewe, for þis case may not happe, but yf
þ{ere} come figures significatyf aft{er} the cifers. Whan þ{o}u hast
borowede þ{a}t 1 of the next figure significatyf, sett þ{a}t on ou{er}
þe hede of þ{a}t fig{ur}e of þe quych þ{o}u wold haue draw þe neþ{er}
figure out yf þ{o}u hadest myȝt, & reken it for ten as þo{u} diddest
i{n} þe oþ{er} case her{e}-a-for{e}. Whaɳ þ{o}u hast þus y-do loke how
mony cifers þ{ere} wer{e} bye-twene þat figur{e} significatyf, & þe
fig{ur}e of þe quych þ{o}u woldest haue y-draw the [*leaf 145a] neþ{er}
figure, and of eu{er}y of þe ylke cifers make a figur{e} of 9. lo an
Ensampull{e} after.
+-----+
|40002|
|10004|
+-----+
Take 4 out of 2. it wel not be. borow 1 out of be next figure
significatyf, þe quych is 4, & þen leues 3. do away þ{a}t figur{e} of 4
& write þ{ere} 3. & sett þ{a}t 1 vppon þe fig{ur}e of 2 hede, & þan take
4 out of ten, & þan þere leues 6. Cast 6 to the fig{ur}e of 2, þ{a}t wol
be 8. do away þat 6 & write þ{er}e 8. Whan þ{o}u hast þus y-do make of
eu{er}y 0 betweyn 3 & 8 a figure of 9, & þan worch forth in goddes name.
& yf þ{o}u hast wel y-do þ{o}u[{5}] schalt haue þis nomb{er}
+-----+
|39998| Sic.
|10004|
+-----+
[Headnote: How to prove the Subtraction.]
¶ Si subt{ra}cc{i}o sit b{e}n{e} facta p{ro}bar{e} valebis
Quas s{u}btraxisti p{ri}mas addendo figuras.
[Sidenote: How to prove a subtraction sum. Here is an example.
He works his proof through, and brings out a result.]
¶ Her{e} he teches þe Craft how þ{o}u schalt know, whan þ{o}u hast
subt{ra}yd, wheþ{er} þou hast wel ydo or no. And þe Craft is þis, ryght
as þ{o}u subtrayd þe neþ{er} figures fro þe hier figures, ryȝt so adde
þe same neþ{er} figures to þe hier figures. And yf þ{o}u haue well
y-wroth a-for{e} þou schalt haue þe hier nombre þe same þ{o}u haddest or
þou be-gan to worch. as for þis I bade þou schulde kepe þe neþ{er}
figures stylle. lo an [*leaf 145b] Ensampull{e} of all{e} þe 4 cases
toged{re}. worche well{e} þis case
+--------+
|40003468|.
|20004664|
+--------+
And yf þou worch well{e} whan þou hast all{e} subtrayd þe þ{a}t hier
nombr{e} her{e}, þis schall{e} be þe nombre here foloyng whan þ{o}u hast
subtrayd.
+--------+
|39998804|. [Sidenote: Our author makes a slip here (3 for 1).]
|20004664|
+--------+
And þou schalt know þ{us}. adde þe neþ{er} rowe of þe same nombre to þe
hier rewe as þus, cast 4 to 4. þat wol be 8. do away þe 4 & write þ{ere}
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, þ{a}t wel be 14. do away 8 &
write þ{ere} a fig{ur}e of 4, þat is þe digit, and write a fig{ur}e of
1. þ{a}t schall be-token ten. þ{a}t is þe articul vpon þe hed of 8 next
aft{er}, þan reken þ{a}t 1. for 1. & cast it to 8. þat schal be 9. cast
to þat 9 þe neþ{er} fig{ur}e vnd{er} þat þe quych is 4, & þat schall{e}
be 13. do away þat 9 & sett þ{er}e 3, & sett a figure of 1. þ{a}t schall
be 10 vpon þe next figur{is} hede þe quych is 9. by þe secu{n}de case
þ{a}t þ{o}u hadest in addicioɳ. þan cast 1 to 9. & þat wol be 10. do
away þe 9. & þat 1. And write þ{ere} a cifer. and write þe articull{e}
þat is 1. betokenyng{e} 10. vpon þe hede of þe next figur{e} toward þe
lyft side, þe quych [*leaf 146a] is 9, & so do forth tyl þ{o}u come to
þe last 9. take þe figur{e} of þat 1. þe quych þ{o}u schalt fynde ou{er}
þe hed of 9. & sett it ou{er} þe next figures hede þat schal be 3.
¶ Also do away þe 9. & set þ{ere} a cifer, & þen cast þat 1 þat stondes
vpon þe hede of 3 to þe same 3, & þ{a}t schall{e} make 4, þen caste to
þe ylke 4 the figur{e} in þe neyþ{er} rewe, þe quych is 2, and þat
schall{e} be 6. And þen schal þ{o}u haue an Ensampull{e} aȝeyɳ, loke &
se, & but þ{o}u haue þis same þ{o}u hase myse-wroȝt.
+--------+
|60003468|
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Sequit{ur} de duplac{i}one
[Headnote: The Craft of Duplation.]
++Si vis duplar{e} num{er}u{m}, sic i{n}cipe p{rim}o
Scribe fig{ur}ar{um} serie{m} q{ua}mcu{n}q{ue} vel{is} tu.
[Sidenote: Four things must be known in Duplation. Here they are.
Mind where you begin. Remember your rules.]
¶ This is the Chaptur{e} of duplacioɳ, in þe quych craft þ{o}u most haue
& know 4 thing{es}. ¶ Þe first þ{a}t þ{o}u most know is what is
duplacioɳ. þe secu{n}de is how mony rewes of fig{ur}es þ{o}u most haue
to þis craft. ¶ þe thryde is how many cases may[{6}] happe in þis craft.
¶ þe fourte is what is þe p{ro}fet of þe craft. ¶ As for þe first.
duplacioɳ is a doublyng{e} of a nombre. ¶ As for þe secu{n}de þ{o}u most
[*leaf 146b] haue on nombre or on rewe of figures, the quych called
nu{merus} dupland{us}. As for þe thrid þ{o}u most know þat 3 diu{er}se
cases may hap in þis craft. As for þe fourte. qwat is þe p{ro}fet of þis
craft, & þ{a}t is to know what a-risyȝt of a nombre I-doublyde.
¶ fforþ{er}-mor{e}, þ{o}u most know & take gode hede in quych side þ{o}u
schall{e} be-gyn in þis craft, or ellis þ{o}u mayst spyl all{e} þ{i}
lab{er} þ{er}e aboute. c{er}teyn þ{o}u schalt begyɳ in the lyft side in
þis Craft. thenke wel ou{er} þis verse. ¶ [{7}]A leua dupla, diuide,
m{u}ltiplica.[{7}] [[Subt{ra}has a{u}t addis a dext{ri}s {ve}l
medi{a}b{is}]] The sentens of þes verses afor{e}, as þ{o}u may see if
þ{o}u take hede. As þe text of þis verse, þat is to say, ¶ Si vis
duplare. þis is þe sentence. ¶ If þ{o}u wel double a nombre þus þ{o}u
most be-gynɳ. Write a rewe of figures of what nomb{re} þou welt.
v{er}sus.
Postea p{ro}cedas p{ri}ma{m} duplando figura{m}
Inde q{uo}d excrescit scribas vbi iusserit ordo
Iuxta p{re}cepta tibi que dant{ur} in addic{i}one.
[Sidenote: How to work a sum.]
¶ Her{e} he telles how þ{o}u schalt worch in þis Craft. he says, fyrst,
whan þ{o}u hast writen þe nombre þ{o}u schalt be-gyn at þe first
figur{e} in the lyft side, & doubull{e} þat fig{ur}e, & þe nombre þat
comes þ{ere}-of þ{o}u schalt write as þ{o}u diddyst in addicioɳ, as
¶ I schal telle þe in þe case. v{er}sus.
[Headnote: The Cases of the Craft of Duplation.]
[*leaf 147a]
¶ Nam si sit digitus in primo limite scribas.
[Sidenote: If the answer is a digit, write it in the place of the
top figure.]
¶ Her{e} is þe first case of þis craft, þe quych is þis. yf of duplacioɳ
of a figur{e} arise a digit. what schal þ{o}u do. þus þ{o}u schal do. do
away þe fig{ur}e þat was doublede, & sett þ{ere} þe diget þat comes of
þe duplacioɳ, as þus. 23. double 2, & þ{a}t wel be 4. do away þe
figur{e} of 2 & sett þ{ere} a figur{e} of 4, & so worch forth till{e}
þ{o}u come to þe ende. v{er}sus.
¶ Articul{us} si sit, in p{ri}mo limite cifram,
¶ Articulu{m} v{er}o reliquis inscribe figuris;
¶ Vel p{er} se scribas, si nulla figura sequat{ur}.
[Sidenote: If it is an article, put a cipher in the place, and
‘carry’ the tens. If there is no figure to ‘carry’ them to, write
them down.]
¶ Here is þe secunde case, þe quych is þis yf þ{ere} come an articull{e}
of þe duplacioɳ of a fig{ur}e þ{o}u schalt do ryȝt as þ{o}u diddyst in
addicioɳ, þat is to wete þat þ{o}u schalt do away þe figur{e} þat is
doublet & sett þ{ere} a cifer, & write þe articull{e} ou{er} þe next
figur{is} hede, yf þ{ere} be any aft{er}-warde toward þe lyft side as
þus. 25. begyn at the lyft side, and doubull{e} 2. þat wel be 4. do away
þat 2 & sett þere 4. þan doubul 5. þat wel be 10. do away 5, & sett
þ{ere} a 0, & sett 1 vpon þe next figur{is} hede þe quych is 4. & þen
draw downe 1 to 4 & þat woll{e} be 5, & þen do away þ{a}t 4 & þat 1,
& sett þ{ere} 5. for þat 1 schal be rekened in þe drawyng{e} toged{re}
for 1. wen [*leaf 147b] þou hast ydon þou schalt haue þis nomb{r}e 50.
yf þ{ere} come no figur{e} aft{er} þe fig{ur}e þ{a}t is addit, of þe
quych addicioɳ comes an articull{e}, þ{o}u schalt do away þe figur{e}
þ{a}t is dowblet & sett þ{ere} 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 figur{e} 5 & set þ{ere} a cifer, & sett þe articul next aft{er}
in þe same rewe toward þe lyft side, & þou schalt haue þis nombre 1023.
þen go forth & double þe oþ{er} nombers þe quych is lyȝt y-nowȝt to do.
v{er}sus.
¶ Compositus si sit, in limite sc{ri}be seq{uen}te
Articulu{m}, p{ri}mo digitu{m}; q{uia} sic iubet ordo:
Et sic de reliq{ui}s facie{n}s, si sint tibi plures.
[Sidenote: If it is a Composite, write down the digit, and ‘carry’
the tens. Here is an example.]
¶ Her{e} he putt{es} þe Thryd case, þe quych is þis, yf of duplacioɳ of
a fig{ur}e come a Composit. þ{o}u schalt do away þe fig{u}re þ{a}t is
doublet & set þ{ere} a digit of þe Composit, & sett þe articull{e}
ou{er} þe next figures hede, & aft{er} draw hym downe w{i}t{h} þe
figur{e} ou{er} whos hede he stondes, & make þ{ere}-of an nombre as
þ{o}u hast done afore, & yf þ{ere} come no fig{ur}e aft{er} þat digit
þat þ{o}u hast y-write, þa{n} set þe articull{e} next aft{er} hym in þe
same rewe as þus, 67: double 6 þat wel be 12, do away 6 & write þ{ere}
þe digit [*leaf 148a] of 12, þe quych is 2, and set þe articull{e} next
aft{er} toward þe lyft side in þe same rewe, for þ{ere} comes no
figur{e} aft{er}. þan dowble þat oþ{er} figur{e}, þe quych is 7, þat wel
be 14. the quych is a Composit. þen do away 7 þat þ{o}u doublet & sett
þe þe diget of hy{m}, the quych is 4, sett þe articull{e} ou{er} þe next
figur{es} hed, þe quych is 2, & þen draw to hym þat on, & make on nombre
þe quych schall{e} be 3. And þen yf þ{o}u haue wel y-do þ{o}u schall{e}
haue þis nombre of þe duplacioɳ, 134. v{er}sus.
¶ Si super ext{re}ma{m} nota sit monade{m} dat eid{em}
Quod t{ibi} {con}tingat si p{ri}mo dimidiabis.
[Sidenote: How to double the mark for one-half. This can only stand
over the first figure.]
¶ Her{e} he says, yf ou{er} þe fyrst fig{ur}e in þe ryȝt side be such a
merke as is her{e} made, ʷ, þ{o}u schall{e} fyrst doubull{e} þe
figur{e}, the quych stondes vnd{er} þ{a}t merke, & þen þou schalt doubul
þat merke þe quych stond{es} for haluendel on. for too haluedels makes
on, & so þ{a}t wol be on. cast þ{a}t on to þat duplacioɳ of þe figur{e}
ou{er} whos hed stode þat merke, & write it in þe same place þ{ere} þat
þe figur{e} þe quych was doublet stode, as þus 23ʷ. double 3, þat wol be
6; doubul þat halue on, & þat wol be on. cast on to 6, þ{a}t wel be 7.
do away 6 & þat 1, & sett þ{ere} 7. þan hase þou do. as for þat
figur{e}, þan go [*leaf 148b] to þe oþ{er} fig{ure} & worch forth.
& þ{o}u schall neu{er} haue such a merk but ou{er} þe hed of þe furst
figure in þe ryght side. And ȝet it schal not happe but yf it were
y-halued a-for{e}, þus þ{o}u schalt vnd{er}stonde þe verse. ¶ Si sup{er}
ext{re}ma{m} &c. Et nota, talis fig{ur}a ʷ significans medietate{m},
unitat{is} veniat, {i.e.} contingat u{e}l fiat sup{er} ext{re}ma{m},
{i.e.} sup{er} p{ri}ma{m} figura{m} in ext{re}mo sic v{er}sus dextram
ars dat: {i.e.} reddit monade{m}. {i.e.} vnitate{m} eide{m}. {i.e.}
eidem note & declina{tur} hec monos, d{i}s, di, dem, &c. ¶ Quod {er}g{o}
to{tum} ho{c} dabis monade{m} note {con}ting{et}. {i.e.} eveniet tibi si
dimidiasti, {i.e.} accipisti u{e}l subtulisti medietatem alicuius unius,
in cuius principio sint figura nu{mer}u{m} denotans i{m}pare{m} p{rim}o
{i.e.} principiis.
[Headnote: The Craft of Mediation.]
¶ Sequit{ur} de mediacione.
++Incipe sic, si vis alique{m} nu{me}ru{m} mediar{e}:
Sc{ri}be figurar{um} seriem sola{m}, velut an{te}.
[Sidenote: The four things to be known in mediation: the first the
second; the third; the fourth. Begin thus.]
¶ In þis Chapter is taȝt þe Craft of mediaciouɳ, in þe quych craft þ{o}u
most know 4 thynges. ffurst what is mediacioɳ. the secunde how mony
rewes of figur{es} þ{o}u most haue in þe wyrchyng{e} of þis craft. þe
thryde how mony diu{er}se cases may happ in þis craft.[{8}] [[the .4.
what is þe p{ro}fet of þis craft.]] ¶ As for þe furst, þ{o}u schalt
vndurstonde þat mediacioɳ is a takyng out of halfe a nomber out of a
holle nomber, [*leaf 149a] as yf þ{o}u wolde take 3 out of 6. ¶ As for
þe secunde, þ{o}u schalt know þ{a}t þ{o}u most haue on{e} rewe of
figures, & no moo, as þ{o}u hayst in þe craft of duplacioɳ. ¶ As for the
thryd, þou most vnd{er}stonde þat 5 cases may happe in þis craft. ¶ As
for þe fourte, þou schall{e} know þat the p{ro}fet of þis craft is when
þ{o}u hast take away þe haluendel of a nomb{re} to telle qwat þer{e}
schall{e} leue. ¶ Incipe sic, &c. The sentence of þis verse is þis. yf
þ{o}u wold medye, þat is to say, take halfe out of þe holle, or halfe
out of halfe, þou most begynne þ{us}. Write on{e} rewe of figur{es} of
what nombre þou wolte, as þ{o}u dyddyst be-for{e} in þe Craft of
duplacioɳ. v{er}sus.
¶ Postea p{ro}cedas medians, si p{ri}ma figura
Si par aut i{m}par videas.
[Sidenote: See if the number is even or odd.]
¶ Her{e} he says, when þ{o}u hast write a rewe of figures, þ{o}u schalt
take hede wheþ{er} þe first figur{e} be eueɳ or odde in nombre, &
vnd{er}stonde þ{a}t he spekes of þe first figure in þe ryȝt side. And
i{n} the ryght side þ{o}u schall{e} begynne in þis Craft.
¶ Quia si fu{er}it par,
Dimidiab{is} eam, scribe{n}s quicq{ui}d remanebit:
[Sidenote: If it is even, halve it, and write the answer in its
place.]
¶ Her{e} is the first case of þis craft, þe quych is þis, yf þe first
figur{e} be euen. þou schal take away fro þe figur{e} euen halfe, & do
away þat fig{ur}e and set þ{ere} þat leues ou{er}, as þus, 4. take
[*leaf 149b] halfe out of 4, & þan þ{ere} leues 2. do away 4 & sett
þ{ere} 2. þis is lyght y-nowȝt. v{er}sus.
[Headnote: The Mediation of an Odd Number.]
¶ Impar si fu{er}it vnu{m} demas mediar{e}
Quod no{n} p{re}sumas, s{ed} quod sup{er}est mediabis
Inde sup{er} tractu{m} fac demptu{m} quod no{ta}t vnu{m}.
[Sidenote: If it is odd, halve the even number less than it. Here is
an example. Then write the sign for one-half over it. Put the mark
only over the first figure.]
Her{e} is þe secunde case of þis craft, the quych is þis. yf þe first
figur{e} betoken{e} a nombre þat is odde, the quych odde schal not be
mediete, þen þ{o}u schalt medye þat nombre þat leues, when the odde of
þe same nomb{re} is take away, & write þat þ{a}t leues as þ{o}u diddest
in þe first case of þis craft. Whaɳ þ{o}u hayst write þat. for þ{a}t þat
leues, write such a merke as is her{e} ʷ vpon his hede, þe quych merke
schal betokeɳ halfe of þe odde þat was take away. lo an Ensampull. 245.
the first figur{e} her{e} is betokenyng{e} odde nombre, þe quych is 5,
for 5 is odde; þ{er}e-for{e} do away þat þ{a}t is odde, þe quych is 1.
þen leues 4. þen medye 4 & þen leues 2. do away 4. & sette þ{ere} 2,
& make such a merke ʷ upon his hede, þat is to say ou{er} his hede of 2
as þus. 242.ʷ And þen worch forth in þe oþ{er} figures tyll þ{o}u come
to þe ende. by þe furst case as þ{o}u schalt vnd{er}stonde þat þ{o}u
schalt [*leaf 150a] neu{er} make such a merk but ou{er} þe first
fig{ur}e hed in þe riȝt side. Wheþ{er} þe other fig{ur}es þat comyɳ
aft{er} hym be eueɳ or odde. v{er}sus.
[Headnote: The Cases of the Craft of Mediation.]
¶ Si monos, dele; sit t{ibi} cifra post no{ta} supra.
[Sidenote: If the first figure is one put a cipher.]
¶ Here is þe thryde case, þe quych yf the first figur{e} be a figur{e}
of 1. þ{o}u schalt do away þat 1 & set þ{ere} a cifer, & a merke ou{er}
þe cifer as þus, 241. do away 1, & sett þ{ere} a cifer w{i}t{h} a merke
ou{er} his hede, & þen hast þ{o}u ydo for þat 0. as þus 0ʷ þen worch
forth in þe oþer fig{ur}ys till þ{o}u come to þe ende, for it is lyght
as dyche water. vn{de} v{er}sus.
¶ Postea p{ro}cedas hac condic{i}one secu{n}da:
Imp{ar} si fu{er}it hinc vnu{m} deme p{ri}ori,
Inscribens quinque, nam denos significabit
Monos p{re}d{ict}am.
[Sidenote: What to do if any other figure is odd. Write a figure of
five over the next lower number’s head. Example.]
¶ Her{e} he putt{es} þe fourte case, þe quych is þis. yf it happeɳ the
secunde figur{e} betoken odde nombre, þou schal do away on of þat odde
nombre, þe quych is significatiue by þ{a}t figure 1. þe quych 1 schall
be rekende for 10. Whan þ{o}u hast take away þ{a}t 1 out of þe nombre
þ{a}t is signifiede by þat figur{e}, þ{o}u schalt medie þ{a}t þat leues
ou{er}, & do away þat figur{e} þat is medied, & sette in his styde halfe
of þ{a}t nombre. ¶ Whan þ{o}u hase so done, þ{o}u schalt write [*leaf
150b] a figure of 5 ou{er} þe next figur{es} hede by-for{e} toward þe
ryȝt side, for þat 1, þe quych made odd nombr{e}, schall stonde for ten,
& 5 is halfe of 10; so þ{o}u most write 5 for his haluendell{e}. lo an
Ensampull{e}, 4678. begyɳ in þe ryȝt side as þ{o}u most nedes. medie 8.
þen þ{o}u schalt leue 4. do away þat 8 & sette þ{ere} 4. þen out of 7.
take away 1. þe quych makes odde, & sett 5. vpon þe next figur{es} hede
afor{e} toward þe ryȝt side, þe quych is now 4. but afor{e} it was 8.
for þat 1 schal be rekenet for 10, of þe quych 10, 5 is halfe, as þou
knowest wel. Whan þ{o}u hast þus ydo, medye þ{a}t þe quych leues aft{er}
þe takying{e} away of þat þat is odde, þe quych leuyng{e} schall{e} be
3; do away 6 & sette þ{er}e 3, & þou schalt haue such a nombre
5
4634.
aft{er} go forth to þe next fig{ur}e, & medy þat, & worch forth, for it
is lyȝt ynovȝt to þe c{er}tayɳ.
¶ Si v{er}o s{e}c{un}da dat vnu{m}.
Illa deleta, sc{ri}bat{ur} cifra; p{ri}ori
¶ Tradendo quinque pro denario mediato;
Nec cifra sc{ri}batur, nisi dei{n}de fig{ur}a seq{u}at{ur}:
Postea p{ro}cedas reliq{ua}s mediando figuras
Vt sup{ra} docui, si sint tibi mille figure.
[Sidenote: If the second figure is one, put a cipher, and write
five over the next figure. How to halve fourteen.]
¶ Her{e} he putt{es} þe 5 case, þe quych is [*leaf 151a] þis: yf þe
secunde figur{e} be of 1, as þis is here 12, þou schalt do away þat 1 &
sett þ{ere} a cifer. & sett 5 ou{er} þe next fig{ur}e hede afor{e}
toward þe riȝt side, as þou diddyst afor{e}; & þat 5 schal be haldel of
þat 1, þe quych 1 is rekent for 10. lo an Ensampull{e}, 214. medye 4.
þ{a}t schall{e} be 2. do away 4 & sett þ{ere} 2. þe{n} go forth to þe
next figur{e}. þe quych is bot 1. do away þat 1. & sett þ{ere} a cifer.
& set 5 vpon þe figur{es} hed afor{e}, þe quych is nowe 2, & þen þou
schalt haue þis no{m}b{re}
5
202,
þen worch forth to þe nex fig{ur}e. And also it is no mayst{er}y yf
þ{ere} come no figur{e} after þat on is medyet, þ{o}u schalt write no 0.
ne nowȝt ellis, but set 5 ou{er} þe next fig{ur}e afor{e} toward þe
ryȝt, as þus 14. medie 4 then leues 2, do away 4 & sett þ{ere} 2. þen
medie 1. þe q{ui}ch is rekende for ten, þe halue{n}del þ{ere}-of wel be
5. sett þ{a}t 5 vpon þe hede of þ{a}t figur{e}, þe quych is now 2, & do
away þ{a}t 1, & þou schalt haue þis nombre yf þ{o}u worch wel,
5
2.
vn{de} v{er}sus.
[Headnote: How to prove the Mediation.]
¶ Si mediacio sit b{e}n{e} f{ac}ta p{ro}bar{e} valeb{is}
¶ Duplando num{er}u{m} que{m} p{ri}mo di{m}ediasti
[Sidenote: How to prove your mediation. First example. The second.
The third example. The fourth example. The fifth example.]
¶ Her{e} he telles þe how þou schalt know wheþ{er} þou hase wel ydo or
no. doubul [*leaf 151b] þe nombre þe quych þ{o}u hase mediet, and yf
þ{o}u haue wel y-medyt after þe dupleacioɳ, þou schalt haue þe same
nombre þat þ{o}u haddyst in þe tabull{e} or þ{o}u began to medye, as
þus. ¶ The furst ensampull{e} was þis. 4. þe quych I-mediet was laft 2,
þe whych 2 was write in þe place þ{a}t 4 was write afor{e}. Now
doubull{e} þat 2, & þ{o}u schal haue 4, as þ{o}u hadyst afor{e}. þe
secunde Ensampull{e} was þis, 245. When þ{o}u haddyst mediet all{e} þis
nomb{re}, yf þou haue wel ydo þou schalt haue of þ{a}t mediacioɳ þis
nombre, 122ʷ. Now doubull{e} þis nombre, & begyn in þe lyft side;
doubull{e} 1, þat schal be 2. do away þat 1 & sett þ{ere} 2. þen
doubull{e} þ{a}t oþ{er} 2 & sett þ{ere} 4, þen doubull{e} þat oþ{er} 2,
& þat wel be 4. þe{n} doubul þat merke þat stondes for halue on. & þat
schall{e} be 1. Cast þat on to 4, & it schall{e} be 5. do away þat 2 &
þat merke, & sette þ{ere} 5, & þen þ{o}u schal haue þis nombre 245. &
þis wos þe same nombur þ{a}t þ{o}u haddyst or þ{o}u began to medye, as
þ{o}u mayst se yf þou take hede. The nombre þe quych þou haddist for an
Ensampul in þe 3 case of mediacioɳ to be mediet was þis 241. whan þ{o}u
haddist medied all{e} þis nombur truly [*leaf 152a] by eu{er}y figur{e},
þou schall haue be þ{a}t mediacioɳ þis nombur 120ʷ. Now dowbul þis
nomb{ur}, & begyn in þe lyft side, as I tolde þe in þe Craft of
duplacioɳ. þus doubull{e} þe fig{ur}e of 1, þat wel be 2. do away þat 1
& sett þ{ere} 2, þen doubul þe next figur{e} afore, the quych is 2,
& þat wel be 4; do away 2 & set þ{ere} 4. þen doubul þe cifer, & þat wel
be noȝt, for a 0 is noȝt. And twyes noȝt is but noȝt. þ{ere}for{e}
doubul the merke aboue þe cifers hede, þe quych betokenes þe halue{n}del
of 1, & þat schal be 1. do away þe cifer & þe merke, & sett þ{ere} 1,
& þen þ{o}u schalt haue þis nombur 241. And þis same nombur þ{o}u
haddyst afore or þ{o}u began to medy, & yf þ{o}u take gode hede. ¶ The
next ensampul þat had in þe 4 case of mediacioɳ was þis 4678. Whan þ{o}u
hast truly ymedit all{e} þis nombur fro þe begynnyng{e} to þe endyng{e},
þ{o}u schalt haue of þe mediacioɳ þis nombur
5
2334.
Now doubul this nombur & begyn in þe lyft side, & doubull{e} 2 þat schal
be 4. do away 2 and sette þere 4; þen doubul{e} 3, þ{a}t wol be 6; do
away 3 & sett þ{ere} 6, þen doubul þat oþ{er} 3, & þat wel be 6; do away
3 & set þ{ere} [*leaf 152b] 6, þen doubul þe 4, þat welle be 8; þen
doubul 5. þe quych stondes ou{er} þe hed of 4, & þat wol be 10; cast 10
to 8, & þ{a}t schal be 18; do away 4 & þat 5, & sett þ{ere} 8, & sett
that 1, þe quych is an articul of þe Composit þe quych is 18, ou{er} þe
next figur{es} hed toward þe lyft side, þe quych is 6. drav þ{a}t 1 to
6, þe quych 1 in þe dravyng schal be rekente bot for 1, & þ{a}t 1 &
þ{a}t 6 togedur wel be 7. do away þat 6 & þat 1. the quych stondes
ou{er} his hede, & sett ther 7, & þen þou schalt haue þis nombur 4678.
And þis same nombur þ{o}u hadyst or þ{o}u began to medye, as þ{o}u mayst
see in þe secunde Ensampul þat þou had in þe 4 case of mediacioɳ, þat
was þis: when þ{o}u had mediet truly all{e} the nombur, a p{ri}ncipio
usque ad fine{m}. þ{o}u schalt haue of þat mediacioɳ þis nombur
5
102.
Now doubul 1. þat wel be 2. do away 1 & sett þ{ere} 2. þen doubul 0.
þ{a}t will be noȝt. þ{ere}for{e} take þe 5, þe quych stondes ou{er} þe
next figur{es} hed, & doubul it, & þat wol be 10. do away þe 0 þat
stondes betwene þe two fig{u}r{i}s, & sette þ{ere} in his stid 1, for
þ{a}t 1 now schal stonde in þe secunde place, wher{e} he schal betoken
10; þen doubul 2, þat wol be 4. do away 2 & sett þere 4. & [*leaf 153a]
þou schal haue þus nombur 214. þis is þe same nu{m}bur þat þ{o}u hadyst
or þ{o}u began to medye, as þ{o}u may see. And so do eu{er} mor{e}, yf
þ{o}u wil knowe wheþ{er} þou hase wel ymedyt or no. ¶. doubull{e} þe
nu{m}bur þat comes aft{er} þe mediaciouɳ, & þ{o}u schal haue þe same
nombur þ{a}t þ{o}u hadyst or þ{o}u began to medye, yf þ{o}u haue welle
ydo. or els doute þe noȝt, but yf þ{o}u haue þe same, þ{o}u hase faylide
in þ{i} Craft.
+Sequitur de multiplicatione.+
[Headnote: The Craft of Multiplication.]
[Headnote: To write down a Multiplication Sum.]
++Si tu p{er} num{er}u{m} num{er}u{m} vis m{u}ltiplicar{e}
Scribe duas q{ua}scu{nque} velis series nu{me}ror{um}
Ordo s{er}vet{ur} vt vltima m{u}ltiplicandi
Ponat{ur} sup{er} ant{er}iorem multiplicant{is}
A leua reliq{u}e sint scripte m{u}ltiplicantes.
[Sidenote: Four things to be known of Multiplication: the first:
the second: the third: the fourth. How to set down the sum. Two
sorts of Multiplication: mentally, and on paper.]
¶ Her{e} be-gynnes þe Chapt{r}e of m{u}ltiplicatioɳ, in þe quych þou
most know 4 thynges. ¶ Ffirst, qwat is m{u}ltiplicacioɳ. The secunde,
how mony cases may hap in multiplicacioɳ. The thryde, how mony rewes of
figur{es} þ{ere} most be. ¶ The 4. what is þe p{ro}fet of þis craft.
¶ As for þe first, þ{o}u schal vnd{er}stonde þat m{u}ltiplicacioɳ is a
bryngyng{e} to-ged{er} of 2 thyng{es} in on nombur, þe quych on nombur
{con}tynes so mony tymes on, howe [*leaf 153b] mony tymes þ{ere} ben
vnytees in þe nowmb{re} of þat 2, as twyes 4 is 8. now her{e} ben þe 2
nomb{er}s, of þe quych too nowmbr{e}s on is betokened be an adu{er}be,
þe quych is þe worde twyes, & þis worde thryes, & þis worde four{e}
sythes,[{9}] [[& þis wordes fyue sithe & sex sythes.]] & so furth of
such other lyke wordes. ¶ And tweyn nombres schal be tokenyde be a
nowne, as þis worde four{e} showys þes tweyɳ nombres y-broth in-to on
hole nombur, þat is 8, for twyes 4 is 8, as þ{o}u wost wel. ¶ And þes
nomb{re} 8 conteynes as oft tymes 4 as þ{ere} ben vnites in þ{a}t other
nomb{re}, þe quych is 2, for in 2 ben 2 vnites, & so oft tymes 4 ben in
8, as þ{o}u wottys wel. ¶ ffor þe secu{n}de, þ{o}u most know þat þ{o}u
most haue too rewes of figures. ¶ As for þe thryde, þ{o}u most know
þ{a}t 8 man{er} of diu{er}se case may happe in þis craft. The p{ro}fet
of þis Craft is to telle when a nomb{re} is m{u}ltiplyed be a noþ{er},
qwat co{m}mys þ{ere} of. ¶ fforthermor{e}, as to þe sentence of our{e}
verse, yf þ{o}u wel m{u}ltiply a nombur be a-noþ{er} nomb{ur}, þou
schalt write [*leaf 154a] a rewe of figures of what nomb{ur}s so eu{er}
þ{o}u welt, & þat schal be called Num{erus} m{u}ltiplicand{us}, Anglice,
þe nomb{ur} the quych to be m{u}ltiplied. þen þ{o}u schalt write
a-nother rewe of figur{e}s, by þe quych þ{o}u schalt m{u}ltiplie the
nombre þat is to be m{u}ltiplied, of þe quych nomb{ur} þe furst fig{ur}e
schal be write vnd{er} þe last figur{e} of þe nomb{ur}, þe quych is to
be m{u}ltiplied. And so write forthe toward þe lyft side, as her{e} you
may se,
+----------+
| 67324 |
| 1234 |
+----------+
And þis on{e} nomb{ur} schall{e} be called nu{meru}s m{u}ltiplicans.
An{gli}ce, þe nomb{ur} m{u}ltipliyng{e}, for he schall{e} m{u}ltiply þe
hyer nounb{ur}, as þus on{e} tyme 6. And so forth, as I schal telle the
aft{er}warde. And þou schal begyn in þe lyft side. ¶ ffor-þ{ere}-more
þou schalt vndurstonde þat þ{ere} is two man{ur}s of m{u}ltiplicacioɳ;
one ys of þe wyrchyng{e} of þe boke only in þe mynde of a mon. fyrst he
teches of þe fyrst man{er} of duplacioɳ, þe quych is be wyrchyng{e} of
tabuls. Aft{er}warde he wol teche on þe secunde man{er}. vn{de}
v{er}sus.
[Headnote: To multiply one Digit by another.]
In digitu{m} cures digitu{m} si duc{er}e ma{i}or
[*leaf 154b.]
P{er} qua{n}tu{m} distat a denis respice debes
¶ Namq{ue} suo decuplo totiens deler{e} mi{n}ore{m}
Sitq{ue} tibi nu{meru}s veniens exinde patebit.
[Sidenote: How to multiply two digits. Subtract the greater from ten;
take the less so many times from ten times itself. Example.]
¶ Her{e} he teches a rewle, how þ{o}u schalt fynde þe nounb{r}e þat
comes by þe m{u}ltiplicacioɳ of a digit be anoþ{er}. loke how mony
[vny]tes ben. bytwene þe mor{e} digit and 10. And reken ten for on
vnite. And so oft do away þe lasse nounbre out of his owne decuple, þat
is to say, fro þat nounb{r}e þat is ten tymes so mych is þe nounb{re}
þ{a}t comes of þe m{u}ltiplicacioɳ. As yf þ{o}u wol m{u}ltiply 2 be 4.
loke how mony vnitees ben by-twene þe quych is þe mor{e} nounb{re},
& be-twene ten. C{er}ten þ{ere} wel be vj vnitees by-twene 4 & ten.
yf þ{o}u reken þ{ere} w{i}t{h} þe ten þe vnite, as þou may se. 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 oþ{er} digetes til þ{o}u come to ten; & whan þ{o}u hast y-take
so mony tymes 2 out of twenty, þe quych is sex tymes, þ{o}u schal leue 8
as þ{o}u wost wel, for 6 times 2 is twelue. take [1]2 out of twenty,
& þ{ere} schal leue 8. bot yf bothe þe digett{es} [*leaf 155a] ben
y-lyech mych as her{e}. 222 or too tymes twenty, þen it is no fors quych
of hem tweyn þ{o}u take out of here decuple. als mony tymes as þ{a}t is
fro 10. but neu{er}-þe-lesse, yf þ{o}u haue hast to worch, þ{o}u schalt
haue her{e} a tabul of figures, wher{e}-by þ{o}u schalt se a-nonɳ ryght
what is þe nounbre þ{a}t comes of þe multiplicacioɳ of 2 digittes. þus
þ{o}u schalt worch in þis fig{ur}e.
[Sidenote: Better use this table, though. How to use it. The way to
use the Multiplication table.]
1|
-----
2| 4|
--------
3| 6| 9|
-----------
4| 8|12|16|
--------------
5|10|15|20|25|
-----------------
6|12|18|24|30|36|
--------------------
7|14|21|28|35|42|49|
-----------------------
8|16|24|32|40|48|56|64|
--------------------------
9|18|27|36|45|54|63|72|81|
----------------------------
1| 2| 3| 4| 5| 6| 7| 8| 9|
----------------------------
yf þe fig{ur}e, þe quych schall{e} be m{u}ltiplied, be euen{e} as mych
as þe diget be, þe quych þat oþ{er} figur{e} schal be m{u}ltiplied,
as two tymes twayɳ, or thre tymes 3. or sych other. loke qwer{e} þat
fig{ur}e sittes in þe lyft side of þe t{ri}angle, & loke qwer{e} þe
diget sittes in þe neþ{er} most rewe of þe triangle. & go fro hym
vpwarde in þe same rewe, þe quych rewe gose vpwarde til þ{o}u come
agaynes þe oþ{er} digette þat sittes in þe lyft side of þe t{ri}angle.
And þat nounbre, þe quych þou [*leaf 155b] fyn[*]des þ{ere} is þe
nounbre þat comes of the m{u}ltiplicacioɳ of þe 2 digittes, as yf þou
wold wete qwat is 2 tymes 2. loke quer{e} sittes 2 in þe lyft side i{n}
þe first rewe, he sittes next 1 in þe lyft side al on hye, as þ{o}u may
se; þe[{n}] loke qwer{e} sittes 2 in þe lowyst rewe of þe t{ri}angle,
& go fro hym vpwarde in þe same rewe tyll{e} þou come a-ȝenenes 2 in þe
hyer place, & þer þou schalt fynd ywrite 4, & þat is þe nounb{r}e þat
comes of þe multiplicacioɳ of two tymes tweyn is 4, as þow wotest
well{e}. yf þe diget. the quych is m{u}ltiplied, be mor{e} þan þe
oþ{er}, þou schalt loke qwer{e} þe mor{e} diget sittes in þe lowest rewe
of þe t{ri}angle, & go vpwarde in þe same rewe tyl[{10}] þ{o}u come
a-nendes þe lasse diget in the lyft side. And þ{ere} þ{o}u schalt fynde
þe no{m}b{r}e þat comes of þe m{u}ltiplicacioɳ; but þ{o}u schalt
vnd{er}stonde þat þis rewle, þe quych is in þis v{er}se. ¶ In digitu{m}
cures, &c., noþ{er} þis t{ri}angle schall{e} not s{er}ue, bot to fynde
þe nounbres þ{a}t comes of the m{u}ltiplicacioɳ þat comes of 2 articuls
or {com}posites, þe nedes no craft but yf þou wolt m{u}ltiply in þi
mynde. And [*leaf 156a] þere-to þou schalt haue a craft aft{er}warde,
for þou schall wyrch w{i}t{h} digettes in þe tables, as þou schalt know
aft{er}warde. v{er}sus.
[Headnote: To multiply one Composite by another.]
¶ Postea p{ro}cedas postrema{m} m{u}ltiplica{n}do
[Recte multiplicans per cu{n}ctas i{n}feriores]
Condic{i}onem tamen t{a}li q{uod} m{u}ltiplicant{es}
Scribas in capite quicq{ui}d p{ro}cesserit inde
Sed postq{uam} fuit hec m{u}ltiplicate fig{ur}e
Anteriorent{ur} serei m{u}ltiplica{n}t{is}
Et sic m{u}ltiplica velut isti m{u}ltiplicasti
Qui sequit{ur} nu{mer}u{m} sc{ri}ptu{m} quiscu{n}q{ue} figur{is}.
[Sidenote: How to multiply one number by another. Multiply the ‘last’
figure of the higher by the ‘first’ of the lower number. Set the
answer over the first of the lower: then multiply the second of the
lower, and so on. Then antery the lower number: as thus. Now multiply
by the last but one of the higher: as thus. Antery the figures again,
and multiply by five: Then add all the figures above the line: and
you will have the answer.]
¶ Her{e} he teches how þ{o}u schalt wyrch in þis craft. þou schalt
m{ul}tiplye þe last figur{e} of þe nombre, and quen þ{o}u hast so ydo
þou schalt draw all{e} þe figures of þe neþ{er} nounbre mor{e} taward þe
ryȝt side, so qwe{n} þ{o}u hast m{u}ltiplyed þe last figur{e} of þe
heyer nounbre by all{e} þe neþ{er} figures. And sette þe nounbir þat
comes þer-of ou{er} þe last figur{e} of þe neþ{er} nounb{re}, & þen þou
schalt sette al þe oþ{er} fig{ur}es of þe neþ{er} nounb{re} mor{e}
ner{e} to þe ryȝt side. ¶ And whan þou hast m{u}ltiplied þat figur{e}
þat schal be m{u}ltiplied þe next aft{er} hym by al þe neþ{er} figures.
And worch as þou dyddyst afor{e} til [*leaf 156b] þou come to þe ende.
And þou schalt vnd{er}stonde þat eu{er}y figur{e} of þe hier nounb{re}
schal be m{u}ltiplied be all{e} þe figur{e}s of the neþ{er} nounbre,
yf þe hier nounb{re} be any figur{e} þen on{e}. lo an Ensampul her{e}
folowyng{e}.
+------+
| 2465|.
|232 |
+------+
þou schalt begyne to m{u}ltiplye in þe lyft side. M{u}ltiply 2 be 2, and
twyes 2 is 4. set 4 ou{er} þe hed of þ{a}t 2, þen m{u}ltiplie þe same
hier 2 by 3 of þe nether nounbre, as thryes 2 þat schal be 6. set 6
ou{er} þe hed of 3, þan m{u}ltiplie þe same hier 2 by þat 2 þe quych
stondes vnd{er} hym, þ{a}t wol be 4; do away þe hier 2 & sette þ{ere} 4.
¶ Now þ{o}u most antery þe nether nounbre, þat is to say, þ{o}u most
sett þe neþ{er} nounbre more towarde þe ryȝt side, as þus. Take þe
neþ{er} 2 toward þe ryȝt side, & sette it eueɳ vnd{er} þe 4 of þe hyer
nounb{r}e, & ant{er}y all{e} þe figures þat comes aft{er} þat 2, as þus;
sette 2 vnd{er} þe 4. þen sett þe figur{e} of 3 þ{ere} þat þe figure of
2 stode, þe quych is now vndur þ{a}t 4 in þe hier nounbre; þen sett þe
oþer figur{e} of 2, þe quych is þe last fig{ur}e toward þe lyft side of
þe neþ{er} nomb{er} þ{ere} þe figur{e} of 3 stode. þen þ{o}u schalt haue
such a nombre.
+------+
|464465|
| 232 |
+------+
[*leaf 157a] ¶ Now m{u}ltiply 4, þe quych comes next aft{er} 6, by þe
last 2 of þe neþ{er} nounbur toward þe lyft side. as 2 tymes 4, þat wel
be 8. sette þat 8 ou{er} þe figure the quych stondes ou{er} þe hede of
þat 2, þe quych is þe last figur{e} of þe neþ{er} nounbre; þan multiplie
þat same 4 by 3, þat comes in þe neþ{er} rewe, þat wol be 12. sette þe
digit of þe composyt ou{er} þe figure þe quych stondes ou{er} þe hed of
þat 3, & sette þe articule of þis co{m}posit ou{er} al þe figures þat
stondes ou{er} þe neþ{er} 2 hede. þen m{u}ltiplie þe same 4 by þe 2 in
þe ryȝt side in þe neþ{er} nounbur, þat wol be 8. do away 4. & sette
þ{ere} 8. Eu{er} mor{e} qwen þ{o}u m{u}ltiplies þe hier figur{e} by þat
figur{e} þe quych stondes vnd{er} hym, þou schalt do away þat hier
figur{e}, & sett þer þat nounbre þe quych comes of m{u}ltiplicacioɳ of
ylke digittes. Whan þou hast done as I haue byde þe, þ{o}u schalt haue
suych an ord{er} of figur{e} as is her{e},
+--------+
| 1 |.
| 82 |
|4648[65]|
| 232 |
+--------+
þen take and ant{er}y þi neþ{er} figures. And sett þe fyrst fig{ur}e of
þe neþ{er} figures[{11}] vndre be figur{e} of 6. ¶ And draw al þe oþ{er}
figures of þe same rewe to hym-warde, [*leaf 157b] as þ{o}u diddyst
afore. þen m{u}ltiplye 6 be 2, & sett þat þe quych comes ou{er}
þ{ere}-of ou{er} al þe oþ{er} figures hedes þat stondes ou{er} þat 2.
þen m{u}ltiply 6 be 3, & sett all{e} þat comes þ{ere}-of vpon all{e} þe
figur{e}s hedes þat standes ou{er} þat 3; þa{n} m{u}ltiplye 6 be 2, þe
quych stondes vnd{er} þat 6, þen do away 6 & write þ{ere} þe digitt of
þe composit þat schal come þ{ere}of, & sette þe articull ou{er} all{e}
þe figures þat stondes ou{er} þe hede of þat 3 as her{e},
+------+
| 11 |
| 121 |
| 828 |
|464825|
| 232 |
+------+
þen ant{er}y þi figures as þou diddyst afor{e}, and m{u}ltipli 5 be 2,
þat wol be 10; sett þe 0 ou{er} all þe figures þ{a}t stonden ou{er} þat
2, & sett þ{a}t 1. ou{er} the next figures hedes, all{e} on hye towarde
þe lyft side. þen m{u}ltiplye 5 be 3. þat wol be 15, write 5 ou{er} þe
figures hedes þat stonden ou{er} þ{a}t 3, & sett þat 1 ou{er} þe next
figur{e}s hedes toward þe lyft side. þen m{u}ltiplye 5 be 2, þat wol be
10. do away þat 5 & sett þ{ere} a 0, & sett þat 1 ou{er} þe figures
hedes þat stonden ou{er} 3. And þen þou schalt haue such a nounbre as
here stondes aftur.[*leaf 158a]
+------+
| 11 |
| 1101 |
| 1215 |
| 82820|
|4648 |
| 232|
+------+
¶ Now draw all{e} þese figures downe toged{er} as þus, 6.8.1. & 1 draw
to-gedur; þat wolle be 16, do away all{e} þese figures saue 6. lat hym
stonde, for þow þ{o}u take hym away þou most write þer þe same aȝene.
þ{ere}for{e} late hym stonde, & sett 1 ou{er} þe figur{e} hede of 4
toward þe lyft side; þen draw on to 4, þat woll{e} be 5. do away þat 4 &
þat 1, & sette þ{ere} 5. þen draw 4221 & 1 toged{ur}, þat wol be 10. do
away all{e} þat, & write þere þat 4 & þat 0, & sett þat 1 ou{er} þe next
figur{es} hede toward þe lyft side, þe quych is 6. þen draw þat 6 & þat
1 togedur, & þat wolle be 7; do away 6 & sett þ{ere} 7, þen draw 8810 &
1, & þat wel be 18; do away all{e} þe figures þ{a}t stondes ou{er} þe
hede of þat 8, & lette 8 stonde stil, & write þat 1 ou{er} þe next
fig{u}r{is} hede, þe quych is a 0. þen do away þat 0, & sett þ{ere} 1,
þe quych stondes ou{er} þe 0. hede. þen draw 2, 5, & 1 toged{ur}, þat
woll{e} be 8. þen do away all{e} þat, & write þ{ere} 8. ¶ And þen þou
schalt haue þis nounbre, 571880.
[Headnote: The Cases of this Craft.]
[*leaf 158b]
¶ S{ed} cu{m} m{u}ltiplicabis, p{ri}mo sic e{st} op{er}andu{m},
Si dabit articulu{m} tibi m{u}ltiplicacio solu{m};
P{ro}posita cifra su{m}ma{m} t{ra}nsferre meme{n}to.
[Sidenote: What to do if the first multiplication results in an
article.]
¶ Her{e} he puttes þe fyrst case of þis craft, þe quych is þis: yf
þ{ere} come an articulle of þe m{u}ltiplicacioɳ ysette befor{e} the
articull{e} in þe lyft side as þus
+---+
| 51|.
|23 |
+---+
multiplye 5 by 2, þat wol be 10; sette ou{er} þe hede of þat 2 a 0,
& sett þat on, þat is þe articul, in þe lyft side, þat is next hym, þen
þ{o}u schalt haue þis nounbre
+----+
|1051|.
| 23 |
+----+
¶ And þen worch forth as þou diddist afore. And þ{o}u schalt
vnd{er}stonde þat þ{o}u schalt write no 0. but whan þat place where þou
schal write þat 0 has no figure afore hy{m} noþ{er} aft{er}. v{er}sus.
¶ Si aut{em} digitus excreu{er}it articul{us}q{ue}.
Articul{us}[{12}] sup{ra}p{osit}o digito salit vltra.
[Sidenote: What to do if the result is a composite number.]
¶ Her{e} is þe secunde case, þe quych is þis: yf hit happe þat þ{ere}
come a composyt, þou schalt write þe digitte ou{er} þe hede of þe
neþ{er} figur{e} by þe quych þ{o}u multipliest þe hier figure; and sett
þe articull{e} next hym toward þe lyft side, as þou diddyst afore, as
þ{us}
+---+
| 83|.
|83 |
+---+
Multiply 8 by 8, þat wol be 64. Write þe 4 ou{er} 8, þat is to say,
ou{er} þe hede of þe neþ{er} 8; & set 6, þe quych [*leaf 159a] is an
articul, next aft{er}. And þen þou schalt haue such a nounb{r}e as is
her{e},
+-----------+
| 6483[{13}]|,
| 83 |
+-----------+
And þen worch forth.
¶ Si digitus t{amen} ponas ip{su}m sup{er} ip{s}am.
[Sidenote: What if it be a digit.]
¶ Her{e} is þe thryde case, þe quych is þis: yf hit happe þat of þi
m{u}ltiplicaciouɳ come a digit, þ{o}u schalt write þe digit ou{er} þe
hede of þe neþ{er} figur{e}, by the quych þou m{u}ltipliest þe hier{e}
figur{e}, for þis nedes no Ensampul.
¶ Subdita m{u}ltiplica non hanc que [incidit] illi
Delet ea{m} penit{us} scribens quod p{ro}uenit inde.
[Sidenote: The fourth case of the craft.]
¶ Her{e} is þe 4 case, þe quych is: yf hit be happe þat þe neþ{er}
figur{e} schal multiplye þat figur{e}, þe quych stondes ou{er} þat
figures hede, þou schal do away þe hier figur{e} & sett þ{er}e þat
þ{a}t comys of þ{a}t m{u}ltiplicacioɳ. As yf þ{er}e come of þat
m{u}ltiplicacioɳ an articuls þou schalt write þere þe hier figur{e}
stode a 0. ¶ And write þe articuls in þe lyft side, yf þat hit be a
digit write þ{er}e a digit. yf þat h{i}t be a composit, write þe digit
of þe composit. And þe articul in þe lyft side. al þis is lyȝt y-nowȝt,
þ{er}e-for{e} þer nedes no Ensampul.
¶ S{ed} si m{u}ltiplicat alia{m} ponas sup{er} ip{s}am
Adiu{n}ges num{er}u{m} que{m} p{re}bet duct{us} ear{um}.
[Sidenote: The fifth case of the craft.]
¶ Her{e} is þe 5 case, þe quych is þis: yf [*leaf 159b] þe neþ{er}
figur{e} schul m{u}ltiplie þe hier, and þat hier figur{e} is not recte
ou{er} his hede. And þat neþ{er} figur{e} hase oþ{er} figures, or on
figure ou{er} his hede by m{u}ltiplicacioɳ, þat hase be afor{e}, þou
schalt write þat nounbre, þe quych comes of þat, ou{er} all{e} þe ylke
figures hedes, as þus here:
+-----+
| 236|
|234 |
+-----+
Multiply 2 by 2, þat wol be 4; set 4 ou{er} þe hede of þat 2. þen[{14}]
m{u}ltiplies þe hier 2 by þe neþ{er} 3, þat wol be 6. set ou{er} his
hede 6, multiplie þe hier 2 by þe neþ{er} 4, þat wol be 8. do away þe
hier 2, þe quych stondes ou{er} þe hede of þe figur{e} of 4, and set
þ{er}e 8. And þou schalt haue þis nounb{re} here
+-------+
| 46836 |
| 234 |
+-------+
And antery þi figur{e}s, þat is to say, set þi neþ{er} 4 vnd{er} þe hier
3, and set þi 2 other figures ner{e} hym, so þat þe neþ{er} 2 stonde
vnd{ur} 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 forthermor{e}, And m{u}ltiplye þat