Chapter I: Part 1

[Transcriber’s Note:

This e-text includes characters that will only display in UTF-8 (Unicode) text readers:

ȝ, ſ (yogh, long s)
ɳ, łł (n with curl, crossed l: see below)
φ (Greek phi: see below)
ʷ (small raised “w”)

If any of these characters do not display properly, or if the apostrophes and quotation marks in this paragraph appear as garbage, make sure your text reader’s “character set” or “file encoding” is set to Unicode (UTF-8). You may also need to change the default font.

In _The Crafte of Nombrynge_, final “n” was sometimes written with an extra curl. It has been rendered as ɳ for visual effect; the character is not intended to convey phonetic information. In the same selection, the numeral “0” was sometimes printed as Greek φ (phi); this has been retained for the e-text. Double “l” with a line is shown as łł. The first few occurrences of “d” (for “pence”) were printed with a decorative curl. The letter is shown with the same “d’” used in the remainder of the text.

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

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

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

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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.]