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+Project Gutenberg's The Earliest Arithmetics in English, by Anonymous
+
+This eBook is for the use of anyone anywhere at no cost and with
+almost no restrictions whatsoever.  You may copy it, give it away or
+re-use it under the terms of the Project Gutenberg License included
+with this eBook or online at www.gutenberg.org
+
+
+Title: The Earliest Arithmetics in English
+
+Author: Anonymous
+
+Editor: Robert Steele
+
+Release Date: June 1, 2008 [EBook #25664]
+
+Language: English
+
+Character set encoding: UTF-8
+
+*** START OF THIS PROJECT GUTENBERG EBOOK THE EARLIEST ARITHMETICS IN ENGLISH ***
+
+
+
+
+Produced by Louise Hope, David Starner and the Online
+Distributed Proofreading Team at http://www.pgdp.net
+
+
+
+
+
+[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 [Illustration]
+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.]
+
+
+       *       *       *       *       *
+           *       *       *       *
+       *       *       *       *       *
+
+
+  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|
+  |20004664|
+  +--------+
+
+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 hier 3 by 2, þat wol be 6,
+set þ{a}t 6 þe quych stondes ou{er} þe hede of þat 2, And þen worch as I
+taȝt þe afore.
+
+  [*leaf 160a]
+
+  ¶ Si sup{ra}posita cifra debet m{u}ltiplicar{e}
+     Prorsus ea{m} deles & ibi scribi cifra debet.
+
+  [Sidenote: The sixth case of the craft.]
+
+¶ Her{e} is þe 6 case, þe quych is þis: yf hit happe þat þe figur{e} by
+þe quych þou schal m{u}ltiplye þe hier figur{e}, þe quych stondes ryght
+ou{er} hym by a 0, þou schalt do away þat figur{e}, þe quych ou{er} þat
+cifre hede. ¶ And write þ{ere} þat nounbre þat comes of þe
+m{u}ltiplicacioɳ as þus, 23. do away 2 and sett þ{er}e a 0. vn{de}
+v{er}sus.
+
+  ¶ Si cifra m{u}ltiplicat alia{m} posita{m} sup{er} ip{s}am
+    Sitq{ue} locus sup{ra} vacu{us} sup{er} hanc cifra{m} fiet.
+
+  [Sidenote: The seventh case of the craft.]
+
+¶ Her{e} is þe 7 case, þe quych is þis: yf a 0 schal m{u}ltiply a
+figur{e}, þe quych stondes not recte ou{er} hym, And ou{er} þat 0 stonde
+no thyng, þou schalt write ou{er} þat 0 anoþ{er} 0 as þus:
+
+  +----+
+  |  24|
+  |03  |
+  +----+
+
+multiplye 2 be a 0, it wol be nothyng{e}. write þere a 0 ou{er} þe hede
+of þe neþ{er} 0, And þen worch forth til þou come to þe ende.
+
+  ¶ Si sup{ra}[{15}] fuerit cifra sem{per} e{st} p{re}t{er}eunda.
+
+  [Sidenote: The eighth case of the craft.]
+
+¶ Her{e} is þe 8 case, þe quych is þis: yf þ{ere} be a 0 or mony cifers
+in þe hier rewe, þ{o}u schalt not m{u}ltiplie hem, bot let hem stonde.
+And antery þe figures beneþe to þe next figur{e} sygnificatyf as þus:
+
+  +-----+
+  |00032|.
+  |22   |
+  +-----+
+
+Ou{er}-lepe all{e} þese cifers & sett þat [*leaf 160b] neþ{er} 2 þat
+stondes toward þe ryght side, and sett hym vnd{ur} þe 3, and sett þe
+oþ{er} nether 2 nere hym, so þat he stonde vnd{ur} þe thrydde 0, þe
+quych stondes next 3. And þan worch. vnd{e} v{er}sus.
+
+  ¶ Si dubites, an sit b{e}n{e} m{u}ltiplicac{i}o facta,
+    Diuide totalem nu{mer}u{m} p{er} multiplicante{m}.
+
+  [Sidenote: How to prove the multiplication.]
+
+¶ Her{e} he teches how þou schalt know wheþ{er} þou hase wel I-do or no.
+And he says þat þou schalt deuide all{e} þe nounb{r}e þat comes of þe
+m{u}ltiplicacioɳ by þe neþ{er} figures. And þen þou schalt haue þe same
+nounbur þat þ{o}u hadyst in þe begynnyng{e}. but ȝet þou hast not þe
+craft of dyuisioɳ, but þ{o}u schalt haue hit aft{er}warde.
+
+  ¶ P{er} num{er}u{m} si vis nu{mer}u{m} q{u}oq{ue} m{u}ltiplicar{e}
+  ¶ T{antu}m p{er} normas subtiles absq{ue} figuris
+    Has normas pot{er}is p{er} v{er}sus scir{e} sequentes.
+
+  [Sidenote: Mental multiplication.]
+
+¶ Her{e} he teches þe to m{u}ltiplie be þowȝt figures in þi mynde. And
+þe sentence of þis v{er}se is þis: yf þo{u} wel m{u}ltiplie on nounbre
+by anoþ{er} in þi mynde, þ{o}u schal haue þ{er}eto rewles in þe v{er}ses
+þat schal come aft{er}.
+
+  ¶ Si tu p{er} digitu{m} digitu{m} vis m{u}ltiplicar{e}
+    Re{gula} p{re}cedens dat qualit{er} est op{er}andu{m}.
+
+  [Sidenote: Digit by digit is easy.]
+
+¶ Her{e} he teches a rewle as þou hast afor{e} to m{u}ltiplie a digit be
+anoþ{er}, as yf þou wolde wete qwat is sex tymes 6. þou [*leaf 161a]
+schalt wete by þe rewle þat I taȝt þe befor{e}, yf þou haue mynde
+þ{er}of.
+
+  ¶ Articulu{m} si p{er} reliquu{m} reliquu{m} vis m{u}lti{plica}r{e}
+    In p{ro}p{r}iu{m} digitu{m} debet vt{er}q{ue} resolui.
+  ¶ Articul{us} digitos post se m{u}ltiplicantes
+    Ex digit{us} quociens retenerit m{u}ltipli{ca}r{i}
+    Articuli faciu{n}t tot centu{m} m{u}ltiplicati.
+
+  [Sidenote: The first case of the craft. Article by article; an
+  example: another example:]
+
+    [Headnote: How to work subtly without Figures.]
+
+  [Sidenote: Mental multiplication. Another example. Another example.
+  Notation. Notation again. Mental multiplication.]
+
+¶ Her{e} he teches þe furst rewle, þe quych is þis: yf þou wel
+m{u}ltiplie an articul be anoþ{er}, so þat both þe articuls bene
+w{i}t{h}-Inne an hundreth, þus þ{o}u schalt do. take þe digit of bothe
+the articuls, for eu{er}y articul hase a digit, þen m{u}ltiplye þat on
+digit by þat oþ{er}, and loke how mony vnytes ben in þe nounbre þat
+comes of þe m{u}ltiplicacioɳ of þe 2 digittes, & so mony hundrythes ben
+in þe nounb{re} þat schal come of þe m{u}ltiplicacioɳ of þe ylke 2
+articuls as þus. yf þ{o}u wold wete qwat is ten tymes ten. take þe digit
+of ten, þe quych is 1; take þe digit of þat oþ{er} ten, þe quych is on.
+¶ Also m{u}ltiplie 1 be 1, as on tyme on þat is but 1. In on is but on
+vnite as þou wost welle, þ{er}efor{e} ten tymes ten is but a hundryth.
+¶ Also yf þou wold wete what is twenty tymes 30. take þe digit of
+twenty, þat is 2; & take þe digitt of thrytty, þat is 3. m{u}ltiplie 3
+be 2, þat is 6. Now in 6 ben 6 vnites, ¶ And so mony hundrythes ben in
+20 tymes 30[*leaf 161b], þ{ere}for{e} 20 tymes 30 is 6 hundryth eueɳ.
+loke & se. ¶ But yf it be so þat on{e} articul be w{i}t{h}-Inne an
+hundryth, or by-twene an hundryth and a thowsande, so þat it be not a
+þowsande fully. þen loke how mony vnytes ben in þe nounbur þat comys of
+þe m{u}ltiplicacioɳ [{16}]And so mony tymes[{16}] of 2 digitt{es} of
+ylke articuls, so mony thowsant ben in þe nounbre, the qwych comes of þe
+m{u}ltiplicacioɳ. And so mony tymes ten thowsand schal be in þe nounbre
+þat comes of þe m{u}ltiplicacion of 2 articuls, as yf þ{o}u wold wete
+qwat is 4 hundryth tymes [two hundryth]. Multiply 4 be 2,[{17}] þat wol
+be 8. in 8 ben 8 vnites. ¶ And so mony tymes ten thousand be in 4
+hundryth tymes [2][{17}] hundryth, þ{a}t is 80 thousand. Take hede,
+I schall telle þe a gen{e}rall{e} rewle whan þ{o}u hast 2 articuls, And
+þou wold wete qwat comes of þe m{u}ltiplicacioɳ of hem 2. m{u}ltiplie þe
+digit of þ{a}t on articuls, and kepe þat nounbre, þen loke how mony
+cifers schuld go befor{e} þat on articuls, and he wer{e} write. Als mony
+cifers schuld go befor{e} þat other, & he wer{e} write of cifers. And
+haue all{e} þe ylke cifers toged{ur} in þi mynde, [*leaf 162a] a-rowe
+ychoɳ aftur other, and in þe last plase set þe nounbre þat comes of þe
+m{u}ltiplicacioɳ of þe 2 digittes. And loke in þi mynde in what place he
+stondes, wher{e} in þe secunde, or in þe thryd, or in þe 4, or wher{e}
+ellis, and loke qwat þe figures by-token in þat place; & so mych is þe
+nounbre þat comes of þe 2 articuls y-m{u}ltiplied to-ged{ur} as þus:
+yf þ{o}u wold wete what is 20 thousant tymes 3 þowsande. m{u}ltiply þe
+digit of þat articull{e} þe quych is 2 by þe digitte of þat oþ{er}
+articul þe quych is 3, þat wol be 6. þen loke how mony cifers schal go
+to 20 thousant as hit schuld be write in a tabul. c{er}tainly 4 cifers
+schuld go to 20 þowsant. ffor þis figure 2 in þe fyrst place betokenes
+twene. ¶ In þe secunde place hit betokenes twenty. ¶ In þe 3. place hit
+betokenes 2 hundryth. .¶. In þe 4 place 2 thousant. ¶ In þe 5 place
+h{i}t betokenes twenty þousant. þ{ere}for{e} he most haue 4 cifers
+a-for{e} hym þat he may sto{n}de in þe 5 place. kepe þese 4 cifers in
+thy mynde, þen loke how mony cifers goɳ to 3 thousant. Certayn to 3
+thousante [*leaf 162b] goɳ 3 cifers afor{e}. Now cast ylke 4 cifers þat
+schuld go to twenty thousant, And thes 3 cifers þat schuld go afor{e} 3
+thousant, & sette hem in rewe ychoɳ aft{er} oþ{er} in þi mynde, as þai
+schuld stonde in a tabull{e}. And þen schal þou haue 7 cifers; þen sett
+þat 6 þe quych comes of þe m{u}ltiplicacioɳ of þe 2 digitt{es} aft{u}r
+þe ylke cifers in þe 8 place as yf þat hit stode in a tabul. And loke
+qwat a figur{e} of 6 schuld betoken in þe 8 place. yf hit wer{e} in a
+tabul & so mych it is. & yf þat figure of 6 stonde in þe fyrst place he
+schuld betoken but 6. ¶ In þe 2 place he schuld betoken sexty. ¶ In the
+3 place he schuld betokeɳ sex hundryth. ¶ In þe 4 place sex thousant.
+¶ In þe 5 place sexty þowsant. ¶ In þe sext place sex hundryth þowsant.
+¶ In þe 7 place sex þowsant thousant{es}. ¶ In þe 8 place sexty þowsant
+thousantes. þ{er}for{e} sett 6 in octauo loco, And he schal betoken
+sexty þowsant thousantes. And so mych is twenty þowsant tymes 3
+thousant, ¶ And þis rewle is gen{er}all{e} for all{e} man{er} of
+articuls, Whethir þai be hundryth or þowsant; but þ{o}u most know well
+þe craft of þe wryrchyng{e} in þe tabull{e} [*leaf 163a] or þou know to
+do þus in þi mynde aftur þis rewle. Thou most þat þis rewle holdyþe note
+but wher{e} þ{ere} ben 2 articuls and no mo of þe quych ayther of hem
+hase but on figur{e} significatyf. As twenty tymes 3 thousant or 3
+hundryth, and such oþ{ur}.
+
+  ¶ Articulum digito si m{u}ltiplicare o{portet}
+    Articuli digit[i sumi quo multiplicate]
+    Debem{us} reliquu{m} quod m{u}ltiplicat{ur} ab ill{is}
+    P{er} reliq{u}o decuplu{m} sic su{m}ma{m} later{e} neq{ui}b{i}t.
+
+  [Sidenote: The third case of the craft; an example.]
+
+¶ Her{e} he puttes þe thryde rewle, þe quych is þis. yf þ{o}u wel
+m{u}ltiply in þi mynde, And þe Articul be a digitte, þou schalt loke þat
+þe digitt be w{i}t{h}-Inne an hundryth, þen þou schalt m{u}ltiply the
+digitt of þe Articulle by þe oþer digitte. And eu{er}y vnite in þe
+nounbre þat schall{e} come þ{ere}-of schal betoken ten. As þus: yf þat
+þ{o}u wold wete qwat is twyes 40. m{u}ltiplie þe digitt{e} of 40, þe
+quych is 4, by þe oþ{er} diget, þe quych is 2. And þat wolle be 8. And
+in þe nombre of 8 ben 8 vnites, & eu{er}y of þe ylke vnites schuld
+stonde for 10. þ{ere}-fore þ{ere} schal be 8 tymes 10, þat wol be 4
+score. And so mony is twyes 40. ¶ If þe articul be a hundryth or be 2
+hundryth And a þowsant, so þat hit be notte a thousant, [*leaf 163b]
+worch as þo{u} dyddyst afor{e}, saue þ{o}u schalt rekene eu{er}y vnite
+for a hundryth.
+
+  ¶ In nu{mer}u{m} mixtu{m} digitu{m} si ducer{e} cures
+    Articul{us} mixti sumat{ur} deinde resoluas
+    In digitu{m} post fac respectu de digitis
+    Articul{us}q{ue} docet excrescens in diriua{n}do
+    In digitu{m} mixti post ducas m{u}ltiplica{n}te{m}
+  ¶ De digitis vt norma [{18}][docet] de [hunc]
+    Multiplica si{mu}l et sic postea summa patebit.
+
+  [Sidenote: The fourth case of the craft: Composite by digit. Mental
+  multiplication.]
+
+Here he puttes þe 4 rewle, þe quych is þis: yf þou m{u}ltipliy on
+composit be a digit as 6 tymes 24, [{19}]þen take þe diget of þat
+composit, & m{u}ltiply þ{a}t digitt by þat oþ{er} diget, and kepe þe
+nomb{ur} þat comes þ{ere}-of. þen take þe digit of þat composit,
+& m{u}ltiply þat digit by anoþ{er} diget, by þe quych þ{o}u hast
+m{u}ltiplyed þe diget of þe articul, and loke qwat comes þ{ere}-of. þen
+take þ{o}u þat nounbur, & cast hit to þat other nounbur þat þ{o}u
+secheste as þus yf þou wel wete qwat comes of 6 tymes 4 & twenty.
+multiply þat articull{e} of þe composit by þe digit, þe quych is 6,
+as yn þe thryd rewle þ{o}u was tauȝt, And þat schal be 6 scor{e}. þen
+m{u}ltiply þe diget of þe {com}posit, [*leaf 164a] þe quych is 4, and
+m{u}ltiply þat by þat other diget, þe quych is 6, as þou wast tauȝt in
+þe first rewle, yf þ{o}u haue mynde þ{er}of, & þat wol be 4 & twenty.
+cast all ylke nounburs to-ged{ir}, & hit schal be 144. And so mych is 6
+tymes 4 & twenty.
+
+    [Headnote: How to multiply without Figures.]
+
+  ¶ Duct{us} in articulu{m} num{erus} si {com}posit{us} sit
+    Articulu{m} puru{m} comites articulu{m} q{u}o{que}
+    Mixti pro digit{is} post fiat [et articulus vt]
+    Norma iubet [retinendo quod extra dicta ab illis]
+    Articuli digitu{m} post tu mixtu{m} digitu{m} duc
+    Re{gula} de digitis nec p{re}cipit articul{us}q{ue}
+    Ex quib{us} exc{re}scens su{m}me tu iunge p{ri}ori
+    Sic ma{n}ifesta cito fiet t{ibi} su{m}ma petita.
+
+  [Sidenote: The fifth case of the craft: Article by Composite.
+  An example.]
+
+¶ Her{e} he puttes þe 5 rewle, þe quych is þis: yf þ{o}u wel m{u}ltiply
+an Articul be a composit, m{u}ltiplie þat Articul by þe articul of þe
+composit, and worch as þou wos tauȝt in þe secunde rewle, of þe quych
+rewle þe v{er}se begynnes þus. ¶ Articulu{m} si p{er} Relicu{m} vis
+m{u}ltiplicare. þen m{u}ltiply þe diget of þe composit by þat oþ{ir}
+articul aft{ir} þe doctrine of þe 3 rewle. take þ{er}of gode hede,
+I p{ra}y þe as þus. Yf þ{o}u wel wete what is 24 tymes ten. Multiplie
+ten by 20, þat wel be 2 hundryth. þen m{u}ltiply þe diget of þe 10, þe
+quych is 1, by þe diget of þe composit, þe quych is 4, & þ{a}t [*leaf
+164b] wol be 4. þen reken eu{er}y vnite þat is in 4 for 10, & þat schal
+be 40. Cast 40 to 2 hundryth, & þat wol be 2 hundryth & 40. And so mych
+is 24 tymes ten.
+
+    [Headnote: How to work without Figures.]
+
+  ¶ Compositu{m} num{er}u{m} mixto si[c] m{u}ltiplicabis
+    Vndecies tredeci{m} sic e{st} ex hiis op{er}andum
+    In reliquu{m} p{rimu}m demu{m} duc post in eund{em}
+    Vnu{m} post den{u}m duc in t{ri}a dei{n}de p{er} vnu{m}
+    Multiplices{que} dem{u}m int{ra} o{mn}ia m{u}ltiplicata
+    In su{m}ma decies q{ua}m si fu{er}it t{ibi} doces
+    Multiplicandor{um} de normis sufficiunt h{ec}.
+
+  [Sidenote: The sixth case of the craft: Composite by Composite.
+  Mental multiplication. An example of the sixth case of the craft.]
+
+¶ Here he puttes þe 6 rewle, & þe last of all{e} multiplicacioɳ,
+þe quych is þis: yf þ{o}u wel m{u}ltiplye a {com}posit by a-noþ{er}
+composit, þou schalt do þus. m{u}ltiplie þ{a}t on composit, qwych þ{o}u
+welt of the twene, by þe articul of þe toþ{er} composit, as þ{o}u wer{e}
+tauȝt in þe 5 rewle, þen m{u}ltiplie þ{a}t same composit, þe quych þou
+hast m{u}ltiplied by þe oþ{er} articul, by þe digit of þe oþ{er}
+composit, as þ{o}u was tauȝt in þe 4 rewle. As þus, yf þou wold wete
+what is 11 tymes 13, as þ{o}u was tauȝt in þe 5 rewle, & þat schal be an
+hundryth & ten, aft{er}warde m{u}ltiply þat same co{m}posit þ{a}t þ{o}u
+hast m{u}ltiplied, þe quych is a .11. And m{u}ltiplye hit be þe digit of
+þe oþ{er} composit, þe quych is 3, for 3 is þe digit of 13, And þat wel
+be 30. þen take þe digit of þat composit, þe quych composit þou
+m{u}ltiplied by þe digit of þ{a}t oþ{er} {com}posit, [*leaf 165a] þe
+quych is a 11. ¶ Also of the quych 11 on is þe digit. m{u}ltiplie þat
+digitt by þe digett of þat oth{er} composit, þe quych diget is 3,
+as þ{o}u was tauȝt in þe first rewle i{n} þe begynnyng{e} of þis craft.
+þe quych rewle begynn{es} “In digitu{m} cures.” And of all{e} þe
+m{u}ltiplicacioɳ of þe 2 digitt comys thre, for onys 3 is but 3. Now
+cast all{e} þese nounbers toged{ur}, the quych is þis, a hundryth & ten
+& 30 & 3. And al þat wel be 143. Write 3 first in þe ryght side. And
+cast 10 to 30, þat wol be 40. set 40 next aft{ur} towarde þe lyft side,
+And set aftur a hundryth as her{e} an Ensampull{e}, 143.
+
+(Cetera desunt.)
+
+
+FOOTNOTES (The Crafte of Nombrynge):
+
+  [1: In MS, ‘awiy.’]
+  [2: ‘ben’ repeated in MS.]
+  [3: In MS. ‘thausandes.’]
+  [4: Perhaps “So.”]
+  [5: ‘hali’ marked for erasure in MS.]
+  [6: ‘moy’ in MS.]
+  [7: ‘Subt{ra}has a{u}t addis a dext{ri}s {ve}l medi{a}b{is}’ added
+    on margin of MS.]
+  [8: After ‘craft’ insert ‘the .4. what is þe p{ro}fet of þis craft.’]
+  [9: After ‘sythes’ insert ‘& þis wordes fyue sithe & sex sythes.’]
+  [10: ‘t’l’ marked for erasure before ‘tyl’ in MS.]
+  [11: Here ‘of þe same rew’ is marked for erasure in MS.]
+  [12: ‘s{ed}’ deleted in MS.]
+  [13: 6883 in MS.]
+  [14: ‘þen’ overwritten on ‘þat’ marked for erasure.]
+  [15: ‘Supra’ inserted in MS. in place of ‘cifra’ marked for erasure.]
+  [16--16: Marked for erasure in MS.]
+  [17: 4 in MS.]
+  [18: docet. decet MS.]
+  [19: ‘4 times 4’ in MS.]
+
+
+
+
++The Art of Nombryng.+
+
+A TRANSLATION OF
+
++John of Holywood’s De Arte Numerandi.+
+
+
+[_Ashmole MS. 396, fol. 48._]
+
+  +Boys seying in the begynnyng of his Arsemetrik{e}:--All{e}
+    [*Fol. 48.] thynges that ben{e} fro the first begynnyng of thynges
+    have p{ro}ceded{e}, and come forth{e}, And by reso{u}n of nombre
+    ben formed{e}; And in wise as they ben{e}, So oweth{e} they to be
+    knowen{e}; wherfor in vniu{er}sall{e} knowlechyng of thynges the
+    Art of nombrynge is best, and most operatyf{e}.+
+
+  [Sidenote: The name of the art. Derivation of Algorism. Another.
+  Another. Kinds of numbers. The 9 rules of the Art.]
+
+Therfore sithen the science of the whiche at this tyme we intenden{e} to
+write of standith{e} all{e} and about nombre: ffirst we most se, what is
+the p{ro}pre name therof{e}, and fro whens the name come: Afterward{e}
+what is nombre, And how manye spices of nombre ther ben. The name is
+cleped{e} Algorisme, had{e} out of Algor{e}, other of Algos, in grewe,
+That is clepid{e} in englissh{e} art other craft, And of Rithm{us} that
+is called{e} nombre. So algorisme is cleped{e} the art of nombryng,
+other it is had of{e} en or in, and gogos that is introduccio{u}n, and
+Rithm{us} nombre, that is to say Interduccio{u}n of nombre. And thirdly
+it is had{e} of the name of a kyng that is cleped{e} Algo and Rythm{us};
+So called{e} Algorism{us}. Sothely .2. maner{e} of nombres ben
+notified{e}; Formall{e},[{1}] as nombr{e} i{s} vnitees gadred{e}
+to-gedres; Materiall{e},[{2}] as nombr{e} is a colleccio{u}n of vnitees.
+Other nombr{e} is a multitude had{e} out of vnitees, vnitee is that
+thynge wher-by eu{er}y thynge is called{e} oone, other o thynge. Of
+nombres, that one is cleped{e} digitall{e}, that other{e} Article,
+Another a nombre componed{e} oþ{er} myxt. Another digitall{e} is a
+nombre w{i}t{h}-in .10.; Article is þ{a}t nombre that may be dyvyded{e}
+in .10. p{ar}ties egally, And that there leve no residue; Componed{e} or
+medled{e} is that nombre that is come of a digite and of an article. And
+vndrestand{e} wele that all{e} nombres betwix .2. articles next is a
+nombr{e} componed{e}. Of this art ben{e} .9. spices, that is forto sey,
+num{er}acio{u}n, addicio{u}n, Subtraccio{u}n, Mediac{i}o{u}n,
+Duplacio{u}n, Multipliacio{u}n, Dyvysio{u}n, Progressio{u}n, And of
+Rootes the extraccio{u}n, and that may be had{e} in .2. maners, that is
+to sey in nombres quadrat, and in cubic{es}: Amonge the which{e}, ffirst
+of Num{er}acio{u}n, and aft{er}ward{e} of þe oþ{er}s by ordure,
+y entende to write.
+
+
+    [Headnote: Chapter I. Numeration.]
+
+  [*Fol. 48b]
+
+  +For-soth{e} num{er}acio{u}n is of eu{er}y numbre by
+  competent figures an artificiall{e} rep{re}sentacio{u}n.+
+
+  [Sidenote: Figures, differences, places, and limits. The 9 figures.
+  The cipher. The numeration of digits, of articles, of composites.
+  The value due to position. Numbers are written from right to left.]
+
+Sothly figure, difference, places, and lynes supposen o thyng other the
+same, But they ben sette here for dyue{r}s resons. ffigure is cleped{e}
+for p{ro}traccio{u}n of figuracio{u}n; Difference is called{e} for
+therby is shewed{e} eu{er}y figure, how it hath{e} difference fro the
+figures before them: place by cause of space, where-in me writeth{e}:
+lynees, for that is ordeyned{e} for the p{re}sentacio{u}n of eu{er}y
+figure. And vnderstonde that ther ben .9. lymytes of figures that
+rep{re}senten the .9. digit{es} that ben these. 0. 9. 8. 7. 6. 5. 4. 3.
+2. 1. The .10. is cleped{e} theta, or a cercle, other a cifre, other a
+figure of nought for nought it signyfieth{e}. Nathelesse she holdyng
+that place giveth{e} others for to signyfie; for with{e}-out cifre or
+cifres a pure article may not be writte. And sithen that by these .9.
+figures significatif{es} Ioyned{e} w{i}t{h} cifre or w{i}t{h} cifres
+all{e} nombres ben and may be rep{re}sented{e}, It was, nether is,
+no nede to fynde any more figures. And note wele that eu{er}y digite
+shall{e} be writte w{i}t{h} oo figure allone to it ap{ro}pred{e}. And
+all{e} articles by a cifre, ffor eu{er}y article is named{e} for oone of
+the digitis as .10. of 1.. 20. of. 2. and so of the others, &c. And
+all{e} nombres digitall{e} owen to be sette in the first difference:
+All{e} articles in the seconde. Also all{e} nombres fro .10. til an
+.100. [which] is excluded{e}, with .2. figures mvst be writte; And yf it
+be an article, by a cifre first put, and the figure y-writte toward{e}
+the lift hond{e}, that signifieth{e} the digit of the which{e} the
+article is named{e}; And yf it be a nombre componed{e}, ffirst write the
+digit that is a part of that componed{e}, and write to the lift side the
+article as it is seid{e} be-fore. All{e} nombre that is fro an
+hundred{e} tille a thousand{e} exclused{e}, owith{e} to be writ by .3.
+figures; and all{e} nombre that is fro a thousand{e} til .x. Mł. mvst be
+writ by .4. figures; And so forthe. And vnderstond{e} wele that eu{er}y
+figure sette in the first place signyfieth{e} his digit; In the
+second{e} place .10. tymes his digit; In the .3. place an hundred{e} so
+moche; In the .4. place a thousand{e} so moche; In the .5. place .x.
+thousand{e} so moch{e}; In the .6. place an hundred{e} thousand{e} so
+moch{e}; In the .7. place a thousand{e} thousand{e}. And so infynytly
+mvltiplying by [*Fol. 49.] these .3. 10, 100, 1000. And vnderstand{e}
+wele that competently me may sette vpon figure in the place of a
+thousand{e}, a prik{e} to shewe how many thousand{e} the last figure
+shall{e} rep{re}sent. We writen{e} in this art to the lift side-ward{e},
+as arabien{e} writen{e}, that weren fynders of this science, other{e}
+for this reso{u}n, that for to kepe a custumable ordr{e} in redyng,
+Sette we all{e}-wey the more nombre before.
+
+    [Headnote: Chapter II. Addition.]
+
+  [Sidenote: Definition. How the numbers should be written. The method
+  of working. Begin at the right. The Sum is a digit, or an article,
+  or a composite.]
+
+Addicio{u}n is of nombre other of nombres vnto nombre or to nombres
+aggregacio{u}n, that me may see that that is come therof as
+exc{re}ssent. In addicio{u}n, 2. ordres of figures and .2. nombres ben
+necessary, that is to sey, a nombre to be added{e} and the nombre wherto
+the addic{i}oun shold{e} be made to. The nombre to be added{e} is that
+þat shold{e} be added{e} therto, and shall{e} be vnderwriten; the nombre
+vnto the which{e} addicio{u}n shall{e} be made to is that nombre that
+resceyueth{e} the addicion of þat other, and shall{e} be writen above;
+and it is convenient that the lesse nombre be vnderwrit, and the more
+added{e}, than the contrary. But whether it happ{e} one other other, the
+same comyth{e} of, Therfor, yf þow wilt adde nombre to nombre, write the
+nombre wherto the addicio{u}n shall{e} be made in the omest ordre by his
+differences, so that the first of the lower ordre be vndre the first of
+the omyst ordre, and so of others. That done, adde the first of the
+lower ordre to the first of the omyst ordre. And of such{e} addicio{u}n,
+other þ{er}e grow{i}t{h} therof a digit, An article, other a
+composed{e}. If it be digit{us}, In the place of the omyst shalt thow
+write the digit excrescyng, as thus:--
+
+  +----------------------------+---+
+  |The resultant               | 2 |
+  +----------------------------+---+
+  |To whom it shal be added{e} | 1 |
+  +----------------------------+---+
+  |The nombre to be added{e}   | 1 |
+  +----------------------------+---+
+
+If the article; in the place of the omyst put a-way by a cifre writte,
+and the digit transferred{e}, of þe which{e} the article toke his name,
+toward{e} the lift side, and be it added{e} to the next figure folowyng,
+yf ther be any figure folowyng; or no, and yf it be not, leve it [in
+the] void{e}, as thus:--
+
+  +---------------------------------+----+
+  | The resultant                   | 10 |
+  +---------------------------------+----+
+  | To whom it shall{e} be added{e} |  7 |
+  +---------------------------------+----+
+  | The nombre to be added{e}       |  3 |
+  +---------------------------------+----+
+
+  +----------------------+---+---+---+---+---+
+  | Resultans            | 2 | 7 | 8 | 2 | 7 |
+  +----------------------+---+---+---+---+---+
+  | Cui d{ebet} addi     | 1 | 0 | 0 | 8 | 4 |
+  +----------------------+---+---+---+---+---+
+  | Num{erus} addend{us} | 1 | 7 | 7 | 4 | 3 |
+  +----------------------+---+---+---+---+---+
+
+And yf it happe that the figure folowyng wherto the addicio{u}n shall{e}
+be made by [the cifre of] an article, it sette a-side; In his place
+write the [*Fol. 49b] [digit of the] Article as thus:--
+
+  +---------------------------------+----+
+  | The resultant                   | 17 |
+  +---------------------------------+----+
+  | To whom it shall{e} be added{e} | 10 |
+  +---------------------------------+----+
+  | The nombre to be added{e}       |  7 |
+  +---------------------------------+----+
+
+And yf it happe that a figure of .9. by the figure that me mvst adde
+[one] to, In the place of that 9. put a cifre {and} write þe article
+toward{e} þe lift hond{e} as bifore, and thus:--
+
+  +---------------------------------+----+
+  | The resultant                   | 10 |
+  +---------------------------------+----+
+  | To whom it shall{e} be added{e} |  9 |
+  +---------------------------------+----+
+  | The nombre to be added{e}       |  1 |
+  +---------------------------------+----+
+
+And yf[{3}] [therefrom grow a] nombre componed,[{4}] [in the place of
+the nombre] put a-way[{5}][let] the digit [be][{6}]writ þ{a}t is part of
+þ{a}t co{m}posid{e}, and þan put to þe lift side the article as before,
+and þus:--
+
+  +---------------------------------+----+
+  | The resultant                   | 12 |
+  +---------------------------------+----+
+  | To whom it shall{e} be added{e} |  8 |
+  +---------------------------------+----+
+  | The nombre to be added{e}       |  4 |
+  +---------------------------------+----+
+
+This done, adde the seconde to the second{e}, and write above oþ{er} as
+before.
+
+  [Sidenote: The translator’s note.]
+
+Note wele þ{a}t in addic{i}ons and in all{e} spices folowyng, whan he
+seith{e} one the other shall{e} be writen aboue, and me most vse eu{er}
+figure, as that eu{er}y figure were sette by half{e}, and by
+hym-self{e}.
+
+
+    [Headnote: Chapter III. Subtraction.]
+
+  [Sidenote: Definition of Subtraction. How it may be done. What is
+  required. Write the greater number above. Subtract the first figure
+  if possible. If it is not possible ‘borrow ten,’ and then subtract.]
+
+Subtraccio{u}n is of .2. p{ro}posed{e} nombres, the fyndyng of the
+excesse of the more to the lasse: Other subtraccio{u}n is ablacio{u}n of
+o nombre fro a-nother, that me may see a some left. The lasse of the
+more, or even of even, may be w{i}t{h}draw; The more fro the lesse may
+neu{er} be. And sothly that nombre is more that hath{e} more figures, So
+that the last be signyficatife{s}: And yf ther ben as many in that one
+as in that other, me most deme it by the last, other by the next last.
+More-ou{er} in w{i}t{h}-drawyng .2. nombres ben necessary; A nombre to
+be w{i}t{h}draw, And a nombre that me shall{e} w{i}t{h}-draw of. The
+nombre to be w{i}t{h}-draw shall{e} be writ in the lower ordre by his
+differences; The nombre fro the which{e} me shall{e} with{e}-draw in the
+omyst ordre, so that the first be vnder the first, the second{e} vnder
+the second{e}, And so of all{e} others. With{e}-draw therfor the first
+of the lower{e} ordre fro the first of the ordre above his hede, and
+that wolle be other more or lesse, oþ{er} egall{e}.
+
+  +---------------------------------+----+
+  | The remanent                    | 20 |
+  +---------------------------------+----+
+  | Wherof me shall{e} w{i}t{h}draw | 22 |
+  +---------------------------------+----+
+  | The nombre to be w{i}t{h}draw   |  2 |
+  +---------------------------------+----+
+
+yf it be egall{e} or even the figure sette beside, put in his place a
+cifre. And yf it be more put away þ{er}fro als many of vnitees the lower
+figure conteyneth{e}, and writ the residue as thus
+
+  +----------------------------------+---+---+
+  | The remanent                     | 2 | 2 |
+  +----------------------------------+---+---+
+  | Wherof me shall{e} w{i}t{h}-draw | 2 | 8 |
+  +----------------------------------+---+---+
+  | Þe nombre to be w{i}t{h}draw     |   | 6 |
+  +----------------------------------+---+---+
+
+  [*Fol. 50.]
+
+  +--------------------------+---+---+-----+---+---+---+---+---+---+
+  | Remane{n}s               | 2 | 2 |  1  | 8 | 2 | 9 | 9 | 9 | 8 |
+  +--------------------------+---+---+-----+---+---+---+---+---+---+
+  | A quo sit subtraccio     | 8 | 7 |  2  | 4 | 3 | 0 | 0 | 0 | 4 |
+  +--------------------------+---+---+-----+---+---+---+---+---+---+
+  | Numerus subt{ra}hend{us} | 6 | 5 |[{7}]|[6]| . | . | . | . | 6 |
+  +--------------------------+---+---+-----+---+---+---+---+---+---+
+
+And yf it be lesse, by-cause the more may not be w{i}t{h}-draw ther-fro,
+borow an vnyte of the next figure that is worth{e} 10. Of that .10. and
+of the figure that ye wold{e} have w{i}t{h}-draw fro be-fore to-gedre
+Ioyned{e}, w{i}t{h}-draw þe figure be-nethe, and put the residue in the
+place of the figure put a-side as þ{us}:--
+
+  +----------------------------------+---+---+
+  | The remanent                     | 1 | 8 |
+  +----------------------------------+---+---+
+  | Wherof me shall{e} w{i}t{h}-draw | 2 | 4 |
+  +----------------------------------+---+---+
+  | The nombre to be w{i}t{h}-draw   | 0 | 6 |
+  +----------------------------------+---+---+
+
+  [Sidenote: If the second figure is one.]
+
+And yf the figure wherof me shal borow the vnyte be one, put it a-side,
+and write a cifre in the place þ{er}of, lest the figures folowing faile
+of thair{e} nombre, and þan worch{e} as it shew{i}t{h} in this figure
+here:--
+
+  +--------------------------------+---+---+------+
+  | The remanent                   | 3 | 0 |9[{8}]|
+  +--------------------------------+---+---+------+
+  | Wherof me shal w{i}t{h}-draw   | 3 | 1 |   2  |
+  +--------------------------------+---+---+------+
+  | The nombre to be w{i}t{h}-draw | . | . |   3  |
+  +--------------------------------+---+---+------+
+
+  [Sidenote: If the second figure is a cipher.]
+
+And yf the vnyte wherof me shal borow be a cifre, go ferther to the
+figure signyficatif{e}, and ther borow one, and reto{ur}nyng bak{e}, in
+the place of eu{er}y cifre þ{a}t ye passid{e} ou{er}, sette figures of
+.9. as here it is specified{e}:--
+
+  +----------------------------------+---+---+---+---+---+
+  | The remenaunt                    | 2 | 9 | 9 | 9 | 9 |
+  +----------------------------------+---+---+---+---+---+
+  | Wherof me shall{e} w{i}t{h}-draw | 3 | 0 | 0 | 0 | 3 |
+  +----------------------------------+---+---+---+---+---+
+  | The nombre to be w{i}t{h}-draw   |   |   |   |   | 4 |
+  +----------------------------------+---+---+---+---+---+
+
+  [Sidenote: A justification of the rule given. Why it is better to
+  work from right to left. How to prove subtraction, and addition.]
+
+And whan me cometh{e} to the nombre wherof me intendith{e}, there
+remayneth{e} all{e}-wayes .10. ffor þe which{e} .10. &c. The reson why
+þat for eu{er}y cifre left behynde me setteth figures ther of .9. this
+it is:--If fro the .3. place me borowed{e} an vnyte, that vnyte by
+respect of the figure that he came fro rep{re}sentith an .C., In the
+place of that cifre [passed over] is left .9., [which is worth ninety],
+and yit it remayneth{e} as .10., And the same reson{e} wold{e} be yf me
+had{e} borowed{e} an vnyte fro the .4., .5., .6., place, or ony other so
+vpward{e}. This done, withdraw the second{e} of the lower ordre fro the
+figure above his hede of þe omyst ordre, and wirch{e} as before. And
+note wele that in addicion or in subtracc{i}o{u}n me may wele fro the
+lift side begynne and ryn to the right side, But it wol be more
+p{ro}fitabler to be do, as it is taught. And yf thow wilt p{ro}ve yf
+thow have do wele or no, The figures that thow hast withdraw, adde them
+ayene to the omyst figures, and they wolle accorde w{i}t{h} the first
+that thow haddest yf thow have labored wele; and in like wise in
+addicio{u}n, whan thow hast added{e} all{e} thy figures, w{i}t{h}draw
+them that thow first [*Fol. 50b] addest, and the same wolle reto{ur}ne.
+The subtraccio{u}n is none other but a p{ro}uff{e} of the addicio{u}n,
+and the contrarye in like wise.
+
+    [Headnote: Chapter IV. Mediation.]
+
+  [Sidenote: Definition of mediation. Where to begin. If the first
+  figure is unity. What to do if it is not unity.]
+
+Mediacio{u}n is the fyndyng of the halfyng of eu{er}y nombre, that it
+may be seyn{e} what and how moch{e} is eu{er}y half{e}. In halfyng ay oo
+order of figures and oo nombre is necessary, that is to sey the nombre
+to be halfed{e}. Therfor yf thow wilt half any nombre, write that nombre
+by his differences, and begynne at the right, that is to sey, fro the
+first figure to the right side, so that it be signyficatif{e} other
+rep{re}sent vnyte or eny other digitall{e} nombre. If it be vnyte write
+in his place a cifre for the figures folowyng, [lest they signify less],
+and write that vnyte w{i}t{h}out in the table, other resolue it in .60.
+mynvt{es} and sette a-side half of tho m{inutes} so, and reserve the
+remen{au}nt w{i}t{h}out in the table, as thus .30.; other sette
+w{i}t{h}out thus .{dī}: that kepeth{e} none ordre of place, Nathelesse
+it hath{e} signyficacio{u}n. And yf the other figure signyfie any other
+digital nombre fro vnyte forth{e}, oþ{er} the nombre is od{e} or
+even{e}. If it be even, write this half in this wise:--
+
+  +-----------------+---+---+
+  | Halfed{e}       | 2 | 2 |
+  +-----------------+----+--+
+  | to be halfed{e} | 4 | 4 |
+  +-----------------+---+---+
+
+And if it be odde, Take the next even vndre hym conteyned{e}, and put
+his half in the place of that odde, and of þe vnyte that remayneth{e} to
+be halfed{e} do thus:--
+
+  +-----------------+---+---+
+  | halfed{e}       | 2 | 3 | [di]
+  +-----------------+---+---+
+  | To be halfed{e} | 4 | 7 |
+  +-----------------+---+---+
+
+  [Sidenote: Then halve the second figure. If it is odd, add 5 to the
+  figure before.]
+
+This done, the second{e} is to be halfed{e}, yf it be a cifre put it
+be-side, and yf it be significatif{e}, other it is even or od{e}: If it
+be even, write in the place of þe nombres wiped{e} out the half{e}; yf
+it be od{e}, take the next even vnder it co{n}tenyth{e}, and in the
+place of the Impar sette a-side put half of the even: The vnyte that
+remayneth{e} to be halfed{e}, respect had{e} to them before, is worth{e}
+.10. Dyvide that .10. in .2., 5. is, and sette a-side that one, and adde
+that other to the next figure p{re}cedent as here:--
+
+  +-----------------+---+---+---+
+  | Halfed{e}       |   |   |   |
+  +-----------------+---+---+---+
+  | to be halfed{e} |   |   |   |
+  +-----------------+---+---+---+
+
+And yf þe addicio{u}n shold{e} be made to a cifre, sette it a-side, and
+write in his place .5. And vnder this fo{ur}me me shall{e} write and
+worch{e}, till{e} the totall{e} nombre be halfed{e}.
+
+  +------------------+---+---+---+---+---+----+----+---+
+  | doubled{e}       | 2 | 6 | 8 | 9 | 0 | 10 | 17 | 4 |
+  +------------------+---+---+---+---+---+----+----+---+
+  | to be doubled{e} | 1 | 3 | 4 | 4 | 5 |  5 |  8 | 7 |
+  +------------------+---+---+---+---+---+----+----+---+
+
+    [Headnote: Chapter V. Duplation.]
+
+  [Sidenote: Definition of Duplation. Where to begin. Why. What to do
+  with the result.]
+
+Duplicacio{u}n is ag{re}gacion of nombre [to itself] þat me may se the
+nombre growen. In doublyng{e} ay is but one ordre of figures necessarie.
+And me most be-gynne w{i}t{h} the lift side, other of the more figure,
+And after the nombre of the more figure rep{re}sentith{e}. [*Fol. 51.]
+In the other .3. before we begynne all{e} way fro the right side and fro
+the lasse nombre, In this spice and in all{e} other folowyng we wolle
+begynne fro the lift side, ffor and me bigon th{e} double fro the first,
+omwhile me myght double oo thynge twyes. And how be it that me myght
+double fro the right, that wold{e} be harder in techyng and in workyng.
+Therfor yf thow wolt double any nombre, write that nombre by his
+differences, and double the last. And of that doubly{n}g other
+growith{e} a nombre digital, article, or componed{e}. [If it be a digit,
+write it in the place of the first digit.] If it be article, write in
+his place a cifre and transferre the article toward{e} the lift, as
+thus:--
+
+  +------------------+----+
+  | double           | 10 |
+  +------------------+----+
+  | to be doubled{e} |  5 |
+  +------------------+----+
+
+And yf the nombre be componed{e}, write a digital that is part of his
+composicio{u}n, and sette the article to the lift hand{e}, as thus:--
+
+  +------------------+----+
+  | doubled{e}       | 16 |
+  +------------------+----+
+  | to be doubled{e} |  8 |
+  +------------------+----+
+
+That done, me most double the last save one, and what groweth{e} þ{er}of
+me most worche as before. And yf a cifre be, touch{e} it not. But yf any
+nombre shall{e} be added{e} to the cifre, in þe place of þe figure
+wiped{e} out me most write the nombre to be added{e}, as thus:--
+
+  +------------------+---+---+---+
+  | doubled{e}       | 6 | 0 | 6 |
+  +------------------+---+---+---+
+  | to be doubled{e} | 3 | 0 | 3 |
+  +------------------+---+---+---+
+
+  [Sidenote: How to prove your answer.]
+
+In the same wise me shall{e} wirch{e} of all{e} others. And this
+p{ro}bacio{u}n: If thow truly double the halfis, and truly half the
+doubles, the same nombre and figure shall{e} mete, such{e} as thow
+labo{ur}ed{e} vpon{e} first, And of the contrarie.
+
+  +------------------+---+---+---+
+  | Doubled{e}       | 6 | 1 | 8 |
+  +------------------+---+---+---+
+  | to be doubled{e} | 3 | 0 | 9 |
+  +------------------+---+---+---+
+
+    [Headnote: Chapter VI. Multiplication.]
+
+  [Sidenote: Definition of Multiplication. Multiplier. Multiplicand.
+  Product.]
+
+Multiplicacio{u}n of nombre by hym-self other by a-nother, w{i}t{h}
+p{ro}posid{e} .2. nombres, [is] the fyndyng of the third{e}, That so
+oft conteyneth{e} that other, as ther ben vnytes in the oþ{er}. In
+multiplicacio{u}n .2. nombres pryncipally ben necessary, that is to
+sey, the nombre multiplying and the nombre to be multiplied{e},
+as here;--twies fyve. [The number multiplying] is designed{e}
+adu{er}bially. The nombre to be multiplied{e} resceyveth{e} a
+no{m}i{n}all{e} appellacio{u}n, as twies .5. 5. is the nombre
+multiplied{e}, and twies is the nombre to be multipliede.
+
+  +-----------------+-----+---+---++---+---+---+---+---+---+---+---+---+
+  | Resultans       |[{9}]| 1 | 0 || 1 | 3 | 2 | 6 | 6 | 8 | 0 | 0 | 8 |
+  +-----------------+-----+---+---++---+---+---+---+---+---+---+---+---+
+  | Multiplicand{us}|  .  | . | 5 || . | . | 4 | . | 3 | 4 | 0 | 0 | 4 |
+  +-----------------+-----+---+---++---+---+---+---+---+---+---+---+---+
+  | Multiplicans    |  .  | 2 | 2 || . | 3 | 3 | 2 | 2 | 2 | . | . | . |
+  +-----------------+-----+---+---++---+---+---+---+---+---+---+---+---+
+
+Also me may thervpon{e} to assigne the. 3. nombre, the which{e} is
+[*Fol. 51b] cleped{e} p{ro}duct or p{ro}venient, of takyng out of one
+fro another: as twyes .5 is .10., 5. the nombre to be multiplied{e},
+and .2. the multipliant, and. 10. as before is come therof. And
+vnderstonde wele, that of the multipliant may be made the nombre to
+be multiplied{e}, and of the contrarie, remaynyng eu{er} the same some,
+and herof{e} cometh{e} the comen speche, that seith{e} all nombre is
+converted{e} by Multiplying in hym-self{e}.
+
+  +----+----+----+----+----+--------+----+----+----+-----+
+  |  1 |  2 |  3 |  4 |  5 |   6    |  7 |  8 |  9 |  10 |
+  +----+----+----+----+----+--------+----+----+----+-----+
+  |  2 |  4 |  6 |  8 | 10 |10[{10}]| 14 | 16 | 18 |  20 |
+  +----+----+----+----+----+--------+----+----+----+-----+
+  |  3 |  6 |  9 | 12 | 15 |  18    | 21 | 24 | 27 |  30 |
+  +----+----+----+----+----+--------+----+----+----+-----+
+  |  4 |  8 | 12 | 16 | 20 |  24    | 28 | 32 | 36 |  40 |
+  +----+----+----+----+----+--------+----+----+----+-----+
+  |  5 | 10 | 15 | 20 | 25 |  30    | 35 | 40 | 45 |  50 |
+  +----+----+----+----+----+--------+----+----+----+-----+
+  |  6 | 12 | 18 | 24 | 30 |  36    | 42 | 48 | 56 |  60 |
+  +----+----+----+----+----+--------+----+----+----+-----+
+  |  7 | 14 | 21 | 28 | 35 |  42    | 49 | 56 | 63 |  70 |
+  +----+----+----+----+----+--------+----+----+----+-----+
+  |  8 | 16 | 24 | 32 | 40 |  48    | 56 | 64 | 72 |  80 |
+  +----+----+----+----+----+--------+----+----+----+-----+
+  |  9 | 18 | 27 | 36 | 45 |  54    | 63 | 72 | 81 |  90 |
+  +----+----+----+----+----+--------+----+----+----+-----+
+  | 10 | 20 | 30 | 40 | 50 |  60    | 70 | 80 | 90 | 100 |
+  +----+----+----+----+----+--------+----+----+----+-----+
+
+    [Headnote: The Cases of Multiplication.]
+
+  [Sidenote: There are 6 rules of Multiplication. (1) Digit by digit.
+  See the table above. (2) Digit by article. (3) Composite by digit.]
+
+And ther ben .6 rules of Multiplicacio{u}n; ffirst, yf a digit multiplie
+a digit, considr{e} how many of vnytees ben betwix the digit by
+multiplying and his .10. beth{e} to-gedre accompted{e}, and so oft
+w{i}t{h}-draw the digit multiplying, vnder the article of his
+deno{m}i{n}acio{u}n. Example of grace. If thow wolt wete how moch{e} is
+.4. tymes .8., [{11}]se how many vnytees ben betwix .8.[{12}] and .10.
+to-geder rekened{e}, and it shew{i}t{h} that .2.: withdraw ther-for the
+quat{e}rnary, of the article of his deno{m}i{n}acion twies, of .40., And
+ther remayneth{e} .32., that is, to some of all{e} the
+multiplicacio{u}n. Wher-vpon for more evidence and declaracion the
+seid{e} table is made. Whan a digit multiplieth{e} an article, thow most
+bryng the digit into þe digit, of þe which{e} the article [has][{13}]
+his name, and eu{er}y vnyte shall{e} stond{e} for .10., and eu{er}y
+article an .100. Whan the digit multiplieth{e} a nombre componed{e},
+þ{o}u most bryng the digit into aiþ{er} part of the nombre componed{e},
+so þ{a}t digit be had into digit by the first rule, into an article by
+þe second{e} rule; and aft{er}ward{e} Ioyne the p{ro}duccio{u}n, and
+þ{er}e wol be the some totall{e}.
+
+  +----------------+---+---+--++---+---+--++---+---+--++---+---+---+---+
+  |Resultans       | 1 | 2 | 6|| 7 | 3 | 6|| 1 | 2 | 0|| 1 | 2 | 0 | 8 |
+  +----------------+---+---+--++---+---+--++---+---+--++---+---+---+---+
+  |Multiplicand{us}|   |   | 2||   | 3 | 2||   |   | 6||   |   |   | 4 |
+  +----------------+---+---+--++---+---+--++---+---+--++---+---+---+---+
+  |Multiplicans    |   | 6 | 3|| 2 | 3 |  ||   | 2 | 0||   | 3 | 0 | 2 |
+  +----------------+---+---+--++---+---+--++---+---+--++---+---+---+---+
+
+  [Sidenote: (4) Article by article. (5) Composite by article.
+  (6) Composite by composite. How to set down your numbers. If the
+  result is a digit, an article, or a composite. Multiply next by
+ the last but one, and so on.]
+
+Whan an article multiplieth{e} an article, the digit wherof he is
+named{e} is to be brought Into the digit wherof the oþ{er} is named{e},
+and eu{er}y vnyte wol be worth{e} [*Fol. 52.] an .100., and eu{er}y
+article. a .1000. Whan an article multiplieth{e} a nombre componed{e},
+thow most bryng the digit of the article into aither part of the nombre
+componed{e}; and Ioyne the p{ro}duccio{u}n, and eu{er}y article wol be
+worth{e} .100., and eu{er}y vnyte .10., and so woll{e} the some be
+open{e}. Whan a nombre componed{e} multiplieth{e} a nombre componed{e},
+eu{er}y p{ar}t of the nombre multiplying is to be had{e} into eu{er}y
+p{ar}t of the nombre to be multiplied{e}, and so shall{e} the digit be
+had{e} twies, onys in the digit, that other in the article. The article
+also twies, ones in the digit, that other in the article. Therfor yf
+thow wilt any nombre by hym-self other by any other multiplie, write the
+nombre to be multiplied{e} in the ou{er} ordre by his differences, The
+nombre multiplying in the lower ordre by his differences, so that the
+first of the lower ordre be vnder the last of the ou{er} ordre. This
+done, of the multiplying, the last is to be had{e} into the last of the
+nombre to be multiplied{e}. Wherof than wolle grow a digit, an article,
+other a nombre componed{e}. If it be a digit, even above the figure
+multiplying is hede write his digit that come of, as it appereth{e}
+here:--
+
+  +-----------------------+---+
+  | The resultant         | 6 |
+  +-----------------------+---+
+  | To be multiplied{e}   | 3 |
+  +-----------------------+---+
+  | Þe nombre multipliyng | 2 |
+  +-----------------------+---+
+
+And yf an article had be writ ou{er} the fig{ur}e multiplying his hede,
+put a cifre þ{er} and transferre the article toward{e} the lift hand{e},
+as thus:--
+
+  +-------------------------+---+---+
+  | The resultant           | 1 | 0 |
+  +-------------------------+---+---+
+  | to be multiplied{e}     |   | 5 |
+  +-------------------------+---+---+
+  | þe nombre m{u}ltipliyng |   | 2 |
+  +-------------------------+---+---+
+
+And yf a nombre componed{e} be writ ou{er} the figure multyplying is
+hede, write the digit in the nombre componed{e} is place, and sette the
+article to the lift hand{e}, as thus:--
+
+  +------------------------+---+---+
+  | Resultant              | 1 | 2 |
+  +------------------------+---+---+
+  | to be multiplied{e}    |   | 4 |
+  +------------------------+---+---+
+  | the nombre multipliyng |   | 3 |
+  +------------------------+---+---+
+
+This done, me most bryng the last save one of the multipliyng into the
+last of þe nombre to be multiplied{e}, and se what comyth{e} therof as
+before, and so do w{i}t{h} all{e}, tille me come to the first of the
+nombre multiplying, that must be brought into the last of the nombre to
+be multiplied{e}, wherof growith{e} oþ{er} a digit, an article, [*Fol.
+52b] other a nombre componed{e}. If it be a digit, In the place of the
+ou{er}er, sette a-side, as here:
+
+  +--------------------------+---+---+
+  | Resultant                | 6 | 6 |
+  +--------------------------+---+---+
+  | to be multiplied{e}      |   | 3 |
+  +--------------------------+---+---+
+  | the nombre m{u}ltipliyng | 2 | 2 |
+  +--------------------------+---+---+
+
+If an article happe, there put a cifre in his place, and put hym to the
+lift hand{e}, as here:
+
+  +-------------------------+---+---+---+
+  | The resultant           | 1 | 1 | 0 |
+  +-------------------------+---+---+---+
+  | to be multiplied{e}     |   |   | 5 |
+  +-------------------------+---+---+---+
+  | þe nombre m{u}ltiplying |   | 2 | 2 |
+  +-------------------------+---+---+---+
+
+If it be a nombre componed{e}, in the place of the ou{er}er sette
+a-side, write a digit that[{14}] is a p{ar}t of the componed{e}, and
+sette on the left hond{e} the article, as here:
+
+  +-----------------------------+---+-------+---+
+  | The resultant               | 1 |3[{15}]| 2 |
+  +-----------------------------+---+-------+---+
+  | to be m{u}ltiplied{e}       |   |       | 4 |
+  +-----------------------------+---+-------+---+
+  | þe nombr{e} m{u}ltiplia{n}t |   |   3   | 3 |
+  +-----------------------------+---+-------+---+
+
+  [Sidenote: Then antery the multiplier one place. Work as before.
+  How to deal with ciphers.]
+
+That done, sette forward{e} the figures of the nombre multiplying by oo
+difference, so that the first of the multipliant be vnder the last save
+one of the nombre to be multiplied{e}, the other by o place sette
+forward{e}. Than me shall{e} bryng{e} the last of the m{u}ltipliant in
+hym to be multiplied{e}, vnder the which{e} is the first multipliant.
+And than wolle growe oþ{er} a digit, an article, or a componed{e}
+nombre. If it be a digit, adde hym even above his hede; If it be an
+article, transferre hym to the lift side; And if it be a nombre
+componed{e}, adde a digit to the figure above his hede, and sette to the
+lift hand{e} the article. And all{e}-wayes eu{er}y figure of the nombre
+multipliant is to be brought to the last save one nombre to be
+multiplied{e}, til me come to the first of the multipliant, where me
+shall{e} wirche as it is seid{e} before of the first, and aft{er}ward{e}
+to put forward{e} the figures by o difference and one till{e} they
+all{e} be multiplied{e}. And yf it happe that the first figure of þe
+multipliant be a cifre, and boue it is sette the figure signyficatif{e},
+write a cifre in the place of the figur{e} sette a-side, as thus,
+{et}c.:
+
+  +---------------------+---+---+---+
+  | The resultant       | 1 | 2 | 0 |
+  +---------------------+---+---+---+
+  | to be multiplied{e} |   |   | 6 |
+  +---------------------+---+---+---+
+  | the multipliant     |   | 2 | 0 |
+  +---------------------+---+---+---+
+
+  [Sidenote: How to deal with ciphers.]
+
+And yf a cifre happe in the lower order be-twix the first and the last,
+and even above be sette the fig{ur}e signyficatif, leve it vntouched{e},
+as here:--
+
+  +---------------------+---+---+---+---+---+
+  | The resultant       | 2 | 2 | 6 | 4 | 4 |
+  +---------------------+---+---+---+---+---+
+  | To be multiplied{e} |   |   | 2 | 2 | 2 |
+  +---------------------+---+---+---+---+---+
+  | The multipliant     | 1 | 0 | 2 |   |   |
+  +---------------------+---+---+---+---+---+
+
+And yf the space above sette be void{e}, in that place write thow a
+cifre. And yf the cifre happe betwix þe first and the last to be
+m{u}ltiplied{e}, me most sette forward{e} the ordre of the figures by
+thair{e} differences, for oft of duccio{u}n of figur{e}s in cifres
+nought is the resultant, as here,
+
+  +-----------------------+---+---+---+---+---+
+  | Resultant             | 8 | 0 | 0 | 8 |   |
+  +-----------------------+---+---+---+---+---+
+  | to be m{u}ltiplied{e} | 4 | 0 | 0 | 4 |   |
+  +-----------------------+---+---+---+---+---+
+  | the m{u}ltipliant     | 2 | . | . | . |   |
+  +-----------------------+---+---+---+---+---+
+
+[*Fol. 53.] wherof it is evident and open, yf that the first figure of
+the nombre be to be multiplied{e} be a cifre, vndir it shall{e} be none
+sette as here:--
+
+  +-----------------------+---+---+--------+
+  | Resultant             | 3 | 2 |0[{16}] |
+  +-----------------------+---+---+--------+
+  | To be m{u}ltiplied{e} |   | 8 |   0    |
+  +-----------------------+---+---+--------+
+  | The m{u}ltipliant     |   | 4 |        |
+  +-----------------------+---+---+--------+
+
+  [Sidenote: Leave room between the rows of figures.]
+
+Vnder[stand] also that in multiplicacio{u}n, divisio{u}n, and of rootis
+the extraccio{u}n, competently me may leve a mydel space betwix .2.
+ordres of figures, that me may write there what is come of addyng other
+with{e}-drawyng, lest any thynge shold{e} be ou{er}-hipped{e} and sette
+out of mynde.
+
+    [Headnote: Chapter VII. Division.]
+
+  [Sidenote: Definition of division. Dividend, Divisor, Quotient.
+  How to set down your Sum. An example. Examples.]
+
+For to dyvyde oo nombre by a-nother, it is of .2. nombres p{ro}posed{e},
+It is forto depart the moder nombre into as many p{ar}tis as ben of
+vnytees in the lasse nombre. And note wele that in makyng{e} of
+dyvysio{u}n ther ben .3. nombres necessary: that is to sey, the nombre
+to be dyvyded{e}; the nombre dyvydyng and the nombre exeant, other how
+oft, or quocient. Ay shall{e} the nombre that is to be dyvyded{e} be
+more, other at the lest even{e} w{i}t{h} the nombre the dyvysere, yf the
+nombre shall{e} be mad{e} by hole nombres. Therfor yf thow wolt any
+nombre dyvyde, write the nombre to be dyvyded{e} in þe ou{er}er
+bordur{e} by his differences, the dyviser{e} in the lower ordur{e} by
+his differences, so that the last of the dyviser be vnder the last of
+the nombre to be dyvyde, the next last vnder the next last, and so of
+the others, yf it may competently be done; as here:--
+
+  +------------------+---+---+---+
+  | The residue      |   | 2 | 7 |
+  +------------------+---+---+---+
+  | The quotient     |   |   | 5 |
+  +------------------+---+---+---+
+  | To be dyvyded{e} | 3 | 4 | 2 |
+  +------------------+---+---+---+
+  | The dyvyser      |   | 6 | 3 |
+  +------------------+---+---+---+
+
+  +--------------+---+---+----+---+---++---+---+---++---+---+---+
+  | Residuu{m}   |   |   | 8 ||   |   ||   | 2 | 7 ||   | 2 | 6 |
+  +--------------+---+---+---++---+---++---+---+---++---+---+---+
+  | Quociens     |   | 2 | 1 || 2 | 2 ||   |   | 5 ||   |   | 9 |
+  +--------------+---+---+---++---+---++---+---+---++---+---+---+
+  | Diuidend{us} | 6 | 8 | 0 || 6 | 6 || 3 | 4 | 2 || 3 | 3 | 2 |
+  +--------------+---+---+---++---+---++---+---+---++---+---+---+
+  | Diuiser      | 3 | 2 |   || 3 |   ||   | 6 | 3 ||   | 3 | 4 |
+  +--------------+---+---+---++---+---++---+---+---++---+---+---+
+
+  [Sidenote: When the last of the divisor must not be set below the
+  last of the dividend. How to begin.]
+
+And ther ben .2. causes whan the last figure may not be sette vnder the
+last, other that the last of the lower nombre may not be w{i}t{h}-draw
+of the last of the ou{er}er nombre for it is lasse than the lower, other
+how be it, that it myght be w{i}t{h}-draw as for hym-self fro the
+ou{er}er the remenaunt may not so oft of them above, other yf þe last of
+the lower be even to the figure above his hede, and þe next last oþ{er}
+the figure be-fore þ{a}t be more þan the figure above sette. [*Fol.
+53^2.] These so ordeyned{e}, me most wirch{e} from the last figure of þe
+nombre of the dyvyser, and se how oft it may be w{i}t{h}-draw of and fro
+the figure aboue his hede, namly so that the remen{au}nt may be take of
+so oft, and to se the residue as here:--
+
+  [Sidenote: An example.]
+
+  +------------------+---+---+---+
+  | The residue      |   | 2 | 6 |
+  +------------------+---+---+---+
+  | The quocient     |   |   | 9 |
+  +------------------+---+---+---+
+  | To be dyvyded{e} | 3 | 3 | 2 |
+  +------------------+---+---+---+
+  | The dyvyser      |   | 3 | 4 |
+  +------------------+---+---+---+
+
+  [Sidenote: Where to set the quotiente. Examples.]
+
+And note wele that me may not with{e}-draw more than .9. tymes nether
+lasse than ones. Therfor se how oft þe figures of the lower ordre may be
+w{i}t{h}-draw fro the figures of the ou{er}er, and the nombre that
+shew{i}t{h} þe q{u}ocient most be writ ou{er} the hede of þat figure,
+vnder the which{e} the first figure is, of the dyviser; And by that
+figure me most with{e}-draw all{e} oþ{er} figures of the lower ordir and
+that of the figures aboue thair{e} hedis. This so don{e}, me most sette
+forward{e} þe figures of the diuiser by o difference toward{es} the
+right hond{e} and worch{e} as before; and thus:--
+
+  +--------------+---+---+---+---+---+---++---+---+---+---+---+---+---+
+  | Residuu{m}   |   |   |   |   |   |   ||   |   |   |   | . | 1 | 2 |
+  +--------------+---+---+---+---+---+---++---+---+---+---+---+---+---+
+  | quo{ciens}   |   |   |   | 6 | 5 | 4 ||   |   |   | 2 | 0 | 0 | 4 |
+  +--------------+---+---+---+---+---+---++---+---+---+---+---+---+---+
+  | Diuidend{us} | 3 | 5 | 5 | 1 | 2 | 2 || 8 | 8 | 6 | 3 | 7 | 0 | 4 |
+  +--------------+---+---+---+---+---+---++---+---+---+---+---+---+---+
+  | Diuisor      |   | 5 | 4 | 3 |   |   || 4 | 4 | 2 | 3 |   |   |   |
+  +--------------+---+---+---+---+---+---++---+---+---+---+---+---+---+
+
+  +------------------+---+---+---+---+---+---+
+  | The quocient     |   |   |   | 6 | 5 | 4 |
+  +------------------+---+---+---+---+---+---+
+  | To be dyvyded{e} | 3 | 5 | 5 | 1 | 2 | 2 |
+  +------------------+---+---+---+---+---+---+
+  | The dyvyser      |   | 5 | 4 | 3 |   |   |
+  +------------------+---+---+---+---+---+---+
+
+  [Sidenote: A special case.]
+
+And yf it happ{e} after þe settyng forward{e} of the fig{ur}es þ{a}t þe
+last of the divisor may not so oft be w{i}t{h}draw of the fig{ur}e above
+his hede, above þat fig{ur}e vnder the which{e} the first of the diuiser
+is writ me most sette a cifre in ordre of the nombre quocient, and sette
+the fig{ur}es forward{e} as be-fore be o difference alone, and so me
+shall{e} do in all{e} nombres to be dyvided{e}, for where the dyviser
+may not be w{i}t{h}-draw me most sette there a cifre, and sette
+forward{e} the figures; as here:--
+
+  +------------------+---+---+---+---+---+---+---+
+  | The residue      |   |   |   |   |   | 1 | 2 |
+  |------------------+---+---+---+---+---+---+---+
+  | The quocient     |   |   |   | 2 | 0 | 0 | 4 |
+  |------------------+---+---+---+---+---+---+---+
+  | To be dyvyded{e} | 8 | 8 | 6 | 3 | 7 | 0 | 4 |
+  |------------------+---+---+---+---+---+---+---+
+  | The dyvyser      | 4 | 4 | 2 | 3 |   |   |   |
+  +------------------+---+---+---+---+---+---+---+
+
+  [Sidenote: Another example. What the quotient shows. How to prove
+  your division, or multiplication.]
+
+And me shall{e} not cesse fro such{e} settyng of fig{ur}es forward{e},
+nether of settyng{e} of þe quocient into the dyviser, neþ{er} of
+subt{ra}ccio{u}n of the dyvyser, till{e} the first of the dyvyser be
+w{i}t{h}-draw fro þe first to be divided{e}. The which{e} don{e}, or
+ought,[{17}] oþ{er} nought shall{e} remayne: and yf it be ought,[{17}]
+kepe it in the tables, And eu{er} vny it to þe diviser. And yf þ{o}u
+wilt wete how many vnytees of þe divisio{u}n [*Fol. 53^3.] wol growe to
+the nombre of the diviser{e}, the nombre quocient wol shewe it: and whan
+such{e} divisio{u}n is made, and þ{o}u lust p{ro}ve yf thow have wele
+done or no, Multiplie the quocient by the diviser, And the same
+fig{ur}es wolle come ayene that thow haddest bifore and none other. And
+yf ought be residue, than w{i}t{h} addicio{u}n therof shall{e} come the
+same figures: And so multiplicacio{u}n p{ro}vith{e} divisio{u}n, and
+dyvisio{u}n multiplicacio{u}n: as thus, yf multiplicacio{u}n be made,
+divide it by the multipliant, and the nombre quocient wol shewe the
+nombre that was to be multiplied{e}, {et}c.
+
+    [Headnote: Chapter VIII. Progression.]
+
+  [Sidenote: Definition of Progression. Natural Progression. Broken
+  Progression. The 1st rule for Natural Progression. The second rule.
+  The first rule of Broken Progression. The second rule.]
+
+Progressio{u}n is of nombre after egall{e} excesse fro oone or tweyn{e}
+take ag{r}egacio{u}n. of p{ro}gressio{u}n one is naturell{e} or
+co{n}tynuell{e}, þ{a}t oþ{er} broken and discontynuell{e}. Naturell{e}
+it is, whan me begynneth{e} w{i}t{h} one, and kepeth{e} ordure
+ou{er}lepyng one; as .1. 2. 3. 4. 5. 6., {et}c., so þ{a}t the nombre
+folowyng{e} passith{e} the other be-fore in one. Broken it is, whan me
+lepith{e} fro o nombre till{e} another, and kepith{e} not the contynuel
+ordir{e}; as 1. 3. 5. 7. 9, {et}c. Ay me may begynne w{i}t{h} .2., as
+þus; .2. 4. 6. 8., {et}c., and the nombre folowyng passeth{e} the others
+by-fore by .2. And note wele, that naturell{e} p{ro}gressio{u}n ay
+begynneth{e} w{i}t{h} one, and Int{er}cise or broken p{ro}gressio{u}n,
+omwhile begynnyth{e} w{i}th one, omwhile w{i}t{h} twayn{e}. Of
+p{ro}gressio{u}n naturell .2. rules ther be yove, of the which{e} the
+first is this; whan the p{ro}gressio{u}n naturell{e} endith{e} in even
+nombre, by the half therof multiplie þe next totall{e} ou{er}er{e}
+nombre; Example of grace: .1. 2. 3. 4. Multiplie .5. by .2. and so .10.
+cometh{e} of, that is the totall{e} nombre þ{er}of. The second{e} rule
+is such{e}, whan the p{ro}gressio{u}n naturell{e} endith{e} in nombre
+od{e}. Take the more porcio{u}n of the oddes, and multiplie therby the
+totall{e} nombre. Example of grace 1. 2. 3. 4. 5., multiplie .5. by .3,
+and thryes .5. shall{e} be resultant. so the nombre totall{e} is .15. Of
+p{ro}gresio{u}n int{er}cise, ther ben also .2.[{18}] rules; and þe first
+is þis: Whan the Int{er}cise p{ro}gression endith{e} in even nombre by
+half therof multiplie the next nombre to þat half{e} as .2.[{18}] 4. 6.
+Multiplie .4. by .3. so þat is thryes .4., and .12. the nombre of all{e}
+the p{ro}gressio{u}n, woll{e} folow. The second{e} rule is this: whan
+the p{ro}gressio{u}n int{er}scise endith{e} in od{e}, take þe more
+porcio{u}n of all{e} þe nombre, [*Fol. 53^4.] and multiplie by
+hym-self{e}; as .1. 3. 5. Multiplie .3. by hym-self{e}, and þe some of
+all{e} wolle be .9., {et}c.
+
+    [Headnote: Chapter IX. Extraction of Roots.]
+
+  [Sidenote: The preamble of the extraction of roots. Linear,
+  superficial, and solid numbers. Superficial numbers. Square numbers.
+  The root of a square number. Notes of some examples of square roots
+  here interpolated. Solid numbers. Three dimensions of solids. Cubic
+  numbers. All cubics are solid numbers. No number may be both linear
+  and solid. Unity is not a number.]
+
+Here folowith{e} the extraccio{u}n of rotis, and first in nombre
+q{ua}drat{es}. Wherfor me shall{e} se what is a nombre quadrat, and what
+is the rote of a nombre quadrat, and what it is to draw out the rote of
+a nombre. And before other note this divisio{u}n: Of nombres one is
+lyneal, anoþ{er} sup{er}ficiall{e}, anoþ{er} quadrat, anoþ{er} cubik{e}
+or hoole. lyneal is that þat is considred{e} after the p{ro}cesse,
+havyng{e} no respect to the direccio{u}n of nombre in nombre, As a lyne
+hath{e} but one dymensio{u}n that is to sey after the length{e}. Nombre
+sup{er}ficial is þ{a}t cometh{e} of ledyng{e} of oo nombre into
+a-nother, wherfor it is called{e} sup{er}ficial, for it hath{e} .2.
+nombres notyng or mesuryng{e} hym, as a sup{er}ficiall{e} thyng{e}
+hath{e} .2. dimensions, þ{a}t is to sey length{e} and brede. And for
+bycause a nombre may be had{e} in a-nother by .2. man{er}s, þ{a}t is to
+sey other in hym-self{e}, oþ{er} in anoþ{er}, Vnderstond{e} yf it be had
+in hym-self, It is a quadrat. ffor dyvisio{u}n write by vnytes, hath{e}
+.4. sides even as a quadrangill{e}. and yf the nombre be had{e} in
+a-noþ{er}, the nombre is sup{er}ficiel and not quadrat, as .2. had{e} in
+.3. maketh{e} .6. that is þe first nombre sup{er}ficiell{e}; wherfor it
+is open þat all{e} nombre quadrat is sup{er}ficiel, and not
+co{n}u{er}tid{e}. The rote of a nombre quadrat is þat nombre that is had
+of hym-self, as twies .2. makith{e} 4. and .4. is the first nombre
+quadrat, and 2. is his rote. 9. 8. 7. 6. 5. 4. 3. 2. 1. / The rote of
+the more quadrat .3. 1. 4. 2. 6. The most nombre quadrat 9. 8. 7. 5.
+9. 3. 4. 7. 6. / the remenent ou{er} the quadrat .6. 0. 8. 4. 5. / The
+first caas of nombre quadrat .5. 4. 7. 5. 6. The rote .2. 3. 4. The
+second{e} caas .3. 8. 4. 5. The rote .6. 2. The third{e} caas .2. 8. 1.
+9. The rote .5. 3. The .4. caas .3. 2. 1. The rote .1. 7. / The 5. caas
+.9. 1. 2. 0. 4. / The rote 3. 0. 2. The solid{e} nombre or cubik{e} is
+þat þ{a}t comytħe of double ledyng of nombre in nombre; And it is
+cleped{e} a solid{e} body that hath{e} þ{er}-in .3 [dimensions] þat is
+to sey, length{e}, brede, and thiknesse. so þ{a}t nombre hath{e} .3.
+nombres to be brought forth{e} in hym. But nombre may be had{e} twies in
+nombre, for other it is had{e} in hym-self{e}, oþ{er} in a-noþ{er}. If a
+nombre be had{e} twies in hym-self, oþ{er} ones in his quadrat, þ{a}t is
+the same, þ{a}t a cubik{e} [*Fol. 54.] is, And is the same that is
+solide. And yf a nombre twies be had{e} in a-noþ{er}, the nombre is
+cleped{e} solide and not cubik{e}, as twies .3. and þ{a}t .2. makith{e}
+.12. Wherfor it is opyn{e} that all{e} cubik{e} nombre is solid{e}, and
+not {con}u{er}tid{e}. Cubik{e} is þ{a}t nombre þat comyth{e} of
+ledyng{e} of hym-self{e} twyes, or ones in his quadrat. And here-by it
+is open that o nombre is the roote of a quadrat and of a cubik{e}.
+Natheles the same nombre is not q{ua}drat and cubik{e}. Opyn{e} it is
+also that all{e} nombres may be a rote to a q{ua}drat and cubik{e}, but
+not all{e} nombre quadrat or cubik{e}. Therfor sithen þe ledyng{e} of
+vnyte in hym-self ones or twies nought cometh{e} but vnytes, Seith{e}
+Boice in Arsemetrik{e}, that vnyte potencially is al nombre, and none in
+act. And vndirstond{e} wele also that betwix euery .2. quadrat{es} ther
+is a meene p{ro}porcionall{e}, That is opened{e} thus; lede the rote of
+o quadrat into the rote of the oþ{er} quadrat, and þan wolle þe meene
+shew.
+
+  [Sidenote: Examples of square roots.]
+
+  +-------------+-+-+-+-++-+-+-+-++-+-+-+-+-++-+---+------+-+
+  | Residuu{m}  | | |0| || | | |4|| | |0| | || |   |   0  | |
+  +-------------+-+-+-+-++-+-+-+-++-+-+-+-+-++-+---+------+-+
+  | Quadrand{e} |4|3|5|6||3|0|2|9||1|7|4|2|4||1| 9 |   3  |6|
+  +-------------+-+-+-+-++-+-+-+-++-+-+-+-+-++-+---+------+-+
+  | Duplum      |1|2| | ||1|0| | ||2| |6| | || |[8]|[{19}]| |
+  +-------------+-+-+-+-++-+-+-+-++-+-+-+-+-++-+---+------+-+
+  | Subduplu{m} | |6| |6|| |5| |5||1| |3| |2|| | 4 |      |4|
+  +-------------+-+-+-+-++-+-+-+-++-+-+-+-+-++-+---+------+-+
+
+  [Sidenote: A note on mean proportionals.]
+
+Also betwix the next .2. cubikis, me may fynde a double meene, that is
+to sey a more meene and a lesse. The more meene thus, as to bryng{e} the
+rote of the lesse into a quadrat of the more. The lesse thus, If the
+rote of the more be brought Into the quadrat of the lesse.
+
+    [Headnote: Chapter X. Extraction of Square Root.]
+
+  [Sidenote: To find a square root. Begin with the last odd place.
+  Find the nearest square root of that number, subtract, double it,
+  and set the double one to the right. Find the second figure by
+  division. Multiply the double by the second figure, and add after
+  it the square of the second figure, and subtract.]
+
+[{20}]To draw a rote of the nombre quadrat it is What-eu{er} nombre be
+p{ro}posed{e} to fynde his rote and to se yf it be quadrat. And yf it be
+not quadrat the rote of the most quadrat fynde out, vnder the nombre
+p{ro}posed{e}. Therfor yf thow wilt the rote of any quadrat nombre draw
+out, write the nombre by his differences, and compt the nombre of the
+figures, and wete yf it be od{e} or even. And yf it be even, than most
+thow begynne worche vnder the last save one. And yf it be od{e} w{i}t{h}
+the last; and forto sey it shortly, al-weyes fro the last od{e} me
+shall{e} begynne. Therfor vnder the last in an od place sette, me most
+fynd{e} a digit, the which{e} lad{e} in hym-self{e} it puttith{e} away
+that, þat is ou{er} his hede, oþ{er} as neigh{e} as me may: suche a
+digit found{e} and w{i}t{h}draw fro his ou{er}er, me most double that
+digit and sette the double vnder the next figure toward{e} the right
+hond{e}, and his vnder double vnder hym. That done, than me most
+fy{n}d{e} a-noþ{er} digit vnder the next figure bifore the doubled{e},
+the which{e} [*Fol. 54b] brought in double setteth{e} a-way all{e} that
+is ou{er} his hede as to reward{e} of the doubled{e}: Than brought into
+hym-self settith{e} all away in respect of hym-self, Other do it as nye
+as it may be do: other me may w{i}t{h}-draw the digit [{21}][last]
+found{e}, and lede hym in double or double hym, and after in
+hym-self{e}; Than Ioyne to-geder the p{ro}duccion{e} of them bothe, So
+that the first figure of the last p{ro}duct be added{e} before the first
+of the first p{ro}duct{es}, the second{e} of the first, {et}c. and so
+forth{e}, subtrahe fro the totall{e} nombre in respect of þe digit.
+
+  [Sidenote: Examples.]
+
+  +------------------+-+-+-+-+-++-+-+-+-+-++---+-+---+-+---+-+-+
+  | The residue      | | | | | || | | | | ||   | |   |5| 4 |3|2|
+  +------------------+-+-+-+-+-++-+-+-+-+-++---+-+---+-+---+-+-+
+  | To be quadred{e} |4|1|2|0|9||1|5|1|3|9|| 9 |0| 0 |5| 4 |3|2|
+  +------------------+-+-+-+-+-++-+-+-+-+-++---+-+---+-+---+-+-+
+  | The double       | |4|0| | || |2| |4| ||   |6|   |0|   | |0|
+  +------------------+-+-+-+-+-++-+-+-+-+-++---+-+---+-+---+-+-+
+  | The vnder double |2| |0| |3||1| |2| |3||[3]| |[0]| |[0]| |0|
+  +------------------+-+-+-+-+-++-+-+-+-+-++---+-+---+-+---+-+-+
+
+  [Sidenote: Special cases. The residue.]
+
+And if it hap þ{a}t no digit may be found{e}, Than sette a cifre vndre
+a cifre, and cesse not till{e} thow fynde a digit; and whan thow hast
+founde it to double it, neþ{er} to sette the doubled{e} forward{e}
+nether the vnder doubled{e}, Till thow fynde vndre the first figure a
+digit, the which{e} lad{e} in all{e} double, settyng away all{e} that is
+ou{er} hym in respect of the doubled{e}: Than lede hym into hym-self{e},
+and put a-way all{e} in regard{e} of hym, other as nygh{e} as thow
+maist. That done, other ought or nought wolle be the residue. If nought,
+than it shewith{e} that a nombre componed{e} was the quadrat, and his
+rote a digit last found{e} w{i}t{h} vnder{e}-double other vndirdoubles,
+so that it be sette be-fore: And yf ought[{22}] remayn{e}, that
+shew{i}t{h} that the nombre p{ro}posed{e} was not quadrat,[{23}]
+[[wher-vpon{e} se the table in the next side of the next leef{e}.]]
+but a digit [last found with the subduple or subduples is]
+
+  [Sidenote: This table is constructed for use in cube root sums,
+  giving the value of ab.^2]
+
+  +---+-----+-----+-----+-----+-----+---------+-----+---------+
+  | 1 |   2 |   3 |   4 |   5 |   6 |    7    |   8 |     9   |
+  +---+-----+-----+-----+-----+-----+---------+-----+---------+
+  | 2 |   8 |  12 |  16 |  20 |  24 |   28    |  32 |    36   |
+  +---+-----+-----+-----+-----+-----+---------+-----+---------+
+  | 3 |  18 |  27 |  36 |  45 |  54 |   63    |  72 |    81   |
+  +---+-----+-----+-----+-----+-----+---------+-----+---------+
+  | 4 |  32 |  48 |  64 |  80 |  96 |112[{24}]| 128 |   144   |
+  +---+-----+-----+-----+-----+-----+---------+-----+---------+
+  | 5 |  50 |  75 | 100 | 125 | 150 |   175   | 200 |   225   |
+  +---+-----+-----+-----+-----+-----+---------+-----+---------+
+  | 6 |  72 | 108 | 144 | 180 | 216 |   252   | 288 |   324   |
+  +---+-----+-----+-----+-----+-----+---------+-----+---------+
+  | 7 |  98 | 147 | 196 | 245 | 294 |   343   | 393 |   441   |
+  +---+-----+-----+-----+-----+-----+---------+-----+---------+
+  | 8 | 128 | 192 | 256 | 320 | 384 |   448   | 512 |   576   |
+  +---+-----+-----+-----+-----+-----+---------+-----+---------+
+  | 9 | 168 | 243 | 324 | 405 | 486 |   567   | 648 |729[{25}]|
+  +---+-----+-----+-----+-----+-----+---------+-----+---------+
+
+  [Sidenote: How to prove the square root without or with a remainder.]
+
+The rote of the most quadrat conteyned{e} vndre the nombre
+p{ro}posed{e}. Therfor yf thow wilt p{ro}ve yf thow have wele do or no,
+Multiplie the digit last found{e} w{i}t{h} the vnder-double oþ{er}
+vnder-doublis, and thow shalt fynde the same figures that thow haddest
+before; And so that nought be the [*Fol. 55.] residue. And yf thow have
+any residue, than w{i}t{h} the addicio{u}n þ{er}of that is res{er}ued{e}
+w{i}t{h}-out in thy table, thow shalt fynd{e} thi first figures as thow
+haddest them before, {et}c.
+
+    [Headnote: Chapter XI. Extraction of Cube Root.]
+
+  [Sidenote: Definition of a cubic number and a cube root. Mark off
+  the places in threes. Find the first digit; treble it and place it
+  under the next but one, and multiply by the digit. Then find the
+  second digit. Multiply the first triplate and the second digit, twice
+  by this digit. Subtract. Examples.]
+
+Heere folowith{e} the extraccio{u}n of rotis in cubik{e} nombres;
+wher-for me most se what is a nombre cubik{e}, and what is his roote,
+And what is the extraccio{u}n of a rote. A nombre cubik{e} it is, as it
+is before declared{e}, that cometh{e} of ledyng of any nombre twies in
+hym-self{e}, other ones in his quadrat. The rote of a nombre cubik{e} is
+the nombre that is twies had{e} in hy{m}-self{e}, or ones in his
+quadrat. Wher-thurgh{e} it is open, that eu{er}y nombre quadrat or
+cubik{e} have the same rote, as it is seid{e} before. And forto draw out
+the rote of a cubik{e}, It is first to fynd{e} þe nombr{e} p{ro}posed{e}
+yf it be a cubik{e}; And yf it be not, than thow most make extraccio{u}n
+of his rote of the most cubik{e} vndre the nombre p{ro}posid{e} his rote
+found{e}. Therfor p{ro}posed{e} some nombre, whos cubical rote þ{o}u
+woldest draw out; First thow most compt the figures by fourthes, that is
+to sey in the place of thousand{es}; And vnder the last thousand{e}
+place, thow most fynde a digit, the which{e} lad{e} in hym-self cubikly
+puttith{e} a-way that þat is ou{er} his hede as in respect of hym, other
+as nygh{e} as thow maist. That done, thow most trebill{e} the digit, and
+that triplat is to be put vnder the .3. next figure toward{e} the right
+hond{e}, And the vnder-trebill{e} vnder the trebill{e}; Than me most
+fynd{e} a digit vndre the next figure bifore the triplat, the which{e}
+w{i}t{h} his vnder-trebill{e} had into a trebill{e}, aft{er}warde other
+vnder[trebille][{26}] had in his p{ro}duccio{u}n, putteth{e} a-way
+all{e} that is ou{er} it in regard{e} of[{27}] [the triplat. Then lade
+in hymself puttithe away that þat is over his hede as in respect of hym,
+other as nyghe as thou maist:] That done, thow most trebill{e} the digit
+ayene, and the triplat is to be sette vnder the next .3. figure as
+before, And the vnder-trebill{e} vnder the trebill{e}: and than most
+thow sette forward{e} the first triplat w{i}t{h} his vndre-trebill{e} by
+.2. differences. And than most thow fynde a digit vnder the next figure
+before the triplat, the which{e} with{e} his vnder-t{r}iplat had in his
+triplat afterward{e}, other vnder-treblis lad in p{ro}duct [*Fol. 55b]
+It sitteth{e} a-way ałł that is ou{er} his hede in respect of the
+triplat than had in hym-self cubikly,[{28}] [[it setteth{e} a-way all{e}
+his respect]] or as nygh{e} as ye may.
+
+  +----------------+--+-+-+-+-+-+---++--+-+-+-+-+--++----+-+--+-+--+
+  | Residuu{m}     |  | | | | | | 5 ||  | | | | | 4||   1|0|1 |9|  |
+  +----------------+--+-+-+-+-+-+---++--+-+-+-+-+--++----+-+--+-+--+
+  | Cubicandu{s}   | 8|3|6|5|4|3| 2 || 3|0|0|7|6| 7|| 1 1|6|6 |7|  |
+  +----------------+--+-+-+-+-+-+---++--+-+-+-+-+--++----+-+--+-+--+
+  | Triplum        |  | |6|0| | |   ||  | | |1|8|  ||    | |4 | |  |
+  +----------------+--+-+-+-+-+-+---++--+-+-+---+--++----+-+--+-+--+
+  | Subt{r}iplu{m} | 2| | |0| | |[3]||  | |6| | | 7||   2| |  |2|  |
+  +----------------+--+-+-+-+-+-+---++--+-+-+-+-+--++----+-+--+-+--+
+
+  [Sidenote: Continue this process till the first figure is reached.
+  Examples. The residue. Special cases. Special case.]
+
+Nother me shall{e} not cesse of the fyndyng{e} of that digit, neither of
+his triplacio{u}n, neþ{er} of the triplat-is [{29}]anteriorac{i}o{u}n,
+that is to sey, settyng forward{e} by .2. differences, Ne therof the
+vndre-triple to be put vndre the triple, Nether of the multiplicacio{u}n
+þ{er}of, Neither of the subtraccio{u}n, till{e} it come to the first
+figure, vnder the which{e} is a digitall{e} nombre to be found{e}, the
+which{e} with{e} his vndre-treblis most be had{e} in tribles,
+After-ward{e} w{i}t{h}out vnder-treblis to be had{e} into produccio{u}n,
+settyng away all{e} that is ou{er} the hed{e} of the triplat nombre,
+After had into hymself{e} cubikly, and sette all{e}-way that is ou{er}
+hym.
+
+  +------------------+---+---+---+---++---+---+---+---+---+
+  | To be cubiced{e} | 1 | 7 | 2 | 8 || 3 | 2 | 7 | 6 | 8 |
+  +------------------+---+---+---+---++---+---+---+---+---+
+  | The triple       |   |   | 3 | 2 ||   |   |   | 9 |   |
+  +------------------+---+---+---+---++---+---+---+---+---+
+  | The vnder triple |   |   | 1 | 2 ||   |[3]|   | 3 | 3 |
+  +------------------+---+---+---+---++---+---+---+---+---+
+
+Also note wele that the p{ro}ducc{i}on comyng{e} of the ledyng of a
+digite found{e}[{30}] [[w{i}t{h} an vndre-triple / other of an
+vndre-triple in a triple or triplat is And after-ward{e} w{i}t{h} out
+vndre-triple other vndre-triplis in the p{ro}duct and ayene that
+p{ro}duct that cometh{e} of the ledyng{e} of a digit found{e} in
+hym-self{e} cubicall{e}]] me may adde to, and also w{i}t{h}-draw fro of
+the totall{e} nombre sette above that digit so found{e}.[{31}] [[as ther
+had be a divisio{u}n made as it is opened{e} before]] That done ought or
+nought most be the residue. If it be nought, It is open that the nombre
+p{ro}posed{e} was a cubik{e} nombre, And his rote a digit founde last
+w{i}t{h} the vnder-triples: If the rote therof wex bad{e} in
+hym-self{e}, and afterward{e} p{ro}duct they shall{e} make the first
+fig{ur}es. And yf ought be in residue, kepe that w{i}t{h}out in the
+table; and it is open{e} that the nombre was not a cubik{e}. but a digit
+last founde w{i}t{h} the vndirtriplis is rote of the most cubik{e} vndre
+the nombre p{ro}posed{e} conteyned{e}, the which{e} rote yf it be had{e}
+in hym-self{e}, And aft{er}ward{e} in a p{ro}duct of that shall{e} growe
+the most cubik{e} vndre the nombre p{ro}posed{e} conteyned{e}, And yf
+that be added{e} to a cubik{e} the residue res{er}ued{e} in the table,
+woll{e} make the same figures that ye had{e} first. [*Fol. 56.] And yf
+no digit after the anterioracio{u}n[{32}] may not be found{e}, than put
+ther{e} a cifre vndre a cifre vndir the third{e} figure, And put
+forward{e} þe fig{ur}es. Note also wele that yf in the nombre
+p{ro}posed{e} ther ben no place of thowsand{es}, me most begynne vnder
+the first figure in the extraccio{u}n of the rote. some vsen forto
+distingue the nombre by threes, and ay begynne forto wirch{e} vndre the
+first of the last t{er}nary other unco{m}plete nombre, the which{e}
+maner of op{er}acio{u}n accordeth{e} w{i}t{h} that before. And this at
+this tyme suffiseth{e} in extraccio{u}n of nombres quadrat or cubik{es}
+{et}c.
+
+  [Sidenote: Examples.]
+
+  +-------------------+---+--+------+--+--+--+--++--+--+--+--+--+--+--+
+  | The residue       |   |  |      |  |  |  | 0||  |  |  |  |  | 1| 1|
+  +-------------------+---+--+------+--+--+--+--++--+--+--+--+--+--+--+
+  | The cubicand{us}  | 8 | 0|   0  | 0| 0| 0| 0|| 8| 2| 4| 2| 4| 1| 9|
+  +-------------------+---+--+------+--+--+--+--++--+--+--+--+--+--+--+
+  | The triple        |   |  |[{33}]| 0| 0|  |  ||  |  | 6|  |  |  |  |
+  +-------------------+---+--+------+--+--+--+--++--+--+--+--+--+--+--+
+  | The vndert{r}iple |[2]|  |      | 0| 0|  |  || 2|  |  | 6| 2|  |  |
+  +-------------------+---+--+------+--+--+--+--++--+--+--+--+--+--+--+
+
+
+    [Headnote: Table of Numbers, &c.]
+
+  [Sidenote: A table of numbers; probably from the Abacus.]
+
+   1   2         3              4               5              6
+  one. x. an. hundred{e}/ a thowsand{e}/ x. thowsand{e}/ An hundred{e}
+                     7
+  thowsand{e}/ A thowsand{e} tymes a thowsand{e}/ x. thousand{e} tymes
+
+  a thousand{e}/ An hundred{e} thousand{e} tymes a thousand{e} A
+
+  thousand{e} thousand{e} tymes a thousand{e}/ this is the x place
+
+    {et}c.
+
+
+[Ende.]
+
+
+FOOTNOTES (The Art of Nombryng):
+
+  [1: MS. Materiall{e}.]
+  [2: MS. Formall{e}.]
+  [3: ‘the’ in MS.]
+  [4: ‘be’ in MS.]
+  [5: ‘and’ in MS.]
+  [6: ‘is’ in MS.]
+  [7: 6 in MS.]
+  [8: 0 in MS.]
+  [9: 2 in MS.]
+  [10: _sic._]
+  [11: ‘And’ inserted in MS.]
+  [12: ‘4 the’ inserted in MS.]
+  [13: ‘to’ in MS.]
+  [14: ‘that’ repeated in MS.]
+  [15: ‘1’ in MS.]
+  [16: Blank in MS.]
+  [17: ‘nought’ in MS.]
+  [18: 3 written for 2 in MS.]
+  [19: 7 in MS.]
+  [20: runs on in MS.]
+  [21: ‘so’ in MS.]
+  [22: ‘nought’ in MS.]
+  [23: MS. adds here: ‘wher-vpon{e} se the table in the next side of
+    the next leef{e}.’]
+  [24: 110 in MS.]
+  [25: 0 in MS.]
+  [26: double in MS.]
+  [27: ‘it hym-self{e}’ in MS.]
+  [28: MS. adds here: ‘it setteth{e} a-way all{e} his respect.’]
+  [29: ‘aucterioracio{u}n’ in MS.]
+  [30: MS. adds here: ’w{i}t{h} an vndre-triple / other of an
+    vndre-triple in a triple or triplat is And after-ward{e} w{i}t{h}
+    out vndre-triple other vndre-triplis in the p{ro}duct and ayene
+    that p{ro}duct that cometh{e} of the ledyng{e} of a digit found{e}
+    in hym-self{e} cubicall{e}’ /]
+  [31: MS. adds here: ‘as ther had be a divisio{u}n made as it is
+    opened{e} before.’]
+  [32: MS. anteriocacio{u}n.]
+  [33: 4 in MS.]
+
+
+
+
+Accomptynge by counters.
+
+  [Transcriber’s Note:
+
+  The original text was printed as a single continuous paragraph, with
+  no break between speakers; all examples were shown inline. It has been
+  broken up for this e-text.]
+
+
+  [*116b]
+
+  ¶ The seconde dialoge of accomptynge by counters.
+
+_Mayster._
+
+Nowe that you haue learned the commen kyndes of Arithmetyke with the
+penne, you shall se the same art in cou{n}ters: whiche feate doth not
+only serue for them that can not write and rede, but also for them that
+can do bothe, but haue not at some tymes theyr penne or tables redye
+with them. This sorte is in two fourmes co{m}menly. The one by lynes,
+and the other without lynes: in that y^t hath lynes, the lynes do stande
+for the order of places: and in y^t that hath no lynes, there must be
+sette in theyr stede so many counters as shall nede, for eche lyne one,
+and they shall supplye the stede of the lynes.
+
+_S._ By examples I shuld better p{er}ceaue your meanynge.
+
+_M._ For example of the [*117a.] ly[*]nes:
+
+  ----1-0-0-0-0-0--
+  ----1-0-0-0-0----
+  -X--1-0-0-0------
+  ----1-0-0--------
+  ----1-0----------
+  ----1------------
+
+  [Sidenote: Numeration.]
+
+Lo here you se .vi. lynes whiche stande for syxe places so that the
+nethermost standeth for y^e fyrst place, and the next aboue it, for the
+second: and so vpward tyll you come to the hyghest, which is the syxte
+lyne, and standeth for the syxte place. Now what is the valewe of euery
+place or lyne, you may perceaue by the figures whiche I haue set on
+them, which is accordynge as you learned before in the Numeration of
+figures by the penne: for the fyrste place is the place of vnities or
+ones, and euery counter set in that lyne betokeneth but one: {and} the
+seconde lyne is the place of 10, for euery counter there, standeth for
+10. The thyrd lyne the place of hundredes: the fourth of thousandes:
+{and} so forth.
+
+_S._ Syr I do perceaue that the same order is here of lynes, as was in
+the other figures [*117b] by places, so that you shall not nede longer
+to stande about Numeration, excepte there be any other difference.
+
+_M._ Yf you do vndersta{n}de it, then how wyll you set 1543?
+
+_S._ Thus, as I suppose.
+
+  -------
+  -X--1--
+  ----5--
+  ----4--
+  ----3--
+
+_M._ You haue set y^e places truely, but your figures be not mete for
+this vse: for the metest figure in this behalfe, is the figure of a
+cou{n}ter round, as you se here, where I haue expressed that same summe.
+
+  -------------
+
+  -X--o--------
+       o
+  -------------
+
+  ----o-o-o-o--
+
+  ----o-o-o----
+
+_S._ So that you haue not one figure for 2, nor 3, nor 4, and so forth,
+but as many digettes as you haue, you set in the lowest lyne: and for
+euery 10 you set one in the second line: and so of other. But I know not
+by what reason you set that one counter for 500 betwene two lynes.
+
+_M._ you shall remember this, that when so euer you nede to set downe 5,
+50, or 500, or 5000, or so forth any other nomber, whose numerator
+[*118a] is 5, you shall set one counter for it, in the next space aboue
+the lyne that it hath his denomination of, as in this example of that
+500, bycause the numerator is 5, it must be set in a voyd space: and
+bycause the denominator is hundred, I knowe that his place is the voyde
+space next aboue hundredes, that is to say, aboue the thyrd lyne. And
+farther you shall marke, that in all workynge by this sorte, yf you
+shall sette downe any summe betwene 4 and 10, for the fyrste parte of
+that nomber you shall set downe 5, & then so many counters more, as
+there reste no{m}bers aboue 5. And this is true bothe of digettes and
+articles. And for example I wyll set downe this su{m}me 287965,
+
+  -X-----------
+
+  ------o-o----
+           o
+  ------o-o-o--
+           o
+  -X----o-o----
+         o
+  ----o-o-o-o--
+       o
+  ----o--------
+       o
+  -------------
+
+which su{m}me yf you marke well, you nede none other exa{m}ples for to
+lerne the numeration of [*118b] this forme. But this shal you marke,
+that as you dyd in the other kynde of arithmetike, set a pricke in the
+places of thousa{n}des, in this worke you shall sette a starre, as you
+se here.
+
+    [Headnote: Addition on the Counting Board.]
+
+  [Sidenote: Addition.]
+
+_S._ Then I perceave numeration, but I praye you, howe shall I do in
+this arte to adde two summes or more together?
+
+_M._ The easyest way in this arte is, to adde but 2 su{m}mes at ones
+together: how be it you may adde more, as I wyll tell you anone.
+Therfore when you wyll adde two su{m}mes, you shall fyrst set downe one
+of them, it forseth not whiche, {and} then by it drawe a lyne crosse the
+other lynes. And afterward set downe the other su{m}me, so that that
+lyne may be betwene them, as yf you wolde adde 2659 to 8342, you must
+set your su{m}mes as you se
+
+  -------------|-----------
+       o       |
+  -X--o-o-o----|--o-o------
+               |   o
+  ----o-o-o----|--o--------
+               |   o
+  ----o-o-o-o--|-----------
+               |   o
+  ----o-o------|--o-o-o-o--
+
+here. And then yf you lyst, you [*119a] may adde the one to the other in
+the same place, or els you may adde them both together in a newe place:
+which waye, bycause it is moste playnest, I wyll showe you fyrst.
+Therfore wyl I begynne at the vnites, whiche in the fyrst su{m}me is but
+2, {and} in y^e second su{m}me 9, that maketh 11, those do I take vp,
+and for them I set 11 in the new roume, thus,
+
+  -------------|-------|-------
+       o       |       |
+  -X--o-o-o----|--o-o--|-------
+               |   o   |
+  ----o-o-o----|--o----|-------
+               |    o  |
+  ----o-o-o-o--|-------|-o-----
+               |       |
+  -------------|-------|-o-----
+
+Then do I take vp all y^e articles vnder a hundred, which in the fyrst
+su{m}me are 40, and in the second summe 50, that maketh 90: or you may
+saye better, that in the fyrste summe there are 4 articles of 10, and in
+the seconde summe 5, which make 9, but then take hede that you sette
+them in theyr [*119b] ryght lynes as you se here.
+
+  -----------|----------|-------------
+       o     |          |
+  -X--o-o-o--|--o-o-----|-------------
+             |   o      |
+  ----o-o-o--|--o-------|-------------
+             |          |   o
+  -----------|----------|--o-o-o-o-o--
+             |          |
+  -----------|----------|--o----------
+
+Where I haue taken awaye 40 fro{m} the fyrste su{m}me, and 50 from y^e
+second, and in theyr stede I haue set 90 in the thyrde, whiche I haue
+set playnely y^t you myght well perceaue it: how be it seynge that 90
+with the 10 that was in y^e thyrd roume all redy, doth make 100,
+I myghte better for those 6 cou{n}ters set 1 in the thyrde lyne, thus:
+
+  ----------
+
+  -X--------
+
+  ----o-----
+
+  ----------
+
+  ----o-----
+
+For it is all one summe as you may se, but it is beste, neuer to set 5
+cou{n}ters in any line, for that may be done with 1 cou{n}ter in a
+hygher place.
+
+_S._ I iudge that good reaso{n}, for many are vnnedefull, where one wyll
+serue.
+
+_M._ Well, then [*120a] wyll I adde forth of hundredes: I fynde 3 in the
+fyrste summe, and 6 in the seconde, whiche make 900, them do I take vp
+{and} set in the thyrd roume where is one hundred all redy, to whiche I
+put 900, and it wyll be 1000, therfore I set one cou{n}ter in the fourth
+lyne for them all, as you se here.
+
+  -----------|-------|--------
+       o     |       |
+  -X--o-o-o--|--o-o--|--o-----
+             |       |
+  -----------|-------|--------
+             |       |
+  -----------|-------|--------
+             |       |
+  -----------|-------|--o-----
+
+Then adde I y^e thousandes together, whiche in the fyrst su{m}me are
+8000, {and} in y^e second 2000, that maketh 10000: them do I take vp
+fro{m} those two places, and for them I set one counter in the fyfte
+lyne, and then appereth as you se, to be 11001, for so many doth amount
+of the addition of 8342 to 2659.
+
+  ----o-----
+
+  -X--o-----
+
+  ----------
+
+  ----------
+
+  ----o-----
+
+[*120b] _S._ Syr, this I do perceave: but how shall I set one su{m}me to
+an other, not chaungynge them to a thyrde place?
+
+_M._ Marke well how I do it: I wyll adde together 65436, and 3245,
+whiche fyrste I set downe thus.
+
+  -------------|--------------
+               |      o
+  -------------|--o-----------
+               |   o
+  -X--o-o-o----|--------------
+               |
+  ----o-o------|--o-o-o-o-----
+               |
+  ----o-o-o-o--|--o-o-o-------
+       o       |       o
+  -------------|--o-----------
+
+Then do I begynne with the smalest, which in the fyrst summe is 5, that
+do I take vp, and wold put to the other 5 in the seconde summe, sauynge
+that two counters can not be set in a voyd place of 5, but for them
+bothe I must set 1 in the seconde lyne, which is the place of 10,
+therfore I take vp the 5 of the fyrst su{m}me, {and} the 5 of the
+seco{n}de, and for them I set 1 in the seco{n}d lyne, [*121a] as you se
+here.
+
+  -------------|--------------
+               |   o
+  -------------|--o-----------
+               |   o
+  -X--o-o-o----|--------------
+               |
+  ----o-o------|--o-o-o-o-----
+               |
+  ----o-o-o-o--|--o-o-o-o-----
+               |
+  -------------|--o-----------
+
+Then do I lyke wayes take vp the 4 counters of the fyrste su{m}me {and}
+seconde lyne (which make 40) and adde them to the 4 counters of the same
+lyne, in the second su{m}me, and it maketh 80, But as I sayde I maye not
+conueniently set aboue 4 cou{n}ters in one lyne, therfore to those 4
+that I toke vp in the fyrst su{m}me, I take one also of the seconde
+su{m}me, and then haue I taken vp 50, for whiche 5 counters I sette
+downe one in the space ouer y^e second lyne, as here doth appere.
+
+  -----------|--------------
+             |   o
+  -----------|--o-----------
+             |   o
+  -X--o-o-o--|--------------
+             |
+  ----o-o----|--o-o-o-o-----
+             |   o
+  -----------|--o-o-o-------
+             |
+  -----------|--o-----------
+
+[*121b.] and then is there 80, as well w^t those 4 counters, as yf I had
+set downe y^e other 4 also. Now do I take the 200 in the fyrste su{m}me,
+and adde them to the 400 in the seconde summe, and it maketh 600,
+therfore I take vp the 2 counters in the fyrste summe, and 3 of them in
+the seconde summe, and for them 5 I set 1 in y^e space aboue, thus.
+
+  -----------|------------
+             |   o
+  -----------|--o---------
+             |   o
+  -X--o-o-o--|------------
+             |   o
+  -----------|--o---------
+             |   o
+  -----------|--o-o-o-----
+             |
+  -----------|--o---------
+
+Then I take y^e 3000 in y^e fyrste su{m}me, vnto whiche there are none
+in the second summe agreynge, therfore I do onely remoue those 3
+counters from the fyrste summe into the seconde, as here doth appere.
+
+  ----|-------------
+      |  o
+  ----|---o---------
+      |  o
+  -X--|---o-o-o-----
+      |  o
+  ----|-o-----------
+      |  o
+  ----|---o-o-o-----
+      |
+  ----|---o---------
+
+[*122a.] And so you see the hole su{m}me, that amou{n}teth of the
+addytio{n} of 65436 with 3245 to be 6868[1]. And yf you haue marked
+these two exa{m}ples well, you nede no farther enstructio{n} in Addition
+of 2 only summes: but yf you haue more then two summes to adde, you may
+adde them thus. Fyrst adde two of them, and then adde the thyrde, and
+y^e fourth, or more yf there be so many: as yf I wolde adde 2679 with
+4286 and 1391. Fyrste I adde the two fyrste summes thus.
+
+  -------------|-----------|--------------
+               |           |   o
+  -X--o-o------|--o-o-o-o--|--o-----------
+       o       |           |   o
+  ----o--------|--o-o------|--o-o-o-o-----
+       o       |   o       |   o
+  ----o-o------|--o-o-o----|--o-----------
+       o       |   o       |   o
+  ----o-o-o-o--|--o--------|--------------
+
+[*122b.] And then I adde the thyrde thereto thus. And so of more yf you
+haue them.
+
+  -------------|-----------|------------
+               |   o       |       o
+  -X--o--------|--o--------|--o-o-o-----
+               |   o       |
+  ----o-o-o----|--o-o-o-o--|--o-o-o-----
+       o       |   o       |   o
+  ----o-o-o-o--|----o------|------------
+               |   o       |   o
+  ----o--------|-----------|--o---------
+
+
+    [Headnote: Subtraction on the Counting Board.]
+
+  [Sidenote: Subtraction.]
+
+_S._ Nowe I thynke beste that you passe forth to Subtraction, except
+there be any wayes to examyn this maner of Addition, then I thynke that
+were good to be knowen nexte.
+
+_M._ There is the same profe here that is in the other Addition by the
+penne, I meane Subtraction, for that onely is a sure waye: but
+consyderynge that Subtraction must be fyrste knowen, I wyl fyrste teache
+you the arte of Subtraction, and that by this example: I wolde subtracte
+2892 out of 8746. These summes must I set downe as I dyd in Addition:
+but here it is best [*116a (_sic_).] to set the lesser no{m}ber fyrste,
+thus.
+
+  -------------|--------------
+               |     o
+  -X--o-o------|--o-o-o-------
+       o       |   o
+  ----o-o-o----|--o-o---------
+       o       |
+  ----o-o-o-o--|--o-o-o-o-----
+               |   o
+  ----o-o------|--o-----------
+
+Then shall I begynne to subtracte the greatest nombres fyrste (contrary
+to the vse of the penne) y^t is the thousandes in this exa{m}ple:
+therfore I fynd amongest the thousandes 2, for which I withdrawe so many
+fro{m} the seconde summe (where are 8) and so remayneth there 6, as this
+exa{m}ple showeth.
+
+  -------------+--------------
+               |   o
+  -+-----------+--o-----------
+       o       |   o
+  ----o-o-o----+--o-o---------
+       o       |
+  ----o-o-o-o--+--o-o-o-o-----
+               |   o
+  ----o-o------+--o-----------
+
+Then do I lyke wayes with the hundredes, of whiche in the fyrste summe
+[*116b] I fynde 8, and is the seconde summe but 7, out of whiche I can
+not take 8, therfore thus muste I do: I muste loke how moche my summe
+dyffereth from 10, whiche I fynde here to be 2, then must I bate for my
+su{m}me of 800, one thousande, and set downe the excesse of hundredes,
+that is to saye 2, for so moche 100[0] is more then I shuld take vp.
+Therfore fro{m} the fyrste su{m}me I take that 800, and from the second
+su{m}me where are 6000, I take vp one thousande, and leue 5000; but then
+set I downe the 200 unto the 700 y^t are there all redye, and make them
+900 thus.
+
+  -------------+--------------
+               |   o
+  -+-----------+--------------
+               |   o
+  -------------+--o-o-o-o-----
+         o     |
+  ----o-o-o-o--+--o-o-o-o-----
+               |   o
+  ----o-o------+--o-----------
+
+Then come I to the articles of te{n}nes where in the fyrste su{m}me I
+fynde 90, [*117a] and in the seconde su{m}me but only 40: Now
+consyderyng that 90 can not be bated from 40, I loke how moche y^t 90
+doth dyffer from the next summe aboue it, that is 100 (or elles whiche
+is all to one effecte, I loke how moch 9 doth dyffer fro{m} 10) {and} I
+fynd it to be 1, then in the stede of that 90, I do take from the second
+summe 100: but consyderynge that it is 10 to moche, I set downe 1 in y^e
+nexte lyne beneth for it, as you se here.
+
+  ---------+------------
+           |   o
+  -+-------+------------
+           |   o
+  ---------+--o-o-o-----
+           |   o
+  ---------+------------
+           |   o
+  ----o-o--+--o---------
+
+Sauynge that here I haue set one counter in y^e space in stede of 5 in
+y^e nexte lyne. And thus haue I subtracted all saue two, which I must
+bate from the 6 in the second summe, and there wyll remayne 4, thus.
+
+  ----+--------------
+      |   o
+  -+--+--------------
+      |   o
+  ----+--o-o-o-------
+      |   o
+  ----+--------------
+      |
+  ----+--o-o-o-o-----
+
+So y^t yf I subtracte 2892 fro{m} 8746, the remayner wyll be 5854,
+[*117b] And that this is truely wrought, you maye proue by Addition: for
+yf you adde to this remayner the same su{m}me that you dyd subtracte,
+then wyll the formar su{m}me 8746 amount agayne.
+
+_S._ That wyll I proue: and fyrst I set the su{m}me that was subtracted,
+which was 2892, {and} the{n} the remayner 5854, thus.
+
+  --------------+--------------
+                |       o
+  -||--o-o------+--------------
+        o       |     o
+  -----o-o-o----+--o-o-o-------
+        o       |   o
+  -----o-o-o-o--+--------------
+                |
+  -----o-o------+--o-o-o-o-----
+
+Then do I adde fyrst y^e 2 to 4, whiche maketh 6, so take I vp 5 of
+those counters, and in theyr stede I sette 1 in the space, as here
+appereth.
+
+  --------------+------------
+                |     o
+  -||--o-o------+------------
+        o       |   o
+  -----o-o-o----+--o-o-o-----
+        o       |   o
+  -----o-o-o-o--+------------
+                |   o
+  --------------+--o---------
+
+[*118a] Then do I adde the 90 nexte aboue to the 50, and it maketh 140,
+therfore I take vp those 6 counters, and for them I sette 1 to the
+hundredes in y^e thyrde lyne, {and} 4 in y^e second lyne, thus.
+
+  ------------+--------------
+              |   o
+  -||--o-o----+--------------
+        o     |   o
+  -----o-o-o--+--o-o-o-o-----
+              |
+  ------------+--o-o-o-o-----
+              |   o
+  ------------+----o---------
+
+Then do I come to the hundredes, of whiche I fynde 8 in the fyrst summe,
+and 9 in y^e second, that maketh 1700, therfore I take vp those 9
+counters, and in theyr stede I sette 1 in the .iiii. lyne, and 1 in the
+space nexte beneth, and 2 in the thyrde lyne, as you se here.
+
+  ----------+--------------
+            |   o
+  -||--o-o--+--o-----------
+            |   o
+  ----------+--o-o---------
+            |
+  ----------+--o-o-o-o-----
+            |   o
+  ----------+--o-----------
+
+Then is there lefte in the fyrste summe but only 2000, whiche I shall
+take vp from thence, and set [*118b] in the same lyne in y^e second
+su{m}me, to y^e one y^t is there all redy: {and} then wyll the hole
+su{m}me appere (as you may wel se) to be 8746, which was y^e fyrst
+grosse summe, {and} therfore I do perceaue, that I hadde well subtracted
+before. And thus you may se how Subtraction maye be tryed by Addition.
+
+  ----+--------------
+      |     o
+  -X--+--o-o-o-------
+      |     o
+  ----+--o-o---------
+      |
+  ----+--o-o-o-o-----
+      |   o
+  ----+----o---------
+
+_S._ I perceaue the same order here w^t cou{n}ters, y^t I lerned before
+in figures.
+
+_M._ Then let me se howe can you trye Addition by Subtraction.
+
+_S._ Fyrste I wyl set forth this exa{m}ple of Additio{n} where I haue
+added 2189 to 4988, and the hole su{m}me appereth to be 7177,
+
+  --------------+-----------+----------
+                |           |   o
+  -||--o-o------+--o-o-o-o--+--o-o-----
+                |   o       |
+  -----o--------+--o-o-o-o--+--o-------
+        o       |   o       |   o
+  -----o-o-o----+--o-o-o----+--o-o-----
+        o       |   o       |   o
+  -----o-o-o-o--+--o-o-o----+--o-o-----
+
+[*119a] Nowe to trye whether that su{m}me be well added or no, I wyll
+subtract one of the fyrst two su{m}mes from the thyrd, and yf I haue
+well done y^e remayner wyll be lyke that other su{m}me. As for example:
+I wyll subtracte the fyrste summe from the thyrde, whiche I set thus in
+theyr order.
+
+  --------------+----------
+                |   o
+  -||--o-o------+--o-o-----
+                |
+  -----o--------+--o-------
+        o       |   o
+  -----o-o-o----+--o-o-----
+        o       |   o
+  -----o-o-o-o--+--o-o-----
+
+Then do I subtract 2000 of the fyrste summe fro{m} y^e second su{m}me,
+and then remayneth there 5000 thus.
+
+  -------------+----------
+               |   o
+  -X-----------+-----------
+               |
+  ----o--------+--o-------
+       o       |   o
+  ----o-o-o----+--o-o-----
+       o       |   o
+  ----o-o-o-o--+--o-o-----
+
+Then in the thyrd lyne, I subtract y^e 100 of the fyrste summe, fro{m}
+the second su{m}me, where is onely 100 also, and then in y^e thyrde lyne
+resteth nothyng. Then in the second lyne with his space ouer hym,
+I fynde 80, which I shuld subtract [*119b] from the other su{m}me, then
+seyng there are but only 70 I must take it out of some hygher summe,
+which is here only 5000, therfore I take vp 5000, and seyng that it is
+to moch by 4920, I sette downe so many in the seconde roume, whiche with
+the 70 beynge there all redy do make 4990, & then the summes doth stande
+thus.
+
+  --------------+--------------
+                |
+  -||-----------+--o-o-o-o-----
+                |   o
+  --------------+--o-o-o-o-----
+                |   o
+  --------------+--o-o-o-o-----
+        o       |   o
+  -----o-o-o-o--+--o-o---------
+
+Yet remayneth there in the fyrst su{m}me 9, to be bated from the second
+summe, where in that place of vnities dothe appere only 7, then I muste
+bate a hygher su{m}me, that is to saye 10, but seynge that 10 is more
+then 9 (which I shulde abate) by 1, therfore shall I take vp one counter
+from the seconde lyne, {and} set downe the same in the fyrst [*120a] or
+lowest lyne, as you se here.
+
+  -----+--------------
+       |
+  -||--+--o-o-o-o-----
+       |   o
+  -----+--o-o-o-o-----
+       |   o
+  -----+--o-o-o-------
+       |   o
+  -----+--o-o-o-------
+
+And so haue I ended this worke, {and} the su{m}me appereth to be y^e
+same, whiche was y^e seconde summe of my addition, and therfore I
+perceaue, I haue wel done.
+
+_M._ To stande longer about this, it is but folye: excepte that this you
+maye also vnderstande, that many do begynne to subtracte with counters,
+not at the hyghest su{m}me, as I haue taught you, but at the
+nethermoste, as they do vse to adde: and when the summe to be abatyd,
+in any lyne appeareth greater then the other, then do they borowe one of
+the next hygher roume, as for example: yf they shuld abate 1846 from
+2378, they set y^e summes thus.
+
+  --------------+------------
+                |
+  -||--o--------+--o-o-------
+        o       |
+  -----o-o-o----+--o-o-o-----
+                |   o
+  -----o-o-o-o--+--o-o-------
+        o       |   o
+  -----o--------+--o-o-o-----
+
+[*120b] And fyrste they take 6 whiche is in the lower lyne, and his
+space from 8 in the same roumes, in y^e second su{m}me, and yet there
+remayneth 2 counters in the lowest lyne. Then in the second lyne must 4
+be subtracte from 7, and so remayneth there 3. Then 8 in the thyrde lyne
+and his space, from 3 of the second summe can not be, therfore do they
+bate it from a hygher roume, that is, from 1000, and bycause that 1000
+is to moch by 200, therfore must I sette downe 200 in the thyrde lyne,
+after I haue taken vp 1000 from the fourth lyne: then is there yet 1000
+in the fourth lyne of the fyrst summe, whiche yf I withdrawe from the
+seconde summe, then doth all y^e figures stande in this order.
+
+  -----+------------
+       |
+  -||--+------------
+       |   o
+  -----+------------
+       |
+  -----+--o-o-o-----
+       |
+  -----+--o-o-------
+
+So that (as you se) it differeth not greatly whether you begynne
+subtractio{n} at the hygher lynes, or at [*121a] the lower. How be it,
+as some menne lyke the one waye beste, so some lyke the other: therfore
+you now knowyng bothe, may vse whiche you lyst.
+
+    [Headnote: Multiplication by Counters.]
+
+  [Sidenote: Multiplication.]
+
+But nowe touchynge Multiplicatio{n}: you shall set your no{m}bers in two
+roumes, as you dyd in those two other kyndes, but so that the multiplier
+be set in the fyrste roume. Then shall you begyn with the hyghest
+no{m}bers of y^e seconde roume, and multiply them fyrst after this sort.
+Take that ouermost lyne in your fyrst workynge, as yf it were the lowest
+lyne, setting on it some mouable marke, as you lyste, and loke how many
+counters be in hym, take them vp, and for them set downe the hole
+multyplyer, so many tymes as you toke vp counters, reckenyng, I saye
+that lyne for the vnites: {and} when you haue so done with the hygheest
+no{m}ber then come to the nexte lyne beneth, {and} do euen so with it,
+and so with y^e next, tyll you haue done all. And yf there be any nomber
+in a space, then for it [*121b] shall you take y^e multiplyer 5 tymes,
+and then must you recken that lyne for the vnites whiche is nexte beneth
+that space: or els after a shorter way, you shall take only halfe the
+multyplyer, but then shall you take the lyne nexte aboue that space, for
+the lyne of vnites: but in suche workynge, yf chau{n}ce your multyplyer
+be an odde nomber, so that you can not take the halfe of it iustly, then
+muste you take the greater halfe, and set downe that, as if that it were
+the iuste halfe, and farther you shall set one cou{n}ter in the space
+beneth that line, which you recken for the lyne of vnities, or els only
+remoue forward the same that is to be multyplyed.
+
+_S._ Yf you set forth an example hereto I thynke I shal perceaue you.
+
+_M._ Take this exa{m}ple: I wold multiply 1542 by 365, therfore I set
+y^e nombers thus.
+
+  ------------+--------------
+              |
+  -||---------+--o-----------
+              |   o
+  -----o-o-o--+--------------
+        o     |
+  -----o------+--o-o-o-o-----
+        o     |
+  ------------+--o-o---------
+
+[*122a] Then fyrste I begynne at the 1000 in y^e hyghest roume, as yf it
+were y^e fyrst place, & I take it vp, settynge downe for it so often
+(that is ones) the multyplyer, which is 365, thus, as you se here:
+
+  -----------+-----------+------------
+             |           |
+  -----------+-----------+--o-o-o-----
+             |           |   o
+  -----------+-----------+--o---------
+             |           |   o
+  -X---------+-----------+------------ [<-]
+             |   o       |
+  ----o-o-o--+-----------+------------
+       o     |           |
+  ----o------+--o-o-o-o--+------------
+       o     |           |
+  -----------+--o-o------+------------
+
+where for the one counter taken vp from the fourth lyne, I haue sette
+downe other 6, whiche make y^e su{m}me of the multyplyer, reckenynge
+that fourth lyne, as yf it were the fyrste: whiche thyng I haue marked
+by the hand set at the begynnyng of y^e same,
+
+_S._ I perceaue this well: for in dede, this summe that you haue set
+downe is 365000, for so moche doth amount [*122b] of 1000, multiplyed by
+365.
+
+_M._ Well the{n} to go forth, in the nexte space I fynde one counter
+which I remoue forward but take not vp, but do (as in such case I must)
+set downe the greater halfe of my multiplier (seyng it is an odde
+no{m}ber) which is 182, {and} here I do styll let that fourth place
+stand, as yf it were y^e fyrst:
+
+  ------------+-----------+--o-o-o--+--o---------
+              |           |   o     |   o
+  ------------+-----------+--o------+--o-o-o-----
+              |           |   o     |
+  -||---------+-----------+---------+--o-o------- [<-]
+              |           |         |   o
+  -----o-o-o--+-----------+---------+------------
+        o     |           |         |
+  -----o------+--o-o-o-o--+---------+------------
+        o     |           |         |
+  ------------+--o-o------+---------+------------
+
+as in this fourme you se, where I haue set this multiplycatio{n} with
+y^e other: but for the ease of your vndersta{n}dynge, I haue set a
+lytell lyne betwene them: now shulde they both in one su{m}me stand
+thus.
+
+  ------------+-----------+--o-o-o-o-o-----
+              |           |
+  ------------+-----------+--o-o-o-o-------
+              |           |   o
+  -||---------+-----------+--o-o----------- [<-]
+              |           |   o
+  -----o-o-o--+-----------+----------------
+        o     |           |
+  -----o------+--o-o-o-o--+----------------
+        o     |           |
+  ------------+--o-o------+----------------
+
+[*123a] Howe be it an other fourme to multyplye suche cou{n}ters i{n}
+space is this: Fyrst to remoue the fynger to the lyne nexte benethe y^e
+space, {and} then to take vp y^e cou{n}ter, {and} to set downe y^e
+multiplyer .v. tymes, as here you se.
+
+      ---------+---------+-o-o-o-+------+------+------+------+------+-
+               |         |  o    |      |      |      |      |      |
+      ---------+---------+-o-----+o-o-o-+o-o-o-+o-o-o-+o-o-o-+o-o-o-+-
+               |         |  o    | o    | o    | o    | o    | o    |
+      ---------+---------+-------+o-----+------+o-----+o-----+o-----+-
+               |         |       | o    | o    | o    | o    | o    |
+  [->]-X-o-o-o-+---------+-------+------+------+------+------+------+-
+          o    |         |       |      |      |      |      |      |
+      ---o-----+-o-o-o-o-+-------+------+------+------+------+------+-
+          o    |         |       |      |      |      |      |      |
+      ---------+-o-o-----+-------+------+------+------+------+------+-
+
+Which su{m}mes yf you do adde together into one su{m}me, you shal
+p{er}ceaue that it wyll be y^e same y^t appeareth of y^e other worki{n}g
+before, so that [*123b] bothe sortes are to one entent, but as the other
+is much shorter, so this is playner to reason, for suche as haue had
+small exercyse in this arte. Not withstandynge you maye adde them in
+your mynde before you sette them downe, as in this exa{m}ple, you myghte
+haue sayde 5 tymes 300 is 1500, {and} 5 tymes 60 is 300, also 5 tymes 5
+is 25, whiche all put together do make 1825, which you maye at one tyme
+set downe yf you lyste. But nowe to go forth, I must remoue the hand to
+the nexte counters, whiche are in the second lyne, and there must I take
+vp those 4 counters, settynge downe for them my multiplyer 4 tymes,
+whiche thynge other I maye do at 4 tymes seuerally, or elles I may
+gather that hole summe in my mynde fyrste, and then set it downe: as to
+saye 4 tymes 300 is 1200: 4 tymes 60 are 240: and 4 tymes 5 make 20: y^t
+is in all 1460, y^t shall I set downe also: as here you se.
+                                   o
+       -----------+-------+-----------+--------------
+                  |       |           |
+       -----------+-------+--o-o-o-o--+--o-----------
+                  |       |   o       |
+       -X---------+-------+--o-o------+--o-o-o-o-----
+                  |       |   o       |   o
+       ----o-o-o--+-------+-----------+--o-----------
+            o     |       |           |
+  [->] ----o------+-------+-----------+--------------
+            o     |       |           |
+       -----------+--o-o--+-----------+--------------
+
+[*124a] whiche yf I ioyne in one summe with the formar nombers, it wyll
+appeare thus.
+                           o
+      ---------+-------+----------
+               |       |   o
+      ---------+-------+--o-------
+               |       |
+      ---------+-------+--o-o-----
+               |       |
+      --o-o-o--+-------+-o--------
+         o     |       |
+ [->] --o------+-------+----------
+         o     |       |
+      ---------+--o-o--+----------
+
+Then to ende this multiplycation, I remoue the fynger to the lowest
+lyne, where are onely 2, them do I take vp, and in theyr stede do I set
+downe twyse 365, that is 730, for which I set [*124b] one in the space
+aboue the thyrd lyne for 500, and 2 more in the thyrd lyne with that one
+that is there all redye, and the reste in theyr order, {and} so haue I
+ended the hole summe thus.
+                     o
+  ---------+-----+------------
+           |     |   o
+  ---------+-----+--o---------
+           |     |
+  ---------+-----+--o-o-------
+           |     |   o
+  --o-o-o--+-----+--o-o-o-----
+     o     |     |
+  --o------+-----+--o-o-o-----
+     o     |     |
+  ---------+-----+------------
+
+Wherby you se, that 1542 (which is the nomber of yeares syth Ch[r]ystes
+incarnation) beyng multyplyed by 365 (which is the nomber of dayes in
+one yeare) dothe amounte vnto 562830, which declareth y^e no{m}ber of
+daies sith Chrystes incarnatio{n} vnto the ende of 1542[{1}] yeares.
+(besyde 385 dayes and 12 houres for lepe yeares).
+
+_S._ Now wyll I proue by an other exa{m}ple, as this: 40 labourers
+(after 6 d. y^e day for eche man) haue wrought 28 dayes, I wold [*125a]
+know what theyr wages doth amou{n}t vnto: In this case muste I worke
+doublely: fyrst I must multyplye the nomber of the labourers by y^e
+wages of a man for one day, so wyll y^e charge of one daye amount: then
+secondarely shall I multyply that charge of one daye, by the hole nomber
+of dayes, {and} so wyll the hole summe appeare: fyrst therefore I shall
+set the su{m}mes thus.
+
+  ------+--------------
+        |
+  ------+--------------
+        |
+  ------+--------------
+        |
+  ------+--o-o-o-o-----
+     o  |
+  --o---+--------------
+
+Where in the fyrste space is the multyplyer (y^t is one dayes wages for
+one man) {and} in the second space is set the nomber of the worke men to
+be multyplyed: the{n} saye I, 6 tymes 4 (reckenynge that second lyne as
+the lyne of vnites) maketh 24, for whiche summe I shulde set 2 counters
+in the thyrde lyne, and 4 in the seconde, therfore do I set 2 in the
+thyrde lyne, and let the 4 stand styll in the seconde lyne, thus.[*125b]
+
+  -----+--------------
+       |
+  -----+--------------
+       |
+  -----+--o-o---------
+       |
+  -----+--o-o-o-o-----
+       |
+  -----+--------------
+
+So apwereth the hole dayes wages to be 240d’. that is 20 s. Then do I
+multiply agayn the same summe by the no{m}ber of dayes and fyrste I
+sette the nombers, thus.
+
+  ---------+--------------
+           |
+  ---------+--------------
+           |
+  ---------+--o-o---------
+           |
+  --o-o----+--o-o-o-o-----
+     o     |
+  --o-o-o--+-------------
+
+The{n} bycause there are counters in dyuers lynes, I shall begynne with
+the hyghest, and take them vp, settynge for them the multyplyer so many
+tymes, as I toke vp counters, y^t is twyse, then wyll y^e su{m}me stande
+thus.
+
+  -----+--------------
+       |   o
+  -----+--------------
+       |   o
+  -----+--o-----------
+       |
+  -----+--o-o-o-o-----
+       |
+  -----+--------------
+
+Then come I to y^e seconde lyne, and take vp those 4 cou{n}ters,
+settynge for them the multiplyer foure tymes, so wyll the hole summe
+appeare thus.[*126a]
+
+  -----+----------
+       |   o
+  -----+--o-------
+       |   o
+  -----+--o-o-----
+       |
+  -----+--o-o-----
+       |
+  -----+----------
+
+So is the hole wages of 40 workeme{n}, for 28 dayes (after 6d’. eche
+daye for a man) 6720d’. that is 560 s. or 28 l’i.
+
+    [Headnote: Division on the Counting Board.]
+
+  [Sidenote: Diuision.]
+
+_M._ Now if you wold proue Multiplycatio{n}, the surest way is by
+Dyuision: therfore wyll I ouer passe it tyll I haue taught you y^e arte
+of Diuision, whiche you shall worke thus. Fyrste sette downe the Diuisor
+for feare of forgettynge, and then set the nomber that shalbe deuided,
+at y^e ryghte syde, so farre from the diuisor, that the quotient may be
+set betwene them: as for exa{m}ple: Yf 225 shepe cost 45 l’i. what dyd
+euery shepe cost? To knowe this, I shulde diuide the hole summe, that is
+45 l’i. by 225, but that can not be, therfore must I fyrste reduce that
+45 l’i. into a lesser denomination, as into shyllynges: then I multiply
+45 by 20, and it is 900, that summe shall I diuide by the no{m}ber of
+[*126b] shepe, whiche is 225, these two nombers therfore I sette thus.
+
+  -------+-----+--------------
+         |     |
+  -------+-----+--------------
+         |     |   o
+  --o-o--+-----+--o-o-o-o-----
+         |     |
+  --o-o--+-----+--------------
+     o   |     |
+  -------+-----+--------------
+
+Then begynne I at the hyghest lyne of the diuident, and seke how often I
+may haue the diuisor therin, and that maye I do 4 tymes, then say I,
+4 tymes 2 are 8, whyche yf I take from 9, there resteth but 1, thus
+
+  -------+-----------+--------
+         |           |
+  -------+-----------+--------
+         |           |
+  --o-o--+-----------+--o-----
+         |           |
+  --o-o--+-----------+--------
+     o   |           |
+  -------+--o-o-o-o--+--------
+
+And bycause I founde the diuisor 4 tymes in the diuidente, I haue set
+(as you se) 4 in the myddle roume, which [*127a] is the place of the
+quotient: but now must I take the reste of the diuisor as often out of
+the remayner: therfore come I to the seconde lyne of the diuisor, sayeng
+2 foure tymes make 8, take 8 from 10, {and} there resteth 2, thus.
+
+  ----------+-----------+----------
+            |           |
+  -||-------+-----------+----------
+            |           |
+  -----o-o--+-----------+----------
+            |           |
+  -----o-o--+-----------+--o-o-----
+        o   |           |
+  ----------+--o-o-o-o--+----------
+
+Then come I to the lowest nomber, which is 5, and multyply it 4 tymes,
+so is it 20, that take I from 20, and there remayneth nothynge, so that
+I se my quotient to be 4, whiche are in valewe shyllynges, for so was
+the diuident: and therby I knowe, that yf 225 shepe dyd coste 45 l’i.
+euery shepe coste 4 s.
+
+_S._ This can I do, as you shall perceaue by this exa{m}ple: Yf 160
+sowldyars do spende euery moneth 68 l’i. what spendeth eche man? Fyrst
+[*127b] bycause I can not diuide the 68 by 160, therfore I wyll turne
+the pou{n}des into pennes by multiplicacio{n}, so shall there be
+16320 d’. Nowe muste I diuide this su{m}me by the nomber of sowldyars,
+therfore I set the{m} i{n} order, thus.
+
+  ---------+-----+--o---------
+           |     |   o
+  -||------+-----+--o---------
+           |     |
+  -----o---+-----+--o-o-o-----
+        o  |     |
+  -----o---+-----+--o-o-------
+           |     |
+  ---------+-----+------------
+
+Then begyn I at the hyghest place of the diuidente, sekynge my diuisor
+there, whiche I fynde ones, Therfore set I 1 in the nether lyne.
+
+_M._ Not in the nether line of the hole summe, but in the nether lyne of
+that worke, whiche is the thyrde lyne.
+
+_S._ So standeth it with reason.
+
+_M._ Then thus do they stande.[*128a]
+
+  ---------+-----+------------
+           |     |
+  -||------+-----+------------
+           |     |
+  -----o---+--o--+--o-o-o-----
+        o  |     |
+  -----o---+-----+--o-o-------
+           |     |
+  ---------+-----+------------
+
+Then seke I agayne in the reste, how often I may fynde my diuisor, and I
+se that in the 300 I myghte fynde 100 thre tymes, but then the 60 wyll
+not be so often founde in 20, therfore I take 2 for my quotient: then
+take I 100 twyse from 300, and there resteth 100, out of whiche with the
+20 (that maketh 120) I may take 60 also twyse, and then standeth the
+nombers thus,
+
+  ---------+-------+-----
+           |       |
+  -||------+-------+-----
+           |       |
+  -----o---+--o----+-----
+        o  |       |
+  -----o---+-------+-----
+           |       |
+  ---------+--o-o--+-----
+
+[*128b] where I haue sette the quotient 2 in the lowest lyne: So is
+euery sowldyars portion 102 d’. that is 8 s. 6 d’.
+
+_M._ But yet bycause you shall perceaue iustly the reason of Diuision,
+it shall be good that you do set your diuisor styll agaynst those
+nombres fro{m} whiche you do take it: as by this example I wyll declare.
+Yf y^e purchace of 200 acres of ground dyd coste 290 l’i. what dyd one
+acre coste? Fyrst wyl I turne the poundes into pennes, so wyll there be
+69600 d’· Then in settynge downe these nombers I shall do thus.
+
+  ---------+-----+--------------
+           |     |   o
+  ----o-o--+-----+--o-----------
+           |     |   o
+  -X-------+-----+--o-o-o-o-----
+           |     |     o
+  ---------+-----+--o-----------
+           |     |
+  ---------+-----+--------------
+           |     |
+  ---------+-----+--------------
+
+Fyrst set the diuident on the ryghte hande as it oughte, and then
+[*129a] the diuisor on the lefte hande agaynst those nombers, fro{m}
+which I entende to take hym fyrst as here you se, wher I haue set the
+diuisor two lynes hygher the{n} is theyr owne place.
+
+_S._ This is lyke the order of diuision by the penne.
+
+_M._ Truth you say, and nowe must I set y^e quotient of this worke in
+the thyrde lyne, for that is the lyne of vnities in respecte to the
+diuisor in this worke. Then I seke howe often the diuisor maye be founde
+in the diuident, {and} that I fynde 3 tymes, then set I 3 in the thyrde
+lyne for the quotient, and take awaye that 60000 fro{m} the diuident,
+and farther I do set the diuisor one line lower, as yow se here.
+
+  ----------+---------+--------------
+            |         |   o
+  -||--o-o--+---------+--o-o-o-o-----
+            |         |   o
+  ----------+--o-o-o--+----o---------
+            |         |
+  ----------+---------+--------------
+            |         |
+  ----------+---------+--------------
+
+[*129b] And then seke I how often the diuisor wyll be taken from the
+nomber agaynste it, whiche wyll be 4 tymes and 1 remaynynge.
+
+_S._ But what yf it chaunce that when the diuisor is so remoued, it can
+not be ones taken out of the diuident agaynste it?
+
+_M._ Then must the diuisor be set in an other line lower.
+
+_S._ So was it in diuision by the penne, and therfore was there a cypher
+set in the quotient: but howe shall that be noted here?
+
+_M._ Here nedeth no token, for the lynes do represente the places: onely
+loke that you set your quotient in that place which standeth for vnities
+in respecte of the diuisor: but now to returne to the example, I fynde
+the diuisor 4 tymes in the diuidente, and 1 remaynynge, for 4 tymes 2
+make 8, which I take from 9, and there resteth 1, as this figure
+sheweth:
+
+  ----------+-----------+---------
+            |           |
+  -||--o-o--+-----------+--o------
+            |           |   o
+  ----------+--o-o-o----+--o------
+            |           |
+  ----------+--o-o-o-o--+---------
+            |           |
+  ----------+-----------+---------
+
+and in the myddle space for the quotient I set 4 in the seconde lyne,
+whiche is in this worke the place of vnities.[*130a] Then remoue I y^e
+diuisor to the next lower line, and seke how often I may haue it in the
+dyuident, which I may do here 8 tymes iust, and nothynge remayne, as in
+this fourme,
+
+  ----------+-----------+-----
+            |           |
+  -||--o-o--+-----------+-----
+            |           |
+  ----------+--o-o-o----+-----
+            |           |
+  ----------+--o-o-o-o--+-----
+            |   o       |
+  ----------+--o-o-o----+-----
+
+where you may se that the hole quotient is 348 d’, that is 29 s. wherby
+I knowe that so moche coste the purchace of one aker.
+
+_S._ Now resteth the profes of Multiplycatio{n}, and also of Diuisio{n}.
+
+_M._ Ther best profes are eche [*130b] one by the other, for
+Multyplication is proued by Diuision, and Diuision by Multiplycation,
+as in the worke by the penne you learned.
+
+_S._ Yf that be all, you shall not nede to repete agayne that, y^t was
+sufficye{n}tly taughte all redye: and excepte you wyll teache me any
+other feate, here maye you make an ende of this arte I suppose.
+
+_M._ So wyll I do as touchynge hole nomber, and as for broken nomber,
+I wyll not trouble your wytte with it, tyll you haue practised this so
+well, y^t you be full perfecte, so that you nede not to doubte in any
+poynte that I haue taught you, and thenne maye I boldly enstructe you in
+y^e arte of fractions or broken no{m}ber, wherin I wyll also showe you
+the reasons of all that you haue nowe learned. But yet before I make an
+ende, I wyll showe you the order of co{m}men castyng, wher in are bothe
+pennes, shyllynges, and poundes, procedynge by no grounded reason, but
+onely by a receaued [*131a] fourme, and that dyuersly of dyuers men: for
+marchau{n}tes vse one fourme, and auditors an other:
+
+    [Headnote: Merchants’ Casting Counters.]
+
+  [Sidenote: Merchants’ casting.]
+
+But fyrste for marchauntes fourme marke this example here,
+
+  o   o o o o
+       o
+  o   o o o
+       o
+  o   o o o o
+       o
+      o o o o o
+
+in which I haue expressed this summe 198 l’i.[{2}] 19 s. 11 d’. So that
+you maye se that the lowest lyne serueth for pe{n}nes, the next aboue
+for shyllynges, the thyrde for poundes, and the fourth for scores of
+pou{n}des. And farther you maye se, that the space betwene pennes and
+shyllynges may receaue but one counter (as all other spaces lyke wayes
+do) and that one standeth in that place for 6 d’. Lyke wayes betwene the
+shyllynges {and} the pou{n}des, one cou{n}ter standeth for 10 s. And
+betwene the poundes and 20 l’i. one counter standeth for 10 pou{n}des.
+But besyde those you maye see at the left syde of shyllynges, that one
+counter standeth alone, {and} betokeneth 5 s. [*131b] So agaynste the
+poundes, that one cou{n}ter standeth for 5 l’i. And agaynst the 20
+poundes, the one counter standeth for 5 score pou{n}des, that is
+100 l’i. so that euery syde counter is 5 tymes so moch as one of them
+agaynst whiche he standeth.
+
+  [Sidenote: Auditors’ casting.]
+
+Now for the accompt of auditors take this example.
+
+      o   o   o   o   o     o
+  o o o   o o o   o o o   o o o
+  o               o       o o
+
+where I haue expressed y^e same su{m}me 198 l’i. 19 s. 11 d’. But here
+you se the pe{n}nes stande toward y^e ryght hande, and the other
+encreasynge orderly towarde the lefte hande. Agayne you maye se, that
+auditours wyll make 2 lynes (yea and more) for pennes, shyllynges, {and}
+all other valewes, yf theyr summes extende therto. Also you se, that
+they set one counter at the ryght ende of eche rowe, whiche so set there
+standeth for 5 of that roume: and on [*132a] the lefte corner of the
+rowe it sta{n}deth for 10, of y^e same row. But now yf you wold adde
+other subtracte after any of both those sortes, yf you marke y^e order
+of y^t other feate which I taught you, you may easely do the same here
+without moch teachynge: for in Additio{n} you must fyrst set downe one
+su{m}me and to the same set the other orderly, and lyke maner yf you
+haue many: but in Subtraction you must sette downe fyrst the greatest
+summe, and from it must you abate that other euery denominatio{n} from
+his dewe place.
+
+_S._ I do not doubte but with a lytell practise I shall attayne these
+bothe: but how shall I multiply and diuide after these fourmes?
+
+_M._ You can not duely do none of both by these sortes, therfore in
+suche case, you must resort to your other artes.
+
+_S._ Syr, yet I se not by these sortes how to expresse hu{n}dreddes,
+yf they excede one hundred, nother yet thousandes.
+
+_M._ They that vse such accomptes that it excede 200 [*132b] in one
+summe, they sette no 5 at the lefte hande of the scores of poundes, but
+they set all the hundredes in an other farther rowe {and} 500 at the
+lefte hand therof, and the thousandes they set in a farther rowe yet,
+{and} at the lefte syde therof they sette the 5000, and in the space
+ouer they sette the 10000, and in a hygher rowe 20000, whiche all I haue
+expressed in this exa{m}ple,
+
+      o o o o
+       o
+  o   o o
+  o   o o o
+      o o o
+  o   o o o o
+        o
+      o o
+        o
+      o o o
+               o o
+               o
+
+which is 97869 l’i. 12 s. 9 d’ ob. q. for I had not told you before
+where, nother how you shuld set downe farthynges, which (as you se here)
+must be set in a voyde space sydelynge beneth the pennes: for q one
+counter: for ob. 2 counters: for ob. q. 3 counters: {and} more there can
+not be, for 4 farthynges [*133a] do make 1 d’. which must be set in his
+dewe place.
+
+    [Headnote: Auditors’ Casting Counters.]
+
+And yf you desyre y^e same summe after audytors maner, lo here it is.
+
+           o o        o               o   o       o
+  o o o   o o     o o o   o o o   o o o   o o   o o o
+  o                               o                      o o
+                                                         o
+
+But in this thyng, you shall take this for suffycyent, and the reste you
+shall obserue as you maye se by the working of eche sorte: for the
+dyuers wittes of men haue inuented dyuers and sundry wayes almost
+vnnumerable. But one feate I shall teache you, whiche not only for the
+straungenes and secretnes is moche pleasaunt, but also for the good
+co{m}moditie of it ryghte worthy to be well marked. This feate hath ben
+vsed aboue 2000 yeares at the leaste, and yet was it neuer come{n}ly
+knowen, especyally in Englysshe it was neuer taughte yet. This is the
+arte of nombrynge on the hand, with diuers gestures of the fyngers,
+expressynge any summe conceaued in the [*133b] mynde. And fyrst to
+begynne, yf you wyll expresse any summe vnder 100, you shall expresse it
+with your lefte hande: and from 100 vnto 10000, you shall expresse it
+with your ryght hande, as here orderly by this table folowynge you may
+perceaue.
+
+  +¶ Here foloweth the table
+    of the arte of the
+      hande+
+
+
+
+
+The arte of nombrynge by the hande.
+
+  [Transcriber’s Note:
+
+  Footnote 3 reads:
+    “Bracket ([) denotes new paragraph in original.”
+  For this e-text, the brackets have been omitted in favor of restoring
+  the paragraph breaks. Changes of speaker (M, S) are also marked by
+  paragraphs, as in the previous selection.
+
+  The illustration includes the printed page number 134; there is
+  therefore no sidenote *134a. The sidenote for “4” is missing.]
+
+
+[Illustration: (Numbers as described in text)]
+
+  [Sidenote: 1]
+
+[*134b] In which as you may se 1 is expressed by y^e lyttle fynger of
+y^e lefte hande closely and harde croked.
+
+  [Sidenote: 2]
+
+[{3}]2 is declared by lyke bowynge of the weddynge fynger (whiche is the
+nexte to the lyttell fynger) together with the lytell fynger.
+
+  [Sidenote: 3]
+
+3 is signified by the myddle fynger bowed in lyke maner, with those
+other two.
+
+4 is declared by the bowyng of the myddle fynger and the rynge fynger,
+or weddynge fynger, with the other all stretched forth.
+
+  [Sidenote: 5, 6]
+
+5 is represented by the myddle fynger onely bowed.
+
+And 6 by the weddynge fynger only crooked: and this you may marke in
+these a certayne order. But now 7, 8, and 9, are expressed w{i}t{h} the
+bowynge of the same fyngers as are 1, 2, and 3, but after an other
+fourme.
+
+  [Sidenote: 7]
+
+For 7 is declared by the bowynge of the lytell fynger, as is 1, saue
+that for 1 the fynger is clasped in, harde {and} [*135a] rounde, but for
+to expresse 7, you shall bowe the myddle ioynte of the lytell fynger
+only, and holde the other ioyntes streyght.
+
+_S._ Yf you wyll geue me leue to expresse it after my rude maner, thus I
+vnderstand your meanyng: that 1 is expressed by crookynge in the lyttell
+fynger lyke the head of a bysshoppes bagle: and 7 is declared by the
+same fynger bowed lyke a gybbet.
+
+_M._ So I perceaue, you vnderstande it.
+
+  [Sidenote: 8]
+
+Then to expresse 8, you shall bowe after the same maner both the lyttell
+fynger and the rynge fynger.
+
+  [Sidenote: 9, 10]
+
+And yf you bowe lyke wayes with them the myddle fynger, then doth it
+betoken 9.
+
+Now to expresse 10, you shall bowe your fore fynger rounde, and set the
+ende of it on the hyghest ioynte of the thombe.
+
+  [Sidenote: 20]
+
+And for to expresse 20, you must set your fyngers streyght, and the ende
+of your thombe to the partitio{n} of the [*135b] fore moste and myddle
+fynger.
+
+  [Sidenote: 30]
+
+30 is represented by the ioynynge together of y^e headdes of the
+foremost fynger and the thombe.
+
+  [Sidenote: 40]
+
+40 is declared by settynge of the thombe crossewayes on the foremost
+fynger.
+
+  [Sidenote: 50]
+
+50 is signified by ryght stretchyng forth of the fyngers ioyntly, and
+applyenge of the thombes ende to the partition of the myddle fynger
+{and} the rynge fynger, or weddynge fynger.
+
+  [Sidenote: 60]
+
+60 is formed by bendynge of the thombe croked and crossynge it with the
+fore fynger.
+
+  [Sidenote: 70]
+
+70 is expressed by the bowynge of the foremost fynger, and settynge the
+ende of the thombe between the 2 foremost or hyghest ioyntes of it.
+
+  [Sidenote: 80]
+
+80 is expressed by settynge of the foremost fynger crossewayes on the
+thombe, so that 80 dyffereth thus fro{m} 40, that for 80 the forefynger
+is set crosse on the thombe, and for 40 the thombe is set crosse ouer
+y^e forefinger.
+
+  [Sidenote: 90]
+
+[*136a] 90 is signified, by bendynge the fore fynger, and settyng the
+ende of it in the innermost ioynte of y^e thombe, that is euen at the
+foote of it. And thus are all the no{m}bers ended vnder 100.
+
+[Sidenote: 11, 12, 13, 21, 22, 23]
+
+_S._ In dede these be all the nombers fro{m} 1 to 10, {and} then all the
+tenthes within 100, but this teacyed me not how to expresse 11, 12, 13,
+{et}c. 21, 22, 23, {et}c. and such lyke.
+
+_M._ You can lytell vnderstande, yf you can not do that without
+teachynge: what is 11? is it not 10 and 1? then expresse 10 as you were
+taught, and 1 also, and that is 11: and for 12 expresse 10 and 2: for 23
+set 20 and 3: and so for 68 you muste make 60 and there to 8: and so of
+all other sortes.
+
+  [Sidenote: 100]
+
+But now yf you wolde represente 100 other any nomber aboue it, you muste
+do that with the ryghte hande, after this maner. [You must expresse 100
+in the ryght hand, with the lytell fynger so bowed as you dyd expresse 1
+in the left hand.
+
+  [Sidenote: 200]
+
+[*136b] And as you expressed 2 in the lefte hande, the same fasshyon in
+the ryght hande doth declare 200.
+
+  [Sidenote: 300]
+
+The fourme of 3 in the ryght hand standeth for 300.
+
+  [Sidenote: 400]
+
+The fourme of 4, for 400.
+
+  [Sidenote: 500]
+
+Lykewayes the fourme of 5, for 500.
+
+  [Sidenote: 600]
+
+The fourme of 6, for 600. And to be shorte: loke how you did expresse
+single vnities and tenthes in the lefte hande, so must you expresse
+vnities {and} tenthes of hundredes, in the ryghte hande.
+
+  [Sidenote: 900]
+
+_S._ I vnderstande you thus: that yf I wold represent 900, I must so
+fourme the fyngers of my ryghte hande, as I shuld do in my left hand to
+expresse 9,
+
+  [Sidenote: 1000]
+
+And as in my lefte hand I expressed 10, so in my ryght hande must I
+expresse 1000.
+
+And so the fourme of euery tenthe in the lefte hande serueth to expresse
+lyke no{m}ber of thousa{n}des,
+
+  [Sidenote: 4000]
+
+so y^e fourme of 40 standeth for 4000.
+
+  [Sidenote: 8000]
+
+The fourme of 80 for 8000.
+
+  [Sidenote: 9000]
+
+  [*137a]
+
+  And the fourme of 90 (whiche is
+  the greatest) for 9000, and aboue that
+  I can not expresse any nomber. _M._
+  No not with one fynger: how be it,
+  w{i}t{h} dyuers fyngers you maye expresse
+  9999, and all at one tyme, and that lac
+  keth but 1 of 10000. So that vnder
+  10000 you may by your fyngers ex-
+  presse any summe. And this shal suf-
+  fyce for Numeration on the fyngers.
+  And as for Addition, Subtraction,
+  Multiplicatio{n}, and Diuision (which
+  yet were neuer taught by any man as
+  farre as I do knowe) I wyll enstruct
+  you after the treatyse of fractions.
+  And now for this tyme fare well,
+  and loke that you cease not to
+  practyse that you haue lear
+  ned. _S._ Syr, with moste
+  harty mynde I thanke
+  you, bothe for your
+  good learnyng, {and}
+  also your good
+  cou{ns}el, which
+  (god wyllyng) I truste to folow.
+
+
+  Finis.
+
+
+  FOOTNOTES (Accomptynge by counters
+  _and_ The arte of nombrynge by the hande):
+
+  [1: 1342 in original.]
+  [2: 168 in original.]
+  [3: Bracket ([) denotes new paragraph in original.]
+
+
+
+
+APPENDIX I.
+
++A Treatise on the Numeration of Algorism.+
+
+
+[_From a MS. of the 14th Century._]
+
+To alle suche even nombrys the most have cifrys as to ten. twenty.
+thirtty. an hundred. an thousand and suche other. but ye schal
+vnderstonde that a cifre tokeneth nothinge but he maketh other the more
+significatyf that comith after hym. Also ye schal vnderstonde that in
+nombrys composyt and in alle other nombrys that ben of diverse figurys
+ye schal begynne in the ritht syde and to rekene backwarde and so he
+schal be wryte as thus--1000. the sifre in the ritht side was first
+wryte and yit he tokeneth nothinge to the secunde no the thridde but
+thei maken that figure of 1 the more signyficatyf that comith after hem
+by as moche as he born oute of his first place where he schuld yf he
+stode ther tokene but one. And there he stondith nowe in the ferye place
+he tokeneth a thousand as by this rewle. In the first place he tokeneth
+but hymself. In the secunde place he tokeneth ten times hymself. In the
+thridde place he tokeneth an hundred tymes himself. In the ferye he
+tokeneth a thousand tymes himself. In the fyftye place he tokeneth ten
+thousand tymes himself. In the sexte place he tokeneth an hundred
+thousand tymes hymself. In the seveth place he tokeneth ten hundred
+thousand tymes hymself, &c. And ye schal vnderstond that this worde
+nombre is partyd into thre partyes. Somme is callyd nombre of digitys
+for alle ben digitys that ben withine ten as ix, viii, vii, vi, v, iv,
+iii, ii, i. Articules ben alle thei that mow be devyded into nombrys of
+ten as xx, xxx, xl, and suche other. Composittys be alle nombrys that
+ben componyd of a digyt and of an articule as fourtene fyftene thrittene
+and suche other. Fourtene is componyd of four that is a digyt and of ten
+that is an articule. Fyftene is componyd of fyve that is a digyt and of
+ten that is an articule and so of others . . . . . . But as to this
+rewle. In the firste place he tokeneth but himself that is to say he
+tokeneth but that and no more. If that he stonde in the secunde place he
+tokeneth ten tymes himself as this figure 2 here 21. this is oon and
+twenty. This figure 2 stondith in the secunde place and therfor he
+tokeneth ten tymes himself and ten tymes 2 is twenty and so forye of
+every figure and he stonde after another toward the lest syde he schal
+tokene ten tymes as moche more as he schuld token and he stode in that
+place ther that the figure afore him stondeth: lo an example as thus
+9634. This figure of foure that hath this schape 4 tokeneth but himself
+for he stondeth in the first place. The figure of thre that hath this
+schape 3 tokeneth ten tyme himself for he stondeth in the secunde place
+and that is thritti. The figure of sexe that hath this schape 6 tokeneth
+ten tyme more than he schuld and he stode in the place yer the figure of
+thre stondeth for ther he schuld tokene but sexty. And now he tokeneth
+ten tymes that is sexe hundrid. The figure of nyne that hath this schape
+9 tokeneth ten tymes more than he schulde and he stode in the place ther
+the figure of 6 stondeth inne for thanne he schuld tokene but nyne
+hundryd. And in the place that he stondeth inne nowe he tokeneth nine
+thousand. Alle the hole nombre of these foure figurys. Nine thousand
+sexe hundrid and foure and thritti.
+
+
+
+
+APPENDIX II.
+
+Carmen de Algorismo.
+
+
+[_From a B.M. MS., 8 C. iv., with additions from 12 E. 1 & Eg. 2622._]
+
+  Hec algorismus ars presens dicitur[{1}]; in qua
+  Talibus Indorum[{2}] fruimur his quinque figuris.
+    0. 9. 8. 7. 6. 5. 4. 3. 2. 1.
+  Prima significat unum: duo vero secunda:
+  Tercia significat tria: sic procede sinistre                       4
+  Donec ad extremam venies, qua cifra vocatur;
+  [{3}][Que nil significat; dat significare sequenti.]
+  Quelibet illarum si primo limite ponas,
+  Simpliciter se significat: si vero secundo,                        8
+  Se decies: sursum procedas multiplicando.[{4}]
+  [Namque figura sequens quevis signat decies plus,
+  Ipsa locata loco quam significet pereunte:                        12
+  Nam precedentes plus ultima significabit.]
+  [{5}]Post predicta scias quod tres breuiter numerorum
+  Distincte species sunt; nam quidam digiti sunt;
+  Articuli quidam; quidam quoque compositi sunt.                    16
+  [Sunt digiti numeri qui citra denarium sunt;
+  Articuli decupli degitorum; compositi sunt
+  Illi qui constant ex articulis digitisque.]
+  Ergo, proposito numero tibi scribere, primo                       20
+  Respicias quis sit numerus; quia si digitus sit,
+  [{5}][Una figura satis sibi; sed si compositus sit,]
+  Primo scribe loco digitum post articulum fac
+  Articulus si sit, cifram post articulum sit,                      24
+  [Articulum vero reliquenti in scribe figure.]
+  Quolibet in numero, si par sit prima figura,
+  Par erit et totum, quicquid sibi continetur;
+  Impar si fuerit, totum sibi fiet et impar.                        28
+  Septem[{6}] sunt partes, non plures, istius artis;
+  Addere, subtrahere, duplare, dimidiare;
+  Sexta est diuidere, set quinta est multiplicare;
+  Radicem extrahere pars septima dicitur esse.                      32
+  Subtrahis aut addis a dextris vel mediabis;
+  A leua dupla, diuide, multiplicaque;
+  Extrahe radicem semper sub parte sinistra.
+
+  [Sidenote: Addition.]
+
+  Addere si numero numerum vis, ordine tali                         36
+  Incipe; scribe duas primo series numerorum
+  Prima sub prima recte ponendo figuram,
+  Et sic de reliquis facias, si sint tibi plures.
+  Inde duas adde primas hac condicione;                             40
+  Si digitus crescat ex addicione priorum,
+  Primo scribe loco digitum, quicunque sit ille;
+  Si sit compositus, in limite scribe sequenti
+  Articulum, primo digitum; quia sic iubet ordo.                    44
+  Articulus si sit, in primo limite cifram,
+  Articulum vero reliquis inscribe figuris;
+  Vel per se scribas si nulla figura sequatur.
+  Si tibi cifra superueniens occurrerit, illam                      48
+  Deme suppositam; post illic scribe figuram:
+  Postea procedas reliquas addendo figuras.
+
+  [Sidenote: Subtraction.]
+
+  A numero numerum si sit tibi demere cura,
+  Scribe figurarum series, vt in addicione;                         52
+  Maiori numero numerum suppone minorem,
+  Siue pari numero supponatur numerus par.
+  Postea si possis a prima subtrahe primam,
+  Scribens quod remanet, cifram si nil remanebit.                   56
+  Set si non possis a prima demere primam;
+  Procedens, vnum de limite deme sequenti;
+  Et demptum pro denario reputabis ab illo,
+  Subtrahe totaliter numerum quem proposuisti.                      60
+  Quo facto, scribe supra quicquit remanebit,
+  Facque novenarios de cifris, cum remanebis,
+  Occurrant si forte cifre, dum demseris vnum;
+  Postea procedas reliquas demendo figuras.                         64
+
+  [Sidenote: Proof.]
+
+  [{7}][Si subtracio sit bene facta probare valebis,
+  Quas subtraxisti primas addendo figuras.
+  Nam, subtractio si bene sit, primas retinebis,
+  Et subtractio facta tibi probat additionem.]                      68
+
+  [Sidenote: Duplation.]
+
+  Si vis duplare numerum, sic incipe; solam
+  Scribe figurarum seriem, quamcumque voles que
+  Postea procedas primam duplando figuram;
+  Inde quod excrescet, scribens, vbi iusserit ordo,                 72
+  Juxta precepta que dantur in addicione.
+  Nam si sit digitus, in primo limite scribe;
+  Articulus si sit, in primo limite cifram,
+  Articulum vero reliquis inscribe figuris;                         76
+  Vel per se scribas, si nulla figura sequatur:
+  Compositus si sit, in limite scribe sequenti
+  Articulum primo, digitum; quia sic jubet ordo:
+  Et sic de reliquis facias, si sint tibi plures.                   80
+  [{8}][Si super extremam nota sit, monadem dat eidem,
+  Quod tibi contingit, si primo dimidiabis.]
+
+  [Sidenote: Mediation.]
+
+  Incipe sic, si vis aliquem numerum mediare:
+  Scribe figurarum seriem solam, velud ante;                        84
+  Postea procedens medias, et prima figura
+  Si par aut impar videas; quia si fuerit par,
+  Dimidiabis eam, scribens quicquit remanebit;
+  Impar si fuerit, vnum demas, mediare,                             88
+  Nonne presumas, sed quod superest mediabis;
+  Inde super tractum, fac demptum quod notat unum;
+  Si monos, dele; sit ibi cifra post nota supra.
+  Postea procedas hac condicione secunda:[{9}]                      92
+  Impar[{10}] si fuerit hic vnum deme priori,
+  Inscribens quinque, nam denos significabit
+  Monos prædictam: si vero secunda dat vnam,
+  Illa deleta, scribatur cifra; priori                              96
+  Tradendo quinque pro denario mediato;
+  Nec cifra scribatur, nisi inde figura sequatur:
+  Postea procedas reliquas mediando figuras,
+  Quin supra docui, si sint tibi mille figure.                     100
+  [{11}][Si mediatio sit bene facta probare valebis,
+  Duplando numerum quem primo dimidiasti.]
+  Si super extremam nota sit monades dat eidem
+  Quod contingat cum primo dimiabis
+  Atque figura prior nuper fuerit mediando.]
+
+  [Sidenote: Multiplication.]
+
+  Si tu per numerum numerum vis multiplicare,
+  Scribe duas, quascunque volis, series numerorum;                 104
+  Ordo tamen seruetur vt vltima multiplicandi
+  Ponatur super anteriorem multiplicantis;
+  [{12}][A leua relique sint scripte multiplicantes.]
+  In digitum cures digitum si ducere, major                        108
+  Per quantes distat a denis respice, debes
+  Namque suo decuplo tociens delere minorem;
+  Sicque tibi numerus veniens exinde patebit.
+  Postea procedas postremam multiplicando,                         112
+  Juste multiplicans per cunctas inferiores,
+  Condicione tamen tali; quod multiplicantis
+  Scribas in capite, quicquid processerit inde;
+  Set postquam fuerit hec multiplicata, figure                     116
+  Anteriorentur seriei multiplicantis;
+  Et sic multiplica, velut istam multiplicasti,
+  Qui sequitur numerum scriptum quicunque figuris.
+  Set cum multiplicas, primo sic est operandum,                    120
+  Si dabit articulum tibi multiplicacio solum;
+  Proposita cifra, summam transferre memento.
+  Sin autem digitus excrescerit articulusque,
+  Articulus supraposito digito salit ultra;                        124
+  Si digitus tamen, ponas illum super ipsam,
+  Subdita multiplicans hanc que super incidit illi
+  Delet eam penitus, scribens quod provenit inde;
+  Sed si multiplices illam posite super ipsam,                     128
+  Adiungens numerum quem prebet ductus earum;
+  Si supraimpositam cifra debet multiplicare,
+  Prorsus eam delet, scribi que loco cifra debet,
+  [{12}][Si cifra multiplicat aliam positam super ipsam,           132
+  Sitque locus supra vacuus super hanc cifra fiet;]
+  Si supra fuerit cifra semper pretereunda est;
+  Si dubites, an sit bene multiplicando secunda,
+  Diuide totalem numerum per multiplicantem,                       136
+  Et reddet numerus emergens inde priorem.
+
+  [Sidenote: Mental Multiplication.]
+
+  [{13}][Per numerum si vis numerum quoque multiplicare
+  Tantum per normas subtiles absque figuris
+  Has normas poteris per versus scire sequentes.                   140
+  Si tu per digitum digitum quilibet multiplicabis
+  Regula precedens dat qualiter est operandum
+  Articulum si per reliquum vis multiplicare
+  In proprium digitum debebit uterque resolvi                      144
+  Articulus digitos post per se multiplicantes
+  Ex digitis quociens teneret multiplicatum
+  Articuli faciunt tot centum multiplicati.
+  Articulum digito si multiplicamus oportet                        148
+  Articulum digitum sumi quo multiplicare
+  Debemus reliquum quod multiplicaris ab illis
+  Per reliquo decuplum sic omne latere nequibit
+  In numerum mixtum digitum si ducere cures                        152
+  Articulus mixti sumatur deinde resolvas
+  In digitum post hec fac ita de digitis nec
+  Articulusque docet excrescens in detinendo
+  In digitum mixti post ducas multiplicantem                       156
+  De digitis ut norma docet sit juncta secundo
+  Multiplica summam et postea summa patebit
+  Junctus in articulum purum articulumque
+  [{14}][Articulum purum comittes articulum que]                   160
+  Mixti pro digitis post fiat et articulus vt
+  Norma jubet retinendo quod egreditur ab illis
+  Articuli digitum post in digitum mixti duc
+  Regula de digitis ut percipit articulusque                       164
+  Ex quibus excrescens summe tu junge priori
+  Sic manifesta cito fiet tibi summa petita.
+  Compositum numerum mixto sic multiplicabis
+  Vndecies tredecem sic est ex hiis operandum                      168
+  In reliquum primum demum duc post in eundem
+  Unum post deinde duc in tercia deinde per unum
+  Multiplices tercia demum tunc omnia multiplicata
+  In summa duces quam que fuerit te dices                          172
+  Hic ut hic mixtus intentus est operandum
+  Multiplicandorum de normis sufficiunt hec.]
+
+  [Sidenote: Division.]
+
+  Si vis dividere numerum, sic incipe primo;
+  Scribe duas, quascunque voles, series numerorum;                 176
+  Majori numero numerum suppone minorem,
+  [{15}][Nam docet ut major teneat bis terve minorem;]
+  Et sub supprima supprimam pone figuram,
+  Sic reliquis reliquas a dextra parte locabis;                    180
+  Postea de prima primam sub parte sinistra
+  Subtrahe, si possis, quociens potes adminus istud,
+  Scribens quod remanet sub tali conditione;
+  Ut totiens demas demendas a remanente,                           184
+  Que serie recte ponentur in anteriori,
+  Unica si, tantum sit ibi decet operari;
+  Set si non possis a prima demere primam,
+  Procedas, et eam numero suppone sequenti;                        188
+  Hanc uno retrahendo gradu quo comites retrahantur,
+  Et, quotiens poteris, ab eadem deme priorem,
+  Ut totiens demas demendas a remanenti,
+  Nec plus quam novies quicquam tibi demere debes,                 192
+  Nascitur hinc numerus quociens supraque sequentem
+  Hunc primo scribas, retrahas exinde figuras,
+  Dum fuerit major supra positus inferiori,
+  Et rursum fiat divisio more priori;                              196
+  Et numerum quotiens supra scribas pereunti,
+  Si fiat saliens retrahendo, cifra locetur,
+  Et pereat numero quotiens, proponas eidem
+  Cifram, ne numerum pereat vis, dum locus illic                   200
+  Restat, et expletis divisio non valet ultra:
+  Dum fuerit numerus numerorum inferiore seorsum
+  Illum servabis; hinc multiplicando probabis,
+
+  [Sidenote: Proof.]
+
+  Si bene fecisti, divisor multiplicetur                           204
+  Per numerum quotiens; cum multiplicaveris, adde
+  Totali summæ, quod servatum fuit ante,
+  Reddeturque tibi numerus quem proposuisti;
+  Et si nil remanet, hunc multiplicando reddet,                    208
+
+  [Sidenote: Square Numbers.]
+
+  Cum ducis numerum per se, qui provenit inde
+  Sit tibi quadratus, ductus radix erit hujus,
+  Nec numeros omnes quadratos dicere debes,
+  Est autem omnis numerus radix alicujus.                          212
+  Quando voles numeri radicem querere, scribi
+  Debet; inde notes si sit locus ulterius impar,
+  Estque figura loco talis scribenda sub illo,
+  Que, per se dicta, numerum tibi destruat illum,                  216
+  Vel quantum poterit ex inde delebis eandem;
+  Vel retrahendo duples retrahens duplando sub ista
+  Que primo sequitur, duplicatur per duplacationem,
+  Post per se minuens pro posse quod est minuendum.                220
+  [{16}]Post his propones digitum, qui, more priori
+  Per precedentes, post per se multiplicatus,
+  Destruat in quantum poterit numerum remanentem,
+  Et sic procedens retrahens duplando figuram,                     224
+  Preponendo novam donec totum peragatur,
+  Subdupla propriis servare docetque duplatis;
+  Si det compositum numerum duplacio, debet
+  Inscribi digitus a parte dextra parte propinqua,                 228
+  Articulusque loco quo non duplicata resessit;
+  Si dabit articulum, sit cifra loco pereunte
+  Articulusque locum tenet unum, de duplicata resessit;
+  Si donet digitum, sub prima pone sequente,                       232
+  Si supraposita fuerit duplicata figura
+  Major proponi debet tantummodo cifra,
+  Has retrahens solito propones more figuram,
+  Usque sub extrema ita fac retrahendo figuras,                    236
+  Si totum deles numerum quem proposuisti,
+  Quadratus fuerit, de dupla quod duplicasti,
+  Sicque tibi radix illius certa patebit,
+  Si de duplatis fit juncta supprima figura;                       240
+  Radicem per se multiplices habeasque
+  Primo propositum, bene te fecisse probasti;
+  Non est quadratus, si quis restat, sed habentur
+  Radix quadrati qui stat major sub eadem;                         244
+  Vel quicquid remanet tabula servare memento;
+  Hoc casu radix per se quoque multiplicetur,
+  Vel sic quadratus sub primo major habetur,
+  Hinc addas remanens, et prius debes haberi;                      248
+  Si locus extremus fuerit par, scribe figuram
+  Sub pereunte loco per quam debes operari,
+  Que quantum poterit supprimas destruat ambas,
+  Vel penitus legem teneas operando priorem,                       252
+  Si suppositum digitus suo fine repertus,
+  Omnino delet illic scribi cifra debet,
+  A leva si qua sit ei sociata figura;
+  Si cifre remanent in fine pares decet harum                      256
+  Radices, numero mediam proponere partem,
+  Tali quesita radix patet arte reperta.
+  Per numerum recte si nosti multiplicare
+  Ejus quadratum, numerus qui pervenit inde                        260
+  Dicetur cubicus; primus radix erit ejus;
+  Nec numeros omnes cubicatos dicere debes,
+  Est autem omnis numerus radix alicujus;
+
+  [Sidenote: Cube Root.]
+
+  Si curas cubici radicem quærere, primo                           264
+  Inscriptum numerum distinguere per loca debes;
+  Que tibi mille notant a mille notante suprema
+  Initiam, summa operandi parte sinistra,
+  Illic sub scribas digitum, qui multiplicatus                     268
+  In semet cubice suprapositum sibi perdat,
+  Et si quid fuerit adjunctum parte sinistra
+  Si non omnino, quantum poteris minuendo,
+  Hinc triplans retrahe saltum, faciendo sub illa                  272
+  Que manet a digito deleto terna, figuram
+  Illi propones quo sub triplo asocietur,
+  Ut cum subtriplo per eam tripla multiplicatur;
+  Hinc per eam solam productum multiplicabis,                      276
+  Postea totalem numerum, qui provenit inde
+  A suprapositis respectu tolle triplate
+  Addita supprimo cubice tunc multiplicetur,
+  Respectu cujus, numerus qui progredietur                         280
+  Ex cubito ductu, supra omnes adimetur;
+  Tunc ipsam delens triples saltum faciendo,
+  Semper sub ternas, retrahens alias triplicatas
+  Ex hinc triplatis aliam propone figuram,                         284
+  Que per triplatas ducatur more priori;
+  Primo sub triplis sibi junctis, postea per se,
+  In numerum ducta, productum de triplicatis:
+  Utque prius dixi numerus qui provenit inde                       288
+  A suprapositis has respiciendo trahatur,
+  Huic cubice ductum sub primo multiplicabis,
+  Respectumque sui, removebis de remanenti,
+  Et sic procedas retrahendo triplando figuram.                    292
+  Et proponendo nonam, donec totum peragatur,
+  Subtripla sub propriis servare decet triplicatis;
+  Si nil in fine remanet, numerus datus ante
+  Est cubicus; cubicam radicem sub tripla prebent,                 296
+  Cum digito juncto quem supprimo posuisti,
+  Hec cubice ducta, numerum reddant tibi primum.
+  Si quid erit remanens non est cubicus, sed habetur
+  Major sub primo qui stat radix cubicam,                          300
+  Servari debet quicquid radice remansit,
+  Extracto numero, decet hec addi cubicato.
+  Quo facto, numerus reddi debet tibi primus.
+  Nam debes per se radicem multiplicare                            304
+  Ex hinc in numerum duces, qui provenit inde
+  Sub primo cubicus major sic invenietur;
+  Illi jungatur remanens, et primus habetur,
+  Si per triplatum numerum nequeas operari;                        308
+  Cifram propones, nil vero per hanc operare
+  Set retrahens illam cum saltu deinde triplata,
+  Propones illi digitum sub lege priori,
+  Cumque cifram retrahas saliendo, non triplicabis,                312
+  Namque nihil cifre triplacio dicitur esse;
+  At tu cum cifram protraxeris aut triplicata,
+  Hanc cum subtriplo semper servare memento:
+  Si det compositum, digiti triplacio debet                        316
+  Illius scribi, digitus saliendo sub ipsam;
+  Digito deleto, que terna dicitur esse;
+  Jungitur articulus cum triplata pereunte,
+  Set facit hunc scribi per se triplacio prima,                    320
+  Que si det digitum per se scribi facit illum;
+  Consumpto numero, si sole fuit tibi cifre
+  Triplato, propone cifram saltum faciendo,
+  Cumque cifram retrahe triplam, scribendo figuram,                324
+  Preponas cifre, sic procedens operare,
+  Si tres vel duo serie in sint, pone sub yma,
+  A dextris digitum servando prius documentum.
+  Si sit continua progressio terminus nuper                        328
+  Per majus medium totalem multiplicato;
+  Si par, per medium tunc multiplicato sequentem.
+  Set si continua non sit progressio finis:
+  Impar, tunc majus medium si multiplicabis,                       332
+  Si par per medium sibi multiplicato propinquum.                  333
+
+
+FOOTNOTES (Appendix II, Carmen de Algorismo):
+
+  [1: “Hec præsens ars dicitur algorismus ab Algore rege ejus
+  inventore, vel dicitur ab _algos_ quod est ars, et _rodos_ quod est
+  numerus; quæ est ars numerorum vel numerandi, ad quam artem bene
+  sciendum inveniebantur apud Indos bis quinque (id est decem)
+  figuræ.” --_Comment. Thomæ de Novo-Mercatu._ MS. Bib. Reg. Mus.
+  Brit. 12 E. 1.]
+
+  [2: “Hæ necessariæ figuræ sunt Indorum characteros.” _MS. de
+  numeratione._ Bib. Sloan. Mus. Brit. 513, fol. 58. “Cum vidissem
+  Yndos constituisse IX literas in universo numero suo propter
+  dispositionem suam quam posuerunt, volui patefacere de opere quod
+  sit per eas aliquidque esset levius discentibus, si Deus voluerit.
+  Si autem Indi hoc voluerunt et intentio illorum nihil novem literis
+  fuit, causa que mihi potuit. Deus direxit me ad hoc. Si vero alia
+  dicam preter eam quam ego exposui, hoc fecerunt per hoc quod ego
+  exposui, eadem tam certissime et absque ulla dubitatione poterit
+  inveniri. Levitasque patebit aspicientibus et discentibus.” MS.
+  U.L.C., Ii. vi. 5, f. 102.]
+
+  [3: From Eg. 2622.]
+
+  [4: 8 C. iv. inserts
+    Nullum cipa significat: dat significare sequenti.]
+
+  [5: From 12 E. 1.]
+
+  [6:
+    En argorisme devon prendre
+    Vii especes . . . .
+    Adision subtracion
+    Doubloison mediacion
+    Monteploie et division
+    Et de radix eustracion
+    A chez vii especes savoir
+    Doit chascun en memoire avoir
+    Letres qui figures sont dites
+    Et qui excellens sont ecrites. --MS. _Seld. Arch._ B. 26.]
+
+  [7: From 12 E. 1.]
+
+  [8: From 12 E. 1.]
+
+  [9: 8 C. iv. inserts
+    Atque figura prior nuper fuerit mediando.]
+
+  [10: _I.e._ figura secundo loco posita.]
+
+  [11: So 12 E. 1; 8 C. iv. inserts--
+
+  [12: 12 E. 1 inserts.]
+
+  [13: 12 E. 1 inserts to l. 174.]
+
+  [14: 12 E. 1 omits, Eg. 2622 inserts.]
+
+  [15: 12 E. 1 inserts.]
+
+  [16: 8 C. iv. inserts--
+    Hinc illam dele duplans sub ei psalliendo
+    Que sequitur retrahens quicquid fuerit duplicatum.]
+
+
+
+
+INDEX OF TECHNICAL TERMS[1*]
+
+    [Footnote 1*: This Index has been kindly prepared by Professor
+    J. B. Dale, of King’s College, University of London, and the
+    best thanks of the Society are due to him for his valuable
+    contribution.]
+
+  [Transcriber’s Note:
+  The Technical Terms and Glossary (following) refer to page and line
+  numbers in the printed book. Information in [[double brackets]] has
+  been added by the transcriber to aid in text searching.]
+
+
+  +algorisme+, 33/12; +algorym+, +augrym+, 3/3; the art of computing,
+    using the so-called Arabic numerals.
+  The word in its various forms is derived from the Arabic
+   _al-Khowarazmi_ (i.e. the native of Khwarazm (Khiva)). This was the
+    surname of Ja’far Mohammad ben Musa, who wrote a treatise early in
+    the 9th century (see p. xiv).
+  The form _algorithm_ is also found, being suggested by a supposed
+    derivation from the Greek ἀριθμός (number).
+
+  +antery+, 24/11; to move figures to the right of the position in
+    which they are first written. This operation is performed repeatedly
+    upon the multiplier in multiplication, and upon certain figures
+    which arise in the process of root extraction.
+
+  +anterioracioun+, 50/5; the operation of moving figures to the
+    right.  [[written anteriorac{i}o{u}n or anterioracio{u}n]]
+
+  +article+, 34/23; +articul+, 5/31; +articuls+, 9/36, 29/7,8;
+    a number divisible by ten without remainder.  [[also articull{e}]]
+
+  +cast+, 8/12; to add one number to another.
+  ‘Addition is a _casting_ together of two numbers into one number,’
+    8/10.
+
+  +cifre+, 4/1; the name of the figure 0. The word is derived from the
+    Arabic _sifr_ = empty, nothing. Hence _zero_.
+  A cipher is the symbol of the absence of number or of zero quantity.
+    It may be used alone or in conjunction with digits or other ciphers,
+    and in the latter case, according to the position which it occupies
+    relative to the other figures, indicates the absence of units, or
+    tens, or hundreds, etc. The great superiority of the Arabic to all
+    other systems of notation resides in the employment of this symbol.
+    When the cipher is not used, the place value of digits has to be
+    indicated by writing them in assigned rows or columns. Ciphers,
+    however, may be interpolated amongst the significant figures used,
+    and as they sufficiently indicate the positions of the empty rows or
+    columns, the latter need not be indicated in any other way. The
+    practical performance of calculations is thus enormously facilitated
+    (see p. xvi).
+
+  +componede+, 33/24; +composyt+, 5/35; with reference to numbers, one
+    compounded of a multiple of ten and a digit.
+    [[written componed{e}]]
+
+  +conuertide+ = conversely, 46/29, 47/9.
+    [[written co{n}u{er}tid{e} or {con}u{er}tid{e}]]
+
+  +cubicede+, 50/13; +to be c.+, to have its cube root found.
+    [[written cubiced{e}]]
+
+  +cubike nombre+, 47/8; a number formed by multiplying a given number
+    twice by itself, _e.g._ 27 = 3 × 3 × 3. Now called simply a cube.
+    [[written cubik{e} ...]]
+
+  +decuple+, 22/12; the product of a number by ten. Tenfold.
+
+  +departys+ = divides, 5/29.  [[written dep{ar}tys]]
+
+  +digit+, 5/30; +digitalle+, 33/24; a number less than ten,
+    represented by one of the nine Arabic numerals.
+    [[written digitall{e}]]
+
+  +dimydicion+, 7/23; the operation of dividing a number by two.
+    Halving.  [[written dimydicioñ]]
+
+  +duccioun+, multiplication, 43/9.  [[written duccio{u}n]]
+
+  +duplacion+, 7/23, 14/15; the operation of multiplying a number by
+    two. Doubling.
+    [[written duplacioñ or duplacioɳ with fancy “n”]]
+
+  +i-mediet+ = halved, 19/23.
+
+  +intercise+ = broken, 46/2; intercise Progression is the name given
+    to either of the Progressions 1, 3, 5, 7, etc.; 2, 4, 6, 8, etc.,
+    in which the common difference is 2.  [[written int{er}cise]]
+
+  +lede into+, multiply by, 47/18.
+    [[words always separated, as “lede ... into”]]
+
+  +lyneal nombre+, 46/14; a number such as that which expresses the
+    measure of the length of a line, and therefore is not _necessarily_
+    the product of two or more numbers (_vide_ Superficial, Solid). This
+    appears to be the meaning of the phrase as used in _The Art of
+    Nombryng_. It is possible that the numbers so designated are the
+    prime numbers, that is, numbers not divisible by any other number
+    except themselves and unity, but it is not clear that this
+    limitation is intended.
+
+  +mediacioun+, 16/36, 38/16; dividing by two (see also +dimydicion+).
+    [[written mediacioɳ with fancy “n”, generally without “u”]]
+
+  +medlede nombre+, 34/1; a number formed of a multiple of ten and a
+    digit (_vide_ componede, composyt).  [[written medled{e} ...]]
+
+  +medye+, 17/8, to halve; +mediete+, halved, 17/30; +ymedit+, 20/9.
+
+  +naturelle progressioun+, 45/22; the series of numbers 1, 2, 3, etc.
+    [[written naturell{e} p{ro}gressio{u}n]]
+
+  +produccioun+, multiplication, 50/11.  [[written produccio{u}n]]
+
+  +quadrat nombre+, 46/12; a number formed by multiplying a given
+    number by itself, _e.g._ 9 = 3 × 3, a square.
+
+  +rote+, 7/25; +roote+, 47/11; root. The roots of squares and cubes
+    are the numbers from which the squares and cubes are derived by
+    multiplication into themselves.
+
+  +significatyf+, significant, 5/14; The significant figures of a
+    number are, strictly speaking, those other than zero, _e.g._ in 3 6
+    5 0 4 0 0, the significant figures are 3, 6, 5, 4. Modern usage,
+    however, regards all figures between the two extreme significant
+    figures as significant, even when some are zero. Thus, in the above
+    example, 3 6 5 0 4 are considered significant.
+
+  +solide nombre+, 46/37; a number which is the product of three other
+    numbers, _e.g._ 66 = 11 × 2 × 3.  [[usually written solid{e}]]
+
+  +superficial nombre+, 46/18; a number which is the product of two
+    other numbers, _e.g._ 6 = 2 × 3.
+    [[written sup{er}ficial or sup{er}ficiall{e}]]
+
+  +ternary+, consisting of three digits, 51/7.
+    [[written t{er}nary]]
+
+  +vnder double+, a digit which has been doubled, 48/3.
+
+  +vnder-trebille+, a digit which has been trebled, 49/28;
+    +vnder-triplat+, 49/39.
+    [[written vnder-trebill{e}, vnder-t{r}iplat]]
+
+  +w+, a symbol used to denote half a unit, 17/33
+    [[shown in e-text as superscript ʷ]]
+
+
+
+
+GLOSSARY
+
+  [Transcriber’s Note:
+
+  Words whose first appearance is earlier than the page cited in the
+  Glossary are identified in double-bracketed notes. To aid in text
+  searching, words written with internal {italics} are also noted,
+  and context is given for common words.]
+
+
+  +ablacioun+, taking away, 36/21  [[written ablacio{u}n]]
+  +addyst+, haddest, 10/37
+  +agregacioun+, addition, 45/22. (First example in N.E.D., 1547.)
+    [[written ag{r}egacio{u}n]]
+  +a-ȝenenes+, against, 23/10
+  +allgate+, always, 8/39
+  +als+, as, 22/24
+  +and+, if, 29/8;
+    +&+, 4/27;
+    +& yf+, 20/7
+  +a-nendes+, towards, 23/15
+  +aproprede+, appropriated, 34/27  [[written ap{ro}pred{e}]]
+  +apwereth+, appears, 61/8
+  +a-risyȝt+, arises, 14/24
+  +a-rowe+, in a row, 29/10
+  +arsemetrike+, arithmetic, 33/1  [[written arsemetrik{e}]]
+  +ayene+, again, 45/15
+
+  +bagle+, crozier, 67/12
+  +bordure+ = ordure, row, 43/30  [[written bordur{e}]]
+  +borro+, _inf._ borrow, 11/38;
+    _imp. s._ +borowe+, 12/20;
+    _pp._ +borwed+, 12/15;
+    +borred+, 12/19
+  +boue+, above, 42/34
+
+  +caputule+, chapter, 7/26  [[written caputul{e}]]
+  +certayn+, assuredly, 18/34  [[written c{er}tayɳ]]
+  +clepede+, called, 47/7  [[written cleped{e}]]
+  +competently+, conveniently, 35/8
+  +compt+, count, 47/29
+  +contynes+, contains, 21/12;  [[written {con}tynes]]
+    _pp._ +contenythe+, 38/39  [[written co{n}tenyth{e}]]
+  +craft+, art, 3/4
+
+  +distingue+, divide, 51/5
+
+  +egalle+, equal, 45/21  [[written egall{e}]]
+  +excep+, except, 5/16]
+  +exclusede+, excluded, 34/37  [[written exclused{e}]]
+  +excressent+, resulting, 35/16  [[written exc{re}ssent]]
+  +exeant+, resulting, 43/26
+  +expone+, expound, 3/23
+
+  +ferye+ = ferþe, fourth, 70/12
+  +figure+ = figures, 5/1  [[written fig{ure}]]
+  +for-by+, past, 12/11
+  +fors; no f.+, no matter, 22/24
+  +forseth+, matters, 53/30
+  +forye+ = forþe, forth, 71/8]
+  +fyftye+ = fyftþe, fifth, 70/16
+
+  +grewe+, Greek, 33/13
+
+  +haluendel+, half, 16/16;
+    +haldel+, 19/4;
+    _pl._ +haluedels+, 16/16
+  +hayst+, hast, 17/3, 32
+  +hast+, haste, 22/25  [[in “haue hast to”]]
+  +heer+, higher, 9/35
+  +here+, their, 7/26  [[in “in her{e} caputul{e}”]]
+  +here-a-fore+, heretofore, 13/7  [[written her{e}-a-for{e}]]
+  +heyth+, was called, 3/5
+  +hole+, whole, 4/39;
+    +holle+, 17/1;
+    +hoole+, of three dimensions, 46/15
+  +holdyþe+, holds good, 30/5
+  +how be it that+, although, 44/4
+
+  +lede+ = lete, let, 8/37
+  +lene+, lend, 12/39
+  +lest+, least, 43/27  [[in “at the lest”]]
+  +lest+ = left, 71/9  [[in “the lest syde”]]
+  +leue+, leave, 6/5;
+    _pr. 3 s._ +leues+, remains, 11/19;  [[first in 10/40]]
+    +leus+, 11/28;
+    _pp._ +laft+, left, 19/24
+  +lewder+, more ignorant, 3/3  [[written lewd{er}]]
+  +lust+, desirest to, 45/13
+  +lyȝt+, easy, 15/31
+  +lymytes+, limits, 34/18;
+    +lynes+, 34/12;
+    +lynees+, 34/17;
+    Lat. limes, _pl._ limites.
+
+  +maystery+, achievement;  [[written mayst{er}y]]
+    +no m.+, no achievement, i.e. easy, 19/10
+  +me+, _indef. pron._ one, 42/1  [[first in 34/16]]
+  +mo+, more, 9/16
+    +moder+ = more (Lat. majorem), 43/22
+  +most+, must, 30/3  [[first in 3/12 and many more]]
+  +multipliede+, +to be m.+ = multiplying, 40/9
+  +mynvtes+, the sixty parts into which a unit is divided, 38/25
+    [[written mynvt{es}]]
+  +myse-wroȝt+, mis-wrought, 14/11
+
+  +nether+, nor, 34/25  [[in “It was, nether is”]]
+  +nex+, next, 19/9
+  +noȝt+, nought, 5/7  [[first in 4/8]]
+  +note+, not, 30/5
+
+  +oo+, one, 42/20; +o+, 42/21  [[first in 34/27; 33/22]]
+  +omest+, uppermost, higher, 35/26;
+    +omyst+, 35/28
+  +omwhile+, sometimes, 45/31  [[first in 39/17]]
+  +on+, one, 8/29  [[in “on vnder an-oþ{er}”]]
+  +opyne+, plain, 47/8  [[written opyn{e}]]
+  +or+, before, 13/25  [[in “or þou be-gan”]]
+  +or+ = þe oþ{er}, the other, 28/34  [[in “or by-twene”]]
+  +ordure+, order, 34/9;
+    row, 43/1  [[word form is “order”]]
+  +other+, or, 33/13, 43/26;
+    [[in “art other craft” on 33/13, “other how oft” on 43/26;
+    note also “one other other” on 35/24]]
+    +other . . . or+, either . . . or, 38/37
+    [[in “other it is even or od{e}” on 38/37;
+    there are earlier occurrences]]
+  +ouerer+, upper, 42/15  [[written ou{er}er]]
+  +ouer-hippede+, passed over, 43/19  [[written ou{er}-hipped{e}]]
+
+  +recte+, directly, 27/20  [[in “stondes not recte”;
+    also on 26/31 in “recte ou{er} his hede”]]
+  +remayner+, remainder, 56/28
+  +representithe+, represented, 39/14  [[written rep{re}sentith{e}]]
+  +resteth+, remains, 63/29  [[first in 57/29 and others]]
+  +rewarde+, regard, 48/6  [[written reward{e}]]
+  +rew+, row, 4/8
+  +rewle+, row, 4/20, 7/12;
+    [[in “place of þe rewle”, “þe rewle of fig{ure}s”]]
+    +rewele+, 4/18;
+    +rewles+, rules, 5/33
+
+  +s.+ = scilicet, 3/8  [[in “s. Algorism{us}”]]
+  +sentens+, meaning, 14/29
+  +signifye(tyf)+, 5/13. The last three letters are added above the
+    line, evidently because of the word ‘significatyf’ in l. 14.
+    But the ‘Solucio,’ which contained the word, has been omitted.
+  +sithen+, since, 33/8
+  +some+, sum, result, 40/17, 32
+    [[first in 36/21 in “me may see a some”, then in “the same some”
+    and “to some of”]]
+  +sowne+, pronounce, 6/29
+  +singillatim+, singly, 7/25
+  +spices+, species, kinds, 34/4  [[first in 5/34 and others]]
+  +spyl+, waste, 14/26
+  +styde+, stead, 18/20
+  +subtrahe+, subtract, 48/12;
+    _pp._ +subtrayd+, 13/21
+  +sythes+, times, 21/16
+
+  +taȝt+, taught, 16/36
+  +take+, _pp._ taken;
+    +t. fro+, starting from, 45/22  [[in “fro oone or tweyn{e} take”]]
+  +taward+, toward, 23/34
+  +thouȝt+, though, 5/20
+  +trebille+, multiply by three, 49/26  [[written trebill{e}]]
+  +twene+, two, 8/11  [[first in 4/23]]
+  +þow+, though, 25/15  [[in “þow þ{o}u take”]]
+  +þowȝt+, thought;
+    +be þ.+, mentally, 28/4
+  +þus+ = þis, this, 20/33  [[in “þus nombur 214”]]
+
+  +vny+, unite, 45/10
+
+  +wel+, wilt, 14/31  [[in “If þ{o}u wel”]]
+  +wete+, wit, 15/16;
+    +wyte+, know, 8/38;
+    _pr. 2 s._ +wost+, 12/38
+  +wex+, become, 50/18
+  +where+, whether, 29/12
+    [[written wher{e} in “wher{e} in þe secunde, or”]]
+  +wher-thurghe+, whence, 49/15  [[written Wher-thurgh{e}]]
+  +worch+, work, 8/19;  [[first in 7/35]]
+    +wrich+, 8/35;
+    +wyrch+, 6/19;
+    _imp. s._ +worch+, 15/9;  [[first in 9/6]]
+    _pp._ +y-wroth+, 13/24
+  +write+, written, 29/19;
+    [[first in 6/37 in “hast write”, “be write”]]
+    +y-write+, 16/1
+  +wryrchynge+ = wyrchynge, working, 30/4  [[written wryrchyng{e}]]
+  +w^t+, with, 55/8
+
+  +y-broth+, brought, 21/18
+  +ychon+, each one, 29/10  [[written ychoɳ]]
+  +ydo+, done, added, 9/6
+    [[first in 8/37 in “haue ydo”; 9/6 in “ydo all to-ged{er}”]]
+  +ylke+, same, 5/12
+  +y-lyech+, alike, 22/23
+  +y-myȝt+, been able, 12/2
+  +y-nowȝt+, enough, 15/31;
+    +ynovȝt+, 18/34
+  +yove+, given, 45/33
+  +y^t+, that, 52/8
+  +y-write+, _v._ +write.+
+  +y-wroth+, _v._ +worch.+
+
+
+       *       *       *       *       *
+           *       *       *       *
+       *       *       *       *       *
+
+
+MARGINAL NOTES:
+
++Headnotes+ have been moved to the beginning of the appropriate
+paragraph. Headnotes were omitted from the two Appendixes, as sidenotes
+give the same information.
+
++Line Numbers+ are cited in the Index and Glossary. They have been
+omitted from the e-text except in the one verse selection (App. II,
+_Carmen de Algorismo_). Instead, the Index and Glossary include
+supplemental information to help locate each word.
+
++Numbered Notes+:
+
+  Numbered sidenotes show page or leaf numbers from the original MSS.
+  In the e-text, the page number is shown as [*123b] inline; mid-word
+  page breaks are marked with a supplemental asterisk [*]. Numbers are
+  not used.
+
+  Footnotes give textual information such as variant readings. They
+  have been numbered sequentially within each title, with numbers
+  shown as [{1}] to avoid confusion with bracked text--including
+  single numerals--in the original. Editorial notes are shown as [1*].
+  When a footnote calls for added text, the addition is shown in the
+  body text with [[double brackets]].
+
++Sidenotes+ giving a running synopsis of the text have been moved to the
+beginning of each paragraph, where they are shown as a single note.
+
+
+ERRORS AND ANOMALIES (Noted by Transcriber):
+
+Introduction:
+
+  dated Mij^c
+    [_In this and the remainder of the paragraph, the letter shown as
+    ^c is printed directly above the preceding j._]
+
+The Crafte of Nombrynge:
+
+  sursu{m} {pr}ocedas m{u}ltiplicando
+    [_Italicized as shown: error for “p{ro}cedas”?_]
+  Sidenote: Our author makes a slip here
+    [_Elsewhere in the book, numerical errors are corrected in the
+    body text, with a footnote giving the original form._]
+  ten tymes so mych is þe nounb{re}
+    [_text unchanged: error for “as”?_]
+  6 tymes 24, [{19}]þen take
+    [_misplaced footnote anchor in original:
+    belongs with “6 times 24”_]
+  Fn. 7: ‘Subt{ra}has a{u}t addis a dext{ri}s  [_open quote missing_]
+
+The Art of Nombryng:
+
+  oone of the digitis as .10. of 1.. 20. of. 2.
+    [_text unchanged: error for “as .10. of .1. 20. of .2.”?_]
+  sette a-side half of tho m{inutes}
+    [_text unchanged: error for “the”?_]
+  and. 10. as before is come therof
+    [_text unchanged: error for “and .10.”?_]
+  Sidenote: Where to set the quotiente  [_spelling (1922) unchanged_]
+  Sidenote: Definition of Progression.  [_f in “of” illegible_]
+  Sidenote: ... giving the value of ab.^2  [_That is, “a(b^2).”_]
+
+Accomptynge by counters:
+
+  For example of the [*117a.] ly[*]nes
+    [_final . in sidenote missing or invisible_]
+    [_also in 121b, 122a]
+  which in the fyrst summe is 5
+    [_invisible “5” supplied by transcriber_]
+  [*116a (_sic_).]
+    [_Editor’s “sic”: page numbering jumps back to 116 instead of the
+    expected 123, and continues from 116._]
+  [*123a] ... set downe y^e multiplyer .v. tymes, as here you se
+    [_Diagram shown as printed, with 35500 for 36500 in one column,
+    and apparent misplaced “thousands” marker_]
+  365 (which is the nomber of dayes ...  [_open ( missing_]
+
+The arte of nombrynge by the hande:
+
+  for 1 the fynger is clasped in
+    [_In at least one printing of the text, “clasped” is misprinted
+    as “elasped”_]
+  but this teacyed me not  [_text unchanged_]
+
+Appendix I: A Treatise on the Numeration of Algorism:
+
+  _See Introduction and Glossary for ſ:f and þ:y errors_
+
+Appendix II: Carmen de Algorismo:
+
+  _In this selection, errors that are not explained in footnotes were
+  assumed to be typographic._
+
+  l. 99 Postea procedas  [procdeas]
+  l. 163 Articuli digitum post in digitum mixti duc  [post iu]
+
+
+
+
+
+
+End of Project Gutenberg's The Earliest Arithmetics in English, by Anonymous
+
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+Project Gutenberg's The Earliest Arithmetics in English, by Anonymous
+
+This eBook is for the use of anyone anywhere at no cost and with
+almost no restrictions whatsoever.  You may copy it, give it away or
+re-use it under the terms of the Project Gutenberg License included
+with this eBook or online at www.gutenberg.org
+
+
+Title: The Earliest Arithmetics in English
+
+Author: Anonymous
+
+Editor: Robert Steele
+
+Release Date: June 1, 2008 [EBook #25664]
+
+Language: English
+
+Character set encoding: ISO-8859-1
+
+*** START OF THIS PROJECT GUTENBERG EBOOK THE EARLIEST ARITHMETICS IN ENGLISH ***
+
+
+
+
+Produced by Louise Hope, David Starner and the Online
+Distributed Proofreading Team at http://www.pgdp.net
+
+
+
+
+
+[Transcriber's Note:
+
+This text is intended for users whose text readers cannot use the "real"
+(unicode/utf-8) version of the file. Characters that could not be fully
+displayed have been "unpacked" and shown in brackets:
+
+  [gh] (yogh)
+  [n~], [l~l] (n with curl, crossed l: see below)
+  +0+ (Greek phi: see below)
+
+In _The Crafte of Nombrynge_, final "n" was sometimes written with an
+extra curl. In this Latin-1 text it is shown as [n~]. In the same
+selection, the numeral "0" was sometimes printed as the Greek letter
+phi. It is shown here as +0+ rather than the usual +ph+ because the
+physical form is more significant than the sound of the letter. Double
+"l" with a line is shown as [l~l]. 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 [Illustration]
+markers and similar, and the letters noted above, 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.]
+
+
+       *       *       *       *       *
+           *       *       *       *
+       *       *       *       *       *
+
+
+  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"�נ5", 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 11�"�נ17�", 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 f�r 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 2�. 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 נ23.
+
+  +-----------+-----------+
+  | 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�נ3.
+  |   | 1 | 2 |   |   |   | 4000�נ3.
+  |   | 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�נ7. The difference of 8 is 2, of 7 is 3, the
+next article being 10; 7�-�2 is�5. 5�נ10 = 50; 2�נ3 = 6. 50�+�6 = 56
+answer. The rule will hold in such cases as 17�נ15 where the article
+next higher is the same for both, _i.e._, 20; but in such a case as
+17�נ9 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 M�nchener
+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 �e boke of algorym, or Augrym aft{er} lewd{er} vse.
+And �is boke tretys �e Craft of Nombryng, �e quych crafte is called also
+Algorym. Ther was a kyng of Inde, �e quich heyth Algor, &�he made �is
+craft. And aft{er} his name he called hit algory{m}; or els ano�{er}
+cause is quy it is called Algorym, for �e latyn word of hit s.
+Algorism{us} com{es} of Algos, grece, q{uid} e{st} ars, latine, craft
+o[n~] englis, and rides, q{uid} e{st} {nu}me{rus}, latine, A�nomb{ur}
+o[n~] 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} [gh]e most vnd{ir}stonde �{a}t in �is craft ben vsid
+teen figurys, as here ben{e} writen for ensampul, +0+�9 8 7 6 5 4 3 2 1.
+��Expone �e too v{er}sus afor{e}: this p{re}sent craft ys called
+Algorism{us}, in �e quych we vse teen signys of Inde. Questio. ��Why
+te[n~] fyguris of Inde? Solucio. for as I haue sayd afore �ai wer{e}
+fonde fyrst in Inde of a kyng{e} of �at Cuntre, �{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 �is verse is notifide �e significac{i}on of �ese figur{is}. And �us
+expone the verse. �e first signifiyth on{e}, �e secu{n}de [*leaf 136b]
+signi[*]fiyth tweyn{e}, �e thryd signifiyth thre, &�the fourte
+signifiyth�4. ��And so forthe towarde �e lyft syde of �e tabul or of �e
+boke �{a}t �e figures ben{e} writen{e} in, til �at �{o}u come to the
+last figure, �{a}t is called a cifre. ��Questio. In quych syde sittes �e
+first figur{e}? Soluc{io}, forsothe loke quich figure is first in �e
+ry[gh]t side of �e bok or of �e tabul, &��{a}t same is �e first
+figur{e}, for �{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 �e first, for he sittes
+o[n~] �e ri[gh]t syde. And the fig{ur}e of 3 is last. ��Neu{er}-�e-les
+wen he says ��P{ri}ma sig{nifica}t vnu{m} &c., �at is to say, �e first
+betokenes on{e}, �e secu{n}de. 2. & fore-�{er}-mor{e}, he vnd{ir}stondes
+no[gh]t of �e first fig{ur}e of eu{er}y rew. ��But he vnd{ir}stondes �e
+first figure �{a}t is in �e nomb{ur} of �e forsayd teen figuris, �e
+quych is on{e} of �{e}se. 1. And �e 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 �is v{er}se �us. Eu{er}y of �ese figuris bitokens hym selfe &
+no mor{e}, yf he stonde in �e first place of �e rewele / this worde
+Simplicit{er} in �at verse it is no more to say but �at, & no mor{e}.
+��If it stonde in the secu{n}de place of �e rewle, he betokens ten{e}
+tymes hym selfe, as �is figur{e} 2 here 20 tokens ten tyme hym selfe,
+[*leaf 137a] �at is twenty, for he hym selfe betokenes twey{ne}, &�ten
+tymes twene is twenty. And for he stondis o[n~] �e lyft side & in �e
+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-no�{er} toward the lyft
+side, he schal betoken{e} ten tymes as mich mor{e} as he schul betoken &
+he stode in �e place �{ere} �at �e fig{ure} a-for{e} hym stondes. loo an
+ensampull{e}. 9. 6. 3. 4. �e fig{ure} of 4. �{a}t hase �is schape +4.+
+betokens bot hymselfe, for he stondes in �e first place. The fig{ure} of
+3. �at hase �is schape +3.+ betokens ten tymes mor{e} �en he schuld & he
+stode �{ere} �{a}t �e fig{ure} of 4. stondes, �{a}t is thretty. The
+fig{ure} of 6, �{a}t hase �is schape +6+, betokens ten tymes mor{e} �an
+he schuld & he stode �{ere} as �e fig{ure} of +3.+ stondes, for �{ere}
+he schuld tokyn{e} bot sexty, &�now he betokens ten tymes mor{e}, �at is
+sex hundryth. The fig{ure} of 9. �{a}t hase �is schape +9.+ betokens ten
+tymes mor{e} �an{e} he schuld & he stode in �e place �{ere} �e fig{ure}
+of sex stondes, for �en he schuld betoken to 9. hundryth, and in �e
+place �{ere} he stondes now he betokens 9. �ousande. Al �e hole nomb{ur}
+is 9 thousande sex hundryth & four{e} & thretty. ��fforthermor{e}, when
+�{o}u schalt rede a nomb{ur} of fig{ure}, �{o}u schalt begyn{e} at �e
+last fig{ure} in the lyft side, &�rede so forth to �e ri[gh]t side as
+her{e} 9.�6. 3.�4. Thou schal begyn to rede at �e fig{ure} of 9. & rede
+forth �us.�9. [*leaf 137b] thousand sex hundryth thritty & foure. But
+when �{o}u schall{e} write, �{o}u schalt be-gynne to write at �e ry[gh]t
+side.
+
+  � Nil cifra sig{nifica}t s{ed} dat signa{re} sequenti.
+
+  [Sidenote: The meaning and use of the cipher.]
+
+Expone �is v{er}se. A�cifre tokens no[gh]t, bot he makes �e fig{ure} to
+betoken �at comes aft{ur} hym mor{e} �an he schuld & he wer{e} away, as
+�us 1+0+. her{e} �e fig{ure} of on{e} tokens ten, &�yf �e cifre wer{e}
+away[{1}] & no fig{ure} by-for{e} hym he schuld token bot on{e}, for
+�an he sch{ul}d stonde in �e first place. ��And �e cifre tokens nothyng
+hym selfe. for al �e nomb{ur} of �e ylke too fig{ure}s is bot ten.
+��Questio. Why says he �at a cifre makys a fig{ure} to signifye (tyf)
+mor{e} &c. ��I�speke for �is worde significatyf, ffor sothe it may happe
+aft{ur} a cifre schuld come a-no�{ur} cifre, as �us 2+0++0+. And [gh]et
+�e secunde cifre shuld token neu{er} �e mor{e} excep he schuld kepe �e
+ord{er} of �e 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 �is v{er}se �us. �e last figu{re} schal token mor{e} �an all{e}
+�e o�{er} afor{e}, thou[gh]t �{ere} wer{e} a hundryth thousant figures
+afor{e}, as �us, 16798. �e last fig{ure} �at is 1. betokens ten
+thousant. And all{e} �e o�{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} �en
+all{e} �at nomb{ur}, {er}go �e last figu{re} tokens mor{e} �an all �e
+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 �is tretis dep{ar}tys �is 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 �is v{er}se. know
+�{o}u aft{ur} �e forsayd rewles �{a}t I sayd afore, �at �{ere} ben thre
+spices of nomb{ur}. Oon{e} is a digit, Ano�{er} is an Articul, &��e
+to�{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 �at 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} �{a}t may be deuidyt into
+nomb{urs} of ten & nothyng{e} leue ou{er}, as twenty, thretty, fourty,
+a�hundryth, a�thousand, &�such o�{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 be[n~] nomb{ur}s �at bene componyt of a digyt &
+of an articull{e} as fouretene, fyftene, sextene, &�such o�{er}.
+ffortene is co{m}ponyd of four{e} �at is a digit & of ten �at is an
+articull{e}. ffiftene is componyd of 5 & ten, &�so of all o�{er}, what
+�at �ai ben. Short-lych eu{er}y nomb{ur} �at be-gynnes w{i}t{h} a digit
+& endyth in a articull{e} is a composyt, as fortene bygennyng{e} by
+four{e} �at 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 �{o}u schalt wyrch whan �{o}u schalt write a
+nomb{ur}. Expone v{er}su{m} sic, &�fac iuxta expon{ent}is sentencia{m};
+whan �{o}u hast a nomb{ur} to write, loke fyrst what man{er} nomb{ur} it
+ys �{a}t �{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 �{a}t digit in �e first place toward �e ryght side. If it
+be a composyt, write �e digit of �e composit in �e first place & write
+�e articul of �at digit in �e secunde place next toward �e lyft side. As
+yf �{o}u schal write sex & twenty. write �e digit of �e nomb{ur} in �e
+first place �at is sex, and write �e articul next aft{ur} �at is twenty,
+as �us 26. But whan �{o}u schalt sowne or speke [*leaf 139a] or rede an
+Composyt �ou schalt first sowne �e articul & aft{ur} �e digit, as �{o}u
+seyst by �e comyn{e} speche, Sex & twenty & nou[gh]t 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 �{o}u schal write when �e nombre �{a}t �{o}u hase to
+write is an Articul. Expone v{er}sus sic & fac s{ecundu}m sentenciam.
+Ife �e nomb{ur} �{a}t �{o}u hast write be an Articul, write first a
+cifre & aft{ur} �e cifer write an Articull{e} �us. 2+0+. fforthermor{e}
+�{o}u schalt vnd{ir}stonde yf �{o}u haue an Articul, loke how mych he
+is, yf he be w{i}t{h}-ynne an hundryth, �{o}u schalt write bot on{e}
+cifre, afore, as her{e} .9+0+. If �e articull{e} be by hym-silfe & be an
+hundrid euen{e}, �en schal �{o}u write .1. & 2 cifers afor{e}, �at he
+may stonde in �e thryd place, for eu{er}y fig{ure} in �e thryd place
+schal token a hundrid tymes hym selfe. If �e articul be a thousant or
+thousandes[{3}] and he stonde by hy{m} selfe, write afor{e} 3 cifers &
+so for� of al o�{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 �{a}t yf �e first fig{ure} in
+�e rewle of fig{ure}s token a nomb{ur} �at is euen{e} al �{a}t nomb{ur}
+of fig{ur}ys in �at rewle schal be euen{e}, as her{e} �{o}u may see 6.
+7. 3. 5.�4. Computa & p{ro}ba. ��If �e first [*leaf 139b] fig{ur}e token
+an nomb{ur} �at is ode, all{e} �at nomb{ur} in �at 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 �{a}t �{er} be[n~] .7. spices or p{ar}tes of �is craft.
+The first is called addicio�, �e secunde is called subtraccio�. The
+thryd is called duplacio�. The 4. is called dimydicio�. The 5. is called
+m{u}ltiplicacio�. The 6 is called diuisio�. The 7. is called extraccio�
+of �e Rote. What all �ese 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 �e ryght side of �e boke or of a tabul. loke
+wer{e} �{o}u wul be-gynne to write latyn or englys in a boke, & �{a}t
+schall{e} be called �e lyft side of the boke, �at �{o}u writest toward
+�{a}t side schal be called �e ryght side of �e boke. V{er}sus.
+
+  A leua dupla, diuide, m{u}ltiplica.
+
+  [Sidenote: Multiply or divide from left to right.]
+
+Here he telles �e in quych side of �e boke or of �e tabul �{o}u
+schall{e} be-gyn{e} to wyrch duplacio�, diuisio�, and m{u}ltiplicacio�.
+Thou schal begyn{e} to worch in �e lyft side of �e boke or of �e tabul,
+but yn what wyse �{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 �e craft of Addicio�. In �is craft �{o}u most knowe
+foure thyng{es}. ��Fyrst �{ou} most know what is addicio�. Next �{o}u
+most know how mony rewles of figurys �ou most haue. ��Next �{o}u most
+know how mony diue{r}s casys happes in �is craft of addicio�. ��And next
+qwat is �e p{ro}fet of �is craft. ��As for �e first �ou most know �at
+addicio� is a castyng to-ged{ur} of twoo nomburys in-to on{e} nombr{e}.
+As yf I aske qwat is twene & thre. �{o}u wyl cast �ese twene nomb{re}s
+to-ged{ur} & say �{a}t it is fyue. � As for �e secunde �ou most know
+�{a}t �ou schall{e} haue tweyne rewes of figures, on{e} vndur a-nother,
+as her{e} �{o}u mayst�se.
+
+  1234
+  2168.
+
+� As for �e thryd �ou most know �{a}t ther{e} ben foure diu{er}se cases.
+As for �e forthe �{o}u most know �{a}t �e p{ro}fet of �is craft is to
+telle what is �e hole nomb{ur} �{a}t comes of diu{er}se nomburis. Now as
+to �e texte of oure verse, he teches ther{e} how �{o}u schal worch in
+�is craft. ��He says yf �{o}u wilt cast on{e} nomb{ur} to ano�{er}
+nomb{ur}, �ou most by-gynne on �is wyse. ��ffyrst write [*leaf 140b] two
+rewes of figuris & nombris so �at �{o}u write �e first figur{e} of �e
+hyer nomb{ur} euen{e} vnd{ir} the first fig{ure} of �e nether nomb{ur},
+And �e secunde of �e nether nomb{ur} euen{e} vnd{ir} �e secunde of �e
+hyer, & so forthe of eu{er}y fig{ur}e of both �e rewes as �{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 �{o}u schalt do when �{o}u hast write too rewes of
+figuris on vnder an-o�{er}, as I sayd be-for{e}. ��He says �{o}u schalt
+take �e first fig{ur}e of �e heyer nomb{re} & �e fyrst figur{e} of �e
+ne�{er} nombre, &�cast hem to-ged{er} vp-on �is condicio[n~]. Thou schal
+loke qwe�{er} �e nombe{r} �at comys �{ere}-of be a digit or no. ��If he
+be a digit �{o}u schalt do away �e first fig{ur}e of �e hyer nomb{re},
+and write �{ere} in his stede �at he stode Inne �e digit, �{a}t comes of
+�e ylke 2 fig{ur}es, &�so wrich forth o[n~] o�{er} figures yf �{ere} be
+ony moo, til �{o}u come to �e ende toward �e lyft side. And lede �e
+nether fig{ure} stonde still eu{er}-mor{e} til �{o}u haue ydo. ffor
+�{ere}-by �{o}u schal wyte whe�{er} �{o}u hast don{e} wel or no, as I
+schal tell �e aft{er}ward in �e ende of �is Chapt{er}. ��And loke
+allgate �at �ou be-gynne to worch in �is Craft of [*leaf 141a]
+Addi[*]cio[n~] in �e ry[gh]t side, here is an ensampul of �is case.
+
+  1234
+  2142.
+
+Caste 2 to four{e} & �at wel be sex, do away 4. & write in �e same place
+�e fig{ur}e of sex. ��And lete �e fig{ur}e of 2 in �e nether rewe stonde
+stil. When �{o}u hast do so, cast 3 & 4 to-ged{ur} and �at wel be seuen
+�{a}t is a digit. Do away �e 3, &�set �{ere} seue[n~], and lete �e
+ne�{er} fig{ure} stonde still{e}, &�so worch forth bakward til �{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 �e secunde case �{a}t may happe in �is craft. And �e case is
+�is, yf of �e casting of 2 nomburis to-ged{er}, as of �e fig{ur}e of �e
+hyer rewe & of �e figure of �e ne�{er} rewe come a Composyt, how schalt
+�{ou} worch. �{us} �{o}u schalt worch. Thou shalt do away �e fig{ur}e of
+�e hyer nomb{er} �at was cast to �e figure of �e ne�{er} nomber. ��And
+write �{ere} �e digit of �e Composyt. And set �e articul of �e composit
+next aft{er} �e digit in �e same rewe, yf �{ere} be no mo fig{ur}es
+aft{er}. But yf �{ere} be mo figuris aft{er} �at digit. And �ere he
+schall be rekend for hym selfe. And when �{o}u schalt adde �{a}t ylke
+figure �{a}t berys �e articull{e} ou{er} his hed to �e figur{e} vnd{er}
+hym, �{o}u schalt cast �at articul to �e figure �{a}t hase hym ou{er}
+his hed, &��{ere} �at Articul schal toke[n~] hym selfe. lo an Ensampull
+[*leaf 141b] of all.
+
+  326
+  216.
+
+Cast 6 to 6, & �{ere}-of wil arise twelue. do away �e hyer 6 & write
+�{ere} 2, �{a}t is �e digit of �is composit. And �e{n} write �e
+articull{e} �at is ten ou{er} �e figuris hed of twene as �{us}.
+
+   1
+  322
+  216.
+
+Now cast �e articull{e} �{a}t standus vpon �e fig{ur}is of twene hed to
+�e same fig{ur}e, &�reken �at articul bot for on{e}, and �an �{ere} wil
+arise thre. �an cast �at thre to �e ne�{er} figure, �at is on{e}, &��at
+wul be four{e}. do away �e fig{ur}e of 3, and write �{ere} a fig{ur}e of
+foure. and lete �e ne�{er} fig{ur}e stonde stil, &��an 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 �e thryde case of �e craft of Addicio[n~]. & �e case
+is �is. yf of Addiciou[n~] of 2 figuris a-ryse an Articull{e}, how schal
+�{o}u do. thou most do away �e heer fig{ur}e �{a}t was addid to �e
+ne�{er}, &�write �{ere} a cifre, and sett �e articuls on �e figuris
+hede, yf �{a}t �{ere} come ony aft{er}. And wyrch �an as I haue tolde �e
+in �e secunde case. An ensampull.
+
+  25.
+  15
+
+Cast 5 to 5, �at wylle be ten. now do away �e hyer 5, &�write �{ere} a
+cifer. And sette ten vpon �e figuris hed of 2. And reken it but for on
+�us.] lo an Ensampull{e}
+
+  +----+
+  | 1  |
+  | 2+0+ |
+  | 15 |
+  +----+
+
+And [*leaf 142a] �an worch forth. But yf �{ere} come no figure aft{er}
+�e cifre, write �e articul next hym in �e same rewe as here
+
+  +---+
+  | 5 |
+  | 5 |
+  +---+
+
+cast 5 to 5, and it wel be ten. do away 5. �at is �e hier 5. and write
+�{ere} a cifre, &�write aft{er} hym �e articul as �us
+
+  +----+
+  | 1+0+ |
+  |  5 |
+  +----+
+
+And �an �{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} �e fourt case, &�it is �is, �at yf �{ere} come a
+cifer in �e hier rewe, how �{o}u schal do. �us �{o}u schalt do. do away
+�e cifer, &�sett �{ere} �e digit �{a}t comes of �e addiciou{n} as �us
+
+  1+0++0+84.
+  17743
+
+In �is ensampul ben all{e} �e four{e} cases. Cast 3 to foure, �{a}t wol
+be seue[n~]. do away 4. & write �{ere} seue[n~]; �an cast 4 to �e
+figur{e} of 8. �{a}t wel be 12. do away 8, &�sett �{ere} 2. �at is a
+digit, and sette �e articul of �e composit, �at is ten, vpon �e cifers
+hed, &�reken it for hym selfe �at is on. �an cast on{e} to a cifer, &
+hit wull{e} be but on, for no[gh]t & on makes but on{e}. �an cast 7.
+�{a}t stondes vnd{er} �at on to hym, &��at wel be 8. do away �e cifer &
+�at 1. & sette �{ere} 8. �an go forthermor{e}. cast �e o�{er} 7 to �e
+cifer �{a}t stondes ou{er} hy{m}. �{a}t wul be bot seuen, for �e cifer
+betokens no[gh]t. do away �e cifer & sette �{ere} seue[n~], [*leaf 142b]
+& �en go for�{er}mor{e} & cast 1 to 1, &��at wel be 2. do away �e hier
+1, &�sette �{ere} 2. �an hast �{o}u do. And yf �{o}u haue wel ydo �is
+nomber �at is sett her{e}-aft{er} wel be �e nomber �at schall{e} aryse
+of all{e} �e addicio[n~] 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 �e Chapt{er} of subtraccio[n~], in the quych �ou most know
+foure nessessary thyng{es}. the first what is subtraccio[n~]. �e secunde
+is how mony nombers �ou most haue to subt{ra}ccio[n~], the thryd is how
+mony maners of cases �{ere} may happe in �is craft of subtraccio[n~].
+The fourte is qwat is �e p{ro}fet of �is craft. ��As for �e first, �{o}u
+most know �{a}t subtraccio[n~] is drawyng{e} of on{e} nowmb{er} oute of
+ano�{er} nomber. As for �e secunde, �ou most knowe �{a}t �ou most haue
+two rewes of figuris on{e} vnd{er} ano�{er}, as �{o}u addyst in
+addicio[n~]. As for �e thryd, �{o}u moyst know �{a}t four{e} man{er} of
+diu{er}se casis mai happe in �is craft. ��As for �e fourt, �ou most know
+�{a}t �e p{ro}fet of �is craft is whenne �{o}u hasse taken �e lasse
+nomber out of �e mor{e} to telle what �{ere} leues ou{er} �{a}t. & �{o}u
+most be-gynne to wyrch in �{is} craft in �e ryght side of �e boke, as
+�{o}u diddyst in addicio[n~]. 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 �at �e hier nomber most be mor{e} �en �e
+ne�{er}, or els eue[n~] as mych. but he may not be lasse. And �e case is
+�is, �ou schalt drawe �e ne�{er} nomber out of �e hyer, &��ou mayst not
+do �{a}t yf �e hier nomber wer{e} lasse �an �at. ffor �{o}u mayst not
+draw sex out of 2. But �{o}u mast draw 2 out of sex. And �ou maiste draw
+twene out of twene, for �ou schal leue no[gh]t of �e 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 �e first case put of subtraccio[n~], &�he says �ou schalt
+begynne in �e ryght side, &�draw �e first fig{ure} of �e ne�{er} rewe
+out of �e first fig{ure} of �e hier rewe. qwether �e hier fig{ur}e be
+mor{e} �en �e ne�{er}, or eue[n~] as mych. And �at is notified in �e
+vers when he says "Si possis." Whan �{o}u has �us ydo, do away �e hiest
+fig{ur}e & sett �{ere} �at leues of �e subtraccio[n~], lo an
+Ensampull{e}
+
+  +-----+
+  | 234 |
+  | 122 |
+  +-----+
+
+draw 2 out of 4. �an leues 2. do away 4 & write �{ere} 2, & latte �e
+ne�{er} figur{e} sto{n}de stille, &�so go for-by o�{er} figuris till
+�{o}u come to �e ende, �an hast �{o}u�do.
+
+  � Cifram si nil remanebit.
+
+  [Sidenote: Put a cipher if nothing remains. Here is an example.]
+
+��Her{e} he putt{es} �e secunde case, &�hit is �is. yf it happe �{a}t
+qwen �{o}u hast draw on ne�{er} fig{ure} out of a hier, &��{er}e leue
+no[gh]t aft{er} �e subt{ra}ccio[n~], �us [*leaf 143b] �ou schalt do.
+�{o}u schall{e} do away �e hier fig{ur}e & write �{ere} a cifer, as lo
+an Ensampull
+
+  +----+
+  | 24 |
+  | 24 |
+  +----+
+
+Take four{e} out of four{e} �an leus no[gh]t. �{er}efor{e} do away �e
+hier 4 & set �{ere} a cifer, �an take 2 out of 2, �an leues no[gh]t. do
+away �e hier 2, &�set �{ere} a cifer, and so worch whar{e} so eu{er} �is
+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 �e thryd case, �e quych is �is. yf it happe �at �e
+ne�{er} fig{ur}e be mor{e} �en �e hier fig{ur}e �at he schall{e} be draw
+out of. how schall{e} �ou do. �us �{o}u schall{e} do. �ou schall{e}
+borro .1. oute of �e next fig{ur}e �at comes aft{er} in �e same rewe,
+for �is case may neu{er} happ but yf �{ere} come figures aft{er}. �an
+�{o}u schalt sett �at on ou{er} �e hier figur{es} hed, of the quych �ou
+woldist y-draw oute �e ney�{er} fig{ur}e yf �{o}u haddyst y-my[gh]t.
+Whane �ou hase �us ydo �ou schall{e} rekene �{a}t .1. for ten. �. And
+out of �at ten �{o}u schal draw �e ney�ermost fig{ur}e, And all{e} �{a}t
+leues �ou schall{e} adde to �e figur{e} on whos hed �at .1. stode. And
+�en �{o}u schall{e} do away all{e} �at, &�sett �{ere} all{e} that arisys
+of the addicio[n~] of �e ylke 2 fig{ur}is. And yf yt [*leaf 144a] happe
+�at �e fig{ur}e of �e quych �{o}u schalt borro on be hym self but 1. If
+�{o}u schalt �at on{e} & sett it vppo[n~] �e o�{er} figur{is} hed, and
+sett in �{a}t 1. place a cifer, yf �{ere} come mony figur{es} aft{er}.
+lo an Ensampul.
+
+  +------+
+  | 2122 |
+  | 1134 |
+  +------+
+
+take 4 out of 2. it wyl not be, �erfor{e} borro on{e} of �e next
+figur{e}, �{a}t is 2. and sett �at ou{er} �e hed of �e fyrst 2. & rekene
+it for ten. and �ere �e secunde stondes write 1. for �{o}u tokest on out
+of hy{m}. �an take �e ne�{er} fig{ur}e, �at is 4, out of ten. And �en
+leues 6. cast to 6 �e fig{ur}e of �at 2 �at stode vnd{er} �e hedde of 1.
+�at was borwed & rekened for ten, and �at wylle be 8. do away �{a}t 6 &
+�at 2, & sette �{ere} 8, &�lette �e ne�{er} fig{ur}e stonde stille.
+Whanne �{o}u hast do �us, go to �e next fig{ur}e �{a}t is now bot 1. but
+first yt was 2, & �{ere}-of was borred�1. �an take out of �{a}t �e
+fig{ur}e vnd{er} hym, �{a}t is 3. hit wel not be. �er-for{e} borowe of
+the next fig{ur}e, �e quych is bot 1. Also take & sett hym ou{er} �e
+hede of �e fig{ure} �at �ou woldest haue y-draw oute of �e nether
+figure, �e quych was 3. & �ou my[gh]t not, &�rekene �{a}t borwed 1 for
+ten & sett in �e same place, of �e quych place �{o}u tokest hy{m} of,
+a�cifer, for he was bot 1. Whanne �{o}u hast �{us} ydo, take out of �at
+1. �{a}t is rekent for ten, �e ne�{er} figure of 3. And �{ere} leues�7.
+[*leaf 144b] cast �e ylke 7 to �e fig{ur}e �at had �e ylke ten vpon his
+hed, �e quych fig{ur}e was�1, &��at wol be�8. �an do away �{a}t 1 and
+�{a}t�7, &�write �{ere} 8. & �an wyrch forth in o�{er} figuris til �{o}u
+come to �e ende, &��an �{o}u hast �e 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} �e fourte case, �e quych is �is, yf it happe �at �e
+ne�{er} fig{ur}e, �e quych �{o}u schalt draw out of �e hier fig{ur}e be
+mor{e} pan �e hier figur ou{er} hym, &��e next fig{ur}e of two or of
+thre or of foure, or how mony �{ere} be by cifers, how wold �{o}u do.
+�{o}u wost wel �{o}u most nede borow, &��{o}u mayst not borow of �e
+cifers, for �ai haue no[gh]t �at �ai may lene or spar{e}. Ergo[{4}] how
+woldest �{o}u do. Certay[n~] �us most �{o}u do, �{o}u most borow on of
+�e next figure significatyf in �at rewe, for �is case may not happe, but
+yf �{ere} come figures significatyf aft{er} the cifers. Whan �{o}u hast
+borowede �{a}t 1 of the next figure significatyf, sett �{a}t on ou{er}
+�e hede of �{a}t fig{ur}e of �e quych �{o}u wold haue draw �e ne�{er}
+figure out yf �{o}u hadest my[gh]t, &�reken it for ten as �o{u} diddest
+i{n} �e o�{er} case her{e}-a-for{e}. Wha[n~] �{o}u hast �us y-do loke
+how mony cifers �{ere} wer{e} bye-twene �at figur{e} significatyf, &��e
+fig{ur}e of �e quych �{o}u woldest haue y-draw the [*leaf 145a] ne�{er}
+figure, and of eu{er}y of �e 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, �e quych is 4, &��en leues 3. do away �{a}t figur{e} of 4
+& write �{ere} 3. & sett �{a}t 1 vppon �e fig{ur}e of 2 hede, &��an take
+4 out of ten, &��an �ere leues 6. Cast 6 to the fig{ur}e of 2, �{a}t wol
+be 8. do away �at 6 & write �{er}e 8. Whan �{o}u hast �us y-do make of
+eu{er}y 0 betweyn 3 & 8 a figure of 9, &��an worch forth in goddes name.
+& yf �{o}u hast wel y-do �{o}u[{5}] schalt haue �is 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 �e Craft how �{o}u schalt know, whan �{o}u hast
+subt{ra}yd, whe�{er} �ou hast wel ydo or no. And �e Craft is �is, ryght
+as �{o}u subtrayd �e ne�{er} figures fro �e hier figures, ry[gh]t so
+adde �e same ne�{er} figures to �e hier figures. And yf �{o}u haue well
+y-wroth a-for{e} �ou schalt haue �e hier nombre �e same �{o}u haddest or
+�ou be-gan to worch. as for �is I bade �ou schulde kepe �e ne�{er}
+figures stylle. lo an [*leaf 145b] Ensampull{e} of all{e} �e 4 cases
+toged{re}. worche well{e} �is case
+
+  +--------+
+  |40003468|.
+  |20004664|
+  +--------+
+
+And yf �ou worch well{e} whan �ou hast all{e} subtrayd �e �{a}t hier
+nombr{e} her{e}, �is schall{e} be �e nombre here foloyng whan �{o}u hast
+subtrayd.
+
+  +--------+
+  |39998804|.  [Sidenote: Our author makes a slip here (3 for�1).]
+  |20004664|
+  +--------+
+
+And �ou schalt know �{us}. adde �e ne�{er} rowe of �e same nombre to �e
+hier rewe as �us, cast 4 to 4. �at wol be 8. do away �e 4 & write �{ere}
+8. by �e first case of addicio[n~]. �an cast 6 to 0 �at wol be 6. do
+away �e 0, &�write �ere 6. �an cast 6 to 8, �{a}t wel be 14. do away 8 &
+write �{ere} a fig{ur}e of 4, �at is �e digit, and write a fig{ur}e of
+1. �{a}t schall be-token ten. �{a}t is �e articul vpon �e hed of 8 next
+aft{er}, �an reken �{a}t 1. for 1. & cast it to 8. �at schal be 9. cast
+to �at 9 �e ne�{er} fig{ur}e vnd{er} �at �e quych is 4, &��at schall{e}
+be 13. do away �at 9 & sett �{er}e 3, & sett a figure of 1. �{a}t schall
+be 10 vpon �e next figur{is} hede �e quych is 9. by �e secu{n}de case
+�{a}t �{o}u hadest in addicio[n~]. �an cast 1 to 9. & �at wol be 10. do
+away �e 9. & �at 1. And write �{ere} a cifer. and write �e articull{e}
+�at is 1. betokenyng{e} 10. vpon �e hede of �e next figur{e} toward �e
+lyft side, �e quych [*leaf 146a] is 9, &�so do forth tyl �{o}u come to
+�e last�9. take �e figur{e} of �at�1. �e quych �{o}u schalt fynde ou{er}
+�e hed of 9. & sett it ou{er} �e next figures hede �at schal be�3.
+��Also do away �e 9. & set �{ere} a cifer, &��en cast �at 1 �at stondes
+vpon �e hede of 3 to �e same 3, &��{a}t schall{e} make�4, �en caste to
+�e ylke 4 the figur{e} in �e ney�{er} rewe, �e quych is�2, and �at
+schall{e} be 6. And �en schal �{o}u haue an Ensampull{e} a[gh]ey[n~],
+loke & se, &�but �{o}u haue �is same �{o}u hase myse-wro[gh]t.
+
+  +--------+
+  |60003468|
+  |20004664|
+  +--------+
+
+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 duplacio[n~], in �e quych craft �{o}u most
+haue & know 4 thing{es}. ���e first �{a}t �{o}u most know is what is
+duplacio[n~]. �e secu{n}de is how mony rewes of fig{ur}es �{o}u most
+haue to �is craft. ���e thryde is how many cases may[{6}] happe in �is
+craft. ���e fourte is what is �e p{ro}fet of �e craft. ��As for �e
+first. duplacio[n~] is a doublyng{e} of a nombre. ��As for �e secu{n}de
+�{o}u most [*leaf 146b] haue on nombre or on rewe of figures, the quych
+called nu{merus} dupland{us}. As for �e thrid �{o}u most know �at 3
+diu{er}se cases may hap in �is craft. As for �e fourte. qwat is �e
+p{ro}fet of �is craft, &��{a}t is to know what a-risy[gh]t of a nombre
+I-doublyde. ��ffor�{er}-mor{e}, �{o}u most know & take gode hede in
+quych side �{o}u schall{e} be-gyn in �is craft, or ellis �{o}u mayst
+spyl all{e} �{i} lab{er} �{er}e aboute. c{er}teyn �{o}u schalt begy[n~]
+in the lyft side in �is Craft. thenke wel ou{er} �is 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 �es verses afor{e}, as
+�{o}u may see if �{o}u take hede. As �e text of �is verse, �at is to
+say, ��Si vis duplare. �is is �e sentence. ��If �{o}u wel double a
+nombre �us �{o}u most be-gyn[n~]. Write a rewe of figures of what
+nomb{re} �ou 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 �{o}u schalt worch in �is Craft. he says, fyrst,
+whan �{o}u hast writen �e nombre �{o}u schalt be-gyn at �e first
+figur{e} in the lyft side, &�doubull{e} �at fig{ur}e, &��e nombre �at
+comes �{ere}-of �{o}u schalt write as �{o}u diddyst in addicio[n~], as
+��I�schal telle �e in �e 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 �e first case of �is craft, �e quych is �is. yf of
+duplacio[n~] of a figur{e} arise a digit. what schal �{o}u do. �us �{o}u
+schal do. do away �e fig{ur}e �at was doublede, &�sett �{ere} �e diget
+�at comes of �e duplacio[n~], as �us. 23. double 2, &��{a}t wel be 4. do
+away �e figur{e} of 2 & sett �{ere} a figur{e} of 4, &�so worch forth
+till{e} �{o}u come to �e 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 �e secunde case, �e quych is �is yf �{ere} come an articull{e}
+of �e duplacio[n~] of a fig{ur}e �{o}u schalt do ry[gh]t as �{o}u
+diddyst in addicio[n~], �at is to wete �at �{o}u schalt do away �e
+figur{e} �at is doublet & sett �{ere} a cifer, &�write �e articull{e}
+ou{er} �e next figur{is} hede, yf �{ere} be any aft{er}-warde toward �e
+lyft side as �us. 25. begyn at the lyft side, and doubull{e} 2. �at wel
+be 4. do away �at 2 & sett �ere 4. �an doubul 5. �at wel be 10. do away
+5, & sett �{ere} a 0, &�sett 1 vpon �e next figur{is} hede �e quych is
+4. & �en draw downe 1 to 4 & �at woll{e} be 5, &��en do away �{a}t 4 &
+�at 1, &�sett �{ere} 5. for �at 1 schal be rekened in �e drawyng{e}
+toged{re} for 1. wen [*leaf 147b] �ou hast ydon �ou schalt haue �is
+nomb{r}e 50. yf �{ere} come no figur{e} aft{er} �e fig{ur}e �{a}t is
+addit, of �e quych addicio[n~] comes an articull{e}, �{o}u schalt do
+away �e figur{e} �{a}t is dowblet & sett �{ere} a 0. & write �e articul
+next by in �e same rewe toward �e lyft syde as �us, 523. double 5 �at
+woll be ten. do away �e figur{e} 5 & set �{ere} a cifer, &�sett �e
+articul next aft{er} in �e same rewe toward �e lyft side, &��ou schalt
+haue �is nombre 1023. �en go forth & double �e o�{er} nombers �e quych
+is ly[gh]t y-now[gh]t 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} �e Thryd case, �e quych is �is, yf of duplacio[n~]
+of a fig{ur}e come a Composit. �{o}u schalt do away �e fig{u}re �{a}t is
+doublet & set �{ere} a digit of �e Composit, & sett �e articull{e}
+ou{er} �e next figures hede, &�aft{er} draw hym downe w{i}t{h} �e
+figur{e} ou{er} whos hede he stondes, &�make �{ere}-of an nombre as
+�{o}u hast done afore, &�yf �{ere} come no fig{ur}e aft{er} �at digit
+�at �{o}u hast y-write, �a{n} set �e articull{e} next aft{er} hym in �e
+same rewe as �us, 67: double 6 �at wel be 12, do away 6 & write �{ere}
+�e digit [*leaf 148a] of 12, �e quych is�2, and set �e articull{e} next
+aft{er} toward �e lyft side in �e same rewe, for �{ere} comes no
+figur{e} aft{er}. �an dowble �at o�{er} figur{e}, �e quych is�7, �at wel
+be 14. the quych is a Composit. �en do away 7 �at �{o}u doublet & sett
+�e �e diget of hy{m}, the quych is 4, sett �e articull{e} ou{er} �e next
+figur{es} hed, �e quych is 2, &��en draw to hym �at on, &�make on nombre
+�e quych schall{e} be 3. And �en yf �{o}u haue wel y-do �{o}u schall{e}
+haue �is nombre of �e duplacio[n~], 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} �e fyrst fig{ur}e in �e ry[gh]t side be such
+a merke as is her{e} made, ^w, �{o}u schall{e} fyrst doubull{e} �e
+figur{e}, the quych stondes vnd{er} �{a}t merke, &��en �ou schalt doubul
+�at merke �e quych stond{es} for haluendel on. for too haluedels makes
+on, & so �{a}t wol be on. cast �{a}t on to �at duplacio[n~] of �e
+figur{e} ou{er} whos hed stode �at merke, &�write it in �e same place
+�{ere} �at �e figur{e} �e quych was doublet stode, as �us 23^w. double
+3, �at wol be 6; doubul �at halue on, &��at wol be on. cast on to 6,
+�{a}t wel be 7. do away 6 & �at 1, &�sett �{ere} 7. �an hase �ou do. as
+for �at figur{e}, �an go [*leaf 148b] to �e o�{er} fig{ure} & worch
+forth. &��{o}u schall neu{er} haue such a merk but ou{er} �e hed of �e
+furst figure in �e ryght side. And [gh]et it schal not happe but yf it
+were y-halued a-for{e}, �us �{o}u schalt vnd{er}stonde �e 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 �is Chapter is ta[gh]t �e Craft of mediaciou[n~], in �e quych craft
+�{o}u most know 4 thynges. ffurst what is mediacio[n~]. the secunde how
+mony rewes of figur{es} �{o}u most haue in �e wyrchyng{e} of �is craft.
+�e thryde how mony diu{er}se cases may happ in �is craft.[{8}] [[the .4.
+what is �e p{ro}fet of �is craft.]] ��As for �e furst, �{o}u schalt
+vndurstonde �at mediacio[n~] is a takyng out of halfe a nomber out of a
+holle nomber, [*leaf 149a] as yf �{o}u wolde take 3 out of 6. ��As for
+�e secunde, �{o}u schalt know �{a}t �{o}u most haue on{e} rewe of
+figures, &�no moo, as �{o}u hayst in �e craft of duplacio[n~]. ��As for
+the thryd, �ou most vnd{er}stonde �at 5 cases may happe in �is craft.
+��As for �e fourte, �ou schall{e} know �at the p{ro}fet of �is craft is
+when �{o}u hast take away �e haluendel of a nomb{re} to telle qwat
+�er{e} schall{e} leue. ��Incipe sic, &c. The sentence of �is verse is
+�is. yf �{o}u wold medye, �at is to say, take halfe out of �e holle, or
+halfe out of halfe, �ou most begynne �{us}. Write on{e} rewe of
+figur{es} of what nombre �ou wolte, as �{o}u dyddyst be-for{e} in �e
+Craft of duplacio[n~]. 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 �{o}u hast write a rewe of figures, �{o}u schalt
+take hede whe�{er} �e first figur{e} be eue[n~] or odde in nombre, &
+vnd{er}stonde �{a}t he spekes of �e first figure in �e ry[gh]t side. And
+i{n} the ryght side �{o}u schall{e} begynne in �is 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 �is craft, �e quych is �is, yf �e first
+figur{e} be euen. �ou schal take away fro �e figur{e} euen halfe, &�do
+away �at fig{ur}e and set �{ere} �at leues ou{er}, as �us, 4. take
+[*leaf 149b] halfe out of 4, &��an �{ere} leues 2. do away 4 & sett
+�{ere} 2. �is is lyght y-now[gh]t. 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 �e secunde case of �is craft, the quych is �is. yf �e first
+figur{e} betoken{e} a nombre �at is odde, the quych odde schal not be
+mediete, �en �{o}u schalt medye �at nombre �at leues, when the odde of
+�e same nomb{re} is take away, &�write �at �{a}t leues as �{o}u diddest
+in �e first case of �is craft. Wha[n~] �{o}u hayst write �at. for �{a}t
+�at leues, write such a merke as is her{e} ^w vpon his hede, �e quych
+merke schal betoke[n~] halfe of �e odde �at was take away. lo an
+Ensampull. 245. the first figur{e} her{e} is betokenyng{e} odde nombre,
+�e quych is 5, for 5 is odde; �{er}e-for{e} do away �at �{a}t is odde,
+�e quych is 1. �en leues 4. �en medye 4 & �en leues 2. do away 4. &
+sette �{ere} 2, &�make such a merke ^w upon his hede, �at is to say
+ou{er} his hede of 2 as �us. 242.^w And �en worch forth in �e o�{er}
+figures tyll �{o}u come to �e ende. by �e furst case as �{o}u schalt
+vnd{er}stonde �at �{o}u schalt [*leaf 150a] neu{er} make such a merk but
+ou{er} �e first fig{ur}e hed in �e ri[gh]t side. Whe�{er} �e other
+fig{ur}es �at comy[n~] aft{er} hym be eue[n~] 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 �e thryde case, �e quych yf the first figur{e} be a figur{e}
+of�1. �{o}u schalt do away �at 1 & set �{ere} a cifer, &�a merke ou{er}
+�e cifer as �us, 241. do away�1, &�sett �{ere} a cifer w{i}t{h} a merke
+ou{er} his hede, &��en hast �{o}u ydo for �at 0. as �us 0^w �en worch
+forth in �e o�er fig{ur}ys till �{o}u come to �e 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} �e fourte case, �e quych is �is. yf it happe[n~]
+the secunde figur{e} betoken odde nombre, �ou schal do away on of �at
+odde nombre, �e quych is significatiue by �{a}t figure 1. �e quych 1
+schall be rekende for 10. Whan �{o}u hast take away �{a}t 1 out of �e
+nombre �{a}t is signifiede by �at figur{e}, �{o}u schalt medie �{a}t �at
+leues ou{er}, &�do away �at figur{e} �at is medied, &�sette in his styde
+halfe of �{a}t nombre. ��Whan �{o}u hase so done, �{o}u schalt write
+[*leaf 150b] a figure of 5 ou{er} �e next figur{es} hede by-for{e}
+toward �e ry[gh]t side, for �at 1, �e quych made odd nombr{e}, schall
+stonde for ten, &�5 is halfe of 10; so �{o}u most write 5 for his
+haluendell{e}. lo an Ensampull{e}, 4678. begy[n~] in �e ry[gh]t side as
+�{o}u most nedes. medie 8. �en �{o}u schalt leue 4. do away �at 8 &
+sette �{ere} 4. �en out of 7. take away 1. �e quych makes odde, &�sett
+5. vpon �e next figur{es} hede afor{e} toward �e ry[gh]t side, �e quych
+is now 4. but afor{e} it was 8. for �at 1 schal be rekenet for 10, of �e
+quych 10, 5 is halfe, as �ou knowest wel. Whan �{o}u hast �us ydo, medye
+�{a}t �e quych leues aft{er} �e takying{e} away of �at �at is odde, �e
+quych leuyng{e} schall{e} be 3; do away 6 & sette �{er}e 3, &��ou schalt
+haue such a nombre
+
+     5
+  4634.
+
+aft{er} go forth to �e next fig{ur}e, &�medy �at, & worch forth, for it
+is ly[gh]t ynov[gh]t to �e c{er}tay[n~].
+
+                      � 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} �e 5 case, �e quych is [*leaf 151a] �is: yf �e
+secunde figur{e} be of 1, as �is is here 12, �ou schalt do away �at 1 &
+sett �{ere} a cifer. & sett 5 ou{er} �e next fig{ur}e hede afor{e}
+toward �e ri[gh]t side, as �ou diddyst afor{e}; & �at 5 schal be haldel
+of �at 1, �e quych 1 is rekent for 10. lo an Ensampull{e}, 214. medye 4.
+�{a}t schall{e} be 2. do away 4 & sett �{ere} 2. �e{n} go forth to �e
+next figur{e}. �e quych is bot 1. do away �at 1. & sett �{ere} a cifer.
+& set 5 vpon �e figur{es} hed afor{e}, �e quych is nowe 2, &��en �ou
+schalt haue �is no{m}b{re}
+
+    5
+  202,
+
+�en worch forth to �e nex fig{ur}e. And also it is no mayst{er}y yf
+�{ere} come no figur{e} after �at on is medyet, �{o}u schalt write no 0.
+ne now[gh]t ellis, but set 5 ou{er} �e next fig{ur}e afor{e} toward �e
+ry[gh]t, as �us 14. medie 4 then leues 2, do away 4 & sett �{ere} 2. �en
+medie 1. �e q{ui}ch is rekende for ten, �e halue{n}del �{ere}-of wel be
+5. sett �{a}t 5 vpon �e hede of �{a}t figur{e}, �e quych is now 2, &�do
+away �{a}t 1, &��ou schalt haue �is nombre yf �{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 �e how �ou schalt know whe�{er} �ou hase wel ydo or
+no. doubul [*leaf 151b] �e nombre �e quych �{o}u hase mediet, and yf
+�{o}u haue wel y-medyt after �e dupleacio[n~], �ou schalt haue �e same
+nombre �at �{o}u haddyst in �e tabull{e} or �{o}u began to medye, as
+�us. ��The furst ensampull{e} was �is. 4. �e quych I-mediet was laft�2,
+�e whych 2 was write in �e place �{a}t 4 was write afor{e}. Now
+doubull{e} �at 2, &��{o}u schal haue 4, as �{o}u hadyst afor{e}. �e
+secunde Ensampull{e} was �is, 245. When �{o}u haddyst mediet all{e} �is
+nomb{re}, yf �ou haue wel ydo �ou schalt haue of �{a}t mediacio[n~] �is
+nombre, 122^w. Now doubull{e} �is nombre, &�begyn in �e lyft side;
+doubull{e} 1, �at schal be 2. do away �at 1 & sett �{ere} 2. �en
+doubull{e} �{a}t o�{er} 2 & sett �{ere} 4, �en doubull{e} �at o�{er} 2,
+&��at wel be 4. �e{n} doubul �at merke �at stondes for halue on. & �at
+schall{e} be 1. Cast �at on to 4, &�it schall{e} be 5. do away �at 2 &
+�at merke, &�sette �{ere} 5, &��en �{o}u schal haue �is nombre 245. &
+�is wos �e same nombur �{a}t �{o}u haddyst or �{o}u began to medye, as
+�{o}u mayst se yf �ou take hede. The nombre �e quych �ou haddist for an
+Ensampul in �e 3 case of mediacio[n~] to be mediet was �is 241. whan
+�{o}u haddist medied all{e} �is nombur truly [*leaf 152a] by eu{er}y
+figur{e}, �ou schall haue be �{a}t mediacio[n~] �is nombur 120^w. Now
+dowbul �is nomb{ur}, &�begyn in �e lyft side, as I tolde �e in �e Craft
+of duplacio[n~]. �us doubull{e} �e fig{ur}e of 1, �at wel be 2. do away
+�at 1 & sett �{ere} 2, �en doubul �e next figur{e} afore, the quych is
+2, &��at wel be 4; do away 2 & set �{ere} 4. �en doubul �e cifer, & �at
+wel be no[gh]t, for a 0 is no[gh]t. And twyes no[gh]t is but no[gh]t.
+�{ere}for{e} doubul the merke aboue �e cifers hede, �e quych betokenes
+�e halue{n}del of 1, &��at schal be 1. do away �e cifer & �e merke,
+&�sett �{ere} 1, &��en �{o}u schalt haue �is nombur 241. And �is same
+nombur �{o}u haddyst afore or �{o}u began to medy, & yf �{o}u take gode
+hede. ��The next ensampul �at had in �e 4 case of mediacio[n~] was �is
+4678. Whan �{o}u hast truly ymedit all{e} �is nombur fro �e begynnyng{e}
+to �e endyng{e}, �{o}u schalt haue of �e mediacio[n~] �is nombur
+
+    5
+ 2334.
+
+Now doubul this nombur & begyn in �e lyft side, &�doubull{e} 2 �at schal
+be 4. do away 2 and sette �ere 4; �en doubul{e} 3, �{a}t wol be 6; do
+away 3 & sett �{ere} 6, �en doubul �at o�{er} 3, &��at wel be 6; do away
+3 & set �{ere} [*leaf 152b] 6, �en doubul �e 4, �at welle be 8; �en
+doubul 5. �e quych stondes ou{er} �e hed of 4, &��at wol be 10; cast 10
+to 8, &��{a}t schal be 18; do away 4 & �at 5, &�sett �{ere} 8, &�sett
+that 1, �e quych is an articul of �e Composit �e quych is 18, ou{er} �e
+next figur{es} hed toward �e lyft side, �e quych is 6. drav �{a}t 1 to
+6, �e quych 1 in �e dravyng schal be rekente bot for 1, &��{a}t 1 &
+�{a}t 6 togedur wel be 7. do away �at 6 & �at 1. the quych stondes
+ou{er} his hede, &�sett ther 7, & �en �ou schalt haue �is nombur 4678.
+And �is same nombur �{o}u hadyst or �{o}u began to medye, as �{o}u mayst
+see in �e secunde Ensampul �at �ou had in �e 4 case of mediacio[n~], �at
+was �is: when �{o}u had mediet truly all{e} the nombur, a�p{ri}ncipio
+usque ad fine{m}. �{o}u schalt haue of �at mediacio[n~] �is nombur
+
+    5
+  102.
+
+Now doubul 1. �at wel be 2. do away 1 & sett �{ere} 2. �en doubul 0.
+�{a}t will be no[gh]t. �{ere}for{e} take �e 5, �e quych stondes ou{er}
+�e next figur{es} hed, &�doubul it, &��at wol be 10. do away �e 0 �at
+stondes betwene �e two fig{u}r{i}s, &�sette �{ere} in his stid 1, for
+�{a}t 1 now schal stonde in �e secunde place, wher{e} he schal betoken
+10; �en doubul 2, �at wol be 4. do away 2 & sett �ere 4. & [*leaf 153a]
+�ou schal haue �us nombur 214. �is is �e same nu{m}bur �at �{o}u hadyst
+or �{o}u began to medye, as �{o}u may see. And so do eu{er} mor{e}, yf
+�{o}u wil knowe whe�{er} �ou hase wel ymedyt or no. �.�doubull{e} �e
+nu{m}bur �at comes aft{er} �e mediaciou[n~], &��{o}u schal haue �e same
+nombur �{a}t �{o}u hadyst or �{o}u began to medye, yf �{o}u haue welle
+ydo. or els doute �e no[gh]t, but yf �{o}u haue �e same, �{o}u hase
+faylide in �{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 �e Chapt{r}e of m{u}ltiplicatio[n~], in �e quych �ou
+most know 4 thynges. ��Ffirst, qwat is m{u}ltiplicacio[n~]. The secunde,
+how mony cases may hap in multiplicacio[n~]. The thryde, how mony rewes
+of figur{es} �{ere} most be. ��The 4. what is �e p{ro}fet of �is craft.
+��As for �e first, �{o}u schal vnd{er}stonde �at m{u}ltiplicacio[n~] is
+a bryngyng{e} to-ged{er} of 2 thyng{es} in on nombur, �e quych on nombur
+{con}tynes so mony tymes on, howe [*leaf 153b] mony tymes �{ere} ben
+vnytees in �e nowmb{re} of �at 2, as twyes 4 is 8. now her{e} ben �e 2
+nomb{er}s, of �e quych too nowmbr{e}s on is betokened be an adu{er}be,
+�e quych is �e worde twyes, &��is worde thryes, &��is worde four{e}
+sythes,[{9}] [[& �is wordes fyue sithe & sex sythes.]] &�so furth of
+such other lyke wordes. ��And tweyn nombres schal be tokenyde be a
+nowne, as �is worde four{e} showys �es twey[n~] nombres y-broth in-to on
+hole nombur, �at is 8, for twyes 4 is 8, as �{o}u wost wel. ��And �es
+nomb{re} 8 conteynes as oft tymes 4 as �{ere} ben vnites in �{a}t other
+nomb{re}, �e quych is 2, for in 2 ben 2 vnites, &�so oft tymes 4 ben in
+8, as �{o}u wottys wel. ��ffor �e secu{n}de, �{o}u most know �at �{o}u
+most haue too rewes of figures. ��As for �e thryde, �{o}u most know
+�{a}t 8 man{er} of diu{er}se case may happe in �is craft. The p{ro}fet
+of �is Craft is to telle when a nomb{re} is m{u}ltiplyed be a no�{er},
+qwat co{m}mys �{ere}�of. � fforthermor{e}, as to �e sentence of our{e}
+verse, yf �{o}u wel m{u}ltiply a nombur be a-no�{er} nomb{ur}, �ou
+schalt write [*leaf 154a] a rewe of figures of what nomb{ur}s so eu{er}
+�{o}u welt, &��at schal be called Num{erus} m{u}ltiplicand{us}, Anglice,
+�e nomb{ur} the quych to be m{u}ltiplied. �en �{o}u schalt write
+a-nother rewe of figur{e}s, by �e quych �{o}u schalt m{u}ltiplie the
+nombre �at is to be m{u}ltiplied, of �e quych nomb{ur} �e furst fig{ur}e
+schal be write vnd{er} �e last figur{e} of �e nomb{ur}, �e quych is to
+be m{u}ltiplied. And so write forthe toward �e lyft side, as her{e} you
+may�se,
+
+  +----------+
+  |    67324 |
+  | 1234     |
+  +----------+
+
+And �is on{e} nomb{ur} schall{e} be called nu{meru}s m{u}ltiplicans.
+An{gli}ce, �e nomb{ur} m{u}ltipliyng{e}, for he schall{e} m{u}ltiply �e
+hyer nounb{ur}, as �us on{e} tyme 6. And so forth, as I schal telle the
+aft{er}warde. And �ou schal begyn in �e lyft side. ��ffor-�{ere}-more
+�ou schalt vndurstonde �at �{ere} is two man{ur}s of
+m{u}ltiplicacio[n~]; one ys of �e wyrchyng{e} of �e boke only in �e
+mynde of a mon. fyrst he teches of �e fyrst man{er} of duplacio[n~], �e
+quych is be wyrchyng{e} of tabuls. Aft{er}warde he wol teche on �e
+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 �{o}u schalt fynde �e nounb{r}e �at
+comes by �e m{u}ltiplicacio[n~] of a digit be ano�{er}. loke how mony
+[vny]tes ben. bytwene �e mor{e} digit and 10. And reken ten for on
+vnite. And so oft do away �e lasse nounbre out of his owne decuple, �at
+is to say, fro �at nounb{r}e �at is ten tymes so mych is �e nounb{re}
+�{a}t comes of �e m{u}ltiplicacio[n~]. As yf �{o}u wol m{u}ltiply 2 be
+4. loke how mony vnitees ben by-twene �e quych is �e mor{e} nounb{re},
+&�be-twene ten. C{er}ten �{ere} wel be vj vnitees by-twene 4 & ten. yf
+�{o}u reken �{ere} w{i}t{h} �e ten �e vnite, as �ou may se. so mony
+tymes take 2. out of his decuple, �e quych is 20. for 20 is �e decuple
+of 2, 10 is �e decuple of 1, 30 is �e decuple of 3, 40 is �e decuple of
+4, And �e o�{er} digetes til �{o}u come to ten; & whan �{o}u hast y-take
+so mony tymes 2 out of twenty, �e quych is sex tymes, �{o}u schal leue 8
+as �{o}u wost wel, for 6 times 2 is twelue. take [1]2 out of twenty,
+&��{ere} schal leue 8. bot yf bothe �e digett{es} [*leaf 155a] ben
+y-lyech mych as her{e}. 222 or too tymes twenty, �en it is no fors quych
+of hem tweyn �{o}u take out of here decuple. als mony tymes as �{a}t is
+fro 10. but neu{er}-�e-lesse, yf �{o}u haue hast to worch, �{o}u schalt
+haue her{e} a tabul of figures, wher{e}-by �{o}u schalt se a-non[n~]
+ryght what is �e nounbre �{a}t comes of �e multiplicacio[n~] of 2
+digittes. �us �{o}u schalt worch in �is 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 �e fig{ur}e, �e quych schall{e} be m{u}ltiplied, be euen{e} as mych
+as �e diget be, �e quych �at o�{er} figur{e} schal be m{u}ltiplied,
+as two tymes tway[n~], or thre tymes 3. or sych other. loke qwer{e} �at
+fig{ur}e sittes in �e lyft side of �e t{ri}angle, &�loke qwer{e} �e
+diget sittes in �e ne�{er} most rewe of �e triangle. & go fro hym
+vpwarde in �e same rewe, �e quych rewe gose vpwarde til �{o}u come
+agaynes �e o�{er} digette �at sittes in �e lyft side of �e t{ri}angle.
+And �at nounbre, �e quych �ou [*leaf 155b] fyn[*]des �{ere} is �e
+nounbre �at comes of the m{u}ltiplicacio[n~] of �e 2 digittes, as yf �ou
+wold wete qwat is 2 tymes 2. loke quer{e} sittes 2 in �e lyft side i{n}
+�e first rewe, he sittes next 1 in �e lyft side al on hye, as �{o}u may
+se; �e[{n}] loke qwer{e} sittes 2 in �e lowyst rewe of �e t{ri}angle,
+&�go fro hym vpwarde in �e same rewe tyll{e} �ou come a-[gh]enenes 2 in
+�e hyer place, &��er �ou schalt fynd ywrite 4, & �at is �e nounb{r}e �at
+comes of �e multiplicacio[n~] of two tymes tweyn is 4, as �ow wotest
+well{e}. yf �e diget. the quych is m{u}ltiplied, be mor{e} �an �e
+o�{er}, �ou schalt loke qwer{e} �e mor{e} diget sittes in �e lowest rewe
+of �e t{ri}angle, &�go vpwarde in �e same rewe tyl[{10}] �{o}u come
+a-nendes �e lasse diget in the lyft side. And �{ere} �{o}u schalt fynde
+�e no{m}b{r}e �at comes of �e m{u}ltiplicacio[n~]; but �{o}u schalt
+vnd{er}stonde �at �is rewle, �e quych is in �is v{er}se. ��In digitu{m}
+cures, &c., no�{er} �is t{ri}angle schall{e} not s{er}ue, bot to fynde
+�e nounbres �{a}t comes of the m{u}ltiplicacio[n~] �at comes of 2
+articuls or {com}posites, �e nedes no craft but yf �ou wolt m{u}ltiply
+in �i mynde. And [*leaf 156a] �ere-to �ou schalt haue a craft
+aft{er}warde, for �ou schall wyrch w{i}t{h} digettes in �e tables, as
+�ou 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 �{o}u schalt wyrch in �is craft. �ou schalt
+m{ul}tiplye �e last figur{e} of �e nombre, and quen �{o}u hast so ydo
+�ou schalt draw all{e} �e figures of �e ne�{er} nounbre mor{e} taward �e
+ry[gh]t side, so qwe{n} �{o}u hast m{u}ltiplyed �e last figur{e} of �e
+heyer nounbre by all{e} �e ne�{er} figures. And sette �e nounbir �at
+comes �er-of ou{er} �e last figur{e} of �e ne�{er} nounb{re}, &��en �ou
+schalt sette al �e o�{er} fig{ur}es of �e ne�{er} nounb{re} mor{e}
+ner{e} to �e ry[gh]t side. ��And whan �ou hast m{u}ltiplied �at figur{e}
+�at schal be m{u}ltiplied �e next aft{er} hym by al �e ne�{er} figures.
+And worch as �ou dyddyst afor{e} til [*leaf 156b] �ou come to �e ende.
+And �ou schalt vnd{er}stonde �at eu{er}y figur{e} of �e hier nounb{re}
+schal be m{u}ltiplied be all{e} �e figur{e}s of the ne�{er} nounbre,
+yf �e hier nounb{re} be any figur{e} �en on{e}. lo an Ensampul her{e}
+folowyng{e}.
+
+  +------+
+  |  2465|.
+  |232   |
+  +------+
+
+�ou schalt begyne to m{u}ltiplye in �e lyft side. M{u}ltiply 2 be 2, and
+twyes 2 is 4. set 4 ou{er} �e hed of �{a}t 2, �en m{u}ltiplie �e same
+hier 2 by 3 of �e nether nounbre, as thryes 2 �at schal be 6. set 6
+ou{er} �e hed of 3, �an m{u}ltiplie �e same hier 2 by �at 2 �e quych
+stondes vnd{er} hym, �{a}t wol be 4; do away �e hier 2 & sette �{ere} 4.
+��Now �{o}u most antery �e nether nounbre, �at is to say, �{o}u most
+sett �e ne�{er} nounbre more towarde �e ry[gh]t side, as �us. Take �e
+ne�{er} 2 toward �e ry[gh]t side, &�sette it eue[n~] vnd{er} �e 4 of �e
+hyer nounb{r}e, & ant{er}y all{e} �e figures �at comes aft{er} �at 2, as
+�us; sette 2 vnd{er} �e 4. �en sett �e figur{e} of 3 �{ere} �at �e
+figure of 2 stode, �e quych is now vndur �{a}t 4 in �e hier nounbre; �en
+sett �e o�er figur{e} of 2, �e quych is �e last fig{ur}e toward �e lyft
+side of �e ne�{er} nomb{er} �{ere} �e figur{e} of 3 stode. �en �{o}u
+schalt haue such a nombre.
+
+  +------+
+  |464465|
+  | 232  |
+  +------+
+
+[*leaf 157a] � Now m{u}ltiply 4, �e quych comes next aft{er} 6, by �e
+last 2 of �e ne�{er} nounbur toward �e lyft side. as 2 tymes 4, �at wel
+be 8. sette �at 8 ou{er} �e figure the quych stondes ou{er} �e hede of
+�at 2, �e quych is �e last figur{e} of �e ne�{er} nounbre; �an multiplie
+�at same 4 by 3, �at comes in �e ne�{er} rewe, �at wol be 12. sette �e
+digit of �e composyt ou{er} �e figure �e quych stondes ou{er} �e hed of
+�at 3, &�sette �e articule of �is co{m}posit ou{er} al �e figures �at
+stondes ou{er} �e ne�{er} 2 hede. �en m{u}ltiplie �e same 4 by �e 2 in
+�e ry[gh]t side in �e ne�{er} nounbur, �at wol be 8. do away 4. & sette
+�{ere} 8. Eu{er} mor{e} qwen �{o}u m{u}ltiplies �e hier figur{e} by �at
+figur{e} �e quych stondes vnd{er} hym, �ou schalt do away �at hier
+figur{e}, & sett �er �at nounbre �e quych comes of m{u}ltiplicacio[n~]
+of ylke digittes. Whan �ou hast done as I haue byde �e, �{o}u schalt
+haue suych an ord{er} of figur{e} as is her{e},
+
+  +--------+
+  | 1      |.
+  | 82     |
+  |4648[65]|
+  | 232    |
+  +--------+
+
+�en take and ant{er}y �i ne�{er} figures. And sett �e fyrst fig{ur}e of
+�e ne�{er} figures[{11}] vndre be figur{e} of 6. ��And draw al �e o�{er}
+figures of �e same rewe to hym-warde, [*leaf 157b] as �{o}u diddyst
+afore. �en m{u}ltiplye 6 be 2, &�sett �at �e quych comes ou{er}
+�{ere}-of ou{er} al �e o�{er} figures hedes �at stondes ou{er} �at 2.
+�en m{u}ltiply 6 be 3, &�sett all{e} �at comes �{ere}-of vpon all{e} �e
+figur{e}s hedes �at standes ou{er} �at 3; �a{n} m{u}ltiplye 6 be 2, �e
+quych stondes vnd{er} �at 6, �en do away 6 & write �{ere} �e digitt of
+�e composit �at schal come �{ere}of, &�sette �e articull ou{er} all{e}
+�e figures �at stondes ou{er} �e hede of �at 3 as her{e},
+
+  +------+
+  | 11   |
+  | 121  |
+  | 828  |
+  |464825|
+  |  232 |
+  +------+
+
+�en ant{er}y �i figures as �ou diddyst afor{e}, and m{u}ltipli 5 be 2,
+�at wol be 10; sett �e 0 ou{er} all �e figures �{a}t stonden ou{er} �at
+2, &�sett �{a}t 1. ou{er} the next figures hedes, all{e} on hye towarde
+�e lyft side. �en m{u}ltiplye 5 be 3. �at wol be 15, write 5 ou{er} �e
+figures hedes �at stonden ou{er} �{a}t 3, & sett �at 1 ou{er} �e next
+figur{e}s hedes toward �e lyft side. �en m{u}ltiplye 5 be 2, �at wol be
+10. do away �at 5 & sett �{ere} a 0, & sett �at 1 ou{er} �e figures
+hedes �at stonden ou{er} 3. And �en �ou schalt haue such a nounbre as
+here stondes aftur.[*leaf 158a]
+
+  +------+
+  |  11  |
+  | 1101 |
+  | 1215 |
+  | 82820|
+  |4648  |
+  |   232|
+  +------+
+
+� Now draw all{e} �ese figures downe toged{er} as �us, 6.8.1. & 1 draw
+to-gedur; �at wolle be 16, do away all{e} �ese figures saue 6. lat hym
+stonde, for �ow �{o}u take hym away �ou most write �er �e same a[gh]ene.
+�{ere}for{e} late hym stonde, &�sett 1 ou{er} �e figur{e} hede of 4
+toward �e lyft side; �en draw on to 4, �at woll{e} be 5. do away �at 4 &
+�at 1, &�sette �{ere} 5. �en draw 4221 & 1 toged{ur}, �at wol be 10. do
+away all{e} �at, &�write �ere �at 4 & �at 0, &�sett �at 1 ou{er} �e next
+figur{es} hede toward �e lyft side, �e quych is 6. �en draw �at 6 & �at
+1 togedur, &��at wolle be 7; do away 6 & sett �{ere} 7, �en draw 8810 &
+1, &��at wel be 18; do away all{e} �e figures �{a}t stondes ou{er} �e
+hede of �at 8, &�lette 8 stonde stil, &�write �at 1 ou{er} �e next
+fig{u}r{is} hede, �e quych is a 0. �en do away �at 0, &�sett �{ere} 1,
+�e quych stondes ou{er} �e 0. hede. �en draw 2, 5, &�1 toged{ur}, �at
+woll{e} be 8. �en do away all{e} �at, &�write �{ere}�8. ��And �en �ou
+schalt haue �is 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 �e fyrst case of �is craft, �e quych is �is: yf
+�{ere} come an articulle of �e m{u}ltiplicacio[n~] ysette befor{e} the
+articull{e} in �e lyft side as �us
+
+  +---+
+  | 51|.
+  |23 |
+  +---+
+
+multiplye 5 by 2, �at wol be 10; sette ou{er} �e hede of �at 2 a 0,
+&�sett �at on, �at is �e articul, in �e lyft side, �at is next hym, �en
+�{o}u schalt haue �is nounbre
+
+  +----+
+  |1051|.
+  | 23 |
+  +----+
+
+� And �en worch forth as �ou diddist afore. And �{o}u schalt
+vnd{er}stonde �at �{o}u schalt write no 0. but whan �at place where �ou
+schal write �at 0 has no figure afore hy{m} no�{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 �e secunde case, �e quych is �is: yf hit happe �at �{ere}
+come a composyt, �ou schalt write �e digitte ou{er} �e hede of �e
+ne�{er} figur{e} by �e quych �{o}u multipliest �e hier figure; and sett
+�e articull{e} next hym toward �e lyft side, as �ou diddyst afore, as
+�{us}
+
+  +---+
+  | 83|.
+  |83 |
+  +---+
+
+Multiply 8 by 8, �at wol be 64. Write �e 4 ou{er} 8, �at is to say,
+ou{er} �e hede of �e ne�{er} 8; & set 6, �e quych [*leaf 159a] is an
+articul, next aft{er}. And �en �ou schalt haue such a nounb{r}e as is
+her{e},
+
+  +-----------+
+  | 6483[{13}]|,
+  |  83       |
+  +-----------+
+
+And �en 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 �e thryde case, �e quych is �is: yf hit happe �at of �i
+m{u}ltiplicaciou[n~] come a digit, �{o}u schalt write �e digit ou{er} �e
+hede of �e ne�{er} figur{e}, by the quych �ou m{u}ltipliest �e hier{e}
+figur{e}, for �is 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 �e 4 case, �e quych is: yf hit be happe �at �e ne�{er}
+figur{e} schal multiplye �at figur{e}, �e quych stondes ou{er} �at
+figures hede, �ou schal do away �e hier figur{e} & sett �{er}e �at
+�{a}t comys of �{a}t m{u}ltiplicacio[n~]. As yf �{er}e come of �at
+m{u}ltiplicacio[n~] an articuls �ou schalt write �ere �e hier figur{e}
+stode a 0. ��And write �e articuls in �e lyft side, yf �at hit be a
+digit write �{er}e a digit. yf �at h{i}t be a composit, write �e digit
+of �e composit. And �e articul in �e lyft side. al �is is ly[gh]t
+y-now[gh]t, �{er}e-for{e} �er 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 �e 5 case, �e quych is �is: yf [*leaf 159b] �e ne�{er}
+figur{e} schul m{u}ltiplie �e hier, and �at hier figur{e} is not recte
+ou{er} his hede. And �at ne�{er} figur{e} hase o�{er} figures, or on
+figure ou{er} his hede by m{u}ltiplicacio[n~], �at hase be afor{e}, �ou
+schalt write �at nounbre, �e quych comes of �at, ou{er} all{e} �e ylke
+figures hedes, as �us here:
+
+  +-----+
+  |  236|
+  |234  |
+  +-----+
+
+Multiply 2 by 2, �at wol be 4; set 4 ou{er} �e hede of �at 2. �en[{14}]
+m{u}ltiplies �e hier 2 by �e ne�{er} 3, �at wol be 6. set ou{er} his
+hede 6, multiplie �e hier 2 by �e ne�{er} 4, �at wol be 8. do away �e
+hier 2, �e quych stondes ou{er} �e hede of �e figur{e} of�4, and set
+�{er}e 8. And �ou schalt haue �is nounb{re} here
+
+  +-------+
+  | 46836 |
+  | 234   |
+  +-------+
+
+And antery �i figur{e}s, �at is to say, set �i ne�{er} 4 vnd{er} �e hier
+3, and set �i 2 other figures ner{e} hym, so �at �e ne�{er} 2 stonde
+vnd{ur} �e hier 6, �e quych 6 stondes in �e lyft side. And �at 3 �at
+stondes vndur 8, as �us aftur [gh]e may�se,
+
+  +-------+
+  | 46836 |
+  |  234  |
+  +-------+
+
+Now worch forthermor{e}, And m{u}ltiplye �at hier 3 by 2, �at wol be 6,
+set �{a}t 6 �e quych stondes ou{er} �e hede of �at 2, And �en worch as I
+ta[gh]t �e afore.
+
+  [*leaf 160a]
+
+  � Si sup{ra}posita cifra debet m{u}ltiplicar{e}
+     Prorsus ea{m} deles & ibi scribi cifra debet.
+
+  [Sidenote: The sixth case of the craft.]
+
+��Her{e} is �e 6 case, �e quych is �is: yf hit happe �at �e figur{e} by
+�e quych �ou schal m{u}ltiplye �e hier figur{e}, �e quych stondes ryght
+ou{er} hym by a 0, �ou schalt do away �at figur{e}, �e quych ou{er} �at
+cifre hede. ��And write �{ere} �at nounbre �at comes of �e
+m{u}ltiplicacio[n~] as �us, 23. do away 2 and sett �{er}e a 0. vn{de}
+v{er}sus.
+
+  � Si cifra m{u}ltiplicat alia{m} posita{m} sup{er} ip{s}am
+    Sitq{ue} locus sup{ra} vacu{us} sup{er} hanc cifra{m} fiet.
+
+  [Sidenote: The seventh case of the craft.]
+
+��Her{e} is �e 7 case, �e quych is �is: yf a 0 schal m{u}ltiply a
+figur{e}, �e quych stondes not recte ou{er} hym, And ou{er} �at 0 stonde
+no thyng, �ou schalt write ou{er} �at 0 ano�{er} 0 as �us:
+
+  +----+
+  |  24|
+  |03  |
+  +----+
+
+multiplye 2 be a 0, it wol be nothyng{e}. write �ere a 0 ou{er} �e hede
+of �e ne�{er} 0, And �en worch forth til �ou come to �e ende.
+
+  � Si sup{ra}[{15}] fuerit cifra sem{per} e{st} p{re}t{er}eunda.
+
+  [Sidenote: The eighth case of the craft.]
+
+��Her{e} is �e 8 case, �e quych is �is: yf �{ere} be a 0 or mony cifers
+in �e hier rewe, �{o}u schalt not m{u}ltiplie hem, bot let hem stonde.
+And antery �e figures bene�e to �e next figur{e} sygnificatyf as �us:
+
+  +-----+
+  |00032|.
+  |22   |
+  +-----+
+
+Ou{er}-lepe all{e} �ese cifers & sett �at [*leaf 160b] ne�{er} 2 �at
+stondes toward �e ryght side, and sett hym vnd{ur} �e 3, and sett �e
+o�{er} nether 2 nere hym, so �at he stonde vnd{ur} �e thrydde 0, �e
+quych stondes next 3. And �an worch. vnd{e} v{er}sus.
+
+  � Si dubites, an sit b{e}n{e} m{u}ltiplicac{i}o facta,
+    Diuide totalem nu{mer}u{m} p{er} multiplicante{m}.
+
+  [Sidenote: How to prove the multiplication.]
+
+��Her{e} he teches how �ou schalt know whe�{er} �ou hase wel I-do or no.
+And he says �at �ou schalt deuide all{e} �e nounb{r}e �at comes of �e
+m{u}ltiplicacio[n~] by �e ne�{er} figures. And �en �ou schalt haue �e
+same nounbur �at �{o}u hadyst in �e begynnyng{e}. but [gh]et �ou hast
+not �e craft of dyuisio[n~], but �{o}u schalt haue hit aft{er}warde.
+
+  � P{er} num{er}u{m} si vis nu{mer}u{m} q{u}oq{ue} m{u}ltiplicar{e}
+  � T{antu}m p{er} normas subtiles absq{ue} figuris
+    Has normas pot{er}is p{er} v{er}sus scir{e} sequentes.
+
+  [Sidenote: Mental multiplication.]
+
+� Her{e} he teches �e to m{u}ltiplie be �ow[gh]t figures in �i mynde.
+And �e sentence of �is v{er}se is �is: yf �o{u} wel m{u}ltiplie on
+nounbre by ano�{er} in �i mynde, �{o}u schal haue �{er}eto rewles in �e
+v{er}ses �at schal come aft{er}.
+
+  � Si tu p{er} digitu{m} digitu{m} vis m{u}ltiplicar{e}
+    Re{gula} p{re}cedens dat qualit{er} est op{er}andu{m}.
+
+  [Sidenote: Digit by digit is easy.]
+
+� Her{e} he teches a rewle as �ou hast afor{e} to m{u}ltiplie a digit be
+ano�{er}, as yf �ou wolde wete qwat is sex tymes 6. �ou [*leaf 161a]
+schalt wete by �e rewle �at I ta[gh]t �e befor{e}, yf �ou haue mynde
+�{er}of.
+
+  � Articulu{m} si p{er} reliquu{m} reliquu{m} vis m{u}lti{plica}r{e}
+    In p{ro}p{r}iu{m} digitu{m} debet vt{er}q{ue} resolui.
+  � Articul{us} digitos post se m{u}ltiplicantes
+    Ex digit{us} quociens retenerit m{u}ltipli{ca}r{i}
+    Articuli faciu{n}t tot centu{m} m{u}ltiplicati.
+
+  [Sidenote: The first case of the craft. Article by article; an
+  example: another example:]
+
+    [Headnote: How to work subtly without Figures.]
+
+  [Sidenote: Mental multiplication. Another example. Another example.
+  Notation. Notation again. Mental multiplication.]
+
+��Her{e} he teches �e furst rewle, �e quych is �is: yf �ou wel
+m{u}ltiplie an articul be ano�{er}, so �at both �e articuls bene
+w{i}t{h}-Inne an hundreth, �us �{o}u schalt do. take �e digit of bothe
+the articuls, for eu{er}y articul hase a digit, �en m{u}ltiplye �at on
+digit by �at o�{er}, and loke how mony vnytes ben in �e nounbre �at
+comes of �e m{u}ltiplicacio[n~] of �e 2 digittes, &�so mony hundrythes
+ben in �e nounb{re} �at schal come of �e m{u}ltiplicacio[n~] of �e ylke
+2 articuls as �us. yf �{o}u wold wete qwat is ten tymes ten. take �e
+digit of ten, �e quych is 1; take �e digit of �at o�{er} ten, �e quych
+is on. ��Also m{u}ltiplie 1 be 1, as on tyme on �at is but 1. In on is
+but on vnite as �ou wost welle, �{er}efor{e} ten tymes ten is but a
+hundryth. ��Also yf �ou wold wete what is twenty tymes 30. take �e digit
+of twenty, �at is 2; & take �e digitt of thrytty, �at is 3. m{u}ltiplie
+3 be 2, �at is 6. Now in 6 ben 6 vnites, ��And so mony hundrythes ben in
+20 tymes 30[*leaf 161b], �{ere}for{e} 20 tymes 30 is 6 hundryth eue[n~].
+loke & se. ��But yf it be so �at on{e} articul be w{i}t{h}-Inne an
+hundryth, or by-twene an hundryth and a thowsande, so �at it be not a
+�owsande fully. �en loke how mony vnytes ben in �e nounbur �at comys of
+�e m{u}ltiplicacio[n~] [{16}]And so mony tymes[{16}] of 2 digitt{es} of
+ylke articuls, so mony thowsant ben in �e nounbre, the qwych comes of �e
+m{u}ltiplicacio[n~]. And so mony tymes ten thowsand schal be in �e
+nounbre �at comes of �e m{u}ltiplicacion of 2 articuls, as yf �{o}u wold
+wete qwat is 4 hundryth tymes [two hundryth]. Multiply 4 be 2,[{17}] �at
+wol be 8. in 8 ben 8 vnites. � And so mony tymes ten thousand be in 4
+hundryth tymes [2][{17}] hundryth, �{a}t is 80 thousand. Take hede,
+I�schall telle �e a gen{e}rall{e} rewle whan �{o}u hast 2 articuls, And
+�ou wold wete qwat comes of �e m{u}ltiplicacio[n~] of hem 2. m{u}ltiplie
+�e digit of �{a}t on articuls, and kepe �at nounbre, �en loke how mony
+cifers schuld go befor{e} �at on articuls, and he wer{e} write. Als mony
+cifers schuld go befor{e} �at other, &�he wer{e} write of cifers. And
+haue all{e} �e ylke cifers toged{ur} in �i mynde, [*leaf 162a] a-rowe
+ycho[n~] aftur other, and in �e last plase set �e nounbre �at comes of
+�e m{u}ltiplicacio[n~] of �e 2 digittes. And loke in �i mynde in what
+place he stondes, wher{e} in �e secunde, or in �e thryd, or in �e 4, or
+wher{e} ellis, and loke qwat �e figures by-token in �at place; & so mych
+is �e nounbre �at comes of �e 2 articuls y-m{u}ltiplied to-ged{ur} as
+�us: yf �{o}u wold wete what is 20 thousant tymes 3 �owsande. m{u}ltiply
+�e digit of �at articull{e} �e quych is 2 by �e digitte of �at o�{er}
+articul �e quych is 3, �at wol be 6. �en loke how mony cifers schal go
+to 20 thousant as hit schuld be write in a tabul. c{er}tainly 4 cifers
+schuld go to 20 �owsant. ffor �is figure 2 in �e fyrst place betokenes
+twene. ��In �e secunde place hit betokenes twenty. ��In �e 3. place hit
+betokenes 2 hundryth. .�. In �e 4 place 2 thousant. ��In �e 5 place
+h{i}t betokenes twenty �ousant. �{ere}for{e} he most haue 4 cifers
+a-for{e} hym �at he may sto{n}de in �e 5 place. kepe �ese 4 cifers in
+thy mynde, �en loke how mony cifers go[n~] to 3 thousant. Certayn to 3
+thousante [*leaf 162b] go[n~] 3 cifers afor{e}. Now cast ylke 4 cifers
+�at schuld go to twenty thousant, And thes 3 cifers �at schuld go
+afor{e} 3 thousant, &�sette hem in rewe ycho[n~] aft{er} o�{er} in �i
+mynde, as �ai schuld stonde in a tabull{e}. And �en schal �ou haue 7
+cifers; �en sett �at 6 �e quych comes of �e m{u}ltiplicacio[n~] of �e 2
+digitt{es} aft{u}r �e ylke cifers in �e 8 place as yf �at hit stode in a
+tabul. And loke qwat a figur{e} of 6 schuld betoken in �e 8 place. yf
+hit wer{e} in a tabul & so mych it is. & yf �at figure of 6 stonde in �e
+fyrst place he schuld betoken but 6. ��In �e 2 place he schuld betoken
+sexty. ��In the 3 place he schuld betoke[n~] sex hundryth. ��In �e 4
+place sex thousant. ��In �e 5 place sexty �owsant. ��In �e sext place
+sex hundryth �owsant. ��In �e 7 place sex �owsant thousant{es}. ��In �e
+8 place sexty �owsant thousantes. �{er}for{e} sett 6 in octauo loco, And
+he schal betoken sexty �owsant thousantes. And so mych is twenty �owsant
+tymes 3 thousant, ��And �is rewle is gen{er}all{e} for all{e} man{er} of
+articuls, Whethir �ai be hundryth or �owsant; but �{o}u most know well
+�e craft of �e wryrchyng{e} in �e tabull{e} [*leaf 163a] or �ou know to
+do �us in �i mynde aftur �is rewle. Thou most �at �is rewle holdy�e note
+but wher{e} �{ere} ben 2 articuls and no mo of �e quych ayther of hem
+hase but on figur{e} significatyf. As twenty tymes 3 thousant or 3
+hundryth, and such o�{ur}.
+
+  � Articulum digito si m{u}ltiplicare o{portet}
+    Articuli digit[i sumi quo multiplicate]
+    Debem{us} reliquu{m} quod m{u}ltiplicat{ur} ab ill{is}
+    P{er} reliq{u}o decuplu{m} sic su{m}ma{m} later{e} neq{ui}b{i}t.
+
+  [Sidenote: The third case of the craft; an example.]
+
+��Her{e} he puttes �e thryde rewle, �e quych is �is. yf �{o}u wel
+m{u}ltiply in �i mynde, And �e Articul be a digitte, �ou schalt loke �at
+�e digitt be w{i}t{h}-Inne an hundryth, �en �ou schalt m{u}ltiply the
+digitt of �e Articulle by �e o�er digitte. And eu{er}y vnite in �e
+nounbre �at schall{e} come �{ere}-of schal betoken ten. As �us: yf �at
+�{o}u wold wete qwat is twyes 40. m{u}ltiplie �e digitt{e} of 40, �e
+quych is 4, by �e o�{er} diget, �e quych is 2. And �at wolle be 8. And
+in �e nombre of 8 ben 8 vnites, &�eu{er}y of �e ylke vnites schuld
+stonde for 10. �{ere}-fore �{ere} schal be 8 tymes 10, �at wol be 4
+score. And so mony is twyes 40. ��If �e articul be a hundryth or be 2
+hundryth And a �owsant, so �at hit be notte a thousant, [*leaf 163b]
+worch as �o{u} dyddyst afor{e}, saue �{o}u schalt rekene eu{er}y vnite
+for a hundryth.
+
+  � In nu{mer}u{m} mixtu{m} digitu{m} si ducer{e} cures
+    Articul{us} mixti sumat{ur} deinde resoluas
+    In digitu{m} post fac respectu de digitis
+    Articul{us}q{ue} docet excrescens in diriua{n}do
+    In digitu{m} mixti post ducas m{u}ltiplica{n}te{m}
+  � De digitis vt norma [{18}][docet] de [hunc]
+    Multiplica si{mu}l et sic postea summa patebit.
+
+  [Sidenote: The fourth case of the craft: Composite by digit. Mental
+  multiplication.]
+
+Here he puttes �e 4 rewle, �e quych is �is: yf �ou m{u}ltipliy on
+composit be a digit as 6 tymes 24, [{19}]�en take �e diget of �at
+composit, & m{u}ltiply �{a}t digitt by �at o�{er} diget, and kepe �e
+nomb{ur} �at comes �{ere}-of. �en take �e digit of �at composit,
+&�m{u}ltiply �at digit by ano�{er} diget, by �e quych �{o}u hast
+m{u}ltiplyed �e diget of �e articul, and loke qwat comes �{ere}-of. �en
+take �{o}u �at nounbur, & cast hit to �at other nounbur �at �{o}u
+secheste as �us yf �ou wel wete qwat comes of 6 tymes 4 & twenty.
+multiply �at articull{e} of �e composit by �e digit, �e quych is 6,
+as yn �e thryd rewle �{o}u was tau[gh]t, And �at schal be 6 scor{e}. �en
+m{u}ltiply �e diget of �e {com}posit, [*leaf 164a] �e quych is 4, and
+m{u}ltiply �at by �at other diget, �e quych is 6, as �ou wast tau[gh]t
+in �e first rewle, yf �{o}u haue mynde �{er}of, &��at wol be 4 & twenty.
+cast all ylke nounburs to-ged{ir}, & hit schal be 144. And so mych is 6
+tymes 4 & twenty.
+
+    [Headnote: How to multiply without Figures.]
+
+  � Duct{us} in articulu{m} num{erus} si {com}posit{us} sit
+    Articulu{m} puru{m} comites articulu{m} q{u}o{que}
+    Mixti pro digit{is} post fiat [et articulus vt]
+    Norma iubet [retinendo quod extra dicta ab illis]
+    Articuli digitu{m} post tu mixtu{m} digitu{m} duc
+    Re{gula} de digitis nec p{re}cipit articul{us}q{ue}
+    Ex quib{us} exc{re}scens su{m}me tu iunge p{ri}ori
+    Sic ma{n}ifesta cito fiet t{ibi} su{m}ma petita.
+
+  [Sidenote: The fifth case of the craft: Article by Composite.
+  An example.]
+
+��Her{e} he puttes �e 5 rewle, �e quych is �is: yf �{o}u wel m{u}ltiply
+an Articul be a composit, m{u}ltiplie �at Articul by �e articul of �e
+composit, and worch as �ou wos tau[gh]t in �e secunde rewle, of �e quych
+rewle �e v{er}se begynnes �us. ��Articulu{m} si p{er} Relicu{m} vis
+m{u}ltiplicare. �en m{u}ltiply �e diget of �e composit by �at o�{ir}
+articul aft{ir} �e doctrine of �e 3 rewle. take �{er}of gode hede,
+I�p{ra}y �e as �us. Yf �{o}u wel wete what is 24 tymes ten. Multiplie
+ten by 20, �at wel be 2 hundryth. �en m{u}ltiply �e diget of �e 10, �e
+quych is 1, by �e diget of �e composit, �e quych is 4, & �{a}t [*leaf
+164b] wol be 4. �en reken eu{er}y vnite �at is in 4 for 10, &��at schal
+be 40. Cast 40 to 2 hundryth, &��at wol be 2 hundryth & 40. And so mych
+is 24 tymes ten.
+
+    [Headnote: How to work without Figures.]
+
+  � Compositu{m} num{er}u{m} mixto si[c] m{u}ltiplicabis
+    Vndecies tredeci{m} sic e{st} ex hiis op{er}andum
+    In reliquu{m} p{rimu}m demu{m} duc post in eund{em}
+    Vnu{m} post den{u}m duc in t{ri}a dei{n}de p{er} vnu{m}
+    Multiplices{que} dem{u}m int{ra} o{mn}ia m{u}ltiplicata
+    In su{m}ma decies q{ua}m si fu{er}it t{ibi} doces
+    Multiplicandor{um} de normis sufficiunt h{ec}.
+
+  [Sidenote: The sixth case of the craft: Composite by Composite.
+  Mental multiplication. An example of the sixth case of the craft.]
+
+��Here he puttes �e 6 rewle, &��e last of all{e} multiplicacio[n~],
+�e quych is �is: yf �{o}u wel m{u}ltiplye a {com}posit by a-no�{er}
+composit, �ou schalt do �us. m{u}ltiplie �{a}t on composit, qwych �{o}u
+welt of the twene, by �e articul of �e to�{er} composit, as �{o}u wer{e}
+tau[gh]t in �e 5 rewle, �en m{u}ltiplie �{a}t same composit, �e quych
+�ou hast m{u}ltiplied by �e o�{er} articul, by �e digit of �e o�{er}
+composit, as �{o}u was tau[gh]t in �e 4 rewle. As �us, yf �ou wold wete
+what is 11 tymes 13, as �{o}u was tau[gh]t in �e 5 rewle, &��at schal be
+an hundryth & ten, aft{er}warde m{u}ltiply �at same co{m}posit �{a}t
+�{o}u hast m{u}ltiplied, �e quych is a .11. And m{u}ltiplye hit be �e
+digit of �e o�{er} composit, �e quych is 3, for 3 is �e digit of 13, And
+�at wel be 30. �en take �e digit of �at composit, �e quych composit �ou
+m{u}ltiplied by �e digit of �{a}t o�{er} {com}posit, [*leaf 165a] �e
+quych is a 11. ��Also of the quych 11 on is �e digit. m{u}ltiplie �at
+digitt by �e digett of �at oth{er} composit, �e quych diget is 3, as
+�{o}u was tau[gh]t in �e first rewle i{n} �e begynnyng{e} of �is craft.
+�e quych rewle begynn{es} "In digitu{m} cures." And of all{e} �e
+m{u}ltiplicacio[n~] of �e 2 digitt comys thre, for onys 3 is but 3. Now
+cast all{e} �ese nounbers toged{ur}, the quych is �is, a�hundryth & ten
+& 30 & 3. And al �at wel be 143. Write 3 first in �e ryght side. And
+cast 10 to 30, �at wol be 40. set 40 next aft{ur} towarde �e lyft side,
+And set aftur a hundryth as her{e} an Ensampull{e}, 143.
+
+(Cetera desunt.)
+
+
+FOOTNOTES (The Crafte of Nombrynge):
+
+  [1: In MS, 'awiy.']
+  [2: 'ben' repeated in MS.]
+  [3: In MS. 'thausandes.']
+  [4: Perhaps "So."]
+  [5: 'hali' marked for erasure in MS.]
+  [6: 'moy' in MS.]
+  [7: 'Subt{ra}has a{u}t addis a dext{ri}s {ve}l medi{a}b{is}' added
+    on margin of�MS.]
+  [8: After 'craft' insert 'the .4. what is �e p{ro}fet of �is craft.']
+  [9: After 'sythes' insert '& �is wordes fyue sithe & sex sythes.']
+  [10: 't'l' marked for erasure before 'tyl' in MS.]
+  [11: Here 'of �e same rew' is marked for erasure in MS.]
+  [12: 's{ed}' deleted in MS.]
+  [13: 6883 in MS.]
+  [14: '�en' overwritten on '�at' marked for erasure.]
+  [15: 'Supra' inserted in MS. in place of 'cifra' marked for erasure.]
+  [16--16: Marked for erasure in MS.]
+  [17: 4 in MS.]
+  [18: docet. decet MS.]
+  [19: '4 times 4' in MS.]
+
+
+
+
++The Art of Nombryng.+
+
+A TRANSLATION OF
+
++John of Holywood's De Arte Numerandi.+
+
+
+[_Ashmole MS. 396, fol. 48._]
+
+  +Boys seying in the begynnyng of his Arsemetrik{e}:--All{e}
+    [*Fol. 48.] thynges that ben{e} fro the first begynnyng of thynges
+    have p{ro}ceded{e}, and come forth{e}, And by reso{u}n of nombre
+    ben formed{e}; And in wise as they ben{e}, So oweth{e} they to be
+    knowen{e}; wherfor in vniu{er}sall{e} knowlechyng of thynges the
+    Art of nombrynge is best, and most operatyf{e}.+
+
+  [Sidenote: The name of the art. Derivation of Algorism. Another.
+  Another. Kinds of numbers. The 9 rules of the Art.]
+
+Therfore sithen the science of the whiche at this tyme we intenden{e} to
+write of standith{e} all{e} and about nombre: ffirst we most se, what is
+the p{ro}pre name therof{e}, and fro whens the name come: Afterward{e}
+what is nombre, And how manye spices of nombre ther ben. The name is
+cleped{e} Algorisme, had{e} out of Algor{e}, other of Algos, in grewe,
+That is clepid{e} in englissh{e} art other craft, And of Rithm{us} that
+is called{e} nombre. So algorisme is cleped{e} the art of nombryng,
+other it is had of{e} en or in, and gogos that is introduccio{u}n, and
+Rithm{us} nombre, that is to say Interduccio{u}n of nombre. And thirdly
+it is had{e} of the name of a kyng that is cleped{e} Algo and Rythm{us};
+So called{e} Algorism{us}. Sothely .2. maner{e} of nombres ben
+notified{e}; Formall{e},[{1}] as nombr{e} i{s} vnitees gadred{e}
+to-gedres; Materiall{e},[{2}] as nombr{e} is a colleccio{u}n of vnitees.
+Other nombr{e} is a multitude had{e} out of vnitees, vnitee is that
+thynge wher-by eu{er}y thynge is called{e} oone, other o thynge. Of
+nombres, that one is cleped{e} digitall{e}, that other{e} Article,
+Another a nombre componed{e} o�{er} myxt. Another digitall{e} is a
+nombre w{i}t{h}-in .10.; Article is �{a}t nombre that may be dyvyded{e}
+in .10. p{ar}ties egally, And that there leve no residue; Componed{e} or
+medled{e} is that nombre that is come of a digite and of an article. And
+vndrestand{e} wele that all{e} nombres betwix .2. articles next is a
+nombr{e} componed{e}. Of this art ben{e} .9. spices, that is forto sey,
+num{er}acio{u}n, addicio{u}n, Subtraccio{u}n, Mediac{i}o{u}n,
+Duplacio{u}n, Multipliacio{u}n, Dyvysio{u}n, Progressio{u}n, And of
+Rootes the extraccio{u}n, and that may be had{e} in .2. maners, that is
+to sey in nombres quadrat, and in cubic{es}: Amonge the which{e}, ffirst
+of Num{er}acio{u}n, and aft{er}ward{e} of �e o�{er}s by ordure,
+y�entende to write.
+
+
+    [Headnote: Chapter I. Numeration.]
+
+  [*Fol. 48b]
+
+  +For-soth{e} num{er}acio{u}n is of eu{er}y numbre by
+  competent figures an artificiall{e} rep{re}sentacio{u}n.+
+
+  [Sidenote: Figures, differences, places, and limits. The 9 figures.
+  The cipher. The numeration of digits, of articles, of composites.
+  The value due to position. Numbers are written from right to left.]
+
+Sothly figure, difference, places, and lynes supposen o thyng other the
+same, But they ben sette here for dyue{r}s resons. ffigure is cleped{e}
+for p{ro}traccio{u}n of figuracio{u}n; Difference is called{e} for
+therby is shewed{e} eu{er}y figure, how it hath{e} difference fro the
+figures before them: place by cause of space, where-in me writeth{e}:
+lynees, for that is ordeyned{e} for the p{re}sentacio{u}n of eu{er}y
+figure. And vnderstonde that ther ben .9. lymytes of figures that
+rep{re}senten the .9. digit{es} that ben these. 0. 9. 8. 7. 6. 5. 4. 3.
+2. 1. The .10. is cleped{e} theta, or a cercle, other a cifre, other a
+figure of nought for nought it signyfieth{e}. Nathelesse she holdyng
+that place giveth{e} others for to signyfie; for with{e}-out cifre or
+cifres a pure article may not be writte. And sithen that by these .9.
+figures significatif{es} Ioyned{e} w{i}t{h} cifre or w{i}t{h} cifres
+all{e} nombres ben and may be rep{re}sented{e}, It was, nether is,
+no nede to fynde any more figures. And note wele that eu{er}y digite
+shall{e} be writte w{i}t{h} oo figure allone to it ap{ro}pred{e}. And
+all{e} articles by a cifre, ffor eu{er}y article is named{e} for oone of
+the digitis as .10. of 1.. 20. of. 2. and so of the others, &c. And
+all{e} nombres digitall{e} owen to be sette in the first difference:
+All{e} articles in the seconde. Also all{e} nombres fro .10. til an
+.100. [which] is excluded{e}, with .2. figures mvst be writte; And yf it
+be an article, by a cifre first put, and the figure y-writte toward{e}
+the lift hond{e}, that signifieth{e} the digit of the which{e} the
+article is named{e}; And yf it be a nombre componed{e}, ffirst write the
+digit that is a part of that componed{e}, and write to the lift side the
+article as it is seid{e} be-fore. All{e} nombre that is fro an
+hundred{e} tille a thousand{e} exclused{e}, owith{e} to be writ by .3.
+figures; and all{e} nombre that is fro a thousand{e} til .x. M[~l]. mvst
+be writ by .4. figures; And so forthe. And vnderstond{e} wele that
+eu{er}y figure sette in the first place signyfieth{e} his digit; In the
+second{e} place .10. tymes his digit; In the .3. place an hundred{e} so
+moche; In the .4. place a thousand{e} so moche; In the .5. place .x.
+thousand{e} so moch{e}; In the .6. place an hundred{e} thousand{e} so
+moch{e}; In the .7. place a thousand{e} thousand{e}. And so infynytly
+mvltiplying by [*Fol. 49.] these .3. 10, 100, 1000. And vnderstand{e}
+wele that competently me may sette vpon figure in the place of a
+thousand{e}, a�prik{e} to shewe how many thousand{e} the last figure
+shall{e} rep{re}sent. We writen{e} in this art to the lift side-ward{e},
+as arabien{e} writen{e}, that weren fynders of this science, other{e}
+for this reso{u}n, that for to kepe a custumable ordr{e} in redyng,
+Sette we all{e}-wey the more nombre before.
+
+    [Headnote: Chapter II. Addition.]
+
+  [Sidenote: Definition. How the numbers should be written. The method
+  of working. Begin at the right. The Sum is a digit, or an article,
+  or a composite.]
+
+Addicio{u}n is of nombre other of nombres vnto nombre or to nombres
+aggregacio{u}n, that me may see that that is come therof as
+exc{re}ssent. In addicio{u}n, 2. ordres of figures and .2. nombres ben
+necessary, that is to sey, a�nombre to be added{e} and the nombre wherto
+the addic{i}oun shold{e} be made to. The nombre to be added{e} is that
+�at shold{e} be added{e} therto, and shall{e} be vnderwriten; the nombre
+vnto the which{e} addicio{u}n shall{e} be made to is that nombre that
+resceyueth{e} the addicion of �at other, and shall{e} be writen above;
+and it is convenient that the lesse nombre be vnderwrit, and the more
+added{e}, than the contrary. But whether it happ{e} one other other, the
+same comyth{e} of, Therfor, yf �ow wilt adde nombre to nombre, write the
+nombre wherto the addicio{u}n shall{e} be made in the omest ordre by his
+differences, so that the first of the lower ordre be vndre the first of
+the omyst ordre, and so of others. That done, adde the first of the
+lower ordre to the first of the omyst ordre. And of such{e} addicio{u}n,
+other �{er}e grow{i}t{h} therof a digit, An article, other a
+composed{e}. If it be digit{us}, In the place of the omyst shalt thow
+write the digit excrescyng, as thus:--
+
+  +----------------------------+---+
+  |The resultant               | 2 |
+  +----------------------------+---+
+  |To whom it shal be added{e} | 1 |
+  +----------------------------+---+
+  |The nombre to be added{e}   | 1 |
+  +----------------------------+---+
+
+If the article; in the place of the omyst put a-way by a cifre writte,
+and the digit transferred{e}, of �e which{e} the article toke his name,
+toward{e} the lift side, and be it added{e} to the next figure folowyng,
+yf ther be any figure folowyng; or no, and yf it be not, leve it [in
+the] void{e}, as thus:--
+
+  +---------------------------------+----+
+  | The resultant                   | 10 |
+  +---------------------------------+----+
+  | To whom it shall{e} be added{e} |  7 |
+  +---------------------------------+----+
+  | The nombre to be added{e}       |  3 |
+  +---------------------------------+----+
+
+  +----------------------+---+---+---+---+---+
+  | Resultans            | 2 | 7 | 8 | 2 | 7 |
+  +----------------------+---+---+---+---+---+
+  | Cui d{ebet} addi     | 1 | 0 | 0 | 8 | 4 |
+  +----------------------+---+---+---+---+---+
+  | Num{erus} addend{us} | 1 | 7 | 7 | 4 | 3 |
+  +----------------------+---+---+---+---+---+
+
+And yf it happe that the figure folowyng wherto the addicio{u}n shall{e}
+be made by [the cifre of] an article, it sette a-side; In his place
+write the [*Fol. 49b] [digit of the] Article as thus:--
+
+  +---------------------------------+----+
+  | The resultant                   | 17 |
+  +---------------------------------+----+
+  | To whom it shall{e} be added{e} | 10 |
+  +---------------------------------+----+
+  | The nombre to be added{e}       |  7 |
+  +---------------------------------+----+
+
+And yf it happe that a figure of .9. by the figure that me mvst adde
+[one] to, In the place of that 9. put a cifre {and} write �e article
+toward{e} �e lift hond{e} as bifore, and thus:--
+
+  +---------------------------------+----+
+  | The resultant                   | 10 |
+  +---------------------------------+----+
+  | To whom it shall{e} be added{e} |  9 |
+  +---------------------------------+----+
+  | The nombre to be added{e}       |  1 |
+  +---------------------------------+----+
+
+And yf[{3}] [therefrom grow a] nombre componed,[{4}] [in the place of
+the nombre] put a-way[{5}][let] the digit [be][{6}]writ �{a}t is part of
+�{a}t co{m}posid{e}, and �an put to �e lift side the article as before,
+and �us:--
+
+  +---------------------------------+----+
+  | The resultant                   | 12 |
+  +---------------------------------+----+
+  | To whom it shall{e} be added{e} |  8 |
+  +---------------------------------+----+
+  | The nombre to be added{e}       |  4 |
+  +---------------------------------+----+
+
+This done, adde the seconde to the second{e}, and write above o�{er} as
+before.
+
+  [Sidenote: The translator's note.]
+
+Note wele �{a}t in addic{i}ons and in all{e} spices folowyng, whan he
+seith{e} one the other shall{e} be writen aboue, and me most vse eu{er}
+figure, as that eu{er}y figure were sette by half{e}, and by
+hym-self{e}.
+
+
+    [Headnote: Chapter III. Subtraction.]
+
+  [Sidenote: Definition of Subtraction. How it may be done. What is
+  required. Write the greater number above. Subtract the first figure
+  if possible. If it is not possible 'borrow ten,' and then subtract.]
+
+Subtraccio{u}n is of .2. p{ro}posed{e} nombres, the fyndyng of the
+excesse of the more to the lasse: Other subtraccio{u}n is ablacio{u}n of
+o nombre fro a-nother, that me may see a some left. The lasse of the
+more, or even of even, may be w{i}t{h}draw; The more fro the lesse may
+neu{er} be. And sothly that nombre is more that hath{e} more figures, So
+that the last be signyficatife{s}: And yf ther ben as many in that one
+as in that other, me most deme it by the last, other by the next last.
+More-ou{er} in w{i}t{h}-drawyng .2. nombres ben necessary; A�nombre to
+be w{i}t{h}draw, And a nombre that me shall{e} w{i}t{h}-draw of. The
+nombre to be w{i}t{h}-draw shall{e} be writ in the lower ordre by his
+differences; The nombre fro the which{e} me shall{e} with{e}-draw in the
+omyst ordre, so that the first be vnder the first, the second{e} vnder
+the second{e}, And so of all{e} others. With{e}-draw therfor the first
+of the lower{e} ordre fro the first of the ordre above his hede, and
+that wolle be other more or lesse, o�{er} egall{e}.
+
+  +---------------------------------+----+
+  | The remanent                    | 20 |
+  +---------------------------------+----+
+  | Wherof me shall{e} w{i}t{h}draw | 22 |
+  +---------------------------------+----+
+  | The nombre to be w{i}t{h}draw   |  2 |
+  +---------------------------------+----+
+
+yf it be egall{e} or even the figure sette beside, put in his place a
+cifre. And yf it be more put away �{er}fro als many of vnitees the lower
+figure conteyneth{e}, and writ the residue as thus
+
+  +----------------------------------+---+---+
+  | The remanent                     | 2 | 2 |
+  +----------------------------------+---+---+
+  | Wherof me shall{e} w{i}t{h}-draw | 2 | 8 |
+  +----------------------------------+---+---+
+  | �e nombre to be w{i}t{h}draw     |   | 6 |
+  +----------------------------------+---+---+
+
+  [*Fol. 50.]
+
+  +--------------------------+---+---+-----+---+---+---+---+---+---+
+  | Remane{n}s               | 2 | 2 |  1  | 8 | 2 | 9 | 9 | 9 | 8 |
+  +--------------------------+---+---+-----+---+---+---+---+---+---+
+  | A quo sit subtraccio     | 8 | 7 |  2  | 4 | 3 | 0 | 0 | 0 | 4 |
+  +--------------------------+---+---+-----+---+---+---+---+---+---+
+  | Numerus subt{ra}hend{us} | 6 | 5 |[{7}]|[6]| . | . | . | . | 6 |
+  +--------------------------+---+---+-----+---+---+---+---+---+---+
+
+And yf it be lesse, by-cause the more may not be w{i}t{h}-draw ther-fro,
+borow an vnyte of the next figure that is worth{e} 10. Of that .10. and
+of the figure that ye wold{e} have w{i}t{h}-draw fro be-fore to-gedre
+Ioyned{e}, w{i}t{h}-draw �e figure be-nethe, and put the residue in the
+place of the figure put a-side as �{us}:--
+
+  +----------------------------------+---+---+
+  | The remanent                     | 1 | 8 |
+  +----------------------------------+---+---+
+  | Wherof me shall{e} w{i}t{h}-draw | 2 | 4 |
+  +----------------------------------+---+---+
+  | The nombre to be w{i}t{h}-draw   | 0 | 6 |
+  +----------------------------------+---+---+
+
+  [Sidenote: If the second figure is one.]
+
+And yf the figure wherof me shal borow the vnyte be one, put it a-side,
+and write a cifre in the place �{er}of, lest the figures folowing faile
+of thair{e} nombre, and �an worch{e} as it shew{i}t{h} in this figure
+here:--
+
+  +--------------------------------+---+---+------+
+  | The remanent                   | 3 | 0 |9[{8}]|
+  +--------------------------------+---+---+------+
+  | Wherof me shal w{i}t{h}-draw   | 3 | 1 |   2  |
+  +--------------------------------+---+---+------+
+  | The nombre to be w{i}t{h}-draw | . | . |   3  |
+  +--------------------------------+---+---+------+
+
+  [Sidenote: If the second figure is a cipher.]
+
+And yf the vnyte wherof me shal borow be a cifre, go ferther to the
+figure signyficatif{e}, and ther borow one, and reto{ur}nyng bak{e}, in
+the place of eu{er}y cifre �{a}t ye passid{e} ou{er}, sette figures of
+.9. as here it is specified{e}:--
+
+  +----------------------------------+---+---+---+---+---+
+  | The remenaunt                    | 2 | 9 | 9 | 9 | 9 |
+  +----------------------------------+---+---+---+---+---+
+  | Wherof me shall{e} w{i}t{h}-draw | 3 | 0 | 0 | 0 | 3 |
+  +----------------------------------+---+---+---+---+---+
+  | The nombre to be w{i}t{h}-draw   |   |   |   |   | 4 |
+  +----------------------------------+---+---+---+---+---+
+
+  [Sidenote: A�justification of the rule given. Why it is better to
+  work from right to left. How to prove subtraction, and addition.]
+
+And whan me cometh{e} to the nombre wherof me intendith{e}, there
+remayneth{e} all{e}-wayes .10. ffor �e which{e} .10. &c. The reson why
+�at for eu{er}y cifre left behynde me setteth figures ther of .9. this
+it is:--If fro the .3. place me borowed{e} an vnyte, that vnyte by
+respect of the figure that he came fro rep{re}sentith an .C., In the
+place of that cifre [passed over] is left .9., [which is worth ninety],
+and yit it remayneth{e} as .10., And the same reson{e} wold{e} be yf me
+had{e} borowed{e} an vnyte fro the .4., .5., .6., place, or ony other so
+vpward{e}. This done, withdraw the second{e} of the lower ordre fro the
+figure above his hede of �e omyst ordre, and wirch{e} as before. And
+note wele that in addicion or in subtracc{i}o{u}n me may wele fro the
+lift side begynne and ryn to the right side, But it wol be more
+p{ro}fitabler to be do, as it is taught. And yf thow wilt p{ro}ve yf
+thow have do wele or no, The figures that thow hast withdraw, adde them
+ayene to the omyst figures, and they wolle accorde w{i}t{h} the first
+that thow haddest yf thow have labored wele; and in like wise in
+addicio{u}n, whan thow hast added{e} all{e} thy figures, w{i}t{h}draw
+them that thow first [*Fol. 50b] addest, and the same wolle reto{ur}ne.
+The subtraccio{u}n is none other but a p{ro}uff{e} of the addicio{u}n,
+and the contrarye in like wise.
+
+    [Headnote: Chapter IV. Mediation.]
+
+  [Sidenote: Definition of mediation. Where to begin. If the first
+  figure is unity. What to do if it is not unity.]
+
+Mediacio{u}n is the fyndyng of the halfyng of eu{er}y nombre, that it
+may be seyn{e} what and how moch{e} is eu{er}y half{e}. In halfyng ay oo
+order of figures and oo nombre is necessary, that is to sey the nombre
+to be halfed{e}. Therfor yf thow wilt half any nombre, write that nombre
+by his differences, and begynne at the right, that is to sey, fro the
+first figure to the right side, so that it be signyficatif{e} other
+rep{re}sent vnyte or eny other digitall{e} nombre. If it be vnyte write
+in his place a cifre for the figures folowyng, [lest they signify less],
+and write that vnyte w{i}t{h}out in the table, other resolue it in .60.
+mynvt{es} and sette a-side half of tho m{inutes} so, and reserve the
+remen{au}nt w{i}t{h}out in the table, as thus .30.; other sette
+w{i}t{h}out thus .{d[-i]}: that kepeth{e} none ordre of place,
+Nathelesse it hath{e} signyficacio{u}n. And yf the other figure signyfie
+any other digital nombre fro vnyte forth{e}, o�{er} the nombre is od{e}
+or even{e}. If it be even, write this half in this wise:--
+
+  +-----------------+---+---+
+  | Halfed{e}       | 2 | 2 |
+  +-----------------+----+--+
+  | to be halfed{e} | 4 | 4 |
+  +-----------------+---+---+
+
+And if it be odde, Take the next even vndre hym conteyned{e}, and put
+his half in the place of that odde, and of �e vnyte that remayneth{e} to
+be halfed{e} do thus:--
+
+  +-----------------+---+---+
+  | halfed{e}       | 2 | 3 | [di]
+  +-----------------+---+---+
+  | To be halfed{e} | 4 | 7 |
+  +-----------------+---+---+
+
+  [Sidenote: Then halve the second figure. If it is odd, add 5 to the
+  figure before.]
+
+This done, the second{e} is to be halfed{e}, yf it be a cifre put it
+be-side, and yf it be significatif{e}, other it is even or od{e}: If it
+be even, write in the place of �e nombres wiped{e} out the half{e}; yf
+it be od{e}, take the next even vnder it co{n}tenyth{e}, and in the
+place of the Impar sette a-side put half of the even: The vnyte that
+remayneth{e} to be halfed{e}, respect had{e} to them before, is worth{e}
+.10. Dyvide that .10. in .2., 5. is, and sette a-side that one, and adde
+that other to the next figure p{re}cedent as here:--
+
+  +-----------------+---+---+---+
+  | Halfed{e}       |   |   |   |
+  +-----------------+---+---+---+
+  | to be halfed{e} |   |   |   |
+  +-----------------+---+---+---+
+
+And yf �e addicio{u}n shold{e} be made to a cifre, sette it a-side, and
+write in his place .5. And vnder this fo{ur}me me shall{e} write and
+worch{e}, till{e} the totall{e} nombre be halfed{e}.
+
+  +------------------+---+---+---+---+---+----+----+---+
+  | doubled{e}       | 2 | 6 | 8 | 9 | 0 | 10 | 17 | 4 |
+  +------------------+---+---+---+---+---+----+----+---+
+  | to be doubled{e} | 1 | 3 | 4 | 4 | 5 |  5 |  8 | 7 |
+  +------------------+---+---+---+---+---+----+----+---+
+
+    [Headnote: Chapter V. Duplation.]
+
+  [Sidenote: Definition of Duplation. Where to begin. Why. What to do
+  with the result.]
+
+Duplicacio{u}n is ag{re}gacion of nombre [to itself] �at me may se the
+nombre growen. In doublyng{e} ay is but one ordre of figures necessarie.
+And me most be-gynne w{i}t{h} the lift side, other of the more figure,
+And after the nombre of the more figure rep{re}sentith{e}. [*Fol. 51.]
+In the other .3. before we begynne all{e} way fro the right side and fro
+the lasse nombre, In this spice and in all{e} other folowyng we wolle
+begynne fro the lift side, ffor and me bigon th{e} double fro the first,
+omwhile me myght double oo thynge twyes. And how be it that me myght
+double fro the right, that wold{e} be harder in techyng and in workyng.
+Therfor yf thow wolt double any nombre, write that nombre by his
+differences, and double the last. And of that doubly{n}g other
+growith{e} a nombre digital, article, or componed{e}. [If it be a digit,
+write it in the place of the first digit.] If it be article, write in
+his place a cifre and transferre the article toward{e} the lift, as
+thus:--
+
+  +------------------+----+
+  | double           | 10 |
+  +------------------+----+
+  | to be doubled{e} |  5 |
+  +------------------+----+
+
+And yf the nombre be componed{e}, write a digital that is part of his
+composicio{u}n, and sette the article to the lift hand{e}, as thus:--
+
+  +------------------+----+
+  | doubled{e}       | 16 |
+  +------------------+----+
+  | to be doubled{e} |  8 |
+  +------------------+----+
+
+That done, me most double the last save one, and what groweth{e} �{er}of
+me most worche as before. And yf a cifre be, touch{e} it not. But yf any
+nombre shall{e} be added{e} to the cifre, in �e place of �e figure
+wiped{e} out me most write the nombre to be added{e}, as thus:--
+
+  +------------------+---+---+---+
+  | doubled{e}       | 6 | 0 | 6 |
+  +------------------+---+---+---+
+  | to be doubled{e} | 3 | 0 | 3 |
+  +------------------+---+---+---+
+
+  [Sidenote: How to prove your answer.]
+
+In the same wise me shall{e} wirch{e} of all{e} others. And this
+p{ro}bacio{u}n: If thow truly double the halfis, and truly half the
+doubles, the same nombre and figure shall{e} mete, such{e} as thow
+labo{ur}ed{e} vpon{e} first, And of the contrarie.
+
+  +------------------+---+---+---+
+  | Doubled{e}       | 6 | 1 | 8 |
+  +------------------+---+---+---+
+  | to be doubled{e} | 3 | 0 | 9 |
+  +------------------+---+---+---+
+
+    [Headnote: Chapter VI. Multiplication.]
+
+  [Sidenote: Definition of Multiplication. Multiplier. Multiplicand.
+  Product.]
+
+Multiplicacio{u}n of nombre by hym-self other by a-nother, w{i}t{h}
+p{ro}posid{e} .2. nombres, [is] the fyndyng of the third{e}, That so
+oft conteyneth{e} that other, as ther ben vnytes in the o�{er}. In
+multiplicacio{u}n .2. nombres pryncipally ben necessary, that is to
+sey, the nombre multiplying and the nombre to be multiplied{e},
+as here;--twies fyve. [The number multiplying] is designed{e}
+adu{er}bially. The nombre to be multiplied{e} resceyveth{e} a
+no{m}i{n}all{e} appellacio{u}n, as twies .5. 5. is the nombre
+multiplied{e}, and twies is the nombre to be multipliede.
+
+  +-----------------+-----+---+---++---+---+---+---+---+---+---+---+---+
+  | Resultans       |[{9}]| 1 | 0 || 1 | 3 | 2 | 6 | 6 | 8 | 0 | 0 | 8 |
+  +-----------------+-----+---+---++---+---+---+---+---+---+---+---+---+
+  | Multiplicand{us}|  .  | . | 5 || . | . | 4 | . | 3 | 4 | 0 | 0 | 4 |
+  +-----------------+-----+---+---++---+---+---+---+---+---+---+---+---+
+  | Multiplicans    |  .  | 2 | 2 || . | 3 | 3 | 2 | 2 | 2 | . | . | . |
+  +-----------------+-----+---+---++---+---+---+---+---+---+---+---+---+
+
+Also me may thervpon{e} to assigne the. 3. nombre, the which{e} is
+[*Fol. 51b] cleped{e} p{ro}duct or p{ro}venient, of takyng out of one
+fro another: as twyes .5 is .10., 5. the nombre to be multiplied{e},
+and .2. the multipliant, and. 10. as before is come therof. And
+vnderstonde wele, that of the multipliant may be made the nombre to
+be multiplied{e}, and of the contrarie, remaynyng eu{er} the same some,
+and herof{e} cometh{e} the comen speche, that seith{e} all nombre is
+converted{e} by Multiplying in hym-self{e}.
+
+  +----+----+----+----+----+--------+----+----+----+-----+
+  |  1 |  2 |  3 |  4 |  5 |   6    |  7 |  8 |  9 |  10 |
+  +----+----+----+----+----+--------+----+----+----+-----+
+  |  2 |  4 |  6 |  8 | 10 |10[{10}]| 14 | 16 | 18 |  20 |
+  +----+----+----+----+----+--------+----+----+----+-----+
+  |  3 |  6 |  9 | 12 | 15 |  18    | 21 | 24 | 27 |  30 |
+  +----+----+----+----+----+--------+----+----+----+-----+
+  |  4 |  8 | 12 | 16 | 20 |  24    | 28 | 32 | 36 |  40 |
+  +----+----+----+----+----+--------+----+----+----+-----+
+  |  5 | 10 | 15 | 20 | 25 |  30    | 35 | 40 | 45 |  50 |
+  +----+----+----+----+----+--------+----+----+----+-----+
+  |  6 | 12 | 18 | 24 | 30 |  36    | 42 | 48 | 56 |  60 |
+  +----+----+----+----+----+--------+----+----+----+-----+
+  |  7 | 14 | 21 | 28 | 35 |  42    | 49 | 56 | 63 |  70 |
+  +----+----+----+----+----+--------+----+----+----+-----+
+  |  8 | 16 | 24 | 32 | 40 |  48    | 56 | 64 | 72 |  80 |
+  +----+----+----+----+----+--------+----+----+----+-----+
+  |  9 | 18 | 27 | 36 | 45 |  54    | 63 | 72 | 81 |  90 |
+  +----+----+----+----+----+--------+----+----+----+-----+
+  | 10 | 20 | 30 | 40 | 50 |  60    | 70 | 80 | 90 | 100 |
+  +----+----+----+----+----+--------+----+----+----+-----+
+
+    [Headnote: The Cases of Multiplication.]
+
+  [Sidenote: There are 6 rules of Multiplication. (1) Digit by digit.
+  See the table above. (2) Digit by article. (3) Composite by digit.]
+
+And ther ben .6 rules of Multiplicacio{u}n; ffirst, yf a digit multiplie
+a digit, considr{e} how many of vnytees ben betwix the digit by
+multiplying and his .10. beth{e} to-gedre accompted{e}, and so oft
+w{i}t{h}-draw the digit multiplying, vnder the article of his
+deno{m}i{n}acio{u}n. Example of grace. If thow wolt wete how moch{e} is
+.4. tymes .8., [{11}]se how many vnytees ben betwix .8.[{12}] and .10.
+to-geder rekened{e}, and it shew{i}t{h} that .2.: withdraw ther-for the
+quat{e}rnary, of the article of his deno{m}i{n}acion twies, of .40., And
+ther remayneth{e} .32., that is, to some of all{e} the
+multiplicacio{u}n. Wher-vpon for more evidence and declaracion the
+seid{e} table is made. Whan a digit multiplieth{e} an article, thow most
+bryng the digit into �e digit, of �e which{e} the article [has][{13}]
+his name, and eu{er}y vnyte shall{e} stond{e} for .10., and eu{er}y
+article an .100. Whan the digit multiplieth{e} a nombre componed{e},
+�{o}u most bryng the digit into ai�{er} part of the nombre componed{e},
+so �{a}t digit be had into digit by the first rule, into an article by
+�e second{e} rule; and aft{er}ward{e} Ioyne the p{ro}duccio{u}n, and
+�{er}e wol be the some totall{e}.
+
+  +----------------+---+---+--++---+---+--++---+---+--++---+---+---+---+
+  |Resultans       | 1 | 2 | 6|| 7 | 3 | 6|| 1 | 2 | 0|| 1 | 2 | 0 | 8 |
+  +----------------+---+---+--++---+---+--++---+---+--++---+---+---+---+
+  |Multiplicand{us}|   |   | 2||   | 3 | 2||   |   | 6||   |   |   | 4 |
+  +----------------+---+---+--++---+---+--++---+---+--++---+---+---+---+
+  |Multiplicans    |   | 6 | 3|| 2 | 3 |  ||   | 2 | 0||   | 3 | 0 | 2 |
+  +----------------+---+---+--++---+---+--++---+---+--++---+---+---+---+
+
+  [Sidenote: (4) Article by article. (5) Composite by article.
+  (6) Composite by composite. How to set down your numbers. If the
+  result is a digit, an article, or a composite. Multiply next by
+ the last but one, and so on.]
+
+Whan an article multiplieth{e} an article, the digit wherof he is
+named{e} is to be brought Into the digit wherof the o�{er} is named{e},
+and eu{er}y vnyte wol be worth{e} [*Fol. 52.] an .100., and eu{er}y
+article. a�.1000. Whan an article multiplieth{e} a nombre componed{e},
+thow most bryng the digit of the article into aither part of the nombre
+componed{e}; and Ioyne the p{ro}duccio{u}n, and eu{er}y article wol be
+worth{e} .100., and eu{er}y vnyte .10., and so woll{e} the some be
+open{e}. Whan a nombre componed{e} multiplieth{e} a nombre componed{e},
+eu{er}y p{ar}t of the nombre multiplying is to be had{e} into eu{er}y
+p{ar}t of the nombre to be multiplied{e}, and so shall{e} the digit be
+had{e} twies, onys in the digit, that other in the article. The article
+also twies, ones in the digit, that other in the article. Therfor yf
+thow wilt any nombre by hym-self other by any other multiplie, write the
+nombre to be multiplied{e} in the ou{er} ordre by his differences, The
+nombre multiplying in the lower ordre by his differences, so that the
+first of the lower ordre be vnder the last of the ou{er} ordre. This
+done, of the multiplying, the last is to be had{e} into the last of the
+nombre to be multiplied{e}. Wherof than wolle grow a digit, an article,
+other a nombre componed{e}. If it be a digit, even above the figure
+multiplying is hede write his digit that come of, as it appereth{e}
+here:--
+
+  +-----------------------+---+
+  | The resultant         | 6 |
+  +-----------------------+---+
+  | To be multiplied{e}   | 3 |
+  +-----------------------+---+
+  | �e nombre multipliyng | 2 |
+  +-----------------------+---+
+
+And yf an article had be writ ou{er} the fig{ur}e multiplying his hede,
+put a cifre �{er} and transferre the article toward{e} the lift hand{e},
+as thus:--
+
+  +-------------------------+---+---+
+  | The resultant           | 1 | 0 |
+  +-------------------------+---+---+
+  | to be multiplied{e}     |   | 5 |
+  +-------------------------+---+---+
+  | �e nombre m{u}ltipliyng |   | 2 |
+  +-------------------------+---+---+
+
+And yf a nombre componed{e} be writ ou{er} the figure multyplying is
+hede, write the digit in the nombre componed{e} is place, and sette the
+article to the lift hand{e}, as thus:--
+
+  +------------------------+---+---+
+  | Resultant              | 1 | 2 |
+  +------------------------+---+---+
+  | to be multiplied{e}    |   | 4 |
+  +------------------------+---+---+
+  | the nombre multipliyng |   | 3 |
+  +------------------------+---+---+
+
+This done, me most bryng the last save one of the multipliyng into the
+last of �e nombre to be multiplied{e}, and se what comyth{e} therof as
+before, and so do w{i}t{h} all{e}, tille me come to the first of the
+nombre multiplying, that must be brought into the last of the nombre to
+be multiplied{e}, wherof growith{e} o�{er} a digit, an article, [*Fol.
+52b] other a nombre componed{e}. If it be a digit, In the place of the
+ou{er}er, sette a-side, as here:
+
+  +--------------------------+---+---+
+  | Resultant                | 6 | 6 |
+  +--------------------------+---+---+
+  | to be multiplied{e}      |   | 3 |
+  +--------------------------+---+---+
+  | the nombre m{u}ltipliyng | 2 | 2 |
+  +--------------------------+---+---+
+
+If an article happe, there put a cifre in his place, and put hym to the
+lift hand{e}, as here:
+
+  +-------------------------+---+---+---+
+  | The resultant           | 1 | 1 | 0 |
+  +-------------------------+---+---+---+
+  | to be multiplied{e}     |   |   | 5 |
+  +-------------------------+---+---+---+
+  | �e nombre m{u}ltiplying |   | 2 | 2 |
+  +-------------------------+---+---+---+
+
+If it be a nombre componed{e}, in the place of the ou{er}er sette
+a-side, write a digit that[{14}] is a p{ar}t of the componed{e}, and
+sette on the left hond{e} the article, as here:
+
+  +-----------------------------+---+-------+---+
+  | The resultant               | 1 |3[{15}]| 2 |
+  +-----------------------------+---+-------+---+
+  | to be m{u}ltiplied{e}       |   |       | 4 |
+  +-----------------------------+---+-------+---+
+  | �e nombr{e} m{u}ltiplia{n}t |   |   3   | 3 |
+  +-----------------------------+---+-------+---+
+
+  [Sidenote: Then antery the multiplier one place. Work as before.
+  How to deal with ciphers.]
+
+That done, sette forward{e} the figures of the nombre multiplying by oo
+difference, so that the first of the multipliant be vnder the last save
+one of the nombre to be multiplied{e}, the other by o place sette
+forward{e}. Than me shall{e} bryng{e} the last of the m{u}ltipliant in
+hym to be multiplied{e}, vnder the which{e} is the first multipliant.
+And than wolle growe o�{er} a digit, an article, or a componed{e}
+nombre. If it be a digit, adde hym even above his hede; If it be an
+article, transferre hym to the lift side; And if it be a nombre
+componed{e}, adde a digit to the figure above his hede, and sette to the
+lift hand{e} the article. And all{e}-wayes eu{er}y figure of the nombre
+multipliant is to be brought to the last save one nombre to be
+multiplied{e}, til me come to the first of the multipliant, where me
+shall{e} wirche as it is seid{e} before of the first, and aft{er}ward{e}
+to put forward{e} the figures by o difference and one till{e} they
+all{e} be multiplied{e}. And yf it happe that the first figure of �e
+multipliant be a cifre, and boue it is sette the figure signyficatif{e},
+write a cifre in the place of the figur{e} sette a-side, as thus,
+{et}c.:
+
+  +---------------------+---+---+---+
+  | The resultant       | 1 | 2 | 0 |
+  +---------------------+---+---+---+
+  | to be multiplied{e} |   |   | 6 |
+  +---------------------+---+---+---+
+  | the multipliant     |   | 2 | 0 |
+  +---------------------+---+---+---+
+
+  [Sidenote: How to deal with ciphers.]
+
+And yf a cifre happe in the lower order be-twix the first and the last,
+and even above be sette the fig{ur}e signyficatif, leve it vntouched{e},
+as here:--
+
+  +---------------------+---+---+---+---+---+
+  | The resultant       | 2 | 2 | 6 | 4 | 4 |
+  +---------------------+---+---+---+---+---+
+  | To be multiplied{e} |   |   | 2 | 2 | 2 |
+  +---------------------+---+---+---+---+---+
+  | The multipliant     | 1 | 0 | 2 |   |   |
+  +---------------------+---+---+---+---+---+
+
+And yf the space above sette be void{e}, in that place write thow a
+cifre. And yf the cifre happe betwix �e first and the last to be
+m{u}ltiplied{e}, me most sette forward{e} the ordre of the figures by
+thair{e} differences, for oft of duccio{u}n of figur{e}s in cifres
+nought is the resultant, as here,
+
+  +-----------------------+---+---+---+---+---+
+  | Resultant             | 8 | 0 | 0 | 8 |   |
+  +-----------------------+---+---+---+---+---+
+  | to be m{u}ltiplied{e} | 4 | 0 | 0 | 4 |   |
+  +-----------------------+---+---+---+---+---+
+  | the m{u}ltipliant     | 2 | . | . | . |   |
+  +-----------------------+---+---+---+---+---+
+
+[*Fol. 53.] wherof it is evident and open, yf that the first figure of
+the nombre be to be multiplied{e} be a cifre, vndir it shall{e} be none
+sette as here:--
+
+  +-----------------------+---+---+--------+
+  | Resultant             | 3 | 2 |0[{16}] |
+  +-----------------------+---+---+--------+
+  | To be m{u}ltiplied{e} |   | 8 |   0    |
+  +-----------------------+---+---+--------+
+  | The m{u}ltipliant     |   | 4 |        |
+  +-----------------------+---+---+--------+
+
+  [Sidenote: Leave room between the rows of figures.]
+
+Vnder[stand] also that in multiplicacio{u}n, divisio{u}n, and of rootis
+the extraccio{u}n, competently me may leve a mydel space betwix .2.
+ordres of figures, that me may write there what is come of addyng other
+with{e}-drawyng, lest any thynge shold{e} be ou{er}-hipped{e} and sette
+out of mynde.
+
+    [Headnote: Chapter VII. Division.]
+
+  [Sidenote: Definition of division. Dividend, Divisor, Quotient.
+  How to set down your Sum. An example. Examples.]
+
+For to dyvyde oo nombre by a-nother, it is of .2. nombres p{ro}posed{e},
+It is forto depart the moder nombre into as many p{ar}tis as ben of
+vnytees in the lasse nombre. And note wele that in makyng{e} of
+dyvysio{u}n ther ben .3. nombres necessary: that is to sey, the nombre
+to be dyvyded{e}; the nombre dyvydyng and the nombre exeant, other how
+oft, or quocient. Ay shall{e} the nombre that is to be dyvyded{e} be
+more, other at the lest even{e} w{i}t{h} the nombre the dyvysere, yf the
+nombre shall{e} be mad{e} by hole nombres. Therfor yf thow wolt any
+nombre dyvyde, write the nombre to be dyvyded{e} in �e ou{er}er
+bordur{e} by his differences, the dyviser{e} in the lower ordur{e} by
+his differences, so that the last of the dyviser be vnder the last of
+the nombre to be dyvyde, the next last vnder the next last, and so of
+the others, yf it may competently be done; as here:--
+
+  +------------------+---+---+---+
+  | The residue      |   | 2 | 7 |
+  +------------------+---+---+---+
+  | The quotient     |   |   | 5 |
+  +------------------+---+---+---+
+  | To be dyvyded{e} | 3 | 4 | 2 |
+  +------------------+---+---+---+
+  | The dyvyser      |   | 6 | 3 |
+  +------------------+---+---+---+
+
+  +--------------+---+---+----+---+---++---+---+---++---+---+---+
+  | Residuu{m}   |   |   | 8 ||   |   ||   | 2 | 7 ||   | 2 | 6 |
+  +--------------+---+---+---++---+---++---+---+---++---+---+---+
+  | Quociens     |   | 2 | 1 || 2 | 2 ||   |   | 5 ||   |   | 9 |
+  +--------------+---+---+---++---+---++---+---+---++---+---+---+
+  | Diuidend{us} | 6 | 8 | 0 || 6 | 6 || 3 | 4 | 2 || 3 | 3 | 2 |
+  +--------------+---+---+---++---+---++---+---+---++---+---+---+
+  | Diuiser      | 3 | 2 |   || 3 |   ||   | 6 | 3 ||   | 3 | 4 |
+  +--------------+---+---+---++---+---++---+---+---++---+---+---+
+
+  [Sidenote: When the last of the divisor must not be set below the
+  last of the dividend. How to begin.]
+
+And ther ben .2. causes whan the last figure may not be sette vnder the
+last, other that the last of the lower nombre may not be w{i}t{h}-draw
+of the last of the ou{er}er nombre for it is lasse than the lower, other
+how be it, that it myght be w{i}t{h}-draw as for hym-self fro the
+ou{er}er the remenaunt may not so oft of them above, other yf �e last of
+the lower be even to the figure above his hede, and �e next last o�{er}
+the figure be-fore �{a}t be more �an the figure above sette. [*Fol.
+53^2.] These so ordeyned{e}, me most wirch{e} from the last figure of �e
+nombre of the dyvyser, and se how oft it may be w{i}t{h}-draw of and fro
+the figure aboue his hede, namly so that the remen{au}nt may be take of
+so oft, and to se the residue as here:--
+
+  [Sidenote: An example.]
+
+  +------------------+---+---+---+
+  | The residue      |   | 2 | 6 |
+  +------------------+---+---+---+
+  | The quocient     |   |   | 9 |
+  +------------------+---+---+---+
+  | To be dyvyded{e} | 3 | 3 | 2 |
+  +------------------+---+---+---+
+  | The dyvyser      |   | 3 | 4 |
+  +------------------+---+---+---+
+
+  [Sidenote: Where to set the quotiente. Examples.]
+
+And note wele that me may not with{e}-draw more than .9. tymes nether
+lasse than ones. Therfor se how oft �e figures of the lower ordre may be
+w{i}t{h}-draw fro the figures of the ou{er}er, and the nombre that
+shew{i}t{h} �e q{u}ocient most be writ ou{er} the hede of �at figure,
+vnder the which{e} the first figure is, of the dyviser; And by that
+figure me most with{e}-draw all{e} o�{er} figures of the lower ordir and
+that of the figures aboue thair{e} hedis. This so don{e}, me most sette
+forward{e} �e figures of the diuiser by o difference toward{es} the
+right hond{e} and worch{e} as before; and thus:--
+
+  +--------------+---+---+---+---+---+---++---+---+---+---+---+---+---+
+  | Residuu{m}   |   |   |   |   |   |   ||   |   |   |   | . | 1 | 2 |
+  +--------------+---+---+---+---+---+---++---+---+---+---+---+---+---+
+  | quo{ciens}   |   |   |   | 6 | 5 | 4 ||   |   |   | 2 | 0 | 0 | 4 |
+  +--------------+---+---+---+---+---+---++---+---+---+---+---+---+---+
+  | Diuidend{us} | 3 | 5 | 5 | 1 | 2 | 2 || 8 | 8 | 6 | 3 | 7 | 0 | 4 |
+  +--------------+---+---+---+---+---+---++---+---+---+---+---+---+---+
+  | Diuisor      |   | 5 | 4 | 3 |   |   || 4 | 4 | 2 | 3 |   |   |   |
+  +--------------+---+---+---+---+---+---++---+---+---+---+---+---+---+
+
+  +------------------+---+---+---+---+---+---+
+  | The quocient     |   |   |   | 6 | 5 | 4 |
+  +------------------+---+---+---+---+---+---+
+  | To be dyvyded{e} | 3 | 5 | 5 | 1 | 2 | 2 |
+  +------------------+---+---+---+---+---+---+
+  | The dyvyser      |   | 5 | 4 | 3 |   |   |
+  +------------------+---+---+---+---+---+---+
+
+  [Sidenote: A special case.]
+
+And yf it happ{e} after �e settyng forward{e} of the fig{ur}es �{a}t �e
+last of the divisor may not so oft be w{i}t{h}draw of the fig{ur}e above
+his hede, above �at fig{ur}e vnder the which{e} the first of the diuiser
+is writ me most sette a cifre in ordre of the nombre quocient, and sette
+the fig{ur}es forward{e} as be-fore be o difference alone, and so me
+shall{e} do in all{e} nombres to be dyvided{e}, for where the dyviser
+may not be w{i}t{h}-draw me most sette there a cifre, and sette
+forward{e} the figures; as here:--
+
+  +------------------+---+---+---+---+---+---+---+
+  | The residue      |   |   |   |   |   | 1 | 2 |
+  |------------------+---+---+---+---+---+---+---+
+  | The quocient     |   |   |   | 2 | 0 | 0 | 4 |
+  |------------------+---+---+---+---+---+---+---+
+  | To be dyvyded{e} | 8 | 8 | 6 | 3 | 7 | 0 | 4 |
+  |------------------+---+---+---+---+---+---+---+
+  | The dyvyser      | 4 | 4 | 2 | 3 |   |   |   |
+  +------------------+---+---+---+---+---+---+---+
+
+  [Sidenote: Another example. What the quotient shows. How to prove
+  your division, or multiplication.]
+
+And me shall{e} not cesse fro such{e} settyng of fig{ur}es forward{e},
+nether of settyng{e} of �e quocient into the dyviser, ne�{er} of
+subt{ra}ccio{u}n of the dyvyser, till{e} the first of the dyvyser be
+w{i}t{h}-draw fro �e first to be divided{e}. The which{e} don{e}, or
+ought,[{17}] o�{er} nought shall{e} remayne: and yf it be ought,[{17}]
+kepe it in the tables, And eu{er} vny it to �e diviser. And yf �{o}u
+wilt wete how many vnytees of �e divisio{u}n [*Fol. 53^3.] wol growe to
+the nombre of the diviser{e}, the nombre quocient wol shewe it: and whan
+such{e} divisio{u}n is made, and �{o}u lust p{ro}ve yf thow have wele
+done or no, Multiplie the quocient by the diviser, And the same
+fig{ur}es wolle come ayene that thow haddest bifore and none other. And
+yf ought be residue, than w{i}t{h} addicio{u}n therof shall{e} come the
+same figures: And so multiplicacio{u}n p{ro}vith{e} divisio{u}n, and
+dyvisio{u}n multiplicacio{u}n: as thus, yf multiplicacio{u}n be made,
+divide it by the multipliant, and the nombre quocient wol shewe the
+nombre that was to be multiplied{e}, {et}c.
+
+    [Headnote: Chapter VIII. Progression.]
+
+  [Sidenote: Definition of Progression. Natural Progression. Broken
+  Progression. The 1st rule for Natural Progression. The second rule.
+  The first rule of Broken Progression. The second rule.]
+
+Progressio{u}n is of nombre after egall{e} excesse fro oone or tweyn{e}
+take ag{r}egacio{u}n. of p{ro}gressio{u}n one is naturell{e} or
+co{n}tynuell{e}, �{a}t o�{er} broken and discontynuell{e}. Naturell{e}
+it is, whan me begynneth{e} w{i}t{h} one, and kepeth{e} ordure
+ou{er}lepyng one; as .1. 2. 3. 4. 5. 6., {et}c., so �{a}t the nombre
+folowyng{e} passith{e} the other be-fore in one. Broken it is, whan me
+lepith{e} fro o nombre till{e} another, and kepith{e} not the contynuel
+ordir{e}; as 1. 3. 5. 7. 9, {et}c. Ay me may begynne w{i}t{h} .2., as
+�us; .2. 4. 6. 8., {et}c., and the nombre folowyng passeth{e} the others
+by-fore by .2. And note wele, that naturell{e} p{ro}gressio{u}n ay
+begynneth{e} w{i}t{h} one, and Int{er}cise or broken p{ro}gressio{u}n,
+omwhile begynnyth{e} w{i}th one, omwhile w{i}t{h} twayn{e}. Of
+p{ro}gressio{u}n naturell .2. rules ther be yove, of the which{e} the
+first is this; whan the p{ro}gressio{u}n naturell{e} endith{e} in even
+nombre, by the half therof multiplie �e next totall{e} ou{er}er{e}
+nombre; Example of grace: .1. 2. 3. 4. Multiplie .5. by .2. and so .10.
+cometh{e} of, that is the totall{e} nombre �{er}of. The second{e} rule
+is such{e}, whan the p{ro}gressio{u}n naturell{e} endith{e} in nombre
+od{e}. Take the more porcio{u}n of the oddes, and multiplie therby the
+totall{e} nombre. Example of grace 1. 2. 3. 4. 5., multiplie .5. by .3,
+and thryes .5. shall{e} be resultant. so the nombre totall{e} is .15. Of
+p{ro}gresio{u}n int{er}cise, ther ben also .2.[{18}] rules; and �e first
+is �is: Whan the Int{er}cise p{ro}gression endith{e} in even nombre by
+half therof multiplie the next nombre to �at half{e} as .2.[{18}] 4. 6.
+Multiplie .4. by .3. so �at is thryes .4., and .12. the nombre of all{e}
+the p{ro}gressio{u}n, woll{e} folow. The second{e} rule is this: whan
+the p{ro}gressio{u}n int{er}scise endith{e} in od{e}, take �e more
+porcio{u}n of all{e} �e nombre, [*Fol. 53^4.] and multiplie by
+hym-self{e}; as .1. 3. 5. Multiplie .3. by hym-self{e}, and �e some of
+all{e} wolle be .9., {et}c.
+
+    [Headnote: Chapter IX. Extraction of Roots.]
+
+  [Sidenote: The preamble of the extraction of roots. Linear,
+  superficial, and solid numbers. Superficial numbers. Square numbers.
+  The root of a square number. Notes of some examples of square roots
+  here interpolated. Solid numbers. Three dimensions of solids. Cubic
+  numbers. All cubics are solid numbers. No number may be both linear
+  and solid. Unity is not a number.]
+
+Here folowith{e} the extraccio{u}n of rotis, and first in nombre
+q{ua}drat{es}. Wherfor me shall{e} se what is a nombre quadrat, and what
+is the rote of a nombre quadrat, and what it is to draw out the rote of
+a nombre. And before other note this divisio{u}n: Of nombres one is
+lyneal, ano�{er} sup{er}ficiall{e}, ano�{er} quadrat, ano�{er} cubik{e}
+or hoole. lyneal is that �at is considred{e} after the p{ro}cesse,
+havyng{e} no respect to the direccio{u}n of nombre in nombre, As a lyne
+hath{e} but one dymensio{u}n that is to sey after the length{e}. Nombre
+sup{er}ficial is �{a}t cometh{e} of ledyng{e} of oo nombre into
+a-nother, wherfor it is called{e} sup{er}ficial, for it hath{e} .2.
+nombres notyng or mesuryng{e} hym, as a sup{er}ficiall{e} thyng{e}
+hath{e} .2. dimensions, �{a}t is to sey length{e} and brede. And for
+bycause a nombre may be had{e} in a-nother by .2. man{er}s, �{a}t is to
+sey other in hym-self{e}, o�{er} in ano�{er}, Vnderstond{e} yf it be had
+in hym-self, It is a quadrat. ffor dyvisio{u}n write by vnytes, hath{e}
+.4. sides even as a quadrangill{e}. and yf the nombre be had{e} in
+a-no�{er}, the nombre is sup{er}ficiel and not quadrat, as .2. had{e} in
+.3. maketh{e} .6. that is �e first nombre sup{er}ficiell{e}; wherfor it
+is open �at all{e} nombre quadrat is sup{er}ficiel, and not
+co{n}u{er}tid{e}. The rote of a nombre quadrat is �at nombre that is had
+of hym-self, as twies .2. makith{e} 4. and .4. is the first nombre
+quadrat, and 2. is his rote. 9. 8. 7. 6. 5. 4. 3. 2. 1. / The rote of
+the more quadrat .3. 1. 4. 2. 6. The most nombre quadrat 9. 8. 7. 5.
+9. 3. 4. 7. 6. / the remenent ou{er} the quadrat .6. 0. 8. 4. 5. / The
+first caas of nombre quadrat .5. 4. 7. 5. 6. The rote .2. 3. 4. The
+second{e} caas .3. 8. 4. 5. The rote .6. 2. The third{e} caas .2. 8. 1.
+9. The rote .5. 3. The .4. caas .3. 2. 1. The rote .1. 7. / The 5. caas
+.9. 1. 2. 0. 4. / The rote 3. 0. 2. The solid{e} nombre or cubik{e} is
+�at �{a}t comyt[-h]e of double ledyng of nombre in nombre; And it is
+cleped{e} a solid{e} body that hath{e} �{er}-in .3 [dimensions] �at is
+to sey, length{e}, brede, and thiknesse. so �{a}t nombre hath{e} .3.
+nombres to be brought forth{e} in hym. But nombre may be had{e} twies in
+nombre, for other it is had{e} in hym-self{e}, o�{er} in a-no�{er}. If a
+nombre be had{e} twies in hym-self, o�{er} ones in his quadrat, �{a}t is
+the same, �{a}t a cubik{e} [*Fol. 54.] is, And is the same that is
+solide. And yf a nombre twies be had{e} in a-no�{er}, the nombre is
+cleped{e} solide and not cubik{e}, as twies .3. and �{a}t .2. makith{e}
+.12. Wherfor it is opyn{e} that all{e} cubik{e} nombre is solid{e}, and
+not {con}u{er}tid{e}. Cubik{e} is �{a}t nombre �at comyth{e} of
+ledyng{e} of hym-self{e} twyes, or ones in his quadrat. And here-by it
+is open that o nombre is the roote of a quadrat and of a cubik{e}.
+Natheles the same nombre is not q{ua}drat and cubik{e}. Opyn{e} it is
+also that all{e} nombres may be a rote to a q{ua}drat and cubik{e}, but
+not all{e} nombre quadrat or cubik{e}. Therfor sithen �e ledyng{e} of
+vnyte in hym-self ones or twies nought cometh{e} but vnytes, Seith{e}
+Boice in Arsemetrik{e}, that vnyte potencially is al nombre, and none in
+act. And vndirstond{e} wele also that betwix euery .2. quadrat{es} ther
+is a meene p{ro}porcionall{e}, That is opened{e} thus; lede the rote of
+o quadrat into the rote of the o�{er} quadrat, and �an wolle �e meene
+shew.
+
+  [Sidenote: Examples of square roots.]
+
+  +-------------+-+-+-+-++-+-+-+-++-+-+-+-+-++-+---+------+-+
+  | Residuu{m}  | | |0| || | | |4|| | |0| | || |   |   0  | |
+  +-------------+-+-+-+-++-+-+-+-++-+-+-+-+-++-+---+------+-+
+  | Quadrand{e} |4|3|5|6||3|0|2|9||1|7|4|2|4||1| 9 |   3  |6|
+  +-------------+-+-+-+-++-+-+-+-++-+-+-+-+-++-+---+------+-+
+  | Duplum      |1|2| | ||1|0| | ||2| |6| | || |[8]|[{19}]| |
+  +-------------+-+-+-+-++-+-+-+-++-+-+-+-+-++-+---+------+-+
+  | Subduplu{m} | |6| |6|| |5| |5||1| |3| |2|| | 4 |      |4|
+  +-------------+-+-+-+-++-+-+-+-++-+-+-+-+-++-+---+------+-+
+
+  [Sidenote: A note on mean proportionals.]
+
+Also betwix the next .2. cubikis, me may fynde a double meene, that is
+to sey a more meene and a lesse. The more meene thus, as to bryng{e} the
+rote of the lesse into a quadrat of the more. The lesse thus, If the
+rote of the more be brought Into the quadrat of the lesse.
+
+    [Headnote: Chapter X. Extraction of Square Root.]
+
+  [Sidenote: To find a square root. Begin with the last odd place.
+  Find the nearest square root of that number, subtract, double it,
+  and set the double one to the right. Find the second figure by
+  division. Multiply the double by the second figure, and add after
+  it the square of the second figure, and subtract.]
+
+[{20}]To draw a rote of the nombre quadrat it is What-eu{er} nombre be
+p{ro}posed{e} to fynde his rote and to se yf it be quadrat. And yf it be
+not quadrat the rote of the most quadrat fynde out, vnder the nombre
+p{ro}posed{e}. Therfor yf thow wilt the rote of any quadrat nombre draw
+out, write the nombre by his differences, and compt the nombre of the
+figures, and wete yf it be od{e} or even. And yf it be even, than most
+thow begynne worche vnder the last save one. And yf it be od{e} w{i}t{h}
+the last; and forto sey it shortly, al-weyes fro the last od{e} me
+shall{e} begynne. Therfor vnder the last in an od place sette, me most
+fynd{e} a digit, the which{e} lad{e} in hym-self{e} it puttith{e} away
+that, �at is ou{er} his hede, o�{er} as neigh{e} as me may: suche a
+digit found{e} and w{i}t{h}draw fro his ou{er}er, me most double that
+digit and sette the double vnder the next figure toward{e} the right
+hond{e}, and his vnder double vnder hym. That done, than me most
+fy{n}d{e} a-no�{er} digit vnder the next figure bifore the doubled{e},
+the which{e} [*Fol. 54b] brought in double setteth{e} a-way all{e} that
+is ou{er} his hede as to reward{e} of the doubled{e}: Than brought into
+hym-self settith{e} all away in respect of hym-self, Other do it as nye
+as it may be do: other me may w{i}t{h}-draw the digit [{21}][last]
+found{e}, and lede hym in double or double hym, and after in
+hym-self{e}; Than Ioyne to-geder the p{ro}duccion{e} of them bothe, So
+that the first figure of the last p{ro}duct be added{e} before the first
+of the first p{ro}duct{es}, the second{e} of the first, {et}c. and so
+forth{e}, subtrahe fro the totall{e} nombre in respect of �e digit.
+
+  [Sidenote: Examples.]
+
+  +------------------+-+-+-+-+-++-+-+-+-+-++---+-+---+-+---+-+-+
+  | The residue      | | | | | || | | | | ||   | |   |5| 4 |3|2|
+  +------------------+-+-+-+-+-++-+-+-+-+-++---+-+---+-+---+-+-+
+  | To be quadred{e} |4|1|2|0|9||1|5|1|3|9|| 9 |0| 0 |5| 4 |3|2|
+  +------------------+-+-+-+-+-++-+-+-+-+-++---+-+---+-+---+-+-+
+  | The double       | |4|0| | || |2| |4| ||   |6|   |0|   | |0|
+  +------------------+-+-+-+-+-++-+-+-+-+-++---+-+---+-+---+-+-+
+  | The vnder double |2| |0| |3||1| |2| |3||[3]| |[0]| |[0]| |0|
+  +------------------+-+-+-+-+-++-+-+-+-+-++---+-+---+-+---+-+-+
+
+  [Sidenote: Special cases. The residue.]
+
+And if it hap �{a}t no digit may be found{e}, Than sette a cifre vndre
+a cifre, and cesse not till{e} thow fynde a digit; and whan thow hast
+founde it to double it, ne�{er} to sette the doubled{e} forward{e}
+nether the vnder doubled{e}, Till thow fynde vndre the first figure a
+digit, the which{e} lad{e} in all{e} double, settyng away all{e} that is
+ou{er} hym in respect of the doubled{e}: Than lede hym into hym-self{e},
+and put a-way all{e} in regard{e} of hym, other as nygh{e} as thow
+maist. That done, other ought or nought wolle be the residue. If nought,
+than it shewith{e} that a nombre componed{e} was the quadrat, and his
+rote a digit last found{e} w{i}t{h} vnder{e}-double other vndirdoubles,
+so that it be sette be-fore: And yf ought[{22}] remayn{e}, that
+shew{i}t{h} that the nombre p{ro}posed{e} was not quadrat,[{23}]
+[[wher-vpon{e} se the table in the next side of the next leef{e}.]]
+but a digit [last found with the subduple or subduples is]
+
+  [Sidenote: This table is constructed for use in cube root sums,
+  giving the value of ab.^2]
+
+  +---+-----+-----+-----+-----+-----+---------+-----+---------+
+  | 1 |   2 |   3 |   4 |   5 |   6 |    7    |   8 |     9   |
+  +---+-----+-----+-----+-----+-----+---------+-----+---------+
+  | 2 |   8 |  12 |  16 |  20 |  24 |   28    |  32 |    36   |
+  +---+-----+-----+-----+-----+-----+---------+-----+---------+
+  | 3 |  18 |  27 |  36 |  45 |  54 |   63    |  72 |    81   |
+  +---+-----+-----+-----+-----+-----+---------+-----+---------+
+  | 4 |  32 |  48 |  64 |  80 |  96 |112[{24}]| 128 |   144   |
+  +---+-----+-----+-----+-----+-----+---------+-----+---------+
+  | 5 |  50 |  75 | 100 | 125 | 150 |   175   | 200 |   225   |
+  +---+-----+-----+-----+-----+-----+---------+-----+---------+
+  | 6 |  72 | 108 | 144 | 180 | 216 |   252   | 288 |   324   |
+  +---+-----+-----+-----+-----+-----+---------+-----+---------+
+  | 7 |  98 | 147 | 196 | 245 | 294 |   343   | 393 |   441   |
+  +---+-----+-----+-----+-----+-----+---------+-----+---------+
+  | 8 | 128 | 192 | 256 | 320 | 384 |   448   | 512 |   576   |
+  +---+-----+-----+-----+-----+-----+---------+-----+---------+
+  | 9 | 168 | 243 | 324 | 405 | 486 |   567   | 648 |729[{25}]|
+  +---+-----+-----+-----+-----+-----+---------+-----+---------+
+
+  [Sidenote: How to prove the square root without or with a remainder.]
+
+The rote of the most quadrat conteyned{e} vndre the nombre
+p{ro}posed{e}. Therfor yf thow wilt p{ro}ve yf thow have wele do or no,
+Multiplie the digit last found{e} w{i}t{h} the vnder-double o�{er}
+vnder-doublis, and thow shalt fynde the same figures that thow haddest
+before; And so that nought be the [*Fol. 55.] residue. And yf thow have
+any residue, than w{i}t{h} the addicio{u}n �{er}of that is res{er}ued{e}
+w{i}t{h}-out in thy table, thow shalt fynd{e} thi first figures as thow
+haddest them before, {et}c.
+
+    [Headnote: Chapter XI. Extraction of Cube Root.]
+
+  [Sidenote: Definition of a cubic number and a cube root. Mark off
+  the places in threes. Find the first digit; treble it and place it
+  under the next but one, and multiply by the digit. Then find the
+  second digit. Multiply the first triplate and the second digit, twice
+  by this digit. Subtract. Examples.]
+
+Heere folowith{e} the extraccio{u}n of rotis in cubik{e} nombres;
+wher-for me most se what is a nombre cubik{e}, and what is his roote,
+And what is the extraccio{u}n of a rote. A�nombre cubik{e} it is, as it
+is before declared{e}, that cometh{e} of ledyng of any nombre twies in
+hym-self{e}, other ones in his quadrat. The rote of a nombre cubik{e} is
+the nombre that is twies had{e} in hy{m}-self{e}, or ones in his
+quadrat. Wher-thurgh{e} it is open, that eu{er}y nombre quadrat or
+cubik{e} have the same rote, as it is seid{e} before. And forto draw out
+the rote of a cubik{e}, It is first to fynd{e} �e nombr{e} p{ro}posed{e}
+yf it be a cubik{e}; And yf it be not, than thow most make extraccio{u}n
+of his rote of the most cubik{e} vndre the nombre p{ro}posid{e} his rote
+found{e}. Therfor p{ro}posed{e} some nombre, whos cubical rote �{o}u
+woldest draw out; First thow most compt the figures by fourthes, that is
+to sey in the place of thousand{es}; And vnder the last thousand{e}
+place, thow most fynde a digit, the which{e} lad{e} in hym-self cubikly
+puttith{e} a-way that �at is ou{er} his hede as in respect of hym, other
+as nygh{e} as thow maist. That done, thow most trebill{e} the digit, and
+that triplat is to be put vnder the .3. next figure toward{e} the right
+hond{e}, And the vnder-trebill{e} vnder the trebill{e}; Than me most
+fynd{e} a digit vndre the next figure bifore the triplat, the which{e}
+w{i}t{h} his vnder-trebill{e} had into a trebill{e}, aft{er}warde other
+vnder[trebille][{26}] had in his p{ro}duccio{u}n, putteth{e} a-way
+all{e} that is ou{er} it in regard{e} of[{27}] [the triplat. Then lade
+in hymself puttithe away that �at is over his hede as in respect of hym,
+other as nyghe as thou maist:] That done, thow most trebill{e} the digit
+ayene, and the triplat is to be sette vnder the next .3. figure as
+before, And the vnder-trebill{e} vnder the trebill{e}: and than most
+thow sette forward{e} the first triplat w{i}t{h} his vndre-trebill{e} by
+.2. differences. And than most thow fynde a digit vnder the next figure
+before the triplat, the which{e} with{e} his vnder-t{r}iplat had in his
+triplat afterward{e}, other vnder-treblis lad in p{ro}duct [*Fol. 55b]
+It sitteth{e} a-way a[l~l] that is ou{er} his hede in respect of the
+triplat than had in hym-self cubikly,[{28}] [[it setteth{e} a-way all{e}
+his respect]] or as nygh{e} as ye may.
+
+  +----------------+--+-+-+-+-+-+---++--+-+-+-+-+--++----+-+--+-+--+
+  | Residuu{m}     |  | | | | | | 5 ||  | | | | | 4||   1|0|1 |9|  |
+  +----------------+--+-+-+-+-+-+---++--+-+-+-+-+--++----+-+--+-+--+
+  | Cubicandu{s}   | 8|3|6|5|4|3| 2 || 3|0|0|7|6| 7|| 1 1|6|6 |7|  |
+  +----------------+--+-+-+-+-+-+---++--+-+-+-+-+--++----+-+--+-+--+
+  | Triplum        |  | |6|0| | |   ||  | | |1|8|  ||    | |4 | |  |
+  +----------------+--+-+-+-+-+-+---++--+-+-+---+--++----+-+--+-+--+
+  | Subt{r}iplu{m} | 2| | |0| | |[3]||  | |6| | | 7||   2| |  |2|  |
+  +----------------+--+-+-+-+-+-+---++--+-+-+-+-+--++----+-+--+-+--+
+
+  [Sidenote: Continue this process till the first figure is reached.
+  Examples. The residue. Special cases. Special case.]
+
+Nother me shall{e} not cesse of the fyndyng{e} of that digit, neither of
+his triplacio{u}n, ne�{er} of the triplat-is [{29}]anteriorac{i}o{u}n,
+that is to sey, settyng forward{e} by .2. differences, Ne therof the
+vndre-triple to be put vndre the triple, Nether of the multiplicacio{u}n
+�{er}of, Neither of the subtraccio{u}n, till{e} it come to the first
+figure, vnder the which{e} is a digitall{e} nombre to be found{e}, the
+which{e} with{e} his vndre-treblis most be had{e} in tribles,
+After-ward{e} w{i}t{h}out vnder-treblis to be had{e} into produccio{u}n,
+settyng away all{e} that is ou{er} the hed{e} of the triplat nombre,
+After had into hymself{e} cubikly, and sette all{e}-way that is ou{er}
+hym.
+
+  +------------------+---+---+---+---++---+---+---+---+---+
+  | To be cubiced{e} | 1 | 7 | 2 | 8 || 3 | 2 | 7 | 6 | 8 |
+  +------------------+---+---+---+---++---+---+---+---+---+
+  | The triple       |   |   | 3 | 2 ||   |   |   | 9 |   |
+  +------------------+---+---+---+---++---+---+---+---+---+
+  | The vnder triple |   |   | 1 | 2 ||   |[3]|   | 3 | 3 |
+  +------------------+---+---+---+---++---+---+---+---+---+
+
+Also note wele that the p{ro}ducc{i}on comyng{e} of the ledyng of a
+digite found{e}[{30}] [[w{i}t{h} an vndre-triple / other of an
+vndre-triple in a triple or triplat is And after-ward{e} w{i}t{h} out
+vndre-triple other vndre-triplis in the p{ro}duct and ayene that
+p{ro}duct that cometh{e} of the ledyng{e} of a digit found{e} in
+hym-self{e} cubicall{e}]] me may adde to, and also w{i}t{h}-draw fro of
+the totall{e} nombre sette above that digit so found{e}.[{31}] [[as ther
+had be a divisio{u}n made as it is opened{e} before]] That done ought or
+nought most be the residue. If it be nought, It is open that the nombre
+p{ro}posed{e} was a cubik{e} nombre, And his rote a digit founde last
+w{i}t{h} the vnder-triples: If the rote therof wex bad{e} in
+hym-self{e}, and afterward{e} p{ro}duct they shall{e} make the first
+fig{ur}es. And yf ought be in residue, kepe that w{i}t{h}out in the
+table; and it is open{e} that the nombre was not a cubik{e}. but a digit
+last founde w{i}t{h} the vndirtriplis is rote of the most cubik{e} vndre
+the nombre p{ro}posed{e} conteyned{e}, the which{e} rote yf it be had{e}
+in hym-self{e}, And aft{er}ward{e} in a p{ro}duct of that shall{e} growe
+the most cubik{e} vndre the nombre p{ro}posed{e} conteyned{e}, And yf
+that be added{e} to a cubik{e} the residue res{er}ued{e} in the table,
+woll{e} make the same figures that ye had{e} first. [*Fol. 56.] And yf
+no digit after the anterioracio{u}n[{32}] may not be found{e}, than put
+ther{e} a cifre vndre a cifre vndir the third{e} figure, And put
+forward{e} �e fig{ur}es. Note also wele that yf in the nombre
+p{ro}posed{e} ther ben no place of thowsand{es}, me most begynne vnder
+the first figure in the extraccio{u}n of the rote. some vsen forto
+distingue the nombre by threes, and ay begynne forto wirch{e} vndre the
+first of the last t{er}nary other unco{m}plete nombre, the which{e}
+maner of op{er}acio{u}n accordeth{e} w{i}t{h} that before. And this at
+this tyme suffiseth{e} in extraccio{u}n of nombres quadrat or cubik{es}
+{et}c.
+
+  [Sidenote: Examples.]
+
+  +-------------------+---+--+------+--+--+--+--++--+--+--+--+--+--+--+
+  | The residue       |   |  |      |  |  |  | 0||  |  |  |  |  | 1| 1|
+  +-------------------+---+--+------+--+--+--+--++--+--+--+--+--+--+--+
+  | The cubicand{us}  | 8 | 0|   0  | 0| 0| 0| 0|| 8| 2| 4| 2| 4| 1| 9|
+  +-------------------+---+--+------+--+--+--+--++--+--+--+--+--+--+--+
+  | The triple        |   |  |[{33}]| 0| 0|  |  ||  |  | 6|  |  |  |  |
+  +-------------------+---+--+------+--+--+--+--++--+--+--+--+--+--+--+
+  | The vndert{r}iple |[2]|  |      | 0| 0|  |  || 2|  |  | 6| 2|  |  |
+  +-------------------+---+--+------+--+--+--+--++--+--+--+--+--+--+--+
+
+
+    [Headnote: Table of Numbers, &c.]
+
+  [Sidenote: A table of numbers; probably from the Abacus.]
+
+   1   2         3              4               5              6
+  one. x. an. hundred{e}/ a thowsand{e}/ x. thowsand{e}/ An hundred{e}
+                     7
+  thowsand{e}/ A thowsand{e} tymes a thowsand{e}/ x. thousand{e} tymes
+
+  a thousand{e}/ An hundred{e} thousand{e} tymes a thousand{e} A
+
+  thousand{e} thousand{e} tymes a thousand{e}/ this is the x place
+
+    {et}c.
+
+
+[Ende.]
+
+
+FOOTNOTES (The Art of Nombryng):
+
+  [1: MS. Materiall{e}.]
+  [2: MS. Formall{e}.]
+  [3: 'the' in MS.]
+  [4: 'be' in MS.]
+  [5: 'and' in MS.]
+  [6: 'is' in MS.]
+  [7: 6 in MS.]
+  [8: 0 in MS.]
+  [9: 2 in MS.]
+  [10: _sic._]
+  [11: 'And' inserted in MS.]
+  [12: '4 the' inserted in MS.]
+  [13: 'to' in MS.]
+  [14: 'that' repeated in MS.]
+  [15: '1' in MS.]
+  [16: Blank in MS.]
+  [17: 'nought' in MS.]
+  [18: 3 written for 2 in MS.]
+  [19: 7 in MS.]
+  [20: runs on in MS.]
+  [21: 'so' in MS.]
+  [22: 'nought' in MS.]
+  [23: MS. adds here: 'wher-vpon{e} se the table in the next side of
+    the next leef{e}.']
+  [24: 110 in MS.]
+  [25: 0 in MS.]
+  [26: double in MS.]
+  [27: 'it hym-self{e}' in MS.]
+  [28: MS. adds here: 'it setteth{e} a-way all{e} his respect.']
+  [29: 'aucterioracio{u}n' in MS.]
+  [30: MS. adds here: 'w{i}t{h} an vndre-triple / other of an
+    vndre-triple in a triple or triplat is And after-ward{e} w{i}t{h}
+    out vndre-triple other vndre-triplis in the p{ro}duct and ayene
+    that p{ro}duct that cometh{e} of the ledyng{e} of a digit found{e}
+    in hym-self{e} cubicall{e}' /]
+  [31: MS. adds here: 'as ther had be a divisio{u}n made as it is
+    opened{e} before.']
+  [32: MS. anteriocacio{u}n.]
+  [33: 4 in MS.]
+
+
+
+
+Accomptynge by counters.
+
+  [Transcriber's Note:
+
+  The original text was printed as a single continuous paragraph, with
+  no break between speakers; all examples were shown inline. It has been
+  broken up for this e-text.]
+
+
+  [*116b]
+
+  � The seconde dialoge of accomptynge by counters.
+
+_Mayster._
+
+Nowe that you haue learned the commen kyndes of Arithmetyke with the
+penne, you shall se the same art in cou{n}ters: whiche feate doth not
+only serue for them that can not write and rede, but also for them that
+can do bothe, but haue not at some tymes theyr penne or tables redye
+with them. This sorte is in two fourmes co{m}menly. The one by lynes,
+and the other without lynes: in that y^t hath lynes, the lynes do stande
+for the order of places: and in y^t that hath no lynes, there must be
+sette in theyr stede so many counters as shall nede, for eche lyne one,
+and they shall supplye the stede of the lynes.
+
+_S._ By examples I shuld better p{er}ceaue your meanynge.
+
+_M._ For example of the [*117a.] ly[*]nes:
+
+  ----1-0-0-0-0-0--
+  ----1-0-0-0-0----
+  -X--1-0-0-0------
+  ----1-0-0--------
+  ----1-0----------
+  ----1------------
+
+  [Sidenote: Numeration.]
+
+Lo here you se .vi. lynes whiche stande for syxe places so that the
+nethermost standeth for y^e fyrst place, and the next aboue it, for the
+second: and so vpward tyll you come to the hyghest, which is the syxte
+lyne, and standeth for the syxte place. Now what is the valewe of euery
+place or lyne, you may perceaue by the figures whiche I haue set on
+them, which is accordynge as you learned before in the Numeration of
+figures by the penne: for the fyrste place is the place of vnities or
+ones, and euery counter set in that lyne betokeneth but one: {and} the
+seconde lyne is the place of 10, for euery counter there, standeth for
+10. The thyrd lyne the place of hundredes: the fourth of thousandes:
+{and} so forth.
+
+_S._ Syr I do perceaue that the same order is here of lynes, as was in
+the other figures [*117b] by places, so that you shall not nede longer
+to stande about Numeration, excepte there be any other difference.
+
+_M._ Yf you do vndersta{n}de it, then how wyll you set 1543?
+
+_S._ Thus, as I suppose.
+
+  -------
+  -X--1--
+  ----5--
+  ----4--
+  ----3--
+
+_M._ You haue set y^e places truely, but your figures be not mete for
+this vse: for the metest figure in this behalfe, is the figure of a
+cou{n}ter round, as you se here, where I haue expressed that same summe.
+
+  -------------
+
+  -X--o--------
+       o
+  -------------
+
+  ----o-o-o-o--
+
+  ----o-o-o----
+
+_S._ So that you haue not one figure for 2, nor 3, nor 4, and so forth,
+but as many digettes as you haue, you set in the lowest lyne: and for
+euery 10 you set one in the second line: and so of other. But I know not
+by what reason you set that one counter for 500 betwene two lynes.
+
+_M._ you shall remember this, that when so euer you nede to set downe 5,
+50, or 500, or 5000, or so forth any other nomber, whose numerator
+[*118a] is 5, you shall set one counter for it, in the next space aboue
+the lyne that it hath his denomination of, as in this example of that
+500, bycause the numerator is 5, it must be set in a voyd space: and
+bycause the denominator is hundred, I�knowe that his place is the voyde
+space next aboue hundredes, that is to say, aboue the thyrd lyne. And
+farther you shall marke, that in all workynge by this sorte, yf you
+shall sette downe any summe betwene 4 and 10, for the fyrste parte of
+that nomber you shall set downe 5, &�then so many counters more, as
+there reste no{m}bers aboue 5. And this is true bothe of digettes and
+articles. And for example I wyll set downe this su{m}me 287965,
+
+  -X-----------
+
+  ------o-o----
+           o
+  ------o-o-o--
+           o
+  -X----o-o----
+         o
+  ----o-o-o-o--
+       o
+  ----o--------
+       o
+  -------------
+
+which su{m}me yf you marke well, you nede none other exa{m}ples for to
+lerne the numeration of [*118b] this forme. But this shal you marke,
+that as you dyd in the other kynde of arithmetike, set a pricke in the
+places of thousa{n}des, in this worke you shall sette a starre, as you
+se here.
+
+    [Headnote: Addition on the Counting Board.]
+
+  [Sidenote: Addition.]
+
+_S._ Then I perceave numeration, but I praye you, howe shall I do in
+this arte to adde two summes or more together?
+
+_M._ The easyest way in this arte is, to adde but 2 su{m}mes at ones
+together: how be it you may adde more, as I wyll tell you anone.
+Therfore when you wyll adde two su{m}mes, you shall fyrst set downe one
+of them, it forseth not whiche, {and} then by it drawe a lyne crosse the
+other lynes. And afterward set downe the other su{m}me, so that that
+lyne may be betwene them, as yf you wolde adde 2659 to 8342, you must
+set your su{m}mes as you se
+
+  -------------|-----------
+       o       |
+  -X--o-o-o----|--o-o------
+               |   o
+  ----o-o-o----|--o--------
+               |   o
+  ----o-o-o-o--|-----------
+               |   o
+  ----o-o------|--o-o-o-o--
+
+here. And then yf you lyst, you [*119a] may adde the one to the other in
+the same place, or els you may adde them both together in a newe place:
+which waye, bycause it is moste playnest, I�wyll showe you fyrst.
+Therfore wyl I begynne at the vnites, whiche in the fyrst su{m}me is but
+2, {and} in y^e second su{m}me 9, that maketh 11, those do I take vp,
+and for them I set 11 in the new roume, thus,
+
+  -------------|-------|-------
+       o       |       |
+  -X--o-o-o----|--o-o--|-------
+               |   o   |
+  ----o-o-o----|--o----|-------
+               |    o  |
+  ----o-o-o-o--|-------|-o-----
+               |       |
+  -------------|-------|-o-----
+
+Then do I take vp all y^e articles vnder a hundred, which in the fyrst
+su{m}me are 40, and in the second summe 50, that maketh 90: or you may
+saye better, that in the fyrste summe there are 4 articles of 10, and in
+the seconde summe 5, which make 9, but then take hede that you sette
+them in theyr [*119b] ryght lynes as you se here.
+
+  -----------|----------|-------------
+       o     |          |
+  -X--o-o-o--|--o-o-----|-------------
+             |   o      |
+  ----o-o-o--|--o-------|-------------
+             |          |   o
+  -----------|----------|--o-o-o-o-o--
+             |          |
+  -----------|----------|--o----------
+
+Where I haue taken awaye 40 fro{m} the fyrste su{m}me, and 50 from y^e
+second, and in theyr stede I haue set 90 in the thyrde, whiche I haue
+set playnely y^t you myght well perceaue it: how be it seynge that 90
+with the 10 that was in y^e thyrd roume all redy, doth make 100,
+I�myghte better for those 6 cou{n}ters set 1 in the thyrde lyne, thus:
+
+  ----------
+
+  -X--------
+
+  ----o-----
+
+  ----------
+
+  ----o-----
+
+For it is all one summe as you may se, but it is beste, neuer to set 5
+cou{n}ters in any line, for that may be done with 1 cou{n}ter in a
+hygher place.
+
+_S._ I iudge that good reaso{n}, for many are vnnedefull, where one wyll
+serue.
+
+_M._ Well, then [*120a] wyll I adde forth of hundredes: I�fynde 3 in the
+fyrste summe, and 6 in the seconde, whiche make 900, them do I take vp
+{and} set in the thyrd roume where is one hundred all redy, to whiche I
+put 900, and it wyll be 1000, therfore I set one cou{n}ter in the fourth
+lyne for them all, as you se here.
+
+  -----------|-------|--------
+       o     |       |
+  -X--o-o-o--|--o-o--|--o-----
+             |       |
+  -----------|-------|--------
+             |       |
+  -----------|-------|--------
+             |       |
+  -----------|-------|--o-----
+
+Then adde I y^e thousandes together, whiche in the fyrst su{m}me are
+8000, {and} in y^e second 2000, that maketh 10000: them do I take vp
+fro{m} those two places, and for them I set one counter in the fyfte
+lyne, and then appereth as you�se, to be 11001, for so many doth amount
+of the addition of 8342 to 2659.
+
+  ----o-----
+
+  -X--o-----
+
+  ----------
+
+  ----------
+
+  ----o-----
+
+[*120b] _S._ Syr, this I do perceave: but how shall I set one su{m}me to
+an other, not chaungynge them to a thyrde place?
+
+_M._ Marke well how I do it: I�wyll adde together 65436, and 3245,
+whiche fyrste I set downe thus.
+
+  -------------|--------------
+               |      o
+  -------------|--o-----------
+               |   o
+  -X--o-o-o----|--------------
+               |
+  ----o-o------|--o-o-o-o-----
+               |
+  ----o-o-o-o--|--o-o-o-------
+       o       |       o
+  -------------|--o-----------
+
+Then do I begynne with the smalest, which in the fyrst summe is 5, that
+do I take vp, and wold put to the other 5 in the seconde summe, sauynge
+that two counters can not be set in a voyd place of 5, but for them
+bothe I must set 1 in the seconde lyne, which is the place of 10,
+therfore I take vp the 5 of the fyrst su{m}me, {and} the 5 of the
+seco{n}de, and for them I set 1 in the seco{n}d lyne, [*121a] as you se
+here.
+
+  -------------|--------------
+               |   o
+  -------------|--o-----------
+               |   o
+  -X--o-o-o----|--------------
+               |
+  ----o-o------|--o-o-o-o-----
+               |
+  ----o-o-o-o--|--o-o-o-o-----
+               |
+  -------------|--o-----------
+
+Then do I lyke wayes take vp the 4 counters of the fyrste su{m}me {and}
+seconde lyne (which make 40) and adde them to the 4 counters of the same
+lyne, in the second su{m}me, and it maketh 80, But as I sayde I maye not
+conueniently set aboue 4 cou{n}ters in one lyne, therfore to those 4
+that I toke vp in the fyrst su{m}me, I�take one also of the seconde
+su{m}me, and then haue I taken vp 50, for whiche 5 counters I sette
+downe one in the space ouer y^e second lyne, as here doth appere.
+
+  -----------|--------------
+             |   o
+  -----------|--o-----------
+             |   o
+  -X--o-o-o--|--------------
+             |
+  ----o-o----|--o-o-o-o-----
+             |   o
+  -----------|--o-o-o-------
+             |
+  -----------|--o-----------
+
+[*121b.] and then is there 80, as well w^t those 4 counters, as yf I had
+set downe y^e other 4 also. Now do I take the 200 in the fyrste su{m}me,
+and adde them to the 400 in the seconde summe, and it maketh 600,
+therfore I take vp the 2 counters in the fyrste summe, and 3 of them in
+the seconde summe, and for them 5 I set 1 in y^e space aboue, thus.
+
+  -----------|------------
+             |   o
+  -----------|--o---------
+             |   o
+  -X--o-o-o--|------------
+             |   o
+  -----------|--o---------
+             |   o
+  -----------|--o-o-o-----
+             |
+  -----------|--o---------
+
+Then I take y^e 3000 in y^e fyrste su{m}me, vnto whiche there are none
+in the second summe agreynge, therfore I do onely remoue those 3
+counters from the fyrste summe into the seconde, as here doth appere.
+
+  ----|-------------
+      |  o
+  ----|---o---------
+      |  o
+  -X--|---o-o-o-----
+      |  o
+  ----|-o-----------
+      |  o
+  ----|---o-o-o-----
+      |
+  ----|---o---------
+
+[*122a.] And so you see the hole su{m}me, that amou{n}teth of the
+addytio{n} of 65436 with 3245 to be 6868[1]. And yf you haue marked
+these two exa{m}ples well, you nede no farther enstructio{n} in Addition
+of 2 only summes: but yf you haue more then two summes to adde, you may
+adde them thus. Fyrst adde two of them, and then adde the thyrde, and
+y^e fourth, or more yf there be so many: as yf I wolde adde 2679 with
+4286 and 1391. Fyrste I adde the two fyrste summes thus.
+
+  -------------|-----------|--------------
+               |           |   o
+  -X--o-o------|--o-o-o-o--|--o-----------
+       o       |           |   o
+  ----o--------|--o-o------|--o-o-o-o-----
+       o       |   o       |   o
+  ----o-o------|--o-o-o----|--o-----------
+       o       |   o       |   o
+  ----o-o-o-o--|--o--------|--------------
+
+[*122b.] And then I adde the thyrde thereto thus. And so of more yf you
+haue them.
+
+  -------------|-----------|------------
+               |   o       |       o
+  -X--o--------|--o--------|--o-o-o-----
+               |   o       |
+  ----o-o-o----|--o-o-o-o--|--o-o-o-----
+       o       |   o       |   o
+  ----o-o-o-o--|----o------|------------
+               |   o       |   o
+  ----o--------|-----------|--o---------
+
+
+    [Headnote: Subtraction on the Counting Board.]
+
+  [Sidenote: Subtraction.]
+
+_S._ Nowe I thynke beste that you passe forth to Subtraction, except
+there be any wayes to examyn this maner of Addition, then I thynke that
+were good to be knowen nexte.
+
+_M._ There is the same profe here that is in the other Addition by the
+penne, I�meane Subtraction, for that onely is a sure waye: but
+consyderynge that Subtraction must be fyrste knowen, I�wyl fyrste teache
+you the arte of Subtraction, and that by this example: I�wolde subtracte
+2892 out of 8746. These summes must I set downe as I dyd in Addition:
+but here it is best [*116a (_sic_).] to set the lesser no{m}ber fyrste,
+thus.
+
+  -------------|--------------
+               |     o
+  -X--o-o------|--o-o-o-------
+       o       |   o
+  ----o-o-o----|--o-o---------
+       o       |
+  ----o-o-o-o--|--o-o-o-o-----
+               |   o
+  ----o-o------|--o-----------
+
+Then shall I begynne to subtracte the greatest nombres fyrste (contrary
+to the vse of the penne) y^t is the thousandes in this exa{m}ple:
+therfore I fynd amongest the thousandes 2, for which I withdrawe so many
+fro{m} the seconde summe (where are�8) and so remayneth there 6, as this
+exa{m}ple showeth.
+
+  -------------+--------------
+               |   o
+  -+-----------+--o-----------
+       o       |   o
+  ----o-o-o----+--o-o---------
+       o       |
+  ----o-o-o-o--+--o-o-o-o-----
+               |   o
+  ----o-o------+--o-----------
+
+Then do I lyke wayes with the hundredes, of whiche in the fyrste summe
+[*116b] I fynde 8, and is the seconde summe but 7, out of whiche I can
+not take 8, therfore thus muste I do: I�muste loke how moche my summe
+dyffereth from 10, whiche I fynde here to be 2, then must I bate for my
+su{m}me of 800, one thousande, and set downe the excesse of hundredes,
+that is to saye 2, for so moche 100[0] is more then I shuld take vp.
+Therfore fro{m} the fyrste su{m}me I take that 800, and from the second
+su{m}me where are 6000, I�take vp one thousande, and leue 5000; but then
+set I downe the 200 unto the 700 y^t are there all redye, and make them
+900 thus.
+
+  -------------+--------------
+               |   o
+  -+-----------+--------------
+               |   o
+  -------------+--o-o-o-o-----
+         o     |
+  ----o-o-o-o--+--o-o-o-o-----
+               |   o
+  ----o-o------+--o-----------
+
+Then come I to the articles of te{n}nes where in the fyrste su{m}me I
+fynde 90, [*117a] and in the seconde su{m}me but only 40: Now
+consyderyng that 90 can not be bated from 40, I�loke how moche y^t 90
+doth dyffer from the next summe aboue it, that is 100 (or elles whiche
+is all to one effecte, I�loke how moch 9 doth dyffer fro{m} 10) {and} I
+fynd it to be 1, then in the stede of that 90, I�do take from the second
+summe 100: but consyderynge that it is 10 to moche, I�set downe 1 in y^e
+nexte lyne beneth for it, as you se here.
+
+  ---------+------------
+           |   o
+  -+-------+------------
+           |   o
+  ---------+--o-o-o-----
+           |   o
+  ---------+------------
+           |   o
+  ----o-o--+--o---------
+
+Sauynge that here I haue set one counter in y^e space in stede of 5 in
+y^e nexte lyne. And thus haue I subtracted all saue two, which I must
+bate from the 6 in the second summe, and there wyll remayne 4, thus.
+
+  ----+--------------
+      |   o
+  -+--+--------------
+      |   o
+  ----+--o-o-o-------
+      |   o
+  ----+--------------
+      |
+  ----+--o-o-o-o-----
+
+So y^t yf I subtracte 2892 fro{m} 8746, the remayner wyll be 5854,
+[*117b] And that this is truely wrought, you maye proue by Addition: for
+yf you adde to this remayner the same su{m}me that you dyd subtracte,
+then wyll the formar su{m}me 8746 amount agayne.
+
+_S._ That wyll I proue: and fyrst I set the su{m}me that was subtracted,
+which was 2892, {and} the{n} the remayner 5854, thus.
+
+  --------------+--------------
+                |       o
+  -||--o-o------+--------------
+        o       |     o
+  -----o-o-o----+--o-o-o-------
+        o       |   o
+  -----o-o-o-o--+--------------
+                |
+  -----o-o------+--o-o-o-o-----
+
+Then do I adde fyrst y^e 2 to 4, whiche maketh 6, so take I vp 5 of
+those counters, and in theyr stede I sette 1 in the space, as here
+appereth.
+
+  --------------+------------
+                |     o
+  -||--o-o------+------------
+        o       |   o
+  -----o-o-o----+--o-o-o-----
+        o       |   o
+  -----o-o-o-o--+------------
+                |   o
+  --------------+--o---------
+
+[*118a] Then do I adde the 90 nexte aboue to the 50, and it maketh 140,
+therfore I take vp those 6 counters, and for them I sette 1 to the
+hundredes in y^e thyrde lyne, {and} 4 in y^e second lyne, thus.
+
+  ------------+--------------
+              |   o
+  -||--o-o----+--------------
+        o     |   o
+  -----o-o-o--+--o-o-o-o-----
+              |
+  ------------+--o-o-o-o-----
+              |   o
+  ------------+----o---------
+
+Then do I come to the hundredes, of whiche I fynde 8 in the fyrst summe,
+and 9 in y^e second, that maketh 1700, therfore I take vp those 9
+counters, and in theyr stede I sette 1 in the .iiii. lyne, and 1 in the
+space nexte beneth, and 2 in the thyrde lyne, as you se here.
+
+  ----------+--------------
+            |   o
+  -||--o-o--+--o-----------
+            |   o
+  ----------+--o-o---------
+            |
+  ----------+--o-o-o-o-----
+            |   o
+  ----------+--o-----------
+
+Then is there lefte in the fyrste summe but only 2000, whiche I shall
+take vp from thence, and set [*118b] in the same lyne in y^e second
+su{m}me, to y^e one y^t is there all redy: {and} then wyll the hole
+su{m}me appere (as you may wel se) to be 8746, which was y^e fyrst
+grosse summe, {and} therfore I do perceaue, that I hadde well subtracted
+before. And thus you may se how Subtraction maye be tryed by Addition.
+
+  ----+--------------
+      |     o
+  -X--+--o-o-o-------
+      |     o
+  ----+--o-o---------
+      |
+  ----+--o-o-o-o-----
+      |   o
+  ----+----o---------
+
+_S._ I perceaue the same order here w^t cou{n}ters, y^t I lerned before
+in figures.
+
+_M._ Then let me se howe can you trye Addition by Subtraction.
+
+_S._ Fyrste I wyl set forth this exa{m}ple of Additio{n} where I haue
+added 2189 to 4988, and the hole su{m}me appereth to be 7177,
+
+  --------------+-----------+----------
+                |           |   o
+  -||--o-o------+--o-o-o-o--+--o-o-----
+                |   o       |
+  -----o--------+--o-o-o-o--+--o-------
+        o       |   o       |   o
+  -----o-o-o----+--o-o-o----+--o-o-----
+        o       |   o       |   o
+  -----o-o-o-o--+--o-o-o----+--o-o-----
+
+[*119a] Nowe to trye whether that su{m}me be well added or no, I�wyll
+subtract one of the fyrst two su{m}mes from the thyrd, and yf I haue
+well done y^e remayner wyll be lyke that other su{m}me. As for example:
+I�wyll subtracte the fyrste summe from the thyrde, whiche I set thus in
+theyr order.
+
+  --------------+----------
+                |   o
+  -||--o-o------+--o-o-----
+                |
+  -----o--------+--o-------
+        o       |   o
+  -----o-o-o----+--o-o-----
+        o       |   o
+  -----o-o-o-o--+--o-o-----
+
+Then do I subtract 2000 of the fyrste summe fro{m} y^e second su{m}me,
+and then remayneth there 5000 thus.
+
+  -------------+----------
+               |   o
+  -X-----------+-----------
+               |
+  ----o--------+--o-------
+       o       |   o
+  ----o-o-o----+--o-o-----
+       o       |   o
+  ----o-o-o-o--+--o-o-----
+
+Then in the thyrd lyne, I subtract y^e 100 of the fyrste summe, fro{m}
+the second su{m}me, where is onely 100 also, and then in y^e thyrde lyne
+resteth nothyng. Then in the second lyne with his space ouer hym,
+I�fynde 80, which I shuld subtract [*119b] from the other su{m}me, then
+seyng there are but only 70 I must take it out of some hygher summe,
+which is here only 5000, therfore I take vp 5000, and seyng that it is
+to moch by 4920, I�sette downe so many in the seconde roume, whiche with
+the 70 beynge there all redy do make 4990, &�then the summes doth stande
+thus.
+
+  --------------+--------------
+                |
+  -||-----------+--o-o-o-o-----
+                |   o
+  --------------+--o-o-o-o-----
+                |   o
+  --------------+--o-o-o-o-----
+        o       |   o
+  -----o-o-o-o--+--o-o---------
+
+Yet remayneth there in the fyrst su{m}me 9, to be bated from the second
+summe, where in that place of vnities dothe appere only 7, then I muste
+bate a hygher su{m}me, that is to saye 10, but seynge that 10 is more
+then 9 (which I shulde abate) by 1, therfore shall I take vp one counter
+from the seconde lyne, {and} set downe the same in the fyrst [*120a] or
+lowest lyne, as you se here.
+
+  -----+--------------
+       |
+  -||--+--o-o-o-o-----
+       |   o
+  -----+--o-o-o-o-----
+       |   o
+  -----+--o-o-o-------
+       |   o
+  -----+--o-o-o-------
+
+And so haue I ended this worke, {and} the su{m}me appereth to be y^e
+same, whiche was y^e seconde summe of my addition, and therfore I
+perceaue, I�haue wel done.
+
+_M._ To stande longer about this, it is but folye: excepte that this you
+maye also vnderstande, that many do begynne to subtracte with counters,
+not at the hyghest su{m}me, as I haue taught you, but at the
+nethermoste, as they do vse to adde: and when the summe to be abatyd,
+in any lyne appeareth greater then the other, then do they borowe one of
+the next hygher roume, as for example: yf they shuld abate 1846 from
+2378, they set y^e summes thus.
+
+  --------------+------------
+                |
+  -||--o--------+--o-o-------
+        o       |
+  -----o-o-o----+--o-o-o-----
+                |   o
+  -----o-o-o-o--+--o-o-------
+        o       |   o
+  -----o--------+--o-o-o-----
+
+[*120b] And fyrste they take 6 whiche is in the lower lyne, and his
+space from 8 in the same roumes, in y^e second su{m}me, and yet there
+remayneth 2 counters in the lowest lyne. Then in the second lyne must 4
+be subtracte from 7, and so remayneth there 3. Then 8 in the thyrde lyne
+and his space, from 3 of the second summe can not be, therfore do they
+bate it from a hygher roume, that is, from 1000, and bycause that 1000
+is to moch by 200, therfore must I sette downe 200 in the thyrde lyne,
+after I haue taken vp 1000 from the fourth lyne: then is there yet 1000
+in the fourth lyne of the fyrst summe, whiche yf I withdrawe from the
+seconde summe, then doth all y^e figures stande in this order.
+
+  -----+------------
+       |
+  -||--+------------
+       |   o
+  -----+------------
+       |
+  -----+--o-o-o-----
+       |
+  -----+--o-o-------
+
+So that (as you se) it differeth not greatly whether you begynne
+subtractio{n} at the hygher lynes, or at [*121a] the lower. How be it,
+as some menne lyke the one waye beste, so some lyke the other: therfore
+you now knowyng bothe, may vse whiche you lyst.
+
+    [Headnote: Multiplication by Counters.]
+
+  [Sidenote: Multiplication.]
+
+But nowe touchynge Multiplicatio{n}: you shall set your no{m}bers in two
+roumes, as you dyd in those two other kyndes, but so that the multiplier
+be set in the fyrste roume. Then shall you begyn with the hyghest
+no{m}bers of y^e seconde roume, and multiply them fyrst after this sort.
+Take that ouermost lyne in your fyrst workynge, as yf it were the lowest
+lyne, setting on it some mouable marke, as you lyste, and loke how many
+counters be in hym, take them vp, and for them set downe the hole
+multyplyer, so many tymes as you toke vp counters, reckenyng, I�saye
+that lyne for the vnites: {and} when you haue so done with the hygheest
+no{m}ber then come to the nexte lyne beneth, {and} do euen so with it,
+and so with y^e next, tyll you haue done all. And yf there be any nomber
+in a space, then for it [*121b] shall you take y^e multiplyer 5 tymes,
+and then must you recken that lyne for the vnites whiche is nexte beneth
+that space: or els after a shorter way, you shall take only halfe the
+multyplyer, but then shall you take the lyne nexte aboue that space, for
+the lyne of vnites: but in suche workynge, yf chau{n}ce your multyplyer
+be an odde nomber, so that you can not take the halfe of it iustly, then
+muste you take the greater halfe, and set downe that, as if that it were
+the iuste halfe, and farther you shall set one cou{n}ter in the space
+beneth that line, which you recken for the lyne of vnities, or els only
+remoue forward the same that is to be multyplyed.
+
+_S._ Yf you set forth an example hereto I thynke I shal perceaue you.
+
+_M._ Take this exa{m}ple: I�wold multiply 1542 by 365, therfore I set
+y^e nombers thus.
+
+  ------------+--------------
+              |
+  -||---------+--o-----------
+              |   o
+  -----o-o-o--+--------------
+        o     |
+  -----o------+--o-o-o-o-----
+        o     |
+  ------------+--o-o---------
+
+[*122a] Then fyrste I begynne at the 1000 in y^e hyghest roume, as yf it
+were y^e fyrst place, &�I take it vp, settynge downe for it so often
+(that is ones) the multyplyer, which is 365, thus, as you se here:
+
+  -----------+-----------+------------
+             |           |
+  -----------+-----------+--o-o-o-----
+             |           |   o
+  -----------+-----------+--o---------
+             |           |   o
+  -X---------+-----------+------------ [<-]
+             |   o       |
+  ----o-o-o--+-----------+------------
+       o     |           |
+  ----o------+--o-o-o-o--+------------
+       o     |           |
+  -----------+--o-o------+------------
+
+where for the one counter taken vp from the fourth lyne, I�haue sette
+downe other 6, whiche make y^e su{m}me of the multyplyer, reckenynge
+that fourth lyne, as yf it were the fyrste: whiche thyng I haue marked
+by the hand set at the begynnyng of y^e same,
+
+_S._ I perceaue this well: for in dede, this summe that you haue set
+downe is 365000, for so moche doth amount [*122b] of 1000, multiplyed by
+365.
+
+_M._ Well the{n} to go forth, in the nexte space I fynde one counter
+which I remoue forward but take not vp, but do (as in such case I must)
+set downe the greater halfe of my multiplier (seyng it is an odde
+no{m}ber) which is 182, {and} here I do styll let that fourth place
+stand, as yf it were y^e fyrst:
+
+  ------------+-----------+--o-o-o--+--o---------
+              |           |   o     |   o
+  ------------+-----------+--o------+--o-o-o-----
+              |           |   o     |
+  -||---------+-----------+---------+--o-o------- [<-]
+              |           |         |   o
+  -----o-o-o--+-----------+---------+------------
+        o     |           |         |
+  -----o------+--o-o-o-o--+---------+------------
+        o     |           |         |
+  ------------+--o-o------+---------+------------
+
+as in this fourme you se, where I haue set this multiplycatio{n} with
+y^e other: but for the ease of your vndersta{n}dynge, I�haue set a
+lytell lyne betwene them: now shulde they both in one su{m}me stand
+thus.
+
+  ------------+-----------+--o-o-o-o-o-----
+              |           |
+  ------------+-----------+--o-o-o-o-------
+              |           |   o
+  -||---------+-----------+--o-o----------- [<-]
+              |           |   o
+  -----o-o-o--+-----------+----------------
+        o     |           |
+  -----o------+--o-o-o-o--+----------------
+        o     |           |
+  ------------+--o-o------+----------------
+
+[*123a] Howe be it an other fourme to multyplye suche cou{n}ters i{n}
+space is this: Fyrst to remoue the fynger to the lyne nexte benethe y^e
+space, {and} then to take vp y^e cou{n}ter, {and} to set downe y^e
+multiplyer .v. tymes, as here you�se.
+
+      ---------+---------+-o-o-o-+------+------+------+------+------+-
+               |         |  o    |      |      |      |      |      |
+      ---------+---------+-o-----+o-o-o-+o-o-o-+o-o-o-+o-o-o-+o-o-o-+-
+               |         |  o    | o    | o    | o    | o    | o    |
+      ---------+---------+-------+o-----+------+o-----+o-----+o-----+-
+               |         |       | o    | o    | o    | o    | o    |
+  [->]-X-o-o-o-+---------+-------+------+------+------+------+------+-
+          o    |         |       |      |      |      |      |      |
+      ---o-----+-o-o-o-o-+-------+------+------+------+------+------+-
+          o    |         |       |      |      |      |      |      |
+      ---------+-o-o-----+-------+------+------+------+------+------+-
+
+Which su{m}mes yf you do adde together into one su{m}me, you shal
+p{er}ceaue that it wyll be y^e same y^t appeareth of y^e other worki{n}g
+before, so that [*123b] bothe sortes are to one entent, but as the other
+is much shorter, so this is playner to reason, for suche as haue had
+small exercyse in this arte. Not withstandynge you maye adde them in
+your mynde before you sette them downe, as in this exa{m}ple, you myghte
+haue sayde 5 tymes 300 is 1500, {and} 5 tymes 60 is 300, also 5 tymes 5
+is 25, whiche all put together do make 1825, which you maye at one tyme
+set downe yf you lyste. But nowe to go forth, I�must remoue the hand to
+the nexte counters, whiche are in the second lyne, and there must I take
+vp those 4 counters, settynge downe for them my multiplyer 4 tymes,
+whiche thynge other I maye do at 4 tymes seuerally, or elles I may
+gather that hole summe in my mynde fyrste, and then set it downe: as to
+saye 4 tymes 300 is 1200: 4 tymes 60 are 240: and 4 tymes 5 make 20: y^t
+is in all 1460, y^t shall I set downe also: as here you�se.
+                                   o
+       -----------+-------+-----------+--------------
+                  |       |           |
+       -----------+-------+--o-o-o-o--+--o-----------
+                  |       |   o       |
+       -X---------+-------+--o-o------+--o-o-o-o-----
+                  |       |   o       |   o
+       ----o-o-o--+-------+-----------+--o-----------
+            o     |       |           |
+  [->] ----o------+-------+-----------+--------------
+            o     |       |           |
+       -----------+--o-o--+-----------+--------------
+
+[*124a] whiche yf I ioyne in one summe with the formar nombers, it wyll
+appeare thus.
+                           o
+      ---------+-------+----------
+               |       |   o
+      ---------+-------+--o-------
+               |       |
+      ---------+-------+--o-o-----
+               |       |
+      --o-o-o--+-------+-o--------
+         o     |       |
+ [->] --o------+-------+----------
+         o     |       |
+      ---------+--o-o--+----------
+
+Then to ende this multiplycation, I remoue the fynger to the lowest
+lyne, where are onely 2, them do I take vp, and in theyr stede do I set
+downe twyse 365, that is 730, for which I set [*124b] one in the space
+aboue the thyrd lyne for 500, and 2 more in the thyrd lyne with that one
+that is there all redye, and the reste in theyr order, {and} so haue I
+ended the hole summe thus.
+                     o
+  ---------+-----+------------
+           |     |   o
+  ---------+-----+--o---------
+           |     |
+  ---------+-----+--o-o-------
+           |     |   o
+  --o-o-o--+-----+--o-o-o-----
+     o     |     |
+  --o------+-----+--o-o-o-----
+     o     |     |
+  ---------+-----+------------
+
+Wherby you se, that 1542 (which is the nomber of yeares syth Ch[r]ystes
+incarnation) beyng multyplyed by 365 (which is the nomber of dayes in
+one yeare) dothe amounte vnto 562830, which declareth y^e no{m}ber of
+daies sith Chrystes incarnatio{n} vnto the ende of 1542[{1}] yeares.
+(besyde 385 dayes and 12 houres for lepe yeares).
+
+_S._ Now wyll I proue by an other exa{m}ple, as this: 40 labourers
+(after 6�d. y^e day for eche man) haue wrought 28 dayes, I�wold [*125a]
+know what theyr wages doth amou{n}t vnto: In this case muste I worke
+doublely: fyrst I must multyplye the nomber of the labourers by y^e
+wages of a man for one day, so wyll y^e charge of one daye amount: then
+secondarely shall I multyply that charge of one daye, by the hole nomber
+of dayes, {and} so wyll the hole summe appeare: fyrst therefore I shall
+set the su{m}mes thus.
+
+  ------+--------------
+        |
+  ------+--------------
+        |
+  ------+--------------
+        |
+  ------+--o-o-o-o-----
+     o  |
+  --o---+--------------
+
+Where in the fyrste space is the multyplyer (y^t is one dayes wages for
+one man) {and} in the second space is set the nomber of the worke men to
+be multyplyed: the{n} saye I, 6 tymes 4 (reckenynge that second lyne as
+the lyne of vnites) maketh 24, for whiche summe I shulde set 2 counters
+in the thyrde lyne, and 4 in the seconde, therfore do I set 2 in the
+thyrde lyne, and let the 4 stand styll in the seconde lyne, thus.[*125b]
+
+  -----+--------------
+       |
+  -----+--------------
+       |
+  -----+--o-o---------
+       |
+  -----+--o-o-o-o-----
+       |
+  -----+--------------
+
+So apwereth the hole dayes wages to be 240d�. that is 20�s. Then do I
+multiply agayn the same summe by the no{m}ber of dayes and fyrste I
+sette the nombers, thus.
+
+  ---------+--------------
+           |
+  ---------+--------------
+           |
+  ---------+--o-o---------
+           |
+  --o-o----+--o-o-o-o-----
+     o     |
+  --o-o-o--+-------------
+
+The{n} bycause there are counters in dyuers lynes, I�shall begynne with
+the hyghest, and take them vp, settynge for them the multyplyer so many
+tymes, as I toke vp counters, y^t is twyse, then wyll y^e su{m}me stande
+thus.
+
+  -----+--------------
+       |   o
+  -----+--------------
+       |   o
+  -----+--o-----------
+       |
+  -----+--o-o-o-o-----
+       |
+  -----+--------------
+
+Then come I to y^e seconde lyne, and take vp those 4 cou{n}ters,
+settynge for them the multiplyer foure tymes, so wyll the hole summe
+appeare thus.[*126a]
+
+  -----+----------
+       |   o
+  -----+--o-------
+       |   o
+  -----+--o-o-----
+       |
+  -----+--o-o-----
+       |
+  -----+----------
+
+So is the hole wages of 40 workeme{n}, for 28 dayes (after 6d�. eche
+daye for a man) 6720d�. that is 560�s. or 28�l'i.
+
+    [Headnote: Division on the Counting Board.]
+
+  [Sidenote: Diuision.]
+
+_M._ Now if you wold proue Multiplycatio{n}, the surest way is by
+Dyuision: therfore wyll I ouer passe it tyll I haue taught you y^e arte
+of Diuision, whiche you shall worke thus. Fyrste sette downe the Diuisor
+for feare of forgettynge, and then set the nomber that shalbe deuided,
+at y^e ryghte syde, so farre from the diuisor, that the quotient may be
+set betwene them: as for exa{m}ple: Yf 225 shepe cost 45�l'i. what dyd
+euery shepe cost? To knowe this, I�shulde diuide the hole summe, that is
+45�l'i. by 225, but that can not be, therfore must I fyrste reduce that
+45�l'i. into a lesser denomination, as into shyllynges: then I multiply
+45 by 20, and it is 900, that summe shall I diuide by the no{m}ber of
+[*126b] shepe, whiche is 225, these two nombers therfore I sette thus.
+
+  -------+-----+--------------
+         |     |
+  -------+-----+--------------
+         |     |   o
+  --o-o--+-----+--o-o-o-o-----
+         |     |
+  --o-o--+-----+--------------
+     o   |     |
+  -------+-----+--------------
+
+Then begynne I at the hyghest lyne of the diuident, and seke how often I
+may haue the diuisor therin, and that maye I do 4 tymes, then say I,
+4 tymes 2 are 8, whyche yf I take from 9, there resteth but 1, thus
+
+  -------+-----------+--------
+         |           |
+  -------+-----------+--------
+         |           |
+  --o-o--+-----------+--o-----
+         |           |
+  --o-o--+-----------+--------
+     o   |           |
+  -------+--o-o-o-o--+--------
+
+And bycause I founde the diuisor 4 tymes in the diuidente, I�haue set
+(as you se) 4 in the myddle roume, which [*127a] is the place of the
+quotient: but now must I take the reste of the diuisor as often out of
+the remayner: therfore come I�to the seconde lyne of the diuisor, sayeng
+2 foure tymes make 8, take 8 from 10, {and} there resteth 2, thus.
+
+  ----------+-----------+----------
+            |           |
+  -||-------+-----------+----------
+            |           |
+  -----o-o--+-----------+----------
+            |           |
+  -----o-o--+-----------+--o-o-----
+        o   |           |
+  ----------+--o-o-o-o--+----------
+
+Then come I to the lowest nomber, which is 5, and multyply it 4 tymes,
+so is it 20, that take I from 20, and there remayneth nothynge, so that
+I se my quotient to be 4, whiche are in valewe shyllynges, for so was
+the diuident: and therby I knowe, that yf 225 shepe dyd coste 45�l'i.
+euery shepe coste 4�s.
+
+_S._ This can I do, as you shall perceaue by this exa{m}ple: Yf 160
+sowldyars do spende euery moneth 68�l'i. what spendeth eche man? Fyrst
+[*127b] bycause I can not diuide the 68 by 160, therfore I wyll turne
+the pou{n}des into pennes by multiplicacio{n}, so shall there be
+16320�d�. Nowe muste I diuide this su{m}me by the nomber of sowldyars,
+therfore I set the{m} i{n} order, thus.
+
+  ---------+-----+--o---------
+           |     |   o
+  -||------+-----+--o---------
+           |     |
+  -----o---+-----+--o-o-o-----
+        o  |     |
+  -----o---+-----+--o-o-------
+           |     |
+  ---------+-----+------------
+
+Then begyn I at the hyghest place of the diuidente, sekynge my diuisor
+there, whiche I fynde ones, Therfore set I 1 in the nether lyne.
+
+_M._ Not in the nether line of the hole summe, but in the nether lyne of
+that worke, whiche is the thyrde lyne.
+
+_S._ So standeth it with reason.
+
+_M._ Then thus do they stande.[*128a]
+
+  ---------+-----+------------
+           |     |
+  -||------+-----+------------
+           |     |
+  -----o---+--o--+--o-o-o-----
+        o  |     |
+  -----o---+-----+--o-o-------
+           |     |
+  ---------+-----+------------
+
+Then seke I agayne in the reste, how often I may fynde my diuisor, and I
+se that in the 300 I myghte fynde 100 thre tymes, but then the 60 wyll
+not be so often founde in 20, therfore I take 2 for my quotient: then
+take I 100 twyse from 300, and there resteth 100, out of whiche with the
+20 (that maketh 120) I�may take 60 also twyse, and then standeth the
+nombers thus,
+
+  ---------+-------+-----
+           |       |
+  -||------+-------+-----
+           |       |
+  -----o---+--o----+-----
+        o  |       |
+  -----o---+-------+-----
+           |       |
+  ---------+--o-o--+-----
+
+[*128b] where I haue sette the quotient 2 in the lowest lyne: So is
+euery sowldyars portion 102�d�. that is 8�s. 6�d�.
+
+_M._ But yet bycause you shall perceaue iustly the reason of Diuision,
+it shall be good that you do set your diuisor styll agaynst those
+nombres fro{m} whiche you do take it: as by this example I wyll declare.
+Yf y^e purchace of 200 acres of ground dyd coste 290�l'i. what dyd one
+acre coste? Fyrst wyl I turne the poundes into pennes, so wyll there be
+69600�d�� Then in settynge downe these nombers I shall do thus.
+
+  ---------+-----+--------------
+           |     |   o
+  ----o-o--+-----+--o-----------
+           |     |   o
+  -X-------+-----+--o-o-o-o-----
+           |     |     o
+  ---------+-----+--o-----------
+           |     |
+  ---------+-----+--------------
+           |     |
+  ---------+-----+--------------
+
+Fyrst set the diuident on the ryghte hande as it oughte, and then
+[*129a] the diuisor on the lefte hande agaynst those nombers, fro{m}
+which I entende to take hym fyrst as here you se, wher I haue set the
+diuisor two lynes hygher the{n} is theyr owne place.
+
+_S._ This is lyke the order of diuision by the penne.
+
+_M._ Truth you say, and nowe must I set y^e quotient of this worke in
+the thyrde lyne, for that is the lyne of vnities in respecte to the
+diuisor in this worke. Then I seke howe often the diuisor maye be founde
+in the diuident, {and} that I fynde 3 tymes, then set I 3 in the thyrde
+lyne for the quotient, and take awaye that 60000 fro{m} the diuident,
+and farther I do set the diuisor one line lower, as yow se here.
+
+  ----------+---------+--------------
+            |         |   o
+  -||--o-o--+---------+--o-o-o-o-----
+            |         |   o
+  ----------+--o-o-o--+----o---------
+            |         |
+  ----------+---------+--------------
+            |         |
+  ----------+---------+--------------
+
+[*129b] And then seke I how often the diuisor wyll be taken from the
+nomber agaynste it, whiche wyll be 4 tymes and 1 remaynynge.
+
+_S._ But what yf it chaunce that when the diuisor is so remoued, it can
+not be ones taken out of the diuident agaynste it?
+
+_M._ Then must the diuisor be set in an other line lower.
+
+_S._ So was it in diuision by the penne, and therfore was there a cypher
+set in the quotient: but howe shall that be noted here?
+
+_M._ Here nedeth no token, for the lynes do represente the places: onely
+loke that you set your quotient in that place which standeth for vnities
+in respecte of the diuisor: but now to returne to the example, I�fynde
+the diuisor 4 tymes in the diuidente, and 1 remaynynge, for 4 tymes 2
+make 8, which I take from 9, and there resteth 1, as this figure
+sheweth:
+
+  ----------+-----------+---------
+            |           |
+  -||--o-o--+-----------+--o------
+            |           |   o
+  ----------+--o-o-o----+--o------
+            |           |
+  ----------+--o-o-o-o--+---------
+            |           |
+  ----------+-----------+---------
+
+and in the myddle space for the quotient I set 4 in the seconde lyne,
+whiche is in this worke the place of vnities.[*130a] Then remoue I y^e
+diuisor to the next lower line, and seke how often I may haue it in the
+dyuident, which I may do here 8 tymes iust, and nothynge remayne, as in
+this fourme,
+
+  ----------+-----------+-----
+            |           |
+  -||--o-o--+-----------+-----
+            |           |
+  ----------+--o-o-o----+-----
+            |           |
+  ----------+--o-o-o-o--+-----
+            |   o       |
+  ----------+--o-o-o----+-----
+
+where you may se that the hole quotient is 348�d�, that is 29�s. wherby
+I knowe that so moche coste the purchace of one aker.
+
+_S._ Now resteth the profes of Multiplycatio{n}, and also of Diuisio{n}.
+
+_M._ Ther best profes are eche [*130b] one by the other, for
+Multyplication is proued by Diuision, and Diuision by Multiplycation,
+as in the worke by the penne you learned.
+
+_S._ Yf that be all, you shall not nede to repete agayne that, y^t was
+sufficye{n}tly taughte all redye: and excepte you wyll teache me any
+other feate, here maye you make an ende of this arte I suppose.
+
+_M._ So wyll I do as touchynge hole nomber, and as for broken nomber,
+I�wyll not trouble your wytte with it, tyll you haue practised this so
+well, y^t you be full perfecte, so that you nede not to doubte in any
+poynte that I haue taught you, and thenne maye I boldly enstructe you in
+y^e arte of fractions or broken no{m}ber, wherin I wyll also showe you
+the reasons of all that you haue nowe learned. But yet before I make an
+ende, I�wyll showe you the order of co{m}men castyng, wher in are bothe
+pennes, shyllynges, and poundes, procedynge by no grounded reason, but
+onely by a receaued [*131a] fourme, and that dyuersly of dyuers men: for
+marchau{n}tes vse one fourme, and auditors an other:
+
+    [Headnote: Merchants' Casting Counters.]
+
+  [Sidenote: Merchants' casting.]
+
+But fyrste for marchauntes fourme marke this example here,
+
+  o   o o o o
+       o
+  o   o o o
+       o
+  o   o o o o
+       o
+      o o o o o
+
+in which I haue expressed this summe 198 l'i.[{2}] 19�s. 11�d�. So that
+you maye se that the lowest lyne serueth for pe{n}nes, the next aboue
+for shyllynges, the thyrde for poundes, and the fourth for scores of
+pou{n}des. And farther you maye se, that the space betwene pennes and
+shyllynges may receaue but one counter (as all other spaces lyke wayes
+do) and that one standeth in that place for 6�d�. Lyke wayes betwene the
+shyllynges {and} the pou{n}des, one cou{n}ter standeth for 10�s. And
+betwene the poundes and 20�l'i. one counter standeth for 10 pou{n}des.
+But besyde those you maye see at the left syde of shyllynges, that one
+counter standeth alone, {and} betokeneth 5�s. [*131b] So agaynste the
+poundes, that one cou{n}ter standeth for 5�l'i. And agaynst the 20
+poundes, the one counter standeth for 5 score pou{n}des, that is
+100�l'i. so that euery syde counter is 5 tymes so moch as one of them
+agaynst whiche he standeth.
+
+  [Sidenote: Auditors' casting.]
+
+Now for the accompt of auditors take this example.
+
+      o   o   o   o   o     o
+  o o o   o o o   o o o   o o o
+  o               o       o o
+
+where I haue expressed y^e same su{m}me 198�l'i. 19�s. 11�d�. But here
+you se the pe{n}nes stande toward y^e ryght hande, and the other
+encreasynge orderly towarde the lefte hande. Agayne you maye se, that
+auditours wyll make 2 lynes (yea and more) for pennes, shyllynges, {and}
+all other valewes, yf theyr summes extende therto. Also you se, that
+they set one counter at the ryght ende of eche rowe, whiche so set there
+standeth for 5 of that roume: and on [*132a] the lefte corner of the
+rowe it sta{n}deth for 10, of y^e same row. But now yf you wold adde
+other subtracte after any of both those sortes, yf you marke y^e order
+of y^t other feate which I taught you, you may easely do the same here
+without moch teachynge: for in Additio{n} you must fyrst set downe one
+su{m}me and to the same set the other orderly, and lyke maner yf you
+haue many: but in Subtraction you must sette downe fyrst the greatest
+summe, and from it must you abate that other euery denominatio{n} from
+his dewe place.
+
+_S._ I do not doubte but with a lytell practise I shall attayne these
+bothe: but how shall I multiply and diuide after these fourmes?
+
+_M._ You can not duely do none of both by these sortes, therfore in
+suche case, you must resort to your other artes.
+
+_S._ Syr, yet I se not by these sortes how to expresse hu{n}dreddes,
+yf they excede one hundred, nother yet thousandes.
+
+_M._ They that vse such accomptes that it excede 200 [*132b] in one
+summe, they sette no 5 at the lefte hande of the scores of poundes, but
+they set all the hundredes in an other farther rowe {and} 500 at the
+lefte hand therof, and the thousandes they set in a farther rowe yet,
+{and} at the lefte syde therof they sette the 5000, and in the space
+ouer they sette the 10000, and in a hygher rowe 20000, whiche all I haue
+expressed in this exa{m}ple,
+
+      o o o o
+       o
+  o   o o
+  o   o o o
+      o o o
+  o   o o o o
+        o
+      o o
+        o
+      o o o
+               o o
+               o
+
+which is 97869�l'i. 12�s. 9 d� ob. q. for I had not told you before
+where, nother how you shuld set downe farthynges, which (as you se here)
+must be set in a voyde space sydelynge beneth the pennes: for q one
+counter: for ob. 2 counters: for ob. q. 3 counters: {and} more there can
+not be, for 4 farthynges [*133a] do make 1�d�. which must be set in his
+dewe place.
+
+    [Headnote: Auditors' Casting Counters.]
+
+And yf you desyre y^e same summe after audytors maner, lo here it�is.
+
+           o o        o               o   o       o
+  o o o   o o     o o o   o o o   o o o   o o   o o o
+  o                               o                      o o
+                                                         o
+
+But in this thyng, you shall take this for suffycyent, and the reste you
+shall obserue as you maye se by the working of eche sorte: for the
+dyuers wittes of men haue inuented dyuers and sundry wayes almost
+vnnumerable. But one feate I shall teache you, whiche not only for the
+straungenes and secretnes is moche pleasaunt, but also for the good
+co{m}moditie of it ryghte worthy to be well marked. This feate hath ben
+vsed aboue 2000 yeares at the leaste, and yet was it neuer come{n}ly
+knowen, especyally in Englysshe it was neuer taughte yet. This is the
+arte of nombrynge on the hand, with diuers gestures of the fyngers,
+expressynge any summe conceaued in the [*133b] mynde. And fyrst to
+begynne, yf you wyll expresse any summe vnder 100, you shall expresse it
+with your lefte hande: and from 100 vnto 10000, you shall expresse it
+with your ryght hande, as here orderly by this table folowynge you may
+perceaue.
+
+  +� Here foloweth the table
+    of the arte of the
+      hande+
+
+
+
+
+The arte of nombrynge by the hande.
+
+  [Transcriber's Note:
+
+  Footnote 3 reads:
+    "Bracket ([) denotes new paragraph in original."
+  For this e-text, the brackets have been omitted in favor of restoring
+  the paragraph breaks. Changes of speaker (M, S) are also marked by
+  paragraphs, as in the previous selection.
+
+  The illustration includes the printed page number 134; there is
+  therefore no sidenote *134a. The sidenote for "4" is missing.]
+
+
+[Illustration: (Numbers as described in text)]
+
+  [Sidenote: 1]
+
+[*134b] In which as you may se 1 is expressed by y^e lyttle fynger of
+y^e lefte hande closely and harde croked.
+
+  [Sidenote: 2]
+
+[{3}]2 is declared by lyke bowynge of the weddynge fynger (whiche is the
+nexte to the lyttell fynger) together with the lytell fynger.
+
+  [Sidenote: 3]
+
+3 is signified by the myddle fynger bowed in lyke maner, with those
+other two.
+
+4 is declared by the bowyng of the myddle fynger and the rynge fynger,
+or weddynge fynger, with the other all stretched forth.
+
+  [Sidenote: 5, 6]
+
+5 is represented by the myddle fynger onely bowed.
+
+And 6 by the weddynge fynger only crooked: and this you may marke in
+these a certayne order. But now 7, 8, and 9, are expressed w{i}t{h} the
+bowynge of the same fyngers as are 1, 2, and 3, but after an other
+fourme.
+
+  [Sidenote: 7]
+
+For 7 is declared by the bowynge of the lytell fynger, as is 1, saue
+that for 1 the fynger is clasped in, harde {and} [*135a] rounde, but for
+to expresse 7, you shall bowe the myddle ioynte of the lytell fynger
+only, and holde the other ioyntes streyght.
+
+_S._ Yf you wyll geue me leue to expresse it after my rude maner, thus I
+vnderstand your meanyng: that 1 is expressed by crookynge in the lyttell
+fynger lyke the head of a bysshoppes bagle: and 7 is declared by the
+same fynger bowed lyke a gybbet.
+
+_M._ So I perceaue, you vnderstande it.
+
+  [Sidenote: 8]
+
+Then to expresse 8, you shall bowe after the same maner both the lyttell
+fynger and the rynge fynger.
+
+  [Sidenote: 9, 10]
+
+And yf you bowe lyke wayes with them the myddle fynger, then doth it
+betoken 9.
+
+Now to expresse 10, you shall bowe your fore fynger rounde, and set the
+ende of it on the hyghest ioynte of the thombe.
+
+  [Sidenote: 20]
+
+And for to expresse 20, you must set your fyngers streyght, and the ende
+of your thombe to the partitio{n} of the [*135b] fore moste and myddle
+fynger.
+
+  [Sidenote: 30]
+
+30 is represented by the ioynynge together of y^e headdes of the
+foremost fynger and the thombe.
+
+  [Sidenote: 40]
+
+40 is declared by settynge of the thombe crossewayes on the foremost
+fynger.
+
+  [Sidenote: 50]
+
+50 is signified by ryght stretchyng forth of the fyngers ioyntly, and
+applyenge of the thombes ende to the partition of the myddle fynger
+{and} the rynge fynger, or weddynge fynger.
+
+  [Sidenote: 60]
+
+60 is formed by bendynge of the thombe croked and crossynge it with the
+fore fynger.
+
+  [Sidenote: 70]
+
+70 is expressed by the bowynge of the foremost fynger, and settynge the
+ende of the thombe between the 2 foremost or hyghest ioyntes of it.
+
+  [Sidenote: 80]
+
+80 is expressed by settynge of the foremost fynger crossewayes on the
+thombe, so that 80 dyffereth thus fro{m} 40, that for 80 the forefynger
+is set crosse on the thombe, and for 40 the thombe is set crosse ouer
+y^e forefinger.
+
+  [Sidenote: 90]
+
+[*136a] 90 is signified, by bendynge the fore fynger, and settyng the
+ende of it in the innermost ioynte of y^e thombe, that is euen at the
+foote of it. And thus are all the no{m}bers ended vnder 100.
+
+[Sidenote: 11, 12, 13, 21, 22, 23]
+
+_S._ In dede these be all the nombers fro{m} 1 to 10, {and} then all the
+tenthes within 100, but this teacyed me not how to expresse 11, 12, 13,
+{et}c. 21, 22, 23, {et}c. and such lyke.
+
+_M._ You can lytell vnderstande, yf you can not do that without
+teachynge: what is 11? is it not 10 and 1? then expresse 10 as you were
+taught, and 1 also, and that is 11: and for 12 expresse 10 and 2: for 23
+set 20 and 3: and so for 68 you muste make 60 and there to 8: and so of
+all other sortes.
+
+  [Sidenote: 100]
+
+But now yf you wolde represente 100 other any nomber aboue it, you muste
+do that with the ryghte hande, after this maner. [You must expresse 100
+in the ryght hand, with the lytell fynger so bowed as you dyd expresse 1
+in the left hand.
+
+  [Sidenote: 200]
+
+[*136b] And as you expressed 2 in the lefte hande, the same fasshyon in
+the ryght hande doth declare 200.
+
+  [Sidenote: 300]
+
+The fourme of 3 in the ryght hand standeth for 300.
+
+  [Sidenote: 400]
+
+The fourme of 4, for 400.
+
+  [Sidenote: 500]
+
+Lykewayes the fourme of 5, for 500.
+
+  [Sidenote: 600]
+
+The fourme of 6, for 600. And to be shorte: loke how you did expresse
+single vnities and tenthes in the lefte hande, so must you expresse
+vnities {and} tenthes of hundredes, in the ryghte hande.
+
+  [Sidenote: 900]
+
+_S._ I vnderstande you thus: that yf I wold represent 900, I�must so
+fourme the fyngers of my ryghte hande, as I shuld do in my left hand to
+expresse 9,
+
+  [Sidenote: 1000]
+
+And as in my lefte hand I expressed 10, so in my ryght hande must I
+expresse 1000.
+
+And so the fourme of euery tenthe in the lefte hande serueth to expresse
+lyke no{m}ber of thousa{n}des,
+
+  [Sidenote: 4000]
+
+so y^e fourme of 40 standeth for 4000.
+
+  [Sidenote: 8000]
+
+The fourme of 80 for 8000.
+
+  [Sidenote: 9000]
+
+  [*137a]
+
+  And the fourme of 90 (whiche is
+  the greatest) for 9000, and aboue that
+  I can not expresse any nomber. _M._
+  No not with one fynger: how be it,
+  w{i}t{h} dyuers fyngers you maye expresse
+  9999, and all at one tyme, and that lac
+  keth but 1 of 10000. So that vnder
+  10000 you may by your fyngers ex-
+  presse any summe. And this shal suf-
+  fyce for Numeration on the fyngers.
+  And as for Addition, Subtraction,
+  Multiplicatio{n}, and Diuision (which
+  yet were neuer taught by any man as
+  farre as I do knowe) I wyll enstruct
+  you after the treatyse of fractions.
+  And now for this tyme fare well,
+  and loke that you cease not to
+  practyse that you haue lear
+  ned. _S._ Syr, with moste
+  harty mynde I thanke
+  you, bothe for your
+  good learnyng, {and}
+  also your good
+  cou{ns}el, which
+  (god wyllyng) I truste to folow.
+
+
+  Finis.
+
+
+  FOOTNOTES (Accomptynge by counters
+  _and_ The arte of nombrynge by the hande):
+
+  [1: 1342 in original.]
+  [2: 168 in original.]
+  [3: Bracket ([) denotes new paragraph in original.]
+
+
+
+
+APPENDIX I.
+
++A Treatise on the Numeration of Algorism.+
+
+
+[_From a MS. of the 14th Century._]
+
+To alle suche even nombrys the most have cifrys as to ten. twenty.
+thirtty. an hundred. an thousand and suche other. but ye schal
+vnderstonde that a cifre tokeneth nothinge but he maketh other the more
+significatyf that comith after hym. Also ye schal vnderstonde that in
+nombrys composyt and in alle other nombrys that ben of diverse figurys
+ye schal begynne in the ritht syde and to rekene backwarde and so he
+schal be wryte as thus--1000. the sifre in the ritht side was first
+wryte and yit he tokeneth nothinge to the secunde no the thridde but
+thei maken that figure of 1 the more signyficatyf that comith after hem
+by as moche as he born oute of his first place where he schuld yf he
+stode ther tokene but one. And there he stondith nowe in the ferye place
+he tokeneth a thousand as by this rewle. In the first place he tokeneth
+but hymself. In the secunde place he tokeneth ten times hymself. In the
+thridde place he tokeneth an hundred tymes himself. In the ferye he
+tokeneth a thousand tymes himself. In the fyftye place he tokeneth ten
+thousand tymes himself. In the sexte place he tokeneth an hundred
+thousand tymes hymself. In the seveth place he tokeneth ten hundred
+thousand tymes hymself, &c. And ye schal vnderstond that this worde
+nombre is partyd into thre partyes. Somme is callyd nombre of digitys
+for alle ben digitys that ben withine ten as ix, viii, vii, vi, v, iv,
+iii, ii, i. Articules ben alle thei that mow be devyded into nombrys of
+ten as xx, xxx, xl, and suche other. Composittys be alle nombrys that
+ben componyd of a digyt and of an articule as fourtene fyftene thrittene
+and suche other. Fourtene is componyd of four that is a digyt and of ten
+that is an articule. Fyftene is componyd of fyve that is a digyt and of
+ten that is an articule and so of others . . . . . . But as to this
+rewle. In the firste place he tokeneth but himself that is to say he
+tokeneth but that and no more. If that he stonde in the secunde place he
+tokeneth ten tymes himself as this figure 2 here 21. this is oon and
+twenty. This figure 2 stondith in the secunde place and therfor he
+tokeneth ten tymes himself and ten tymes 2 is twenty and so forye of
+every figure and he stonde after another toward the lest syde he schal
+tokene ten tymes as moche more as he schuld token and he stode in that
+place ther that the figure afore him stondeth: lo an example as thus
+9634. This figure of foure that hath this schape 4 tokeneth but himself
+for he stondeth in the first place. The figure of thre that hath this
+schape 3 tokeneth ten tyme himself for he stondeth in the secunde place
+and that is thritti. The figure of sexe that hath this schape 6 tokeneth
+ten tyme more than he schuld and he stode in the place yer the figure of
+thre stondeth for ther he schuld tokene but sexty. And now he tokeneth
+ten tymes that is sexe hundrid. The figure of nyne that hath this schape
+9 tokeneth ten tymes more than he schulde and he stode in the place ther
+the figure of 6 stondeth inne for thanne he schuld tokene but nyne
+hundryd. And in the place that he stondeth inne nowe he tokeneth nine
+thousand. Alle the hole nombre of these foure figurys. Nine thousand
+sexe hundrid and foure and thritti.
+
+
+
+
+APPENDIX II.
+
+Carmen de Algorismo.
+
+
+[_From a B.M. MS., 8 C. iv., with additions from 12 E. 1 & Eg. 2622._]
+
+  Hec algorismus ars presens dicitur[{1}]; in qua
+  Talibus Indorum[{2}] fruimur his quinque figuris.
+    0. 9. 8. 7. 6. 5. 4. 3. 2. 1.
+  Prima significat unum: duo vero secunda:
+  Tercia significat tria: sic procede sinistre                       4
+  Donec ad extremam venies, qua cifra vocatur;
+  [{3}][Que nil significat; dat significare sequenti.]
+  Quelibet illarum si primo limite ponas,
+  Simpliciter se significat: si vero secundo,                        8
+  Se decies: sursum procedas multiplicando.[{4}]
+  [Namque figura sequens quevis signat decies plus,
+  Ipsa locata loco quam significet pereunte:                        12
+  Nam precedentes plus ultima significabit.]
+  [{5}]Post predicta scias quod tres breuiter numerorum
+  Distincte species sunt; nam quidam digiti sunt;
+  Articuli quidam; quidam quoque compositi sunt.                    16
+  [Sunt digiti numeri qui citra denarium sunt;
+  Articuli decupli degitorum; compositi sunt
+  Illi qui constant ex articulis digitisque.]
+  Ergo, proposito numero tibi scribere, primo                       20
+  Respicias quis sit numerus; quia si digitus sit,
+  [{5}][Una figura satis sibi; sed si compositus sit,]
+  Primo scribe loco digitum post articulum fac
+  Articulus si sit, cifram post articulum sit,                      24
+  [Articulum vero reliquenti in scribe figure.]
+  Quolibet in numero, si par sit prima figura,
+  Par erit et totum, quicquid sibi continetur;
+  Impar si fuerit, totum sibi fiet et impar.                        28
+  Septem[{6}] sunt partes, non plures, istius artis;
+  Addere, subtrahere, duplare, dimidiare;
+  Sexta est diuidere, set quinta est multiplicare;
+  Radicem extrahere pars septima dicitur esse.                      32
+  Subtrahis aut addis a dextris vel mediabis;
+  A leua dupla, diuide, multiplicaque;
+  Extrahe radicem semper sub parte sinistra.
+
+  [Sidenote: Addition.]
+
+  Addere si numero numerum vis, ordine tali                         36
+  Incipe; scribe duas primo series numerorum
+  Prima sub prima recte ponendo figuram,
+  Et sic de reliquis facias, si sint tibi plures.
+  Inde duas adde primas hac condicione;                             40
+  Si digitus crescat ex addicione priorum,
+  Primo scribe loco digitum, quicunque sit ille;
+  Si sit compositus, in limite scribe sequenti
+  Articulum, primo digitum; quia sic iubet ordo.                    44
+  Articulus si sit, in primo limite cifram,
+  Articulum vero reliquis inscribe figuris;
+  Vel per se scribas si nulla figura sequatur.
+  Si tibi cifra superueniens occurrerit, illam                      48
+  Deme suppositam; post illic scribe figuram:
+  Postea procedas reliquas addendo figuras.
+
+  [Sidenote: Subtraction.]
+
+  A numero numerum si sit tibi demere cura,
+  Scribe figurarum series, vt in addicione;                         52
+  Maiori numero numerum suppone minorem,
+  Siue pari numero supponatur numerus par.
+  Postea si possis a prima subtrahe primam,
+  Scribens quod remanet, cifram si nil remanebit.                   56
+  Set si non possis a prima demere primam;
+  Procedens, vnum de limite deme sequenti;
+  Et demptum pro denario reputabis ab illo,
+  Subtrahe totaliter numerum quem proposuisti.                      60
+  Quo facto, scribe supra quicquit remanebit,
+  Facque novenarios de cifris, cum remanebis,
+  Occurrant si forte cifre, dum demseris vnum;
+  Postea procedas reliquas demendo figuras.                         64
+
+  [Sidenote: Proof.]
+
+  [{7}][Si subtracio sit bene facta probare valebis,
+  Quas subtraxisti primas addendo figuras.
+  Nam, subtractio si bene sit, primas retinebis,
+  Et subtractio facta tibi probat additionem.]                      68
+
+  [Sidenote: Duplation.]
+
+  Si vis duplare numerum, sic incipe; solam
+  Scribe figurarum seriem, quamcumque voles que
+  Postea procedas primam duplando figuram;
+  Inde quod excrescet, scribens, vbi iusserit ordo,                 72
+  Juxta precepta que dantur in addicione.
+  Nam si sit digitus, in primo limite scribe;
+  Articulus si sit, in primo limite cifram,
+  Articulum vero reliquis inscribe figuris;                         76
+  Vel per se scribas, si nulla figura sequatur:
+  Compositus si sit, in limite scribe sequenti
+  Articulum primo, digitum; quia sic jubet ordo:
+  Et sic de reliquis facias, si sint tibi plures.                   80
+  [{8}][Si super extremam nota sit, monadem dat eidem,
+  Quod tibi contingit, si primo dimidiabis.]
+
+  [Sidenote: Mediation.]
+
+  Incipe sic, si vis aliquem numerum mediare:
+  Scribe figurarum seriem solam, velud ante;                        84
+  Postea procedens medias, et prima figura
+  Si par aut impar videas; quia si fuerit par,
+  Dimidiabis eam, scribens quicquit remanebit;
+  Impar si fuerit, vnum demas, mediare,                             88
+  Nonne presumas, sed quod superest mediabis;
+  Inde super tractum, fac demptum quod notat unum;
+  Si monos, dele; sit ibi cifra post nota supra.
+  Postea procedas hac condicione secunda:[{9}]                      92
+  Impar[{10}] si fuerit hic vnum deme priori,
+  Inscribens quinque, nam denos significabit
+  Monos pr�dictam: si vero secunda dat vnam,
+  Illa deleta, scribatur cifra; priori                              96
+  Tradendo quinque pro denario mediato;
+  Nec cifra scribatur, nisi inde figura sequatur:
+  Postea procedas reliquas mediando figuras,
+  Quin supra docui, si sint tibi mille figure.                     100
+  [{11}][Si mediatio sit bene facta probare valebis,
+  Duplando numerum quem primo dimidiasti.]
+  Si super extremam nota sit monades dat eidem
+  Quod contingat cum primo dimiabis
+  Atque figura prior nuper fuerit mediando.]
+
+  [Sidenote: Multiplication.]
+
+  Si tu per numerum numerum vis multiplicare,
+  Scribe duas, quascunque volis, series numerorum;                 104
+  Ordo tamen seruetur vt vltima multiplicandi
+  Ponatur super anteriorem multiplicantis;
+  [{12}][A leua relique sint scripte multiplicantes.]
+  In digitum cures digitum si ducere, major                        108
+  Per quantes distat a denis respice, debes
+  Namque suo decuplo tociens delere minorem;
+  Sicque tibi numerus veniens exinde patebit.
+  Postea procedas postremam multiplicando,                         112
+  Juste multiplicans per cunctas inferiores,
+  Condicione tamen tali; quod multiplicantis
+  Scribas in capite, quicquid processerit inde;
+  Set postquam fuerit hec multiplicata, figure                     116
+  Anteriorentur seriei multiplicantis;
+  Et sic multiplica, velut istam multiplicasti,
+  Qui sequitur numerum scriptum quicunque figuris.
+  Set cum multiplicas, primo sic est operandum,                    120
+  Si dabit articulum tibi multiplicacio solum;
+  Proposita cifra, summam transferre memento.
+  Sin autem digitus excrescerit articulusque,
+  Articulus supraposito digito salit ultra;                        124
+  Si digitus tamen, ponas illum super ipsam,
+  Subdita multiplicans hanc que super incidit illi
+  Delet eam penitus, scribens quod provenit inde;
+  Sed si multiplices illam posite super ipsam,                     128
+  Adiungens numerum quem prebet ductus earum;
+  Si supraimpositam cifra debet multiplicare,
+  Prorsus eam delet, scribi que loco cifra debet,
+  [{12}][Si cifra multiplicat aliam positam super ipsam,           132
+  Sitque locus supra vacuus super hanc cifra fiet;]
+  Si supra fuerit cifra semper pretereunda est;
+  Si dubites, an sit bene multiplicando secunda,
+  Diuide totalem numerum per multiplicantem,                       136
+  Et reddet numerus emergens inde priorem.
+
+  [Sidenote: Mental Multiplication.]
+
+  [{13}][Per numerum si vis numerum quoque multiplicare
+  Tantum per normas subtiles absque figuris
+  Has normas poteris per versus scire sequentes.                   140
+  Si tu per digitum digitum quilibet multiplicabis
+  Regula precedens dat qualiter est operandum
+  Articulum si per reliquum vis multiplicare
+  In proprium digitum debebit uterque resolvi                      144
+  Articulus digitos post per se multiplicantes
+  Ex digitis quociens teneret multiplicatum
+  Articuli faciunt tot centum multiplicati.
+  Articulum digito si multiplicamus oportet                        148
+  Articulum digitum sumi quo multiplicare
+  Debemus reliquum quod multiplicaris ab illis
+  Per reliquo decuplum sic omne latere nequibit
+  In numerum mixtum digitum si ducere cures                        152
+  Articulus mixti sumatur deinde resolvas
+  In digitum post hec fac ita de digitis nec
+  Articulusque docet excrescens in detinendo
+  In digitum mixti post ducas multiplicantem                       156
+  De digitis ut norma docet sit juncta secundo
+  Multiplica summam et postea summa patebit
+  Junctus in articulum purum articulumque
+  [{14}][Articulum purum comittes articulum que]                   160
+  Mixti pro digitis post fiat et articulus vt
+  Norma jubet retinendo quod egreditur ab illis
+  Articuli digitum post in digitum mixti duc
+  Regula de digitis ut percipit articulusque                       164
+  Ex quibus excrescens summe tu junge priori
+  Sic manifesta cito fiet tibi summa petita.
+  Compositum numerum mixto sic multiplicabis
+  Vndecies tredecem sic est ex hiis operandum                      168
+  In reliquum primum demum duc post in eundem
+  Unum post deinde duc in tercia deinde per unum
+  Multiplices tercia demum tunc omnia multiplicata
+  In summa duces quam que fuerit te dices                          172
+  Hic ut hic mixtus intentus est operandum
+  Multiplicandorum de normis sufficiunt hec.]
+
+  [Sidenote: Division.]
+
+  Si vis dividere numerum, sic incipe primo;
+  Scribe duas, quascunque voles, series numerorum;                 176
+  Majori numero numerum suppone minorem,
+  [{15}][Nam docet ut major teneat bis terve minorem;]
+  Et sub supprima supprimam pone figuram,
+  Sic reliquis reliquas a dextra parte locabis;                    180
+  Postea de prima primam sub parte sinistra
+  Subtrahe, si possis, quociens potes adminus istud,
+  Scribens quod remanet sub tali conditione;
+  Ut totiens demas demendas a remanente,                           184
+  Que serie recte ponentur in anteriori,
+  Unica si, tantum sit ibi decet operari;
+  Set si non possis a prima demere primam,
+  Procedas, et eam numero suppone sequenti;                        188
+  Hanc uno retrahendo gradu quo comites retrahantur,
+  Et, quotiens poteris, ab eadem deme priorem,
+  Ut totiens demas demendas a remanenti,
+  Nec plus quam novies quicquam tibi demere debes,                 192
+  Nascitur hinc numerus quociens supraque sequentem
+  Hunc primo scribas, retrahas exinde figuras,
+  Dum fuerit major supra positus inferiori,
+  Et rursum fiat divisio more priori;                              196
+  Et numerum quotiens supra scribas pereunti,
+  Si fiat saliens retrahendo, cifra locetur,
+  Et pereat numero quotiens, proponas eidem
+  Cifram, ne numerum pereat vis, dum locus illic                   200
+  Restat, et expletis divisio non valet ultra:
+  Dum fuerit numerus numerorum inferiore seorsum
+  Illum servabis; hinc multiplicando probabis,
+
+  [Sidenote: Proof.]
+
+  Si bene fecisti, divisor multiplicetur                           204
+  Per numerum quotiens; cum multiplicaveris, adde
+  Totali summ�, quod servatum fuit ante,
+  Reddeturque tibi numerus quem proposuisti;
+  Et si nil remanet, hunc multiplicando reddet,                    208
+
+  [Sidenote: Square Numbers.]
+
+  Cum ducis numerum per se, qui provenit inde
+  Sit tibi quadratus, ductus radix erit hujus,
+  Nec numeros omnes quadratos dicere debes,
+  Est autem omnis numerus radix alicujus.                          212
+  Quando voles numeri radicem querere, scribi
+  Debet; inde notes si sit locus ulterius impar,
+  Estque figura loco talis scribenda sub illo,
+  Que, per se dicta, numerum tibi destruat illum,                  216
+  Vel quantum poterit ex inde delebis eandem;
+  Vel retrahendo duples retrahens duplando sub ista
+  Que primo sequitur, duplicatur per duplacationem,
+  Post per se minuens pro posse quod est minuendum.                220
+  [{16}]Post his propones digitum, qui, more priori
+  Per precedentes, post per se multiplicatus,
+  Destruat in quantum poterit numerum remanentem,
+  Et sic procedens retrahens duplando figuram,                     224
+  Preponendo novam donec totum peragatur,
+  Subdupla propriis servare docetque duplatis;
+  Si det compositum numerum duplacio, debet
+  Inscribi digitus a parte dextra parte propinqua,                 228
+  Articulusque loco quo non duplicata resessit;
+  Si dabit articulum, sit cifra loco pereunte
+  Articulusque locum tenet unum, de duplicata resessit;
+  Si donet digitum, sub prima pone sequente,                       232
+  Si supraposita fuerit duplicata figura
+  Major proponi debet tantummodo cifra,
+  Has retrahens solito propones more figuram,
+  Usque sub extrema ita fac retrahendo figuras,                    236
+  Si totum deles numerum quem proposuisti,
+  Quadratus fuerit, de dupla quod duplicasti,
+  Sicque tibi radix illius certa patebit,
+  Si de duplatis fit juncta supprima figura;                       240
+  Radicem per se multiplices habeasque
+  Primo propositum, bene te fecisse probasti;
+  Non est quadratus, si quis restat, sed habentur
+  Radix quadrati qui stat major sub eadem;                         244
+  Vel quicquid remanet tabula servare memento;
+  Hoc casu radix per se quoque multiplicetur,
+  Vel sic quadratus sub primo major habetur,
+  Hinc addas remanens, et prius debes haberi;                      248
+  Si locus extremus fuerit par, scribe figuram
+  Sub pereunte loco per quam debes operari,
+  Que quantum poterit supprimas destruat ambas,
+  Vel penitus legem teneas operando priorem,                       252
+  Si suppositum digitus suo fine repertus,
+  Omnino delet illic scribi cifra debet,
+  A leva si qua sit ei sociata figura;
+  Si cifre remanent in fine pares decet harum                      256
+  Radices, numero mediam proponere partem,
+  Tali quesita radix patet arte reperta.
+  Per numerum recte si nosti multiplicare
+  Ejus quadratum, numerus qui pervenit inde                        260
+  Dicetur cubicus; primus radix erit ejus;
+  Nec numeros omnes cubicatos dicere debes,
+  Est autem omnis numerus radix alicujus;
+
+  [Sidenote: Cube Root.]
+
+  Si curas cubici radicem qu�rere, primo                           264
+  Inscriptum numerum distinguere per loca debes;
+  Que tibi mille notant a mille notante suprema
+  Initiam, summa operandi parte sinistra,
+  Illic sub scribas digitum, qui multiplicatus                     268
+  In semet cubice suprapositum sibi perdat,
+  Et si quid fuerit adjunctum parte sinistra
+  Si non omnino, quantum poteris minuendo,
+  Hinc triplans retrahe saltum, faciendo sub illa                  272
+  Que manet a digito deleto terna, figuram
+  Illi propones quo sub triplo asocietur,
+  Ut cum subtriplo per eam tripla multiplicatur;
+  Hinc per eam solam productum multiplicabis,                      276
+  Postea totalem numerum, qui provenit inde
+  A suprapositis respectu tolle triplate
+  Addita supprimo cubice tunc multiplicetur,
+  Respectu cujus, numerus qui progredietur                         280
+  Ex cubito ductu, supra omnes adimetur;
+  Tunc ipsam delens triples saltum faciendo,
+  Semper sub ternas, retrahens alias triplicatas
+  Ex hinc triplatis aliam propone figuram,                         284
+  Que per triplatas ducatur more priori;
+  Primo sub triplis sibi junctis, postea per�se,
+  In numerum ducta, productum de triplicatis:
+  Utque prius dixi numerus qui provenit inde                       288
+  A suprapositis has respiciendo trahatur,
+  Huic cubice ductum sub primo multiplicabis,
+  Respectumque sui, removebis de remanenti,
+  Et sic procedas retrahendo triplando figuram.                    292
+  Et proponendo nonam, donec totum peragatur,
+  Subtripla sub propriis servare decet triplicatis;
+  Si nil in fine remanet, numerus datus ante
+  Est cubicus; cubicam radicem sub tripla prebent,                 296
+  Cum digito juncto quem supprimo posuisti,
+  Hec cubice ducta, numerum reddant tibi primum.
+  Si quid erit remanens non est cubicus, sed habetur
+  Major sub primo qui stat radix cubicam,                          300
+  Servari debet quicquid radice remansit,
+  Extracto numero, decet hec addi cubicato.
+  Quo facto, numerus reddi debet tibi primus.
+  Nam debes per se radicem multiplicare                            304
+  Ex hinc in numerum duces, qui provenit inde
+  Sub primo cubicus major sic invenietur;
+  Illi jungatur remanens, et primus habetur,
+  Si per triplatum numerum nequeas operari;                        308
+  Cifram propones, nil vero per hanc operare
+  Set retrahens illam cum saltu deinde triplata,
+  Propones illi digitum sub lege priori,
+  Cumque cifram retrahas saliendo, non triplicabis,                312
+  Namque nihil cifre triplacio dicitur esse;
+  At tu cum cifram protraxeris aut triplicata,
+  Hanc cum subtriplo semper servare memento:
+  Si det compositum, digiti triplacio debet                        316
+  Illius scribi, digitus saliendo sub ipsam;
+  Digito deleto, que terna dicitur esse;
+  Jungitur articulus cum triplata pereunte,
+  Set facit hunc scribi per se triplacio prima,                    320
+  Que si det digitum per se scribi facit illum;
+  Consumpto numero, si sole fuit tibi cifre
+  Triplato, propone cifram saltum faciendo,
+  Cumque cifram retrahe triplam, scribendo figuram,                324
+  Preponas cifre, sic procedens operare,
+  Si tres vel duo serie in sint, pone sub yma,
+  A dextris digitum servando prius documentum.
+  Si sit continua progressio terminus nuper                        328
+  Per majus medium totalem multiplicato;
+  Si par, per medium tunc multiplicato sequentem.
+  Set si continua non sit progressio finis:
+  Impar, tunc majus medium si multiplicabis,                       332
+  Si par per medium sibi multiplicato propinquum.                  333
+
+
+FOOTNOTES (Appendix II, Carmen de Algorismo):
+
+  [1: "Hec pr�sens ars dicitur algorismus ab Algore rege ejus
+  inventore, vel dicitur ab _algos_ quod est ars, et _rodos_ quod est
+  numerus; qu� est ars numerorum vel numerandi, ad quam artem bene
+  sciendum inveniebantur apud Indos bis quinque (id est decem)
+  figur�." --_Comment. Thom� de Novo-Mercatu._ MS. Bib. Reg. Mus.
+  Brit. 12 E.�1.]
+
+  [2: "H� necessari� figur� sunt Indorum characteros." _MS. de
+  numeratione._ Bib. Sloan. Mus. Brit. 513, fol. 58. "Cum vidissem
+  Yndos constituisse IX literas in universo numero suo propter
+  dispositionem suam quam posuerunt, volui patefacere de opere quod
+  sit per eas aliquidque esset levius discentibus, si Deus voluerit.
+  Si autem Indi hoc voluerunt et intentio illorum nihil novem literis
+  fuit, causa que mihi potuit. Deus direxit me ad hoc. Si vero alia
+  dicam preter eam quam ego exposui, hoc fecerunt per hoc quod ego
+  exposui, eadem tam certissime et absque ulla dubitatione poterit
+  inveniri. Levitasque patebit aspicientibus et discentibus." MS.
+  U.L.C., Ii. vi. 5, f.�102.]
+
+  [3: From Eg. 2622.]
+
+  [4: 8 C. iv. inserts
+    Nullum cipa significat: dat significare sequenti.]
+
+  [5: From 12 E. 1.]
+
+  [6:
+    En argorisme devon prendre
+    Vii especes . . . .
+    Adision subtracion
+    Doubloison mediacion
+    Monteploie et division
+    Et de radix eustracion
+    A chez vii especes savoir
+    Doit chascun en memoire avoir
+    Letres qui figures sont dites
+    Et qui excellens sont ecrites. --MS. _Seld. Arch._ B.�26.]
+
+  [7: From 12 E. 1.]
+
+  [8: From 12 E. 1.]
+
+  [9: 8 C. iv. inserts
+    Atque figura prior nuper fuerit mediando.]
+
+  [10: _I.e._ figura secundo loco posita.]
+
+  [11: So 12 E. 1; 8 C. iv. inserts--
+
+  [12: 12 E. 1 inserts.]
+
+  [13: 12 E. 1 inserts to l. 174.]
+
+  [14: 12 E. 1 omits, Eg. 2622 inserts.]
+
+  [15: 12 E. 1 inserts.]
+
+  [16: 8 C. iv. inserts--
+    Hinc illam dele duplans sub ei psalliendo
+    Que sequitur retrahens quicquid fuerit duplicatum.]
+
+
+
+
+INDEX OF TECHNICAL TERMS[1*]
+
+    [Footnote 1*: This Index has been kindly prepared by Professor
+    J.�B. Dale, of King's College, University of London, and the
+    best thanks of the Society are due to him for his valuable
+    contribution.]
+
+  [Transcriber's Note:
+  The Technical Terms and Glossary (following) refer to page and line
+  numbers in the printed book. Information in [[double brackets]] has
+  been added by the transcriber to aid in text searching.]
+
+
+  +algorisme+, 33/12; +algorym+, +augrym+, 3/3; the art of computing,
+    using the so-called Arabic numerals.
+  The word in its various forms is derived from the Arabic
+   _al-Khowarazmi_ (i.e. the native of Khwarazm (Khiva)). This was the
+    surname of Ja'far Mohammad ben Musa, who wrote a treatise early in
+    the 9th century (see p.�xiv).
+  The form _algorithm_ is also found, being suggested by a supposed
+    derivation from the Greek +arithmos+ (number).
+
+  +antery+, 24/11; to move figures to the right of the position in
+    which they are first written. This operation is performed repeatedly
+    upon the multiplier in multiplication, and upon certain figures
+    which arise in the process of root extraction.
+
+  +anterioracioun+, 50/5; the operation of moving figures to the
+    right.  [[written anteriorac{i}o{u}n or anterioracio{u}n]]
+
+  +article+, 34/23; +articul+, 5/31; +articuls+, 9/36, 29/7,8;
+    a�number divisible by ten without remainder.   [[also articull{e}]]
+
+  +cast+, 8/12; to add one number to another.
+  'Addition is a _casting_ together of two numbers into one number,'
+    8/10.
+
+  +cifre+, 4/1; the name of the figure 0. The word is derived from the
+    Arabic _sifr_ = empty, nothing. Hence _zero_.
+  A cipher is the symbol of the absence of number or of zero quantity.
+    It may be used alone or in conjunction with digits or other ciphers,
+    and in the latter case, according to the position which it occupies
+    relative to the other figures, indicates the absence of units, or
+    tens, or hundreds, etc. The great superiority of the Arabic to all
+    other systems of notation resides in the employment of this symbol.
+    When the cipher is not used, the place value of digits has to be
+    indicated by writing them in assigned rows or columns. Ciphers,
+    however, may be interpolated amongst the significant figures used,
+    and as they sufficiently indicate the positions of the empty rows or
+    columns, the latter need not be indicated in any other way. The
+    practical performance of calculations is thus enormously facilitated
+    (see p.�xvi).
+
+  +componede+, 33/24; +composyt+, 5/35; with reference to numbers, one
+    compounded of a multiple of ten and a digit.
+    [[written componed{e}]]
+
+  +conuertide+ = conversely, 46/29, 47/9.
+    [[written co{n}u{er}tid{e} or {con}u{er}tid{e}]]
+
+  +cubicede+, 50/13; +to be c.+, to have its cube root found.
+    [[written cubiced{e}]]
+
+  +cubike nombre+, 47/8; a�number formed by multiplying a given number
+    twice by itself, _e.g._ 27 = 3�נ3�נ3. Now called simply a cube.
+    [[written cubik{e} ...]]
+
+  +decuple+, 22/12; the product of a number by ten. Tenfold.
+
+  +departys+ = divides, 5/29.  [[written dep{ar}tys]]
+
+  +digit+, 5/30; +digitalle+, 33/24; a�number less than ten,
+    represented by one of the nine Arabic numerals.
+    [[written digitall{e}]]
+
+  +dimydicion+, 7/23; the operation of dividing a number by two.
+    Halving.  [[written dimydicio�]]
+
+  +duccioun+, multiplication, 43/9.  [[written duccio{u}n]]
+
+  +duplacion+, 7/23, 14/15; the operation of multiplying a number by
+    two. Doubling.
+    [[written duplacio� or duplacio[n~] with fancy "n"]]
+
+  +i-mediet+ = halved, 19/23.
+
+  +intercise+ = broken, 46/2; intercise Progression is the name given
+    to either of the Progressions 1, 3, 5, 7, etc.; 2, 4, 6, 8, etc.,
+    in which the common difference is�2.  [[written int{er}cise]]
+
+  +lede into+, multiply by, 47/18.
+    [[words always separated, as "lede ... into"]]
+
+  +lyneal nombre+, 46/14; a number such as that which expresses the
+    measure of the length of a line, and therefore is not _necessarily_
+    the product of two or more numbers (_vide_ Superficial, Solid). This
+    appears to be the meaning of the phrase as used in _The Art of
+    Nombryng_. It is possible that the numbers so designated are the
+    prime numbers, that is, numbers not divisible by any other number
+    except themselves and unity, but it is not clear that this
+    limitation is intended.
+
+  +mediacioun+, 16/36, 38/16; dividing by two (see also +dimydicion+).
+    [[written mediacio[n~] with fancy "n", generally without "u"]]
+
+  +medlede nombre+, 34/1; a�number formed of a multiple of ten and a
+    digit (_vide_ componede, composyt).  [[written medled{e} ...]]
+
+  +medye+, 17/8, to halve; +mediete+, halved, 17/30; +ymedit+, 20/9.
+
+  +naturelle progressioun+, 45/22; the series of numbers 1, 2, 3, etc.
+    [[written naturell{e} p{ro}gressio{u}n]]
+
+  +produccioun+, multiplication, 50/11.  [[written produccio{u}n]]
+
+  +quadrat nombre+, 46/12; a number formed by multiplying a given
+    number by itself, _e.g._ 9 = 3�נ3, a�square.
+
+  +rote+, 7/25; +roote+, 47/11; root. The roots of squares and cubes
+    are the numbers from which the squares and cubes are derived by
+    multiplication into themselves.
+
+  +significatyf+, significant, 5/14; The significant figures of a
+    number are, strictly speaking, those other than zero, _e.g._ in 3 6
+    5 0 4 0 0, the significant figures are 3, 6, 5, 4. Modern usage,
+    however, regards all figures between the two extreme significant
+    figures as significant, even when some are zero. Thus, in the above
+    example, 3 6 5 0 4 are considered significant.
+
+  +solide nombre+, 46/37; a�number which is the product of three other
+    numbers, _e.g._ 66 = 11�נ2�נ3.  [[usually written solid{e}]]
+
+  +superficial nombre+, 46/18; a�number which is the product of two
+    other numbers, _e.g._ 6 = 2�נ3.
+    [[written sup{er}ficial or sup{er}ficiall{e}]]
+
+  +ternary+, consisting of three digits, 51/7.
+    [[written t{er}nary]]
+
+  +vnder double+, a digit which has been doubled, 48/3.
+
+  +vnder-trebille+, a digit which has been trebled, 49/28;
+    +vnder-triplat+, 49/39.
+    [[written vnder-trebill{e}, vnder-t{r}iplat]]
+
+  +w+, a symbol used to denote half a unit, 17/33
+    [[printed as superscript�^w]]
+
+
+
+
+GLOSSARY
+
+  [Transcriber's Note:
+
+  Words whose first appearance is earlier than the page cited in the
+  Glossary are identified in double-bracketed notes. To aid in text
+  searching, words written with internal {italics} are also noted,
+  and context is given for common words.]
+
+
+  +ablacioun+, taking away, 36/21  [[written ablacio{u}n]]
+  +addyst+, haddest, 10/37
+  +agregacioun+, addition, 45/22. (First example in N.E.D., 1547.)
+    [[written ag{r}egacio{u}n]]
+  +a-[gh]enenes+, against, 23/10
+  +allgate+, always, 8/39
+  +als+, as, 22/24
+  +and+, if, 29/8;
+    +&+, 4/27;
+    +& yf+, 20/7
+  +a-nendes+, towards, 23/15
+  +aproprede+, appropriated, 34/27  [[written ap{ro}pred{e}]]
+  +apwereth+, appears, 61/8
+  +a-risy[gh]t+, arises, 14/24
+  +a-rowe+, in a row, 29/10
+  +arsemetrike+, arithmetic, 33/1  [[written arsemetrik{e}]]
+  +ayene+, again, 45/15
+
+  +bagle+, crozier, 67/12
+  +bordure+ = ordure, row, 43/30  [[written bordur{e}]]
+  +borro+, _inf._ borrow, 11/38;
+    _imp. s._ +borowe+, 12/20;
+    _pp._ +borwed+, 12/15;
+    +borred+, 12/19
+  +boue+, above, 42/34
+
+  +caputule+, chapter, 7/26  [[written caputul{e}]]
+  +certayn+, assuredly, 18/34  [[written c{er}tay[n~]]]
+  +clepede+, called, 47/7  [[written cleped{e}]]
+  +competently+, conveniently, 35/8
+  +compt+, count, 47/29
+  +contynes+, contains, 21/12;  [[written {con}tynes]]
+    _pp._ +contenythe+, 38/39  [[written co{n}tenyth{e}]]
+  +craft+, art, 3/4
+
+  +distingue+, divide, 51/5
+
+  +egalle+, equal, 45/21  [[written egall{e}]]
+  +excep+, except, 5/16]
+  +exclusede+, excluded, 34/37  [[written exclused{e}]]
+  +excressent+, resulting, 35/16  [[written exc{re}ssent]]
+  +exeant+, resulting, 43/26
+  +expone+, expound, 3/23
+
+  +ferye+ = fer�e, fourth, 70/12
+  +figure+ = figures, 5/1  [[written fig{ure}]]
+  +for-by+, past, 12/11
+  +fors; no f.+, no matter, 22/24
+  +forseth+, matters, 53/30
+  +forye+ = for�e, forth, 71/8]
+  +fyftye+ = fyft�e, fifth, 70/16
+
+  +grewe+, Greek, 33/13
+
+  +haluendel+, half, 16/16;
+    +haldel+, 19/4;
+    _pl._ +haluedels+, 16/16
+  +hayst+, hast, 17/3, 32
+  +hast+, haste, 22/25  [[in "haue hast�to"]]
+  +heer+, higher, 9/35
+  +here+, their, 7/26  [[in "in her{e} caputul{e}"]]
+  +here-a-fore+, heretofore, 13/7  [[written her{e}-a-for{e}]]
+  +heyth+, was called, 3/5
+  +hole+, whole, 4/39;
+    +holle+, 17/1;
+    +hoole+, of three dimensions, 46/15
+  +holdy�e+, holds good, 30/5
+  +how be it that+, although, 44/4
+
+  +lede+ = lete, let, 8/37
+  +lene+, lend, 12/39
+  +lest+, least, 43/27  [[in "at the lest"]]
+  +lest+ = left, 71/9  [[in "the lest syde"]]
+  +leue+, leave, 6/5;
+    _pr. 3 s._ +leues+, remains, 11/19;  [[first in 10/40]]
+    +leus+, 11/28;
+    _pp._ +laft+, left, 19/24
+  +lewder+, more ignorant, 3/3  [[written lewd{er}]]
+  +lust+, desirest to, 45/13
+  +ly[gh]t+, easy, 15/31
+  +lymytes+, limits, 34/18;
+    +lynes+, 34/12;
+    +lynees+, 34/17;
+    Lat. limes, _pl._ limites.
+
+  +maystery+, achievement;  [[written mayst{er}y]]
+    +no m.+, no achievement, i.e. easy, 19/10
+  +me+, _indef. pron._ one, 42/1  [[first in 34/16]]
+  +mo+, more, 9/16
+    +moder+ = more (Lat. majorem), 43/22
+  +most+, must, 30/3  [[first in 3/12 and many more]]
+  +multipliede+, +to be m.+ = multiplying, 40/9
+  +mynvtes+, the sixty parts into which a unit is divided, 38/25
+    [[written mynvt{es}]]
+  +myse-wro[gh]t+, mis-wrought, 14/11
+
+  +nether+, nor, 34/25  [[in "It was, nether�is"]]
+  +nex+, next, 19/9
+  +no[gh]t+, nought, 5/7  [[first in 4/8]]
+  +note+, not, 30/5
+
+  +oo+, one, 42/20; +o+, 42/21  [[first in 34/27; 33/22]]
+  +omest+, uppermost, higher, 35/26;
+    +omyst+, 35/28
+  +omwhile+, sometimes, 45/31  [[first in 39/17]]
+  +on+, one, 8/29  [[in "on vnder an-o�{er}"]]
+  +opyne+, plain, 47/8  [[written opyn{e}]]
+  +or+, before, 13/25  [[in "or �ou be-gan"]]
+  +or+ = �e o�{er}, the other, 28/34  [[in "or by-twene"]]
+  +ordure+, order, 34/9;
+    row, 43/1  [[word form is "order"]]
+  +other+, or, 33/13, 43/26;
+    [[in "art other craft" on 33/13, "other how oft" on 43/26;
+    note also "one other other" on 35/24]]
+    +other .�.�. or+, either .�. . or, 38/37
+    [[in "other it is even or od{e}" on 38/37;
+    there are earlier occurrences]]
+  +ouerer+, upper, 42/15  [[written ou{er}er]]
+  +ouer-hippede+, passed over, 43/19  [[written ou{er}-hipped{e}]]
+
+  +recte+, directly, 27/20  [[in "stondes not recte";
+    also on 26/31 in "recte ou{er} his hede"]]
+  +remayner+, remainder, 56/28
+  +representithe+, represented, 39/14  [[written rep{re}sentith{e}]]
+  +resteth+, remains, 63/29  [[first in 57/29 and others]]
+  +rewarde+, regard, 48/6  [[written reward{e}]]
+  +rew+, row, 4/8
+  +rewle+, row, 4/20, 7/12;
+    [[in "place of �e rewle", "�e rewle of fig{ure}s"]]
+    +rewele+, 4/18;
+    +rewles+, rules, 5/33
+
+  +s.+ = scilicet, 3/8  [[in "s. Algorism{us}"]]
+  +sentens+, meaning, 14/29
+  +signifye(tyf)+, 5/13. The last three letters are added above the
+    line, evidently because of the word 'significatyf' in l.�14.
+    But the 'Solucio,' which contained the word, has been omitted.
+  +sithen+, since, 33/8
+  +some+, sum, result, 40/17, 32
+    [[first in 36/21 in "me may see a some", then in "the same some"
+    and "to some�of"]]
+  +sowne+, pronounce, 6/29
+  +singillatim+, singly, 7/25
+  +spices+, species, kinds, 34/4  [[first in 5/34 and others]]
+  +spyl+, waste, 14/26
+  +styde+, stead, 18/20
+  +subtrahe+, subtract, 48/12;
+    _pp._ +subtrayd+, 13/21
+  +sythes+, times, 21/16
+
+  +ta[gh]t+, taught, 16/36
+  +take+, _pp._ taken;
+    +t. fro+, starting from, 45/22  [[in "fro oone or tweyn{e} take"]]
+  +taward+, toward, 23/34
+  +thou[gh]t+, though, 5/20
+  +trebille+, multiply by three, 49/26  [[written trebill{e}]]
+  +twene+, two, 8/11  [[first in 4/23]]
+  +�ow+, though, 25/15  [[in "�ow �{o}u take"]]
+  +�ow[gh]t+, thought;
+    +be �.+, mentally, 28/4
+  +�us+ = �is, this, 20/33  [[in "�us nombur 214"]]
+
+  +vny+, unite, 45/10
+
+  +wel+, wilt, 14/31  [[in "If �{o}u wel"]]
+  +wete+, wit, 15/16;
+    +wyte+, know, 8/38;
+    _pr. 2 s._ +wost+, 12/38
+  +wex+, become, 50/18
+  +where+, whether, 29/12
+    [[written wher{e} in "wher{e} in �e secunde,�or"]]
+  +wher-thurghe+, whence, 49/15  [[written Wher-thurgh{e}]]
+  +worch+, work, 8/19;  [[first in 7/35]]
+    +wrich+, 8/35;
+    +wyrch+, 6/19;
+    _imp. s._ +worch+, 15/9;  [[first in 9/6]]
+    _pp._ +y-wroth+, 13/24
+  +write+, written, 29/19;
+    [[first in 6/37 in "hast write", "be write"]]
+    +y-write+, 16/1
+  +wryrchynge+ = wyrchynge, working, 30/4  [[written wryrchyng{e}]]
+  +w^t+, with, 55/8
+
+  +y-broth+, brought, 21/18
+  +ychon+, each one, 29/10  [[written ycho[n~]]]
+  +ydo+, done, added, 9/6
+    [[first in 8/37 in "haue ydo"; 9/6 in "ydo all to-ged{er}"]]
+  +ylke+, same, 5/12
+  +y-lyech+, alike, 22/23
+  +y-my[gh]t+, been able, 12/2
+  +y-now[gh]t+, enough, 15/31;
+    +ynov[gh]t+, 18/34
+  +yove+, given, 45/33
+  +y^t+, that, 52/8
+  +y-write+, _v._ +write.+
+  +y-wroth+, _v._ +worch.+
+
+
+       *       *       *       *       *
+           *       *       *       *
+       *       *       *       *       *
+
+
+MARGINAL NOTES:
+
++Headnotes+ have been moved to the beginning of the appropriate
+paragraph. Headnotes were omitted from the two Appendixes, as sidenotes
+give the same information.
+
++Line Numbers+ are cited in the Index and Glossary. They have been
+omitted from the e-text except in the one verse selection (App. II,
+_Carmen de Algorismo_). Instead, the Index and Glossary include
+supplemental information to help locate each word.
+
++Numbered Notes+:
+
+  Numbered sidenotes show page or leaf numbers from the original MSS.
+  In the e-text, the page number is shown as [*123b] inline; mid-word
+  page breaks are marked with a supplemental asterisk [*]. Numbers are
+  not used.
+
+  Footnotes give textual information such as variant readings. They
+  have been numbered sequentially within each title, with numbers
+  shown as [{1}] to avoid confusion with bracked text--including
+  single numerals--in the original. Editorial notes are shown as [1*].
+  When a footnote calls for added text, the addition is shown in the
+  body text with [[double brackets]].
+
++Sidenotes+ giving a running synopsis of the text have been moved to the
+beginning of each paragraph, where they are shown as a single note.
+
+
+ERRORS AND ANOMALIES (Noted by Transcriber):
+
+Introduction:
+
+  dated Mij^c
+    [_In this and the remainder of the paragraph, the letter shown as
+    ^c is printed directly above the preceding j._]
+
+The Crafte of Nombrynge:
+
+  sursu{m} {pr}ocedas m{u}ltiplicando
+    [_Italicized as shown: error for "p{ro}cedas"?_]
+  Sidenote: Our author makes a slip here
+    [_Elsewhere in the book, numerical errors are corrected in the
+    body text, with a footnote giving the original form._]
+  ten tymes so mych is �e nounb{re}
+    [_text unchanged: error for "as"?_]
+  6 tymes 24, [{19}]�en take
+    [_misplaced footnote anchor in original:
+    belongs with "6 times 24"_]
+  Fn. 7: 'Subt{ra}has a{u}t addis a dext{ri}s  [_open quote missing_]
+
+The Art of Nombryng:
+
+  oone of the digitis as .10. of 1.. 20. of. 2.
+    [_text unchanged: error for "as .10. of .1. 20. of .2."?_]
+  sette a-side half of tho m{inutes}
+    [_text unchanged: error for "the"?_]
+  and. 10. as before is come therof
+    [_text unchanged: error for "and .10."?_]
+  Sidenote: Where to set the quotiente  [_spelling (1922) unchanged_]
+  Sidenote: Definition of Progression.  [_f in "of" illegible_]
+  Sidenote: ... giving the value of ab.^2  [_That is, "a(b^2)."_]
+
+Accomptynge by counters:
+
+  For example of the [*117a.] ly[*]nes
+    [_final . in sidenote missing or invisible_]
+    [_also in 121b, 122a]
+  which in the fyrst summe is 5
+    [_invisible "5" supplied by transcriber_]
+  [*116a (_sic_).]
+    [_Editor's "sic": page numbering jumps back to 116 instead of the
+    expected 123, and continues from 116._]
+  [*123a] ... set downe y^e multiplyer .v. tymes, as here you�se
+    [_Diagram shown as printed, with 35500 for 36500 in one column,
+    and apparent misplaced "thousands" marker_]
+  365 (which is the nomber of dayes ...  [_open ( missing_]
+
+The arte of nombrynge by the hande:
+
+  for 1 the fynger is clasped in
+    [_In at least one printing of the text, "clasped" is misprinted
+    as "elasped"_]
+  but this teacyed me not  [_text unchanged_]
+
+Appendix I: A Treatise on the Numeration of Algorism:
+
+  _See Introduction and Glossary for long s:f and �:y errors_
+
+Appendix II: Carmen de Algorismo:
+
+  _In this selection, errors that are not explained in footnotes were
+  assumed to be typographic._
+
+  l. 99 Postea procedas  [procdeas]
+  l. 163 Articuli digitum post in digitum mixti duc  [post iu]
+
+
+
+
+
+
+End of Project Gutenberg's The Earliest Arithmetics in English, by Anonymous
+
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+
+<pre>
+
+Project Gutenberg's The Earliest Arithmetics in English, by Anonymous
+
+This eBook is for the use of anyone anywhere at no cost and with
+almost no restrictions whatsoever.  You may copy it, give it away or
+re-use it under the terms of the Project Gutenberg License included
+with this eBook or online at www.gutenberg.org
+
+
+Title: The Earliest Arithmetics in English
+
+Author: Anonymous
+
+Editor: Robert Steele
+
+Release Date: June 1, 2008 [EBook #25664]
+
+Language: English
+
+Character set encoding: UTF-8
+
+*** START OF THIS PROJECT GUTENBERG EBOOK THE EARLIEST ARITHMETICS IN ENGLISH ***
+
+
+
+
+Produced by Louise Hope, David Starner and the Online
+Distributed Proofreading Team at http://www.pgdp.net
+
+
+
+
+
+
+</pre>
+
+
+<div class = "mynote">
+<p><a name = "start" id = "start">This text</a> includes characters that
+will only display in UTF-8 (Unicode) file encoding:</p>
+
+<p class = "inset">
+ȝ, ſ (yogh, long s)<br />
+ɳ, łł (n with curl, crossed l: see below)<br />
+φ (Greek phi, sometimes used in printed text for 0)
+</p>
+
+<p>If any of these characters do not display properly, or if the
+apostrophes and quotation marks in this paragraph appear as garbage, you
+may have an incompatible browser or unavailable fonts. First, make sure
+that the browser’s “character set” or “file encoding” is set to Unicode
+(UTF-8). You may also need to change your browser’s default font.</p>
+
+<p>In <i>The Crafte of Nombrynge</i>, final <b>n</b> was sometimes
+written with an extra curl as
+<img src = "images/n_curl.gif" width = "17" height = "18"
+alt = "n with curl" />.
+It has been rendered as <b>ɳ</b> 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 <b>l</b> with a line
+<img src = "images/l_pair.gif" width = "15" height = "19"
+alt = "joined ll" />
+is shown as <b>łł</b>. The first few occurrences of <b>d</b>
+(for “pence”) were printed with a curl as
+<img src = "images/d_curl.gif" width = "15" height = "19"
+alt = "d with curl" />.
+The letter is shown with the same <b>d’</b> used in the
+remainder of the text.</p>
+
+<p>The word “withdraw” or “w<i>i</i>t<i>h</i>draw” was inconsistently
+hyphenated; it was left as printed, and line-end hyphens were retained.
+All brackets [&nbsp;] are in the original.</p>
+
+<p>The diagrams in “Accomptynge by Counters” may not line up perfectly
+in all browsers, but the contents should still be intelligible.</p>
+
+<p>The original text contained at least five types of marginal note.
+Details are given at the <a href = "#endnote">end of the e-text</a>.</p>
+
+<p>Typographical errors are shown in the text with <ins class =
+"correction" title = "like this">mouse-hover popups</ins>. Other
+underlined words are cross-references to the <a class = "terms" href =
+"#terms">Index of Technical Terms</a> and the <a class = "gloss" href =
+"#glossary">Glossary</a>.</p>
+</div>
+
+<div class = "titlepage">
+
+<p class = "illustration">
+<img src = "images/title_main.png" width = "375" height = "219" alt =
+"The Earliest Arithmetics in English / Early English Text Society. /
+Extra Series, No. CXVIII. / 1922 (for 1916)." />
+</p>
+
+</div>
+
+
+<hr class = "mid" />
+<div class = "contents">
+<h4><a name = "contents" id = "contents">Contents</a><br />
+<span class = "smaller">(added by transcriber)</span></h4>
+
+<table class = "toc" summary = "table of contents">
+<tr>
+<td>
+<a href = "#intro">Introduction</a>
+</td>
+<td class = "number">v</td>
+</tr>
+<tr>
+<td>
+<a href = "#crafte">The Crafte of Nombrynge</a>
+</td>
+<td class = "number">3</td>
+</tr>
+<tr>
+<td>
+<a href = "#art">The Art of Nombryng</a>
+</td>
+<td class = "number">33</td>
+</tr>
+<tr>
+<td>
+<a href = "#count">Accomptynge by Counters</a>
+</td>
+<td class = "number">52</td>
+</tr>
+<tr>
+<td>
+<p><a href = "#hand">The arte of nombrynge by the hande</a></p>
+</td>
+<td class = "number">66</td>
+</tr>
+<tr>
+<td>
+<p><a href = "#app1"><span class = "smallcaps">App. I.</span> A Treatise
+on the Numeration of Algorism</a></p>
+</td>
+<td class = "number">70</td>
+</tr>
+<tr>
+<td>
+<p><a href = "#app2"><span class = "smallcaps">App. II.</span> Carmen de
+Algorismo</a></p>
+</td>
+<td class = "number">72</td>
+</tr>
+<tr>
+<td>
+<a href = "#terms">Index of Technical Terms</a>
+</td>
+<td class = "number">81</td>
+</tr>
+<tr>
+<td>
+<a href = "#glossary">Glossary</a>
+</td>
+<td class = "number">83</td>
+</tr>
+</table>
+</div>
+
+<hr class = "mid" />
+
+<div class = "titlepage">
+<h1>The Earliest Arithmetics<br />
+in English</h1>
+
+<p>&nbsp;<br />&nbsp;</p>
+
+<h2 class = "two">EDITED WITH INTRODUCTION</h2>
+
+<h2 class = "three">BY</h2>
+
+<h2 class = "one">ROBERT STEELE</h2>
+
+<p>&nbsp;<br />&nbsp;</p>
+
+<p>&nbsp;<br />&nbsp;</p>
+
+<h2 class = "two">LONDON:</h2>
+<h2 class = "three">PUBLISHED FOR THE EARLY ENGLISH TEXT SOCIETY</h2>
+<h2 class = "two">BY HUMPHREY MILFORD, OXFORD UNIVERSITY PRESS,</h2>
+<h2 class = "three">AMEN CORNER, E.C. 4.<br />
+1922.</h2>
+
+</div>
+
+<div class = "intro">
+
+<span class = "pagenum">v</span>
+<a name = "page_v" id = "page_v"> </a>
+<h3><a name = "intro" id = "intro">INTRODUCTION</a></h3>
+
+
+<p><span class = "firstword">The</span> 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&nbsp;knowledge of which latter was fairly
+common from the thirteenth to the sixteenth centuries.</p>
+
+<p>The arithmetics in English known to me are:&mdash;</p>
+
+<div class = "inset hanging">
+<p>(1) Bodl. 790 G. VII. (2653) f. 146-154 (15th c.) <i>inc.</i> “Of
+angrym ther be IX figures in numbray&nbsp;.&nbsp;.&nbsp;.” A&nbsp;mere
+unfinished fragment, only getting as far as Duplation.</p>
+
+<p>(2) Camb. Univ. LI. IV. 14 (III.) f. 121-142 (15th c.) <i>inc.</i>
+“Al maner of thyngis that prosedeth ffro the frist
+begynnyng&nbsp;.&nbsp;.&nbsp;.”</p>
+
+<p>(3) Fragmentary passages or diagrams in Sloane 213 f.&nbsp;120-3
+(a&nbsp;fourteenth-century counting board), Egerton 2852 f.&nbsp;5-13,
+Harl. 218 f.&nbsp;147 and</p>
+
+<p>(4) The two MSS. here printed; Eg. 2622 f.&nbsp;136 and Ashmole 396
+f.&nbsp;48. All of these, as the language shows, are of the fifteenth
+century.</p>
+</div>
+
+<p>The <span class = "smallcaps">Crafte of Nombrynge</span> 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”&nbsp;×&nbsp;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 <i>de algorismo</i> of Alexander de Villa
+Dei (c. 1220), such as that of
+<span class = "pagenum">vi</span>
+<a name = "page_vi" id = "page_vi"> </a>
+Thomas of Newmarket contained in the British Museum MS. Reg. 12,
+E.&nbsp;1. A&nbsp;fragment of another translation of the same gloss was
+printed by Halliwell in his <i>Rara Mathematica</i> (1835) p.&nbsp;29.<a
+class = "tag" name = "tag_intro1" id = "tag_intro1" href =
+"#note_intro1">1</a> It corresponds, as far as p.&nbsp;71, l.&nbsp;2,
+roughly to p.&nbsp;3 of our version, and from thence to the end
+p.&nbsp;2, ll.&nbsp;16-40.</p>
+
+<p>The <span class = "smallcaps">Art of Nombryng</span> is one of the
+treatises bound up in the Bodleian MS. Ashmole 396. It measures
+11½”&nbsp;×&nbsp;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 <i>de arte numerandi</i> attributed to John
+of Holywood (Sacrobosco) and the translator had obviously a poor MS.
+before him. The <i>de arte numerandi</i> was printed in 1488, 1490
+(<i>s.n.</i>), 1501, 1503, 1510, 1517, 1521, 1522, 1523, 1582, and by
+Halliwell separately and in his two editions of <i>Rara Mathematica</i>,
+1839 and 1841, and reprinted by Curze in 1897.</p>
+
+<p>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 <i>Archiv für die
+Geschichte der Naturwissenschaften und der Technik</i>, 1909, and
+invited the scientific world to take up the “not unpleasant task” of
+editing&nbsp;it.</p>
+
+<p><span class = "smallcaps">Accomptynge by Counters</span> is reprinted
+from the 1543 edition of Robert Record’s Arithmetic, printed by
+R.&nbsp;Wolfe. It has been reprinted within the last few years by Mr.
+F.&nbsp;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&nbsp;system of digital numeration which
+seems of great antiquity and almost world-wide extension.</p>
+
+<p>After the fragment already referred to, I&nbsp;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
+<i>Rara Mathemathica</i>, but I have added a number of stanzas from
+<span class = "pagenum">vii</span>
+<a name = "page_vii" id = "page_vii"> </a>
+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.</p>
+
+
+<h4><a name = "intro_subject" id = "intro_subject">
+The Subject Matter.</a></h4>
+
+<p>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,<a class = "tag" name = "tag_intro2" id =
+"tag_intro2" href = "#note_intro2">2</a> were unable to devise methods
+for its use.</p>
+
+
+<h4><a name = "intro_instruments" id = "intro_instruments">
+The Pre-Medieval Instruments Used in Calculation.</a></h4>
+
+<p>The following are known:&mdash;</p>
+
+<p>(1) A flat polished surface or tablets, strewn with sand, on which
+figures were inscribed with a stylus.</p>
+
+<p>(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.</p>
+
+<p>(3) Tablets or boxes containing nine grooves or wires, in or on which
+ran beads.</p>
+
+<p>(4) Tablets on which nine (or more) horizontal lines were marked,
+each third being marked off.</p>
+
+<p>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
+<span class = "pagenum">viii</span>
+<a name = "page_viii" id = "page_viii"> </a>
+rows of arithmetical symbols. This board could only have been used with
+counters (<i>calculi</i>), preferably unmarked, as in our treatise of
+<i>Accomptynge by Counters</i>.</p>
+
+
+<h4><a name = "intro_roman" id = "intro_roman">
+Classical Roman Methods of Calculation.</a></h4>
+
+<p>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 (<i>calculi</i>, or <i>lapilli</i>),
+which were kept in boxes (<i>loculi</i>), were used in calculation, as
+we learn from Horace’s schoolboys (Sat.&nbsp;1. vi. 74). For the sand
+see Persius I.&nbsp;131, “Nec qui abaco numeros et secto in pulvere
+metas scit risisse,” Apul. Apolog. 16 (pulvisculo), Mart. Capella, lib.
+vii. 3,&nbsp;4, etc. Cicero says of an expert calculator “eruditum
+attigisse pulverem,” (de nat. Deorum, ii.&nbsp;18). Tertullian calls a
+teacher of arithmetic “primus numerorum arenarius” (de Pallio, <i>in
+fine</i>). 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,&nbsp;33).</p>
+
+<p>There are, however, still in existence four Roman counting boards of
+a kind which does not appear to come into literature. A&nbsp;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 <i>Accomptynge by Counters</i>, the right-hand five bead
+replacing the counter between the lines.</p>
+
+
+<h4><a name = "intro_abacus" id = "intro_abacus">
+The Boethian Abacus.</a></h4>
+
+<p>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.&nbsp;II. of his <i>Geometria</i> 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
+<span class = "pagenum">ix</span>
+<a name = "page_ix" id = "page_ix"> </a>
+nine numbers, known as the Boethian “apices” or “notae” (from whence our
+word “notation”). To these we shall return later&nbsp;on.</p>
+
+
+<h4><a name = "intro_abacists" id = "intro_abacists">
+The Abacists.</a></h4>
+
+<p>It would seem probable that writers on the calendar like Bede (<span
+class = "smallroman">A.D.</span> 721) and Helpericus (<span class =
+"smallroman">A.D.</span> 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&nbsp;II. in 999. His
+treatise on the use of the Abacus was written (c.&nbsp;980) to a friend
+Constantine, and was first printed among the works of Bede in the Basle
+(1563) edition of his works, I.&nbsp;159, in a somewhat enlarged form.
+Another tenth century treatise is that of Abbo of Fleury (c.&nbsp;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.</p>
+
+<p>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.</p>
+
+<p>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
+<span class = "pagenum">x</span>
+<a name = "page_x" id = "page_x"> </a>
+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?”<a class = "tag" name = "tag_intro3" id =
+"tag_intro3" href = "#note_intro3">3</a> Exchequer methods up to the
+sixteenth century were founded on the abacus, though when we have
+details later on, a&nbsp;different and simpler form was used.</p>
+
+<p>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&nbsp;zero
+is obviously not needed, and the result may be written down in
+words.</p>
+
+<h5>(<i>a</i>) <span class = "smallcaps">Multiplication.</span> 4600
+×&nbsp;23.</h5>
+
+<table class = "grid center" summary = "example">
+<col class = "outline" />
+<col class = "outline" />
+<col class = "outline" />
+<col class = "outline" />
+<col class = "outline" />
+<col class = "outline" />
+<col />
+<tr class = "outline">
+<td colspan = "3">Thousands</td>
+<td colspan = "3">&nbsp;</td>
+<td class = "edge">&nbsp;</td>
+</tr>
+<tr class = "outline middle">
+<td>H<br />u<br />n<br />d<br />r<br />e<br />d<br />s</td>
+<td>T<br />e<br />n<br />s</td>
+<td>U<br />n<br />i<br />t<br />s</td>
+<td>H<br />u<br />n<br />d<br />r<br />e<br />d<br />s</td>
+<td>T<br />e<br />n<br />s</td>
+<td>U<br />n<br />i<br />t<br />s</td>
+<td class = "edge">&nbsp;</td>
+</tr>
+<tr class = "outline">
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>4</td>
+<td>6</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td class = "edge"><b>Multiplicand.</b></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>1</td>
+<td>8</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td class = "edge">600 ×&nbsp;3.</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>1</td>
+<td>2</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td class = "edge">4000 ×&nbsp;3.</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>1</td>
+<td>2</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td class = "edge">600 × 20.</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>8</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td class = "edge">4000 × 20.</td>
+</tr>
+<tr class = "outline">
+<td>1</td>
+<td>&nbsp;</td>
+<td>5</td>
+<td>8</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td class = "edge">Total&nbsp;product.</td>
+</tr>
+<tr class = "outline">
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>2</td>
+<td>3</td>
+<td class = "edge"><b>Multiplier.</b></td>
+</tr>
+</table>
+
+<span class = "pagenum">xi</span>
+<a name = "page_xi" id = "page_xi"> </a>
+
+<h5>(<i>b</i>) <span class = "smallcaps">Division: direct.</span>
+100,000 ÷ 20,023. Here each counter in turn is a separate divisor.</h5>
+
+<table class = "grid center" summary = "example">
+<col class = "outline" />
+<col class = "outline" />
+<col class = "outline" />
+<col class = "outline" />
+<col class = "outline" />
+<col class = "outline" />
+<col />
+<tr class = "outline">
+<td>H.</td>
+<td>T.</td>
+<td>U.</td>
+<td>H.</td>
+<td>T.</td>
+<td>U.</td>
+<td class = "edge">&nbsp;</td>
+</tr>
+<tr class = "outline">
+<td>&nbsp;</td>
+<td>2</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>2</td>
+<td>3</td>
+<td class = "edge"><b>Divisors.</b></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>2</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td class = "edge"><p>Place greatest divisor to right of
+dividend.</p></td>
+</tr>
+<tr>
+<td>1</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td class = "edge"><b>Dividend.</b></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>2</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td class = "edge">Remainder.</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>1</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td class = "edge">&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>1</td>
+<td>9</td>
+<td>9</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td class = "edge">Another form of same.</td>
+</tr>
+<tr class = "underline">
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>8</td>
+<td>&nbsp;</td>
+<td class = "edge"><p>Product of 1st Quotient and&nbsp;20.</p></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>1</td>
+<td>9</td>
+<td>9</td>
+<td>2</td>
+<td>&nbsp;</td>
+<td class = "edge">Remainder.</td>
+</tr>
+<tr class = "underline">
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>1</td>
+<td>2</td>
+<td class = "edge"><p>Product of 1st Quotient and&nbsp;3.</p></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>1</td>
+<td>9</td>
+<td>9</td>
+<td>&nbsp;</td>
+<td>8</td>
+<td class = "edge"><b>Final remainder.</b></td>
+</tr>
+<tr class = "underline">
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>4</td>
+<td class = "edge">Quotient.</td>
+</tr>
+</table>
+
+<h5>(<i>c</i>) <span class = "smallcaps">Division by Differences.</span>
+900 ÷ 8. Here we divide by (10-2).</h5>
+
+<table class = "grid center" summary = "example">
+<col class = "outline" />
+<col class = "outline" />
+<col class = "outline" />
+<col class = "outline" />
+<col class = "outline" />
+<col class = "outline" />
+<col />
+<tr class = "outline">
+<td>&nbsp;&nbsp;&nbsp;</td>
+<td>&nbsp;&nbsp;&nbsp;</td>
+<td>&nbsp;&nbsp;&nbsp;</td>
+<td>H.</td>
+<td>T.</td>
+<td>U.</td>
+<td class = "edge">&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>2</td>
+<td class = "edge">Difference.</td>
+</tr>
+<tr class = "underline">
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>8</td>
+<td class = "edge">Divisor.</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td><a class = "tag" name = "tag_intro4" id = "tag_intro4" href =
+"#note_intro4">4</a>9</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td class = "edge"><b>Dividend.</b></td>
+</tr>
+<tr class = "underline">
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td><a class = "tag" href = "#note_intro4">4</a>1</td>
+<td>8</td>
+<td>&nbsp;</td>
+<td class = "edge"><p>Product of difference by 1st
+Quotient&nbsp;(9).</p></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>2</td>
+<td>&nbsp;</td>
+<td class = "edge"><p>Product of difference by 2nd
+Quotient&nbsp;(1).</p></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td><a class = "tag" href = "#note_intro4">4</a>1</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td class = "edge">Sum of 8 and&nbsp;2.</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>2</td>
+<td>&nbsp;</td>
+<td class = "edge"><p>Product of difference by 3rd
+Quotient&nbsp;(1).</p></td>
+</tr>
+<tr class = "underline">
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>4</td>
+<td class = "edge"><p>Product of difference by 4th Quot. (2).
+<b>Remainder.</b></p></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>2</td>
+<td class = "edge">4th Quotient.</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>1</td>
+<td>&nbsp;</td>
+<td class = "edge">3rd Quotient.</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>1</td>
+<td>&nbsp;</td>
+<td class = "edge">2nd Quotient.</td>
+</tr>
+<tr class = "underline">
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>9</td>
+<td>&nbsp;</td>
+<td class = "edge">1st Quotient.</td>
+</tr>
+<tr class = "underline">
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>1</td>
+<td>1</td>
+<td>2</td>
+<td class = "edge"><p><b>Quotient.</b> (<b>Total of all
+four.</b>)</p></td>
+</tr>
+</table>
+
+<span class = "pagenum">xii</span>
+<a name = "page_xii" id = "page_xii"> </a>
+
+<h5><span class = "smallcaps">Division.</span> 7800 ÷ 166.</h5>
+
+<table class = "grid center" summary = "example">
+<col class = "outline" />
+<col class = "outline" />
+<col class = "outline" />
+<col class = "outline" />
+<col class = "outline" />
+<col class = "outline" />
+<col />
+<tr class = "outline">
+<td colspan = "3">Thousands</td>
+<td colspan = "3">&nbsp;</td>
+<td class = "edge">&nbsp;</td>
+</tr>
+<tr class = "outline">
+<td>H.</td>
+<td>T.</td>
+<td>U.</td>
+<td>H.</td>
+<td>T.</td>
+<td>U.</td>
+<td class = "edge">&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>3</td>
+<td>4</td>
+<td class = "edge"><p>Differences (making 200 trial divisor).</p></td>
+</tr>
+<tr class = "underline">
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>1</td>
+<td>6</td>
+<td>6</td>
+<td class = "edge">Divisors.</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td><a class = "tag" href = "#note_intro4">4</a>7</td>
+<td>8</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td class = "edge"><b>Dividends.</b></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>1</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td class = "edge"><p>Remainder of greatest dividend.</p></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>1</td>
+<td>2</td>
+<td>&nbsp;</td>
+<td class = "edge"><p>Product of 1st difference (4) by 1st
+Quotient&nbsp;(3).</p></td>
+</tr>
+<tr class = "underline">
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>9</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td class = "edge"><p>Product of 2nd difference (3) by 1st
+Quotient&nbsp;(3).</p></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td><a class = "tag" href = "#note_intro4">4</a>2</td>
+<td>8</td>
+<td>2</td>
+<td>&nbsp;</td>
+<td class = "edge">New dividends.</td>
+</tr>
+<tr class = "underline">
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>3</td>
+<td>4</td>
+<td>&nbsp;</td>
+<td class = "edge"><p>Product of 1st and 2nd difference by 2nd
+Quotient&nbsp;(1).</p></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td><a class = "tag" href = "#note_intro4">4</a>1</td>
+<td>1</td>
+<td>6</td>
+<td>&nbsp;</td>
+<td class = "edge">New dividends.</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>2</td>
+<td>&nbsp;</td>
+<td class = "edge"><p>Product of 1st difference by 3rd
+Quotient&nbsp;(5).</p></td>
+</tr>
+<tr class = "underline">
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>1</td>
+<td>5</td>
+<td>&nbsp;</td>
+<td class = "edge"><p>Product of 2nd difference by 3rd
+Quotient&nbsp;(5).</p></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td><a class = "tag" href = "#note_intro4">4</a>3</td>
+<td>3</td>
+<td>&nbsp;</td>
+<td class = "edge">New dividends.</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>1</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td class = "edge"><p>Remainder of greatest dividend.</p></td>
+</tr>
+<tr class = "underline">
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>3</td>
+<td>4</td>
+<td class = "edge"><p>Product of 1st and 2nd difference by 4th
+Quotient&nbsp;(1).</p></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>1</td>
+<td>6</td>
+<td>4</td>
+<td class = "edge"><p><b>Remainder</b> (less than divisor).</p></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>1</td>
+<td class = "edge">4th Quotient.</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>5</td>
+<td class = "edge">3rd Quotient.</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>1</td>
+<td>&nbsp;</td>
+<td class = "edge">2nd Quotient.</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>3</td>
+<td>&nbsp;</td>
+<td class = "edge">1st Quotient.</td>
+</tr>
+<tr class = "outline">
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>4</td>
+<td>6</td>
+<td class = "edge"><b>Quotient.</b></td>
+</tr>
+</table>
+
+
+<span class = "pagenum">xiii</span>
+<a name = "page_xiii" id = "page_xiii"> </a>
+
+<h5><span class = "smallcaps">Division.</span> 8000 ÷ 606.</h5>
+
+<table class = "grid center" summary = "example">
+<col class = "outline" />
+<col class = "outline" />
+<col class = "outline" />
+<col class = "outline" />
+<col class = "outline" />
+<col class = "outline" />
+<col />
+<tr class = "outline">
+<td colspan = "3">Thousands</td>
+<td colspan = "3">&nbsp;</td>
+<td class = "edge">&nbsp;</td>
+</tr>
+<tr class = "outline">
+<td>H.</td>
+<td>T.</td>
+<td>U.</td>
+<td>H.</td>
+<td>T.</td>
+<td>U.</td>
+<td class = "edge">&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>9</td>
+<td>&nbsp;</td>
+<td class = "edge"><p>Difference (making 700 trial divisor).</p></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>4</td>
+<td class = "edge">Difference.</td>
+</tr>
+<tr class = "underline">
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>6</td>
+<td>&nbsp;</td>
+<td>6</td>
+<td class = "edge">Divisors.</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td><a class = "tag" href = "#note_intro4">4</a>8</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td class = "edge"><b>Dividend.</b></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>1</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td class = "edge">Remainder of dividend.</td>
+</tr>
+<tr class = "underline">
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>9</td>
+<td>4</td>
+<td>&nbsp;</td>
+<td class = "edge"><p>Product of difference 1 and 2 with 1st
+Quotient&nbsp;(1).</p></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td><a class = "tag" href = "#note_intro4">4</a>1</td>
+<td>9</td>
+<td>4</td>
+<td>&nbsp;</td>
+<td class = "edge">New dividends.</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>3</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td class = "edge"><p>Remainder of greatest dividend.</p></td>
+</tr>
+<tr class = "underline">
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>9</td>
+<td>4</td>
+<td class = "edge"><p>Product of difference 1 and 2 with 2nd
+Quotient&nbsp;(1).</p></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td><a class = "tag" href = "#note_intro4">4</a>1</td>
+<td>3</td>
+<td>3</td>
+<td>4</td>
+<td class = "edge">New dividends.</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>3</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td class = "edge"><p>Remainder of greatest dividend.</p></td>
+</tr>
+<tr class = "underline">
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>9</td>
+<td>4</td>
+<td class = "edge"><p>Product of difference 1 and 2 with 3rd
+Quotient&nbsp;(1).</p></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>7</td>
+<td>2</td>
+<td>8</td>
+<td class = "edge">New dividends.</td>
+</tr>
+<tr class = "underline">
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>6</td>
+<td>&nbsp;</td>
+<td>6</td>
+<td class = "edge"><p>Product of divisors by 4th
+Quotient&nbsp;(1).</p></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>1</td>
+<td>2</td>
+<td>2</td>
+<td class = "edge"><b>Remainder.</b></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>1</td>
+<td class = "edge">4th Quotient.</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>1</td>
+<td class = "edge">3rd Quotient.</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>1</td>
+<td class = "edge">2nd Quotient.</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>1</td>
+<td>&nbsp;</td>
+<td class = "edge">1st Quotient.</td>
+</tr>
+<tr class = "outline">
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>1</td>
+<td>3</td>
+<td class = "edge"><b>Quotient.</b></td>
+</tr>
+</table>
+
+<p>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.&nbsp;.&nbsp;.&nbsp;. “the
+best of living computers,” Gislebert, and Simonus de Rotellis (Simon of
+the Rolls). They flourished most probably in the
+<span class = "pagenum">xiv</span>
+<a name = "page_xiv" id = "page_xiv"> </a>
+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.</p>
+
+<p>Finally, the term Abacists came to be applied to computers by manual
+arithmetic. A&nbsp;MS. Algorithm of the thirteenth century (Sl. 3281,
+f.&nbsp;6,&nbsp;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.”</p>
+
+<p>In a composite treatise containing tracts written <span class =
+"smallroman">A.D.</span> 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&nbsp;×&nbsp;7. The difference of 8 is 2, of 7 is 3, the next article
+being 10; 7&nbsp;-&nbsp;2 is 5. 5&nbsp;×&nbsp;10 = 50; 2&nbsp;×&nbsp;3 =
+6. 50&nbsp;+&nbsp;6 = 56 answer. The rule will hold in such cases as
+17&nbsp;×&nbsp;15 where the article next higher is the same for both,
+<i>i.e.</i>, 20; but in such a case as 17&nbsp;×&nbsp;9 the difference
+for each number must be taken from the higher article, <i>i.e.</i>, the
+difference of 9 will be&nbsp;11.</p>
+
+
+<h4><a name = "intro_algorists" id = "intro_algorists">
+The Algorists.</a></h4>
+
+<p>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.&nbsp;.&nbsp;.&nbsp;.” 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&nbsp;few lines of this translation, as copied by Halliwell, are
+given on p.&nbsp;72, note&nbsp;2. Another translation still seems to
+have been made by Johannes Hispalensis.</p>
+
+<p>Algorism is distinguished from Abacist computation by recognising
+seven rules, Addition, Subtraction, Duplation, Mediation,
+Multiplication, Division, and Extraction of Roots, to which were
+afterwards
+<span class = "pagenum">xv</span>
+<a name = "page_xv" id = "page_xv"> </a>
+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.<a class =
+"tag" name = "tag_intro5" id = "tag_intro5" href = "#note_intro5">5</a>
+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.</p>
+
+<p>The number of Latin Algorisms still in manuscript is comparatively
+large, but we are here only concerned with two&mdash;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 <i>Rara Mathematica</i>, is adapted for use on a board covered with
+sand, a&nbsp;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.&nbsp;.&nbsp;.&nbsp;.” “Si voluerimus depingere in
+pulvere predictos digitos secundum consuetudinem
+algorismi&nbsp;.&nbsp;.&nbsp;.” “et sciendum est quod in nullo loco
+minutorum sive secundorum .&nbsp;.&nbsp;. in pulvere debent scribi
+plusquam sexaginta.”</p>
+
+
+<h4><a name = "intro_modern" id = "intro_modern">Modern
+Arithmetic.</a></h4>
+
+<p>Modern Arithmetic begins with Leonardi Fibonacci’s treatise “de
+Abaco,” written in 1202 and re-written in 1228. It is modern
+<span class = "pagenum">xvi</span>
+<a name = "page_xvi" id = "page_xvi"> </a>
+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 <i>The arte of nombrynge by the
+hand</i> in mental arithmetic, calling it “modus Indorum.” Leonardo also
+introduces the method of proof by “casting out the nines.”</p>
+
+
+<h4><a name = "intro_digital" id = "intro_digital">Digital
+Arithmetic.</a></h4>
+
+<p>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&nbsp;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 <i>Zeitschrift des
+Münchener Alterthumsvereins</i> a set, almost complete, of them with a
+Byzantine treatise; a&nbsp;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.</p>
+
+<p>Methods of calculation by means of these signs undoubtedly have
+existed, but they were too involved and liable to error to be much
+used.</p>
+
+
+<h4><a name = "intro_arabic" id = "intro_arabic">
+The Use of “Arabic” Figures.</a></h4>
+
+<p>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.&nbsp;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
+<span class = "pagenum">xvii</span>
+<a name = "page_xvii" id = "page_xvii"> </a>
+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 <ins class = "correction"
+title = "c printed directly above j">Mij<sup>c</sup></ins>. lviii.,
+the second Mij<sup>c</sup>. lxi., the third Mij<sup>c</sup>. 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.</p>
+
+
+<h4><a name = "intro_board" id = "intro_board">
+The Counting Board.</a></h4>
+
+<p>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.</p>
+
+<p>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,
+<i>Anthropology</i> (1892), p.&nbsp;314.</p>
+
+<span class = "pagenum">xviii</span>
+<a name = "page_xviii" id = "page_xviii"> </a>
+<p>Record’s treatise also mentions another method of counter notation
+(p.&nbsp;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&nbsp;counting board for merchants’ use is printed by Halliwell in
+<i>Rara Mathematica</i> (p.&nbsp;72) from Sloane MS. 213, and two others
+are figured in Egerton 2622 f.&nbsp;82 and f.&nbsp;83. The latter is
+said to be “novus modus computandi secundum inventionem Magistri Thome
+Thorleby,” and is in principle, the same as the “Swanpan.”</p>
+
+<p>The Exchequer table is described in the <i>Dialogus de Scaccario</i>
+(Oxford, 1902), p.&nbsp;38.</p>
+
+<div class = "footnote">
+
+<p>
+<a name = "note_intro1" id = "note_intro1" href = "#tag_intro1">1.</a>
+Halliwell printed the two sides of his leaf in the wrong order. This and
+some obvious errors of transcription&mdash;‘ferye’ for ‘ferthe,’ ‘lest’
+for ‘left,’ etc., have not been corrected in the reprint on
+pp.&nbsp;70-71.</p>
+
+<p>
+<a name = "note_intro2" id = "note_intro2" href = "#tag_intro2">2.</a>
+For Egyptian use see Herodotus, ii.&nbsp;36, Plato, <i>de Legibus</i>,
+VII.</p>
+
+<p>
+<a name = "note_intro3" id = "note_intro3" href = "#tag_intro3">3.</a>
+See on this Dr. Poole, <i>The Exchequer in the Twelfth Century</i>,
+Chap. III., and Haskins, <i>Eng. Hist. Review</i>, 27, 101. The hidage
+of Essex in 1130 was 2364 hides.</p>
+
+<p>
+<a name = "note_intro4" id = "note_intro4" href = "#tag_intro4">4.</a>
+These figures are removed at the next step.</p>
+
+<p>
+<a name = "note_intro5" id = "note_intro5" href = "#tag_intro5">5.</a>
+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.</p>
+
+</div>
+
+</div> <!-- end div intro -->
+
+<div class = "titlepage">
+
+<p>&nbsp;</p>
+
+<p class = "illustration">
+<img src = "images/title_inner.png" width = "324" height = "71"
+alt = "The Earliest Arithmetics in English."
+title = "The Earliest Arithmetics in English." /></p>
+
+<p>&nbsp;</p>
+
+</div>
+
+<div class = "maintext">
+
+<div class = "craft">
+
+<span class = "pagenum">3</span>
+<a name = "page3" id = "page3"> </a>
+
+<p class = "illustration">
+<a name = "crafte" id = "crafte">
+<img src = "images/title_craft.png" width = "350" height = "36"
+alt = "The Crafte of Nombrynge."
+title = "The Crafte of Nombrynge." /></a></p>
+
+<hr class = "tiny" />
+
+<p class = "subhead"><i>Egerton</i> 2622.</p>
+
+<span class = "linenum">leaf 136 <i>a</i>.</span>
+
+<div class = "verse">
+<p><span class = "dropcap">H</span>Ec algorism<i>us</i> ars
+p<i>re</i>sens dicit<i>ur</i>; in qua</p>
+<p>Talib<i>us</i> indor<i>um</i> fruim<i>ur</i> bis
+qui<i>n</i>q<i>ue</i> figuris.</p>
+</div>
+
+<p>
+<span class = "sidenote">A derivation of Algorism.</span>
+This boke is called þe boke of <a class = "terms" name = "algorym" id =
+"algorym" href = "#terms_algorisme">algorym</a>, or Augrym aft<i>er</i>
+<a class = "gloss" name = "lewder" id = "lewder" href =
+"#gloss_lewder">lewd<i>er</i></a> vse. And þis boke tretys þe <a class =
+"gloss" name = "craft" id = "craft" href = "#gloss_craft">Craft</a> of
+Nombryng, þe quych crafte is called also Algorym. Ther was a kyng of
+Inde, þe quich <a class = "gloss" name = "heyth" id = "heyth" href =
+"#gloss_heyth">heyth</a> Algor, &amp; he made þis craft. And
+aft<i>er</i> his name he called hit algory<i>m</i>;
+<span class = "sidenote">Another derivation of the word.</span>
+or els anoþ<i>er</i> cause is quy it is called Algorym, for þe latyn
+word of hit <a class = "gloss" name = "sc" id = "sc" href =
+"#gloss_sc">s.</a> Algorism<i>us</i> com<i>es</i> of Algos, grece,
+q<i>uid</i> e<i>st</i> ars, latine, craft oɳ englis, and rides,
+q<i>uid</i> e<i>st</i> <i>nu</i>me<i>rus</i>, latine,
+A&nbsp;nomb<i>ur</i> oɳ englys, inde d<i>icitu</i>r Algorism<i>us</i>
+p<i>er</i> addic<i>i</i>one<i>m</i> hui<i>us</i> sillabe m<i>us</i>
+&amp; subtracc<i>i</i>onem d &amp; e, q<i>ua</i>si ars num<i>er</i>andi.
+¶&nbsp;fforthermor<i>e</i> ȝe <a class = "gloss" name = "most1" id =
+"most1" href = "#gloss_most">most</a> vnd<i>ir</i>stonde þ<i>a</i>t in
+þis craft ben vsid teen figurys, as here ben<i>e</i> writen for
+ensampul, φ&nbsp;9 8 7 6 5 4 3 2&nbsp;1. ¶&nbsp;<a class = "gloss" name
+= "expone" id = "expone" href = "#gloss_expone">Expone</a> þe too
+v<i>er</i>sus afor<i>e</i>: this p<i>re</i>sent craft ys called
+Algorism<i>us</i>, in þe quych we vse teen signys of Inde. Questio.
+¶&nbsp;Why teɳ fyguris of Inde? Solucio. for as I haue sayd afore þai
+wer<i>e</i> fonde fyrst in Inde of a kyng<i>e</i> of þat Cuntre,
+þ<i>a</i>t was called Algor.</p>
+
+<p class = "headnote"><span class = "headnote">
+Notation and Numeration.</span></p>
+
+<span class = "sidenote">v<i>ersus</i> [in&nbsp;margin].</span>
+
+<div class = "verse">
+<p class = "pilcrow">
+¶ Prima sig<i>nifica</i>t unu<i>m</i>; duo ve<i>r</i>o
+s<i>e</i>c<i>un</i>da:</p>
+<p class = "pilcrow">
+¶ Tercia sig<i>nifica</i>t tria; sic procede sinistre.</p>
+<p class = "pilcrow">
+¶ Don<i>e</i>c ad extrema<i>m</i> venias, que cifra voca<i>tur</i>.</p>
+</div>
+
+<h5>¶ Cap<i>itulu</i>m primum de significac<i>i</i>o<i>n</i>e
+figurar<i>um</i>.</h5>
+
+<p>
+<span class = "sidenote original">Expo<i>sitio</i> v<i>ersus</i>.</span>
+In þis verse is notifide þe significac<i>i</i>on of þese figur<i>is</i>.
+And þus expone the verse.
+<span class = "sidenote">The meaning and place of the figures.</span>
+Þe first signifiyth on<i>e</i>, þe secu<i>n</i>de
+<span class = "linenum">leaf 136 <i>b</i>.</span>
+signi*fiyth tweyn<i>e</i>, þe thryd signifiyth thre, &amp; the fourte
+signifiyth 4. ¶&nbsp;And so forthe towarde þe lyft syde of þe tabul or
+of þe boke þ<i>a</i>t þe figures ben<i>e</i> writen<i>e</i> in, til þat
+þ<i>o</i>u come to the last figure, þ<i>a</i>t is
+<span class = "pagenum">4</span>
+<a name = "page4" id = "page4"> </a>
+called a <a class = "terms" name = "cifre" id = "cifre" href =
+"#terms_cifre">cifre</a>. ¶&nbsp;Questio. In quych syde sittes þe first
+figur<i>e</i>? Soluc<i>io</i>, forsothe loke quich figure is first in þe
+ryȝt side of þe bok or of þe tabul, &amp; þ<i>a</i>t same is þe first
+figur<i>e</i>, for þ<i>o</i>u schal write bakeward, as here, 3. 2. 6. 4.
+1. 2. 5.
+<span class = "sidenote">Which figure is read first.</span>
+The fig<i>ur</i>e of 5. was first <a class = "gloss" name = "write1" id
+= "write1" href = "#gloss_write">write</a>, &amp; he is þe first, for he
+sittes oɳ þe riȝt syde. And the fig<i>ur</i>e of 3 is last.
+¶&nbsp;Neu<i>er</i>-þe-les wen he says ¶&nbsp;P<i>ri</i>ma
+sig<i>nifica</i>t vnu<i>m</i> &amp;c., þat is to say, þe first betokenes
+on<i>e</i>, þe secu<i>n</i>de. 2. &amp; fore-þ<i>er</i>-mor<i>e</i>, he
+vnd<i>ir</i>stondes <a class = "gloss" name = "noght1" id = "noght1"
+href = "#gloss_noght">noȝt</a> of þe first fig<i>ur</i>e of eu<i>er</i>y
+<a class = "gloss" name = "rew" id = "rew" href = "#gloss_rew">rew</a>.
+¶&nbsp;But he vnd<i>ir</i>stondes þe first figure þ<i>a</i>t is in þe
+nomb<i>ur</i> of þe forsayd teen figuris, þe quych is on<i>e</i> of
+þ<i>e</i>se. 1. And þe secu<i>n</i>de 2. &amp; so forth.</p>
+
+<span class = "sidenote">v<i>ersus</i> [in&nbsp;margin].</span>
+
+<div class = "verse">
+<p class = "pilcrow">
+¶ Quelib<i>et</i> illar<i>um</i> si pr<i>im</i>o limite ponas,</p>
+<p class = "pilcrow">
+¶ Simplicite<i>r</i> se significat: si v<i>er</i>o se<i>cun</i>do,</p>
+<p>Se decies: sursu<i>m</i> <ins class = "correction" title = "italics as shown: error for ‘p{ro}’?"><i>pr</i>ocedas</ins>
+m<i>u</i>ltiplicando.</p>
+<p class = "pilcrow">
+¶ Na<i>m</i>q<i>ue</i> figura seque<i>n</i>s q<i>uam</i>uis signat
+decies pl<i>us.</i></p>
+<p class = "pilcrow">
+¶ Ipsa locata loco quam sign<i>ific</i>at p<i>ertin</i>ente.</p>
+</div>
+
+<p><span class = "sidenote">Expo<i>sitio</i> [in margin].</span>
+¶&nbsp;Expone þis v<i>er</i>se þus. Eu<i>er</i>y of þese figuris
+bitokens hym selfe &amp; no mor<i>e</i>, yf he stonde in þe first place
+of þe <a class = "gloss" name = "rewele" id = "rewele" href =
+"#gloss_rewle">rewele</a> / this worde Simplicit<i>er</i> in þat verse
+it is no more to say but þat, &amp; no mor<i>e</i>.
+<span class = "sidenote">An explanation of the principles of
+notation.</span>
+¶&nbsp;If it stonde in the secu<i>n</i>de place of þe <a class = "gloss"
+name = "rewle" id = "rewle" href = "#gloss_rewle">rewle</a>, he betokens
+ten<i>e</i> tymes hym selfe, as þis figur<i>e</i> 2 here 20 tokens ten
+tyme hym selfe,
+<span class = "linenum">leaf 137 <i>a</i>.</span>
+*þat is twenty, for he hym selfe betokenes twey<i>ne</i>, &amp; ten
+tymes <a class = "gloss" name = "twene1" id = "twene1" href =
+"#gloss_twene">twene</a> is twenty. And for he stondis oɳ þe lyft side
+&amp; in þe secu<i>n</i>de place, he betokens ten tyme hy<i>m</i> selfe.
+And so go forth. ¶&nbsp;ffor eu<i>er</i>y <a class = "gloss" name =
+"figure" id = "figure" href = "#gloss_figure">fig<i>ure</i></a>, &amp;
+he stonde aft<i>ur</i> a-noþ<i>er</i> toward the lyft side, he schal
+betoken<i>e</i> ten tymes as mich mor<i>e</i> as he schul betoken &amp;
+he stode in þe place þ<i>ere</i> þat þe fig<i>ure</i> a-for<i>e</i> hym
+stondes.
+<span class = "sidenote">An example:</span>
+loo an ensampull<i>e</i>. 9.&nbsp;6.&nbsp;3.&nbsp;4. Þe fig<i>ure</i> of
+4. þ<i>a</i>t hase þis schape <a href = "images/num4_full.png" target =
+"_blank"><img src = "images/num4_full.png" width = "11" height = "14"
+alt = "{4}" /></a>. betokens bot hymselfe, for he stondes in þe first
+place.
+<span class = "sidenote">units,</span>
+The fig<i>ure</i> of 3. þat hase þis schape <a href =
+"images/num3_full.png" target = "_blank"><img src =
+"images/num3_full.png" width = "7" height = "15" alt = "{3}" /></a>.
+betokens ten tymes mor<i>e</i> þen he schuld <a class = "gloss" name =
+"amp" id = "amp" href = "#gloss_and">&amp;</a> he stde þ<i>ere</i>
+þ<i>a</i>t þe fig<i>ure</i> of 4. stondes, þ<i>a</i>t is thretty.
+<span class = "sidenote">tens,</span>
+The fig<i>ure</i> of 6, þ<i>a</i>t hase þis schape <a href =
+"images/num6_full.png" target = "_blank"><img src =
+"images/num6_full.png" width = "8" height = "14" alt = "{6}" /></a>,
+betokens ten tymes mor<i>e</i> þan he schuld &amp; he stode þ<i>ere</i>
+as þe fig<i>ure</i> of <a href = "images/num3_full.png" target =
+"_blank"><img src = "images/num3_full.png" width = "7" height = "15" alt
+= "{3}" /></a>. stondes, for þ<i>ere</i> he schuld tokyn<i>e</i> bot
+sexty, &amp; now he betokens ten tymes mor<i>e</i>, þat is sex hundryth.
+<span class = "sidenote">hundreds,</span>
+The fig<i>ure</i> of 9. þ<i>a</i>t hase þis schape <a href =
+"images/num9_full.png" target = "_blank"><img src =
+"images/num9_full.png" width = "8" height = "15" alt = "{9}" /></a>.
+betokens ten tymes mor<i>e</i> þan<i>e</i> he schuld &amp; he stode in
+þe place þ<i>ere</i> þe fig<i>ure</i> of sex stondes, for þen he schuld
+betoken to 9. hundryth, and in þe place þ<i>ere</i> he stondes now he
+betokens 9. þousande.
+<span class = "sidenote">thousands.</span>
+Al þe <a class = "gloss" name = "hole" id = "hole" href =
+"#gloss_hole">hole</a> nomb<i>ur</i> is 9 thousande sex hundryth &amp;
+four<i>e</i> &amp; thretty. ¶&nbsp;fforthermor<i>e</i>, when
+<span class = "pagenum">5</span>
+<a name = "page5" id = "page5"> </a>
+þ<i>o</i>u schalt rede a nomb<i>ur</i> of fig<i>ure</i>,
+<span class = "sidenote">How to read the number.</span>
+þ<i>o</i>u schalt begyn<i>e</i> at þe last fig<i>ure</i> in the lyft
+side, &amp; rede so forth to þe riȝt side as her<i>e</i> 9.&nbsp;6.
+3.&nbsp;4. Thou schal begyn to rede at þe fig<i>ure</i> of 9. &amp; rede
+forth þus.&nbsp;9.
+<span class = "linenum">leaf 137 <i>b</i>.</span>
+*thousand sex hundryth thritty &amp; foure. But when þ<i>o</i>u
+schall<i>e</i> write, þ<i>o</i>u schalt be-gynne to write at þe ryȝt
+side.</p>
+
+<div class = "verse">
+<p class = "pilcrow">
+¶ Nil cifra sig<i>nifica</i>t s<i>ed</i> dat signa<i>re</i>
+sequenti.</p>
+</div>
+
+<p><span class = "sidenote">The meaning and use of the cipher.</span>
+Expone þis v<i>er</i>se. A&nbsp;cifre tokens <a class = "gloss" name =
+"noght" id = "noght" href = "#gloss_noght">noȝt</a>, bot he makes þe
+fig<i>ure</i> to betoken þat comes aft<i>ur</i> hym mor<i>e</i> þan he
+schuld &amp; he wer<i>e</i> away, as þus 1φ. her<i>e</i> þe
+fig<i>ure</i> of on<i>e</i> tokens ten, &amp; yf þe cifre wer<i>e</i>
+away<a class = "tag" name = "tag_craft1" id = "tag_craft1" href =
+"#note_craft1">1</a> &amp; no fig<i>ure</i> by-for<i>e</i> hym he schuld
+token bot on<i>e</i>, for þan he sch<i>ul</i>d stonde in þe first place.
+¶&nbsp;And þe cifre tokens nothyng hym selfe. for al þe nomb<i>ur</i> of
+þe <a class = "gloss" name = "ylke" id = "ylke" href =
+"#gloss_ylke">ylke</a> too fig<i>ure</i>s is bot ten. ¶&nbsp;Questio.
+Why says he þat a cifre makys a fig<i>ure</i> to <a class = "gloss" name
+= "signifyetyf" id = "signifyetyf" href = "#gloss_signifyetyf">signifye
+(tyf)</a> mor<i>e</i> &amp;c. ¶&nbsp;I speke for þis worde <a class =
+"terms" name = "significatyf" id = "significatyf" href =
+"#terms_significatyf">significatyf</a>, ffor sothe it may happe
+aft<i>ur</i> a cifre schuld come a-noþ<i>ur</i> cifre, as þus 2φφ. And
+ȝet þe secunde cifre shuld token neu<i>er</i> þe mor<i>e</i> <a class =
+"gloss" name = "excep" id = "excep" href = "#gloss_excep">excep</a> he
+schuld kepe þe ord<i>er</i> of þe place. and a cifre is no fig<i>ure</i>
+significatyf.</p>
+
+<div class = "verse">
+<p class = "pilcrow">
+¶ Q<i>ua</i>m p<i>re</i>cedentes plus ulti<i>m</i>a significabit /</p>
+</div>
+
+<p><span class = "sidenote">The last figure means more than all the
+others, since it is of the highest value.</span>
+Expone þis v<i>er</i>se þus. Þe last figu<i>re</i> schal token
+mor<i>e</i> þan all<i>e</i> þe oþ<i>er</i> afor<i>e</i>, <a class =
+"gloss" name = "thought" id = "thought" href =
+"#gloss_thought">thouȝt</a> þ<i>ere</i> wer<i>e</i> a hundryth thousant
+figures afor<i>e</i>, as þus, 16798. Þe last fig<i>ure</i> þat is 1.
+betokens ten thousant. And all<i>e</i> þe oþ<i>er</i> fig<i>ure</i>s
+b<i>e</i>n bot betoken<i>e</i> bot sex thousant seuyn<i>e</i>
+h<i>u</i>ndryth nynty &amp; 8. ¶&nbsp;And ten thousant is mor<i>e</i>
+þen all<i>e</i> þat nomb<i>ur</i>, <i>er</i>go þe last figu<i>re</i>
+tokens mor<i>e</i> þan all þe nomb<i>ur</i> afor<i>e</i>.</p>
+
+<p class = "headnote"><span class = "headnote">
+The Three Kinds of Numbers</span></p>
+
+<div class = "verse">
+<span class = "linenum">leaf 138 <i>a</i>.</span>
+<p class = "pilcrow plus">
+* ¶ Post p<i>re</i>dicta scias breuit<i>er</i> q<i>uod</i> tres
+num<i>er</i>or<i>um</i></p>
+<p>Distincte species sunt; nam quidam digiti sunt;</p>
+<p>Articuli quidam; quidam q<i>uoque</i> compositi sunt.</p>
+</div>
+
+<h5>¶ Capit<i>ulu</i>m 2<sup>m</sup> de t<i>ri</i>plice divisione
+nu<i>mer</i>or<i>um</i>.</h5>
+
+<p>¶ The auctor of þis tretis <a class = "terms" name = "departys" id =
+"departys" href = "#terms_departys">dep<i>ar</i>tys</a> þis worde a
+nomb<i>ur</i> into 3 p<i>ar</i>tes. Some nomb<i>ur</i> is called
+digit<i>us</i> latine, a&nbsp;<a class = "terms" name = "digit" id =
+"digit" href = "#terms_digit">digit</a> in englys.
+<span class = "sidenote">Digits.</span>
+So<i>m</i>me nomb<i>ur</i> is called articul<i>us</i> latine. An
+<a class = "terms" name = "articul" id = "articul" href =
+"#terms_article">Articul</a> in englys.
+<span class = "sidenote">Articles.</span>
+Some nomb<i>ur</i> is called a <a class = "terms" name = "composyt" id =
+"composyt" href = "#terms_componede">composyt</a> in englys.
+<span class = "sidenote">Composites.</span>
+¶&nbsp;Expone þis v<i>er</i>se. know þ<i>o</i>u aft<i>ur</i> þe forsayd
+<a class = "gloss" name = "rewles" id = "rewles" href =
+"#gloss_rewle">rewles</a> þ<i>a</i>t I sayd afore, þat þ<i>ere</i> ben
+thre <a class = "gloss" name = "spices1" id = "spices1" href =
+"#gloss_spices">spices</a> of nomb<i>ur</i>. Oon<i>e</i> is a digit,
+Anoþ<i>er</i> is an Articul, &amp; þe toþ<i>er</i> a Composyt.
+v<i>er</i>sus.</p>
+
+<p class = "headnote"><span class = "headnote">
+Digits, Articles, and Composites.</span></p>
+
+<div class = "verse">
+<p class = "pilcrow">
+¶ Sunt digiti num<i>er</i>i qui cit<i>ra</i> denariu<i>m</i>
+s<i>u</i>nt.</p>
+</div>
+
+<p><span class = "sidenote">What are digits.</span>
+¶ Her<i>e</i> he telles qwat is a digit, Expone v<i>er</i>su<i>s</i>
+sic. Nomb<i>ur</i>s digitus ben<i>e</i> all<i>e</i> nomb<i>ur</i>s þat
+ben w<i>i</i>t<i>h</i>-inne ten, as nyne, 8. 7. 6. 5. 4. 3.
+2.&nbsp;1.</p>
+
+<span class = "pagenum">6</span>
+<a name = "page6" id = "page6"> </a>
+<div class = "verse">
+<p class = "pilcrow">
+¶ Articupli decupli degito<i>rum</i>; compositi s<i>u</i>nt</p>
+<p>Illi qui constant ex articulis degitisq<i>ue</i>.</p>
+</div>
+
+<p>¶ Her<i>e</i> he telles what is a composyt and what is an<i>e</i>
+articul. Expone sic v<i>er</i>sus.
+<span class = "sidenote">What are articles.</span>
+¶&nbsp;Articulis ben<a class = "tag" name = "tag_craft2" id =
+"tag_craft2" href = "#note_craft2">2</a>
+all<i>e</i> þ<i>a</i>t may be deuidyt into nomb<i>urs</i> of ten &amp;
+nothyng<i>e</i> <a class = "gloss" name = "leue" id = "leue" href =
+"#gloss_leue">leue</a> ou<i>er</i>, as twenty, thretty, fourty,
+a&nbsp;hundryth, a&nbsp;thousand, &amp; such oþ<i>er</i>, ffor twenty
+may be dep<i>ar</i>tyt in-to 2 nomb<i>ur</i>s of ten, fforty in to
+four<i>e</i> nomb<i>ur</i>s of ten, &amp; so forth.</p>
+
+<p><span class = "linenum">leaf 138 <i>b</i>.</span>
+<span class = "sidenote">What numbers are composites.</span>
+*Compositys beɳ nomb<i>ur</i>s þat bene componyt of a digyt &amp; of an
+articull<i>e</i> as fouretene, fyftene, sextene, &amp; such oþ<i>er</i>.
+ffortene is co<i>m</i>ponyd of four<i>e</i> þat is a digit &amp; of ten
+þat is an articull<i>e</i>. ffiftene is componyd of 5 &amp; ten, &amp;
+so of all oþ<i>er</i>, what þat þai ben. Short-lych eu<i>er</i>y
+nomb<i>ur</i> þat be-gynnes w<i>i</i>t<i>h</i> a digit &amp; endyth in a
+articull<i>e</i> is a composyt, as fortene bygennyng<i>e</i> by
+four<i>e</i> þat is a digit, &amp; endes in ten.</p>
+
+<div class = "verse">
+<p class = "pilcrow">
+¶ Ergo, p<i>ro</i>posito nu<i>mer</i>o tibi scriber<i>e</i>,
+p<i>ri</i>mo</p>
+<p>Respicias quid sit nu<i>merus</i>; si digitus sit</p>
+<p>P<i>ri</i>mo scribe loco digitu<i>m</i>, si compositus sit</p>
+<p>P<i>ri</i>mo scribe loco digitu<i>m</i> post articulu<i>m</i>;
+sic.</p>
+</div>
+
+<p><span class = "sidenote">How to write a number,</span>
+¶ here he telles how þ<i>o</i>u schalt <a class = "gloss" name = "wyrch"
+id = "wyrch" href = "#gloss_worch">wyrch</a> whan þ<i>o</i>u schalt
+write a nomb<i>ur</i>. Expone v<i>er</i>su<i>m</i> sic, &amp; fac iuxta
+expon<i>ent</i>is sentencia<i>m</i>; whan þ<i>o</i>u hast a
+nomb<i>ur</i> to write, loke fyrst what man<i>er</i> nomb<i>ur</i> it ys
+þ<i>a</i>t þ<i>o</i>u schalt write, whether it be a digit or a composit
+or an Articul.
+<span class = "sidenote">if it is a digit;</span>
+¶&nbsp;If he be a digit, write a digit, as yf it be seuen, write seuen
+&amp; write þ<i>a</i>t digit in þe first place toward þe ryght side.
+<span class = "sidenote">if it is a composite.</span>
+If it be a composyt, write þe digit of þe composit in þe first place
+&amp; write þe articul of þat digit in þe secunde place next toward þe
+lyft side. As yf þ<i>o</i>u schal write sex &amp; twenty. write þe digit
+of þe nomb<i>ur</i> in þe first place þat is sex, and write þe articul
+next aft<i>ur</i> þat is twenty, as þus 26.
+<span class = "sidenote">How to read it.</span>
+But whan þ<i>o</i>u schalt <a class = "gloss" name = "sowne" id =
+"sowne" href = "#gloss_sowne">sowne</a> or speke
+<span class = "linenum">leaf 139 <i>a</i>.</span>
+*or rede an Composyt þou schalt first sowne þe articul &amp;
+aft<i>ur</i> þe digit, as þ<i>o</i>u seyst by þe comyn<i>e</i> speche,
+Sex &amp; twenty &amp; nouȝt twenty &amp; sex. v<i>er</i>sus.</p>
+
+<div class = "verse">
+<p class = "pilcrow">
+¶ Articul<i>us</i> si sit, in p<i>ri</i>mo limite cifram,</p>
+<p>Articulu<i>m vero</i> reliq<i>ui</i>s insc<i>ri</i>be
+figur<i>is</i>.</p>
+</div>
+
+<p><span class = "sidenote">How to write Articles:</span>
+¶ Here he tells how þ<i>o</i>u schal write when þe nombre þ<i>a</i>t
+þ<i>o</i>u hase to write is an Articul. Expone v<i>er</i>sus sic &amp;
+fac s<i>ecundu</i>m sentenciam. Ife þe nomb<i>ur</i> þ<i>a</i>t
+þ<i>o</i>u hast write be an Articul, write first a cifre &amp;
+aft<i>ur</i> þe cifer write an Articull<i>e</i> þus. 2φ.
+<span class = "sidenote">tens,</span>
+fforthermor<i>e</i> þ<i>o</i>u schalt vnd<i>ir</i>stonde yf þ<i>o</i>u
+haue an Articul, loke how
+<span class = "pagenum">7</span>
+<a name = "page7" id = "page7"> </a>
+mych he is, yf he be w<i>i</i>t<i>h</i>-ynne an hundryth, þ<i>o</i>u
+schalt write bot on<i>e</i> cifre, afore, as her<i>e</i> .9φ.
+<span class = "sidenote">hundreds,</span>
+If þe articull<i>e</i> be by hym-silfe &amp; be an hundrid euen<i>e</i>,
+þen schal þ<i>o</i>u write .1. &amp; 2 cifers afor<i>e</i>, þat he may
+stonde in þe thryd place, for eu<i>er</i>y fig<i>ure</i> in þe thryd
+place schal token a hundrid tymes hym selfe.
+<span class = "sidenote">thousands, &amp;c.</span>
+If þe articul be a thousant or thousandes<a class = "tag" name =
+"tag_craft3" id = "tag_craft3" href = "#note_craft3">3</a>
+and he stonde by hy<i>m</i> selfe, write afor<i>e</i> 3 cifers &amp; so
+forþ of al oþ<i>er</i>.</p>
+
+<div class = "verse">
+<p class = "pilcrow">
+¶ Quolib<i>et</i> in nu<i>mer</i>o, si par sit p<i>ri</i>ma figura,</p>
+<p>Par erit &amp; to<i>tu</i>m, quicquid sibi
+co<i>n</i>ti<i>nua</i>t<i>ur</i>;</p>
+<p>Imp<i>ar</i> si fu<i>er</i>it, totu<i>m</i> tu<i>n</i>c fiet
+<i>et</i> impar.</p>
+</div>
+
+<p><span class = "sidenote">To tell an even number</span>
+¶ Her<i>e</i> he teches a gen<i>er</i>all<i>e</i> rewle þ<i>a</i>t yf þe
+first fig<i>ure</i> in þe <a class = "gloss" name = "rewle2" id =
+"rewle2" href = "#gloss_rewle">rewle</a> of fig<i>ure</i>s token a
+nomb<i>ur</i> þat is euen<i>e</i> al þ<i>a</i>t nomb<i>ur</i> of
+fig<i>ur</i>ys in þat rewle schal be euen<i>e</i>, as her<i>e</i>
+þ<i>o</i>u may see 6. 7. 3. 5.&nbsp;4. Computa &amp; p<i>ro</i>ba.
+<span class = "sidenote">or an odd.</span>
+¶&nbsp;If þe first
+<span class = "linenum">leaf 139 <i>b</i>.</span>
+*fig<i>ur</i>e token an nomb<i>ur</i> þat is ode, all<i>e</i> þat
+nomb<i>ur</i> in þat rewle schall<i>e</i> be ode, as her<i>e</i> 5 6 7 8
+6&nbsp;7. Computa &amp; p<i>ro</i>ba. v<i>er</i>sus.</p>
+
+<div class = "verse">
+<p class = "pilcrow">
+¶ Septe<i>m</i> su<i>n</i>t partes, no<i>n</i> pl<i>u</i>res, istius
+artis;</p>
+<p class = "pilcrow">
+¶ Adder<i>e</i>, subt<i>ra</i>her<i>e</i>, duplar<i>e</i>,
+dimidiar<i>e</i>,</p>
+<p>Sextaq<i>ue</i> diuider<i>e</i>, s<i>ed</i> qui<i>n</i>ta
+m<i>u</i>ltiplicar<i>e</i>;</p>
+<p>Radice<i>m</i> ext<i>ra</i>her<i>e</i> p<i>ar</i>s septi<i>m</i>a
+dicitur esse.</p>
+</div>
+
+<p class = "headnote"><span class = "headnote">
+The Seven Rules of Arithmetic.</span></p>
+
+<p><span class = "sidenote">The seven rules.</span>
+¶ Her<i>e</i> telles þ<i>a</i>t þ<i>er</i> beɳ .7. spices or
+p<i>ar</i>tes of þis craft. The first is called addicioñ, þe secunde is
+called subtraccioñ. The thryd is called <a class = "terms" name =
+"duplacion" id = "duplacion" href = "#terms_duplacion">duplacioñ</a>.
+The 4. is called <a class = "terms" name = "dimydicion" id =
+"dimydicion" href = "#terms_dimydicion">dimydicioñ</a>. The 5. is called
+m<i>u</i>ltiplicacioñ. The 6 is called diuisioñ. The 7. is called
+extraccioñ of þe <a class = "terms" name = "rote" id = "rote" href =
+"#terms_rote">Rote</a>. What all þese spices ben<i>e</i> hit
+schall<i>e</i> be tolde singillati<i>m</i> in <a class = "gloss" name =
+"here" id = "here" href = "#gloss_here">her<i>e</i></a> <a class =
+"gloss" name = "caputule" id = "caputule" href =
+"#gloss_caputule">caputul<i>e</i></a>.</p>
+
+<div class = "verse">
+<p class = "pilcrow">
+¶ Subt<i>ra</i>his aut addis a dext<i>ri</i>s vel mediabis:</p>
+</div>
+
+<p><span class = "sidenote">Add, subtract, or halve, from right to
+left.</span>
+Thou schal be-gynne in þe ryght side of þe boke or of a tabul. loke
+wer<i>e</i> þ<i>o</i>u wul be-gynne to write latyn or englys in a boke,
+&amp; þ<i>a</i>t schall<i>e</i> be called þe lyft side of the boke, þat
+þ<i>o</i>u writest toward þ<i>a</i>t side schal be called þe ryght side
+of þe boke. V<i>er</i>sus.</p>
+
+<div class = "verse">
+<p>A leua dupla, diuide, m<i>u</i>ltiplica.</p>
+</div>
+
+<p>Here he telles þe in quych side of þe boke or of þe tabul þ<i>o</i>u
+schall<i>e</i> be-gyn<i>e</i> to wyrch duplacioñ, diuisioñ, and
+m<i>u</i>ltiplicacioñ.
+<span class = "sidenote">Multiply or divide from left to right.</span>
+Thou schal begyn<i>e</i> to <a class = "gloss" name = "worch1" id =
+"worch1" href = "#gloss_worch">worch</a> in þe lyft side of þe boke or
+of þe tabul, but yn what wyse þ<i>o</i>u schal wyrch in hym <b>dicetur
+<a class = "gloss" name = "singillatim" id = "singillatim" href =
+"#gloss_singillatim">singillatim</a> in seque<i>n</i>tib<i>us</i>
+capi<i>tulis</i> et de vtilitate cui<i>us</i>li<i>bet</i> art<i>is</i>
+&amp; sic Completur</b>
+<span class = "linenum">leaf <ins class = "correction" title = "error for ‘140a’?">140</ins>.</span>
+*<b>p<i>ro</i>hemi<i>um</i> &amp; sequit<i>ur</i> tractat<i>us</i> &amp;
+p<i>ri</i>mo de arte addic<i>ion</i>is que p<i>ri</i>ma ars est in
+ordine.</b></p>
+
+<span class = "pagenum">8</span>
+<a name = "page8" id = "page8"> </a>
+
+<p class = "headnote"><span class = "headnote">
+The Craft of Addition.</span></p>
+
+<div class = "verse">
+<p><span class = "dropcap">A</span>dder<i>e</i> si nu<i>mer</i>o
+num<i>e</i>ru<i>m</i> vis, ordine tali</p>
+<p>Incipe; scribe duas p<i>rim</i>o series nu<i>mer</i>or<i>um</i></p>
+<p>P<i>ri</i>ma<i>m</i> sub p<i>ri</i>ma recte pone<i>n</i>do
+figura<i>m</i>,</p>
+<p>Et sic de reliq<i>ui</i>s facias, si sint tibi plures.</p>
+</div>
+
+<p><span class = "sidenote">Four things must be known:</span>
+¶ Her<i>e</i> by-gynnes þe craft of Addicioñ. In þis craft þ<i>o</i>u
+most knowe foure thyng<i>es</i>. ¶&nbsp;Fyrst þ<i>ou</i> most know what
+is addicioñ. Next þ<i>o</i>u most know how mony rewles of figurys þou
+most haue. ¶&nbsp;Next þ<i>o</i>u most know how mony diue<i>r</i>s casys
+happes in þis craft of addicioñ. ¶&nbsp;And next qwat is þe
+p<i>ro</i>fet of þis craft.
+<span class = "sidenote">what it is;</span>
+¶&nbsp;As for þe first þou most know þat addicioñ is a <a class =
+"terms" name = "castyng" id = "castyng" href = "#terms_cast">castyng</a>
+to-ged<i>ur</i> of twoo nomburys in-to on<i>e</i> nombr<i>e</i>. As yf I
+aske qwat is <a class = "gloss" name = "twene" id = "twene" href =
+"#gloss_twene">twene</a> &amp; thre. Þ<i>o</i>u wyl <a class = "terms"
+name = "cast" id = "cast" href = "#terms_cast">cast</a> þese twene
+nomb<i>re</i>s to-ged<i>ur</i> &amp; say þ<i>a</i>t it is fyue.
+<span class = "sidenote">how many rows of figures;</span>
+¶ As for þe secunde þou most know þ<i>a</i>t þou schall<i>e</i> haue
+tweyne rewes of figures, on<i>e</i> vndur a-nother, as her<i>e</i>
+þ<i>o</i>u mayst&nbsp;se.
+<span class = "float">
+1234<br />
+2168.</span>
+<span class = "sidenote">how many cases;</span>
+¶ As for þe thryd þou most know þ<i>a</i>t ther<i>e</i> ben foure
+diu<i>er</i>se cases.
+<span class = "sidenote">what is its result.</span>
+As for þe forthe þ<i>o</i>u most know þ<i>a</i>t þe p<i>ro</i>fet of þis
+craft is to telle what is þe hole nomb<i>ur</i> þ<i>a</i>t comes of
+diu<i>er</i>se nomburis. Now as to þe texte of oure verse, he teches
+ther<i>e</i> how þ<i>o</i>u schal <a class = "gloss" name = "worch" id =
+"worch" href = "#gloss_worch">worch</a> in þis craft. ¶&nbsp;He says yf
+þ<i>o</i>u wilt cast on<i>e</i> nomb<i>ur</i> to anoþ<i>er</i>
+nomb<i>ur</i>, þou most by-gynne on þis wyse.
+<span class = "sidenote">How to set down the sum.</span>
+¶&nbsp;ffyrst write
+<span class = "linenum">leaf 140 <i>b</i>.</span>
+*two rewes of figuris &amp; nombris so þat þ<i>o</i>u write þe first
+figur<i>e</i> of þe hyer nomb<i>ur</i> euen<i>e</i> vnd<i>ir</i> the
+first fig<i>ure</i> of þe nether nomb<i>ur</i>,
+<span class = "float">
+123<br />
+234.</span>
+And þe secunde of þe nether nomb<i>ur</i> euen<i>e</i> vnd<i>ir</i> þe
+secunde of þe hyer, &amp; so forthe of eu<i>er</i>y fig<i>ur</i>e of
+both þe rewes as þ<i>o</i>u mayst&nbsp;se.</p>
+
+<p class = "headnote"><span class = "headnote">
+The Cases of the Craft of Addition.</span></p>
+
+<div class = "verse">
+<p class = "pilcrow">
+¶ Inde duas adde p<i>ri</i>mas hac condic<i>i</i>one:</p>
+<p>Si digitus crescat ex addic<i>i</i>one prior<i>um</i>;</p>
+<p>P<i>ri</i>mo scribe loco digitu<i>m</i>, quicu<i>n</i>q<i>ue</i> sit
+ille.</p>
+</div>
+
+<p>¶ Here he teches what þ<i>o</i>u schalt do when þ<i>o</i>u hast write
+too rewes of figuris <a class = "gloss" name = "on" id = "on" href =
+"#gloss_on">on</a> vnder an-oþ<i>er</i>, as I sayd be-for<i>e</i>.
+<span class = "sidenote">Add the first figures;</span>
+¶&nbsp;He says þ<i>o</i>u schalt take þe first fig<i>ur</i>e of þe heyer
+nomb<i>re</i> &amp; þe fyrst figur<i>e</i> of þe neþ<i>er</i> nombre,
+&amp; cast hem to-ged<i>er</i> vp-on þis condicioɳ. Thou schal loke
+qweþ<i>er</i> þe nombe<i>r</i> þat comys þ<i>ere</i>-of be a digit or
+no.
+<span class = "sidenote">rub out the top figure;</span>
+¶&nbsp;If he be a digit þ<i>o</i>u schalt do away þe first fig<i>ur</i>e
+of þe hyer nomb<i>re</i>, and write þ<i>ere</i> in his stede þat he
+stode Inne þe digit, þ<i>a</i>t comes of þe ylke 2 fig<i>ur</i>es, &amp;
+so
+<span class = "sidenote">write the result in its place.</span>
+<a class = "gloss" name = "wrich" id = "wrich" href =
+"#gloss_worch">wrich</a> forth oɳ oþ<i>er</i> figures yf þ<i>ere</i> be
+ony moo, til þ<i>o</i>u come to þe ende toward þe lyft side. And
+<a class = "gloss" name = "lede" id = "lede" href = "#gloss_lede">lede</a>
+þe nether fig<i>ure</i> stonde still eu<i>er</i>-mor<i>e</i> til
+þ<i>o</i>u haue <a class = "gloss" name = "ydo1" id = "ydo1" href =
+"#gloss_ydo">ydo</a>. ffor þ<i>ere</i>-by þ<i>o</i>u schal <a class =
+"gloss" name = "wyte" id = "wyte" href = "#gloss_wete">wyte</a>
+wheþ<i>er</i> þ<i>o</i>u hast don<i>e</i> wel or no, as I schal tell þe
+aft<i>er</i>ward in þe ende of þis Chapt<i>er</i>. ¶&nbsp;And loke
+<a class = "gloss" name = "allgate" id = "allgate" href =
+"#gloss_allgate">allgate</a>
+<span class = "linenum">leaf 141 <i>a</i>.</span>
+þat þou be-gynne to worch in þis Craft of Addi*cioɳ in þe ryȝt side,
+<span class = "pagenum">9</span>
+<a name = "page9" id = "page9"> </a>
+<span class = "sidenote">Here is an example.</span>
+here is an ensampul of þis case.
+<span class = "float">
+1234<br />
+2142.</span>
+Caste 2 to four<i>e</i> &amp; þat wel be sex, do away 4. &amp; write in
+þe same place þe fig<i>ur</i>e of sex. ¶&nbsp;And lete þe fig<i>ur</i>e
+of 2 in þe nether rewe stonde stil. When þ<i>o</i>u hast do so, cast 3
+&amp; 4 to-ged<i>ur</i> and þat wel be seuen þ<i>a</i>t is a digit. Do
+away þe 3, &amp; set þ<i>ere</i> seueɳ, and lete þe neþ<i>er</i>
+fig<i>ure</i> stonde still<i>e</i>, &amp; so <a class = "gloss" name =
+"worch_imp1" id = "worch_imp1" href = "#gloss_worch">worch</a> forth
+bakward til þ<i>o</i>u hast <a class = "gloss" name = "ydo" id = "ydo"
+href = "#gloss_ydo">ydo</a> all to-ged<i>er</i>.</p>
+
+<div class = "verse">
+<p>Et si composit<i>us</i>, in limite scribe seque<i>n</i>te</p>
+<p>Articulum, p<i>ri</i>mo digitum; q<i>uia</i> sic iubet ordo.</p>
+</div>
+
+<p>¶ Here is þe secunde case þ<i>a</i>t may happe in þis craft. And þe
+case is þis,
+<span class = "sidenote">Suppose it is a Composite, set down the digit,
+and carry the tens.</span>
+yf of þe casting of 2 nomburis to-ged<i>er</i>, as of þe fig<i>ur</i>e
+of þe hyer rewe &amp; of þe figure of þe neþ<i>er</i> rewe come a
+Composyt, how schalt þ<i>ou</i> worch. Þ<i>us</i> þ<i>o</i>u schalt
+worch. Thou shalt do away þe fig<i>ur</i>e of þe hyer nomb<i>er</i> þat
+was cast to þe figure of þe neþ<i>er</i> nomber. ¶&nbsp;And write
+þ<i>ere</i> þe digit of þe Composyt. And set þe articul of þe composit
+next aft<i>er</i> þe digit in þe same rewe, yf þ<i>ere</i> be no
+<a class = "gloss" name = "mo" id = "mo" href = "#gloss_mo">mo</a>
+fig<i>ur</i>es aft<i>er</i>. But yf þ<i>ere</i> be mo figuris
+aft<i>er</i> þat digit. And þere he schall be rekend for hym selfe. And
+when þ<i>o</i>u schalt adde þ<i>a</i>t ylke figure þ<i>a</i>t berys þe
+articull<i>e</i> ou<i>er</i> his hed to þe figur<i>e</i> vnd<i>er</i>
+hym, þ<i>o</i>u schalt cast þat articul to þe figure þ<i>a</i>t hase hym
+ou<i>er</i> his hed, &amp; þ<i>ere</i> þat Articul schal tokeɳ hym
+selfe.
+<span class = "sidenote">Here is an example.</span>
+lo an Ensampull
+<span class = "linenum">leaf 141 <i>b</i>.</span>
+*of all.
+<span class = "float">
+326<br />
+216.</span>
+Cast 6 to 6, &amp; þ<i>ere</i>-of wil arise twelue. do away þe hyer 6
+&amp; write þ<i>ere</i> 2, þ<i>a</i>t is þe digit of þis composit. And
+þe<i>n</i> write þe articull<i>e</i> þat is ten ou<i>er</i> þe figuris
+hed of twene as þ<i>us</i>.
+<span class = "float">
+&nbsp; 1<br />
+322<br />
+216.</span>
+Now cast þe articull<i>e</i> þ<i>a</i>t standus vpon þe fig<i>ur</i>is
+of twene hed to þe same fig<i>ur</i>e, &amp; reken þat articul bot for
+on<i>e</i>, and þan þ<i>ere</i> wil arise thre. Þan cast þat thre to þe
+neþ<i>er</i> figure, þat is on<i>e</i>, &amp; þat wul be four<i>e</i>.
+do away þe fig<i>ur</i>e of 3, and write þ<i>ere</i> a fig<i>ur</i>e of
+foure. and lete þe neþ<i>er</i> fig<i>ur</i>e stonde stil, &amp; þan
+worch forth. vn<i>de</i> <i>ver</i>sus.</p>
+
+
+<div class = "verse">
+<p class = "pilcrow">
+¶ Articulus si sit, in p<i>ri</i>mo limite cifram,</p>
+<p class = "pilcrow">
+¶ Articulu<i>m</i> v<i>er</i>o reliquis inscribe figuris,</p>
+<p>Vel p<i>er</i> se scribas si nulla figura sequat<i>ur</i>.</p>
+</div>
+
+<p>¶ Her<i>e</i> he puttes þe thryde case of þe craft of Addicioɳ. &amp;
+þe case is þis.
+<span class = "sidenote">Suppose it is an Article, set down a cipher and
+carry the tens.</span>
+yf of Addiciouɳ of 2 figuris a-ryse an Articull<i>e</i>, how schal
+þ<i>o</i>u do. thou most do away þe <a class = "gloss" name = "heer" id
+= "heer" href = "#gloss_heer">heer</a> fig<i>ur</i>e þ<i>a</i>t was
+addid to þe neþ<i>er</i>, &amp; write þ<i>ere</i> a cifre, and sett þe
+<a class = "terms" name = "articuls" id = "articuls" href =
+"#terms_article">articuls</a> on þe figuris hede, yf þ<i>a</i>t
+þ<i>ere</i> come ony aft<i>er</i>. And wyrch þan as I haue tolde þe in
+þe secunde case. An ensampull.
+<span class = "float">
+25.<br />
+15</span>
+Cast 5 to 5, þat wylle be ten. now do away þe hyer 5, &amp; write
+þ<i>ere</i> a cifer. And sette ten vpon þe figuris hed of 2. And reken
+it but for on þus. lo
+<span class = "sidenote">Here is an example.</span>
+<span class = "pagenum">10</span>
+<a name = "page10" id = "page10"> </a>
+an Ensampull<i>e</i>
+<span class = "float box">
+1 &nbsp;<br />
+2φ<br />
+15</span>
+And
+<span class = "linenum">leaf 142 <i>a</i>.</span>
+*þan worch forth. But yf þ<i>ere</i> come no figure aft<i>er</i> þe
+cifre, write þe articul next hym in þe same rewe as here
+<span class = "float box">
+5<br />
+5</span>
+cast 5 to 5, and it wel be ten. do away 5. þat is þe hier 5. and write
+þ<i>ere</i> a cifre, &amp; write aft<i>er</i> hym þe articul as þus
+<span class = "float box">
+1φ<br />
+&nbsp; 5</span>
+And þan þ<i>o</i>u hast done.</p>
+
+<div class = "verse">
+<p class = "pilcrow">
+¶ Si tibi cifra sup<i>er</i>ueniens occurrerit, illa<i>m</i></p>
+<p>Dele sup<i>er</i>posita<i>m</i>; fac illic scribe figura<i>m</i>,</p>
+<p>Postea procedas reliquas addendo figuras.</p>
+</div>
+
+<p><span class = "sidenote">What to do when you have a cipher in the top
+row.</span>
+¶&nbsp;Her<i>e</i> he putt<i>es</i> þe fourt case, &amp; it is þis, þat
+yf þ<i>ere</i> come a cifer in þe hier rewe, how þ<i>o</i>u schal do.
+þus þ<i>o</i>u schalt do. do away þe cifer, &amp; sett þ<i>ere</i> þe
+digit þ<i>a</i>t comes of þe addiciou<i>n</i> as þus
+<span class = "float">
+1φφ84.<br />
+17743</span>
+<span class = "sidenote">An example of all the difficulties.</span>
+In þis ensampul ben all<i>e</i> þe four<i>e</i> cases. Cast 3 to foure,
+þ<i>a</i>t wol be seueɳ. do away 4. &amp; write þ<i>ere</i> seueɳ; þan
+cast 4 to þe figur<i>e</i> of 8. þ<i>a</i>t wel be 12. do away 8, &amp;
+sett þ<i>ere</i> 2. þat is a digit, and sette þe articul of þe composit,
+þat is ten, vpon þe cifers hed, &amp; reken it for hym selfe þat is on.
+þan cast on<i>e</i> to a cifer, &amp; hit wull<i>e</i> be but on, for
+noȝt &amp; on makes but on<i>e</i>. þan cast 7. þ<i>a</i>t stondes
+vnd<i>er</i> þat on to hym, &amp; þat wel be 8. do away þe cifer &amp;
+þat 1. &amp; sette þ<i>ere</i> 8. þan go forthermor<i>e</i>. cast þe
+oþ<i>er</i> 7 to þe cifer þ<i>a</i>t stondes ou<i>er</i> hy<i>m</i>.
+þ<i>a</i>t wul be bot seuen, for þe cifer betokens noȝt. do away þe
+cifer &amp; sette þ<i>ere</i> seueɳ,
+<span class = "linenum">leaf 142 <i>b</i>.</span>
+*&amp; þen go forþ<i>er</i>mor<i>e</i> &amp; cast 1 to 1, &amp; þat wel
+be 2. do away þe hier 1, &amp; sette þ<i>ere</i> 2. þan hast þ<i>o</i>u
+do. And yf þ<i>o</i>u haue wel ydo þis nomber þat is sett
+her<i>e</i>-aft<i>er</i> wel be þe nomber þat schall<i>e</i> aryse of
+all<i>e</i> þe addicioɳ as her<i>e</i> 27827. ¶&nbsp;Sequi<i>tu</i>r
+alia sp<i>eci</i>es.</p>
+
+<p class = "headnote"><span class = "headnote">
+The Craft of Subtraction.</span></p>
+
+<div class = "verse">
+<p>A nu<i>mer</i>o num<i>er</i>u<i>m</i> si sit tibi demer<i>e</i>
+cura</p>
+<p>Scribe figurar<i>um</i> series, vt in addicione.</p>
+</div>
+
+<p><span class = "sidenote">Four things to know about
+subtraction:</span>
+¶&nbsp;This is þe Chapt<i>er</i> of subtraccioɳ, in the quych þou most
+know foure nessessary thyng<i>es</i>. the first what is subtraccioɳ. þe
+secunde is how mony nombers þou most haue to subt<i>ra</i>ccioɳ, the
+thryd is how mony maners of cases þ<i>ere</i> may happe in þis craft of
+subtraccioɳ. The fourte is qwat is þe p<i>ro</i>fet of þis craft.
+¶&nbsp;As for
+<span class = "sidenote">the first;</span>
+þe first, þ<i>o</i>u most know þ<i>a</i>t subtraccioɳ is drawyng<i>e</i>
+of on<i>e</i> nowmb<i>er</i> oute of anoþ<i>er</i> nomber.
+<span class = "sidenote">the second;</span>
+As for þe secunde, þou most knowe þ<i>a</i>t þou most haue two rewes of
+figuris on<i>e</i> vnd<i>er</i> anoþ<i>er</i>, as þ<i>o</i>u <a class =
+"gloss" name = "addyst" id = "addyst" href = "#gloss_addyst">addyst</a>
+in addicioɳ.
+<span class = "sidenote">the third;</span>
+As for þe thryd, þ<i>o</i>u moyst know þ<i>a</i>t four<i>e</i>
+man<i>er</i> of diu<i>er</i>se casis mai happe in þis craft.
+<span class = "sidenote">the fourth.</span>
+¶&nbsp;As for þe fourt, þou most know þ<i>a</i>t þe p<i>ro</i>fet of þis
+craft is whenne þ<i>o</i>u hasse taken þe lasse nomber out of þe
+mor<i>e</i> to telle what þ<i>ere</i> <a class = "gloss" name = "leues1"
+id = "leues1" href = "#gloss_leue">leues</a> ou<i>er</i>
+<span class = "pagenum">11</span>
+<a name = "page11" id = "page11"> </a>
+þ<i>a</i>t. &amp; þ<i>o</i>u most be-gynne to wyrch in þ<i>is</i> craft
+in þe ryght side of þe boke, as þ<i>o</i>u diddyst in addicioɳ.
+V<i>er</i>sus.</p>
+
+<div class = "verse">
+<p class = "pilcrow">
+¶ Maiori nu<i>mer</i>o num<i>er</i>u<i>m</i> suppone minorem,</p>
+<p class = "pilcrow">
+¶ Siue pari nu<i>mer</i>o supponat<i>ur</i> num<i>er</i>us par.</p>
+</div>
+
+<p><span class = "linenum">leaf 143 <i>a</i>.</span>
+* ¶ Her<i>e</i> he telles þat
+<span class = "sidenote">Put the greater number above the less.</span>
+þe hier nomber most be mor<i>e</i> þen þe neþ<i>er</i>, or els eueɳ as
+mych. but he may not be lasse. And þe case is þis, þou schalt drawe þe
+neþ<i>er</i> nomber out of þe hyer, &amp; þou mayst not do þ<i>a</i>t yf
+þe hier nomber wer<i>e</i> lasse þan þat. ffor þ<i>o</i>u mayst not draw
+sex out of 2. But þ<i>o</i>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<i>de</i>
+v<i>er</i>sus.</p>
+
+<p class = "headnote"><span class = "headnote">
+The Cases of the Craft of Subtraction.</span></p>
+
+<div class = "verse">
+<p class = "pilcrow">
+¶ Postea si possis a prima subt<i>ra</i>he p<i>ri</i>ma<i>m</i></p>
+<p>Scribens quod remanet.</p>
+</div>
+
+<p><span class = "sidenote">The first case of subtraction.</span>
+Her<i>e</i> is þe first case put of subtraccioɳ, &amp; he says þou
+schalt begynne in þe ryght side, &amp; draw þe first fig<i>ure</i> of þe
+neþ<i>er</i> rewe out of þe first fig<i>ure</i> of þe hier rewe. qwether
+þe hier fig<i>ur</i>e be mor<i>e</i> þen þe neþ<i>er</i>, or eueɳ as
+mych. And þat is notified in þe vers when he says “Si possis.” Whan
+þ<i>o</i>u has þus ydo, do away þe hiest fig<i>ur</i>e &amp; sett
+þ<i>ere</i> þat <a class = "gloss" name = "leues" id = "leues" href =
+"#gloss_leue">leues</a> of þe subtraccioɳ,
+<span class = "sidenote">Here is an example.</span>
+lo an Ensampull<i>e</i>
+<span class = "float box">
+234<br />
+122</span>
+draw 2 out of 4. þan leues 2. do away 4 &amp; write þ<i>ere</i> 2, &amp;
+latte þe neþ<i>er</i> figur<i>e</i> sto<i>n</i>de stille, &amp; so go
+<a class = "gloss" name = "forby" id = "forby" href =
+"#gloss_forby">for-by</a> oþ<i>er</i> figuris till þ<i>o</i>u come to þe
+ende, þan hast þ<i>o</i>u&nbsp;do.</p>
+
+<div class = "verse">
+<p class = "halfline">
+¶ Cifram si nil remanebit.</p>
+</div>
+
+<p><span class = "sidenote">Put a cipher if nothing remains.</span>
+¶&nbsp;Her<i>e</i> he putt<i>es</i> þe secunde case, &amp; hit is þis.
+yf it happe þ<i>a</i>t qwen þ<i>o</i>u hast draw on neþ<i>er</i>
+fig<i>ure</i> out of a hier, &amp; þ<i>er</i>e leue noȝt aft<i>er</i> þe
+subt<i>ra</i>ccioɳ, þus
+<span class = "linenum">leaf 143 <i>b</i>.</span>
+*þou schalt do. þ<i>o</i>u schall<i>e</i> do away þe hier fig<i>ur</i>e
+&amp; write þ<i>ere</i> a cifer, as
+<span class = "sidenote">Here is an example.</span>
+lo an Ensampull
+<span class = "float box">
+24<br />
+24</span>
+Take four<i>e</i> out of four<i>e</i> þan <a class = "gloss" name =
+"leus" id = "leus" href = "#gloss_leue">leus</a> noȝt.
+þ<i>er</i>efor<i>e</i> do away þe hier 4 &amp; set þ<i>ere</i> a cifer,
+þan take 2 out of 2, þan leues noȝt. do away þe hier 2, &amp; set
+þ<i>ere</i> a cifer, and so worch whar<i>e</i> so eu<i>er</i> þis
+happe.</p>
+
+<div class = "verse">
+<p>Sed si no<i>n</i> possis a p<i>ri</i>ma dem<i>er</i>e
+p<i>ri</i>ma<i>m</i></p>
+<p>P<i>re</i>cedens vnu<i>m</i> de limite deme seque<i>n</i>te,</p>
+<p>Quod demptu<i>m</i> p<i>ro</i> denario reputabis ab illo</p>
+<p>Subt<i>ra</i>he to<i>ta</i>lem num<i>er</i>u<i>m</i> qu<i>em</i>
+p<i>ro</i>posuisti</p>
+<p>Quo facto sc<i>ri</i>be super quicquid remaneb<i>i</i>t.</p>
+</div>
+
+<p><span class = "sidenote">Suppose you cannot take the lower figure
+from the top one, borrow ten;</span>
+Her<i>e</i> he puttes þe thryd case, þe quych is þis. yf it happe þat þe
+neþ<i>er</i> fig<i>ur</i>e be mor<i>e</i> þen þe hier fig<i>ur</i>e þat
+he schall<i>e</i> be draw out of. how schall<i>e</i> þou do. þus
+þ<i>o</i>u schall<i>e</i> do. þou schall<i>e</i> <a class = "gloss" name
+= "borro" id = "borro" href = "#gloss_borro">borro</a> .1. oute of þe
+next fig<i>ur</i>e þat comes aft<i>er</i> in þe same rewe, for þis case
+may neu<i>er</i> happ but yf þ<i>ere</i> come figures aft<i>er</i>. þan
+þ<i>o</i>u schalt sett
+<span class = "pagenum">12</span>
+<a name = "page12" id = "page12"> </a>
+þat on ou<i>er</i> þe hier figur<i>es</i> hed, of the quych þou woldist
+y-draw oute þe neyþ<i>er</i> fig<i>ur</i>e yf þ<i>o</i>u haddyst
+<a class = "gloss" name = "ymyght" id = "ymyght" href =
+"#gloss_ymyght">y-myȝt</a>. Whane þou hase þus ydo þou schall<i>e</i>
+rekene þ<i>a</i>t .1. for ten.
+<span class = "sidenote">take the lower number from ten;</span>
+¶. And out of þat ten þ<i>o</i>u schal draw þe neyþermost fig<i>ur</i>e,
+And all<i>e</i> þ<i>a</i>t leues þou schall<i>e</i>
+<span class = "sidenote">add the answer to the top number.</span>
+adde to þe figur<i>e</i> on whos hed þat .1. stode. And þen þ<i>o</i>u
+schall<i>e</i> do away all<i>e</i> þat, &amp; sett þ<i>ere</i>
+all<i>e</i> that arisys of the addicioɳ of þe ylke 2 fig<i>ur</i>is. And
+yf yt
+<span class = "linenum">leaf 144 <i>a</i>.</span>
+*happe þat þe fig<i>ur</i>e of þe quych þ<i>o</i>u schalt borro on be
+hym self but 1. If þ<i>o</i>u schalt þat on<i>e</i> &amp; sett it vppoɳ
+þe oþ<i>er</i> figur<i>is</i> hed, and sett in þ<i>a</i>t 1. place a
+cifer, yf þ<i>ere</i> come mony figur<i>es</i> aft<i>er</i>.
+<span class = "sidenote">Example.</span>
+lo an Ensampul.
+<span class = "float box">
+2122<br />
+1134</span>
+take 4 out of 2. it wyl not be, þerfor<i>e</i> borro on<i>e</i> of þe
+next figur<i>e</i>, þ<i>a</i>t is 2. and sett þat ou<i>er</i> þe hed of
+þe fyrst 2. &amp; rekene it for ten. and þere þe secunde stondes write
+1. for þ<i>o</i>u tokest on out of hy<i>m</i>. þan take þe neþ<i>er</i>
+fig<i>ur</i>e, þat is 4, out of ten. And þen leues 6. cast to 6 þe
+fig<i>ur</i>e of þat 2 þat stode vnd<i>er</i> þe hedde of 1. þat was
+<a class = "gloss" name = "borwed" id = "borwed" href =
+"#gloss_borro">borwed</a> &amp; rekened for ten, and þat wylle be 8. do
+away þ<i>a</i>t 6 &amp; þat 2, &amp; sette þ<i>ere</i> 8, &amp; lette þe
+neþ<i>er</i> fig<i>ur</i>e stonde stille. Whanne þ<i>o</i>u hast do þus,
+go to þe next fig<i>ur</i>e þ<i>a</i>t is now bot 1. but first yt was 2,
+&amp; þ<i>ere</i>-of was <a class = "gloss" name = "borred" id =
+"borred" href = "#gloss_borro">borred</a>&nbsp;1.
+<span class = "sidenote">How to ‘Pay back’ the borrowed ten.</span>
+þan take out of þ<i>a</i>t þe fig<i>ur</i>e vnd<i>er</i> hym, þ<i>a</i>t
+is 3. hit wel not be. þer-for<i>e</i> <a class = "gloss" name = "borowe"
+id = "borowe" href = "#gloss_borro">borowe</a> of the next
+fig<i>ur</i>e, þe quych is bot 1. Also take &amp; sett hym ou<i>er</i>
+þe hede of þe fig<i>ure</i> þat þou woldest haue y-draw oute of þe
+nether figure, þe quych was 3. &amp; þou myȝt not, &amp; rekene
+þ<i>a</i>t borwed 1 for ten &amp; sett in þe same place, of þe quych
+place þ<i>o</i>u tokest hy<i>m</i> of, a&nbsp;cifer, for he was bot 1.
+Whanne þ<i>o</i>u hast þ<i>us</i> ydo, take out of þat 1. þ<i>a</i>t is
+rekent for ten, þe neþ<i>er</i> figure of 3. And þ<i>ere</i>
+leues&nbsp;7.
+<span class = "linenum">leaf 144 <i>b</i>.</span>
+*cast þe ylke 7 to þe fig<i>ur</i>e þat had þe ylke ten vpon his hed, þe
+quych fig<i>ur</i>e was&nbsp;1, &amp; þat wol be&nbsp;8. þan do away
+þ<i>a</i>t 1 and þ<i>a</i>t&nbsp;7, &amp; write þ<i>ere</i> 8. &amp; þan
+wyrch forth in oþ<i>er</i> figuris til þ<i>o</i>u come to þe ende, &amp;
+þan þ<i>o</i>u hast þe do. V<i>er</i>sus.</p>
+
+<div class = "verse">
+<p class = "pilcrow">
+¶ Facque nonenarios de cifris, cu<i>m</i> remeabis</p>
+<p class = "pilcrow">
+¶ Occ<i>ur</i>rant si forte cifre; dum demps<i>er</i>is vnum</p>
+<p class = "pilcrow">
+¶ Postea p<i>ro</i>cedas reliquas deme<i>n</i>do figuras.</p>
+</div>
+
+<p><span class = "sidenote">A&nbsp;very hard case is put.</span>
+¶ Her<i>e</i> he putt<i>es</i> þe fourte case, þe quych is þis, yf it
+happe þat þe neþ<i>er</i> fig<i>ur</i>e, þe quych þ<i>o</i>u schalt draw
+out of þe hier fig<i>ur</i>e be mor<i>e</i> pan þe hier figur
+ou<i>er</i> hym, &amp; þe next fig<i>ur</i>e of two or of thre or of
+foure, or how mony þ<i>ere</i> be by cifers, how wold þ<i>o</i>u do.
+Þ<i>o</i>u <a class = "gloss" name = "wost" id = "wost" href =
+"#gloss_wete">wost</a> wel þ<i>o</i>u most nede borow, &amp; þ<i>o</i>u
+mayst not borow of þe cifers, for þai haue noȝt þat þai may <a class =
+"gloss" name = "lene" id = "lene" href = "#gloss_lene">lene</a> or
+spar<i>e</i>. Ergo<a class = "tag" name = "tag_craft4" id = "tag_craft4"
+href = "#note_craft4">4</a>
+how
+<span class = "pagenum">13</span>
+<a name = "page13" id = "page13"> </a>
+woldest þ<i>o</i>u do. Certayɳ þus most þ<i>o</i>u do, þ<i>o</i>u most
+borow on of þe next figure significatyf in þat rewe, for þis case may
+not happe, but yf þ<i>ere</i> come figures significatyf aft<i>er</i> the
+cifers. Whan þ<i>o</i>u hast borowede þ<i>a</i>t 1 of the next figure
+significatyf, sett þ<i>a</i>t on ou<i>er</i> þe hede of þ<i>a</i>t
+fig<i>ur</i>e of þe quych þ<i>o</i>u wold haue draw þe neþ<i>er</i>
+figure out yf þ<i>o</i>u hadest myȝt, &amp; reken it for ten as
+þo<i>u</i> diddest i<i>n</i> þe oþ<i>er</i> case <a class = "gloss" name
+= "hereafore" id = "hereafore" href =
+"#gloss_hereafore">her<i>e</i>-a-for<i>e</i></a>. Whaɳ þ<i>o</i>u hast
+þus y-do loke how mony cifers þ<i>ere</i> wer<i>e</i> bye-twene þat
+figur<i>e</i> significatyf, &amp; þe fig<i>ur</i>e of þe quych
+þ<i>o</i>u woldest haue y-draw the
+<span class = "linenum">leaf 145 <i>a</i>.</span>
+*neþ<i>er</i> figure, and of eu<i>er</i>y of þe ylke cifers make a
+figur<i>e</i> of 9.
+<span class = "sidenote">Here is an example.</span>
+lo an Ensampull<i>e</i> after.
+<span class = "float box">
+40002<br />
+10004</span>
+Take 4 out of 2. it wel not be. borow 1 out of be next figure
+significatyf, þe quych is 4, &amp; þen leues 3. do away þ<i>a</i>t
+figur<i>e</i> of 4 &amp; write þ<i>ere</i> 3. &amp; sett þ<i>a</i>t 1
+vppon þe fig<i>ur</i>e of 2 hede, &amp; þan take 4 out of ten, &amp; þan
+þere leues 6. Cast 6 to the fig<i>ur</i>e of 2, þ<i>a</i>t wol be 8. do
+away þat 6 &amp; write þ<i>er</i>e 8. Whan þ<i>o</i>u hast þus y-do make
+of eu<i>er</i>y 0 betweyn 3 &amp; 8 a figure of 9, &amp; þan worch forth
+in goddes name.
+<span class = "sidenote">Sic.</span>
+<span class = "float box">
+39998<br />
+10004</span>
+&amp; yf þ<i>o</i>u hast wel y-do þ<i>o</i>u<a class = "tag" name =
+"tag_craft5" id = "tag_craft5" href = "#note_craft5">5</a>
+schalt haue þis nomb<i>er</i></p>
+
+
+<p class = "headnote"><span class = "headnote">
+How to prove the Subtraction.</span></p>
+
+<div class = "verse">
+<p class = "pilcrow">
+¶ Si subt<i>ra</i>cc<i>i</i>o sit b<i>e</i>n<i>e</i> facta
+p<i>ro</i>bar<i>e</i> valebis</p>
+<p>Quas s<i>u</i>btraxisti p<i>ri</i>mas addendo figuras.</p>
+</div>
+
+<p><span class = "sidenote">How to prove a subtraction sum.</span>
+¶&nbsp;Her<i>e</i> he teches þe Craft how þ<i>o</i>u schalt know, whan
+þ<i>o</i>u hast subt<i>ra</i>yd, wheþ<i>er</i> þou hast wel ydo or no.
+And þe Craft is þis, ryght as þ<i>o</i>u <a class = "gloss" name =
+"subtrayd" id = "subtrayd" href = "#gloss_subtrahe">subtrayd</a> þe
+neþ<i>er</i> figures fro þe hier figures, ryȝt so adde þe same
+neþ<i>er</i> figures to þe hier figures. And yf þ<i>o</i>u haue well
+<a class = "gloss" name = "ywroth" id = "ywroth" href =
+"#gloss_worch">y-wroth</a> a-for<i>e</i> þou schalt haue þe hier nombre
+þe same þ<i>o</i>u haddest <a class = "gloss" name = "or" id = "or" href
+= "#gloss_or">or</a> þou be-gan to worch. as for þis I bade þou schulde
+kepe þe neþ<i>er</i> figures stylle.
+<span class = "sidenote">Here is an example.</span>
+lo an
+<span class = "linenum">leaf 145 <i>b</i>.</span>
+*Ensampull<i>e</i> of all<i>e</i> þe 4 cases toged<i>re</i>. worche
+well<i>e</i> þis case
+<span class = "float box">
+40003468<br />
+20004664</span>
+And yf þou worch well<i>e</i> whan þou hast all<i>e</i> subtrayd þe
+þ<i>a</i>t hier nombr<i>e</i> her<i>e</i>, þis schall<i>e</i> be þe
+nombre here foloyng whan þ<i>o</i>u hast subtrayd.
+<span class = "float box">
+39998804<br />
+20004664</span>
+<span class = "sidenote">Our author makes a slip here (3
+for&nbsp;1).</span>
+And þou schalt know þ<i>us</i>. adde þe neþ<i>er</i> rowe of þe same
+nombre to þe hier rewe as þus, cast 4 to 4. þat wol be 8. do away þe 4
+&amp; write þ<i>ere</i> 8. by þe first case of addicioɳ. þan cast 6 to 0
+þat wol be 6. do away þe 0, &amp; write þere 6. þan cast 6 to 8,
+þ<i>a</i>t wel be 14. do away 8 &amp; write þ<i>ere</i> a fig<i>ur</i>e
+of 4, þat is þe digit, and write a fig<i>ur</i>e of 1. þ<i>a</i>t schall
+be-token ten. þ<i>a</i>t is þe articul vpon þe hed of 8 next
+aft<i>er</i>, þan reken þ<i>a</i>t 1. for 1. &amp; cast it to 8. þat
+schal be 9. cast to þat 9 þe neþ<i>er</i> fig<i>ur</i>e vnd<i>er</i> þat
+þe quych is 4, &amp; þat schall<i>e</i> be 13. do away þat 9 &amp; sett
+þ<i>er</i>e 3, &amp; sett a figure of 1. þ<i>a</i>t schall be 10 vpon þe
+next figur<i>is</i> hede þe
+<span class = "pagenum">14</span>
+<a name = "page14" id = "page14"> </a>
+quych is 9. by þe secu<i>n</i>de case þ<i>a</i>t þ<i>o</i>u hadest in
+addicioɳ. þan cast 1 to 9. &amp; þat wol be 10. do away þe 9. &amp; þat
+1. And write þ<i>ere</i> a cifer. and write þe articull<i>e</i> þat is
+1. betokenyng<i>e</i> 10. vpon þe hede of þe next figur<i>e</i> toward
+þe lyft side, þe quych
+<span class = "linenum">leaf 146 <i>a</i>.</span>
+*is 9, &amp; so do forth tyl þ<i>o</i>u come to þe last&nbsp;9.
+<span class = "sidenote">He works his proof through,</span>
+take þe figur<i>e</i> of þat&nbsp;1. þe quych þ<i>o</i>u schalt fynde
+ou<i>er</i> þe hed of 9. &amp; sett it ou<i>er</i> þe next figures hede
+þat schal be&nbsp;3. ¶&nbsp;Also do away þe 9. &amp; set þ<i>ere</i> a
+cifer, &amp; þen cast þat 1 þat stondes vpon þe hede of 3 to þe same 3,
+&amp; þ<i>a</i>t schall<i>e</i> make&nbsp;4, þen caste to þe ylke 4 the
+figur<i>e</i> in þe neyþ<i>er</i> rewe, þe quych is&nbsp;2, and þat
+schall<i>e</i> be 6.
+<span class = "sidenote">and brings out a result.</span>
+<span class = "float box">
+60003468<br />
+20004664</span>
+And þen schal þ<i>o</i>u haue an Ensampull<i>e</i> aȝeyɳ, loke &amp; se,
+&amp; but þ<i>o</i>u haue þis same þ<i>o</i>u hase <a class = "gloss"
+name = "mysewroght" id = "mysewroght" href =
+"#gloss_mysewroght">myse-wroȝt</a>.</p>
+
+
+<p class = "headnote"><span class = "headnote">
+The Craft of Duplation.</span></p>
+
+<h5>Sequit<i>ur</i> de duplac<i>i</i>one</h5>
+
+<div class = "verse">
+<p><span class = "dropcap">S</span>i vis duplar<i>e</i>
+num<i>er</i>u<i>m</i>, sic i<i>n</i>cipe p<i>rim</i>o</p>
+<p>Scribe fig<i>ur</i>ar<i>um</i> serie<i>m</i>
+q<i>ua</i>mcu<i>n</i>q<i>ue</i> vel<i>is</i> tu.</p>
+</div>
+
+<p><span class = "sidenote">Four things must be known in
+Duplation.</span>
+¶&nbsp;This is the Chaptur<i>e</i> of <a class = "terms" name =
+"duplacion2" id = "duplacion2" href = "#terms_duplacion">duplacioɳ</a>,
+in þe quych craft þ<i>o</i>u most haue &amp; know 4 thing<i>es</i>.
+¶&nbsp;Þe first þ<i>a</i>t þ<i>o</i>u most know is what is duplacioɳ. þe
+secu<i>n</i>de is how mony rewes of fig<i>ur</i>es þ<i>o</i>u most haue
+to þis craft. ¶&nbsp;þe thryde is how many cases may<a class = "tag"
+name = "tag_craft6" id = "tag_craft6" href = "#note_craft6">6</a>
+happe in þis craft. ¶&nbsp;þe fourte is what is þe p<i>ro</i>fet of þe
+craft.
+<span class = "sidenote">Here they are.</span>
+¶&nbsp;As for þe first. duplacioɳ is a doublyng<i>e</i> of a nombre.
+¶&nbsp;As for þe secu<i>n</i>de þ<i>o</i>u most
+<span class = "linenum">leaf 146 <i>b</i>.</span>
+*haue on nombre or on rewe of figures, the quych called nu<i>merus</i>
+dupland<i>us</i>. As for þe thrid þ<i>o</i>u most know þat 3
+diu<i>er</i>se cases may hap in þis craft. As for þe fourte. qwat is þe
+p<i>ro</i>fet of þis craft, &amp; þ<i>a</i>t is to know what <a class =
+"gloss" name = "arisyght" id = "arisyght" href =
+"#gloss_arisyght">a-risyȝt</a> of a nombre I-doublyde.
+<span class = "sidenote">Mind where you begin.</span>
+¶&nbsp;fforþ<i>er</i>-mor<i>e</i>, þ<i>o</i>u most know &amp; take gode
+hede in quych side þ<i>o</i>u schall<i>e</i> be-gyn in þis craft, or
+ellis þ<i>o</i>u mayst <a class = "gloss" name = "spyl" id = "spyl" href
+= "#gloss_spyl">spyl</a> all<i>e</i> þ<i>i</i> lab<i>er</i> þ<i>er</i>e
+aboute. c<i>er</i>teyn þ<i>o</i>u schalt begyɳ in the lyft side in þis
+Craft. thenke wel ou<i>er</i> þis verse. ¶&nbsp;<a class = "tag" name =
+"tag_craft7" id = "tag_craft7" href = "#note_craft7">7</a>A leua dupla,
+diuide, m<i>u</i>ltiplica.<a class = "tag" href =
+"#note_craft7">7</a></p>
+
+<p>The <a class = "gloss" name = "sentens" id = "sentens" href =
+"#gloss_sentens">sentens</a> of þes verses afor<i>e</i>, as þ<i>o</i>u
+may see if þ<i>o</i>u take hede.
+<span class = "sidenote">Remember your rules.</span>
+As þe text of þis verse, þat is to say, ¶&nbsp;Si vis duplare. þis is þe
+sentence. ¶&nbsp;If þ<i>o</i>u <a class = "gloss" name = "wel" id =
+"wel" href = "#gloss_wel">wel</a> double a nombre þus þ<i>o</i>u most
+be-gynɳ. Write a rewe of figures of what nomb<i>re</i> þou welt.
+v<i>er</i>sus.</p>
+
+<div class = "verse">
+<p>Postea p<i>ro</i>cedas p<i>ri</i>ma<i>m</i> duplando
+figura<i>m</i></p>
+<p>Inde q<i>uo</i>d excrescit scribas vbi iusserit ordo</p>
+<p>Iuxta p<i>re</i>cepta tibi que dant<i>ur</i> in addic<i>i</i>one.</p>
+</div>
+
+<p><span class = "sidenote">How to work a sum.</span>
+¶ Her<i>e</i> he telles how þ<i>o</i>u schalt worch in þis Craft. he
+says, fyrst, whan þ<i>o</i>u hast writen þe nombre þ<i>o</i>u schalt
+be-gyn at þe first
+<span class = "pagenum">15</span>
+<a name = "page15" id = "page15"> </a>
+figur<i>e</i> in the lyft side, &amp; doubull<i>e</i> þat fig<i>ur</i>e,
+&amp; þe nombre þat comes þ<i>ere</i>-of þ<i>o</i>u schalt write as
+þ<i>o</i>u diddyst in addicioɳ, as ¶&nbsp;I schal telle þe in þe case.
+v<i>er</i>sus.</p>
+
+<p class = "headnote"><span class = "headnote">
+The Cases of the Craft of Duplation.</span></p>
+
+<span class = "linenum">leaf 147 <i>a</i>.</span>
+<div class = "verse">
+<p class = "pilcrow plus">* ¶ Nam si sit digitus in primo limite
+scribas.</p>
+</div>
+
+<p><span class = "sidenote">If the answer is a digit,</span>
+¶ Her<i>e</i> is þe first case of þis craft, þe quych is þis. yf of
+duplacioɳ of a figur<i>e</i> arise a digit. what schal þ<i>o</i>u do.
+þus þ<i>o</i>u schal do.
+<span class = "sidenote">write it in the place of the top figure.</span>
+do away þe fig<i>ur</i>e þat was doublede, &amp; sett þ<i>ere</i> þe
+diget þat comes of þe duplacioɳ, as þus. 23. double 2, &amp; þ<i>a</i>t
+wel be 4. do away þe figur<i>e</i> of 2 &amp; sett þ<i>ere</i> a
+figur<i>e</i> of 4, &amp; so <a class = "gloss" name = "worch_imp" id =
+"worch_imp" href = "#gloss_worch">worch</a> forth till<i>e</i>
+þ<i>o</i>u come to þe ende. v<i>er</i>sus.</p>
+
+<div class = "verse">
+<p class = "pilcrow">
+¶ Articul<i>us</i> si sit, in p<i>ri</i>mo limite cifram,</p>
+<p class = "pilcrow">
+¶ Articulu<i>m</i> v<i>er</i>o reliquis inscribe figuris;</p>
+<p class = "pilcrow">
+¶ Vel p<i>er</i> se scribas, si nulla figura sequat<i>ur</i>.</p>
+</div>
+
+<p><span class = "sidenote">If it is an article,</span>
+¶ Here is þe secunde case, þe quych is þis yf þ<i>ere</i> come an
+articull<i>e</i> of þe duplacioɳ of a fig<i>ur</i>e þ<i>o</i>u schalt do
+ryȝt as þ<i>o</i>u diddyst in addicioɳ, þat is to <a class = "gloss"
+name = "wete" id = "wete" href = "#gloss_wete">wete</a> þat þ<i>o</i>u
+schalt do away þe figur<i>e</i> þat is doublet &amp;
+<span class = "sidenote">put a cipher in the place, and ‘carry’ the
+tens.</span>
+sett þ<i>ere</i> a cifer, &amp; write þe articull<i>e</i> ou<i>er</i> þe
+next figur<i>is</i> hede, yf þ<i>ere</i> be any aft<i>er</i>-warde
+toward þe lyft side as þus. 25. begyn at the lyft side, and
+doubull<i>e</i> 2. þat wel be 4. do away þat 2 &amp; sett þere 4. þan
+doubul 5. þat wel be 10. do away 5, &amp; sett þ<i>ere</i> a 0, &amp;
+sett 1 vpon þe next figur<i>is</i> hede þe quych is 4. &amp; þen draw
+downe 1 to 4 &amp; þat woll<i>e</i> be 5, &amp; þen do away þ<i>a</i>t 4
+&amp; þat 1, &amp; sett þ<i>ere</i> 5. for þat 1 schal be rekened in þe
+drawyng<i>e</i> toged<i>re</i> for 1. wen
+<span class = "linenum">leaf 147 <i>b</i>.</span>
+*þou hast ydon þou schalt haue þis nomb<i>r</i>e 50.
+<span class = "sidenote">If there is no figure to ‘carry’ them to, write
+them down.</span>
+yf þ<i>ere</i> come no figur<i>e</i> aft<i>er</i> þe fig<i>ur</i>e
+þ<i>a</i>t is addit, of þe quych addicioɳ comes an articull<i>e</i>,
+þ<i>o</i>u schalt do away þe figur<i>e</i> þ<i>a</i>t is dowblet &amp;
+sett þ<i>ere</i> a 0. &amp; 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<i>e</i> 5 &amp; set þ<i>ere</i> a cifer, &amp; sett þe articul
+next aft<i>er</i> in þe same rewe toward þe lyft side, &amp; þou schalt
+haue þis nombre 1023. þen go forth &amp; double þe oþ<i>er</i> nombers
+þe quych is <a class = "gloss" name = "lyght" id = "lyght" href =
+"#gloss_lyght">lyȝt</a> <a class = "gloss" name = "ynowght" id =
+"ynowght" href = "#gloss_ynowght">y-nowȝt</a> to do. v<i>er</i>sus.</p>
+
+<div class = "verse">
+<p class = "pilcrow">
+¶ Compositus si sit, in limite sc<i>ri</i>be seq<i>uen</i>te</p>
+<p>Articulu<i>m</i>, p<i>ri</i>mo digitu<i>m</i>; q<i>uia</i> sic iubet
+ordo:</p>
+<p>Et sic de reliq<i>ui</i>s facie<i>n</i>s, si sint tibi plures.</p>
+</div>
+
+<p><span class = "sidenote">If it is a Composite,</span>
+¶ Her<i>e</i> he putt<i>es</i> þe Thryd case, þe quych is þis, yf of
+duplacioɳ of a fig<i>ur</i>e come a Composit. þ<i>o</i>u schalt do away
+þe fig<i>u</i>re þ<i>a</i>t is doublet &amp; set þ<i>ere</i> a digit of
+þe Composit,
+<span class = "sidenote">write down the digit, and ‘carry’ the
+tens.</span>
+&amp; sett þe articull<i>e</i> ou<i>er</i> þe next figures hede, &amp;
+aft<i>er</i> draw hym downe w<i>i</i>t<i>h</i> þe figur<i>e</i>
+ou<i>er</i> whos hede he stondes, &amp; make þ<i>ere</i>-of an nombre as
+þ<i>o</i>u hast done
+<span class = "pagenum">16</span>
+<a name = "page16" id = "page16"> </a>
+afore, &amp; yf þ<i>ere</i> come no fig<i>ur</i>e aft<i>er</i> þat digit
+þat þ<i>o</i>u hast <a class = "gloss" name = "ywrite" id = "ywrite"
+href = "#gloss_write">y-write</a>, þa<i>n</i> set þe articull<i>e</i>
+next aft<i>er</i> hym in þe same rewe as þus, 67: double 6 þat wel be
+12, do away 6 &amp; write þ<i>ere</i> þe digit
+<span class = "linenum">leaf 148 <i>a</i>.</span>
+*of 12, þe quych is&nbsp;2,
+<span class = "sidenote">Here is an example.</span>
+and set þe articull<i>e</i> next aft<i>er</i> toward þe lyft side in þe
+same rewe, for þ<i>ere</i> comes no figur<i>e</i> aft<i>er</i>. þan
+dowble þat oþ<i>er</i> figur<i>e</i>, þe quych is&nbsp;7, þat wel be 14.
+the quych is a Composit. þen do away 7 þat þ<i>o</i>u doublet &amp; sett
+þe þe diget of hy<i>m</i>, the quych is 4, sett þe articull<i>e</i>
+ou<i>er</i> þe next figur<i>es</i> hed, þe quych is 2, &amp; þen draw to
+hym þat on, &amp; make on nombre þe quych schall<i>e</i> be 3. And þen
+yf þ<i>o</i>u haue wel y-do þ<i>o</i>u schall<i>e</i> haue þis nombre of
+þe duplacioɳ, 134. v<i>er</i>sus.</p>
+
+<div class = "verse">
+<p class = "pilcrow">
+¶ Si super ext<i>re</i>ma<i>m</i> nota sit monade<i>m</i> dat
+eid<i>em</i></p>
+<p>Quod t<i>ibi</i> <i>con</i>tingat si p<i>ri</i>mo dimidiabis.</p>
+</div>
+
+<p><span class = "sidenote">How to double the mark for one-half.</span>
+¶&nbsp;Her<i>e</i> he says, yf ou<i>er</i> þe fyrst fig<i>ur</i>e in þe
+ryȝt side be such a merke as is her<i>e</i> made, <a class = "terms"
+name = "sup_w" id = "sup_w" href = "#terms_sup_w"><sup>w</sup></a>,
+þ<i>o</i>u schall<i>e</i> fyrst doubull<i>e</i> þe figur<i>e</i>, the
+quych stondes vnd<i>er</i> þ<i>a</i>t merke, &amp; þen þou schalt doubul
+þat merke þe quych stond<i>es</i> for <a class = "gloss" name =
+"haluendel" id = "haluendel" href = "#gloss_haluendel">haluendel</a> on.
+for too <a class = "gloss" name = "haluedels" id = "haluedels" href =
+"#gloss_haluendel">haluedels</a> makes on, &amp; so þ<i>a</i>t wol be
+on. cast þ<i>a</i>t on to þat duplacioɳ of þe figur<i>e</i> ou<i>er</i>
+whos hed stode þat merke, &amp; write it in þe same place þ<i>ere</i>
+þat þe figur<i>e</i> þe quych was doublet stode, as þus 23<sup>w</sup>.
+double 3, þat wol be 6; doubul þat halue on, &amp; þat wol be on. cast
+on to 6, þ<i>a</i>t wel be 7. do away 6 &amp; þat 1, &amp; sett
+þ<i>ere</i> 7. þan hase þou do. as for þat figur<i>e</i>, þan go
+<span class = "linenum">leaf 148 <i>b</i>.</span>
+*to þe oþ<i>er</i> fig<i>ure</i> &amp; worch forth.
+<span class = "sidenote">This can only stand over the first
+figure.</span>
+&amp; þ<i>o</i>u schall neu<i>er</i> haue such a merk but ou<i>er</i> þe
+hed of þe furst figure in þe ryght side. And ȝet it schal not happe but
+yf it were y-halued a-for<i>e</i>, þus þ<i>o</i>u schalt
+vnd<i>er</i>stonde þe verse. ¶&nbsp;Si sup<i>er</i>
+ext<i>re</i>ma<i>m</i> &amp;c. Et nota, talis fig<i>ur</i>a <sup>w</sup>
+significans medietate<i>m</i>, unitat<i>is</i> veniat, <i>i.e.</i>
+contingat u<i>e</i>l fiat sup<i>er</i> ext<i>re</i>ma<i>m</i>,
+<i>i.e.</i> sup<i>er</i> p<i>ri</i>ma<i>m</i> figura<i>m</i> in
+ext<i>re</i>mo sic v<i>er</i>sus dextram ars dat: <i>i.e.</i> reddit
+monade<i>m</i>. <i>i.e.</i> vnitate<i>m</i> eide<i>m</i>. <i>i.e.</i>
+eidem note &amp; declina<i>tur</i> hec monos, d<i>i</i>s, di, dem,
+&amp;c. ¶&nbsp;Quod <i>er</i>g<i>o</i> to<i>tum</i> ho<i>c</i> dabis
+monade<i>m</i> note <i>con</i>ting<i>et</i>. <i>i.e.</i> eveniet tibi si
+dimidiasti, <i>i.e.</i> accipisti u<i>e</i>l subtulisti medietatem
+alicuius unius, in cuius principio sint figura nu<i>mer</i>u<i>m</i>
+denotans i<i>m</i>pare<i>m</i> p<i>rim</i>o <i>i.e.</i> principiis.</p>
+
+<p class = "headnote"><span class = "headnote">
+The Craft of Mediation.</span></p>
+
+<h5>¶ Sequit<i>ur</i> de mediacione.</h5>
+
+<div class = "verse">
+<p><span class = "dropcap">I</span>ncipe sic, si vis alique<i>m</i>
+nu<i>me</i>ru<i>m</i> mediar<i>e</i>:</p>
+<p>Sc<i>ri</i>be figurar<i>um</i> seriem sola<i>m</i>, velut
+an<i>te</i>.</p>
+</div>
+
+<p><span class = "sidenote">The four things to be known in
+mediation:</span>
+¶&nbsp;In þis Chapter is <a class = "gloss" name = "taght" id = "taght"
+href = "#gloss_taght">taȝt</a> þe Craft of <a class = "terms" name =
+"mediacioun" id = "mediacioun" href =
+"#terms_mediacioun">mediaciouɳ</a>, in þe quych craft þ<i>o</i>u most
+know 4 thynges. ffurst what is mediacioɳ. the secunde how mony rewes of
+figur<i>es</i> þ<i>o</i>u most haue in þe wyrchyng<i>e</i> of þis craft.
+þe thryde how mony diu<i>er</i>se cases may happ in þis craft.<a class =
+"tag" name = "tag_craft8" id = "tag_craft8" href = "#note_craft8">8</a>
+<span class = "sidenote">the first</span>
+¶&nbsp;As for þe furst, þ<i>o</i>u schalt vndurstonde þat mediacioɳ is a
+<span class = "pagenum">17</span>
+<a name = "page17" id = "page17"> </a>
+takyng out of halfe a nomber out of a <a class = "gloss" name = "holle"
+id = "holle" href = "#gloss_hole">holle</a> nomber,
+<span class = "linenum">leaf 149 <i>a</i>.</span>
+*as yf þ<i>o</i>u
+<span class = "sidenote">the second;</span>
+wolde take 3 out of 6. ¶&nbsp;As for þe secunde, þ<i>o</i>u schalt know
+þ<i>a</i>t þ<i>o</i>u most haue on<i>e</i> rewe of figures, &amp; no
+moo, as þ<i>o</i>u <a class = "gloss" name = "hayst" id = "hayst" href =
+"#gloss_hayst">hayst</a> in þe
+<span class = "sidenote">the third;</span>
+craft of duplacioɳ. ¶&nbsp;As for the thryd, þou most vnd<i>er</i>stonde
+þat
+<span class = "sidenote">the fourth.</span>
+5 cases may happe in þis craft. ¶&nbsp;As for þe fourte, þou
+schall<i>e</i> know þat the p<i>ro</i>fet of þis craft is when
+þ<i>o</i>u hast take away þe haluendel of a nomb<i>re</i> to telle qwat
+þer<i>e</i> schall<i>e</i> leue. ¶&nbsp;Incipe sic, &amp;c. The sentence
+of þis verse is þis. yf þ<i>o</i>u wold <a class = "terms" name =
+"medye" id = "medye" href = "#terms_medye">medye</a>, þat is to say,
+take halfe out of þe holle, or halfe out of halfe, þou most begynne
+þ<i>us</i>.
+<span class = "sidenote">Begin thus.</span>
+Write on<i>e</i> rewe of figur<i>es</i> of what nombre þou wolte, as
+þ<i>o</i>u dyddyst be-for<i>e</i> in þe Craft of duplacioɳ.
+v<i>er</i>sus.</p>
+
+<div class = "verse">
+<p class = "pilcrow">
+¶ Postea p<i>ro</i>cedas medians, si p<i>ri</i>ma figura</p>
+<p>Si par aut i<i>m</i>par videas.</p>
+</div>
+
+<p>¶ Her<i>e</i> he says, when þ<i>o</i>u hast write a rewe of figures,
+þ<i>o</i>u schalt
+<span class = "sidenote">See if the number is even or odd.</span>
+take hede wheþ<i>er</i> þe first figur<i>e</i> be eueɳ or odde in
+nombre, &amp; vnd<i>er</i>stonde þ<i>a</i>t he spekes of þe first figure
+in þe ryȝt side. And i<i>n</i> the ryght side þ<i>o</i>u schall<i>e</i>
+begynne in þis Craft.</p>
+
+<div class = "verse">
+<p class = "halfline">
+¶ Quia si fu<i>er</i>it par,</p>
+<p>Dimidiab<i>is</i> eam, scribe<i>n</i>s quicq<i>ui</i>d remanebit:</p>
+</div>
+
+<p><span class = "sidenote">If it is even, halve it, and write the
+answer in its place.</span>
+¶&nbsp;Her<i>e</i> is the first case of þis craft, þe quych is þis, yf
+þe first figur<i>e</i> be euen. þou schal take away fro þe figur<i>e</i>
+euen halfe, &amp; do away þat fig<i>ur</i>e and set þ<i>ere</i> þat
+leues ou<i>er</i>, as þus, 4. take
+<span class = "linenum">leaf 149 <i>b</i>.</span>
+*halfe out of 4, &amp; þan þ<i>ere</i> leues 2. do away 4 &amp; sett
+þ<i>ere</i> 2. þis is lyght y-nowȝt. v<i>er</i>sus.</p>
+
+<p class = "headnote"><span class = "headnote">
+The Mediation of an Odd Number.</span></p>
+
+<div class = "verse">
+<p class = "pilcrow">
+¶ Impar si fu<i>er</i>it vnu<i>m</i> demas mediar<i>e</i></p>
+<p>Quod no<i>n</i> p<i>re</i>sumas, s<i>ed</i> quod sup<i>er</i>est
+mediabis</p>
+<p>Inde sup<i>er</i> tractu<i>m</i> fac demptu<i>m</i> quod no<i>ta</i>t
+vnu<i>m</i>.</p>
+</div>
+
+<p><span class = "sidenote">If it is odd, halve the even number less
+than it.</span>
+Her<i>e</i> is þe secunde case of þis craft, the quych is þis. yf þe
+first figur<i>e</i> betoken<i>e</i> a nombre þat is odde, the quych odde
+schal not be <a class = "terms" name = "mediete" id = "mediete" href =
+"#terms_medye">mediete</a>, þen þ<i>o</i>u schalt medye þat nombre þat
+leues, when the odde of þe same nomb<i>re</i> is take away, &amp; write
+þat þ<i>a</i>t leues as þ<i>o</i>u diddest in þe first case of þis
+craft. Whaɳ þ<i>o</i>u <a class = "gloss" name = "hayst2" id = "hayst2"
+href = "#gloss_hayst">hayst</a> write þat. for þ<i>a</i>t þat leues,
+<span class = "sidenote">Then write the sign for one-half over
+it.</span>
+write such a merke as is her<i>e</i> <sup>w</sup> vpon his hede, þe
+quych merke schal betokeɳ halfe of þe odde þat was take away.
+<span class = "sidenote">Here is an example.</span>
+lo an Ensampull. 245. the first figur<i>e</i> her<i>e</i> is
+betokenyng<i>e</i> odde nombre, þe quych is 5, for 5 is odde;
+þ<i>er</i>e-for<i>e</i> do away þat þ<i>a</i>t is odde, þe quych is 1.
+þen leues 4. þen medye 4 &amp; þen leues 2. do away 4. &amp; sette
+þ<i>ere</i> 2, &amp; make such a merke <sup>w</sup> upon his hede, þat
+is to say ou<i>er</i> his hede of 2 as þus. 242.<sup>w</sup> And þen
+worch forth in þe oþ<i>er</i> figures tyll þ<i>o</i>u come to þe ende.
+by þe furst case as þ<i>o</i>u schalt
+<span class = "pagenum">18</span>
+<a name = "page18" id = "page18"> </a>
+vnd<i>er</i>stonde þat
+<span class = "sidenote">Put the mark only over the first figure.</span>
+þ<i>o</i>u schalt
+<span class = "linenum">leaf 150 <i>a</i>.</span>
+*neu<i>er</i> make such a merk but ou<i>er</i> þe first fig<i>ur</i>e
+hed in þe riȝt side. Wheþ<i>er</i> þe other fig<i>ur</i>es þat comyɳ
+aft<i>er</i> hym be eueɳ or odde. v<i>er</i>sus.</p>
+
+<p class = "headnote"><span class = "headnote">
+The Cases of the Craft of Mediation.</span></p>
+
+<div class = "verse">
+<p class = "pilcrow">
+¶ Si monos, dele; sit t<i>ibi</i> cifra post no<i>ta</i> supra.</p>
+</div>
+
+<p><span class = "sidenote">If the first figure is one put a
+cipher.</span>
+¶&nbsp;Here is þe thryde case, þe quych yf the first figur<i>e</i> be a
+figur<i>e</i> of 1. þ<i>o</i>u schalt do away þat 1 &amp; set
+þ<i>ere</i> a cifer, &amp; a merke ou<i>er</i> þe cifer as þus, 241. do
+away 1, &amp; sett þ<i>ere</i> a cifer w<i>i</i>t<i>h</i> a merke
+ou<i>er</i> his hede, &amp; þen hast þ<i>o</i>u ydo for þat 0. as þus
+0<sup>w</sup> þen worch forth in þe oþer fig<i>ur</i>ys till þ<i>o</i>u
+come to þe ende, for it is lyght as dyche water. vn<i>de</i>
+v<i>er</i>sus.</p>
+
+<div class = "verse">
+<p class = "pilcrow">
+¶ Postea p<i>ro</i>cedas hac condic<i>i</i>one secu<i>n</i>da:</p>
+<p>Imp<i>ar</i> si fu<i>er</i>it hinc vnu<i>m</i> deme
+p<i>ri</i>ori,</p>
+<p>Inscribens quinque, nam denos significabit</p>
+<p>Monos p<i>re</i>d<i>ict</i>am.</p>
+</div>
+
+<p><span class = "sidenote">What to do if any other figure is
+odd.</span>
+¶&nbsp;Her<i>e</i> he putt<i>es</i> þe fourte case, þe quych is þis. yf
+it happeɳ the secunde figur<i>e</i> betoken odde nombre, þou schal do
+away on of þat odde nombre, þe quych is significatiue by þ<i>a</i>t
+figure 1. þe quych 1 schall be rekende for 10. Whan þ<i>o</i>u hast take
+away þ<i>a</i>t 1 out of þe nombre þ<i>a</i>t is signifiede by þat
+figur<i>e</i>, þ<i>o</i>u schalt medie þ<i>a</i>t þat leues ou<i>er</i>,
+&amp; do away þat figur<i>e</i> þat is medied, &amp; sette in his
+<a class = "gloss" name = "styde" id = "styde" href =
+"#gloss_styde">styde</a> halfe of þ<i>a</i>t nombre.
+<span class = "sidenote">Write a figure of five over the next lower
+number’s head.</span>
+¶&nbsp;Whan þ<i>o</i>u hase so done, þ<i>o</i>u schalt write
+<span class = "linenum">leaf 150 <i>b</i>.</span>
+*a figure of 5 ou<i>er</i> þe next figur<i>es</i> hede by-for<i>e</i>
+toward þe ryȝt side, for þat 1, þe quych made odd nombr<i>e</i>, schall
+stonde for ten, &amp; 5 is halfe of 10; so þ<i>o</i>u most write 5 for
+his haluendell<i>e</i>.
+<span class = "sidenote">Example.</span>
+lo an Ensampull<i>e</i>, 4678. begyɳ in þe ryȝt side as þ<i>o</i>u most
+nedes. medie 8. þen þ<i>o</i>u schalt leue 4. do away þat 8 &amp; sette
+þ<i>ere</i> 4. þen out of 7. take away 1. þe quych makes odde, &amp;
+sett 5. vpon þe next figur<i>es</i> hede afor<i>e</i> toward þe ryȝt
+side, þe quych is now 4. but afor<i>e</i> it was 8. for þat 1 schal be
+rekenet for 10, of þe quych 10, 5 is halfe, as þou knowest wel. Whan
+þ<i>o</i>u hast þus ydo, medye þ<i>a</i>t þe quych leues aft<i>er</i> þe
+takying<i>e</i> away of þat þat is odde, þe quych leuyng<i>e</i>
+schall<i>e</i> be 3;
+<span class = "float">
+&nbsp; &nbsp; &nbsp; <span class = "smaller">5</span><br />
+4634.</span>
+do away 6 &amp; sette þ<i>er</i>e 3, &amp; þou schalt haue such a nombre
+aft<i>er</i> go forth to þe next fig<i>ur</i>e, &amp; medy þat, &amp;
+worch forth, for it is lyȝt <a class = "gloss" name = "ynovght" id =
+"ynovght" href = "#gloss_ynowght">ynovȝt</a> to þe <a class = "gloss"
+name = "certayn" id = "certayn" href =
+"#gloss_certayn">c<i>er</i>tayɳ</a>.</p>
+
+<div class = "verse">
+<p class = "halfline">
+¶ Si v<i>er</i>o s<i>e</i>c<i>un</i>da dat vnu<i>m</i>.</p>
+<p>Illa deleta, sc<i>ri</i>bat<i>ur</i> cifra; p<i>ri</i>ori</p>
+<p class = "pilcrow">
+¶ Tradendo quinque pro denario mediato;</p>
+<p>Nec cifra sc<i>ri</i>batur, nisi dei<i>n</i>de fig<i>ur</i>a
+seq<i>u</i>at<i>ur</i>:</p>
+<p>Postea p<i>ro</i>cedas reliq<i>ua</i>s mediando figuras</p>
+<p>Vt sup<i>ra</i> docui, si sint tibi mille figure.</p>
+</div>
+
+<p><span class = "pagenum">19</span>
+<a name = "page19" id = "page19"> </a>
+¶ Her<i>e</i> he putt<i>es</i> þe 5 case, þe quych is
+<span class = "linenum">leaf 151 <i>a</i>.</span>
+*þis:
+<span class = "sidenote">If the second figure is one, put a cipher, and
+write five over the next figure.</span>
+yf þe secunde figur<i>e</i> be of 1, as þis is here 12, þou schalt do
+away þat 1 &amp; sett þ<i>ere</i> a cifer. &amp; sett 5 ou<i>er</i> þe
+next fig<i>ur</i>e hede afor<i>e</i> toward þe riȝt side, as þou diddyst
+afor<i>e</i>; &amp; þat 5 schal be <a class = "gloss" name = "haldel" id
+= "haldel" href = "#gloss_haluendel">haldel</a> of þat 1, þe quych 1 is
+rekent for 10. lo an Ensampull<i>e</i>, 214. medye 4. þ<i>a</i>t
+schall<i>e</i> be 2. do away 4 &amp; sett þ<i>ere</i> 2. þe<i>n</i> go
+forth to þe next figur<i>e</i>. þe quych is bot 1. do away þat 1. &amp;
+sett þ<i>ere</i> a cifer. &amp; set 5 vpon þe figur<i>es</i> hed
+afor<i>e</i>, þe quych is nowe 2, &amp; þen þou schalt haue þis
+no<i>m</i>b<i>re</i>
+<span class = "float">
+&nbsp; &nbsp; <span class = "smaller">5</span><br />
+202,</span>
+þen worch forth to þe <a class = "gloss" name = "nex" id = "nex" href =
+"#gloss_nex">nex</a> fig<i>ur</i>e. And also it is no <a class = "gloss"
+name = "maystery" id = "maystery" href =
+"#gloss_maystery">mayst<i>er</i>y</a> yf þ<i>ere</i> come no
+figur<i>e</i> after þat on is medyet, þ<i>o</i>u schalt write no 0. ne
+nowȝt ellis, but set 5 ou<i>er</i> þe next fig<i>ur</i>e afor<i>e</i>
+toward þe ryȝt, as þus 14.
+<span class = "sidenote">How to halve fourteen.</span>
+medie 4 then leues 2, do away 4 &amp; sett þ<i>ere</i> 2. þen medie 1.
+þe q<i>ui</i>ch is rekende for ten, þe halue<i>n</i>del þ<i>ere</i>-of
+wel be 5. sett þ<i>a</i>t 5 vpon þe hede of þ<i>a</i>t figur<i>e</i>, þe
+quych is now 2,
+<span class = "float">
+&nbsp;<span class = "smaller">5</span><br />
+2,</span>
+&amp; do away þ<i>a</i>t 1, &amp; þou schalt haue þis nombre yf
+þ<i>o</i>u worch wel, vn<i>de</i> v<i>er</i>sus.</p>
+
+<p class = "headnote"><span class = "headnote">
+How to prove the Mediation.</span></p>
+
+<div class = "verse">
+<p class = "pilcrow">
+¶ Si mediacio sit b<i>e</i>n<i>e</i> f<i>ac</i>ta p<i>ro</i>bar<i>e</i>
+valeb<i>is</i></p>
+<p class = "pilcrow">
+¶ Duplando num<i>er</i>u<i>m</i> que<i>m</i> p<i>ri</i>mo
+di<i>m</i>ediasti</p>
+</div>
+
+<p><span class = "sidenote">How to prove your mediation.</span>
+¶ Her<i>e</i> he telles þe how þou schalt know wheþ<i>er</i> þou hase
+wel ydo or no. doubul
+<span class = "linenum">leaf 151 <i>b</i>.</span>
+*þe nombre þe quych þ<i>o</i>u hase mediet, and yf þ<i>o</i>u haue wel
+y-medyt after þe dupleacioɳ, þou schalt haue þe same nombre þat
+þ<i>o</i>u haddyst in þe tabull<i>e</i> or þ<i>o</i>u began to medye, as
+þus.
+<span class = "sidenote">First example.</span>
+¶&nbsp;The furst ensampull<i>e</i> was þis. 4. þe quych <a class =
+"terms" name = "imediet" id = "imediet" href =
+"#terms_imediet">I-mediet</a> was <a class = "gloss" name = "laft" id =
+"laft" href = "#gloss_leue">laft</a>&nbsp;2, þe whych 2 was write in þe
+place þ<i>a</i>t 4 was write afor<i>e</i>. Now doubull<i>e</i> þat 2,
+&amp; þ<i>o</i>u schal haue 4, as þ<i>o</i>u hadyst afor<i>e</i>.
+<span class = "sidenote">The second.</span>
+þe secunde Ensampull<i>e</i> was þis, 245. When þ<i>o</i>u haddyst
+mediet all<i>e</i> þis nomb<i>re</i>, yf þou haue wel ydo þou schalt
+haue of þ<i>a</i>t mediacioɳ þis nombre, 122<sup>w</sup>. Now
+doubull<i>e</i> þis nombre, &amp; begyn in þe lyft side; doubull<i>e</i>
+1, þat schal be 2. do away þat 1 &amp; sett þ<i>ere</i> 2. þen
+doubull<i>e</i> þ<i>a</i>t oþ<i>er</i> 2 &amp; sett þ<i>ere</i> 4, þen
+doubull<i>e</i> þat oþ<i>er</i> 2, &amp; þat wel be 4. þe<i>n</i> doubul
+þat merke þat stondes for halue on. &amp; þat schall<i>e</i> be 1. Cast
+þat on to 4, &amp; it schall<i>e</i> be 5. do away þat 2 &amp; þat
+merke, &amp; sette þ<i>ere</i> 5, &amp; þen þ<i>o</i>u schal haue þis
+nombre 245. &amp; þis wos þe same nombur þ<i>a</i>t þ<i>o</i>u haddyst
+or þ<i>o</i>u began to medye, as þ<i>o</i>u mayst se yf þou take hede.
+<span class = "sidenote">The third example.</span>
+The nombre þe quych þou haddist for an Ensampul in þe 3 case of
+mediacioɳ to be mediet was þis 241. whan þ<i>o</i>u haddist medied
+all<i>e</i> þis nombur truly
+<span class = "linenum">leaf 152 <i>a</i>.</span>
+*by eu<i>er</i>y figur<i>e</i>, þou schall haue be þ<i>a</i>t mediacioɳ
+þis nombur 120<sup>w</sup>. Now dowbul þis nomb<i>ur</i>, &amp; begyn in
+þe lyft side, as I tolde þe in þe Craft of duplacioɳ. þus
+doubull<i>e</i> þe fig<i>ur</i>e of 1, þat wel be 2. do
+<span class = "pagenum">20</span>
+<a name = "page20" id = "page20"> </a>
+away þat 1 &amp; sett þ<i>ere</i> 2, þen doubul þe next figur<i>e</i>
+afore, the quych is 2, &amp; þat wel be 4; do away 2 &amp; set
+þ<i>ere</i> 4. þen doubul þe cifer, &amp; þat wel be noȝt, for a 0 is
+noȝt. And twyes noȝt is but noȝt. þ<i>ere</i>for<i>e</i> doubul the
+merke aboue þe cifers hede, þe quych betokenes þe halue<i>n</i>del of 1,
+&amp; þat schal be 1. do away þe cifer &amp; þe merke, &amp; sett
+þ<i>ere</i> 1, &amp; þen þ<i>o</i>u schalt haue þis nombur 241. And þis
+same nombur þ<i>o</i>u haddyst afore or þ<i>o</i>u began to medy,
+<a class = "gloss" name = "and_yf" id = "and_yf" href = "#gloss_and">&amp;
+yf</a> þ<i>o</i>u take gode hede.
+<span class = "sidenote">The fourth example.</span>
+¶&nbsp;The next ensampul þat had in þe 4 case of mediacioɳ was þis 4678.
+Whan þ<i>o</i>u hast truly <a class = "terms" name = "ymedit" id =
+"ymedit" href = "#terms_medye">ymedit</a> all<i>e</i> þis nombur fro þe
+begynnyng<i>e</i> to þe endyng<i>e</i>, þ<i>o</i>u schalt haue of þe
+mediacioɳ þis nombur
+<span class = "float">
+&nbsp; &nbsp; &nbsp;<span class = "smaller">5</span><br />
+2334.</span>
+Now doubul this nombur &amp; begyn in þe lyft side, &amp;
+doubull<i>e</i> 2 þat schal be 4. do away 2 and sette þere 4; þen
+doubul<i>e</i> 3, þ<i>a</i>t wol be 6; do away 3 &amp; sett þ<i>ere</i>
+6, þen doubul þat oþ<i>er</i> 3, &amp; þat wel be 6; do away 3 &amp; set
+þ<i>ere</i>
+<span class = "linenum">leaf 152 <i>b</i>.</span>
+*6, þen doubul þe 4, þat welle be 8; þen doubul 5. þe quych stondes
+ou<i>er</i> þe hed of 4, &amp; þat wol be 10; cast 10 to 8, &amp;
+þ<i>a</i>t schal be 18; do away 4 &amp; þat 5, &amp; sett þ<i>ere</i> 8,
+&amp; sett that 1, þe quych is an articul of þe Composit þe quych is 18,
+ou<i>er</i> þe next figur<i>es</i> hed toward þe lyft side, þe quych is
+6. drav þ<i>a</i>t 1 to 6, þe quych 1 in þe dravyng schal be rekente bot
+for 1, &amp; þ<i>a</i>t 1 &amp; þ<i>a</i>t 6 togedur wel be 7. do away
+þat 6 &amp; þat 1. the quych stondes ou<i>er</i> his hede, &amp; sett
+ther 7, &amp; þen þou schalt haue þis nombur 4678. And þis same nombur
+þ<i>o</i>u hadyst or þ<i>o</i>u began to medye, as þ<i>o</i>u mayst see
+in þe secunde Ensampul þat þou had in þe 4 case of mediacioɳ, þat was
+þis:
+<span class = "sidenote">The fifth example.</span>
+when þ<i>o</i>u had mediet truly all<i>e</i> the nombur,
+a&nbsp;p<i>ri</i>ncipio usque ad fine<i>m</i>. þ<i>o</i>u schalt haue of
+þat mediacioɳ þis nombur
+<span class = "float">
+&nbsp; &nbsp; <span class = "smaller">5</span><br />
+102.</span>
+Now doubul 1. þat wel be 2. do away 1 &amp; sett þ<i>ere</i> 2. þen
+doubul 0. þ<i>a</i>t will be noȝt. þ<i>ere</i>for<i>e</i> take þe 5, þe
+quych stondes ou<i>er</i> þe next figur<i>es</i> hed, &amp; doubul it,
+&amp; þat wol be 10. do away þe 0 þat stondes betwene þe two
+fig<i>u</i>r<i>i</i>s, &amp; sette þ<i>ere</i> in his stid 1, for
+þ<i>a</i>t 1 now schal stonde in þe secunde place, wher<i>e</i> he schal
+betoken 10; þen doubul 2, þat wol be 4. do away 2 &amp; sett þere 4.
+&amp;
+<span class = "linenum">leaf 153 <i>a</i>.</span>
+*þou schal haue <a class = "gloss" name = "thus" id = "thus" href =
+"#gloss_thus">þus</a> nombur 214. þis is þe same nu<i>m</i>bur þat
+þ<i>o</i>u hadyst or þ<i>o</i>u began to medye, as þ<i>o</i>u may see.
+And so do eu<i>er</i> mor<i>e</i>, yf þ<i>o</i>u wil knowe wheþ<i>er</i>
+þou hase wel ymedyt or no. ¶.&nbsp;doubull<i>e</i> þe nu<i>m</i>bur þat
+comes aft<i>er</i> þe mediaciouɳ, &amp; þ<i>o</i>u schal haue þe same
+nombur þ<i>a</i>t þ<i>o</i>u hadyst or þ<i>o</i>u began to medye, yf
+þ<i>o</i>u haue welle ydo. or els doute þe noȝt, but yf þ<i>o</i>u haue
+þe same, þ<i>o</i>u hase faylide in þ<i>i</i> Craft.</p>
+
+
+<p class = "headnote"><span class = "headnote">
+The Craft of Multiplication.</span></p>
+
+<h5>Sequitur de multiplicatione.</h5>
+
+<span class = "pagenum">21</span>
+<a name = "page21" id = "page21"> </a>
+
+<p class = "headnote"><span class = "headnote">
+To write down a Multiplication Sum.</span></p>
+
+<div class = "verse">
+<p><span class = "dropcap">S</span>&nbsp;i tu p<i>er</i>
+num<i>er</i>u<i>m</i> num<i>er</i>u<i>m</i> vis
+m<i>u</i>ltiplicar<i>e</i></p>
+<p>Scribe duas q<i>ua</i>scu<i>nque</i> velis series
+nu<i>me</i>ror<i>um</i></p>
+<p>Ordo s<i>er</i>vet<i>ur</i> vt vltima m<i>u</i>ltiplicandi</p>
+<p>Ponat<i>ur</i> sup<i>er</i> ant<i>er</i>iorem
+multiplicant<i>is</i></p>
+<p>A leua reliq<i>u</i>e sint scripte m<i>u</i>ltiplicantes.</p>
+</div>
+
+<p><span class = "sidenote">Four things to be known of
+Multiplication:</span>
+¶&nbsp;Her<i>e</i> be-gynnes þe Chapt<i>r</i>e of m<i>u</i>ltiplicatioɳ,
+in þe quych þou most know 4 thynges. ¶&nbsp;Ffirst, qwat is
+m<i>u</i>ltiplicacioɳ. The secunde, how mony cases may hap in
+multiplicacioɳ. The thryde, how mony rewes of figur<i>es</i> þ<i>ere</i>
+most be. ¶&nbsp;The 4. what is þe p<i>ro</i>fet of þis craft.
+<span class = "sidenote">the first:</span>
+¶&nbsp;As for þe first, þ<i>o</i>u schal vnd<i>er</i>stonde þat
+m<i>u</i>ltiplicacioɳ is a bryngyng<i>e</i> to-ged<i>er</i> of 2
+thyng<i>es</i> in on nombur, þe quych on nombur <a class = "gloss" name
+= "contynes" id = "contynes" href =
+"#gloss_contynes"><i>con</i>tynes</a> so mony tymes on, howe
+<span class = "linenum">leaf 153 <i>b</i>.</span>
+*mony tymes þ<i>ere</i> ben vnytees in þe nowmb<i>re</i> of þat 2, as
+twyes 4 is 8. now her<i>e</i> ben þe 2 nomb<i>er</i>s, of þe quych too
+nowmbr<i>e</i>s on is betokened be an adu<i>er</i>be, þe quych is þe
+worde twyes, &amp; þis worde thryes, &amp; þis worde four<i>e</i>
+<a class = "gloss" name = "sythes" id = "sythes" href =
+"#gloss_sythes">sythes</a>,<a class = "tag" name = "tag_craft9" id =
+"tag_craft9" href = "#note_craft9">9</a>
+&amp; so furth of such other lyke wordes. ¶&nbsp;And tweyn nombres schal
+be tokenyde be a nowne, as þis worde four<i>e</i> showys þes tweyɳ
+nombres <a class = "gloss" name = "ybroth" id = "ybroth" href =
+"#gloss_ybroth">y-broth</a> in-to on hole nombur, þat is 8, for twyes 4
+is 8, as þ<i>o</i>u wost wel. ¶&nbsp;And þes nomb<i>re</i> 8 conteynes
+as oft tymes 4 as þ<i>ere</i> ben vnites in þ<i>a</i>t other
+nomb<i>re</i>, þe quych is 2, for in 2 ben 2 vnites, &amp; so oft tymes
+4 ben in 8, as þ<i>o</i>u wottys wel.
+<span class = "sidenote">the second:</span>
+¶&nbsp;ffor þe secu<i>n</i>de, þ<i>o</i>u most know þat þ<i>o</i>u most
+haue too rewes of figures.
+<span class = "sidenote">the third:</span>
+¶&nbsp;As for þe thryde, þ<i>o</i>u most know þ<i>a</i>t 8 man<i>er</i>
+of diu<i>er</i>se case may happe in þis craft.
+<span class = "sidenote">the fourth.</span>
+The p<i>ro</i>fet of þis Craft is to telle when a nomb<i>re</i> is
+m<i>u</i>ltiplyed be a noþ<i>er</i>, qwat co<i>m</i>mys
+þ<i>ere</i>&nbsp;of. ¶ fforthermor<i>e</i>, as to þe sentence of
+our<i>e</i> verse, yf þ<i>o</i>u wel m<i>u</i>ltiply a nombur be
+a-noþ<i>er</i> nomb<i>ur</i>, þou schalt write
+<span class = "linenum">leaf 154 <i>a</i>.</span>
+*a rewe of figures of what nomb<i>ur</i>s so eu<i>er</i> þ<i>o</i>u
+welt,
+<span class = "sidenote">The multiplicand.</span>
+&amp; þat schal be called Num<i>erus</i> m<i>u</i>ltiplicand<i>us</i>,
+Anglice, þe nomb<i>ur</i> the quych to be m<i>u</i>ltiplied. þen
+þ<i>o</i>u schalt write a-nother rewe of figur<i>e</i>s, by þe quych
+þ<i>o</i>u schalt m<i>u</i>ltiplie the nombre þat is to be
+m<i>u</i>ltiplied, of þe quych nomb<i>ur</i> þe furst fig<i>ur</i>e
+schal be write vnd<i>er</i> þe last figur<i>e</i> of þe nomb<i>ur</i>,
+þe quych is to be m<i>u</i>ltiplied.
+<span class = "sidenote">How to set down the sum.</span>
+And so write forthe toward þe lyft side, as her<i>e</i> you may&nbsp;se,
+<span class = "float box">
+&nbsp; &nbsp; &nbsp;67324<br />
+1234</span>
+And þis on<i>e</i> nomb<i>ur</i> schall<i>e</i> be called nu<i>meru</i>s
+m<i>u</i>ltiplicans. An<i>gli</i>ce, þe nomb<i>ur</i>
+m<i>u</i>ltipliyng<i>e</i>, for he schall<i>e</i> m<i>u</i>ltiply þe
+hyer nounb<i>ur</i>, as þus on<i>e</i> tyme 6. And so forth, as I schal
+telle the aft<i>er</i>warde. And þou schal begyn in þe lyft side.
+<span class = "sidenote">Two sorts of Multiplication: mentally,</span>
+¶&nbsp;ffor-þ<i>ere</i>-more þou schalt vndurstonde þat þ<i>ere</i> is
+two man<i>ur</i>s of m<i>u</i>ltiplicacioɳ; one ys of þe
+wyrchyng<i>e</i> of þe boke only in þe mynde of a mon. fyrst he
+<span class = "pagenum">22</span>
+<a name = "page22" id = "page22"> </a>
+teches of þe fyrst man<i>er</i> of duplacioɳ, þe quych is be
+wyrchyng<i>e</i> of tabuls.
+<span class = "sidenote">and on paper.</span>
+Aft<i>er</i>warde he wol teche on þe secunde man<i>er</i>. vn<i>de</i>
+v<i>er</i>sus.</p>
+
+<p class = "headnote"><span class = "headnote">
+To multiply one Digit by another.</span></p>
+
+<div class = "verse">
+<p>In digitu<i>m</i> cures digitu<i>m</i> si duc<i>er</i>e
+ma<i>i</i>or</p>
+<span class = "linenum">leaf 154 <i>b</i>.</span>
+<p class = "pilcrow">
+* P<i>er</i> qua<i>n</i>tu<i>m</i> distat a denis respice debes</p>
+<p class = "pilcrow">
+¶ Namq<i>ue</i> suo decuplo totiens deler<i>e</i>
+mi<i>n</i>ore<i>m</i></p>
+<p>Sitq<i>ue</i> tibi nu<i>meru</i>s veniens exinde patebit.</p>
+</div>
+
+<p><span class = "sidenote">How to multiply two digits.</span>
+¶&nbsp;Her<i>e</i> he teches a rewle, how þ<i>o</i>u schalt fynde þe
+nounb<i>r</i>e þat comes by þe m<i>u</i>ltiplicacioɳ of a digit be
+anoþ<i>er</i>. loke how mony [vny]tes ben. bytwene þe mor<i>e</i> digit
+and 10. And reken ten for on vnite.
+<span class = "sidenote">Subtract the greater from ten;</span>
+And so oft do away þe lasse nounbre out of his owne <a class = "terms"
+name = "decuple" id = "decuple" href = "#terms_decuple">decuple</a>, þat
+is to say, fro þat nounb<i>r</i>e þat is ten tymes so mych <ins class =
+"correction" title = "error for ‘as’?">is</ins> þe nounb<i>re</i>
+þ<i>a</i>t comes of þe m<i>u</i>ltiplicacioɳ. As yf þ<i>o</i>u wol
+m<i>u</i>ltiply 2 be 4. loke how mony vnitees ben by-twene þe quych is
+þe mor<i>e</i> nounb<i>re</i>, &amp; be-twene ten. C<i>er</i>ten
+þ<i>ere</i> wel be vj vnitees by-twene 4 &amp; ten. yf þ<i>o</i>u reken
+þ<i>ere</i> w<i>i</i>t<i>h</i> þe ten þe vnite, as þou may se.
+<span class = "sidenote">take the less so many times from ten times
+itself.</span>
+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<i>er</i> digetes til þ<i>o</i>u come to ten;
+&amp; whan þ<i>o</i>u
+<span class = "sidenote">Example.</span>
+hast y-take so mony tymes 2 out of twenty, þe quych is sex tymes,
+þ<i>o</i>u schal leue 8 as þ<i>o</i>u wost wel, for 6 times 2 is twelue.
+take [1]2 out of twenty, &amp; þ<i>ere</i> schal leue 8. bot yf bothe þe
+digett<i>es</i>
+<span class = "linenum">leaf 155 <i>a</i>.</span>
+*ben <a class = "gloss" name = "ylyech" id = "ylyech" href =
+"#gloss_ylyech">y-lyech</a> mych as her<i>e</i>. 222 or too tymes
+twenty, þen it is <a class = "gloss" name = "fors" id = "fors" href =
+"#gloss_fors">no fors</a> quych of hem tweyn þ<i>o</i>u take out of here
+decuple. <a class = "gloss" name = "als" id = "als" href =
+"#gloss_als">als</a> mony
+<span class = "sidenote">Better use this table, though.</span>
+tymes as þ<i>a</i>t is fro 10. but neu<i>er</i>-þe-lesse, yf þ<i>o</i>u
+haue <a class = "gloss" name = "hast" id = "hast" href =
+"#gloss_hast">hast</a> to worch, þ<i>o</i>u schalt haue her<i>e</i> a
+tabul of figures, wher<i>e</i>-by þ<i>o</i>u schalt se a-nonɳ ryght what
+is þe nounbre þ<i>a</i>t comes of þe multiplicacioɳ of 2 digittes. þus
+þ<i>o</i>u schalt worch in þis fig<i>ur</i>e.</p>
+
+<table class = "grid right" summary = "example">
+<tr>
+<td class = "grid">&nbsp; 1</td>
+<td colspan = "9"></td>
+</tr>
+<tr>
+<td class = "grid">2</td>
+<td class = "grid">4</td>
+<td colspan = "8"></td>
+</tr>
+<tr>
+<td class = "grid">3</td>
+<td class = "grid">6</td>
+<td class = "grid">9</td>
+<td colspan = "7">&nbsp;</td>
+</tr>
+<tr>
+<td class = "grid">4</td>
+<td class = "grid">8</td>
+<td class = "grid">12</td>
+<td class = "grid">16</td>
+<td colspan = "6">&nbsp;</td>
+</tr>
+<tr>
+<td class = "grid">5</td>
+<td class = "grid">10</td>
+<td class = "grid">15</td>
+<td class = "grid">20</td>
+<td class = "grid">25</td>
+<td colspan = "5">&nbsp;</td>
+</tr>
+<tr>
+<td class = "grid">6</td>
+<td class = "grid">12</td>
+<td class = "grid">18</td>
+<td class = "grid">24</td>
+<td class = "grid">30</td>
+<td class = "grid">36</td>
+<td colspan = "4">&nbsp;</td>
+</tr>
+<tr>
+<td class = "grid">7</td>
+<td class = "grid">14</td>
+<td class = "grid">21</td>
+<td class = "grid">28</td>
+<td class = "grid">35</td>
+<td class = "grid">42</td>
+<td class = "grid">49</td>
+<td colspan = "3">&nbsp;</td>
+</tr>
+<tr>
+<td class = "grid">8</td>
+<td class = "grid">16</td>
+<td class = "grid">24</td>
+<td class = "grid">32</td>
+<td class = "grid">40</td>
+<td class = "grid">48</td>
+<td class = "grid">56</td>
+<td class = "grid">64</td>
+<td colspan = "2">&nbsp;</td>
+</tr>
+<tr>
+<td class = "grid">9</td>
+<td class = "grid">18</td>
+<td class = "grid">27</td>
+<td class = "grid">36</td>
+<td class = "grid">45</td>
+<td class = "grid">54</td>
+<td class = "grid">63</td>
+<td class = "grid">72</td>
+<td class = "grid">81</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td class = "grid">1</td>
+<td class = "grid">2</td>
+<td class = "grid">3</td>
+<td class = "grid">4</td>
+<td class = "grid">5</td>
+<td class = "grid">6</td>
+<td class = "grid">7</td>
+<td class = "grid">8</td>
+<td class = "grid">9</td>
+<td class = "grid">&nbsp; &nbsp;</td>
+</tr>
+</table>
+
+<p><span class = "sidenote">How to use it.</span>
+yf þe fig<i>ur</i>e, þe quych schall<i>e</i> be m<i>u</i>ltiplied, be
+euen<i>e</i> as mych as þe diget be, þe quych þat oþ<i>er</i>
+figur<i>e</i> schal be m<i>u</i>ltiplied, as two tymes twayɳ, or thre
+tymes 3. or sych other.
+<span class = "sidenote">The way to use the Multiplication table.</span>
+loke qwer<i>e</i> þat fig<i>ur</i>e sittes in
+<span class = "pagenum">23</span>
+<a name = "page23" id = "page23"> </a>
+þe lyft side of þe t<i>ri</i>angle, &amp; loke qwer<i>e</i> þe diget
+sittes in þe neþ<i>er</i> most rewe of þe triangle. &amp; go fro hym
+vpwarde in þe same rewe, þe quych rewe gose vpwarde til þ<i>o</i>u come
+agaynes þe oþ<i>er</i> digette þat sittes in þe lyft side of þe
+t<i>ri</i>angle. And þat nounbre, þe quych þou
+<span class = "linenum">leaf 155 <i>b</i>.</span>
+fyn*des þ<i>ere</i> is þe nounbre þat comes of the m<i>u</i>ltiplicacioɳ
+of þe 2 digittes, as yf þou wold wete qwat is 2 tymes 2. loke
+quer<i>e</i> sittes 2 in þe lyft side i<i>n</i> þe first rewe, he sittes
+next 1 in þe lyft side al on hye, as þ<i>o</i>u may se; þe[<i>n</i>]
+loke qwer<i>e</i> sittes 2 in þe lowyst rewe of þe t<i>ri</i>angle,
+&amp; go fro hym vpwarde in þe same rewe tyll<i>e</i> þou come <a class
+= "gloss" name = "aghenenes" id = "aghenenes" href =
+"#gloss_aghenenes">a-ȝenenes</a> 2 in þe hyer place, &amp; þer þou
+schalt fynd ywrite 4, &amp; þat is þe nounb<i>r</i>e þat comes of þe
+multiplicacioɳ of two tymes tweyn is 4, as þow wotest well<i>e</i>. yf
+þe diget. the quych is m<i>u</i>ltiplied, be mor<i>e</i> þan þe
+oþ<i>er</i>, þou schalt loke qwer<i>e</i> þe mor<i>e</i> diget sittes in
+þe lowest rewe of þe t<i>ri</i>angle, &amp; go vpwarde in þe same rewe
+tyl<a class = "tag" name = "tag_craft10" id = "tag_craft10" href =
+"#note_craft10">10</a>
+þ<i>o</i>u come <a class = "gloss" name = "anendes" id = "anendes" href
+= "#gloss_anendes">a-nendes</a> þe lasse diget in the lyft side. And
+þ<i>ere</i> þ<i>o</i>u schalt fynde þe no<i>m</i>b<i>r</i>e þat comes of
+þe m<i>u</i>ltiplicacioɳ; but þ<i>o</i>u schalt vnd<i>er</i>stonde þat
+þis rewle, þe quych is in þis v<i>er</i>se. ¶&nbsp;In digitu<i>m</i>
+cures, &amp;c., noþ<i>er</i> þis t<i>ri</i>angle schall<i>e</i> not
+s<i>er</i>ue, bot to fynde þe nounbres þ<i>a</i>t comes of the
+m<i>u</i>ltiplicacioɳ þat comes of 2 articuls or <i>com</i>posites, þe
+nedes no craft but yf þou wolt m<i>u</i>ltiply in þi mynde. And
+<span class = "linenum">leaf 156 <i>a</i>.</span>
+*þere-to þou schalt haue a craft aft<i>er</i>warde, for þou schall wyrch
+w<i>i</i>t<i>h</i> digettes in þe tables, as þou schalt know
+aft<i>er</i>warde. v<i>er</i>sus.</p>
+
+<p class = "headnote"><span class = "headnote">
+To multiply one Composite by another.</span></p>
+
+<div class = "verse">
+<p class = "pilcrow">
+¶ Postea p<i>ro</i>cedas postrema<i>m</i>
+m<i>u</i>ltiplica<i>n</i>do</p>
+<p>[Recte multiplicans per cu<i>n</i>ctas i<i>n</i>feriores]</p>
+<p>Condic<i>i</i>onem tamen t<i>a</i>li q<i>uod</i>
+m<i>u</i>ltiplicant<i>es</i></p>
+<p>Scribas in capite quicq<i>ui</i>d p<i>ro</i>cesserit inde</p>
+<p>Sed postq<i>uam</i> fuit hec m<i>u</i>ltiplicate fig<i>ur</i>e</p>
+<p>Anteriorent<i>ur</i> serei m<i>u</i>ltiplica<i>n</i>t<i>is</i></p>
+<p>Et sic m<i>u</i>ltiplica velut isti m<i>u</i>ltiplicasti</p>
+<p>Qui sequit<i>ur</i> nu<i>mer</i>u<i>m</i> sc<i>ri</i>ptu<i>m</i>
+quiscu<i>n</i>q<i>ue</i> figur<i>is</i>.</p>
+</div>
+
+<p><span class = "sidenote">How to multiply one number by
+another.</span>
+¶&nbsp;Her<i>e</i> he teches how þ<i>o</i>u schalt wyrch in þis craft.
+þou schalt m<i>ul</i>tiplye þe last figur<i>e</i> of þe nombre, and quen
+þ<i>o</i>u hast so ydo þou schalt draw all<i>e</i> þe figures of þe
+neþ<i>er</i> nounbre mor<i>e</i> <a class = "gloss" name = "taward" id =
+"taward" href = "#gloss_taward">taward</a> þe ryȝt side, so qwe<i>n</i>
+þ<i>o</i>u hast m<i>u</i>ltiplyed þe last figur<i>e</i> of þe heyer
+nounbre by all<i>e</i> þe neþ<i>er</i> figures.
+<span class = "sidenote">Multiply the ‘last’ figure of the higher by the
+‘first’ of the lower number.</span>
+And sette þe nounbir þat comes þer-of ou<i>er</i> þe last figur<i>e</i>
+of þe neþ<i>er</i> nounb<i>re</i>, &amp; þen þou schalt sette al þe
+oþ<i>er</i> fig<i>ur</i>es of þe neþ<i>er</i> nounb<i>re</i> mor<i>e</i>
+ner<i>e</i> to þe ryȝt side. ¶&nbsp;And whan þou hast m<i>u</i>ltiplied
+þat figur<i>e</i> þat schal be m<i>u</i>ltiplied þe next aft<i>er</i>
+<span class = "pagenum">24</span>
+<a name = "page24" id = "page24"> </a>
+hym by al þe neþ<i>er</i> figures. And worch as þou dyddyst afor<i>e</i>
+til
+<span class = "linenum">leaf 156 <i>b</i>.</span>
+*þou come to þe ende. And þou schalt vnd<i>er</i>stonde þat eu<i>er</i>y
+figur<i>e</i> of þe hier nounb<i>re</i> schal be m<i>u</i>ltiplied be
+all<i>e</i> þe figur<i>e</i>s of the neþ<i>er</i> nounbre, yf þe hier
+nounb<i>re</i> be any figur<i>e</i> þen on<i>e</i>.
+<span class = "sidenote">Set the answer over the first of the
+lower:</span>
+lo an Ensampul her<i>e</i> folowyng<i>e</i>.
+<span class = "float box">
+&nbsp; &nbsp; 2465.<br />
+232</span>
+þou schalt begyne to m<i>u</i>ltiplye in þe lyft side. M<i>u</i>ltiply 2
+be 2, and twyes 2 is 4. set 4
+<span class = "sidenote">then multiply the second of the lower, and so
+on.</span>
+ou<i>er</i> þe hed of þ<i>a</i>t 2, þen m<i>u</i>ltiplie þe same hier 2
+by 3 of þe nether nounbre, as thryes 2 þat schal be 6. set 6 ou<i>er</i>
+þe hed of 3, þan m<i>u</i>ltiplie þe same hier 2 by þat 2 þe quych
+stondes vnd<i>er</i> hym, þ<i>a</i>t wol be 4; do away þe hier 2 &amp;
+sette þ<i>ere</i> 4.
+<span class = "sidenote">Then antery the lower number:</span>
+¶&nbsp;Now þ<i>o</i>u most <a class = "terms" name = "antery" id =
+"antery" href = "#terms_antery">antery</a> þe nether nounbre, þat is to
+say, þ<i>o</i>u most sett þe neþ<i>er</i> nounbre more towarde þe ryȝt
+side, as þus. Take þe neþ<i>er</i> 2 toward þe ryȝt side, &amp; sette it
+eueɳ vnd<i>er</i> þe 4 of þe hyer nounb<i>r</i>e, &amp; ant<i>er</i>y
+all<i>e</i> þe figures þat comes aft<i>er</i> þat 2, as þus; sette 2
+vnd<i>er</i> þe 4. þen sett þe figur<i>e</i> of 3 þ<i>ere</i> þat þe
+figure of 2 stode, þe quych is now vndur þ<i>a</i>t 4 in þe hier
+nounbre; þen sett þe oþer figur<i>e</i> of 2, þe quych is þe last
+fig<i>ur</i>e toward þe lyft side of þe neþ<i>er</i> nomb<i>er</i>
+þ<i>ere</i> þe figur<i>e</i> of 3 stode.
+<span class = "sidenote">as thus.</span>
+þen þ<i>o</i>u schalt haue such a nombre.
+<span class = "float box">
+464465<br />
+&nbsp; 232</span>
+<span class = "linenum">leaf 157 <i>a</i>.</span>
+* ¶ Now m<i>u</i>ltiply 4, þe quych comes next aft<i>er</i> 6, by þe
+last 2 of þe neþ<i>er</i> nounbur toward þe lyft side. as 2 tymes 4, þat
+wel be 8. sette þat 8 ou<i>er</i> þe figure the quych stondes
+ou<i>er</i> þe hede of þat 2, þe quych is þe last figur<i>e</i> of þe
+neþ<i>er</i> nounbre; þan multiplie þat same 4 by 3, þat comes in þe
+neþ<i>er</i> rewe, þat wol be 12. sette þe digit of þe composyt
+ou<i>er</i> þe figure þe quych stondes ou<i>er</i> þe hed of þat 3,
+&amp; sette þe articule of þis co<i>m</i>posit ou<i>er</i> al þe figures
+þat stondes ou<i>er</i> þe neþ<i>er</i> 2 hede.
+<span class = "sidenote">Now multiply by the last but one of the
+higher:</span>
+þen m<i>u</i>ltiplie þe same 4 by þe 2 in þe ryȝt side in þe
+neþ<i>er</i> nounbur, þat wol be 8. do away 4. &amp; sette þ<i>ere</i>
+8. Eu<i>er</i> mor<i>e</i> qwen þ<i>o</i>u m<i>u</i>ltiplies þe hier
+figur<i>e</i> by þat figur<i>e</i> þe quych stondes vnd<i>er</i> hym,
+þou schalt do away þat hier figur<i>e</i>, &amp; sett þer þat nounbre þe
+quych comes of m<i>u</i>ltiplicacioɳ of ylke digittes.
+<span class = "sidenote">as thus.</span>
+Whan þou hast done as I haue byde þe, þ<i>o</i>u schalt haue suych an
+ord<i>er</i> of figur<i>e</i> as is her<i>e</i>,
+<span class = "float box">
+&nbsp; <span class = "smaller">1</span><br />
+&nbsp; <span class = "smaller">82</span><br />
+4648[65]<br />
+&nbsp; 232.</span>
+þen take and ant<i>er</i>y þi neþ<i>er</i> figures. And sett þe fyrst
+fig<i>ur</i>e of þe neþ<i>er</i> figures<a class = "tag" name =
+"tag_craft11" id = "tag_craft11" href = "#note_craft11">11</a>
+vndre be figur<i>e</i> of 6. ¶&nbsp;And draw al þe oþ<i>er</i> figures
+of þe same rewe to hym-warde,
+<span class = "linenum">leaf 157 <i>b</i>.</span>
+*as þ<i>o</i>u diddyst afore. þen m<i>u</i>ltiplye 6 be 2, &amp; sett
+þat þe quych comes ou<i>er</i> þ<i>ere</i>-of ou<i>er</i> al þe
+oþ<i>er</i> figures hedes þat stondes ou<i>er</i> þat 2. þen
+m<i>u</i>ltiply 6 be 3, &amp; sett all<i>e</i> þat comes þ<i>ere</i>-of
+vpon all<i>e</i> þe figur<i>e</i>s hedes þat standes ou<i>er</i> þat 3;
+þa<i>n</i> m<i>u</i>ltiplye 6 be 2, þe quych
+<span class = "pagenum">25</span>
+<a name = "page25" id = "page25"> </a>
+stondes vnd<i>er</i> þat 6, þen do away 6 &amp; write þ<i>ere</i> þe
+digitt of þe composit þat schal come þ<i>ere</i>of, &amp; sette þe
+articull ou<i>er</i> all<i>e</i> þe figures þat stondes ou<i>er</i> þe
+hede of þat 3 as her<i>e</i>,
+<span class = "float box">
+&nbsp; 11<br />
+&nbsp; 121<br />
+&nbsp; 828<br />
+464825<br />
+&nbsp; &nbsp; 232</span>
+<span class = "sidenote">Antery the figures again, and multiply by
+five:</span>
+þen ant<i>er</i>y þi figures as þou diddyst afor<i>e</i>, and
+m<i>u</i>ltipli 5 be 2, þat wol be 10; sett þe 0 ou<i>er</i> all þe
+figures þ<i>a</i>t stonden ou<i>er</i> þat 2, &amp; sett þ<i>a</i>t 1.
+ou<i>er</i> the next figures hedes, all<i>e</i> on hye towarde þe lyft
+side. þen m<i>u</i>ltiplye 5 be 3. þat wol be 15, write 5 ou<i>er</i> þe
+figures hedes þat stonden ou<i>er</i> þ<i>a</i>t 3, &amp; sett þat 1
+ou<i>er</i> þe next figur<i>e</i>s hedes toward þe lyft side. þen
+m<i>u</i>ltiplye 5 be 2, þat wol be 10. do away þat 5 &amp; sett
+þ<i>ere</i> a 0, &amp; sett þat 1 ou<i>er</i> þe figures hedes þat
+stonden ou<i>er</i> 3. And þen
+<span class = "linenum">leaf 158 <i>a</i>.</span>
+þou schalt haue such a nounbre as here stondes aftur.*
+<span class = "float box">
+&nbsp; &nbsp; 11<br />
+&nbsp; 1101<br />
+&nbsp; 1215<br />
+&nbsp; 82820<br />
+4648<br />
+&nbsp; &nbsp; &nbsp; 232</span>
+¶ Now draw all<i>e</i> þese figures downe toged<i>er</i> as þus, 6.8.1.
+&amp; 1 draw to-gedur; þat wolle be 16, do away all<i>e</i> þese figures
+saue 6. lat hym stonde, for <a class = "gloss" name = "thow" id = "thow"
+href = "#gloss_thow">þow</a> þ<i>o</i>u take hym away þou most write þer
+þe same aȝene. þ<i>ere</i>for<i>e</i> late hym stonde, &amp; sett 1
+ou<i>er</i> þe figur<i>e</i> hede of 4 toward þe lyft side; þen draw on
+to 4, þat woll<i>e</i> be 5.
+<span class = "sidenote">Then add all the figures above the line:</span>
+do away þat 4 &amp; þat 1, &amp; sette þ<i>ere</i> 5. þen draw 4221
+&amp; 1 toged<i>ur</i>, þat wol be 10. do away all<i>e</i> þat, &amp;
+write þere þat 4 &amp; þat 0, &amp; sett þat 1 ou<i>er</i> þe next
+figur<i>es</i> hede toward þe lyft side, þe quych is 6. þen draw þat 6
+&amp; þat 1 togedur, &amp; þat wolle be 7; do away 6 &amp; sett
+þ<i>ere</i> 7, þen draw 8810 &amp; 1, &amp; þat wel be 18; do away
+all<i>e</i> þe figures þ<i>a</i>t stondes ou<i>er</i> þe hede of þat 8,
+&amp; lette 8 stonde stil, &amp; write þat 1 ou<i>er</i> þe next
+fig<i>u</i>r<i>is</i> hede, þe quych is a 0. þen do away þat 0, &amp;
+sett þ<i>ere</i> 1, þe quych stondes ou<i>er</i> þe 0. hede. þen draw 2,
+5, &amp; 1 toged<i>ur</i>, þat woll<i>e</i> be 8. þen do away
+all<i>e</i> þat, &amp; write þ<i>ere</i>&nbsp;8.
+<span class = "sidenote">and you will have the answer.</span>
+¶&nbsp;And þen þou schalt haue þis nounbre, 571880.</p>
+
+<p class = "headnote"><span class = "headnote">
+The Cases of this Craft.</span></p>
+
+<span class = "linenum">leaf 158 <i>b</i>.</span>
+<div class = "verse">
+<p class = "pilcrow plus">
+* ¶ S<i>ed</i> cu<i>m</i> m<i>u</i>ltiplicabis, p<i>ri</i>mo sic
+e<i>st</i> op<i>er</i>andu<i>m</i>,</p>
+<p>Si dabit articulu<i>m</i> tibi m<i>u</i>ltiplicacio solu<i>m</i>;</p>
+<p>P<i>ro</i>posita cifra su<i>m</i>ma<i>m</i> t<i>ra</i>nsferre
+meme<i>n</i>to.</p>
+</div>
+
+<p><span class = "sidenote">What to do if the first multiplication
+results in an article.</span>
+¶&nbsp;Her<i>e</i> he puttes þe fyrst case of þis craft, þe quych is
+þis: yf þ<i>ere</i> come an articulle of þe m<i>u</i>ltiplicacioɳ ysette
+befor<i>e</i> the articull<i>e</i> in þe lyft side as þus
+<span class = "float box">
+&nbsp; 51<br />
+23.</span>
+multiplye 5 by 2, þat wol be 10; sette ou<i>er</i> þe hede of þat 2 a 0,
+&amp; sett þat on, þat is þe articul, in þe lyft side, þat is next hym,
+þen þ<i>o</i>u schalt haue þis nounbre
+<span class = "float box">
+1051.<br />
+&nbsp; 23</span>
+¶ And þen worch forth as þou diddist afore. And þ<i>o</i>u schalt
+vnd<i>er</i>stonde þat þ<i>o</i>u schalt write no 0. but whan þat place
+where þou schal write þat 0 has no figure afore hy<i>m</i> noþ<i>er</i>
+aft<i>er</i>. v<i>er</i>sus.</p>
+
+<span class = "pagenum">26</span>
+<a name = "page26" id = "page26"> </a>
+<div class = "verse">
+<p class = "pilcrow">
+¶ Si aut<i>em</i> digitus excreu<i>er</i>it
+articul<i>us</i>q<i>ue</i>.</p>
+<p>Articul<i>us</i><a class = "tag" name = "tag_craft12" id =
+"tag_craft12" href = "#note_craft12">12</a>
+sup<i>ra</i>p<i>osit</i>o digito salit vltra.</p>
+</div>
+
+<p><span class = "sidenote">What to do if the result is a composite
+number.</span>
+¶&nbsp;Her<i>e</i> is þe secunde case, þe quych is þis: yf hit happe þat
+þ<i>ere</i> come a composyt, þou schalt write þe digitte ou<i>er</i> þe
+hede of þe neþ<i>er</i> figur<i>e</i> by þe quych þ<i>o</i>u multipliest
+þe hier figure; and sett þe articull<i>e</i> next hym toward þe lyft
+side, as þou diddyst afore, as þ<i>us</i>
+<span class = "float box">
+&nbsp; 83.<br />
+83</span>
+Multiply 8 by 8, þat wol be 64. Write þe 4 ou<i>er</i> 8, þat is to say,
+ou<i>er</i> þe hede of þe neþ<i>er</i> 8; &amp; set 6, þe quych
+<span class = "linenum">leaf 159 <i>a</i>.</span>
+*is an articul, next aft<i>er</i>.
+And þen þou schalt haue such a nounb<i>r</i>e as is her<i>e</i>,
+<span class = "float box">
+6483<a class = "tag" name = "tag_craft13" id = "tag_craft13" href =
+"#note_craft13">13</a>,<br />
+&nbsp; 83</span>
+And þen worch forth.</p>
+
+<div class = "verse">
+<p class = "pilcrow">
+¶ Si digitus t<i>amen</i> ponas ip<i>su</i>m sup<i>er</i>
+ip<i>s</i>am.</p>
+</div>
+
+<p><span class = "sidenote">What if it be a digit.</span>
+¶ Her<i>e</i> is þe thryde case, þe quych is þis: yf hit happe þat of þi
+m<i>u</i>ltiplicaciouɳ come a digit, þ<i>o</i>u schalt write þe digit
+ou<i>er</i> þe hede of þe neþ<i>er</i> figur<i>e</i>, by the quych þou
+m<i>u</i>ltipliest þe hier<i>e</i> figur<i>e</i>, for þis nedes no
+Ensampul.</p>
+
+<div class = "verse">
+<p class = "pilcrow">
+¶ Subdita m<i>u</i>ltiplica non hanc que [incidit] illi</p>
+<p>Delet ea<i>m</i> penit<i>us</i> scribens quod p<i>ro</i>uenit
+inde.</p>
+</div>
+
+<p><span class = "sidenote">The fourth case of the craft.</span>
+¶&nbsp;Her<i>e</i> is þe 4 case, þe quych is: yf hit be happe þat þe
+neþ<i>er</i> figur<i>e</i> schal multiplye þat figur<i>e</i>, þe quych
+stondes ou<i>er</i> þat figures hede, þou schal do away þe hier
+figur<i>e</i> &amp; sett þ<i>er</i>e þat þ<i>a</i>t comys of þ<i>a</i>t
+m<i>u</i>ltiplicacioɳ. As yf þ<i>er</i>e come of þat
+m<i>u</i>ltiplicacioɳ an articuls þou schalt write þere þe hier
+figur<i>e</i> stode a 0. ¶&nbsp;And write þe articuls in þe lyft side,
+yf þat hit be a digit write þ<i>er</i>e a digit. yf þat h<i>i</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<i>er</i>e-for<i>e</i> þer nedes no
+Ensampul.</p>
+
+<div class = "verse">
+<p class = "pilcrow">
+¶ S<i>ed</i> si m<i>u</i>ltiplicat alia<i>m</i> ponas sup<i>er</i>
+ip<i>s</i>am</p>
+<p>Adiu<i>n</i>ges num<i>er</i>u<i>m</i> que<i>m</i> p<i>re</i>bet
+duct<i>us</i> ear<i>um</i>.</p>
+</div>
+
+<span class = "linenum">leaf 159 <i>b</i>.</span>
+
+<p><span class = "sidenote">The fifth case of the craft.</span>
+¶&nbsp;Her<i>e</i> is þe 5 case, þe quych is þis: yf *þe neþ<i>er</i>
+figur<i>e</i> schul m<i>u</i>ltiplie þe hier, and þat hier figur<i>e</i>
+is not <a class = "gloss" name = "recte1" id = "recte1" href =
+"#gloss_recte">recte</a> ou<i>er</i> his hede. And þat neþ<i>er</i>
+figur<i>e</i> hase oþ<i>er</i> figures, or on figure ou<i>er</i> his
+hede by m<i>u</i>ltiplicacioɳ, þat hase be afor<i>e</i>, þou schalt
+write þat nounbre, þe quych comes of þat, ou<i>er</i> all<i>e</i> þe
+ylke figures hedes, as þus here:
+<span class = "float box">
+&nbsp; &nbsp; 236<br />
+234</span>
+Multiply 2 by 2, þat wol be 4; set 4 ou<i>er</i> þe hede of þat 2. þen<a
+class = "tag" name = "tag_craft14" id = "tag_craft14" href =
+"#note_craft14">14</a>
+m<i>u</i>ltiplies þe hier 2 by þe neþ<i>er</i> 3, þat wol be 6. set
+ou<i>er</i> his hede 6, multiplie þe hier 2 by þe neþ<i>er</i> 4, þat
+wol be 8. do away þe hier 2, þe quych stondes ou<i>er</i> þe hede of þe
+figur<i>e</i> of&nbsp;4,
+<span class = "pagenum">27</span>
+<a name = "page27" id = "page27"> </a>
+and set þ<i>er</i>e 8. And þou schalt haue þis nounb<i>re</i> here
+<span class = "float box">
+46836<br />
+234</span>
+And antery þi figur<i>e</i>s, þat is to say, set þi neþ<i>er</i> 4
+vnd<i>er</i> þe hier 3, and set þi 2 other figures ner<i>e</i> hym, so
+þat þe neþ<i>er</i> 2 stonde vnd<i>ur</i> þe hier 6, þe quych 6 stondes
+in þe lyft side. And þat 3 þat stondes vndur 8, as þus aftur ȝe
+may&nbsp;se,
+<span class = "float box">
+46836<br />
+&nbsp; 234</span>
+Now worch forthermor<i>e</i>, And m<i>u</i>ltiplye þat hier 3 by 2, þat
+wol be 6, set þ<i>a</i>t 6 þe quych stondes ou<i>er</i> þe hede of þat
+2, And þen worch as I taȝt þe afore.</p>
+
+<span class = "linenum">leaf 160 <i>a</i>.</span>
+<div class = "verse">
+<p class = "pilcrow plus">
+* ¶ Si sup<i>ra</i>posita cifra debet m<i>u</i>ltiplicar<i>e</i></p>
+<p>Prorsus ea<i>m</i> deles &amp; ibi scribi cifra debet.</p>
+</div>
+
+<p><span class = "sidenote">The sixth case of the craft.</span>
+¶&nbsp;Her<i>e</i> is þe 6 case, þe quych is þis: yf hit happe þat þe
+figur<i>e</i> by þe quych þou schal m<i>u</i>ltiplye þe hier
+figur<i>e</i>, þe quych stondes ryght ou<i>er</i> hym by a 0, þou schalt
+do away þat figur<i>e</i>, þe quych ou<i>er</i> þat cifre hede.
+¶&nbsp;And write þ<i>ere</i> þat nounbre þat comes of þe
+m<i>u</i>ltiplicacioɳ as þus, 23. do away 2 and sett þ<i>er</i>e a 0.
+vn<i>de</i> v<i>er</i>sus.</p>
+
+<div class = "verse">
+<p class = "pilcrow">
+¶ Si cifra m<i>u</i>ltiplicat alia<i>m</i> posita<i>m</i> sup<i>er</i>
+ip<i>s</i>am</p>
+<p>Sitq<i>ue</i> locus sup<i>ra</i> vacu<i>us</i> sup<i>er</i> hanc
+cifra<i>m</i> fiet.</p>
+</div>
+
+<p><span class = "sidenote">The seventh case of the craft.</span>
+¶&nbsp;Her<i>e</i> is þe 7 case, þe quych is þis: yf a 0 schal
+m<i>u</i>ltiply a figur<i>e</i>, þe quych stondes not <a class = "gloss"
+name = "recte" id = "recte" href = "#gloss_recte">recte</a> ou<i>er</i>
+hym, And ou<i>er</i> þat 0 stonde no thyng, þou schalt write ou<i>er</i>
+þat 0 anoþ<i>er</i> 0 as þus:
+<span class = "float box">
+&nbsp; &nbsp; 24<br />
+03</span>
+multiplye 2 be a 0, it wol be nothyng<i>e</i>. write þere a 0
+ou<i>er</i> þe hede of þe neþ<i>er</i> 0, And þen worch forth til þou
+come to þe ende.</p>
+
+
+<div class = "verse">
+<p class = "pilcrow">
+¶ Si sup<i>ra</i><a class = "tag" name = "tag_craft15" id =
+"tag_craft15" href = "#note_craft15">15</a> fuerit cifra sem<i>per</i>
+e<i>st</i> p<i>re</i>t<i>er</i>eunda.</p>
+</div>
+
+<p><span class = "sidenote">The eighth case of the craft.</span>
+¶&nbsp;Her<i>e</i> is þe 8 case, þe quych is þis: yf þ<i>ere</i> be a 0
+or mony cifers in þe hier rewe, þ<i>o</i>u schalt not m<i>u</i>ltiplie
+hem, bot let hem stonde. And antery þe figures beneþe to þe next
+figur<i>e</i> sygnificatyf as þus:
+<span class = "float box">
+00032.<br />
+22</span>
+Ou<i>er</i>-lepe all<i>e</i> þese cifers &amp; sett þat
+<span class = "linenum">leaf 160 <i>b</i>.</span>
+*neþ<i>er</i> 2 þat stondes toward þe ryght side, and sett hym
+vnd<i>ur</i> þe 3, and sett þe oþ<i>er</i> nether 2 nere hym, so þat he
+stonde vnd<i>ur</i> þe thrydde 0, þe quych stondes next 3. And þan
+worch. vnd<i>e</i> v<i>er</i>sus.</p>
+
+<div class = "verse">
+<p class = "pilcrow">
+¶ Si dubites, an sit b<i>e</i>n<i>e</i> m<i>u</i>ltiplicac<i>i</i>o
+facta,</p>
+<p>Diuide totalem nu<i>mer</i>u<i>m</i> p<i>er</i>
+multiplicante<i>m</i>.</p>
+</div>
+
+<p><span class = "sidenote">How to prove the multiplication.</span>
+¶&nbsp;Her<i>e</i> he teches how þou schalt know wheþ<i>er</i> þou hase
+wel I-do or no. And he says þat þou schalt deuide all<i>e</i> þe
+nounb<i>r</i>e þat comes of þe m<i>u</i>ltiplicacioɳ by þe neþ<i>er</i>
+figures. And þen þou schalt haue þe same nounbur þat þ<i>o</i>u hadyst
+in þe begynnyng<i>e</i>. but ȝet þou hast not þe craft of dyuisioɳ, but
+þ<i>o</i>u schalt haue hit aft<i>er</i>warde.</p>
+
+<span class = "pagenum">28</span>
+<a name = "page28" id = "page28"> </a>
+<div class = "verse">
+<p class = "pilcrow">
+¶ P<i>er</i> num<i>er</i>u<i>m</i> si vis nu<i>mer</i>u<i>m</i>
+q<i>u</i>oq<i>ue</i> m<i>u</i>ltiplicar<i>e</i></p>
+<p class = "pilcrow">
+¶ T<i>antu</i>m p<i>er</i> normas subtiles absq<i>ue</i> figuris</p>
+<p>Has normas pot<i>er</i>is p<i>er</i> v<i>er</i>sus scir<i>e</i>
+sequentes.</p>
+</div>
+
+<p><span class = "sidenote">Mental multiplication.</span>
+¶ Her<i>e</i> he teches þe to m<i>u</i>ltiplie <a class = "gloss" name =
+"thowght" id = "thowght" href = "#gloss_thowght">be þowȝt</a> figures in
+þi mynde. And þe sentence of þis v<i>er</i>se is þis: yf þo<i>u</i> wel
+m<i>u</i>ltiplie on nounbre by anoþ<i>er</i> in þi mynde, þ<i>o</i>u
+schal haue þ<i>er</i>eto rewles in þe v<i>er</i>ses þat schal come
+aft<i>er</i>.</p>
+
+<div class = "verse">
+<p class = "pilcrow">
+¶ Si tu p<i>er</i> digitu<i>m</i> digitu<i>m</i> vis
+m<i>u</i>ltiplicar<i>e</i></p>
+<p>Re<i>gula</i> p<i>re</i>cedens dat qualit<i>er</i> est
+op<i>er</i>andu<i>m</i>.</p>
+</div>
+
+<p><span class = "sidenote">Digit by digit is easy.</span>
+¶ Her<i>e</i> he teches a rewle as þou hast afor<i>e</i> to
+m<i>u</i>ltiplie a digit be anoþ<i>er</i>, as yf þou wolde wete qwat is
+sex tymes 6. þou
+<span class = "linenum">leaf 161 <i>a</i>.</span>
+*schalt wete by þe rewle þat I taȝt þe befor<i>e</i>, yf þou haue mynde
+þ<i>er</i>of.</p>
+
+<div class = "verse">
+<p class = "pilcrow">
+¶ Articulu<i>m</i> si p<i>er</i> reliquu<i>m</i> reliquu<i>m</i> vis
+m<i>u</i>lti<i>plica</i>r<i>e</i></p>
+<p>In p<i>ro</i>p<i>r</i>iu<i>m</i> digitu<i>m</i> debet
+vt<i>er</i>q<i>ue</i> resolui.</p>
+<p class = "pilcrow">
+¶ Articul<i>us</i> digitos post se m<i>u</i>ltiplicantes</p>
+<p>Ex digit<i>us</i> quociens retenerit
+m<i>u</i>ltipli<i>ca</i>r<i>i</i></p>
+<p>Articuli faciu<i>n</i>t tot centu<i>m</i> m<i>u</i>ltiplicati.</p>
+</div>
+
+<p><span class = "sidenote">The first case of the craft.</span>
+¶&nbsp;Her<i>e</i> he teches þe furst rewle, þe quych is þis: yf þou wel
+m<i>u</i>ltiplie an articul be anoþ<i>er</i>, so þat both þe articuls
+bene w<i>i</i>t<i>h</i>-Inne an hundreth, þus þ<i>o</i>u schalt do.
+<span class = "sidenote">Article by article;</span>
+take þe digit of bothe the articuls, for eu<i>er</i>y articul hase a
+digit, þen m<i>u</i>ltiplye þat on digit by þat oþ<i>er</i>, and loke
+how mony vnytes ben in þe nounbre þat comes of þe m<i>u</i>ltiplicacioɳ
+of þe 2 digittes, &amp; so mony hundrythes ben in þe nounb<i>re</i> þat
+schal come of þe m<i>u</i>ltiplicacioɳ of þe ylke 2 articuls as þus.
+<span class = "sidenote">an example:</span>
+yf þ<i>o</i>u wold wete qwat is ten tymes ten. take þe digit of ten, þe
+quych is 1; take þe digit of þat oþ<i>er</i> ten, þe quych is on.
+¶&nbsp;Also m<i>u</i>ltiplie 1 be 1, as on tyme on þat is but 1. In on
+is but on vnite as þou wost welle, þ<i>er</i>efor<i>e</i> ten tymes ten
+is but a hundryth.
+<span class = "sidenote">another example:</span>
+¶&nbsp;Also yf þou wold wete what is twenty tymes 30. take þe digit of
+twenty, þat is 2; &amp; take þe digitt of thrytty, þat is 3.
+m<i>u</i>ltiplie 3 be 2, þat is 6. Now in 6 ben 6 vnites, ¶&nbsp;And so
+mony hundrythes ben in 20 tymes 30*,
+<span class = "linenum">leaf 161 <i>b</i>.</span>
+þ<i>ere</i>for<i>e</i> 20 tymes 30 is 6 hundryth eueɳ. loke &amp; se.
+¶&nbsp;But yf it be so þat on<i>e</i> articul be w<i>i</i>t<i>h</i>-Inne
+an hundryth, <a class = "gloss" name = "or2" id = "or2" href =
+"#gloss_or2">or</a> by-twene an hundryth and a thowsande, so þat it be
+not a þowsande fully. þen loke how mony vnytes ben in þe nounbur þat
+comys of þe m<i>u</i>ltiplicacioɳ <a class = "tag" name = "tag_craft16"
+id = "tag_craft16" href = "#note_craft16">16</a>And so mony tymes<a
+class = "tag" href = "#note_craft16">16</a> of 2 digitt<i>es</i> of ylke
+articuls, so mony thowsant ben in þe nounbre, the qwych comes of þe
+m<i>u</i>ltiplicacioɳ. And so mony tymes ten thowsand schal be in þe
+nounbre þat comes of þe m<i>u</i>ltiplicacion of
+<span class = "pagenum">29</span>
+<a name = "page29" id = "page29"> </a>
+2 articuls, as yf þ<i>o</i>u wold wete qwat is 4 hundryth tymes [two
+hundryth]. Multiply 4 be 2,<a class = "tag" name = "tag_craft17" id =
+"tag_craft17" href = "#note_craft17">17</a>
+þat wol be 8. in 8 ben 8 vnites.
+<span class = "headnote float">
+How to work subtly without Figures.</span>
+<span class = "sidenote">Mental multiplication.</span>
+¶ And so mony tymes ten thousand be in 4 hundryth tymes [2]<a class =
+"tag" href = "#note_craft17">17</a> hundryth, þ<i>a</i>t is 80 thousand.
+Take hede, I&nbsp;schall telle þe a
+<span class = "sidenote">Another example.</span>
+gen<i>e</i>rall<i>e</i> rewle whan þ<i>o</i>u hast 2 articuls, And þou
+wold wete qwat comes of þe m<i>u</i>ltiplicacioɳ of hem 2.
+m<i>u</i>ltiplie þe digit of þ<i>a</i>t on <a class = "terms" name =
+"articuls2" id = "articuls2" href = "#terms_article">articuls</a>, and
+kepe þat nounbre, þen loke how mony cifers schuld go befor<i>e</i> þat
+on articuls, <a class = "gloss" name = "and" id = "and" href =
+"#gloss_and">and</a> he wer<i>e</i> <a class = "gloss" name = "write" id
+= "write" href = "#gloss_write">write</a>. Als mony cifers schuld go
+befor<i>e</i> þat other, &amp; he wer<i>e</i> write of cifers. And haue
+all<i>e</i> þe ylke cifers toged<i>ur</i> in þi mynde,
+<span class = "linenum">leaf 162 <i>a</i>.</span>
+*<a class = "gloss" name = "arowe" id = "arowe" href =
+"#gloss_arowe">a-rowe</a> <a class = "gloss" name = "ychon" id = "ychon"
+href = "#gloss_ychon">ychoɳ</a> aftur other, and in þe last plase set þe
+nounbre þat comes of þe m<i>u</i>ltiplicacioɳ of þe 2 digittes. And loke
+in þi mynde in what place he stondes, <a class = "gloss" name = "where"
+id = "where" href = "#gloss_where">wher<i>e</i></a> in þe secunde, or in
+þe thryd, or in þe 4, or wher<i>e</i> ellis, and loke qwat þe figures
+by-token in þat place; &amp; so mych is þe nounbre þat
+<span class = "sidenote">Another example.</span>
+comes of þe 2 articuls y-m<i>u</i>ltiplied to-ged<i>ur</i> as þus: yf
+þ<i>o</i>u wold wete what is 20 thousant tymes 3 þowsande.
+m<i>u</i>ltiply þe digit of þat articull<i>e</i> þe quych is 2 by þe
+digitte of þat oþ<i>er</i> articul þe quych is 3, þat wol be 6. þen loke
+how mony cifers schal go to 20 thousant as hit schuld be write in a
+tabul. c<i>er</i>tainly 4 cifers schuld go to 20 þowsant. ffor þis
+figure 2 in þe fyrst place betokenes twene.
+<span class = "sidenote">Notation.</span>
+¶&nbsp;In þe secunde place hit betokenes twenty. ¶&nbsp;In þe 3. place
+hit betokenes 2 hundryth. .¶. In þe 4 place 2 thousant. ¶&nbsp;In þe 5
+place h<i>i</i>t betokenes twenty þousant. þ<i>ere</i>for<i>e</i> he
+most haue 4 cifers a-for<i>e</i> hym þat he may sto<i>n</i>de in þe 5
+place. kepe þese 4 cifers in thy mynde, þen loke how mony cifers goɳ to
+3 thousant. Certayn to 3 thousante
+<span class = "linenum">leaf 162 <i>b</i>.</span>
+*goɳ 3 cifers afor<i>e</i>. Now cast ylke 4 cifers þat schuld go to
+twenty thousant, And thes 3 cifers þat schuld go afor<i>e</i> 3
+thousant, &amp; sette hem in rewe ychoɳ aft<i>er</i> oþ<i>er</i> in þi
+mynde, as þai schuld stonde in a tabull<i>e</i>. And þen schal þou haue
+7 cifers; þen sett þat 6 þe quych comes of þe m<i>u</i>ltiplicacioɳ of
+þe 2 digitt<i>es</i> aft<i>u</i>r þe ylke cifers in þe 8 place as yf þat
+hit stode in a tabul. And loke qwat a figur<i>e</i> of 6 schuld betoken
+in þe 8 place. yf hit wer<i>e</i> in a tabul &amp; so mych it is. &amp;
+yf þat figure of 6 stonde in þe fyrst place he schuld betoken but 6.
+¶&nbsp;In þe 2 place he schuld betoken sexty. ¶&nbsp;In the 3 place he
+schuld betokeɳ sex hundryth.
+<span class = "sidenote">Notation again.</span>
+¶&nbsp;In þe 4 place sex thousant. ¶&nbsp;In þe 5 place sexty þowsant.
+¶&nbsp;In þe sext place sex hundryth þowsant. ¶&nbsp;In þe 7 place sex
+þowsant thousant<i>es</i>. ¶&nbsp;In þe 8 place sexty þowsant
+thousantes. þ<i>er</i>for<i>e</i> sett 6 in octauo loco, And he schal
+betoken sexty þowsant
+<span class = "pagenum">30</span>
+<a name = "page30" id = "page30"> </a>
+thousantes.
+<span class = "sidenote">Mental multiplication.</span>
+And so mych is twenty þowsant tymes 3 thousant, ¶&nbsp;And þis rewle is
+gen<i>er</i>all<i>e</i> for all<i>e</i> man<i>er</i> of articuls,
+Whethir þai be hundryth or þowsant; but þ<i>o</i>u <a class = "gloss"
+name = "most" id = "most" href = "#gloss_most">most</a> know well þe
+craft of þe <a class = "gloss" name = "wryrchynge" id = "wryrchynge"
+href = "#gloss_wryrchynge">wryrchyng<i>e</i></a> in þe tabull<i>e</i>
+<span class = "linenum">leaf 163 <i>a</i>.</span>
+*or þou know to do þus in þi mynde aftur þis rewle. Thou most þat þis
+rewle <a class = "gloss" name = "holdythe" id = "holdythe" href =
+"#gloss_holdythe">holdyþe</a> <a class = "gloss" name = "note" id =
+"note" href = "#gloss_note">note</a> but wher<i>e</i> þ<i>ere</i> ben 2
+articuls and no mo of þe quych ayther of hem hase but on figur<i>e</i>
+significatyf. As twenty tymes 3 thousant or 3 hundryth, and such
+oþ<i>ur</i>.</p>
+
+<div class = "verse">
+<p class = "pilcrow">
+¶ Articulum digito si m<i>u</i>ltiplicare o<i>portet</i></p>
+<p>Articuli digit[i sumi quo multiplicate]</p>
+<p>Debem<i>us</i> reliquu<i>m</i> quod m<i>u</i>ltiplicat<i>ur</i> ab
+ill<i>is</i></p>
+<p>P<i>er</i> reliq<i>u</i>o decuplu<i>m</i> sic su<i>m</i>ma<i>m</i>
+later<i>e</i> neq<i>ui</i>b<i>i</i>t.</p>
+</div>
+
+<p><span class = "sidenote">The third case of the craft;</span>
+¶&nbsp;Her<i>e</i> he puttes þe thryde rewle, þe quych is þis. yf
+þ<i>o</i>u wel m<i>u</i>ltiply in þi mynde, And þe Articul be a digitte,
+þou schalt loke þat þe digitt be w<i>i</i>t<i>h</i>-Inne an hundryth,
+þen þou schalt m<i>u</i>ltiply the digitt of þe Articulle by þe oþer
+digitte. And eu<i>er</i>y vnite in þe nounbre þat schall<i>e</i> come
+þ<i>ere</i>-of schal betoken ten. As þus:
+<span class = "sidenote">an example.</span>
+yf þat þ<i>o</i>u wold wete qwat is twyes 40. m<i>u</i>ltiplie þe
+digitt<i>e</i> of 40, þe quych is 4, by þe oþ<i>er</i> diget, þe quych
+is 2. And þat wolle be 8. And in þe nombre of 8 ben 8 vnites, &amp;
+eu<i>er</i>y of þe ylke vnites schuld stonde for 10. þ<i>ere</i>-fore
+þ<i>ere</i> schal be 8 tymes 10, þat wol be 4 score. And so mony is
+twyes 40. ¶&nbsp;If þe articul be a hundryth or be 2 hundryth And a
+þowsant, so þat hit be notte a thousant,
+<span class = "linenum">leaf 163 <i>b</i>.</span>
+*worch as þo<i>u</i> dyddyst afor<i>e</i>, saue þ<i>o</i>u schalt rekene
+eu<i>er</i>y vnite for a hundryth.</p>
+
+<div class = "verse">
+<p class = "pilcrow">
+¶ In nu<i>mer</i>u<i>m</i> mixtu<i>m</i> digitu<i>m</i> si ducer<i>e</i>
+cures</p>
+<p>Articul<i>us</i> mixti sumat<i>ur</i> deinde resoluas</p>
+<p>In digitu<i>m</i> post fac respectu de digitis</p>
+<p>Articul<i>us</i>q<i>ue</i> docet excrescens in diriua<i>n</i>do</p>
+<p>In digitu<i>m</i> mixti post ducas
+m<i>u</i>ltiplica<i>n</i>te<i>m</i></p>
+<p class = "pilcrow">
+¶ De digitis vt norma <a class = "tag" name = "tag_craft18" id =
+"tag_craft18" href = "#note_craft18">18</a>[docet] de [hunc]</p>
+<p>Multiplica si<i>mu</i>l et sic postea summa patebit.</p>
+</div>
+
+<p><span class = "sidenote">The fourth case of the craft:</span>
+Here he puttes þe 4 rewle, þe quych is þis: yf þou m<i>u</i>ltipliy on
+composit be a digit as 6 tymes 24, <ins class = "correction" title =
+"footnote tag positioned as shown"><a class = "tag" name = "tag_craft19"
+id = "tag_craft19" href = "#note_craft19">19</a></ins>þen take þe diget
+of þat composit, &amp; m<i>u</i>ltiply þ<i>a</i>t digitt by þat
+oþ<i>er</i> diget, and kepe þe nomb<i>ur</i> þat comes þ<i>ere</i>-of.
+þen take þe digit of þat composit, &amp; m<i>u</i>ltiply þat digit by
+anoþ<i>er</i> diget, by þe quych þ<i>o</i>u hast m<i>u</i>ltiplyed þe
+diget of þe articul, and loke qwat comes þ<i>ere</i>-of.
+<span class = "sidenote">Composite by digit.</span>
+þen take þ<i>o</i>u þat nounbur, &amp; cast hit to þat other nounbur þat
+þ<i>o</i>u secheste as þus yf þou wel
+<span class = "pagenum">31</span>
+<a name = "page31" id = "page31"> </a>
+wete qwat comes of 6 tymes 4 &amp; twenty.
+<span class = "sidenote">Mental multiplication.</span>
+multiply þat articull<i>e</i> of þe composit by þe digit, þe quych is 6,
+as yn þe thryd rewle þ<i>o</i>u was tauȝt, And þat schal be 6
+scor<i>e</i>. þen m<i>u</i>ltiply þe diget of þe <i>com</i>posit,
+<span class = "linenum">leaf 164 <i>a</i>.</span>
+*þe quych is 4, and m<i>u</i>ltiply þat by þat other diget, þe quych is
+6, as þou wast tauȝt in þe first rewle, yf þ<i>o</i>u haue mynde
+þ<i>er</i>of, &amp; þat wol be 4 &amp; twenty. cast all ylke nounburs
+to-ged<i>ir</i>, &amp; hit schal be 144. And so mych is 6 tymes 4 &amp;
+twenty.</p>
+
+<p class = "headnote"><span class = "headnote">
+How to multiply without Figures.</span></p>
+
+<div class = "verse">
+<p class = "pilcrow">
+¶ Duct<i>us</i> in articulu<i>m</i> num<i>erus</i> si
+<i>com</i>posit<i>us</i> sit</p>
+<p>Articulu<i>m</i> puru<i>m</i> comites articulu<i>m</i>
+q<i>u</i>o<i>que</i></p>
+<p>Mixti pro digit<i>is</i> post fiat [et articulus vt]</p>
+<p>Norma iubet [retinendo quod extra dicta ab illis]</p>
+<p>Articuli digitu<i>m</i> post tu mixtu<i>m</i> digitu<i>m</i> duc</p>
+<p>Re<i>gula</i> de digitis nec p<i>re</i>cipit
+articul<i>us</i>q<i>ue</i></p>
+<p>Ex quib<i>us</i> exc<i>re</i>scens su<i>m</i>me tu iunge
+p<i>ri</i>ori</p>
+<p>Sic ma<i>n</i>ifesta cito fiet t<i>ibi</i> su<i>m</i>ma petita.</p>
+</div>
+
+<p><span class = "sidenote">The fifth case of the craft:</span>
+¶&nbsp;Her<i>e</i> he puttes þe 5 rewle, þe quych is þis: yf þ<i>o</i>u
+wel m<i>u</i>ltiply an Articul be a composit, m<i>u</i>ltiplie þat
+Articul by þe articul of þe composit, and worch as þou wos tauȝt in þe
+secunde rewle, of þe quych rewle þe v<i>er</i>se begynnes þus.
+<span class = "sidenote">Article by Composite.</span>
+¶&nbsp;Articulu<i>m</i> si p<i>er</i> Relicu<i>m</i> vis
+m<i>u</i>ltiplicare. þen m<i>u</i>ltiply þe diget of þe composit by þat
+oþ<i>ir</i> articul aft<i>ir</i> þe doctrine of þe 3 rewle. take
+þ<i>er</i>of gode hede, I&nbsp;p<i>ra</i>y þe as þus. Yf þ<i>o</i>u wel
+wete what is 24 tymes ten.
+<span class = "sidenote">An example.</span>
+Multiplie ten by 20, þat wel be 2 hundryth. þen m<i>u</i>ltiply þe diget
+of þe 10, þe quych is 1, by þe diget of þe composit, þe quych is 4,
+&amp; þ<i>a</i>t
+<span class = "linenum">leaf 164 <i>b</i>.</span>
+*wol be 4. þen reken eu<i>er</i>y vnite þat is in 4 for 10, &amp; þat
+schal be 40. Cast 40 to 2 hundryth, &amp; þat wol be 2 hundryth &amp;
+40. And so mych is 24 tymes ten.</p>
+
+<p class = "headnote"><span class = "headnote">
+How to work without Figures.</span></p>
+
+<div class = "verse">
+<p class = "pilcrow">
+¶ Compositu<i>m</i> num<i>er</i>u<i>m</i> mixto si[c]
+m<i>u</i>ltiplicabis</p>
+<p>Vndecies tredeci<i>m</i> sic e<i>st</i> ex hiis op<i>er</i>andum</p>
+<p>In reliquu<i>m</i> p<i>rimu</i>m demu<i>m</i> duc post in
+eund<i>em</i></p>
+<p>Vnu<i>m</i> post den<i>u</i>m duc in t<i>ri</i>a dei<i>n</i>de
+p<i>er</i> vnu<i>m</i></p>
+<p>Multiplices<i>que</i> dem<i>u</i>m int<i>ra</i> o<i>mn</i>ia
+m<i>u</i>ltiplicata</p>
+<p>In su<i>m</i>ma decies q<i>ua</i>m si fu<i>er</i>it t<i>ibi</i>
+doces</p>
+<p>Multiplicandor<i>um</i> de normis sufficiunt h<i>ec</i>.</p>
+</div>
+
+<p><span class = "sidenote">The sixth case of the craft:</span>
+¶&nbsp;Here he puttes þe 6 rewle, &amp; þe last of all<i>e</i>
+multiplicacioɳ, þe quych is þis: yf þ<i>o</i>u wel m<i>u</i>ltiplye a
+<i>com</i>posit by a-noþ<i>er</i> composit, þou schalt do þus.
+<span class = "sidenote">Composite by Composite.</span>
+m<i>u</i>ltiplie þ<i>a</i>t on composit, qwych þ<i>o</i>u welt of the
+twene, by þe articul of þe toþ<i>er</i> composit, as þ<i>o</i>u
+wer<i>e</i> tauȝt in þe 5 rewle, þen m<i>u</i>ltiplie þ<i>a</i>t same
+composit, þe quych þou hast m<i>u</i>ltiplied by þe oþ<i>er</i> articul,
+by þe digit of þe oþ<i>er</i> composit,
+<span class = "sidenote">Mental multiplication.</span>
+as
+<span class = "pagenum">32</span>
+<a name = "page32" id = "page32"> </a>
+þ<i>o</i>u was tauȝt in þe 4 rewle.
+<span class = "sidenote">An example</span>
+As þus, yf þou wold wete what is 11 tymes 13, as þ<i>o</i>u was tauȝt in
+þe 5 rewle, &amp; þat schal be an hundryth &amp; ten, aft<i>er</i>warde
+m<i>u</i>ltiply þat same co<i>m</i>posit þ<i>a</i>t þ<i>o</i>u hast
+m<i>u</i>ltiplied, þe quych is a .11. And m<i>u</i>ltiplye hit be þe
+digit of þe oþ<i>er</i> composit, þe quych is 3, for 3 is þe digit of
+13, And þat wel be 30. þen take þe digit of þat composit, þe quych
+composit þou m<i>u</i>ltiplied by þe digit of þ<i>a</i>t oþ<i>er</i>
+<i>com</i>posit,
+<span class = "linenum">leaf 165 <i>a</i>.</span>
+*þe quych is a 11.
+<span class = "sidenote">of the sixth case of the craft.</span>
+¶&nbsp;Also of the quych 11 on is þe digit. m<i>u</i>ltiplie þat digitt
+by þe digett of þat oth<i>er</i> composit, þe quych diget is 3, as
+þ<i>o</i>u was tauȝt in þe first rewle i<i>n</i> þe begynnyng<i>e</i> of
+þis craft. þe quych rewle begynn<i>es</i> “In digitu<i>m</i> cures.” And
+of all<i>e</i> þe m<i>u</i>ltiplicacioɳ of þe 2 digitt comys thre, for
+onys 3 is but 3. Now cast all<i>e</i> þese nounbers toged<i>ur</i>, the
+quych is þis, a&nbsp;hundryth &amp; ten &amp; 30 &amp; 3. And al þat wel
+be 143. Write 3 first in þe ryght side. And cast 10 to 30, þat wol be
+40. set 40 next aft<i>ur</i> towarde þe lyft side, And set aftur a
+hundryth as her<i>e</i> an Ensampull<i>e</i>, 143.</p>
+
+<p class = "center">(Cetera desunt.)</p>
+
+<hr class = "mid" />
+
+<div class = "footnote">
+
+<p><a name = "note_craft1" id = "note_craft1" href =
+"#tag_craft1">1.</a>
+In MS, ‘awiy.’</p>
+
+<p><a name = "note_craft2" id = "note_craft2" href =
+"#tag_craft2">2.</a>
+‘ben’ repeated in MS.</p>
+
+<p><a name = "note_craft3" id = "note_craft3" href =
+"#tag_craft3">3.</a>
+In MS. ‘thausandes.’</p>
+
+<p><a name = "note_craft4" id = "note_craft4" href =
+"#tag_craft4">4.</a>
+Perhaps “So.”</p>
+
+<p><a name = "note_craft5" id = "note_craft5" href =
+"#tag_craft5">5.</a>
+‘hali’ marked for erasure in MS.</p>
+
+<p><a name = "note_craft6" id = "note_craft6" href =
+"#tag_craft6">6.</a>
+‘moy’ in MS.</p>
+
+<p><a name = "note_craft7" id = "note_craft7" href =
+"#tag_craft7">7.</a>
+<ins class = "correction" title = "open quote missing">‘S</ins>ubt<i>ra</i>has a<i>u</i>t addis a dext<i>ri</i>s
+<i>ve</i>l medi<i>a</i>b<i>is</i>’ added on margin of&nbsp;MS.</p>
+
+<p><a name = "note_craft8" id = "note_craft8" href =
+"#tag_craft8">8.</a>
+After ‘craft’ insert ‘the .4. what is þe p<i>ro</i>fet of þis
+craft.’</p>
+
+<p><a name = "note_craft9" id = "note_craft9" href =
+"#tag_craft9">9.</a>
+After ‘sythes’ insert ‘&amp; þis wordes fyue sithe &amp; sex
+sythes.’</p>
+
+<p><a name = "note_craft10" id = "note_craft10" href =
+"#tag_craft10">10.</a>
+‘t’l’ marked for erasure before ‘tyl’ in MS.</p>
+
+<p><a name = "note_craft11" id = "note_craft11" href =
+"#tag_craft11">11.</a>
+Here ‘of þe same rew’ is marked for erasure in MS.</p>
+
+<p><a name = "note_craft12" id = "note_craft12" href =
+"#tag_craft12">12.</a>
+‘s<i>ed</i>’ deleted in MS.</p>
+
+<p><a name = "note_craft13" id = "note_craft13" href =
+"#tag_craft13">13.</a>
+6883 in MS.</p>
+
+<p><a name = "note_craft14" id = "note_craft14" href =
+"#tag_craft14">14.</a>
+‘þen’ overwritten on ‘þat’ marked for erasure.</p>
+
+<p><a name = "note_craft15" id = "note_craft15" href =
+"#tag_craft15">15.</a>
+‘Supra’ inserted in MS. in place of ‘cifra’ marked for erasure.</p>
+
+<p><a name = "note_craft16" id = "note_craft16" href =
+"#tag_craft16">16&ndash;16.</a>
+Marked for erasure in MS.</p>
+
+<p><a name = "note_craft17" id = "note_craft17" href =
+"#tag_craft17">17.</a>
+4 in MS.</p>
+
+<p><a name = "note_craft18" id = "note_craft18" href =
+"#tag_craft18">18.</a>
+docet. decet MS.</p>
+
+<p><a name = "note_craft19" id = "note_craft19" href =
+"#tag_craft19">19.</a>
+‘4 times 4’ in MS.</p>
+
+</div>
+
+</div> <!-- end div craft -->
+
+<div class = "art">
+
+<span class = "pagenum">33</span>
+<a name = "page33" id = "page33"> </a>
+
+<p class = "illustration">
+<a name = "art" id = "art">
+<img src = "images/title_art.png" width = "411" height = "108"
+alt = "The Art of Nombryng. / a translation of /
+John of Holywood’s De Arte Numerandi."
+title = "The Art of Nombryng. /a translation of /
+John of Holywood’s De Arte Numerandi." /></a></p>
+
+<hr class = "tiny" />
+
+<p class = "subhead">[<i>Ashmole MS. 396, fol. 48.</i>]</p>
+
+<p class = "inset"><b><span class = "dropcap">B</span>oys seying in the
+begynnyng of his <a class = "gloss" name = "arsemetrike" id =
+"arsemetrike" href =
+"#gloss_arsemetrike">Arsemetrik<i>e</i></a>:&mdash;All<i>e</i>
+<span class = "linenum">Fol. 48.</span>
+thynges that ben<i>e</i> fro the first begynnyng of thynges have
+p<i>ro</i>ceded<i>e</i>, and come forth<i>e</i>, And by reso<i>u</i>n of
+nombre ben formed<i>e</i>; And in wise as they ben<i>e</i>, So
+oweth<i>e</i> they to be knowen<i>e</i>; wherfor in
+vniu<i>er</i>sall<i>e</i> knowlechyng of thynges the Art of nombrynge is
+best, and most operatyf<i>e</i>.</b></p>
+
+<p><span class = "dropcap">T</span>herfore <a class = "gloss" name =
+"sithen" id = "sithen" href = "#gloss_sithen">sithen</a> the science of
+the whiche at this tyme we
+<span class = "sidenote">The name of the art.</span>
+intenden<i>e</i> to write of standith<i>e</i> all<i>e</i> and about
+nombre: ffirst we most se, what is the p<i>ro</i>pre name
+therof<i>e</i>, and fro whens the name come: Afterward<i>e</i> what is
+nombre, And how manye spices of nombre ther ben. The name is
+cleped<i>e</i> Algorisme,
+<span class = "sidenote">Derivation of Algorism.</span>
+had<i>e</i> out of Algor<i>e</i>, other of Algos, in <a class = "gloss"
+name = "grewe" id = "grewe" href = "#gloss_grewe">grewe</a>, That is
+clepid<i>e</i> in englissh<i>e</i> art <a class = "gloss" name = "other"
+id = "other" href = "#gloss_other">other</a> craft, And of
+Rithm<i>us</i> that is called<i>e</i> nombre. So <a class = "terms" name
+= "algorisme" id = "algorisme" href = "#terms_algorisme">algorisme</a>
+is cleped<i>e</i> the art of nombryng,
+<span class = "sidenote">Another.</span>
+other it is had of<i>e</i> en or in, and gogos that is
+introduccio<i>u</i>n, and Rithm<i>us</i> nombre, that is to say
+Interduccio<i>u</i>n of nombre.
+<span class = "sidenote">Another.</span>
+And thirdly it is had<i>e</i> of the name of a kyng that is
+cleped<i>e</i> Algo and Rythm<i>us</i>; So called<i>e</i>
+Algorism<i>us</i>.
+<span class = "sidenote">Kinds of numbers.</span>
+Sothely .2. maner<i>e</i> of nombres ben notified<i>e</i>;
+Formall<i>e</i>,<a class = "tag" name = "tag_art1" id = "tag_art1" href
+= "#note_art1">1</a>
+as nombr<i>e</i> i<i>s</i> vnitees gadred<i>e</i> to-gedres;
+Materiall<i>e</i>,<a class = "tag" name = "tag_art2" id = "tag_art2"
+href = "#note_art2">2</a>
+as nombr<i>e</i> is a colleccio<i>u</i>n of vnitees. Other nombr<i>e</i>
+is a multitude had<i>e</i> out of vnitees, vnitee is that thynge wher-by
+eu<i>er</i>y thynge is called<i>e</i> oone, other <a class = "gloss"
+name = "o1" id = "o1" href = "#gloss_oo">o</a> thynge. Of nombres, that
+one is cleped<i>e</i> <a class = "terms" name = "digitalle" id =
+"digitalle" href = "#terms_digit">digitall<i>e</i></a>, that
+other<i>e</i> Article, Another a nombre <a class = "terms" name =
+"componede" id = "componede" href =
+"#terms_componede">componed<i>e</i></a> oþ<i>er</i> myxt. Another
+digitall<i>e</i> is a nombre w<i>i</i>t<i>h</i>-in .10.; Article is
+þ<i>a</i>t nombre that may be dyvyded<i>e</i> in .10. p<i>ar</i>ties
+egally, And that there
+<span class = "pagenum">34</span>
+<a name = "page34" id = "page34"> </a>
+leve no residue; Componed<i>e</i> or <a class = "terms" name = "medlede"
+id = "medlede" href = "#terms_medlede">medled<i>e</i></a> is that nombre
+that is come of a digite and of an <a class = "terms" name = "article"
+id = "article" href = "#terms_article">article</a>. And
+vndrestand<i>e</i> wele that all<i>e</i> nombres betwix .2. articles
+next is a nombr<i>e</i> componed<i>e</i>.
+<span class = "sidenote">The 9 rules of the Art.</span>
+Of this art ben<i>e</i> .9. <a class = "gloss" name = "spices" id =
+"spices" href = "#gloss_spices">spices</a>, that is forto sey,
+num<i>er</i>acio<i>u</i>n, addicio<i>u</i>n, Subtraccio<i>u</i>n,
+Mediac<i>i</i>o<i>u</i>n, Duplacio<i>u</i>n, Multipliacio<i>u</i>n,
+Dyvysio<i>u</i>n, Progressio<i>u</i>n, And of Rootes the
+extraccio<i>u</i>n, and that may be had<i>e</i> in .2. maners, that is
+to sey in nombres quadrat, and in cubic<i>es</i>: Amonge the
+which<i>e</i>, ffirst of Num<i>er</i>acio<i>u</i>n, and
+aft<i>er</i>ward<i>e</i> of þe oþ<i>er</i>s by <a class = "gloss" name =
+"ordure" id = "ordure" href = "#gloss_ordure">ordure</a>, y&nbsp;entende
+to write.</p>
+
+<p class = "headnote"><span class = "headnote">
+Chapter I. Numeration.</span></p>
+
+<span class = "linenum">Fol. 48 <i>b</i>.</span>
+<h5>*For-soth<i>e</i> num<i>er</i>acio<i>u</i>n is of eu<i>er</i>y
+numbre by competent figures an artificiall<i>e</i>
+rep<i>re</i>sentacio<i>u</i>n.</h5>
+
+<p><span class = "sidenote">Figures, differences, places, and
+limits.</span>
+Sothly figure, difference, places, and <a class = "gloss" name = "lynes"
+id = "lynes" href = "#gloss_lymytes">lynes</a> supposen o thyng other
+the same, But they ben sette here for dyue<i>r</i>s resons. ffigure is
+cleped<i>e</i> for p<i>ro</i>traccio<i>u</i>n of figuracio<i>u</i>n;
+Difference is called<i>e</i> for therby is shewed<i>e</i> eu<i>er</i>y
+figure, how it hath<i>e</i> difference fro the figures before them:
+place by cause of space, where-in <a class = "gloss" name = "me1" id =
+"me1" href = "#gloss_me">me</a> writeth<i>e</i>: <a class = "gloss" name
+= "lynees" id = "lynees" href = "#gloss_lymytes">lynees</a>, for that is
+ordeyned<i>e</i> for the p<i>re</i>sentacio<i>u</i>n of eu<i>er</i>y
+figure.
+<span class = "sidenote">The 9 figures.</span>
+And vnderstonde that ther ben .9. <a class = "gloss" name = "lymytes" id
+= "lymytes" href = "#gloss_lymytes">lymytes</a> of figures that
+rep<i>re</i>senten the .9. digit<i>es</i> that ben these. 0. 9. 8. 7. 6.
+5. 4. 3. 2. 1.
+<span class = "sidenote">The cipher.</span>
+The .10. is cleped<i>e</i> theta, or a cercle, other a cifre, other a
+figure of nought for nought it signyfieth<i>e</i>. Nathelesse she
+holdyng that place giveth<i>e</i> others for to signyfie; for
+with<i>e</i>-out cifre or cifres a pure article may not be writte.
+<span class = "sidenote">The numeration</span>
+And sithen that by these .9. figures significatif<i>es</i>
+Ioyned<i>e</i> w<i>i</i>t<i>h</i> cifre or w<i>i</i>t<i>h</i> cifres
+all<i>e</i> nombres ben and may be rep<i>re</i>sented<i>e</i>, It was,
+<a class = "gloss" name = "nether" id = "nether" href =
+"#gloss_nether">nether</a> is, no nede to fynde any more figures.
+<span class = "sidenote">of digits,</span>
+And note wele that eu<i>er</i>y digite shall<i>e</i> be writte
+w<i>i</i>t<i>h</i> <a class = "gloss" name = "oo1" id = "oo1" href =
+"#gloss_oo">oo</a> figure allone to it <a class = "gloss" name =
+"aproprede" id = "aproprede" href =
+"#gloss_aproprede">ap<i>ro</i>pred<i>e</i></a>.
+<span class = "sidenote">of articles,</span>
+And all<i>e</i> articles by a cifre, ffor eu<i>er</i>y article is
+named<i>e</i> for oone of the digitis as <ins class = "correction" title
+= "punctuated as shown: error for ‘.10. of .1. 20. of .2.’?">.10. of 1..
+20. of. 2.</ins> and so of the others, &amp;c. And all<i>e</i> nombres
+digitall<i>e</i> owen to be sette in the first difference: All<i>e</i>
+articles in the seconde. Also all<i>e</i> nombres fro .10. til an .100.
+[which] is excluded<i>e</i>, with .2. figures mvst be writte; And yf it
+be an article, by a cifre first put, and the figure y-writte
+toward<i>e</i> the lift hond<i>e</i>, that signifieth<i>e</i> the digit
+of the which<i>e</i> the article is named<i>e</i>;
+<span class = "sidenote">of composites.</span>
+And yf it be a nombre componed<i>e</i>, ffirst write the digit that is a
+part of that componed<i>e</i>, and write to the lift side the article as
+it is seid<i>e</i> be-fore. All<i>e</i> nombre that is fro an
+hundred<i>e</i> tille a thousand<i>e</i> <a class = "gloss" name =
+"exclusede" id = "exclusede" href =
+"#gloss_exclusede">exclused<i>e</i></a>, owith<i>e</i> to be writ by .3.
+figures; and all<i>e</i> nombre that is fro a thousand<i>e</i>
+<span class = "pagenum">35</span>
+<a name = "page35" id = "page35"> </a>
+til .x. Mł. mvst be writ by .4. figures; And so forthe.
+<span class = "sidenote">The value due to position.</span>
+And vnderstond<i>e</i> wele that eu<i>er</i>y figure sette in the first
+place signyfieth<i>e</i> his digit; In the second<i>e</i> place .10.
+tymes his digit; In the .3. place an hundred<i>e</i> so moche; In the
+.4. place a thousand<i>e</i> so moche; In the .5. place .x.
+thousand<i>e</i> so moch<i>e</i>; In the .6. place an hundred<i>e</i>
+thousand<i>e</i> so moch<i>e</i>; In the .7. place a thousand<i>e</i>
+thousand<i>e</i>. And so infynytly mvltiplying by
+<span class = "linenum">Fol. 49.</span>
+*these .3. 10, 100, 1000. And vnderstand<i>e</i> wele that <a class =
+"gloss" name = "competently" id = "competently" href =
+"#gloss_competently">competently</a> me may sette vpon figure in the
+place of a thousand<i>e</i>, a&nbsp;prik<i>e</i> to shewe how many
+thousand<i>e</i> the last figure shall<i>e</i> rep<i>re</i>sent.
+<span class = "sidenote">Numbers are written from right to left.</span>
+We writen<i>e</i> in this art to the lift side-ward<i>e</i>, as
+arabien<i>e</i> writen<i>e</i>, that weren fynders of this science,
+other<i>e</i> for this reso<i>u</i>n, that for to kepe a custumable
+ordr<i>e</i> in redyng, Sette we all<i>e</i>-wey the more nombre
+before.</p>
+
+<p class = "headnote"><span class = "headnote">
+Chapter II. Addition.</span></p>
+
+<p><span class = "sidenote">Definition.</span>
+<span class = "dropcap">A</span>ddicio<i>u</i>n is of nombre other of
+nombres vnto nombre or to nombres aggregacio<i>u</i>n, that me may see
+that that is come therof as <a class = "gloss" name = "excressent" id =
+"excressent" href = "#gloss_excressent">exc<i>re</i>ssent</a>. In
+addicio<i>u</i>n, 2. ordres of figures and .2. nombres ben necessary,
+that is to sey, a&nbsp;nombre to be added<i>e</i> and the nombre wherto
+the addic<i>i</i>oun shold<i>e</i> be made to. The nombre to be
+added<i>e</i> is that þat shold<i>e</i> be added<i>e</i> therto, and
+shall<i>e</i> be vnderwriten; the nombre vnto the which<i>e</i>
+addicio<i>u</i>n shall<i>e</i> be made to is that nombre that
+resceyueth<i>e</i> the addicion of þat other, and shall<i>e</i> be
+writen above;
+<span class = "sidenote">How the numbers should be written.</span>
+and it is convenient that the lesse nombre be vnderwrit, and the more
+added<i>e</i>, than the contrary. But whether it happ<i>e</i> one
+<a class = "gloss" name = "other1" id = "other1" href =
+"#gloss_other">other</a> other, the same comyth<i>e</i> of, Therfor, yf
+þow wilt adde nombre to nombre, write the nombre wherto the
+addicio<i>u</i>n shall<i>e</i> be made in the <a class = "gloss" name =
+"omest" id = "omest" href = "#gloss_omest">omest</a> ordre by his
+differences, so that the first of the lower ordre be vndre the first of
+the <a class = "gloss" name = "omyst" id = "omyst" href =
+"#gloss_omest">omyst</a> ordre, and so of others.
+<span class = "sidenote">The method of working.</span>
+That done, adde the first of the lower ordre to the first of the omyst
+ordre. And of such<i>e</i> addicio<i>u</i>n, other þ<i>er</i>e
+grow<i>i</i>t<i>h</i> therof a digit, An article, other a
+composed<i>e</i>.
+<span class = "sidenote">Begin at the right.</span>
+If it be digit<i>us</i>, In the place of the omyst shalt thow write the
+digit excrescyng, as thus:&mdash;</p>
+
+<table class = "grid outline float" summary = "example">
+<tr>
+<td class = "words">The resultant</td>
+<td>2</td>
+</tr>
+<tr>
+<td class = "words">To whom it shal be added<i>e</i></td>
+<td>1</td>
+</tr>
+<tr>
+<td class = "words">The nombre to be added<i>e</i></td>
+<td>1</td>
+</tr>
+</table>
+
+<p><span class = "sidenote">The Sum is a digit,</span>
+If the article; in the place of the omyst put a-way by a cifre writte,
+and the digit transferred<i>e</i>, of þe which<i>e</i> the article toke
+his name, toward<i>e</i> the lift side, and be it added<i>e</i> to the
+next figure folowyng, yf ther be any figure folowyng; or no, and yf it
+be not, leve it [in the] void<i>e</i>, as thus:&mdash;</p>
+
+<span class = "pagenum">36</span>
+<a name = "page36" id = "page36"> </a>
+<span class = "sidenote">or an article,</span>
+
+<table class = "grid outline float" summary = "example">
+<tr>
+<td class = "words">The resultant</td>
+<td>10</td>
+</tr>
+<tr>
+<td class = "words">To whom it shall<i>e</i> be added<i>e</i></td>
+<td>7</td>
+</tr>
+<tr>
+<td class = "words">The nombre to be added<i>e</i></td>
+<td>3</td>
+</tr>
+</table>
+
+<table class = "grid outline floatleft" summary = "example">
+<tr>
+<td class = "words">Resultans</td>
+<td>2</td>
+<td>7</td>
+<td>8</td>
+<td>2</td>
+<td>7</td>
+</tr>
+<tr>
+<td class = "words">Cui d<i>ebet</i> addi</td>
+<td>1</td>
+<td>0</td>
+<td>0</td>
+<td>8</td>
+<td>4</td>
+</tr>
+<tr>
+<td class = "words">Num<i>erus</i> addend<i>us</i></td>
+<td>1</td>
+<td>7</td>
+<td>7</td>
+<td>4</td>
+<td>3</td>
+</tr>
+</table>
+
+<p class = "allclear">
+And yf it happe that the figure folowyng wherto the addicio<i>u</i>n
+shall<i>e</i> be made by [the cifre of] an article, it sette a-side;</p>
+
+<table class = "grid outline float" summary = "example">
+<tr>
+<td class = "words">The resultant</td>
+<td>17</td>
+</tr>
+<tr>
+<td class = "words">To whom it shall<i>e</i> be added<i>e</i></td>
+<td>10</td>
+</tr>
+<tr>
+<td class = "words">The nombre to be added<i>e</i></td>
+<td>7</td>
+</tr>
+</table>
+
+<p class = "nospace">
+In his place write the
+<span class = "linenum">Fol. 49 <i>b</i>.</span>
+*[digit of the] Article as thus:&mdash;</p>
+
+<p class = "allclear">
+And yf it happe that a figure of .9. by the figure that me mvst adde
+[one] to,</p>
+
+<table class = "grid outline float" summary = "example">
+<tr>
+<td class = "words">The resultant</td>
+<td>10</td>
+</tr>
+<tr>
+<td class = "words">To whom it shall<i>e</i> be added<i>e</i></td>
+<td>9</td>
+</tr>
+<tr>
+<td class = "words">The nombre to be added<i>e</i></td>
+<td>1</td>
+</tr>
+</table>
+
+<p class = "nospace">
+In the place of that 9. put a cifre <i>and</i> write þe article
+toward<i>e</i> þe lift hond<i>e</i> as bifore, and thus:&mdash;</p>
+
+<p class = "allclear">
+<span class = "sidenote">or a composite.</span>
+And yf<a class = "tag" name = "tag_art3" id = "tag_art3" href =
+"#note_art3">3</a>
+[therefrom grow a] nombre componed,<a class = "tag" name = "tag_art4" id
+= "tag_art4" href = "#note_art4">4</a>
+[in the place of the nombre] put a-way<a class = "tag" name = "tag_art5"
+id = "tag_art5" href = "#note_art5">5</a></p>
+
+<table class = "grid outline float" summary = "example">
+<tr>
+<td class = "words">The resultant</td>
+<td>12</td>
+</tr>
+<tr>
+<td class = "words">To whom it shall<i>e</i> be added<i>e</i></td>
+<td>8</td>
+</tr>
+<tr>
+<td class = "words">The nombre to be added<i>e</i></td>
+<td>4</td>
+</tr>
+</table>
+
+<p class = "nospace">
+[let] the digit [be]<a class = "tag" name = "tag_art6" id = "tag_art6"
+href = "#note_art6">6</a>
+writ þ<i>a</i>t is part of þ<i>a</i>t co<i>m</i>posid<i>e</i>, and þan
+put to þe lift side the article as before, and þus:&mdash;</p>
+
+<p class = "allclear">
+<span class = "sidenote">The translator’s note.</span>
+This done, adde the seconde to the second<i>e</i>, and write above
+oþ<i>er</i> as before. Note wele þ<i>a</i>t in addic<i>i</i>ons and in
+all<i>e</i> spices folowyng, whan he seith<i>e</i> one the other
+shall<i>e</i> be writen aboue, and me most vse eu<i>er</i> figure, as
+that eu<i>er</i>y figure were sette by half<i>e</i>, and by
+hym-self<i>e</i>.</p>
+
+<p class = "headnote"><span class = "headnote">
+Chapter III. Subtraction.</span></p>
+
+<p><span class = "sidenote">Definition of Subtraction.</span>
+<span class = "dropcap">S</span>ubtraccio<i>u</i>n is of .2.
+p<i>ro</i>posed<i>e</i> nombres, the fyndyng of the excesse of the more
+to the lasse: Other subtraccio<i>u</i>n is <a class = "gloss" name =
+"ablacioun" id = "ablacioun" href =
+"#gloss_ablacioun">ablacio<i>u</i>n</a> of o nombre fro a-nother, that
+me may see a <a class = "gloss" name = "some1" id = "some1" href =
+"#gloss_some">some</a> left. The lasse of the more, or even of even, may
+be w<i>i</i>t<i>h</i>draw; The more fro the lesse may neu<i>er</i> be.
+<span class = "sidenote">How it may be done.</span>
+And sothly that nombre is more that hath<i>e</i> more figures, So that
+the last be signyficatife<i>s</i>: And yf ther ben as many in that one
+as in that other, me most deme it by the last, other by the next last.
+<span class = "sidenote">What is required.</span>
+More-ou<i>er</i> in w<i>i</i>t<i>h</i>-drawyng .2. nombres ben
+necessary; A&nbsp;nombre to be w<i>i</i>t<i>h</i>draw, And a nombre that
+me shall<i>e</i> w<i>i</i>t<i>h</i>-draw of. The nombre to be
+w<i>i</i>t<i>h</i>-draw shall<i>e</i> be writ in the lower ordre by his
+differences;
+<span class = "sidenote">Write the greater number above.</span>
+The
+<span class = "pagenum">37</span>
+<a name = "page37" id = "page37"> </a>
+nombre fro the which<i>e</i> me shall<i>e</i> with<i>e</i>-draw in the
+omyst ordre, so that the first be vnder the first, the second<i>e</i>
+vnder the second<i>e</i>, And so of all<i>e</i> others.
+<span class = "sidenote">Subtract the first figure if possible.</span>
+With<i>e</i>-draw therfor the first of the lower<i>e</i> ordre fro the
+first of the ordre above his hede, and that wolle be <a class = "gloss"
+name = "other_or1" id = "other_or1" href = "#gloss_other">other</a> more
+or lesse, oþ<i>er</i> egall<i>e</i>.</p>
+
+<table class = "grid outline float" summary = "example">
+<tr>
+<td class = "words">The remanent</td>
+<td>20</td>
+</tr>
+<tr>
+<td class = "words">Wherof me shall<i>e</i> w<i>i</i>t<i>h</i>draw</td>
+<td>22</td>
+</tr>
+<tr>
+<td class = "words">The nombre to be w<i>i</i>t<i>h</i>draw</td>
+<td>2</td>
+</tr>
+</table>
+
+<table class = "grid outline float" summary = "example">
+<tr>
+<td class = "words">The remanent</td>
+<td>2</td>
+<td>2</td>
+</tr>
+<tr>
+<td class = "words">Wherof me shall<i>e</i> w<i>i</i>t<i>h</i>-draw</td>
+<td>2</td>
+<td>8</td>
+</tr>
+<tr>
+<td class = "words">Þe nombre to be w<i>i</i>t<i>h</i>draw</td>
+<td></td>
+<td>6</td>
+</tr>
+</table>
+
+<p class = "nospace">
+yf it be egall<i>e</i> or even the figure sette beside, put in his place
+a cifre. And yf it be more put away þ<i>er</i>fro als many of vnitees
+the lower figure conteyneth<i>e</i>, and writ the residue as thus</p>
+
+<span class = "linenum">Fol. 50.</span>
+
+<table class = "grid outline floatleft" summary = "example">
+<tr>
+<td class = "words">*Remane<i>n</i>s</td>
+<td>2</td>
+<td>2</td>
+<td>1</td>
+<td>8</td>
+<td>2</td>
+<td>9</td>
+<td>9</td>
+<td>9</td>
+<td>8</td>
+</tr>
+<tr>
+<td class = "words"><p>A quo sit subtraccio</p></td>
+<td>8</td>
+<td>7</td>
+<td>2</td>
+<td>4</td>
+<td>3</td>
+<td>0</td>
+<td>0</td>
+<td>0</td>
+<td>4</td>
+</tr>
+<tr>
+<td class = "words"><p>Numerus subt<i>ra</i>hend<i>us</i></p></td>
+<td>6</td>
+<td>5</td>
+<td><a class = "tag" name = "tag_art7" id = "tag_art7" href =
+"#note_art7">7</a></td>
+<td>[6]</td>
+<td>.</td>
+<td>.</td>
+<td>.</td>
+<td>.</td>
+<td>6</td>
+</tr>
+</table>
+
+<p class = "rightclear">
+<span class = "sidenote">If it is not possible ‘borrow ten,’</span>
+And yf it be lesse, by-cause the more may not be w<i>i</i>t<i>h</i>-draw
+ther-fro, borow an vnyte of the next figure that is worth<i>e</i> 10. Of
+that .10. and of the figure that ye wold<i>e</i> have
+w<i>i</i>t<i>h</i>-draw fro
+<span class = "sidenote">and then subtract.</span>
+be-fore to-gedre Ioyned<i>e</i>,</p>
+
+<table class = "grid outline float" summary = "example">
+<tr>
+<td class = "words">The remanent</td>
+<td>1</td>
+<td>8</td>
+</tr>
+<tr>
+<td class = "words">Wherof me shall<i>e</i> w<i>i</i>t<i>h</i>-draw</td>
+<td>2</td>
+<td>4</td>
+</tr>
+<tr>
+<td class = "words">The nombre to be w<i>i</i>t<i>h</i>-draw</td>
+<td>0</td>
+<td>6</td>
+</tr>
+</table>
+
+<p class = "nospace">
+w<i>i</i>t<i>h</i>-draw þe figure be-nethe, and put the residue in the
+place of the figure put a-side as þ<i>us</i>:&mdash;</p>
+
+<p><span class = "sidenote">If the second figure is one.</span>
+And yf the figure wherof me shal borow the vnyte be one, put it a-side,
+and write a cifre in the place þ<i>er</i>of, lest the figures folowing
+faile of thair<i>e</i> nombre, and þan worch<i>e</i> as it
+shew<i>i</i>t<i>h</i> in this figure here:&mdash;</p>
+
+<table class = "grid outline float" summary = "example">
+<tr>
+<td class = "words">The remanent</td>
+<td>3</td>
+<td>0</td>
+<td>9<a class = "tag" name = "tag_art8" id = "tag_art8" href =
+"#note_art8">8</a></td>
+</tr>
+<tr>
+<td class = "words">Wherof me shal w<i>i</i>t<i>h</i>-draw</td>
+<td>3</td>
+<td>1</td>
+<td>2</td>
+</tr>
+<tr>
+<td class = "words">The nombre to be w<i>i</i>t<i>h</i>-draw</td>
+<td>.</td>
+<td>.</td>
+<td>3</td>
+</tr>
+</table>
+
+<p><span class = "sidenote">If the second figure is a cipher.</span>
+And yf the vnyte wherof me shal borow be a cifre, go ferther to the
+figure signyficatif<i>e</i>, and ther borow one, and reto<i>ur</i>nyng
+bak<i>e</i>, in the place of eu<i>er</i>y cifre þ<i>a</i>t ye
+passid<i>e</i> ou<i>er</i>, sette figures of .9. as here it is
+specified<i>e</i>:&mdash;</p>
+
+<table class = "grid outline float" summary = "example">
+<tr>
+<td class = "words">The remenaunt</td>
+<td>2</td>
+<td>9</td>
+<td>9</td>
+<td>9</td>
+<td>9</td>
+</tr>
+<tr>
+<td class = "words">Wherof me shall<i>e</i> w<i>i</i>t<i>h</i>-draw</td>
+<td>3</td>
+<td>0</td>
+<td>0</td>
+<td>0</td>
+<td>3</td>
+</tr>
+<tr>
+<td class = "words">The nombre to be w<i>i</i>t<i>h</i>-draw</td>
+<td></td>
+<td></td>
+<td></td>
+<td></td>
+<td>4</td>
+</tr>
+</table>
+
+<p>And whan me cometh<i>e</i> to the nombre wherof me intendith<i>e</i>,
+there remayneth<i>e</i> all<i>e</i>-wayes .10. ffor þe which<i>e</i>
+.10. &amp;c.
+<span class = "sidenote">A&nbsp;justification of the rule given.</span>
+The reson why þat for eu<i>er</i>y cifre left behynde me setteth figures
+ther of .9. this it is:&mdash;If fro the .3. place me borowed<i>e</i> an
+vnyte, that vnyte by respect of the figure that he came fro
+rep<i>re</i>sentith an .C., In the
+<span class = "pagenum">38</span>
+<a name = "page38" id = "page38"> </a>
+place of that cifre [passed over] is left .9., [which is worth ninety],
+and yit it remayneth<i>e</i> as .10., And the same reson<i>e</i>
+wold<i>e</i> be yf me had<i>e</i> borowed<i>e</i> an vnyte fro the .4.,
+.5., .6., place, or ony other so vpward<i>e</i>. This done, withdraw the
+second<i>e</i> of the lower ordre fro the figure above his hede of þe
+omyst ordre, and wirch<i>e</i> as before.
+<span class = "sidenote">Why it is better to work from right to
+left.</span>
+And note wele that in addicion or in subtracc<i>i</i>o<i>u</i>n me may
+wele fro the lift side begynne and ryn to the right side, But it wol be
+more p<i>ro</i>fitabler to be do, as it is taught.
+<span class = "sidenote">How to prove subtraction,</span>
+And yf thow wilt p<i>ro</i>ve yf thow have do wele or no, The figures
+that thow hast withdraw, adde them <a class = "gloss" name = "ayene" id
+= "ayene" href = "#gloss_ayene">ayene</a> to the omyst figures, and they
+wolle accorde w<i>i</i>t<i>h</i> the first that thow haddest yf thow
+have labored wele;
+<span class = "sidenote">and addition.</span>
+and in like wise in addicio<i>u</i>n, whan thow hast added<i>e</i>
+all<i>e</i> thy figures, w<i>i</i>t<i>h</i>draw them that thow first
+<span class = "linenum">Fol. 50 <i>b</i>.</span>
+*addest, and the same wolle reto<i>ur</i>ne. The subtraccio<i>u</i>n is
+none other but a p<i>ro</i>uff<i>e</i> of the addicio<i>u</i>n, and the
+contrarye in like wise.</p>
+
+<p class = "headnote"><span class = "headnote">
+Chapter IV. Mediation.</span></p>
+
+<p><span class = "sidenote">Definition of mediation.</span>
+<span class = "dropcap">M</span><a class = "terms" name = "mediacioun2"
+id = "mediacioun2" href = "#terms_mediacioun">ediacio<i>u</i>n</a> is
+the fyndyng of the halfyng of eu<i>er</i>y nombre, that it may be
+seyn<i>e</i> what and how moch<i>e</i> is eu<i>er</i>y half<i>e</i>. In
+halfyng ay oo order of figures and oo nombre is necessary, that is to
+sey the nombre to be halfed<i>e</i>. Therfor yf thow wilt half any
+nombre, write that nombre by his differences, and
+<span class = "sidenote">Where to begin.</span>
+begynne at the right, that is to sey, fro the first figure to the right
+side, so that it be signyficatif<i>e</i> other rep<i>re</i>sent vnyte or
+eny other digitall<i>e</i> nombre. If it be vnyte write in his place a
+cifre for the
+<span class = "sidenote">If the first figure is unity.</span>
+figures folowyng, [lest they signify less], and write that vnyte
+w<i>i</i>t<i>h</i>out in the table, other resolue it in .60. <a class =
+"gloss" name = "mynvtes" id = "mynvtes" href =
+"#gloss_mynvtes">mynvt<i>es</i></a> and sette a-side half of <ins class
+= "correction" title = "error for the?">tho</ins> m<i>inutes</i> so, and
+reserve the remen<i>au</i>nt w<i>i</i>t<i>h</i>out in the table, as thus
+.30.; other sette w<i>i</i>t<i>h</i>out thus .<i>dī</i>: that
+kepeth<i>e</i> none ordre of place, Nathelesse it hath<i>e</i>
+signyficacio<i>u</i>n. And yf the other figure signyfie any other
+digital nombre fro vnyte forth<i>e</i>, oþ<i>er</i> the nombre is
+od<i>e</i> or even<i>e</i>.</p>
+
+<table class = "grid outline float" summary = "example">
+<tr>
+<td class = "words">Halfed<i>e</i></td>
+<td>2</td>
+<td>2</td>
+</tr>
+<tr>
+<td class = "words">to be halfed<i>e</i></td>
+<td>4</td>
+<td>4</td>
+</tr>
+</table>
+
+<table class = "grid outline float" summary = "example">
+<tr>
+<td class = "words">halfed<i>e</i></td>
+<td>2</td>
+<td>3</td>
+<td class = "edge">[di]</td>
+</tr>
+<tr>
+<td class = "words">To be halfed<i>e</i></td>
+<td>4</td>
+<td>7</td>
+</tr>
+</table>
+
+<p class = "nospace">
+<span class = "sidenote">What to do if it is not unity.</span>
+If it be even, write this half in this wise:&mdash;</p>
+
+<p>And if it be odde, Take the next even vndre hym conteyned<i>e</i>,
+and put his half in the place of that odde, and of þe vnyte that
+remayneth<i>e</i> to be halfed<i>e</i> do thus:&mdash;</p>
+
+<p><span class = "sidenote">Then halve the second figure.</span>
+This done, the second<i>e</i> is to be halfed<i>e</i>, yf it be a cifre
+put it be-side, and yf it be significatif<i>e</i>, <a class = "gloss"
+name = "other_or" id = "other_or" href = "#gloss_other">other it is even
+or od<i>e</i></a>: If it be even, write in the place of þe nombres
+wiped<i>e</i> out the half<i>e</i>; yf it be od<i>e</i>, take the next
+even vnder it <a class = "gloss" name = "contenythe" id = "contenythe"
+href = "#gloss_contynes">co<i>n</i>tenyth<i>e</i></a>, and in the place
+of the Impar sette a-side put half of the even: The
+<span class = "pagenum">39</span>
+<a name = "page39" id = "page39"> </a>
+vnyte that remayneth<i>e</i> to be halfed<i>e</i>, respect had<i>e</i>
+to them before, is worth<i>e</i> .10.</p>
+
+<table class = "grid outline float" summary = "example">
+<tr>
+<td class = "words">Halfed<i>e</i></td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td class = "words">to be halfed<i>e</i></td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+</table>
+
+<p class = "nospace">
+<span class = "sidenote">If it is odd, add 5 to the figure
+before.</span>
+Dyvide that .10. in .2., 5. is, and sette a-side that one, and adde that
+other to the next figure p<i>re</i>cedent as here:&mdash;</p>
+
+<p>And yf þe addicio<i>u</i>n shold<i>e</i> be made to a cifre, sette it
+a-side, and write in his place .5.</p>
+
+<table class = "grid outline float" summary = "example">
+<tr>
+<td class = "words">doubled<i>e</i></td>
+<td>2</td>
+<td>6</td>
+<td>8</td>
+<td>9</td>
+<td>0</td>
+<td>10</td>
+<td>17</td>
+<td>4</td>
+</tr>
+<tr>
+<td class = "words">to be doubled<i>e</i></td>
+<td>1</td>
+<td>3</td>
+<td>4</td>
+<td>4</td>
+<td>5</td>
+<td>5</td>
+<td>8</td>
+<td>7</td>
+</tr>
+</table>
+
+<p class = "nospace">
+And vnder this fo<i>ur</i>me me shall<i>e</i> write and worch<i>e</i>,
+till<i>e</i> the totall<i>e</i> nombre be halfed<i>e</i>.</p>
+
+<p class = "headnote"><span class = "headnote">
+Chapter V. Duplation.</span></p>
+
+<p><span class = "sidenote">Definition of Duplation.</span>
+<span class = "dropcap">D</span>uplicacio<i>u</i>n is ag<i>re</i>gacion
+of nombre [to itself] þat me may se the nombre growen. In
+doublyng<i>e</i> ay is but one ordre of figures necessarie. And me most
+be-gynne w<i>i</i>t<i>h</i> the lift side, other of the more figure, And
+after the nombre of the more figure <a class = "gloss" name =
+"representithe" id = "representithe" href =
+"#gloss_representithe">rep<i>re</i>sentith<i>e</i></a>.
+<span class = "linenum">Fol. 51.</span>
+*In the other .3. before we begynne all<i>e</i> way fro the right side
+and fro the lasse nombre,
+<span class = "sidenote">Where to begin.</span>
+In this spice and in all<i>e</i> other folowyng we wolle begynne fro the
+lift side, ffor and me bigon th<i>e</i> double fro the first, <a class =
+"gloss" name = "omwhile1" id = "omwhile1" href =
+"#gloss_omwhile">omwhile</a> me myght double oo thynge twyes.
+<span class = "sidenote">Why.</span>
+And how be it that me myght double fro the right, that wold<i>e</i> be
+harder in techyng and in workyng. Therfor yf thow wolt double any
+nombre, write that nombre by his differences, and double the last. And
+of that doubly<i>n</i>g other growith<i>e</i> a nombre digital, article,
+or componed<i>e</i>. [If it be a digit, write it in the place of the
+first digit.]</p>
+
+<table class = "grid outline float" summary = "example">
+<tr>
+<td class = "words">double</td>
+<td>10</td>
+</tr>
+<tr>
+<td class = "words">to be doubled<i>e</i></td>
+<td>5</td>
+</tr>
+</table>
+
+<p class = "nospace">
+<span class = "sidenote">What to do with the result.</span>
+If it be article, write in his place a cifre and transferre the article
+toward<i>e</i> the lift, as thus:&mdash;</p>
+
+<p>And yf the nombre be componed<i>e</i>,</p>
+
+<table class = "grid outline float" summary = "example">
+<tr>
+<td class = "words">doubled<i>e</i></td>
+<td>16</td>
+</tr>
+<tr>
+<td class = "words">to be doubled<i>e</i></td>
+<td>8</td>
+</tr>
+</table>
+
+<p class = "nospace">
+write a digital that is part of his composicio<i>u</i>n, and sette the
+article to the lift hand<i>e</i>, as thus:&mdash;</p>
+
+<p>That done, me most double the last save one, and what groweth<i>e</i>
+þ<i>er</i>of me most worche as before. And yf a cifre be, touch<i>e</i>
+it not. But yf any nombre shall<i>e</i> be added<i>e</i> to the
+cifre,</p>
+
+<table class = "grid outline float" summary = "example">
+<tr>
+<td class = "words">doubled<i>e</i></td>
+<td>6</td>
+<td>0</td>
+<td>6</td>
+</tr>
+<tr>
+<td class = "words">to be doubled<i>e</i></td>
+<td>3</td>
+<td>0</td>
+<td>3</td>
+</tr>
+</table>
+
+<p class = "nospace">
+in þe place of þe figure wiped<i>e</i> out me most write the nombre to
+be added<i>e</i>, as thus:&mdash;</p>
+
+<p>In the same wise me shall<i>e</i> wirch<i>e</i> of all<i>e</i>
+others.
+<span class = "sidenote">How to prove your answer.</span>
+And this p<i>ro</i>bacio<i>u</i>n:</p>
+
+<table class = "grid outline float" summary = "example">
+<tr>
+<td class = "words">Doubled<i>e</i></td>
+<td>6</td>
+<td>1</td>
+<td>8</td>
+</tr>
+<tr>
+<td class = "words">to be doubled<i>e</i></td>
+<td>3</td>
+<td>0</td>
+<td>9</td>
+</tr>
+</table>
+
+<p class = "nospace">
+If thow truly double the halfis, and truly half the doubles, the same
+nombre and figure shall<i>e</i> mete, such<i>e</i> as thow
+labo<i>ur</i>ed<i>e</i> vpon<i>e</i> first, And of the contrarie.</p>
+
+<span class = "pagenum">40</span>
+<a name = "page40" id = "page40"> </a>
+<p class = "headnote"><span class = "headnote">
+Chapter VI. Multiplication.</span></p>
+
+<p><span class = "sidenote">Definition of Multiplication.</span>
+<span class = "dropcap">M</span>ultiplicacio<i>u</i>n of nombre by
+hym-self other by a-nother, w<i>i</i>t<i>h</i> p<i>ro</i>posid<i>e</i>
+.2. nombres, [is] the fyndyng of the third<i>e</i>, That so oft
+conteyneth<i>e</i> that other, as ther ben vnytes in the oþ<i>er</i>. In
+multiplicacio<i>u</i>n .2. nombres pryncipally ben necessary, that is to
+sey, the nombre multiplying and the nombre to be multiplied<i>e</i>, as
+here;&mdash;twies fyve.
+<span class = "sidenote">Multiplier.</span>
+[The number multiplying] is designed<i>e</i> adu<i>er</i>bially.
+<span class = "sidenote">Multiplicand.</span>
+The nombre to be multiplied<i>e</i> resceyveth<i>e</i> a
+no<i>m</i>i<i>n</i>all<i>e</i> appellacio<i>u</i>n, as twies .5. 5. is
+the nombre multiplied<i>e</i>, and twies is the nombre to be <a class =
+"gloss" name = "multipliede" id = "multipliede" href =
+"#gloss_multipliede">multipliede</a>.</p>
+
+<table class = "grid outline" summary = "example">
+<tr>
+<td class = "words">Resultans</td>
+<td><a class = "tag" name = "tag_art9" id = "tag_art9" href =
+"#note_art9">9</a></td>
+<td>1</td>
+<td>0</td>
+<td class = "double">1</td>
+<td>3</td>
+<td>2</td>
+<td>6</td>
+<td>6</td>
+<td>8</td>
+<td>0</td>
+<td>0</td>
+<td>8</td>
+</tr>
+<tr>
+<td class = "words">Multiplicand<i>us</i></td>
+<td>.</td>
+<td>.</td>
+<td>5</td>
+<td class = "double">.</td>
+<td>.</td>
+<td>4</td>
+<td>.</td>
+<td>3</td>
+<td>4</td>
+<td>0</td>
+<td>0</td>
+<td>4</td>
+</tr>
+<tr>
+<td class = "words">Multiplicans</td>
+<td>.</td>
+<td>2</td>
+<td>2</td>
+<td class = "double">.</td>
+<td>3</td>
+<td>3</td>
+<td>2</td>
+<td>2</td>
+<td>2</td>
+<td>.</td>
+<td>.</td>
+<td>.</td>
+</tr>
+</table>
+
+<p><span class = "sidenote">Product.</span>
+Also me may thervpon<i>e</i> to assigne the. 3. nombre, the
+which<i>e</i> is
+<span class = "linenum">Fol. 51 <i>b</i>.</span>
+*cleped<i>e</i> p<i>ro</i>duct or p<i>ro</i>venient, of takyng out of
+one fro another: as twyes .5 is .10., 5. the nombre to be
+multiplied<i>e</i>, and .2. the multipliant, <ins class = "correction"
+title = "error for ‘and .10.’?">and. 10.</ins> as before is come therof.
+And vnderstonde wele, that of the multipliant may be made the nombre to
+be multiplied<i>e</i>, and of the contrarie, remaynyng eu<i>er</i> the
+same <a class = "gloss" name = "some" id = "some" href =
+"#gloss_some">some</a>, and herof<i>e</i> cometh<i>e</i> the comen
+speche, that seith<i>e</i> all nombre is converted<i>e</i> by
+Multiplying in hym-self<i>e</i>.
+<span class = "headnote float">
+The Cases of Multiplication.</span></p>
+
+<span class = "sidenote">There are 6 rules of Multiplication.</span>
+
+<table class = "grid outline float" summary = "example">
+<tr>
+<td>1</td>
+<td>2</td>
+<td>3</td>
+<td>4</td>
+<td>5</td>
+<td>6</td>
+<td>7</td>
+<td>8</td>
+<td>9</td>
+<td>10</td>
+</tr>
+<tr>
+<td>2</td>
+<td>4</td>
+<td>6</td>
+<td>8</td>
+<td>10</td>
+<td>10<a class = "tag" name = "tag_art10" id = "tag_art10" href =
+"#note_art10">10</a></td>
+<td>14</td>
+<td>16</td>
+<td>18</td>
+<td>20</td>
+</tr>
+<tr>
+<td>3</td>
+<td>6</td>
+<td>9</td>
+<td>12</td>
+<td>15</td>
+<td>18</td>
+<td>21</td>
+<td>24</td>
+<td>27</td>
+<td>30</td>
+</tr>
+<tr>
+<td>4</td>
+<td>8</td>
+<td>12</td>
+<td>16</td>
+<td>20</td>
+<td>24</td>
+<td>28</td>
+<td>32</td>
+<td>36</td>
+<td>40</td>
+</tr>
+<tr>
+<td>5</td>
+<td>10</td>
+<td>15</td>
+<td>20</td>
+<td>25</td>
+<td>30</td>
+<td>35</td>
+<td>40</td>
+<td>45</td>
+<td>50</td>
+</tr>
+<tr>
+<td>6</td>
+<td>12</td>
+<td>18</td>
+<td>24</td>
+<td>30</td>
+<td>36</td>
+<td>42</td>
+<td>48</td>
+<td>56</td>
+<td>60</td>
+</tr>
+<tr>
+<td>7</td>
+<td>14</td>
+<td>21</td>
+<td>28</td>
+<td>35</td>
+<td>42</td>
+<td>49</td>
+<td>56</td>
+<td>63</td>
+<td>70</td>
+</tr>
+<tr>
+<td>8</td>
+<td>16</td>
+<td>24</td>
+<td>32</td>
+<td>40</td>
+<td>48</td>
+<td>56</td>
+<td>64</td>
+<td>72</td>
+<td>80</td>
+</tr>
+<tr>
+<td>9</td>
+<td>18</td>
+<td>27</td>
+<td>36</td>
+<td>45</td>
+<td>54</td>
+<td>63</td>
+<td>72</td>
+<td>81</td>
+<td>90</td>
+</tr>
+<tr>
+<td>10</td>
+<td>20</td>
+<td>30</td>
+<td>40</td>
+<td>50</td>
+<td>60</td>
+<td>70</td>
+<td>80</td>
+<td>90</td>
+<td>100</td>
+</tr>
+</table>
+
+<p class = "nospace">
+And ther ben .6 rules of Multiplicacio<i>u</i>n;
+<span class = "sidenote">(1) Digit by digit.</span>
+ffirst, yf a digit multiplie a digit, considr<i>e</i> how many of
+vnytees ben betwix the digit by multiplying and his .10. beth<i>e</i>
+to-gedre accompted<i>e</i>, and so oft w<i>i</i>t<i>h</i>-draw the digit
+multiplying, vnder the article of his
+deno<i>m</i>i<i>n</i>acio<i>u</i>n. Example of grace. If thow wolt wete
+how moch<i>e</i> is .4. tymes .8.,
+<a class = "tag" name = "tag_art11" id = "tag_art11" href =
+"#note_art11">11</a>se how many vnytees ben betwix .8.<a class = "tag"
+name = "tag_art12" id = "tag_art12" href = "#note_art12">12</a>
+and .10. to-geder rekened<i>e</i>, and it shew<i>i</i>t<i>h</i> that
+.2.: withdraw ther-for the quat<i>e</i>rnary, of the article of his
+deno<i>m</i>i<i>n</i>acion twies, of .40., And ther remayneth<i>e</i>
+.32., that is, to <a class = "gloss" name = "some2" id = "some2" href =
+"#gloss_some">some</a> of all<i>e</i> the multiplicacio<i>u</i>n.
+<span class = "sidenote">See the table above.</span>
+Wher-vpon for more evidence and declaracion the seid<i>e</i> table is
+made.
+<span class = "sidenote">(2) Digit by article.</span>
+Whan a digit multiplieth<i>e</i> an article, thow most bryng the digit
+into þe digit, of þe which<i>e</i> the article [has]<a class = "tag"
+name = "tag_art13" id = "tag_art13" href = "#note_art13">13</a>
+his name, and eu<i>er</i>y vnyte
+<span class = "pagenum">41</span>
+<a name = "page41" id = "page41"> </a>
+shall<i>e</i> stond<i>e</i> for .10., and eu<i>er</i>y article an .100.
+<span class = "sidenote">(3) Composite by digit.</span>
+Whan the digit multiplieth<i>e</i> a nombre componed<i>e</i>, þ<i>o</i>u
+most bryng the digit into aiþ<i>er</i> part of the nombre
+componed<i>e</i>, so þ<i>a</i>t digit be had into digit by the first
+rule, into an article by þe second<i>e</i> rule; and
+aft<i>er</i>ward<i>e</i> Ioyne the p<i>ro</i>duccio<i>u</i>n, and
+þ<i>er</i>e wol be the some totall<i>e</i>.</p>
+
+<table class = "grid outline" summary = "example">
+<tr>
+<td class = "words">Resultans</td>
+<td>1</td>
+<td>2</td>
+<td class = "double">6</td>
+<td>7</td>
+<td>3</td>
+<td class = "double">6</td>
+<td>1</td>
+<td>2</td>
+<td class = "double">0</td>
+<td>1</td>
+<td>2</td>
+<td>0</td>
+<td>8</td>
+</tr>
+<tr>
+<td class = "words">Multiplicand<i>us</i></td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td class = "double">2</td>
+<td>&nbsp;</td>
+<td>3</td>
+<td class = "double">2</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td class = "double">6</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>4</td>
+</tr>
+<tr>
+<td class = "words">Multiplicans</td>
+<td>&nbsp;</td>
+<td>6</td>
+<td class = "double">3</td>
+<td>2</td>
+<td>3</td>
+<td class = "double">&nbsp;</td>
+<td>&nbsp;</td>
+<td>2</td>
+<td class = "double">0</td>
+<td>&nbsp;</td>
+<td>3</td>
+<td>0</td>
+<td>2</td>
+</tr>
+</table>
+
+<p><span class = "sidenote">(4) Article by article.</span>
+Whan an article multiplieth<i>e</i> an article, the digit wherof he is
+named<i>e</i> is to be brought Into the digit wherof the oþ<i>er</i> is
+named<i>e</i>, and eu<i>er</i>y vnyte wol be worth<i>e</i>
+<span class = "linenum">Fol. 52.</span>
+*an .100., and eu<i>er</i>y article. a&nbsp;.1000.
+<span class = "sidenote">(5) Composite by article.</span>
+Whan an article multiplieth<i>e</i> a nombre componed<i>e</i>, thow most
+bryng the digit of the article into aither part of the nombre
+componed<i>e</i>; and Ioyne the p<i>ro</i>duccio<i>u</i>n, and
+eu<i>er</i>y article wol be worth<i>e</i> .100., and eu<i>er</i>y vnyte
+.10., and so woll<i>e</i> the some be open<i>e</i>.
+<span class = "sidenote">(6) Composite by composite.</span>
+Whan a nombre componed<i>e</i> multiplieth<i>e</i> a nombre
+componed<i>e</i>, eu<i>er</i>y p<i>ar</i>t of the nombre multiplying is
+to be had<i>e</i> into eu<i>er</i>y p<i>ar</i>t of the nombre to be
+multiplied<i>e</i>, and so shall<i>e</i> the digit be had<i>e</i> twies,
+onys in the digit, that other in the article. The article also twies,
+ones in the digit, that other in the article. Therfor yf thow wilt any
+nombre by hym-self other by any other multiplie, write the nombre to be
+multiplied<i>e</i> in the ou<i>er</i> ordre by his differences,
+<span class = "sidenote">How to set down your numbers.</span>
+The nombre multiplying in the lower ordre by his differences, so that
+the first of the lower ordre be vnder the last of the ou<i>er</i> ordre.
+This done, of the multiplying, the last is to be had<i>e</i> into the
+last of the nombre to be multiplied<i>e</i>. Wherof than wolle grow a
+digit, an article, other a nombre componed<i>e</i>.
+<span class = "sidenote">If the result is a digit,</span></p>
+
+<table class = "grid outline float" summary = "example">
+<tr>
+<td class = "words">The resultant</td>
+<td>6</td>
+</tr>
+<tr>
+<td class = "words">To be multiplied<i>e</i></td>
+<td>3</td>
+</tr>
+<tr>
+<td class = "words">Þe nombre multipliyng</td>
+<td>2</td>
+</tr>
+</table>
+
+<p class = "nospace">
+If it be a digit, even above the figure multiplying is hede write his
+digit that come of, as it appereth<i>e</i> here:&mdash;</p>
+
+<p><span class = "sidenote">an article,</span>
+And yf an article had be writ ou<i>er</i> the fig<i>ur</i>e multiplying
+his hede, put a cifre þ<i>er</i> and transferre the article
+toward<i>e</i> the lift hand<i>e</i>, as thus:&mdash;</p>
+
+<table class = "grid outline float" summary = "example">
+<tr>
+<td class = "words">The resultant</td>
+<td>1</td>
+<td>0</td>
+</tr>
+<tr>
+<td class = "words">to be multiplied<i>e</i></td>
+<td></td>
+<td>5</td>
+</tr>
+<tr>
+<td class = "words">þe nombre m<i>u</i>ltipliyng</td>
+<td></td>
+<td>2</td>
+</tr>
+</table>
+
+<p><span class = "sidenote">or a composite.</span>
+And yf a nombre componed<i>e</i> be writ ou<i>er</i> the figure
+multyplying is hede, write the digit in the nombre componed<i>e</i> is
+place, and sette the article to the lift hand<i>e</i>, as
+thus:&mdash;</p>
+
+<span class = "pagenum">42</span>
+<a name = "page42" id = "page42"> </a>
+
+<table class = "grid outline floatleft" summary = "example">
+<tr>
+<td class = "words">Resultant</td>
+<td>1</td>
+<td>2</td>
+</tr>
+<tr>
+<td class = "words">to be multiplied<i>e</i></td>
+<td></td>
+<td>4</td>
+</tr>
+<tr>
+<td class = "words">the nombre multipliyng</td>
+<td></td>
+<td>3</td>
+</tr>
+</table>
+
+<p><span class = "sidenote">Multiply next by the last but one, and so
+on.</span>
+This done, <a class = "gloss" name = "me" id = "me" href =
+"#gloss_me">me</a> most bryng the last save one of the multipliyng into
+the last of þe nombre to be multiplied<i>e</i>, and se what
+comyth<i>e</i> therof as before, and so do w<i>i</i>t<i>h</i>
+all<i>e</i>, tille me come to the first of the nombre multiplying, that
+must be brought into the last of the nombre to be multiplied<i>e</i>,
+wherof growith<i>e</i> oþ<i>er</i> a digit, an article,
+<span class = "linenum">Fol. 52 <i>b</i>.</span>
+*other a nombre componed<i>e</i>.</p>
+
+<table class = "grid outline float" summary = "example">
+<tr>
+<td class = "words">Resultant</td>
+<td>6</td>
+<td>6</td>
+</tr>
+<tr>
+<td class = "words">to be multiplied<i>e</i></td>
+<td></td>
+<td>3</td>
+</tr>
+<tr>
+<td class = "words">the nombre m<i>u</i>ltipliyng</td>
+<td>2</td>
+<td>2</td>
+</tr>
+</table>
+
+<table class = "grid outline float" summary = "example">
+<tr>
+<td class = "words">The resultant</td>
+<td>1</td>
+<td>1</td>
+<td>0</td>
+</tr>
+<tr>
+<td class = "words">to be multiplied<i>e</i></td>
+<td></td>
+<td></td>
+<td>5</td>
+</tr>
+<tr>
+<td class = "words">þe nombre m<i>u</i>ltiplying</td>
+<td></td>
+<td>2</td>
+<td>2</td>
+</tr>
+</table>
+
+<table class = "grid outline float" summary = "example">
+<tr>
+<td class = "words">The resultant</td>
+<td>1</td>
+<td>3<a class = "tag" name = "tag_art15" id = "tag_art15" href =
+"#note_art15">15</a></td>
+<td>2</td>
+</tr>
+<tr>
+<td class = "words">to be m<i>u</i>ltiplied<i>e</i></td>
+<td></td>
+<td></td>
+<td>4</td>
+</tr>
+<tr>
+<td class = "words">þe nombr<i>e</i> m<i>u</i>ltiplia<i>n</i>t</td>
+<td></td>
+<td>3</td>
+<td>3</td>
+</tr>
+</table>
+
+<p class = "nospace">
+If it be a digit, In the place of the <a class = "gloss" name = "ouerer"
+id = "ouerer" href = "#gloss_ouerer">ou<i>er</i>er</a>, sette a-side, as
+here:</p>
+
+<p>If an article happe, there put a cifre in his place, and put hym to
+the lift hand<i>e</i>, as here:</p>
+
+<p>If it be a nombre componed<i>e</i>, in the place of the ou<i>er</i>er
+sette a-side, write a digit that<a class = "tag" name = "tag_art14" id =
+"tag_art14" href = "#note_art14">14</a>
+is a p<i>ar</i>t of the componed<i>e</i>, and sette on the left
+hond<i>e</i> the article, as here:</p>
+
+<p><span class = "sidenote">Then antery the multiplier one place.</span>
+That done, sette forward<i>e</i> the figures of the nombre multiplying
+by <a class = "gloss" name = "oo" id = "oo" href = "#gloss_oo">oo</a>
+difference, so that the first of the multipliant be vnder the last save
+one of the nombre to be multiplied<i>e</i>, the other by <a class =
+"gloss" name = "o" id = "o" href = "#gloss_oo">o</a> place sette
+forward<i>e</i>. Than me shall<i>e</i> bryng<i>e</i> the last of the
+m<i>u</i>ltipliant in hym to be multiplied<i>e</i>, vnder the
+which<i>e</i> is the first multipliant.
+<span class = "sidenote">Work as before.</span>
+And than wolle growe oþ<i>er</i> a digit, an article, or a
+componed<i>e</i> nombre. If it be a digit, adde hym even above his hede;
+If it be an article, transferre hym to the lift side; And if it be a
+nombre componed<i>e</i>, adde a digit to the figure above his hede, and
+sette to the lift hand<i>e</i> the article. And all<i>e</i>-wayes
+eu<i>er</i>y figure of the nombre multipliant is to be brought to the
+last save one nombre to be multiplied<i>e</i>, til me come to the first
+of the multipliant, where me shall<i>e</i> wirche as it is seid<i>e</i>
+before of the first, and aft<i>er</i>ward<i>e</i> to put forward<i>e</i>
+the figures by o difference and one till<i>e</i> they all<i>e</i> be
+multiplied<i>e</i>.
+<span class = "sidenote">How to deal with ciphers.</span>
+And yf it happe that the first figure of þe multipliant be a cifre, and
+<a class = "gloss" name = "boue" id = "boue" href =
+"#gloss_boue">boue</a> it is sette the figure signyficatif<i>e</i>,
+write a cifre in the place of the figur<i>e</i> sette a-side, as thus,
+<i>et</i>c.:</p>
+
+<table class = "grid outline" summary = "example">
+<tr>
+<td class = "words">The resultant</td>
+<td>1</td>
+<td>2</td>
+<td>0</td>
+</tr>
+<tr>
+<td class = "words">to be multiplied<i>e</i></td>
+<td></td>
+<td></td>
+<td>6</td>
+</tr>
+<tr>
+<td class = "words">the multipliant</td>
+<td></td>
+<td>2</td>
+<td>0</td>
+</tr>
+</table>
+
+<span class = "pagenum">43</span>
+<a name = "page43" id = "page43"> </a>
+
+<p><span class = "sidenote">How to deal with ciphers.</span>
+And yf a cifre happe in the lower <a class = "gloss" name = "order" id =
+"order" href = "#gloss_ordure">order</a> be-twix the first and the last,
+and even above be sette the fig<i>ur</i>e signyficatif,</p>
+
+<table class = "grid outline float" summary = "example">
+<tr>
+<td class = "words">The resultant</td>
+<td>2</td>
+<td>2</td>
+<td>6</td>
+<td>4</td>
+<td>4</td>
+</tr>
+<tr>
+<td class = "words">To be multiplied<i>e</i></td>
+<td></td>
+<td></td>
+<td>2</td>
+<td>2</td>
+<td>2</td>
+</tr>
+<tr>
+<td class = "words">The multipliant</td>
+<td>1</td>
+<td>0</td>
+<td>2</td>
+<td></td>
+<td></td>
+</tr>
+</table>
+
+<p class = "nospace">
+leve it vntouched<i>e</i>, as here:&mdash;</p>
+
+<p>And yf the space above sette be void<i>e</i>, in that place write
+thow a cifre. And yf the cifre happe betwix þe first and the last to be
+m<i>u</i>ltiplied<i>e</i>, me most sette forward<i>e</i> the ordre of
+the figures by thair<i>e</i> differences, for oft of <a class = "terms"
+name = "duccioun" id = "duccioun" href =
+"#terms_duccioun">duccio<i>u</i>n</a> of figur<i>e</i>s in cifres nought
+is the resultant, as here,</p>
+
+<table class = "grid outline float" summary = "example">
+<tr>
+<td class = "words">Resultant</td>
+<td>8</td>
+<td>0</td>
+<td>0</td>
+<td>8</td>
+<td>&nbsp;&nbsp;</td>
+</tr>
+<tr>
+<td class = "words">to be m<i>u</i>ltiplied<i>e</i></td>
+<td>4</td>
+<td>0</td>
+<td>0</td>
+<td>4</td>
+<td>&nbsp;&nbsp;</td>
+</tr>
+<tr>
+<td class = "words">the m<i>u</i>ltipliant</td>
+<td>2</td>
+<td>.</td>
+<td>.</td>
+<td>.</td>
+<td>&nbsp;&nbsp;</td>
+</tr>
+</table>
+
+<p class = "nospace">
+<span class = "linenum">Fol. 53.</span>
+*wherof it is evident and open, yf that the first figure of the nombre
+be to be multiplied<i>e</i> be a cifre, vndir it shall<i>e</i> be none
+sette as here:&mdash;</p>
+
+<table class = "grid outline floatleft" summary = "example">
+<tr>
+<td class = "words">Resultant</td>
+<td>3</td>
+<td>2</td>
+<td>0<a class = "tag" name = "tag_art16" id = "tag_art16" href =
+"#note_art16">16</a></td>
+</tr>
+<tr>
+<td class = "words">To be m<i>u</i>ltiplied<i>e</i></td>
+<td></td>
+<td>8</td>
+<td>0</td>
+</tr>
+<tr>
+<td class = "words">The m<i>u</i>ltipliant</td>
+<td></td>
+<td>4</td>
+<td></td>
+</tr>
+</table>
+
+<p><span class = "sidenote">Leave room between the rows of
+figures.</span>
+Vnder[stand] also that in multiplicacio<i>u</i>n, divisio<i>u</i>n, and
+of rootis the extraccio<i>u</i>n, competently me may leve a mydel space
+betwix .2. ordres of figures, that me may write there what is come of
+addyng other with<i>e</i>-drawyng, lest any thynge shold<i>e</i> be
+<a class = "gloss" name = "ouerhippede" id = "ouerhippede" href =
+"#gloss_ouerhippede">ou<i>er</i>-hipped<i>e</i></a> and sette out of
+mynde.</p>
+
+<p class = "headnote"><span class = "headnote">
+Chapter VII. Division.</span></p>
+
+<p><span class = "sidenote">Definition of division.</span>
+<span class = "dropcap">F</span>or to dyvyde oo nombre by a-nother, it
+is of .2. nombres p<i>ro</i>posed<i>e</i>, It is forto depart the
+<a class = "gloss" name = "moder" id = "moder" href = "#gloss_mo">moder</a>
+nombre into as many p<i>ar</i>tis as ben of vnytees in the lasse nombre.
+And note wele that in makyng<i>e</i> of dyvysio<i>u</i>n ther ben .3.
+nombres necessary:
+<span class = "sidenote">Dividend, Divisor, Quotient.</span>
+that is to sey, the nombre to be dyvyded<i>e</i>; the nombre dyvydyng
+and the nombre <a class = "gloss" name = "exeant" id = "exeant" href =
+"#gloss_exeant">exeant</a>, <a class = "gloss" name = "other2" id =
+"other2" href = "#gloss_other">other</a> how oft, or quocient. Ay
+shall<i>e</i> the nombre that is to be dyvyded<i>e</i> be more, other at
+the <a class = "gloss" name = "lest" id = "lest" href =
+"#gloss_lest">lest</a> even<i>e</i> w<i>i</i>t<i>h</i> the nombre the
+dyvysere, yf the nombre shall<i>e</i> be mad<i>e</i> by hole nombres.
+<span class = "sidenote">How to set down your Sum.</span>
+Therfor yf thow wolt any nombre dyvyde, write the nombre to be
+dyvyded<i>e</i> in þe ou<i>er</i>er <a class = "gloss" name = "bordure"
+id = "bordure" href = "#gloss_bordure">bordur<i>e</i></a> by his
+differences, the dyviser<i>e</i> in the lower ordur<i>e</i> by his
+differences, so that the last of the dyviser be vnder the last of the
+nombre to be dyvyde, the next last vnder the next last, and so of the
+others, yf it may competently be done;
+<span class = "sidenote">An example.</span>
+as here:&mdash;</p>
+
+<table class = "grid outline" summary = "example">
+<tr>
+<td class = "words">The residue</td>
+<td></td>
+<td>2</td>
+<td>7</td>
+</tr>
+<tr>
+<td class = "words">The quotient</td>
+<td></td>
+<td></td>
+<td>5</td>
+</tr>
+<tr>
+<td class = "words">To be dyvyded<i>e</i></td>
+<td>3</td>
+<td>4</td>
+<td>2</td>
+</tr>
+<tr>
+<td class = "words">The dyvyser</td>
+<td></td>
+<td>6</td>
+<td>3</td>
+</tr>
+</table>
+
+<span class = "pagenum">44</span>
+<a name = "page44" id = "page44"> </a>
+<span class = "sidenote">Examples.</span>
+
+<table class = "grid outline" summary = "example">
+<tr>
+<td class = "words">Residuu<i>m</i></td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td class = "double">8</td>
+<td>&nbsp;</td>
+<td class = "double">&nbsp;</td>
+<td>&nbsp;</td>
+<td>2</td>
+<td class = "double">7</td>
+<td>&nbsp;</td>
+<td>2</td>
+<td>6</td>
+</tr>
+<tr>
+<td class = "words">Quociens</td>
+<td>&nbsp;</td>
+<td>2</td>
+<td class = "double">1</td>
+<td>2</td>
+<td class = "double">2</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td class = "double">5</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>9</td>
+</tr>
+<tr>
+<td class = "words">Diuidend<i>us</i></td>
+<td>6</td>
+<td>8</td>
+<td class = "double">0</td>
+<td>6</td>
+<td class = "double">6</td>
+<td>3</td>
+<td>4</td>
+<td class = "double">2</td>
+<td>3</td>
+<td>3</td>
+<td>2</td>
+</tr>
+<tr>
+<td class = "words">Diuiser</td>
+<td>3</td>
+<td>2</td>
+<td class = "double">&nbsp;</td>
+<td>3</td>
+<td class = "double">&nbsp;</td>
+<td>&nbsp;</td>
+<td>6</td>
+<td class = "double">3</td>
+<td>&nbsp;</td>
+<td>3</td>
+<td>4</td>
+</tr>
+</table>
+
+<p><span class = "sidenote">When the last of the divisor must not be set
+below the last of the dividend.</span>
+And ther ben .2. causes whan the last figure may not be sette vnder the
+last, other that the last of the lower nombre may not be
+w<i>i</i>t<i>h</i>-draw of the last of the ou<i>er</i>er nombre for it
+is lasse than the lower, other <a class = "gloss" name =
+"how_be_it_that" id = "how_be_it_that" href =
+"#gloss_how_be_it_that">how be it, that</a> it myght be
+w<i>i</i>t<i>h</i>-draw as for hym-self fro the ou<i>er</i>er the
+remenaunt may not so oft of them above, other yf þe last of the lower be
+even to the figure above his hede, and þe next last oþ<i>er</i> the
+figure be-fore þ<i>a</i>t be more þan the figure above sette.
+<span class = "linenum">Fol. 53<sup>2</sup>.</span>
+*These so ordeyned<i>e</i>, me most wirch<i>e</i> from the last figure
+of þe nombre of the dyvyser, and se how oft it may be
+w<i>i</i>t<i>h</i>-draw of
+<span class = "sidenote">How to begin.</span>
+and fro the figure aboue his hede, namly so that the remen<i>au</i>nt
+may be take of so oft, and to se the residue as here:&mdash;</p>
+
+<table class = "grid outline floatleft" summary = "example">
+<tr>
+<td class = "words">The residue</td>
+<td></td>
+<td>2</td>
+<td>6</td>
+</tr>
+<tr>
+<td class = "words">The quocient</td>
+<td></td>
+<td></td>
+<td>9</td>
+</tr>
+<tr>
+<td class = "words">To be dyvyded<i>e</i></td>
+<td>3</td>
+<td>3</td>
+<td>2</td>
+</tr>
+<tr>
+<td class = "words">The dyvyser</td>
+<td></td>
+<td>3</td>
+<td>4</td>
+</tr>
+</table>
+
+<p><span class = "sidenote">An example.</span>
+And note wele that me may not with<i>e</i>-draw more than .9. tymes
+nether lasse than ones. Therfor se how oft þe figures of the lower ordre
+may be w<i>i</i>t<i>h</i>-draw fro the figures of the ou<i>er</i>er, and
+the nombre that shew<i>i</i>t<i>h</i> þe q<i>u</i>ocient most be writ
+ou<i>er</i> the hede of þat figure, vnder the which<i>e</i> the first
+figure is, of the dyviser;
+<span class = "sidenote">Where to set the <ins class = "correction"
+title = "spelling (1922) unchanged">quotiente</ins></span>
+And by that figure me most with<i>e</i>-draw all<i>e</i> oþ<i>er</i>
+figures of the lower ordir and that of the figures aboue thair<i>e</i>
+hedis. This so don<i>e</i>, me most sette forward<i>e</i> þe figures of
+the diuiser by o difference toward<i>es</i> the right hond<i>e</i> and
+worch<i>e</i> as before; and thus:&mdash;
+<span class = "sidenote">Examples.</span></p>
+
+<table class = "grid outline" summary = "example">
+<tr>
+<td class = "words">Residuu<i>m</i></td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td class = "double">&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>.</td>
+<td>1</td>
+<td>2</td>
+</tr>
+<tr>
+<td class = "words">quo<i>ciens</i></td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>6</td>
+<td>5</td>
+<td class = "double">4</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>2</td>
+<td>0</td>
+<td>0</td>
+<td>4</td>
+</tr>
+<tr>
+<td class = "words">Diuidend<i>us</i></td>
+<td>3</td>
+<td>5</td>
+<td>5</td>
+<td>1</td>
+<td>2</td>
+<td class = "double">2</td>
+<td>8</td>
+<td>8</td>
+<td>6</td>
+<td>3</td>
+<td>7</td>
+<td>0</td>
+<td>4</td>
+</tr>
+<tr>
+<td class = "words">Diuisor</td>
+<td>&nbsp;</td>
+<td>5</td>
+<td>4</td>
+<td>3</td>
+<td>&nbsp;</td>
+<td class = "double">&nbsp;</td>
+<td>4</td>
+<td>4</td>
+<td>2</td>
+<td>3</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+</table>
+
+<table class = "grid outline" summary = "example">
+<tr>
+<td class = "words">The quocient</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>6</td>
+<td>5</td>
+<td>4</td>
+</tr>
+<tr>
+<td class = "words">To be dyvyded<i>e</i></td>
+<td>3</td>
+<td>5</td>
+<td>5</td>
+<td>1</td>
+<td>2</td>
+<td>2</td>
+</tr>
+<tr>
+<td class = "words">The dyvyser</td>
+<td>&nbsp;</td>
+<td>5</td>
+<td>4</td>
+<td>3</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+</table>
+
+<p><span class = "sidenote">A special case.</span>
+And yf it happ<i>e</i> after þe settyng forward<i>e</i> of the
+fig<i>ur</i>es þ<i>a</i>t þe last of the divisor may not so oft be
+w<i>i</i>t<i>h</i>draw of the fig<i>ur</i>e above his hede, above þat
+fig<i>ur</i>e vnder the which<i>e</i> the first of the diuiser is writ
+me most sette a cifre in ordre of the nombre quocient, and sette the
+fig<i>ur</i>es forward<i>e</i> as be-fore be o difference alone, and so
+me shall<i>e</i> do in all<i>e</i> nombres to be dyvided<i>e</i>, for
+where the dyviser may
+<span class = "pagenum">45</span>
+<a name = "page45" id = "page45"> </a>
+not be w<i>i</i>t<i>h</i>-draw me most sette there a cifre, and sette
+forward<i>e</i> the figures; as here:&mdash;</p>
+
+<table class = "grid outline floatleft" summary = "example">
+<tr>
+<td class = "words">The residue</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>1</td>
+<td>2</td>
+</tr>
+<tr>
+<td class = "words">The quocient</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>2</td>
+<td>0</td>
+<td>0</td>
+<td>4</td>
+</tr>
+<tr>
+<td class = "words">To be dyvyded<i>e</i></td>
+<td>8</td>
+<td>8</td>
+<td>6</td>
+<td>3</td>
+<td>7</td>
+<td>0</td>
+<td>4</td>
+</tr>
+<tr>
+<td class = "words">The dyvyser</td>
+<td>4</td>
+<td>4</td>
+<td>2</td>
+<td>3</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+</table>
+
+<p><span class = "sidenote">Another example.</span>
+And me shall<i>e</i> not cesse fro such<i>e</i> settyng of
+fig<i>ur</i>es forward<i>e</i>, nether of settyng<i>e</i> of þe quocient
+into the dyviser, neþ<i>er</i> of subt<i>ra</i>ccio<i>u</i>n of the
+dyvyser, till<i>e</i> the first of the dyvyser be
+w<i>i</i>t<i>h</i>-draw fro þe first to be divided<i>e</i>. The
+which<i>e</i> don<i>e</i>, or ought,<a class = "tag" name = "tag_art17"
+id = "tag_art17" href = "#note_art17">17</a> oþ<i>er</i> nought
+shall<i>e</i> remayne: and yf it be ought,<a class = "tag" href =
+"#note_art17">17</a>
+kepe it in the tables, And eu<i>er</i> <a class = "gloss" name = "vny"
+id = "vny" href = "#gloss_vny">vny</a> it to þe diviser. And yf
+þ<i>o</i>u wilt wete how many vnytees of þe divisio<i>u</i>n
+<span class = "linenum">Fol. 53<sup>3</sup>.</span>
+*wol growe to the nombre of the diviser<i>e</i>,
+<span class = "sidenote">What the quotient shows.</span>
+the nombre quocient wol shewe it: and whan such<i>e</i> divisio<i>u</i>n
+is made, and þ<i>o</i>u <a class = "gloss" name = "lust" id = "lust"
+href = "#gloss_lust">lust</a> p<i>ro</i>ve yf thow have wele done or
+<span class = "sidenote">How to prove your division,</span>
+no, Multiplie the quocient by the diviser, And the same fig<i>ur</i>es
+wolle come ayene that thow haddest bifore and none other. And yf ought
+be residue, than w<i>i</i>t<i>h</i> addicio<i>u</i>n therof
+shall<i>e</i> come the same figures: And so multiplicacio<i>u</i>n
+p<i>ro</i>vith<i>e</i> divisio<i>u</i>n, and dyvisio<i>u</i>n
+multiplicacio<i>u</i>n:
+<span class = "sidenote">or multiplication.</span>
+as thus, yf multiplicacio<i>u</i>n be made, divide it by the
+multipliant, and the nombre quocient wol shewe the nombre that was to be
+multiplied<i>e</i>, <i>et</i>c.</p>
+
+<p class = "headnote"><span class = "headnote">
+Chapter VIII. Progression.</span></p>
+
+<p><span class = "sidenote">Definition <ins class = "correction" title =
+"f illegible">of</ins> Progression.</span>
+<span class = "dropcap">P</span>rogressio<i>u</i>n is of nombre after
+<a class = "gloss" name = "egalle" id = "egalle" href =
+"#gloss_egalle">egall<i>e</i></a> excesse fro oone or tweyn<i>e</i>
+<a class = "gloss" name = "take" id = "take" href = "#gloss_take">take</a>
+<a class = "gloss" name = "agregacioun" id = "agregacioun" href =
+"#gloss_agregacioun">ag<i>r</i>egacio<i>u</i>n</a>. of
+p<i>ro</i>gressio<i>u</i>n one is <a class = "terms" name = "naturelle"
+id = "naturelle" href = "#terms_naturelle">naturell<i>e</i></a> or
+co<i>n</i>tynuell<i>e</i>, þ<i>a</i>t oþ<i>er</i> broken and
+discontynuell<i>e</i>.
+<span class = "sidenote">Natural Progression.</span>
+Naturell<i>e</i> it is, whan me begynneth<i>e</i> w<i>i</i>t<i>h</i>
+one, and kepeth<i>e</i> ordure ou<i>er</i>lepyng one; as .1. 2. 3. 4. 5.
+6., <i>et</i>c., so þ<i>a</i>t the nombre folowyng<i>e</i>
+passith<i>e</i> the other be-fore in one.
+<span class = "sidenote">Broken Progression.</span>
+Broken it is, whan me lepith<i>e</i> fro o nombre till<i>e</i> another,
+and kepith<i>e</i> not the contynuel ordir<i>e</i>; as 1. 3. 5. 7. 9,
+<i>et</i>c. Ay me may begynne w<i>i</i>t<i>h</i> .2., as þus; .2. 4. 6.
+8., <i>et</i>c., and the nombre folowyng passeth<i>e</i> the others
+by-fore by .2. And note wele, that naturell<i>e</i>
+p<i>ro</i>gressio<i>u</i>n ay begynneth<i>e</i> w<i>i</i>t<i>h</i> one,
+and Int<i>er</i>cise or broken p<i>ro</i>gressio<i>u</i>n, <a class =
+"gloss" name = "omwhile" id = "omwhile" href =
+"#gloss_omwhile">omwhile</a> begynnyth<i>e</i> w<i>i</i>th one, omwhile
+w<i>i</i>t<i>h</i> twayn<i>e</i>. Of p<i>ro</i>gressio<i>u</i>n naturell
+.2. rules ther be <a class = "gloss" name = "yove" id = "yove" href =
+"#gloss_yove">yove</a>, of the which<i>e</i> the first is this;
+<span class = "sidenote">The 1st rule for Natural Progression.</span>
+whan the p<i>ro</i>gressio<i>u</i>n naturell<i>e</i> endith<i>e</i> in
+even nombre, by the half therof multiplie þe next totall<i>e</i>
+ou<i>er</i>er<i>e</i> nombre; Example of grace: .1. 2. 3. 4. Multiplie
+.5. by .2. and so .10. cometh<i>e</i> of, that is the totall<i>e</i>
+nombre þ<i>er</i>of.
+<span class = "sidenote">The second rule.</span>
+The second<i>e</i> rule is such<i>e</i>, whan the
+p<i>ro</i>gressio<i>u</i>n naturell<i>e</i> endith<i>e</i> in nombre
+od<i>e</i>. Take the more porcio<i>u</i>n of the oddes, and multiplie
+therby the totall<i>e</i> nombre. Example of grace 1. 2. 3. 4. 5.,
+multiplie
+<span class = "pagenum">46</span>
+<a name = "page46" id = "page46"> </a>
+.5. by .3, and thryes .5. shall<i>e</i> be resultant. so the nombre
+totall<i>e</i> is .15.
+<span class = "sidenote">The first rule of Broken Progression.</span>
+Of p<i>ro</i>gresio<i>u</i>n <a class = "terms" name = "intercise" id =
+"intercise" href = "#terms_intercise">int<i>er</i>cise</a>, ther ben
+also .2.<a class = "tag" name = "tag_art18" id = "tag_art18" href =
+"#note_art18">18</a>
+rules; and þe first is þis: Whan the Int<i>er</i>cise p<i>ro</i>gression
+endith<i>e</i> in even nombre by half therof multiplie the next nombre
+to þat half<i>e</i> as .2.<a class = "tag" href = "#note_art18">18</a>
+4. 6. Multiplie .4. by .3. so þat is thryes .4., and .12. the nombre of
+all<i>e</i> the p<i>ro</i>gressio<i>u</i>n, woll<i>e</i> folow.
+<span class = "sidenote">The second rule.</span>
+The second<i>e</i> rule is this: whan the p<i>ro</i>gressio<i>u</i>n
+int<i>er</i>scise endith<i>e</i> in od<i>e</i>, take þe more
+porcio<i>u</i>n of all<i>e</i> þe nombre,
+<span class = "linenum">Fol. 53<sup>4</sup>.</span>
+*and multiplie by hym-self<i>e</i>; as .1. 3. 5. Multiplie .3. by
+hym-self<i>e</i>, and þe some of all<i>e</i> wolle be .9.,
+<i>et</i>c.</p>
+
+<p class = "headnote"><span class = "headnote">
+Chapter IX. Extraction of Roots.</span></p>
+
+<p><span class = "sidenote">The preamble of the extraction of
+roots.</span>
+<span class = "dropcap">H</span>ere folowith<i>e</i> the
+extraccio<i>u</i>n of rotis, and first in nombre
+q<i>ua</i>drat<i>es</i>. Wherfor me shall<i>e</i> se what is a <a class
+= "terms" name = "quadrat" id = "quadrat" href = "#terms_quadrat">nombre
+quadrat</a>, and what is the rote of a nombre quadrat, and what it is to
+draw out the rote of a nombre. And before other note this
+divisio<i>u</i>n:
+<span class = "sidenote">Linear, superficial, and solid numbers.</span>
+Of nombres one is <a class = "terms" name = "lyneal" id = "lyneal" href
+= "#terms_lyneal">lyneal</a>, anoþ<i>er</i> <a class = "terms" name =
+"superficial" id = "superficial" href =
+"#terms_superficial">sup<i>er</i>ficiall<i>e</i></a>, anoþ<i>er</i>
+quadrat, anoþ<i>er</i> cubik<i>e</i> or <a class = "gloss" name =
+"hoole" id = "hoole" href = "#gloss_hole">hoole</a>. lyneal is that þat
+is considred<i>e</i> after the p<i>ro</i>cesse, havyng<i>e</i> no
+respect to the direccio<i>u</i>n of nombre in nombre, As a lyne
+hath<i>e</i> but one dymensio<i>u</i>n that is to sey after the
+length<i>e</i>.
+<span class = "sidenote">Superficial numbers.</span>
+Nombre sup<i>er</i>ficial is þ<i>a</i>t cometh<i>e</i> of ledyng<i>e</i>
+of oo nombre into a-nother, wherfor it is called<i>e</i>
+sup<i>er</i>ficial, for it hath<i>e</i> .2. nombres notyng or
+mesuryng<i>e</i> hym, as a sup<i>er</i>ficiall<i>e</i> thyng<i>e</i>
+hath<i>e</i> .2. dimensions, þ<i>a</i>t is to sey length<i>e</i> and
+brede.
+<span class = "sidenote">Square numbers.</span>
+And for bycause a nombre may be had<i>e</i> in a-nother by .2.
+man<i>er</i>s, þ<i>a</i>t is to sey other in hym-self<i>e</i>,
+oþ<i>er</i> in anoþ<i>er</i>, Vnderstond<i>e</i> yf it be had in
+hym-self, It is a quadrat. ffor dyvisio<i>u</i>n write by vnytes,
+hath<i>e</i> .4. sides even as a quadrangill<i>e</i>. and yf the nombre
+be had<i>e</i> in a-noþ<i>er</i>, the nombre is sup<i>er</i>ficiel and
+not quadrat, as .2. had<i>e</i> in .3. maketh<i>e</i> .6. that is þe
+first nombre sup<i>er</i>ficiell<i>e</i>; wherfor it is open þat
+all<i>e</i> nombre quadrat is sup<i>er</i>ficiel, and not <a class =
+"terms" name = "conuertide" id = "conuertide" href =
+"#terms_conuertide">co<i>n</i>u<i>er</i>tid<i>e</i></a>.
+<span class = "sidenote">The root of a square number.</span>
+The rote of a nombre quadrat is þat nombre that is had of hym-self, as
+twies .2. makith<i>e</i> 4. and .4. is the first nombre quadrat, and 2.
+is his rote. 9. 8. 7. 6. 5. 4. 3. 2. 1. / The rote of the more quadrat
+.3. 1. 4. 2. 6.
+<span class = "sidenote">Notes of some examples of square roots here
+interpolated.</span>
+The most nombre quadrat 9. 8. 7. 5. 9. 3. 4. 7. 6. / the remenent
+ou<i>er</i> the quadrat .6. 0. 8. 4. 5. / The first caas of nombre
+quadrat .5. 4. 7. 5. 6. The rote .2. 3. 4. The second<i>e</i> caas .3.
+8. 4. 5. The rote .6. 2. The third<i>e</i> caas .2. 8. 1. 9. The rote
+.5. 3. The .4. caas .3. 2. 1. The rote .1. 7. / The 5. caas .9. 1. 2. 0.
+4. / The rote 3. 0. 2.
+<span class = "sidenote">Solid numbers.</span>
+The <a class = "terms" name = "solide" id = "solide" href =
+"#terms_solide">solid<i>e</i> nombre</a> or cubik<i>e</i> is þat
+þ<i>a</i>t comytħe of double ledyng of nombre in nombre;
+<span class = "sidenote">Three dimensions of solids.</span>
+And it is cleped<i>e</i> a solid<i>e</i> body that hath<i>e</i>
+þ<i>er</i>-in .3
+<span class = "pagenum">47</span>
+<a name = "page47" id = "page47"> </a>
+[dimensions] þat is to sey, length<i>e</i>, brede, and thiknesse. so
+þ<i>a</i>t nombre hath<i>e</i> .3. nombres to be brought forth<i>e</i>
+in hym. But nombre may be had<i>e</i> twies in nombre, for other it is
+had<i>e</i> in hym-self<i>e</i>, oþ<i>er</i> in a-noþ<i>er</i>.
+<span class = "sidenote">Cubic numbers.</span>
+If a nombre be had<i>e</i> twies in hym-self, oþ<i>er</i> ones in his
+quadrat, þ<i>a</i>t is the same, þ<i>a</i>t a <a class = "terms" name =
+"cubike" id = "cubike" href = "#terms_cubike">cubik<i>e</i></a>
+<span class = "linenum">Fol. 54.</span>
+*is, And is the same that is solide. And yf a nombre twies be
+had<i>e</i> in a-noþ<i>er</i>, the nombre is <a class = "gloss" name =
+"clepede" id = "clepede" href = "#gloss_clepede">cleped<i>e</i></a>
+solide and not cubik<i>e</i>, as twies .3. and þ<i>a</i>t .2.
+makith<i>e</i> .12.
+<span class = "sidenote">All cubics are solid numbers.</span>
+Wherfor it is <a class = "gloss" name = "opyne" id = "opyne" href =
+"#gloss_opyne">opyn<i>e</i></a> that all<i>e</i> cubik<i>e</i> nombre is
+solid<i>e</i>, and not <a class = "terms" name = "conuertide2" id =
+"conuertide2" href =
+"#terms_conuertide"><i>con</i>u<i>er</i>tid<i>e</i></a>. Cubik<i>e</i>
+is þ<i>a</i>t nombre þat comyth<i>e</i> of ledyng<i>e</i> of
+hym-self<i>e</i> twyes, or ones in his quadrat. And here-by it is open
+that o nombre is the <a class = "terms" name = "roote" id = "roote" href
+= "#terms_rote">roote</a> of a quadrat and of a cubik<i>e</i>. Natheles
+the same nombre is not q<i>ua</i>drat and cubik<i>e</i>.
+<span class = "sidenote">No number may be both linear and solid.</span>
+Opyn<i>e</i> it is also that all<i>e</i> nombres may be a rote to a
+q<i>ua</i>drat and cubik<i>e</i>, but not all<i>e</i> nombre quadrat or
+cubik<i>e</i>. Therfor sithen þe ledyng<i>e</i> of vnyte in hym-self
+ones or twies nought cometh<i>e</i> but vnytes, Seith<i>e</i> Boice in
+Arsemetrik<i>e</i>,
+<span class = "sidenote">Unity is not a number.</span>
+that vnyte potencially is al nombre, and none in act. And
+vndirstond<i>e</i> wele also that betwix euery .2. quadrat<i>es</i> ther
+is a meene p<i>ro</i>porcionall<i>e</i>,
+<span class = "sidenote">Examples of square roots.</span>
+That is opened<i>e</i> thus; <a class = "terms" name = "lede_into" id =
+"lede_into" href = "#terms_lede_into">lede the rote of o quadrat
+into</a> the rote of the oþ<i>er</i> quadrat, and þan wolle þe meene
+shew.</p>
+
+<table class = "grid outline" summary = "example">
+<tr>
+<td class = "words">Residuu<i>m</i></td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>0</td>
+<td class = "double">&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td class = "double">4</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>0</td>
+<td>&nbsp;</td>
+<td class = "double">&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>0</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td class = "words">Quadrand<i>e</i></td>
+<td>4</td>
+<td>3</td>
+<td>5</td>
+<td class = "double">6</td>
+<td>3</td>
+<td>0</td>
+<td>2</td>
+<td class = "double">9</td>
+<td>1</td>
+<td>7</td>
+<td>4</td>
+<td>2</td>
+<td class = "double">4</td>
+<td>1</td>
+<td>9&nbsp;</td>
+<td>3</td>
+<td>6</td>
+</tr>
+<tr>
+<td class = "words">Duplum</td>
+<td>1</td>
+<td>2</td>
+<td>&nbsp;</td>
+<td class = "double">&nbsp;</td>
+<td>1</td>
+<td>0</td>
+<td>&nbsp;</td>
+<td class = "double">&nbsp;</td>
+<td>2</td>
+<td>&nbsp;</td>
+<td>6</td>
+<td>&nbsp;</td>
+<td class = "double">&nbsp;</td>
+<td>&nbsp;</td>
+<td>[8]</td>
+<td><a class = "tag" name = "tag_art19" id = "tag_art19" href =
+"#note_art19">19</a></td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td class = "words">Subduplu<i>m</i></td>
+<td>&nbsp;</td>
+<td>6</td>
+<td>&nbsp;</td>
+<td class = "double">6</td>
+<td>&nbsp;</td>
+<td>5</td>
+<td>&nbsp;</td>
+<td class = "double">5</td>
+<td>1</td>
+<td>&nbsp;</td>
+<td>3</td>
+<td>&nbsp;</td>
+<td class = "double">2</td>
+<td>&nbsp;</td>
+<td>4&nbsp;</td>
+<td>&nbsp;</td>
+<td>4</td>
+</tr>
+</table>
+
+<p><span class = "sidenote">A note on mean proportionals.</span>
+Also betwix the next .2. cubikis, me may fynde a double meene, that is
+to sey a more meene and a lesse. The more meene thus, as to
+bryng<i>e</i> the rote of the lesse into a quadrat of the more. The
+lesse thus, If the rote of the more be brought Into the quadrat of the
+lesse.</p>
+
+<p class = "headnote"><span class = "headnote">
+Chapter X. Extraction of Square Root.</span></p>
+
+<p><span class = "dropcap">T</span>o<a class = "tag" name = "tag_art20"
+id = "tag_art20" href = "#note_art20">20</a> draw a rote of the nombre
+quadrat it is What-eu<i>er</i> nombre be p<i>ro</i>posed<i>e</i> to
+fynde his rote and to se yf it be quadrat.
+<span class = "sidenote">To find a square root.</span>
+And yf it be not quadrat the rote of the most quadrat fynde out, vnder
+the nombre p<i>ro</i>posed<i>e</i>. Therfor yf thow wilt the rote of any
+quadrat nombre draw out, write the nombre by his differences, and
+<a class = "gloss" name = "compt" id = "compt" href =
+"#gloss_compt">compt</a> the nombre of the figures, and wete yf it be
+od<i>e</i> or even. And yf
+<span class = "sidenote">Begin with the last odd place.</span>
+it be even, than most thow begynne worche vnder the last save one. And
+yf it be od<i>e</i> w<i>i</i>t<i>h</i> the last; and forto sey it
+shortly, al-weyes fro the last od<i>e</i> me shall<i>e</i> begynne.
+Therfor vnder the last in an od place sette,
+<span class = "sidenote">Find the nearest square root of that number,
+subtract,</span>
+me most fynd<i>e</i> a digit, the which<i>e</i> lad<i>e</i> in
+hym-self<i>e</i> it puttith<i>e</i> away that, þat is ou<i>er</i> his
+hede, oþ<i>er</i> as neigh<i>e</i> as me
+<span class = "pagenum">48</span>
+<a name = "page48" id = "page48"> </a>
+may: suche a digit found<i>e</i> and w<i>i</i>t<i>h</i>draw fro his
+ou<i>er</i>er, me most double that digit and sette the double vnder the
+next figure toward<i>e</i> the right hond<i>e</i>, and his <a class =
+"terms" name = "vnder_double" id = "vnder_double" href =
+"#terms_vnder_double">vnder double</a> vnder hym.
+<span class = "sidenote">double it,</span>
+That done, than me most fy<i>n</i>d<i>e</i> a-noþ<i>er</i> digit vnder
+the next figure bifore the doubled<i>e</i>,
+<span class = "sidenote">and set the double one to the right.</span>
+the which<i>e</i>
+<span class = "linenum">Fol. 54 <i>b</i>.</span>
+*brought in double setteth<i>e</i> a-way all<i>e</i> that is ou<i>er</i>
+his hede as to <a class = "gloss" name = "rewarde" id = "rewarde" href =
+"#gloss_rewarde">reward<i>e</i></a> of the doubled<i>e</i>: Than brought
+into hym-self settith<i>e</i> all away in respect of hym-self,
+<span class = "sidenote">Find the second figure by division.</span>
+Other do it as nye as it may be do: other me may w<i>i</i>t<i>h</i>-draw
+the digit
+<a class = "tag" name = "tag_art21" id = "tag_art21" href =
+"#note_art21">21</a>[last] found<i>e</i>, and lede hym in double or
+double hym, and after in hym-self<i>e</i>;
+<span class = "sidenote">Multiply the double by the second figure, and
+add after it the square of the second figure, and subtract.</span>
+Than Ioyne to-geder the p<i>ro</i>duccion<i>e</i> of them bothe, So that
+the first figure of the last p<i>ro</i>duct be added<i>e</i> before the
+first of the first p<i>ro</i>duct<i>es</i>, the second<i>e</i> of the
+first, <i>et</i>c. and so forth<i>e</i>, <a class = "gloss" name =
+"subtrahe" id = "subtrahe" href = "#gloss_subtrahe">subtrahe</a> fro the
+totall<i>e</i> nombre in respect of þe digit.</p>
+
+<span class = "sidenote">Examples.</span>
+
+<table class = "grid outline" summary = "example">
+<tr>
+<td class = "words">The residue</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td class = "double">&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td class = "double">&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>5</td>
+<td>4</td>
+<td>3</td>
+<td>2</td>
+</tr>
+<tr>
+<td class = "words"><p>To be quadred<i>e</i></p></td>
+<td>4</td>
+<td>1</td>
+<td>2</td>
+<td>0</td>
+<td class = "double">9</td>
+<td>1</td>
+<td>5</td>
+<td>1</td>
+<td>3</td>
+<td class = "double">9</td>
+<td>9</td>
+<td>0</td>
+<td>0</td>
+<td>5</td>
+<td>4</td>
+<td>3</td>
+<td>2</td>
+</tr>
+<tr>
+<td class = "words">The double</td>
+<td>&nbsp;</td>
+<td>4</td>
+<td>0</td>
+<td>&nbsp;</td>
+<td class = "double">&nbsp;</td>
+<td>&nbsp;</td>
+<td>2</td>
+<td>&nbsp;</td>
+<td>4</td>
+<td class = "double">&nbsp;</td>
+<td>&nbsp;</td>
+<td>6</td>
+<td>&nbsp;</td>
+<td>0</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>0</td>
+</tr>
+<tr>
+<td class = "words"><p>The vnder double</p></td>
+<td>2</td>
+<td>&nbsp;</td>
+<td>0</td>
+<td>&nbsp;</td>
+<td class = "double">3</td>
+<td>1</td>
+<td>&nbsp;</td>
+<td>2</td>
+<td>&nbsp;</td>
+<td class = "double">3</td>
+<td>[3]</td>
+<td>&nbsp;</td>
+<td>[0]</td>
+<td>&nbsp;</td>
+<td>[0]</td>
+<td>&nbsp;</td>
+<td>0</td>
+</tr>
+</table>
+
+<p>And if it hap þ<i>a</i>t no digit may be found<i>e</i>, Than sette a
+cifre vndre a cifre, and cesse not till<i>e</i> thow fynde a digit; and
+whan thow hast founde it to double it, neþ<i>er</i> to sette the
+doubled<i>e</i> forward<i>e</i> nether the vnder doubled<i>e</i>,
+<span class = "sidenote">Special cases.</span>
+Till thow fynde vndre the first figure a digit, the which<i>e</i>
+lad<i>e</i> in all<i>e</i> double, settyng away all<i>e</i> that is
+ou<i>er</i> hym in respect of the doubled<i>e</i>: Than lede hym into
+hym-self<i>e</i>, and put a-way all<i>e</i> in regard<i>e</i> of hym,
+other as nygh<i>e</i> as thow maist.
+<span class = "sidenote">The residue.</span>
+That done, other ought or nought wolle be the residue. If nought, than
+it shewith<i>e</i> that a nombre componed<i>e</i> was the quadrat, and
+his rote a digit last found<i>e</i> w<i>i</i>t<i>h</i>
+vnder<i>e</i>-double other vndirdoubles, so that it be sette be-fore:
+And yf ought<a class = "tag" name = "tag_art22" id = "tag_art22" href =
+"#note_art22">22</a>
+remayn<i>e</i>, that shew<i>i</i>t<i>h</i> that the nombre
+p<i>ro</i>posed<i>e</i> was not quadrat,<a class = "tag" name =
+"tag_art23" id = "tag_art23" href = "#note_art23">23</a>
+but a digit [last found with the subduple or subduples
+<span class = "pagenum">49</span>
+<a name = "page49" id = "page49"> </a>
+is]</p>
+
+<span class = "sidenote">This table is constructed for use in cube root
+sums, giving the value of <ins class = "correction" title = "that is, ‘a(bˆ2).’">ab.<sup>2</sup></ins></span>
+
+<table class = "grid outline right" summary = "example">
+<tr>
+<td class = "words">1</td> <!-- for extra spacing -->
+<td>2</td>
+<td>3</td>
+<td>4</td>
+<td>5</td>
+<td>6</td>
+<td>7 &nbsp;</td>
+<td>8</td>
+<td>9 &nbsp;</td>
+</tr>
+<tr>
+<td class = "words">2</td>
+<td>8</td>
+<td>12</td>
+<td>16</td>
+<td>20</td>
+<td>24</td>
+<td>28 &nbsp;</td>
+<td>32</td>
+<td>36 &nbsp;</td>
+</tr>
+<tr>
+<td class = "words">3</td>
+<td>18</td>
+<td>27</td>
+<td>36</td>
+<td>45</td>
+<td>54</td>
+<td>63 &nbsp;</td>
+<td>72</td>
+<td>81 &nbsp;</td>
+</tr>
+<tr>
+<td class = "words">4</td>
+<td>32</td>
+<td>48</td>
+<td>64</td>
+<td>80</td>
+<td>96</td>
+<td>112<a class = "tag" name = "tag_art24" id = "tag_art24" href =
+"#note_art24">24</a></td>
+<td>128</td>
+<td>144 &nbsp;</td>
+</tr>
+<tr>
+<td class = "words">5</td>
+<td>50</td>
+<td>75</td>
+<td>100</td>
+<td>125</td>
+<td>150</td>
+<td>175 &nbsp;</td>
+<td>200</td>
+<td>225 &nbsp;</td>
+</tr>
+<tr>
+<td class = "words">6</td>
+<td>72</td>
+<td>108</td>
+<td>144</td>
+<td>180</td>
+<td>216</td>
+<td>252 &nbsp;</td>
+<td>288</td>
+<td>324 &nbsp;</td>
+</tr>
+<tr>
+<td class = "words">7</td>
+<td>98</td>
+<td>147</td>
+<td>196</td>
+<td>245</td>
+<td>294</td>
+<td>343 &nbsp;</td>
+<td>393</td>
+<td>441 &nbsp;</td>
+</tr>
+<tr>
+<td class = "words">8</td>
+<td>128</td>
+<td>192</td>
+<td>256</td>
+<td>320</td>
+<td>384</td>
+<td>448 &nbsp;</td>
+<td>512</td>
+<td>576 &nbsp;</td>
+</tr>
+<tr>
+<td class = "words">9</td>
+<td>168</td>
+<td>243</td>
+<td>324</td>
+<td>405</td>
+<td>486</td>
+<td>567 &nbsp;</td>
+<td>648</td>
+<td>729<a class = "tag" name = "tag_art25" id = "tag_art25" href =
+"#note_art25">25</a></td>
+</tr>
+</table>
+
+<p>The rote of the most quadrat conteyned<i>e</i> vndre the nombre
+p<i>ro</i>posed<i>e</i>.
+<span class = "sidenote">How to prove the square root without or with a
+remainder.</span>
+Therfor yf thow wilt p<i>ro</i>ve yf thow have wele do or no, Multiplie
+the digit last found<i>e</i> w<i>i</i>t<i>h</i> the vnder-double
+oþ<i>er</i> vnder-doublis, and thow shalt fynde the same figures that
+thow haddest before; And so that nought be the
+<span class = "linenum">Fol. 55.</span>
+*residue. And yf thow have any residue, than w<i>i</i>t<i>h</i> the
+addicio<i>u</i>n þ<i>er</i>of that is res<i>er</i>ued<i>e</i>
+w<i>i</i>t<i>h</i>-out in thy table, thow shalt fynd<i>e</i> thi first
+figures as thow haddest them before, <i>et</i>c.</p>
+
+<p class = "headnote"><span class = "headnote">
+Chapter XI. Extraction of Cube Root.</span></p>
+
+<p><span class = "sidenote">Definition of a cubic number and a cube
+root.</span>
+<span class = "dropcap">H</span>eere folowith<i>e</i> the
+extraccio<i>u</i>n of rotis in cubik<i>e</i> nombres; wher-for me most
+se what is a nombre cubik<i>e</i>, and what is his roote, And what is
+the extraccio<i>u</i>n of a rote. A&nbsp;nombre cubik<i>e</i> it is, as
+it is before declared<i>e</i>, that cometh<i>e</i> of ledyng of any
+nombre twies in hym-self<i>e</i>, other ones in his quadrat. The rote of
+a nombre cubik<i>e</i> is the nombre that is twies had<i>e</i> in
+hy<i>m</i>-self<i>e</i>, or ones in his quadrat. <a class = "gloss" name
+= "wherthurghe" id = "wherthurghe" href =
+"#gloss_wherthurghe">Wher-thurgh<i>e</i></a> it is open, that
+eu<i>er</i>y nombre quadrat or cubik<i>e</i> have the same rote, as it
+is seid<i>e</i> before. And forto draw out the rote of a cubik<i>e</i>,
+It is first to fynd<i>e</i> þe nombr<i>e</i> p<i>ro</i>posed<i>e</i> yf
+it be a cubik<i>e</i>; And yf it be not, than thow most make
+extraccio<i>u</i>n of his rote of the most cubik<i>e</i> vndre the
+nombre p<i>ro</i>posid<i>e</i> his rote found<i>e</i>. Therfor
+p<i>ro</i>posed<i>e</i> some nombre, whos cubical rote þ<i>o</i>u
+woldest draw out;
+<span class = "sidenote">Mark off the places in threes.</span>
+First thow most compt the figures by fourthes, that is to sey in the
+place of thousand<i>es</i>;
+<span class = "sidenote">Find the first digit;</span>
+And vnder the last thousand<i>e</i> place, thow most fynde a digit, the
+which<i>e</i> lad<i>e</i> in hym-self cubikly puttith<i>e</i> a-way that
+þat is ou<i>er</i> his hede as in respect of hym, other as nygh<i>e</i>
+as thow maist.
+<span class = "sidenote">treble it and place it under the next but one,
+and multiply by the digit.</span>
+That done, thow most <a class = "gloss" name = "trebille" id =
+"trebille" href = "#gloss_trebille">trebill<i>e</i></a> the digit, and
+that triplat is to be put vnder the .3. next figure toward<i>e</i> the
+right hond<i>e</i>, And the <a class = "terms" name = "vnder_trebille"
+id = "vnder_trebille" href =
+"#terms_vnder_trebille">vnder-trebill<i>e</i></a> vnder the
+trebill<i>e</i>;
+<span class = "sidenote">Then find the second digit.</span>
+Than me most fynd<i>e</i> a digit vndre the next figure bifore the
+triplat, the which<i>e</i> w<i>i</i>t<i>h</i> his vnder-trebill<i>e</i>
+had into a trebill<i>e</i>, aft<i>er</i>warde other vnder[trebille]<a
+class = "tag" name = "tag_art26" id = "tag_art26" href =
+"#note_art26">26</a>
+had in his p<i>ro</i>duccio<i>u</i>n, putteth<i>e</i> a-way all<i>e</i>
+that is ou<i>er</i> it in regard<i>e</i> of<a class = "tag" name =
+"tag_art27" id = "tag_art27" href = "#note_art27">27</a>
+[the triplat. Then lade in hymself puttithe away that þat is over his
+hede as in respect of hym, other as nyghe as thou maist:]
+<span class = "sidenote">Multiply the first triplate and the second
+digit, twice by this digit.</span>
+That done, thow most trebill<i>e</i> the digit ayene, and the triplat is
+to be sette vnder the next .3. figure as before, And the
+vnder-trebill<i>e</i> vnder the trebill<i>e</i>: and than most thow
+sette forward<i>e</i> the first triplat w<i>i</i>t<i>h</i> his
+vndre-trebill<i>e</i> by .2. differences. And than most thow fynde a
+digit vnder the next figure before the triplat, the which<i>e</i>
+with<i>e</i> his <a class = "terms" name = "vnder_triplat" id =
+"vnder_triplat" href = "#terms_vnder_trebille">vnder-t<i>r</i>iplat</a>
+had in his triplat afterward<i>e</i>,
+<span class = "pagenum">50</span>
+<a name = "page50" id = "page50"> </a>
+<span class = "sidenote">Subtract.</span>
+other vnder-treblis lad in p<i>ro</i>duct
+<span class = "linenum">Fol. 55 <i>b</i>.</span>
+*It sitteth<i>e</i> a-way ałł that is ou<i>er</i> his hede in respect of
+the triplat than had in hym-self cubikly,<a class = "tag" name =
+"tag_art28" id = "tag_art28" href = "#note_art28">28</a>
+or as nygh<i>e</i> as ye may.</p>
+
+<span class = "sidenote">Examples.</span>
+
+<table class = "grid outline" summary = "example">
+<tr>
+<td class = "words">Residuu<i>m</i></td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td class = "double">5&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td class = "double">4</td>
+<td>&nbsp;</td>
+<td>1</td>
+<td>0</td>
+<td>1</td>
+<td>9</td>
+<td>&nbsp;&nbsp;</td>
+</tr>
+<tr>
+<td class = "words">Cubicandu<i>s</i></td>
+<td>8</td>
+<td>3</td>
+<td>6</td>
+<td>5</td>
+<td>4</td>
+<td>3</td>
+<td class = "double">2&nbsp;</td>
+<td>3</td>
+<td>0</td>
+<td>0</td>
+<td>7</td>
+<td>6</td>
+<td class = "double">7</td>
+<td>1</td>
+<td>1</td>
+<td>6</td>
+<td>6</td>
+<td>7</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td class = "words">Triplum</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>6</td>
+<td>0</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td class = "double">&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>1</td>
+<td>8</td>
+<td class = "double">&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>4</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td class = "words">Subt<i>r</i>iplu<i>m</i></td>
+<td>2</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>0</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td class = "double">[3]</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>6</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td class = "double">7</td>
+<td>&nbsp;</td>
+<td>2</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>2</td>
+<td>&nbsp;</td>
+</tr>
+</table>
+
+<p><span class = "sidenote">Continue this process till the first figure
+is reached.</span>
+Nother me shall<i>e</i> not cesse of the fyndyng<i>e</i> of that digit,
+neither of his triplacio<i>u</i>n, neþ<i>er</i> of the triplat-is
+<a class = "tag" name = "tag_art29" id = "tag_art29" href =
+"#note_art29">29</a><a class = "terms" name = "anterioracioun" id =
+"anterioracioun" href =
+"#terms_anterioracioun">anteriorac<i>i</i>o<i>u</i>n</a>, that is to
+sey, settyng forward<i>e</i> by .2. differences, Ne therof the
+vndre-triple to be put vndre the triple, Nether of the
+multiplicacio<i>u</i>n þ<i>er</i>of, Neither of the subtraccio<i>u</i>n,
+till<i>e</i> it come to the first figure, vnder the which<i>e</i> is a
+digitall<i>e</i> nombre to be found<i>e</i>, the which<i>e</i>
+with<i>e</i> his vndre-treblis most be had<i>e</i> in tribles,
+After-ward<i>e</i> w<i>i</i>t<i>h</i>out vnder-treblis to be had<i>e</i>
+into <a class = "terms" name = "produccioun" id = "produccioun" href =
+"#terms_produccioun">produccio<i>u</i>n</a>, settyng away all<i>e</i>
+that is ou<i>er</i> the hed<i>e</i> of the triplat nombre, After had
+into hymself<i>e</i> cubikly,
+<span class = "sidenote">Examples.</span>
+and sette all<i>e</i>-way that is ou<i>er</i> hym.</p>
+
+<table class = "grid outline float" summary = "example">
+<tr>
+<td class = "words">To be <a class = "terms" name = "cubicede" id =
+"cubicede" href = "#terms_cubicede">cubiced<i>e</i></a></td>
+<td>1</td>
+<td>7</td>
+<td>2</td>
+<td class = "double">8</td>
+<td>3</td>
+<td>2&nbsp;</td>
+<td>7</td>
+<td>6</td>
+<td>8</td>
+</tr>
+<tr>
+<td class = "words">The triple</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>3</td>
+<td class = "double">2</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>9</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td class = "words">The vnder triple</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>1</td>
+<td class = "double">2</td>
+<td>&nbsp;</td>
+<td>[3]</td>
+<td>&nbsp;</td>
+<td>3</td>
+<td>3</td>
+</tr>
+</table>
+
+<p class = "nospace">
+Also note wele that the p<i>ro</i>ducc<i>i</i>on comyng<i>e</i> of the
+ledyng of a digite found<i>e</i><a class = "tag" name = "tag_art30" id =
+"tag_art30" href = "#note_art30">30</a>
+me may adde to, and also w<i>i</i>t<i>h</i>-draw fro of the
+totall<i>e</i> nombre sette above that digit so found<i>e</i>.<a class =
+"tag" name = "tag_art31" id = "tag_art31" href = "#note_art31">31</a>
+<span class = "sidenote">The residue.</span>
+That done ought or nought most be the residue. If it be nought, It is
+open that the nombre p<i>ro</i>posed<i>e</i> was a cubik<i>e</i> nombre,
+And his rote a digit founde last w<i>i</i>t<i>h</i> the vnder-triples:
+If the rote therof <a class = "gloss" name = "wex" id = "wex" href =
+"#gloss_wex">wex</a> bad<i>e</i> in hym-self<i>e</i>, and
+afterward<i>e</i> p<i>ro</i>duct they shall<i>e</i> make the first
+fig<i>ur</i>es. And yf ought be in residue, kepe that
+w<i>i</i>t<i>h</i>out in the table; and it is open<i>e</i> that the
+nombre was not a cubik<i>e</i>. but a digit last founde
+w<i>i</i>t<i>h</i> the vndirtriplis is rote of the most cubik<i>e</i>
+vndre the nombre p<i>ro</i>posed<i>e</i> conteyned<i>e</i>, the
+which<i>e</i> rote yf it be had<i>e</i> in hym-self<i>e</i>,
+<span class = "sidenote">Special cases.</span>
+And aft<i>er</i>ward<i>e</i> in a p<i>ro</i>duct of that shall<i>e</i>
+growe the most cubik<i>e</i> vndre the nombre p<i>ro</i>posed<i>e</i>
+conteyned<i>e</i>, And yf that be added<i>e</i> to a cubik<i>e</i> the
+residue res<i>er</i>ued<i>e</i> in the table, woll<i>e</i> make the same
+figures that ye had<i>e</i> first.
+<span class = "sidenote">Special case.</span>
+<span class = "linenum">Fol. 56.</span>
+*And
+<span class = "pagenum">51</span>
+<a name = "page51" id = "page51"> </a>
+yf no digit after the anterioracio<i>u</i>n<a class = "tag" name =
+"tag_art32" id = "tag_art32" href = "#note_art32">32</a>
+may not be found<i>e</i>, than put ther<i>e</i> a cifre vndre a cifre
+vndir the third<i>e</i> figure, And put forward<i>e</i> þe
+fig<i>ur</i>es. Note also wele that yf in the nombre
+p<i>ro</i>posed<i>e</i> ther ben no place of thowsand<i>es</i>, me most
+begynne vnder the first figure in the extraccio<i>u</i>n of the rote.
+some vsen forto <a class = "gloss" name = "distingue" id = "distingue"
+href = "#gloss_distingue">distingue</a> the nombre by threes, and ay
+begynne forto wirch<i>e</i> vndre the first of the last <a class =
+"terms" name = "ternary" id = "ternary" href =
+"#terms_ternary">t<i>er</i>nary</a> other unco<i>m</i>plete nombre, the
+which<i>e</i> maner of op<i>er</i>acio<i>u</i>n accordeth<i>e</i>
+w<i>i</i>t<i>h</i> that before.</p>
+
+<span class = "sidenote">Examples.</span>
+
+<table class = "grid outline" summary = "example">
+<tr>
+<td class = "words">The residue</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td class = "double">0</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>1</td>
+<td>1</td>
+</tr>
+<tr>
+<td class = "words">The cubicand<i>us</i></td>
+<td>8&nbsp;</td>
+<td>0</td>
+<td>0</td>
+<td>0</td>
+<td>0</td>
+<td>0</td>
+<td class = "double">0</td>
+<td>8</td>
+<td>2</td>
+<td>4</td>
+<td>2</td>
+<td>4</td>
+<td>1</td>
+<td>9</td>
+</tr>
+<tr>
+<td class = "words">The triple</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td><a class = "tag" name = "tag_art33" id = "tag_art33" href =
+"#note_art33">33</a>&nbsp;</td>
+<td>0</td>
+<td>0</td>
+<td>&nbsp;</td>
+<td class = "double">&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>6</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td class = "words">The vndert<i>r</i>iple</td>
+<td>[2]</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>0</td>
+<td>0</td>
+<td>&nbsp;</td>
+<td class = "double">&nbsp;</td>
+<td>2</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>6</td>
+<td>2</td>
+<td>&nbsp;</td>
+<td></td>
+</tr>
+</table>
+
+<p class = "nospace">
+And this at this tyme suffiseth<i>e</i> in extraccio<i>u</i>n of nombres
+quadrat or cubik<i>es</i> <i>et</i>c.</p>
+
+<p class = "headnote"><span class = "headnote">
+Table of Numbers, &amp;c.</span></p>
+
+<span class = "sidenote">A table of numbers; probably from the
+Abacus.</span>
+
+<table class = "gloss" summary = "table of numbers">
+<tr class = "gloss">
+<td>1</td>
+<td>2</td>
+<td>3</td>
+<td>4</td>
+<td>5</td>
+</tr>
+<tr class = "orig">
+<td>one.</td>
+<td>x.</td>
+<td>an. hundred<i>e</i>&nbsp;/</td>
+<td>a thowsand<i>e</i>&nbsp;/</td>
+<td>x. thowsand<i>e</i>&nbsp;/</td>
+</tr>
+</table>
+
+<table class = "gloss" summary = "table of numbers">
+<tr class = "gloss">
+<td>6</td>
+<td>7</td>
+</tr>
+<tr class = "orig">
+<td>An hundred<i>e</i> thowsand<i>e</i>&nbsp;/</td>
+<td>A thowsand<i>e</i> tymes a thowsand<i>e</i>&nbsp;/</td>
+</tr>
+</table>
+
+<p class = "wide">x. thousand<i>e</i> tymes a thousand<i>e</i> / An
+hundred<i>e</i> thousand<i>e</i> tymes a thousand<i>e</i>
+A&nbsp;thousand<i>e</i> thousand<i>e</i> tymes a thousand<i>e</i> / this
+is the x place <i>et</i>c.</p>
+
+<p class = "center">[Ende.]</p>
+
+<hr class = "mid" />
+
+<div class = "footnote">
+
+<p><a name = "note_art1" id = "note_art1" href = "#tag_art1">1.</a>
+MS. Materiall<i>e</i>.</p>
+
+<p><a name = "note_art2" id = "note_art2" href = "#tag_art2">2.</a>
+MS. Formall<i>e</i>.</p>
+
+<p><a name = "note_art3" id = "note_art3" href = "#tag_art3">3.</a>
+‘the’ in MS.</p>
+
+<p><a name = "note_art4" id = "note_art4" href = "#tag_art4">4.</a>
+‘be’ in MS.</p>
+
+<p><a name = "note_art5" id = "note_art5" href = "#tag_art5">5.</a>
+‘and’ in MS.</p>
+
+<p><a name = "note_art6" id = "note_art6" href = "#tag_art6">6.</a>
+‘is’ in MS.</p>
+
+<p><a name = "note_art7" id = "note_art7" href = "#tag_art7">7.</a>
+6 in MS.</p>
+
+<p><a name = "note_art8" id = "note_art8" href = "#tag_art8">8.</a>
+0 in MS.</p>
+
+<p><a name = "note_art9" id = "note_art9" href = "#tag_art9">9.</a>
+2 in MS.</p>
+
+<p><a name = "note_art10" id = "note_art10" href = "#tag_art10">10.</a>
+<i>sic.</i></p>
+
+<p><a name = "note_art11" id = "note_art11" href = "#tag_art11">11.</a>
+‘And’ inserted in MS.</p>
+
+<p><a name = "note_art12" id = "note_art12" href = "#tag_art12">12.</a>
+‘4 the’ inserted in MS.</p>
+
+<p><a name = "note_art13" id = "note_art13" href = "#tag_art13">13.</a>
+‘to’ in MS.</p>
+
+<p><a name = "note_art14" id = "note_art14" href = "#tag_art14">14.</a>
+‘that’ repeated in MS.</p>
+
+<p><a name = "note_art15" id = "note_art15" href = "#tag_art15">15.</a>
+‘1’ in MS.</p>
+
+<p><a name = "note_art16" id = "note_art16" href = "#tag_art16">16.</a>
+Blank in MS.</p>
+
+<p><a name = "note_art17" id = "note_art17" href = "#tag_art17">17.</a>
+‘nought’ in MS.</p>
+
+<p><a name = "note_art18" id = "note_art18" href = "#tag_art18">18.</a>
+3 written for 2 in MS.</p>
+
+<p><a name = "note_art19" id = "note_art19" href = "#tag_art19">19.</a>
+7 in MS.</p>
+
+<p><a name = "note_art20" id = "note_art20" href = "#tag_art20">20.</a>
+runs on in MS.</p>
+
+<p><a name = "note_art21" id = "note_art21" href = "#tag_art21">21.</a>
+‘so’ in MS.</p>
+
+<p><a name = "note_art22" id = "note_art22" href = "#tag_art22">22.</a>
+‘nought’ in MS.</p>
+
+<p><a name = "note_art23" id = "note_art23" href = "#tag_art23">23.</a>
+MS. adds here: ‘wher-vpon<i>e</i> se the table in the next side of the
+next leef<i>e</i>.’</p>
+
+<p><a name = "note_art24" id = "note_art24" href = "#tag_art24">24.</a>
+110 in MS.</p>
+
+<p><a name = "note_art25" id = "note_art25" href = "#tag_art25">25.</a>
+0 in MS.</p>
+
+<p><a name = "note_art26" id = "note_art26" href = "#tag_art26">26.</a>
+double in MS.</p>
+
+<p><a name = "note_art27" id = "note_art27" href = "#tag_art27">27.</a>
+‘it hym-self<i>e</i>’ in MS.</p>
+
+<p><a name = "note_art28" id = "note_art28" href = "#tag_art28">28.</a>
+MS. adds here: ‘it setteth<i>e</i> a-way all<i>e</i> his respect.’</p>
+
+<p><a name = "note_art29" id = "note_art29" href = "#tag_art29">29.</a>
+‘aucterioracio<i>u</i>n’ in MS.</p>
+
+<p><a name = "note_art30" id = "note_art30" href = "#tag_art30">30.</a>
+MS. adds here: ’w<i>i</i>t<i>h</i> an vndre-triple / other of an
+vndre-triple in a triple or triplat is And after-ward<i>e</i>
+w<i>i</i>t<i>h</i> out vndre-triple other vndre-triplis in the
+p<i>ro</i>duct and ayene that p<i>ro</i>duct that cometh<i>e</i> of the
+ledyng<i>e</i> of a digit found<i>e</i> in hym-self<i>e</i>
+cubicall<i>e</i>’ /</p>
+
+<p><a name = "note_art31" id = "note_art31" href = "#tag_art31">31.</a>
+MS. adds here: ‘as ther had be a divisio<i>u</i>n made as it is
+opened<i>e</i> before.’</p>
+
+<p><a name = "note_art32" id = "note_art32" href = "#tag_art32">32.</a>
+MS. anteriocacio<i>u</i>n.</p>
+
+<p><a name = "note_art33" id = "note_art33" href = "#tag_art33">33.</a>
+4 in MS.</p>
+
+</div>
+
+</div> <!-- end div art -->
+
+
+<div class = "count">
+
+<span class = "pagenum">52</span>
+<a name = "page52" id = "page52"> </a>
+
+<p class = "illustration">
+<a name = "count" id = "count">
+<img src = "images/title_count.png" width = "323" height = "36"
+alt = "Accomptynge by counters."
+title = "Accomptynge by counters." /></a></p>
+
+<p class = "mynote">
+The original text was printed as a single continuous paragraph, with
+no break between speakers; all examples were shown inline. It has been
+broken up for this e-text.</p>
+
+<span class = "linenum">116 <i>b</i>.</span>
+
+<h4>* ¶ The seconde dialoge of accomptynge by counters.</h4>
+
+<p class = "inset"><i>Mayster.</i></p>
+
+<p><span class = "dropcap">N</span>owe that you haue learned the commen
+kyndes of Arithmetyke with the penne, you shall se the same art in
+cou<i>n</i>ters: whiche feate doth not only serue for them that can not
+write and rede, but also for them that can do bothe, but haue not at
+some tymes theyr penne or tables redye with them. This sorte is in two
+fourmes co<i>m</i>menly. The one by lynes, and the other without lynes:
+in that <a class = "gloss" name = "yt" id = "yt" href =
+"#gloss_yt">y<sup>t</sup></a> hath lynes, the lynes do stande for the
+order of places: and in y<sup>t</sup> that hath no lynes, there must be
+sette in theyr stede so many counters as shall nede, for eche lyne one,
+and they shall supplye the stede of the lynes.</p>
+
+<p><i>S.</i> By examples I shuld better p<i>er</i>ceaue your
+meanynge.</p>
+
+<p><i>M.</i> For example of the
+<span class = "linenum">117 <i>a</i></span>
+ly*nes:</p>
+
+<table class = "backline small float" summary = "counters example">
+<tr>
+<td>&nbsp;</td>
+<td>1</td>
+<td>0 0 0 0 0</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>1</td>
+<td>0 0 0 0</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>X</td>
+<td>1</td>
+<td>0 0 0</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>1</td>
+<td>0 0</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>1</td>
+<td>0</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>1</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+</table>
+
+<p class = "nospace">
+Lo here you se .vi. lynes whiche stande for syxe places so that the
+nethermost standeth for y<sup>e</sup> fyrst place, and the next aboue
+it, for the second: and so vpward tyll you come to the hyghest, which is
+the syxte lyne, and standeth for the syxte place.
+<span class = "sidenote">Numeration.</span>
+Now what is the valewe of euery place or lyne, you may perceaue by the
+figures whiche I haue set on them, which is accordynge as you learned
+before in the Numeration of figures by the penne: for the fyrste place
+is the place of vnities or ones, and euery counter set in that lyne
+betokeneth but one: <i>and</i> the seconde lyne is the place of 10, for
+euery counter there, standeth for 10. The thyrd lyne the place of
+hundredes: the fourth of thousandes: <i>and</i> so forth.</p>
+
+<p><i>S.</i> Syr I do perceaue that the same order is here of lynes, as
+was in the other figures
+<span class = "linenum">117 <i>b</i>.</span>
+*by places, so that you shall not nede longer to stande about
+Numeration, excepte there be any other difference.</p>
+
+<p><i>M.</i> Yf you do vndersta<i>n</i>de it, then how wyll you set
+1543?</p>
+
+<table class = "backline small float" summary = "counters example">
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>X</td>
+<td>1</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>5</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>4</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>3</td>
+<td>&nbsp;</td>
+</tr>
+</table>
+
+<p><i>S.</i> Thus, as I suppose.</p>
+
+<p><i>M.</i> You haue set y<sup>e</sup> places truely, but your figures
+be not mete for this vse:
+<span class = "pagenum">53</span>
+<a name = "page53" id = "page53"> </a>
+for the metest figure in this behalfe, is the figure of a cou<i>n</i>ter
+round, as you se here, where I haue expressed that same summe.</p>
+
+<table class = "backline float" summary = "counters example">
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>×</td>
+<td>o</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp; <span class = "five">°</span></td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o o o o</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o o o</td>
+<td>&nbsp;</td>
+</tr>
+</table>
+
+<p><i>S.</i> So that you haue not one figure for 2, nor 3, nor 4, and so
+forth, but as many digettes as you haue, you set in the lowest lyne: and
+for euery 10 you set one in the second line: and so of other. But I know
+not by what reason you set that one counter for 500 betwene two
+lynes.</p>
+
+<p><i>M.</i> you shall remember this, that when so euer you nede to set
+downe 5, 50, or 500, or 5000, or so forth any other nomber, whose
+numerator
+<span class = "linenum">118 <i>a</i>.</span>
+*is 5, you shall set one counter for it, in the next space aboue the
+lyne that it hath his denomination of, as in this example of that 500,
+bycause the numerator is 5, it must be set in a voyd space: and bycause
+the denominator is hundred, I&nbsp;knowe that his place is the voyde
+space next aboue hundredes, that is to say, aboue the thyrd lyne. And
+farther you shall marke, that in all workynge by this sorte, yf you
+shall sette downe any summe betwene 4 and 10, for the fyrste parte of
+that nomber you shall set downe 5, &amp; then so many counters more, as
+there reste no<i>m</i>bers aboue 5. And this is true bothe of digettes
+and articles. And for example I wyll set downe this su<i>m</i>me
+287965,</p>
+
+<table class = "backline float" summary = "counters example">
+<tr>
+<td>X</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp; &nbsp;o o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp; &nbsp;o o<span class = "five">°</span>o</td>
+</tr>
+<tr>
+<td>X</td>
+<td>&nbsp; &nbsp;o o<span class = "five">°</span></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o o<span class = "five">°</span>o o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+</table>
+
+<p class = "nospace">
+which su<i>m</i>me yf you marke well, you nede none other
+exa<i>m</i>ples for to lerne the numeration of
+<span class = "linenum">118 <i>b</i>.</span>
+*this forme. But this shal you marke, that as you dyd in the other kynde
+of arithmetike, set a pricke in the places of thousa<i>n</i>des, in this
+worke you shall sette a starre, as you se here.</p>
+
+<p class = "headnote"><span class = "headnote">
+Addition on the Counting Board.</span></p>
+
+<p><span class = "sidenote">Addition.</span>
+<i>S.</i> Then I perceave numeration, but I praye you, howe shall I do
+in this arte to adde two summes or more together?</p>
+
+<p><i>M.</i> The easyest way in this arte is, to adde but 2
+su<i>m</i>mes at ones together: how be it you may adde more, as I wyll
+tell you anone. Therfore when you wyll adde two su<i>m</i>mes, you shall
+fyrst set downe one of them, it <a class = "gloss" name = "forseth" id =
+"forseth" href = "#gloss_forseth">forseth</a> not whiche, <i>and</i>
+then by it drawe a lyne crosse the other lynes. And afterward set downe
+the other su<i>m</i>me, so that that lyne may be betwene them, as yf you
+wolde adde 2659 to 8342, you must set your su<i>m</i>mes as you se
+here.</p>
+
+<table class = "backline float" summary = "counters example">
+<col />
+<col class = "rightline" />
+<col />
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>X</td>
+<td>o<span class = "five">°</span>o o</td>
+<td>o o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o o o</td>
+<td>o<span class = "five">°</span></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o o o o</td>
+<td>&nbsp; <span class = "five">°</span></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o o</td>
+<td>o<span class = "five">°</span>o o o</td>
+</tr>
+</table>
+
+<p class = "nospace">
+And then yf you lyst, you
+<span class = "linenum">119 <i>a</i>.</span>
+*may adde the one to the other in the same place, or els you may adde
+them both together in a newe place: which waye, bycause it is moste
+playnest, I&nbsp;wyll showe you fyrst. Therfore wyl I begynne at the
+vnites, whiche in the fyrst su<i>m</i>me is but 2, <i>and</i> in
+y<sup>e</sup> second su<i>m</i>me 9, that maketh 11, those do I take vp,
+and for them I set 11 in the new roume, thus,</p>
+
+<span class = "pagenum">54</span>
+<a name = "page54" id = "page54"> </a>
+
+<table class = "backline float" summary = "counters example">
+<col />
+<col class = "rightline" />
+<col class = "rightline" />
+<col />
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>X</td>
+<td>o<span class = "five">°</span>o o</td>
+<td>o o</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o o o</td>
+<td>o<span class = "five">°</span></td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o o o o</td>
+<td>&nbsp; &nbsp; <span class = "five">°</span></td>
+<td>o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>o</td>
+</tr>
+</table>
+
+<p class = "nospace">
+Then do I take vp all y<sup>e</sup> articles vnder a hundred, which in
+the fyrst su<i>m</i>me are 40, and in the second summe 50, that maketh
+90: or you may saye better, that in the fyrste summe there are 4
+articles of 10, and in the seconde summe 5, which make 9, but then take
+hede that you sette them in theyr
+<span class = "linenum">119 <i>b</i>.</span>
+*ryght lynes as you se here.</p>
+
+<table class = "backline float" summary = "counters example">
+<col />
+<col class = "rightline" />
+<col class = "rightline" />
+<col />
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>X</td>
+<td>o<span class = "five">°</span>o o</td>
+<td>o o</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o o o</td>
+<td>o<span class = "five">°</span></td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span>o o o o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>o</td>
+</tr>
+</table>
+
+<p class = "nospace">
+Where I haue taken awaye 40 fro<i>m</i> the fyrste su<i>m</i>me, and 50
+from y<sup>e</sup> second, and in theyr stede I haue set 90 in the
+thyrde, whiche I haue set playnely y<sup>t</sup> you myght well perceaue
+it: how be it seynge that 90 with the 10 that was in y<sup>e</sup> thyrd
+roume all redy, doth make 100, I&nbsp;myghte better for those 6
+cou<i>n</i>ters set 1 in the thyrde lyne, thus:</p>
+
+<table class = "backline floatleft" summary = "counters example">
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>X</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o</td>
+</tr>
+</table>
+
+<p class = "nospace">
+For it is all one summe as you may se, but it is beste, neuer to set 5
+cou<i>n</i>ters in any line, for that may be done with 1 cou<i>n</i>ter
+in a hygher place.</p>
+
+<p><i>S.</i> I iudge that good reaso<i>n</i>, for many are vnnedefull,
+where one wyll serue.</p>
+
+<p><i>M.</i> Well, then
+<span class = "linenum">120 <i>a</i>.</span>
+*wyll I adde forth of hundredes: I&nbsp;fynde 3 in the fyrste summe, and
+6 in the seconde, whiche make 900, them do I take vp <i>and</i> set in
+the thyrd roume where is one hundred all redy, to whiche I put 900, and
+it wyll be 1000, therfore I set one cou<i>n</i>ter in the fourth lyne
+for them all, as you se here.</p>
+
+<table class = "backline float" summary = "counters example">
+<col />
+<col class = "rightline" />
+<col class = "rightline" />
+<col />
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>X</td>
+<td>o<span class = "five">°</span>o o</td>
+<td>o o</td>
+<td>o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>o</td>
+</tr>
+</table>
+
+<p class = "nospace">
+Then adde I y<sup>e</sup> thousandes together, whiche in the fyrst
+su<i>m</i>me are 8000, <i>and</i> in y<sup>e</sup> second 2000, that
+maketh 10000: them do I take vp fro<i>m</i> those two places, and for
+them I set one counter in the fyfte lyne, and then appereth as
+you&nbsp;se,</p>
+
+<table class = "backline floatleft" summary = "counters example">
+<tr>
+<td>&nbsp;</td>
+<td>o</td>
+</tr>
+<tr>
+<td>X</td>
+<td>o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o</td>
+</tr>
+</table>
+
+<p class = "nospace">
+to be 11001, for so many doth amount of the addition of 8342 to
+2659.</p>
+
+<p><span class = "linenum">120 <i>b</i>.</span>
+*<i>S.</i> Syr, this I do perceave: but how shall I set one su<i>m</i>me
+to an other, not chaungynge them to a thyrde place?</p>
+
+<p><i>M.</i> Marke well how I do it: I&nbsp;wyll adde together 65436,
+and 3245, whiche fyrste I set downe thus.</p>
+
+<table class = "backline float" summary = "counters example">
+<col />
+<col class = "rightline" />
+<col />
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>o&nbsp; &nbsp;<span class = "five">°</span></td>
+</tr>
+<tr>
+<td>X</td>
+<td>o o o</td>
+<td>&nbsp; <span class = "five">°</span></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o o</td>
+<td>o o o o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o o o o</td>
+<td>o o o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp; <span class = "five">°</span></td>
+<td>o&nbsp; &nbsp;<span class = "five">°</span></td>
+</tr>
+</table>
+
+<p class = "nospace">
+Then do I begynne with the smalest, which in the fyrst summe is <ins
+class = "correction" title = "invisible ‘5’ supplied by transcriber">5</ins>, that do I take vp, and wold put to the other 5 in
+the seconde summe, sauynge that two counters can not be set in a voyd
+place of 5, but for them bothe I must set 1 in the seconde lyne, which
+is the place of 10, therfore I take vp the 5 of the fyrst su<i>m</i>me,
+<i>and</i> the 5 of the seco<i>n</i>de, and for them I set 1 in the
+seco<i>n</i>d lyne,
+<span class = "linenum">121 <i>a</i>.</span>
+*as you se here.</p>
+
+<table class = "backline float" summary = "counters example">
+<col />
+<col class = "rightline" />
+<col />
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span></td>
+</tr>
+<tr>
+<td>X</td>
+<td>o o o</td>
+<td>&nbsp; <span class = "five">°</span></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o o</td>
+<td>o o o o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o o o o</td>
+<td>o o o o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>o</td>
+</tr>
+</table>
+
+<p class = "nospace">
+Then do I lyke wayes take vp the 4 counters of the fyrste su<i>m</i>me
+<i>and</i>
+<span class = "pagenum">55</span>
+<a name = "page55" id = "page55"> </a>
+seconde lyne (which make 40) and adde them to the 4 counters of the same
+lyne, in the second su<i>m</i>me, and it maketh 80, But as I sayde I
+maye not conueniently set aboue 4 cou<i>n</i>ters in one lyne, therfore
+to those 4 that I toke vp in the fyrst su<i>m</i>me, I&nbsp;take one
+also of the seconde su<i>m</i>me, and then haue I taken vp 50, for
+whiche 5 counters I sette downe one in the space ouer y<sup>e</sup>
+second lyne, as here doth appere.</p>
+
+<table class = "backline float" summary = "counters example">
+<col />
+<col class = "rightline" />
+<col />
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span></td>
+</tr>
+<tr>
+<td>X</td>
+<td>o o o</td>
+<td>&nbsp; <span class = "five">°</span></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o o</td>
+<td>o o o o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span>o o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>o</td>
+</tr>
+</table>
+
+<p class = "nospace">
+<span class = "linenum">121 <i>b</i></span>
+*and then is there 80, as well <a class = "gloss" name = "wt" id = "wt"
+href = "#gloss_wt">w<sup>t</sup></a> those 4 counters, as yf I had set
+downe y<sup>e</sup> other 4 also. Now do I take the 200 in the fyrste
+su<i>m</i>me, and adde them to the 400 in the seconde summe, and it
+maketh 600, therfore I take vp the 2 counters in the fyrste summe, and 3
+of them in the seconde summe, and for them 5 I set 1 in y<sup>e</sup>
+space aboue, thus.</p>
+
+<table class = "backline float" summary = "counters example">
+<col />
+<col class = "rightline" />
+<col />
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span></td>
+</tr>
+<tr>
+<td>X</td>
+<td>o o o</td>
+<td>&nbsp; <span class = "five">°</span></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span>o o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>o</td>
+</tr>
+</table>
+
+<p class = "nospace">
+Then I take y<sup>e</sup> 3000 in y<sup>e</sup> fyrste su<i>m</i>me,
+vnto whiche there are none in the second summe agreynge, therfore I do
+onely remoue those 3 counters from the fyrste summe into the seconde, as
+here doth appere.</p>
+
+<table class = "backline floatleft" summary = "counters example">
+<col class = "rightline" />
+<col />
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;o<span class = "five">°</span></td>
+</tr>
+<tr>
+<td>X</td>
+<td>&nbsp;o<span class = "five">°</span>o o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o&nbsp;<span class = "five">°</span></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;o<span class = "five">°</span>o o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;o</td>
+</tr>
+</table>
+
+<p class = "nospace">
+<span class = "linenum">122 <i>a</i></span>
+*And so you see the hole su<i>m</i>me, that amou<i>n</i>teth of the
+addytio<i>n</i> of 65436 with 3245 to be 6868[1]. And yf you haue marked
+these two exa<i>m</i>ples well, you nede no farther enstructio<i>n</i>
+in Addition of 2 only summes: but yf you haue more then two summes to
+adde, you may adde them thus.</p>
+
+<table class = "backline float" summary = "counters example">
+<col />
+<col class = "rightline" />
+<col class = "rightline" />
+<col />
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>X</td>
+<td>o o</td>
+<td>o o o o</td>
+<td>o<span class = "five">°</span></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span></td>
+<td>o o</td>
+<td>o<span class = "five">°</span>o o o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span>o</td>
+<td>o<span class = "five">°</span>o o</td>
+<td>o<span class = "five">°</span></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span>o o o</td>
+<td>o<span class = "five">°</span></td>
+<td>&nbsp; <span class = "five">°</span></td>
+</tr>
+</table>
+
+<p class = "nospace">
+Fyrst adde two of them, and then adde the thyrde, and y<sup>e</sup>
+fourth, or more yf there be so many: as yf I wolde adde 2679 with 4286
+and 1391. Fyrste I adde the two fyrste summes thus.
+<span class = "linenum">122 <i>b</i>.</span>
+*And then I adde the thyrde thereto thus.</p>
+
+<table class = "backline float" summary = "counters example">
+<col />
+<col class = "rightline" />
+<col class = "rightline" />
+<col />
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>X</td>
+<td>o</td>
+<td>o<span class = "five">°</span></td>
+<td>o o o<span class = "five">°</span></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o o o</td>
+<td>o<span class = "five">°</span>o o o</td>
+<td>o o o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span>o o o</td>
+<td>&nbsp; <span class = "five">°</span>o</td>
+<td>&nbsp; <span class = "five">°</span></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o</td>
+<td>&nbsp; <span class = "five">°</span></td>
+<td>o<span class = "five">°</span></td>
+</tr>
+</table>
+
+<p class = "nospace">
+And so of more yf you haue them.</p>
+
+<p class = "headnote"><span class = "headnote">
+Subtraction on the Counting Board.</span></p>
+
+<p><i>S.</i> Nowe I thynke beste that you passe forth to Subtraction,
+except there be any wayes to examyn this maner of Addition, then I
+thynke that were good to be knowen nexte.</p>
+
+<p><i>M.</i> There is the same profe here that is
+<span class = "sidenote">Subtraction.</span>
+in the other Addition by the penne, I&nbsp;meane Subtraction, for that
+onely is a sure waye: but consyderynge that Subtraction must be fyrste
+knowen, I&nbsp;wyl fyrste teache you the arte of Subtraction, and that
+by this example: I&nbsp;wolde subtracte 2892 out of 8746. These summes
+must I set downe as I dyd in Addition: but here it is best
+<span class = "linenum">116 <i>a</i> <ins class = "correction" title =
+"editor’s ‘sic’ for jump back in line numbering">(<i>sic</i>)</ins>.</span>
+*to set the lesser no<i>m</i>ber fyrste, thus.</p>
+
+<table class = "backline float" summary = "counters example">
+<col />
+<col class = "rightline" />
+<col />
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>X</td>
+<td>o o</td>
+<td>o o<span class = "five">°</span>o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span>o o</td>
+<td>o<span class = "five">°</span>o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span>o o o</td>
+<td>o o o o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o o</td>
+<td>o<span class = "five">°</span></td>
+</tr>
+</table>
+
+<p class = "nospace">
+Then shall I begynne to subtracte the greatest nombres fyrste (contrary
+to the vse of the penne)
+<span class = "pagenum">56</span>
+<a name = "page56" id = "page56"> </a>
+y<sup>t</sup> is the thousandes in this exa<i>m</i>ple: therfore I fynd
+amongest the thousandes 2, for which I withdrawe so many fro<i>m</i> the
+seconde summe (where are&nbsp;8) and so remayneth there 6, as this
+exa<i>m</i>ple showeth.</p>
+
+<table class = "backline float" summary = "counters example">
+<col />
+<col class = "rightline" />
+<col />
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span></td>
+</tr>
+<tr>
+<td>+</td>
+<td>o<span class = "five">°</span>o o</td>
+<td>o<span class = "five">°</span>o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span>o o o</td>
+<td>o o o o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o o</td>
+<td>o<span class = "five">°</span></td>
+</tr>
+</table>
+
+<p class = "nospace">
+Then do I lyke wayes with the hundredes, of whiche in the fyrste summe
+<span class = "linenum">116 <i>b</i>.</span>
+*I fynde 8, and is the seconde summe but 7, out of whiche I can not take
+8, therfore thus muste I do: I&nbsp;muste loke how moche my summe
+dyffereth from 10, whiche I fynde here to be 2, then must I bate for my
+su<i>m</i>me of 800, one thousande, and set downe the excesse of
+hundredes, that is to saye 2, for so moche 100[0] is more then I shuld
+take vp. Therfore fro<i>m</i> the fyrste su<i>m</i>me I take that 800,
+and from the second su<i>m</i>me where are 6000, I&nbsp;take vp one
+thousande, and leue 5000; but then set I downe the 200 unto the 700
+y<sup>t</sup> are there all redye, and make them 900 thus.</p>
+
+<table class = "backline float" summary = "counters example">
+<col />
+<col class = "rightline" />
+<col />
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>+</td>
+<td>&nbsp;</td>
+<td>&nbsp; <span class = "five">°</span></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span>o o o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o o<span class = "five">°</span>o o</td>
+<td>o o o o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o o</td>
+<td>o<span class = "five">°</span></td>
+</tr>
+</table>
+
+<p class = "nospace">
+Then come I to the articles of te<i>n</i>nes where in the fyrste
+su<i>m</i>me I fynde 90,
+<span class = "linenum">117 <i>a</i>.</span>
+*and in the seconde su<i>m</i>me but only 40: Now consyderyng that 90
+can not be bated from 40, I&nbsp;loke how moche y<sup>t</sup> 90 doth
+dyffer from the next summe aboue it, that is 100 (or elles whiche is all
+to one effecte, I&nbsp;loke how moch 9 doth dyffer fro<i>m</i> 10)
+<i>and</i> I fynd it to be 1, then in the stede of that 90, I&nbsp;do
+take from the second summe 100: but consyderynge that it is 10 to moche,
+I&nbsp;set downe 1 in y<sup>e</sup> nexte lyne beneth for it, as you se
+here.</p>
+
+<table class = "backline float" summary = "counters example">
+<col />
+<col class = "rightline" />
+<col />
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>+</td>
+<td>&nbsp;</td>
+<td>&nbsp; <span class = "five">°</span></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span>o o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp; <span class = "five">°</span></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o o</td>
+<td>o<span class = "five">°</span></td>
+</tr>
+</table>
+
+<p class = "nospace">
+Sauynge that here I haue set one counter in y<sup>e</sup> space in stede
+of 5 in y<sup>e</sup> nexte lyne. And thus haue I subtracted all saue
+two, which I must bate from the 6 in the second summe, and there wyll
+remayne 4, thus.</p>
+
+<table class = "backline float" summary = "counters example">
+<col class = "rightline" />
+<col />
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>=</td>
+<td>&nbsp; <span class = "five">°</span></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span>o o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp; <span class = "five">°</span></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o o o o</td>
+</tr>
+</table>
+
+<p class = "nospace">
+So y<sup>t</sup> yf I subtracte 2892 fro<i>m</i> 8746, the <a class =
+"gloss" name = "remayner" id = "remayner" href =
+"#gloss_remayner">remayner</a> wyll be 5854,
+<span class = "linenum">117 <i>b</i>.</span>
+*And that this is truely wrought, you maye proue by Addition: for yf you
+adde to this remayner the same su<i>m</i>me that you dyd subtracte, then
+wyll the formar su<i>m</i>me 8746 amount agayne.</p>
+
+<p><i>S.</i> That wyll I proue: and fyrst I set the su<i>m</i>me that
+was subtracted, which was 2892, <i>and</i> the<i>n</i> the remayner
+5854, thus.</p>
+
+<table class = "backline float" summary = "counters example">
+<col />
+<col class = "rightline" />
+<col />
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>||</td>
+<td>o o</td>
+<td>&nbsp; &nbsp; &nbsp; <span class = "five">°</span></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span>o o</td>
+<td>o o<span class = "five">°</span>o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o o<span class = "five">°</span>o o</td>
+<td>&nbsp; <span class = "five">°</span></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o o</td>
+<td>o o o o</td>
+</tr>
+</table>
+
+<p class = "nospace">
+Then do I adde fyrst y<sup>e</sup> 2 to 4, whiche maketh 6, so take I vp
+5 of those counters, and in theyr stede I sette 1 in the space, as here
+appereth.</p>
+
+<table class = "backline float" summary = "counters example">
+<col />
+<col class = "rightline" />
+<col />
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>||</td>
+<td>o o</td>
+<td>&nbsp; &nbsp; &nbsp; <span class = "five">°</span></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span>o o</td>
+<td>o<span class = "five">°</span>o o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span>o o o</td>
+<td>&nbsp; <span class = "five">°</span></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span></td>
+</tr>
+</table>
+
+<p class = "nospace">
+<span class = "linenum">118 <i>a</i>.</span>
+*Then do I adde the 90 nexte aboue to the 50, and it maketh 140,
+therfore I take vp those 6 counters, and for them I sette 1 to the
+hundredes in y<sup>e</sup> thyrde lyne, <i>and</i> 4 in y<sup>e</sup>
+<span class = "pagenum">57</span>
+<a name = "page57" id = "page57"> </a>
+second lyne, thus.</p>
+
+<table class = "backline float" summary = "counters example">
+<col />
+<col class = "rightline" />
+<col />
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>||</td>
+<td>o o</td>
+<td>&nbsp; <span class = "five">°</span></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span>o o</td>
+<td>o<span class = "five">°</span>o o o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>o o o o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp; <span class = "five">°</span>o</td>
+</tr>
+</table>
+
+<p class = "nospace">
+Then do I come to the hundredes, of whiche I fynde 8 in the fyrst summe,
+and 9 in y<sup>e</sup> second, that maketh 1700, therfore I take vp
+those 9 counters, and in theyr stede I sette 1 in the .iiii. lyne, and 1
+in the space nexte beneth, and 2 in the thyrde lyne, as you se here.</p>
+
+<table class = "backline float" summary = "counters example">
+<col />
+<col class = "rightline" />
+<col />
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>||</td>
+<td>o o</td>
+<td>o<span class = "five">°</span></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span>o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>o o o o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span></td>
+</tr>
+</table>
+
+<p class = "nospace">
+Then is there lefte in the fyrste summe but only 2000, whiche I shall
+take vp from thence, and set
+<span class = "linenum">118 <i>b</i>.</span>
+*in the same lyne in y<sup>e</sup> second su<i>m</i>me, to y<sup>e</sup>
+one y<sup>t</sup> is there all redy: <i>and</i> then wyll the hole
+su<i>m</i>me appere (as you may wel se) to be 8746, which was
+y<sup>e</sup> fyrst grosse summe, <i>and</i> therfore I do perceaue,
+that I hadde well subtracted before.</p>
+
+<table class = "backline float" summary = "counters example">
+<col class = "rightline" />
+<col />
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>X</td>
+<td>o o<span class = "five">°</span>o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o o<span class = "five">°</span></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o o o o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp; <span class = "five">°</span>o</td>
+</tr>
+</table>
+
+<p class = "nospace">
+And thus you may se how Subtraction maye be tryed by Addition.</p>
+
+<p><i>S.</i> I perceaue the same order here w<sup>t</sup>
+cou<i>n</i>ters, y<sup>t</sup> I lerned before in figures.</p>
+
+<p><i>M.</i> Then let me se howe can you trye Addition by
+Subtraction.</p>
+
+<p><i>S.</i> Fyrste I wyl set forth this exa<i>m</i>ple of
+Additio<i>n</i> where I haue added 2189 to 4988, and the hole
+su<i>m</i>me appereth to be 7177,</p>
+
+<table class = "backline float" summary = "counters example">
+<col />
+<col class = "rightline" />
+<col class = "rightline" />
+<col />
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>||</td>
+<td>o o</td>
+<td>o o o o</td>
+<td>o<span class = "five">°</span>o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o</td>
+<td>o<span class = "five">°</span>o o o</td>
+<td>o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span>o o</td>
+<td>o<span class = "five">°</span>o o</td>
+<td>o<span class = "five">°</span>o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span>o o o</td>
+<td>o<span class = "five">°</span>o o</td>
+<td>o<span class = "five">°</span>o</td>
+</tr>
+</table>
+
+<p class = "nospace">
+<span class = "linenum">119 <i>a</i>.</span>
+*Nowe to trye whether that su<i>m</i>me be well added or no, I&nbsp;wyll
+subtract one of the fyrst two su<i>m</i>mes from the thyrd, and yf I
+haue well done y<sup>e</sup> remayner wyll be lyke that other
+su<i>m</i>me. As for example: I&nbsp;wyll subtracte the fyrste summe
+from the thyrde, whiche I set thus in theyr order.</p>
+
+<table class = "backline float" summary = "counters example">
+<col />
+<col class = "rightline" />
+<col />
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>||</td>
+<td>o o</td>
+<td>o<span class = "five">°</span>o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o</td>
+<td>o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span>o o</td>
+<td>o<span class = "five">°</span>o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span>o o o</td>
+<td>o<span class = "five">°</span>o</td>
+</tr>
+</table>
+
+<p class = "nospace">
+Then do I subtract 2000 of the fyrste summe fro<i>m</i> y<sup>e</sup>
+second su<i>m</i>me, and then remayneth there 5000 thus.</p>
+
+<table class = "backline float" summary = "counters example">
+<col />
+<col class = "rightline" />
+<col />
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>X</td>
+<td>&nbsp;</td>
+<td>&nbsp; <span class = "five">°</span></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o</td>
+<td>o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span>o o</td>
+<td>o<span class = "five">°</span>o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span>o o o</td>
+<td>o<span class = "five">°</span>o</td>
+</tr>
+</table>
+
+<p class = "nospace">
+Then in the thyrd lyne, I subtract y<sup>e</sup> 100 of the fyrste
+summe, fro<i>m</i> the second su<i>m</i>me, where is onely 100 also, and
+then in y<sup>e</sup> thyrde lyne <a class = "gloss" name = "resteth1"
+id = "resteth1" href = "#gloss_resteth">resteth</a> nothyng. Then in the
+second lyne with his space ouer hym, I&nbsp;fynde 80, which I shuld
+subtract
+<span class = "linenum">119 <i>b</i>.</span>
+*from the other su<i>m</i>me, then seyng there are but only 70 I must
+take it out of some hygher summe, which is here only 5000, therfore I
+take vp 5000, and seyng that it is to moch by 4920, I&nbsp;sette downe
+so many in the seconde roume, whiche with the 70 beynge there all redy
+do make 4990, &amp; then the summes doth stande thus.</p>
+
+<table class = "backline float" summary = "counters example">
+<col />
+<col class = "rightline" />
+<col />
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>||</td>
+<td>&nbsp;</td>
+<td>o o o o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span>o o o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span>o o o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span>o o o</td>
+<td>o<span class = "five">°</span>o</td>
+</tr>
+</table>
+
+<p class = "nospace">
+Yet remayneth there in the fyrst su<i>m</i>me 9, to be bated from the
+second summe, where in that place of vnities dothe appere only 7, then I
+muste bate a hygher su<i>m</i>me, that is to saye 10, but seynge that 10
+is more then 9 (which I shulde abate) by 1, therfore shall I take vp one
+counter from the seconde lyne, <i>and</i> set downe the same in the
+fyrst
+<span class = "linenum">120 <i>a</i>.</span>
+*or
+<span class = "pagenum">58</span>
+<a name = "page58" id = "page58"> </a>
+lowest lyne, as you se here.</p>
+
+<table class = "backline float" summary = "counters example">
+<col class = "rightline" />
+<col />
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>||</td>
+<td>o o o o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span>o o o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span>o o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span>o o</td>
+</tr>
+</table>
+
+<p class = "nospace">
+And so haue I ended this worke, <i>and</i> the su<i>m</i>me appereth to
+be y<sup>e</sup> same, whiche was y<sup>e</sup> seconde summe of my
+addition, and therfore I perceaue, I&nbsp;haue wel done.</p>
+
+<p><i>M.</i> To stande longer about this, it is but folye: excepte that
+this you maye also vnderstande, that many do begynne to subtracte with
+counters, not at the hyghest su<i>m</i>me, as I haue taught you, but at
+the nethermoste, as they do vse to adde: and when the summe to be
+abatyd, in any lyne appeareth greater then the other, then do they
+borowe one of the next hygher roume, as for example: yf they shuld abate
+1846 from 2378, they set y<sup>e</sup> summes thus.</p>
+
+<table class = "backline float" summary = "counters example">
+<col />
+<col class = "rightline" />
+<col />
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>||</td>
+<td>o</td>
+<td>o o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span>o o</td>
+<td>o o o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o o o o</td>
+<td>o<span class = "five">°</span>o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span></td>
+<td>o<span class = "five">°</span>o o</td>
+</tr>
+</table>
+
+<p class = "nospace">
+<span class = "linenum">120 <i>b</i>.</span>
+*And fyrste they take 6 whiche is in the lower lyne, and his space from
+8 in the same roumes, in y<sup>e</sup> second su<i>m</i>me, and yet
+there remayneth 2 counters in the lowest lyne. Then in the second lyne
+must 4 be subtracte from 7, and so remayneth there 3. Then 8 in the
+thyrde lyne and his space, from 3 of the second summe can not be,
+therfore do they bate it from a hygher roume, that is, from 1000, and
+bycause that 1000 is to moch by 200, therfore must I sette downe 200 in
+the thyrde lyne, after I haue taken vp 1000 from the fourth lyne: then
+is there yet 1000 in the fourth lyne of the fyrst summe, whiche yf I
+withdrawe from the seconde summe, then doth all y<sup>e</sup> figures
+stande in this order.</p>
+
+<table class = "backline float" summary = "counters example">
+<col class = "rightline" />
+<col />
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>||</td>
+<td></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp; <span class = "five">°</span></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o o o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o o</td>
+</tr>
+</table>
+
+<p class = "nospace">
+So that (as you se) it differeth not greatly whether you begynne
+subtractio<i>n</i> at the hygher lynes, or at
+<span class = "linenum">121 <i>a</i>.</span>
+*the lower. How be it, as some menne lyke the one waye beste, so some
+lyke the other: therfore you now knowyng bothe, may vse whiche you
+lyst.</p>
+
+<p class = "headnote"><span class = "headnote">
+Multiplication by Counters.</span></p>
+
+<p><span class = "sidenote">Multiplication.</span>
+But nowe touchynge Multiplicatio<i>n</i>: you shall set your
+no<i>m</i>bers in two roumes, as you dyd in those two other kyndes, but
+so that the multiplier be set in the fyrste roume. Then shall you begyn
+with the hyghest no<i>m</i>bers of y<sup>e</sup> seconde roume, and
+multiply them fyrst after this sort. Take that ouermost lyne in your
+fyrst workynge, as yf it were the lowest lyne, setting on it some
+mouable marke, as you lyste, and loke how many counters be in hym, take
+them vp, and for them set downe the hole multyplyer, so many tymes as
+you toke vp counters, reckenyng, I&nbsp;saye that lyne for the vnites:
+<i>and</i> when you haue so done with the hygheest no<i>m</i>ber then
+come to the nexte lyne beneth, <i>and</i> do euen so with it, and so
+with y<sup>e</sup> next, tyll you haue done all. And yf there be any
+nomber in a space, then for it
+<span class = "linenum">121 <i>b</i>.</span>
+*shall you take y<sup>e</sup> multiplyer 5 tymes, and then must you
+recken that lyne for the vnites whiche is nexte beneth that space: or
+els
+<span class = "pagenum">59</span>
+<a name = "page59" id = "page59"> </a>
+after a shorter way, you shall take only halfe the multyplyer, but then
+shall you take the lyne nexte aboue that space, for the lyne of vnites:
+but in suche workynge, yf chau<i>n</i>ce your multyplyer be an odde
+nomber, so that you can not take the halfe of it iustly, then muste you
+take the greater halfe, and set downe that, as if that it were the iuste
+halfe, and farther you shall set one cou<i>n</i>ter in the space beneth
+that line, which you recken for the lyne of vnities, or els only remoue
+forward the same that is to be multyplyed.</p>
+
+<p><i>S.</i> Yf you set forth an example hereto I thynke I shal perceaue
+you.</p>
+
+<p><i>M.</i> Take this exa<i>m</i>ple: I&nbsp;wold multiply 1542 by 365,
+therfore I set y<sup>e</sup> nombers thus.</p>
+
+<table class = "backline float" summary = "counters example">
+<col />
+<col class = "rightline" />
+<col />
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>||</td>
+<td>&nbsp;</td>
+<td>o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o o o</td>
+<td>&nbsp; <span class = "five">°</span></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span></td>
+<td>o o o o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp; <span class = "five">°</span></td>
+<td>o o</td>
+</tr>
+</table>
+
+<p class = "nospace">
+<span class = "linenum">122 <i>a</i>.</span>
+*Then fyrste I begynne at the 1000 in y<sup>e</sup> hyghest roume, as yf
+it were y<sup>e</sup> fyrst place, &amp; I take it vp, settynge downe
+for it so often (that is ones) the multyplyer, which is 365, thus, as
+you se here:</p>
+
+<table class = "backline float" summary = "counters example">
+<col />
+<col class = "rightline" />
+<col class = "rightline" />
+<col />
+<col />
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td class = "plain">&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>o o o</td>
+<td class = "plain">&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span></td>
+<td class = "plain">&nbsp;</td>
+</tr>
+<tr>
+<td>X</td>
+<td>&nbsp;</td>
+<td>&nbsp; <span class = "five">°</span></td>
+<td>&nbsp;</td>
+<td class = "plain">
+<img src = "images/finger_left.gif" width = "30" height = "13"
+alt = "&lt;--" /></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o o o</td>
+<td>&nbsp;</td>
+<td>&nbsp; <span class = "five">°</span></td>
+<td class = "plain">&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span></td>
+<td>o o o o</td>
+<td>&nbsp;</td>
+<td class = "plain">&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp; <span class = "five">°</span></td>
+<td>o o</td>
+<td>&nbsp;</td>
+<td class = "plain">&nbsp;</td>
+</tr>
+</table>
+
+<p class = "nospace">
+where for the one counter taken vp from the fourth lyne, I&nbsp;haue
+sette downe other 6, whiche make y<sup>e</sup> su<i>m</i>me of the
+multyplyer, reckenynge that fourth lyne, as yf it were the fyrste:
+whiche thyng I haue marked by the hand set at the begynnyng of
+y<sup>e</sup> same,</p>
+
+<p><i>S.</i> I perceaue this well: for in dede, this summe that you haue
+set downe is 365000, for so moche doth amount
+<span class = "linenum">122 <i>b</i>.</span>
+*of 1000, multiplyed by 365.</p>
+
+<p><i>M.</i> Well the<i>n</i> to go forth, in the nexte space I fynde
+one counter which I remoue forward but take not vp, but do (as in such
+case I must) set downe the greater halfe of my multiplier (seyng it is
+an odde no<i>m</i>ber) which is 182, <i>and</i> here I do styll let that
+fourth place stand, as yf it were y<sup>e</sup> fyrst:</p>
+
+<table class = "backline float" summary = "counters example">
+<col />
+<col class = "rightline" />
+<col class = "rightline" />
+<col class = "rightline" />
+<col />
+<col />
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>o o o</td>
+<td>o</td>
+<td class = "plain">&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span></td>
+<td>o<span class = "five">°</span>o o</td>
+<td class = "plain">&nbsp;</td>
+</tr>
+<tr>
+<td>||</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp; <span class = "five">°</span></td>
+<td>o<span class = "five">°</span>o</td>
+<td class = "plain">
+<img src = "images/finger_left.gif" width = "30" height = "13"
+alt = "&lt;--" /></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o o o</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp; <span class = "five">°</span></td>
+<td class = "plain">&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span></td>
+<td>o o o o</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td class = "plain">&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp; <span class = "five">°</span></td>
+<td>o o</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td class = "plain">&nbsp;</td>
+</tr>
+</table>
+
+<p class = "nospace">
+as in this fourme you se, where I haue set this multiplycatio<i>n</i>
+with y<sup>e</sup> other: but for the ease of your
+vndersta<i>n</i>dynge, I&nbsp;haue set a lytell lyne betwene them: now
+shulde they both in one su<i>m</i>me stand thus.</p>
+
+<table class = "backline float" summary = "counters example">
+<col />
+<col class = "rightline" />
+<col class = "rightline" />
+<col />
+<col />
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>o o o o o</td>
+<td class = "plain">&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>o o o o</td>
+<td class = "plain">&nbsp;</td>
+</tr>
+<tr>
+<td>||</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span>o</td>
+<td class = "plain">
+<img src = "images/finger_left.gif" width = "30" height = "13"
+alt = "&lt;--" /></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o o o</td>
+<td>&nbsp;</td>
+<td>&nbsp; <span class = "five">°</span></td>
+<td class = "plain">&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span></td>
+<td>o o o o</td>
+<td>&nbsp;</td>
+<td class = "plain">&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp; <span class = "five">°</span></td>
+<td>o o</td>
+<td>&nbsp;</td>
+<td class = "plain">&nbsp;</td>
+</tr>
+</table>
+
+<p class = "nospace">
+<span class = "linenum">123 <i>a</i>.</span>
+*Howe be it an other fourme to multyplye suche cou<i>n</i>ters i<i>n</i>
+space is this: Fyrst to remoue the fynger to the lyne nexte benethe
+y<sup>e</sup> space, <i>and</i> then to take vp y<sup>e</sup>
+cou<i>n</i>ter, <i>and</i> to set downe y<sup>e</sup> multiplyer .v.
+tymes, as here you&nbsp;se.</p>
+
+<!-- 5th column typo 35500 for 36500 -->
+
+<table class = "backline" summary = "counters example">
+<col />
+<col class = "rightline" />
+<col class = "rightline" />
+<col class = "rightline" />
+<col class = "rightline" />
+<col class = "rightline" />
+<col class = "rightline" />
+<col class = "rightline" />
+<col class = "rightline" />
+<col />
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>o o o</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span></td>
+<td>o o o</td>
+<td>o o o</td>
+<td>o o o</td>
+<td>o o o</td>
+<td>o o o</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp; <span class = "five">°</span></td>
+<td>o<span class = "five">°</span></td>
+<td><ins class = "correction" title = "shown as printed">&nbsp; <span
+class = "five">°</span></ins></td>
+<td>o<span class = "five">°</span></td>
+<td>o<span class = "five">°</span></td>
+<td>o<span class = "five">°</span></td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>
+<span class = "finger left">
+<img src = "images/finger.gif" width = "30" height = "13"
+alt = "--&gt;" /></span>
+X</td>
+<td>o o o</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp; <span class = "five">°</span></td>
+<td>&nbsp; <span class = "five">°</span></td>
+<td>&nbsp; <span class = "five">°</span></td>
+<td>&nbsp; <span class = "five">°</span></td>
+<td>&nbsp; <span class = "five">°</span></td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span></td>
+<td>o o o o</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp; <span class = "five">°</span></td>
+<td>o o</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+</table>
+
+<p class = "nospace">
+Which su<i>m</i>mes yf you do adde together into one su<i>m</i>me, you
+shal p<i>er</i>ceaue that it wyll be y<sup>e</sup>
+<span class = "pagenum">60</span>
+<a name = "page60" id = "page60"> </a>
+same y<sup>t</sup> appeareth of y<sup>e</sup> other worki<i>n</i>g
+before, so that
+<span class = "linenum">123 <i>b</i>.</span>
+*bothe sortes are to one entent, but as the other is much shorter, so
+this is playner to reason, for suche as haue had small exercyse in this
+arte. Not withstandynge you maye adde them in your mynde before you
+sette them downe, as in this exa<i>m</i>ple, you myghte haue sayde 5
+tymes 300 is 1500, <i>and</i> 5 tymes 60 is 300, also 5 tymes 5 is 25,
+whiche all put together do make 1825, which you maye at one tyme set
+downe yf you lyste. But nowe to go forth, I&nbsp;must remoue the hand to
+the nexte counters, whiche are in the second lyne, and there must I take
+vp those 4 counters, settynge downe for them my multiplyer 4 tymes,
+whiche thynge other I maye do at 4 tymes seuerally, or elles I may
+gather that hole summe in my mynde fyrste, and then set it downe: as to
+saye 4 tymes 300 is 1200: 4 tymes 60 are 240: and 4 tymes 5 make 20:
+y<sup>t</sup> is in all 1460, y<sup>t</sup> shall I set downe also: as
+here you&nbsp;se.</p>
+
+<table class = "backline float" summary = "counters example">
+<col />
+<col class = "rightline" />
+<col class = "rightline" />
+<col class = "rightline" />
+<col />
+<col />
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp; <span class = "five">°</span></td>
+<td>&nbsp;</td>
+<td class = "plain">&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>o o o o</td>
+<td>o</td>
+<td class = "plain">&nbsp;</td>
+</tr>
+<tr>
+<td>X</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span>o</td>
+<td>o o o o</td>
+<td class = "plain">&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o o o</td>
+<td>&nbsp;</td>
+<td>&nbsp; <span class = "five">°</span></td>
+<td>o<span class = "five">°</span></td>
+<td class = "plain">&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span></td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td class = "plain"><img src = "images/finger_left.gif" width = "30"
+height = "13"
+alt = "&lt;--" /></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp; <span class = "five">°</span></td>
+<td>o o</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td class = "plain">&nbsp;</td>
+</tr>
+</table>
+
+<p class = "nospace">
+<span class = "linenum">124 <i>a</i>.</span>
+*whiche yf I ioyne in one summe with the formar nombers, it wyll appeare
+thus.</p>
+
+<table class = "backline floatleft" summary = "counters example">
+<col />
+<col class = "rightline" />
+<col class = "rightline" />
+<col />
+<tr>
+<td class = "plain">&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp; <span class = "five">°</span></td>
+</tr>
+<tr>
+<td class = "plain">&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span></td>
+</tr>
+<tr>
+<td class = "plain">&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>o o</td>
+</tr>
+<tr>
+<td class = "plain">
+<img src = "images/finger.gif" width = "30" height = "13"
+alt = "&lt;--" /></td>
+<td>o o o</td>
+<td>&nbsp;</td>
+<td>o</td>
+</tr>
+<tr>
+<td class = "plain">&nbsp;</td>
+<td>o<span class = "five">°</span></td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td class = "plain">&nbsp;</td>
+<td>&nbsp; <span class = "five">°</span></td>
+<td>o o</td>
+<td>&nbsp;</td>
+</tr>
+</table>
+
+<p class = "nospace">
+Then to ende this multiplycation, I remoue the fynger to the lowest
+lyne, where are onely 2, them do I take vp, and in theyr stede do I set
+downe twyse 365, that is 730, for which I set
+<span class = "linenum">124 <i>b</i>.</span>
+*one in the space aboue the thyrd lyne for 500, and 2 more in the thyrd
+lyne with that one that is there all redye, and the reste in theyr
+order, <i>and</i> so haue I ended the hole summe thus.</p>
+
+<table class = "backline float" summary = "counters example">
+<col class = "rightline" />
+<col class = "rightline" />
+<col />
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp; &nbsp;</td>
+<td>&nbsp; <span class = "five">°</span></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>o o</td>
+</tr>
+<tr>
+<td>o o o</td>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span>o o</td>
+</tr>
+<tr>
+<td>o<span class = "five">°</span></td>
+<td>&nbsp;</td>
+<td>o o o</td>
+</tr>
+<tr>
+<td>&nbsp; <span class = "five">°</span></td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+</table>
+
+<p class = "nospace">
+Wherby you se, that 1542 (which is the nomber of yeares syth Ch[r]ystes
+incarnation) beyng multyplyed by 365 <ins class = "correction" title =
+"open parenthesis missing">(which</ins> is the nomber of dayes in one
+yeare) dothe amounte vnto 562830, which declareth y<sup>e</sup>
+no<i>m</i>ber of daies sith Chrystes incarnatio<i>n</i> vnto the ende of
+1542<a class = "tag" name = "tag_count1" id = "tag_count1" href =
+"#note_count1">1</a>
+yeares. (besyde 385 dayes and 12 houres for lepe yeares).</p>
+
+<p><i>S.</i> Now wyll I proue by an other exa<i>m</i>ple, as this: 40
+labourers (after 6&nbsp;d. y<sup>e</sup> day for eche man) haue wrought
+28 dayes, I&nbsp;wold
+<span class = "linenum">125 <i>a</i>.</span>
+*know what theyr wages doth amou<i>n</i>t vnto: In this case muste I
+worke doublely: fyrst I must multyplye the nomber of the labourers by
+y<sup>e</sup> wages of a man for one day, so wyll y<sup>e</sup> charge
+of one daye amount: then secondarely shall I multyply that charge of one
+daye, by the hole nomber of dayes, <i>and</i> so wyll the hole summe
+appeare: fyrst therefore I shall set the su<i>m</i>mes thus.</p>
+
+<table class = "backline floatleft" summary = "counters example">
+<col class = "rightline" />
+<col />
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o o o o</td>
+</tr>
+<tr>
+<td>o<span class = "five">°</span></td>
+<td>&nbsp;</td>
+</tr>
+</table>
+
+<span class = "pagenum">61</span>
+<a name = "page61" id = "page61"> </a>
+<p class = "nospace">
+Where in the fyrste space is the multyplyer (y<sup>t</sup> is one dayes
+wages for one man) <i>and</i> in the second space is set the nomber of
+the worke men to be multyplyed: the<i>n</i> saye I, 6 tymes 4
+(reckenynge that second lyne as the lyne of vnites) maketh 24, for
+whiche summe I shulde set 2 counters in the thyrde lyne, and 4 in the
+seconde, therfore do I set 2 in the thyrde lyne, and let the 4 stand
+styll in the seconde lyne, thus.*
+<span class = "linenum">125 <i>b</i>.</span></p>
+
+<table class = "backline float" summary = "counters example">
+<col class = "rightline" />
+<col />
+<tr>
+<td>&nbsp; &nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o o o o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+</table>
+
+<p class = "nospace">
+So <a class = "gloss" name = "apwereth" id = "apwereth" href =
+"#gloss_apwereth">apwereth</a> the hole dayes wages to be 240d’. that is
+20&nbsp;s. Then do I multiply agayn the same summe by the no<i>m</i>ber
+of dayes and fyrste I sette the nombers, thus.</p>
+
+<table class = "backline floatleft" summary = "counters example">
+<col class = "rightline" />
+<col />
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o o</td>
+</tr>
+<tr>
+<td>o o</td>
+<td>o o o o</td>
+</tr>
+<tr>
+<td>o<span class = "five">°</span>o o</td>
+<td>&nbsp;</td>
+</tr>
+</table>
+
+<p class = "nospace">
+The<i>n</i> bycause there are counters in dyuers lynes, I&nbsp;shall
+begynne with the hyghest, and take them vp, settynge for them the
+multyplyer so many tymes, as I toke vp counters, y<sup>t</sup> is twyse,
+then wyll y<sup>e</sup> su<i>m</i>me stande thus.</p>
+
+<table class = "backline float" summary = "counters example">
+<col class = "rightline" />
+<col />
+<tr>
+<td>&nbsp; &nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp; <span class = "five">°</span></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o o o o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+</table>
+
+<p class = "nospace">
+Then come I to y<sup>e</sup> seconde lyne, and take vp those 4
+cou<i>n</i>ters, settynge for them the multiplyer foure tymes, so wyll
+the hole summe appeare thus.*
+<span class = "linenum">126 <i>a</i>.</span></p>
+
+<table class = "backline floatleft" summary = "counters example">
+<col class = "rightline" />
+<col />
+<tr>
+<td>&nbsp; &nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span>o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+</table>
+
+<p class = "nospace">
+So is the hole wages of 40 workeme<i>n</i>, for 28 dayes (after 6d’.
+eche daye for a man) 6720d’. that is 560&nbsp;s. or 28&nbsp;l’i.</p>
+
+<p class = "headnote"><span class = "headnote">
+Division on the Counting Board.</span></p>
+
+<p><span class = "sidenote">Diuision.</span>
+<i>M.</i> Now if you wold proue Multiplycatio<i>n</i>, the surest way is
+by Dyuision: therfore wyll I ouer passe it tyll I haue taught you
+y<sup>e</sup> arte of Diuision, whiche you shall worke thus. Fyrste
+sette downe the Diuisor for feare of forgettynge, and then set the
+nomber that shalbe deuided, at y<sup>e</sup> ryghte syde, so farre from
+the diuisor, that the quotient may be set betwene them: as for
+exa<i>m</i>ple: Yf 225 shepe cost 45&nbsp;l’i. what dyd euery shepe
+cost? To knowe this, I&nbsp;shulde diuide the hole summe, that is
+45&nbsp;l’i. by 225, but that can not be, therfore must I fyrste reduce
+that 45&nbsp;l’i. into a lesser denomination, as into shyllynges: then I
+multiply 45 by 20, and it is 900, that summe shall I diuide by the
+no<i>m</i>ber of
+<span class = "linenum">126 <i>b</i>.</span>
+*shepe, whiche is 225, these two nombers therfore I sette thus.</p>
+
+<table class = "backline float" summary = "counters example">
+<col class = "rightline" />
+<col class = "rightline" />
+<col />
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp; &nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>o o</td>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span>o o o</td>
+</tr>
+<tr>
+<td>o o</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp; <span class = "five">°</span></td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+</table>
+
+<p class = "nospace">
+Then begynne I at the hyghest lyne of the diuident, and seke how often I
+may haue the diuisor therin, and that maye I do 4 tymes, then say I, 4
+tymes 2 are 8, whyche yf I take from 9, there resteth but 1, thus</p>
+
+<table class = "backline float" summary = "counters example">
+<col class = "rightline" />
+<col class = "rightline" />
+<col />
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>o o</td>
+<td>&nbsp;</td>
+<td>o</td>
+</tr>
+<tr>
+<td>o o</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp; <span class = "five">°</span></td>
+<td>o o o o</td>
+<td>&nbsp;</td>
+</tr>
+</table>
+
+<p class = "nospace">
+And bycause I founde the diuisor 4 tymes in the diuidente, I&nbsp;haue
+set (as you se) 4 in the myddle roume, which
+<span class = "linenum">127 <i>a</i>.</span>
+*is the place of the quotient: but now must I take the reste of the
+diuisor as often out of the remayner: therfore come
+<span class = "pagenum">62</span>
+<a name = "page62" id = "page62"> </a>
+I&nbsp;to the seconde lyne of the diuisor, sayeng 2 foure tymes make 8,
+take 8 from 10, <i>and</i> there resteth 2, thus.</p>
+
+<table class = "backline float" summary = "counters example">
+<col />
+<col class = "rightline" />
+<col class = "rightline" />
+<col />
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>||</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o o</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o o</td>
+<td>&nbsp;</td>
+<td>o o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp; <span class = "five">°</span></td>
+<td>o o o o</td>
+<td>&nbsp;</td>
+</tr>
+</table>
+
+<p class = "nospace">
+Then come I to the lowest nomber, which is 5, and multyply it 4 tymes,
+so is it 20, that take I from 20, and there remayneth nothynge, so that
+I se my quotient to be 4, whiche are in valewe shyllynges, for so was
+the diuident: and therby I knowe, that yf 225 shepe dyd coste
+45&nbsp;l’i. euery shepe coste 4&nbsp;s.</p>
+
+<p><i>S.</i> This can I do, as you shall perceaue by this
+exa<i>m</i>ple: Yf 160 sowldyars do spende euery moneth 68&nbsp;l’i.
+what spendeth eche man? Fyrst
+<span class = "linenum">127 <i>b</i>.</span>
+*bycause I can not diuide the 68 by 160, therfore I wyll turne the
+pou<i>n</i>des into pennes by multiplicacio<i>n</i>, so shall there be
+16320&nbsp;d’. Nowe muste I diuide this su<i>m</i>me by the nomber of
+sowldyars, therfore I set the<i>m</i> i<i>n</i> order, thus.</p>
+
+<table class = "backline float" summary = "counters example">
+<col />
+<col class = "rightline" />
+<col class = "rightline" />
+<col />
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp; &nbsp;</td>
+<td>o</td>
+</tr>
+<tr>
+<td>||</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o</td>
+<td>&nbsp;</td>
+<td>o o o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span></td>
+<td>&nbsp;</td>
+<td>o o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+</table>
+
+<p class = "nospace">
+Then begyn I at the hyghest place of the diuidente, sekynge my diuisor
+there, whiche I fynde ones, Therfore set I 1 in the nether lyne.</p>
+
+<p><i>M.</i> Not in the nether line of the hole summe, but in the nether
+lyne of that worke, whiche is the thyrde lyne.</p>
+
+<p><i>S.</i> So standeth it with reason.</p>
+
+<p><i>M.</i> Then thus do they stande.*
+<span class = "linenum">128 <i>a</i>.</span></p>
+
+<table class = "backline float" summary = "counters example">
+<col />
+<col class = "rightline" />
+<col class = "rightline" />
+<col />
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>||</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o</td>
+<td>o</td>
+<td>o o o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span></td>
+<td>&nbsp;</td>
+<td>o o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+</table>
+
+<p class = "nospace">
+Then seke I agayne in the reste, how often I may fynde my diuisor, and I
+se that in the 300 I myghte fynde 100 thre tymes, but then the 60 wyll
+not be so often founde in 20, therfore I take 2 for my quotient: then
+take I 100 twyse from 300, and there resteth 100, out of whiche with the
+20 (that maketh 120) I&nbsp;may take 60 also twyse, and then standeth
+the nombers thus,</p>
+
+<table class = "backline float" summary = "counters example">
+<col />
+<col class = "rightline" />
+<col class = "rightline" />
+<col />
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp; &nbsp;</td>
+</tr>
+<tr>
+<td>||</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o</td>
+<td>o</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span></td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>o o</td>
+<td>&nbsp;</td>
+</tr>
+</table>
+
+<p class = "nospace">
+<span class = "linenum">128 <i>b</i>.</span>
+*where I haue sette the quotient 2 in the lowest lyne: So is euery
+sowldyars portion 102&nbsp;d’. that is 8&nbsp;s. 6&nbsp;d’.</p>
+
+<p><i>M.</i> But yet bycause you shall perceaue iustly the reason of
+Diuision, it shall be good that you do set your diuisor styll agaynst
+those nombres fro<i>m</i> whiche you do take it: as by this example I
+wyll declare. Yf y<sup>e</sup> purchace of 200 acres of ground dyd coste
+290&nbsp;l’i. what dyd one acre coste? Fyrst wyl I turne the poundes
+into pennes, so wyll there be 69600&nbsp;d’· Then in settynge downe
+these nombers I shall do thus.</p>
+
+<table class = "backline float" summary = "counters example">
+<col />
+<col class = "rightline" />
+<col class = "rightline" />
+<col />
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp; &nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o o</td>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span></td>
+</tr>
+<tr>
+<td>X</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span>o o o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>o &nbsp; <span class = "five">°</span></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+</table>
+
+<p class = "nospace">
+Fyrst set the diuident on the ryghte hande as it oughte, and then
+<span class = "linenum">129 <i>a</i>.</span>
+*the diuisor on the lefte hande agaynst those nombers, fro<i>m</i> which
+I entende to take hym fyrst as here you se, wher I haue set the diuisor
+two lynes hygher the<i>n</i> is theyr owne place.</p>
+
+<p><i>S.</i> This is lyke the order of diuision by the penne.
+<span class = "pagenum">63</span>
+<a name = "page63" id = "page63"> </a></p>
+
+<p><i>M.</i> Truth you say, and nowe must I set y<sup>e</sup> quotient
+of this worke in the thyrde lyne, for that is the lyne of vnities in
+respecte to the diuisor in this worke. Then I seke howe often the
+diuisor maye be founde in the diuident, <i>and</i> that I fynde 3 tymes,
+then set I 3 in the thyrde lyne for the quotient, and take awaye that
+60000 fro<i>m</i> the diuident, and farther I do set the diuisor one
+line lower, as yow se here.</p>
+
+<table class = "backline float" summary = "counters example">
+<col />
+<col class = "rightline" />
+<col class = "rightline" />
+<col />
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>||</td>
+<td>o o</td>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span>o o o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>o o o</td>
+<td>&nbsp; <span class = "five">°</span>o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+</table>
+
+<p class = "nospace">
+<span class = "linenum">129 <i>b</i>.</span>
+*And then seke I how often the diuisor wyll be taken from the nomber
+agaynste it, whiche wyll be 4 tymes and 1 remaynynge.</p>
+
+<p><i>S.</i> But what yf it chaunce that when the diuisor is so remoued,
+it can not be ones taken out of the diuident agaynste it?</p>
+
+<p><i>M.</i> Then must the diuisor be set in an other line lower.</p>
+
+<p><i>S.</i> So was it in diuision by the penne, and therfore was there
+a cypher set in the quotient: but howe shall that be noted here?</p>
+
+<p><i>M.</i> Here nedeth no token, for the lynes do represente the
+places: onely loke that you set your quotient in that place which
+standeth for vnities in respecte of the diuisor: but now to returne to
+the example, I&nbsp;fynde the diuisor 4 tymes in the diuidente, and 1
+remaynynge, for 4 tymes 2 make 8, which I take from 9, and there
+<a class = "gloss" name = "resteth" id = "resteth" href =
+"#gloss_resteth">resteth</a> 1, as this figure sheweth:</p>
+
+<table class = "backline float" summary = "counters example">
+<col />
+<col class = "rightline" />
+<col class = "rightline" />
+<col />
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>||</td>
+<td>o o</td>
+<td>&nbsp;</td>
+<td>o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>o o o</td>
+<td>o<span class = "five">°</span></td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>o o o o</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+</table>
+
+<p class = "nospace">
+and in the myddle space for the quotient I set 4 in the seconde lyne,
+whiche is in this worke the place of vnities.*
+<span class = "linenum">130 <i>a</i>.</span>
+Then remoue I y<sup>e</sup> diuisor to the next lower line, and seke how
+often I may haue it in the dyuident, which I may do here 8 tymes iust,
+and nothynge remayne, as in this fourme,</p>
+
+<table class = "backline float" summary = "counters example">
+<col />
+<col class = "rightline" />
+<col class = "rightline" />
+<col />
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>||</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o o</td>
+<td>o o o</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>o o o o</td>
+<td>&nbsp;</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp;</td>
+<td>o<span class = "five">°</span>o o</td>
+<td>&nbsp;</td>
+</tr>
+</table>
+
+<p class = "nospace">
+where you may se that the hole quotient is 348&nbsp;d’, that is
+29&nbsp;s. wherby I knowe that so moche coste the purchace of one
+aker.</p>
+
+<p><i>S.</i> Now resteth the profes of Multiplycatio<i>n</i>, and also
+of Diuisio<i>n</i>.</p>
+
+<p><i>M.</i> Ther best profes are eche
+<span class = "linenum">130 <i>b</i>.</span>
+*one by the other, for Multyplication is proued by Diuision, and
+Diuision by Multiplycation, as in the worke by the penne you
+learned.</p>
+
+<p><i>S.</i> Yf that be all, you shall not nede to repete agayne that,
+y<sup>t</sup> was sufficye<i>n</i>tly taughte all redye: and excepte you
+wyll teache me any other feate, here maye you make an ende of this arte
+I suppose.</p>
+
+<p><i>M.</i> So wyll I do as touchynge hole nomber, and as for broken
+nomber, I&nbsp;wyll not trouble your wytte with it, tyll you haue
+practised this so well, y<sup>t</sup> you be full perfecte, so that you
+nede not to doubte in any poynte that I haue taught you, and thenne maye
+I boldly enstructe you in y<sup>e</sup> arte of fractions or broken
+no<i>m</i>ber, wherin I
+<span class = "pagenum">64</span>
+<a name = "page64" id = "page64"> </a>
+wyll also showe you the reasons of all that you haue nowe learned. But
+yet before I make an ende, I&nbsp;wyll showe you the order of
+co<i>m</i>men castyng, wher in are bothe pennes, shyllynges, and
+poundes, procedynge by no grounded reason, but onely by a receaued
+<span class = "linenum">131 <i>a</i>.</span>
+*fourme, and that dyuersly of dyuers men: for marchau<i>n</i>tes vse one
+fourme, and auditors an other:</p>
+
+<p class = "headnote"><span class = "headnote">
+Merchants’ Casting Counters.</span></p>
+
+<p><span class = "sidenote">Merchants’ casting.</span>
+But fyrste for marchauntes fourme marke this example here,</p>
+
+<table class = "noline float" summary = "merchant counters example">
+<tr>
+<td>o</td>
+<td>o o o o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp; o</td>
+</tr>
+<tr>
+<td>o</td>
+<td>o o o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp; o</td>
+</tr>
+<tr>
+<td>o</td>
+<td>o o o o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>&nbsp; o</td>
+</tr>
+<tr>
+<td>&nbsp;</td>
+<td>o o o o o</td>
+</tr>
+</table>
+
+<p class = "nospace">
+in which I haue expressed this summe 198 l’i.<a class = "tag" name =
+"tag_count2" id = "tag_count2" href = "#note_count2">2</a> 19&nbsp;s.
+11&nbsp;d’. So that you maye se that the lowest lyne serueth for
+pe<i>n</i>nes, the next aboue for shyllynges, the thyrde for poundes,
+and the fourth for scores of pou<i>n</i>des. And farther you maye se,
+that the space betwene pennes and shyllynges may receaue but one counter
+(as all other spaces lyke wayes do) and that one standeth in that place
+for 6&nbsp;d’. Lyke wayes betwene the shyllynges <i>and</i> the
+pou<i>n</i>des, one cou<i>n</i>ter standeth for 10&nbsp;s. And betwene
+the poundes and 20&nbsp;l’i. one counter standeth for 10 pou<i>n</i>des.
+But besyde those you maye see at the left syde of shyllynges, that one
+counter standeth alone, <i>and</i> betokeneth 5&nbsp;s.
+<span class = "linenum">131 <i>b</i>.</span>
+*So agaynste the poundes, that one cou<i>n</i>ter standeth for
+5&nbsp;l’i. And agaynst the 20 poundes, the one counter standeth for 5
+score pou<i>n</i>des, that is 100&nbsp;l’i. so that euery syde counter
+is 5 tymes so moch as one of them agaynst whiche he standeth.
+<span class = "sidenote">Auditors’ casting.</span>
+Now for the accompt of auditors take this example.</p>
+
+<table class = "noline float" summary = "merchant counters example">
+<tr>
+<td>&nbsp; &nbsp; &nbsp; o</td>
+<td>o &nbsp; &nbsp;o</td>
+<td>o &nbsp; &nbsp;o</td>
+<td>&nbsp; &nbsp;o</td>
+</tr>
+<tr>
+<td>o o o</td>
+<td>o o o</td>
+<td>o o o</td>
+<td>o o o</td>
+</tr>
+<tr>
+<td>o</td>
+<td></td>
+<td>o</td>
+<td>o o</td>
+</tr>
+</table>
+
+<p class = "nospace">
+where I haue expressed y<sup>e</sup> same su<i>m</i>me 198&nbsp;l’i.
+19&nbsp;s. 11&nbsp;d’. But here you se the pe<i>n</i>nes stande toward
+y<sup>e</sup> ryght hande, and the other encreasynge orderly towarde the
+lefte hande. Agayne you maye se, that auditours wyll make 2 lynes (yea
+and more) for pennes, shyllynges, <i>and</i> all other valewes, yf theyr
+summes extende therto. Also you se, that they set one counter at the
+ryght ende of eche rowe, whiche so set there standeth for 5 of that
+roume: and on
+<span class = "linenum">132 <i>a</i>.</span>
+*the lefte corner of the rowe it sta<i>n</i>deth for 10, of
+y<sup>e</sup> same row. But now yf you wold adde other subtracte after
+any of both those sortes, yf you marke y<sup>e</sup> order of
+y<sup>t</sup> other feate which I taught you, you may easely do the same
+here without moch teachynge: for in Additio<i>n</i> you must fyrst set
+downe one su<i>m</i>me and to the same set the other orderly, and lyke
+maner yf you haue many: but in Subtraction you must sette downe fyrst
+the greatest summe, and from it must you abate that other euery
+denominatio<i>n</i> from his dewe place.</p>
+
+<p><i>S.</i> I do not doubte but with a
+<span class = "pagenum">65</span>
+<a name = "page65" id = "page65"> </a>
+lytell practise I shall attayne these bothe: but how shall I multiply
+and diuide after these fourmes?</p>
+
+<p><i>M.</i> You can not duely do none of both by these sortes, therfore
+in suche case, you must resort to your other artes.</p>
+
+<p><i>S.</i> Syr, yet I se not by these sortes how to expresse
+hu<i>n</i>dreddes, yf they excede one hundred, nother yet
+thousandes.</p>
+
+<p><i>M.</i> They that vse such accomptes that it excede 200
+<span class = "linenum">132 <i>b</i>.</span>
+*in one summe, they sette no 5 at the lefte hande of the scores of
+poundes, but they set all the hundredes in an other farther rowe
+<i>and</i> 500 at the lefte hand therof, and the thousandes they set in
+a farther rowe yet, <i>and</i> at the lefte syde therof they sette the
+5000, and in the space ouer they sette the 10000, and in a hygher rowe
+20000, whiche all I haue expressed in this exa<i>m</i>ple,</p>
+
+<table class = "noline float" summary = "merchant counters example">
+<tr>
+<td></td>
+<td>o o o o</td>
+</tr>
+<tr>
+<td></td>
+<td>&nbsp; o</td>
+</tr>
+<tr>
+<td>o</td>
+<td>o o</td>
+</tr>
+<tr>
+<td>o</td>
+<td>o o o</td>
+</tr>
+<tr>
+<td></td>
+<td>o o o</td>
+</tr>
+<tr>
+<td>o</td>
+<td>o o o o</td>
+</tr>
+<tr>
+<td></td>
+<td>&nbsp; o</td>
+</tr>
+<tr>
+<td></td>
+<td>o o</td>
+</tr>
+<tr>
+<td></td>
+<td>&nbsp; o</td>
+</tr>
+<tr>
+<td></td>
+<td>o o o</td>
+</tr>
+<tr>
+<td></td>
+<td>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; o o</td>
+</tr>
+<tr>
+<td></td>
+<td>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; o</td>
+</tr>
+</table>
+
+<p class = "nospace">
+which is 97869&nbsp;l’i. 12&nbsp;s. 9 d’ ob. q. for I had not told you
+before where, nother how you shuld set downe farthynges, which (as you
+se here) must be set in a voyde space sydelynge beneth the pennes: for q
+one counter: for ob. 2 counters: for ob. q. 3 counters: <i>and</i> more
+there can not be, for 4 farthynges
+<span class = "linenum">133 <i>a</i>.</span>
+*do make 1&nbsp;d’. which must be set in his dewe place.</p>
+
+<p class = "headnote"><span class = "headnote">
+Auditors’ Casting Counters.</span></p>
+
+<p>And yf you desyre y<sup>e</sup> same summe after audytors maner, lo
+here it&nbsp;is.</p>
+
+<table class = "noline" summary = "merchant counters example">
+<tr>
+<td></td>
+<td>&nbsp; o o</td>
+<td>&nbsp; &nbsp; &nbsp; o</td>
+<td></td>
+<td>&nbsp; &nbsp; &nbsp; o</td>
+<td>o</td>
+<td>&nbsp; o</td>
+</tr>
+<tr>
+<td>o o o</td>
+<td>o o</td>
+<td>o o o</td>
+<td>o o o</td>
+<td>o o o</td>
+<td>o o</td>
+<td>o o o</td>
+<td></td>
+</tr>
+<tr>
+<td>o</td><td></td><td></td><td></td>
+<td>o</td><td></td><td></td><td>o o</td>
+</tr>
+<tr>
+<td></td><td></td><td></td><td></td>
+<td></td><td></td><td></td><td>o</td>
+</tr>
+</table>
+
+<p class = "nospace">
+But in this thyng, you shall take this for suffycyent, and the reste you
+shall obserue as you maye se by the working of eche sorte: for the
+dyuers wittes of men haue inuented dyuers and sundry wayes almost
+vnnumerable. But one feate I shall teache you, whiche not only for the
+straungenes and secretnes is moche pleasaunt, but also for the good
+co<i>m</i>moditie of it ryghte worthy to be well marked. This feate hath
+ben vsed aboue 2000 yeares at the leaste, and yet was it neuer
+come<i>n</i>ly knowen, especyally in Englysshe it was neuer taughte yet.
+This is the arte of nombrynge on the hand, with diuers gestures of the
+fyngers, expressynge any summe conceaued in the
+<span class = "linenum">133 <i>b</i>.</span>
+*mynde. And fyrst to begynne, yf you wyll expresse any summe vnder 100,
+you shall expresse it with your lefte hande: and from 100 vnto 10000,
+you shall expresse it with your ryght hande, as here orderly by this
+table folowynge you may perceaue.</p>
+
+<h5>¶ Here foloweth the table<br />
+of the arte of the<br />
+hande</h5>
+
+<p>&nbsp;</p>
+
+<span class = "pagenum">66</span>
+<a name = "page66" id = "page66"> </a>
+
+<p class = "illustration">
+<a name = "hand" id = "hand">
+<img src = "images/title_hand.png" width = "472" height = "35"
+alt = "The arte of nombrynge by the hande."
+title = "The arte of nombrynge by the hande." /></a></p>
+
+<span class = "linenum">
+<img src = "images/num_134.png" width = "32" height = "17"
+alt = "page number ‘134’ from original illustration"
+title = "page number ‘134’ from original illustration" /></span>
+
+<p class = "illustration">
+<img src = "images/hand_count.png" width = "410" height = "612"
+alt = "hand numbering as described in text" /></p>
+
+<p><span class = "sidenote">1</span>
+<span class = "linenum">134 <i>b</i>.</span>
+*In which as you may se 1 is expressed by y<sup>e</sup> lyttle fynger of
+y<sup>e</sup> lefte hande closely and harde croked.</p>
+
+<p><span class = "sidenote">2</span>
+<a class = "tag" name = "tag_count3" id = "tag_count3" href =
+"#note_count3">3</a>2 is declared by lyke bowynge of the weddynge fynger
+(whiche is the nexte to the lyttell fynger) together with the lytell
+fynger.</p>
+
+<p><span class = "sidenote">3</span>
+3 is signified by the myddle fynger bowed in lyke maner, with those
+other two.</p>
+
+<p><span class = "sidenote"><ins class = "correction" title = "missing sidenote added by transcriber"><i>&nbsp;4&nbsp;</i></ins></span>
+4 is declared by the bowyng of the myddle fynger and the rynge
+<span class = "pagenum">67</span>
+<a name = "page67" id = "page67"> </a>
+fynger, or weddynge fynger, with the other all stretched forth.</p>
+
+<p><span class = "sidenote">5</span>
+5 is represented by the myddle fynger onely bowed.</p>
+
+<p><span class = "sidenote">6</span>
+And 6 by the weddynge fynger only crooked: and this you may marke in
+these a certayne order. But now 7, 8, and 9, are expressed
+w<i>i</i>t<i>h</i> the bowynge of the same fyngers as are 1, 2, and 3,
+but after an other fourme.</p>
+
+<p><span class = "sidenote">7</span>
+For 7 is declared by the bowynge of the lytell fynger, as is 1, saue
+that for 1 the fynger is <ins class = "correction" title = "in at least one printing, text reads ‘elapsed’">clasped</ins> in, harde <i>and</i>
+<span class = "linenum">135 <i>a</i>.</span>
+*rounde, but for to expresse 7, you shall bowe the myddle ioynte of the
+lytell fynger only, and holde the other ioyntes streyght.</p>
+
+<p><i>S.</i> Yf you wyll geue me leue to expresse it after my rude
+maner, thus I vnderstand your meanyng: that 1 is expressed by crookynge
+in the lyttell fynger lyke the head of a bysshoppes <a class = "gloss"
+name = "bagle" id = "bagle" href = "#gloss_bagle">bagle</a>: and 7 is
+declared by the same fynger bowed lyke a gybbet.</p>
+
+<p><i>M.</i> So I perceaue, you vnderstande it.</p>
+
+<p><span class = "sidenote">8</span>
+Then to expresse 8, you shall bowe after the same maner both the lyttell
+fynger and the rynge fynger.</p>
+
+<p><span class = "sidenote">9</span>
+And yf you bowe lyke wayes with them the myddle fynger, then doth it
+betoken 9.</p>
+
+<p><span class = "sidenote">10</span>
+Now to expresse 10, you shall bowe your fore fynger rounde, and set the
+ende of it on the hyghest ioynte of the thombe.</p>
+
+<p><span class = "sidenote">20</span>
+And for to expresse 20, you must set your fyngers streyght, and the ende
+of your thombe to the partitio<i>n</i> of the
+<span class = "linenum">135 <i>b</i>.</span>
+*fore moste and myddle fynger.</p>
+
+<p><span class = "sidenote">30</span>
+30 is represented by the ioynynge together of y<sup>e</sup> headdes of
+the foremost fynger and the thombe.</p>
+
+<p><span class = "sidenote">40</span>
+40 is declared by settynge of the thombe crossewayes on the foremost
+fynger.</p>
+
+<p><span class = "sidenote">50</span>
+50 is signified by ryght stretchyng forth of the fyngers ioyntly, and
+applyenge of the thombes ende to the partition of the myddle fynger
+<i>and</i> the rynge fynger, or weddynge fynger.</p>
+
+<p><span class = "sidenote">60</span>
+60 is formed by bendynge of the thombe croked and crossynge it with the
+fore fynger.</p>
+
+<p><span class = "sidenote">70</span>
+70 is expressed by the bowynge of the foremost fynger, and settynge the
+ende of the thombe between the 2 foremost or hyghest ioyntes of it.</p>
+
+<p><span class = "sidenote">80</span>
+80 is expressed by settynge of the foremost fynger crossewayes on the
+thombe, so that 80 dyffereth thus fro<i>m</i> 40, that for 80 the
+forefynger is set crosse on the thombe, and for 40 the thombe is set
+crosse ouer y<sup>e</sup> forefinger.</p>
+
+<p><span class = "sidenote">90</span>
+<span class = "linenum">136 <i>a</i>.</span>
+*90 is signified, by bendynge the fore fynger, and settyng the ende of
+it in the innermost ioynte of y<sup>e</sup> thombe, that is euen at the
+foote of it. And thus are all the no<i>m</i>bers ended vnder 100.</p>
+
+<p><i>S.</i> In dede these be all the nombers fro<i>m</i> 1 to 10,
+<i>and</i> then all the
+tenthes within 100,
+<span class = "sidenote">11, 12, 13,<br />
+21, 22, 23</span>
+but this <ins class = "correction" title = "spelling unchanged">teacyed</ins> me not how to expresse 11, 12, 13, <i>et</i>c.
+21, 22, 23, <i>et</i>c. and such lyke.</p>
+
+<p><i>M.</i> You can lytell vnderstande, yf you can not do that without
+teachynge: what is
+<span class = "pagenum">68</span>
+<a name = "page68" id = "page68"> </a>
+11? is it not 10 and 1? then expresse 10 as you were taught, and 1 also,
+and that is 11: and for 12 expresse 10 and 2: for 23 set 20 and 3: and
+so for 68 you muste make 60 and there to 8: and so of all other
+sortes.</p>
+
+<p><span class = "sidenote">100</span>
+But now yf you wolde represente 100 other any nomber aboue it, you muste
+do that with the ryghte hande, after this maner.</p>
+
+<p>You must expresse 100 in the ryght hand, with the lytell fynger so
+bowed as you dyd expresse 1 in the left hand.</p>
+
+<p><span class = "sidenote">200</span>
+<span class = "linenum">136 <i>b</i>.</span>
+*And as you expressed 2 in the lefte hande, the same fasshyon in the
+ryght hande doth declare 200.</p>
+
+<p><span class = "sidenote">300</span>
+The fourme of 3 in the ryght hand standeth for 300.</p>
+
+<p><span class = "sidenote">400</span>
+The fourme of 4, for 400.</p>
+
+<p><span class = "sidenote">500</span>
+Lykewayes the fourme of 5, for 500.</p>
+
+<p><span class = "sidenote">600</span>
+The fourme of 6, for 600. And to be shorte: loke how you did expresse
+single vnities and tenthes in the lefte hande, so must you expresse
+vnities <i>and</i> tenthes of hundredes, in the ryghte hande.</p>
+
+<p><span class = "sidenote">900</span>
+<i>S.</i> I vnderstande you thus: that yf I wold represent 900,
+I&nbsp;must so fourme the fyngers of my ryghte hande, as I shuld do in
+my left hand to expresse 9,
+<span class = "sidenote">1000</span>
+And as in my lefte hand I expressed 10, so in my ryght hande must I
+expresse 1000.</p>
+
+<p>And so the fourme of euery tenthe in the lefte hande serueth to
+expresse lyke no<i>m</i>ber of thousa<i>n</i>des,
+<span class = "sidenote">4000</span>
+so y<sup>e</sup> fourme of 40 standeth for 4000.</p>
+
+<p><span class = "sidenote">8000</span>
+The fourme of 80 for 8000.</p>
+
+<span class = "sidenote">9000</span>
+<span class = "linenum">137 <i>a</i>.</span>
+<p class = "center">*And the fourme of 90 (whiche is<br />
+the greatest) for 9000, and aboue that<br />
+I can not expresse any nomber. <i>M.</i><br />
+No not with one fynger: how be it,<br />
+w<i>i</i>t<i>h</i> dyuers fyngers you maye expresse<br />
+9999, and all at one tyme, and that lac<br />
+keth but 1 of 10000. So that vnder<br />
+10000 you may by your fyngers ex-<br />
+presse any summe. And this shal suf-<br />
+fyce for Numeration on the fyngers.<br />
+And as for Addition, Subtraction,<br />
+Multiplicatio<i>n</i>, and Diuision (which<br />
+yet were neuer taught by any man as<br />
+farre as I do knowe) I wyll enstruct<br />
+you after the treatyse of fractions.<br />
+And now for this tyme fare well,<br />
+<span class = "pagenum">69</span>
+<a name = "page69" id = "page69"> </a>
+and loke that you cease not to<br />
+practyse that you haue lear<br />
+ned. <i>S.</i> Syr, with moste<br />
+harty mynde I thanke<br />
+you, bothe for your<br />
+good learnyng, <i>and</i><br />
+also your good<br />
+cou<i>ns</i>el, which<br />
+(god wyllyng) I truste to folow.</p>
+
+<p class = "center">Finis.</p>
+
+<hr class = "mid" />
+
+<div class = "footnote">
+
+<p><a name = "note_count1" id = "note_count1" href =
+"#tag_count1">1.</a>
+1342 in original.</p>
+
+<p><a name = "note_count2" id = "note_count2" href =
+"#tag_count2">2.</a>
+168 in original.</p>
+
+<p><a name = "note_count3" id = "note_count3" href =
+"#tag_count3">3.</a>
+Bracket ([) denotes new paragraph in original.</p>
+<p class = "mynote">
+For this e-text, the brackets have been omitted in favor of restoring
+the paragraph breaks. Numbers 200 and up were printed as separate
+paragraphs and are unchanged. Sidenote 4 was missing and has been
+supplied by the transcriber; the pairs 5, 6 and 9, 10 (originally on one
+line) have been separated.</p>
+
+</div>
+
+</div> <!-- end div count -->
+
+<div class = "app">
+
+<span class = "pagenum">70</span>
+<a name = "page70" id = "page70"> </a>
+
+<h3><a name = "app1" id = "app1">APPENDIX I.</a></h3>
+
+<hr class = "tiny" />
+
+<p class = "illustration">
+<img src = "images/title_app1.png" width = "441" height = "71"
+alt = "A Treatise on the Numeration of Algorism."
+title = "A Treatise on the Numeration of Algorism." /></p>
+
+
+<p class = "subhead">[<i>From a MS. of the 14th Century.</i>]</p>
+
+<p>To alle suche even nombrys the most have cifrys as to ten. twenty.
+thirtty. an hundred. an thousand and suche other. but ye schal
+vnderstonde that a cifre tokeneth nothinge but he maketh other the more
+significatyf that comith after hym. Also ye schal vnderstonde that in
+nombrys composyt and in alle other nombrys that ben of diverse figurys
+ye schal begynne in the ritht syde and to rekene backwarde and so he
+schal be wryte as thus&mdash;1000. the sifre in the ritht side was first
+wryte and yit he tokeneth nothinge to the secunde no the thridde but
+thei maken that figure of 1 the more signyficatyf that comith after hem
+by as moche as he born oute of his first place where he schuld yf he
+stode ther tokene but one. And there he stondith nowe in the <a class =
+"gloss" name = "ferye" id = "ferye" href = "#gloss_ferye">ferye</a>
+place he tokeneth a thousand as by this rewle. In the first place he
+tokeneth but hymself. In the secunde place he tokeneth ten times
+hymself. In the thridde place he tokeneth an hundred tymes himself. In
+the ferye he tokeneth a thousand tymes himself. In the <a class =
+"gloss" name = "fyftye" id = "fyftye" href = "#gloss_fyftye">fyftye</a>
+place he tokeneth ten thousand tymes himself. In the sexte place he
+tokeneth an hundred thousand tymes hymself. In the seveth place he
+tokeneth ten hundred thousand tymes hymself, &amp;c. And ye schal
+vnderstond that this worde nombre is partyd into thre partyes. Somme is
+callyd nombre of digitys for alle ben digitys that ben withine ten as
+ix, viii, vii, vi, v, iv, iii, ii, i. Articules ben alle thei that mow
+be devyded into nombrys of ten as xx, xxx, xl, and suche other.
+Composittys be alle nombrys that ben componyd of a digyt and of an
+articule as fourtene fyftene thrittene and suche other. Fourtene is
+componyd of four that is a digyt
+<span class = "pagenum">71</span>
+<a name = "page71" id = "page71"> </a>
+and of ten that is an articule. Fyftene is componyd of fyve that is a
+digyt and of ten that is an articule and so of others
+.&nbsp;.&nbsp;.&nbsp;.&nbsp;.&nbsp;. But as to this rewle. In the firste
+place he tokeneth but himself that is to say he tokeneth but that and no
+more. If that he stonde in the secunde place he tokeneth ten tymes
+himself as this figure 2 here 21. this is oon and twenty. This figure 2
+stondith in the secunde place and therfor he tokeneth ten tymes himself
+and ten tymes 2 is twenty and so <a class = "gloss" name = "forye" id =
+"forye" href = "#gloss_forye">forye</a> of every figure and he stonde
+after another toward the <a class = "gloss" name = "lest2" id = "lest2"
+href = "#gloss_lest2">lest</a> syde he schal tokene ten tymes as moche
+more as he schuld token and he stode in that place ther that the figure
+afore him stondeth: lo an example as thus 9634. This figure of foure
+that hath this schape 4 tokeneth but himself for he stondeth in the
+first place. The figure of thre that hath this schape 3 tokeneth ten
+tyme himself for he stondeth in the secunde place and that is thritti.
+The figure of sexe that hath this schape 6 tokeneth ten tyme more than
+he schuld and he stode in the place yer the figure of thre stondeth for
+ther he schuld tokene but sexty. And now he tokeneth ten tymes that is
+sexe hundrid. The figure of nyne that hath this schape 9 tokeneth ten
+tymes more than he schulde and he stode in the place ther the figure of
+6 stondeth inne for thanne he schuld tokene but nyne hundryd. And in the
+place that he stondeth inne nowe he tokeneth nine thousand. Alle the
+hole nombre of these foure figurys. Nine thousand sexe hundrid and foure
+and thritti.</p>
+
+
+
+
+<span class = "pagenum">72</span>
+<a name = "page72" id = "page72"> </a>
+
+<h3><a name = "app2" id = "app2">APPENDIX II.</a></h3>
+
+<hr class = "tiny" />
+
+<p class = "illustration">
+<img src = "images/title_app2.png" width = "288" height = "35"
+alt = "Carmen de Algorismo."
+title = "Carmen de Algorismo." /></p>
+
+<p class = "subhead">[<i>From a B.M. MS., </i>8<i> C. </i>iv.<i>, with
+additions from </i>12<i> E. </i>1<i> &amp; Eg. </i>2622<i>.</i>]</p>
+
+<div class = "verse">
+<p>Hec algorismus ars presens dicitur<a class = "tag" name = "tag_app_1"
+id = "tag_app_1" href = "#note_app_1">1</a>; in qua</p>
+<p>Talibus Indorum<a class = "tag" name = "tag_app_2" id = "tag_app_2"
+href = "#note_app_2">2</a> fruimur his quinque figuris.</p>
+<p><span class = "gap">0. 9. 8. 7. 6. 5. 4. 3. 2. 1.</span></p>
+<p>Prima significat unum: duo vero secunda:</p>
+<span class = "linenum">4</span>
+<p>Tercia significat tria: sic procede sinistre</p>
+<p>Donec ad extremam venies, qua cifra vocatur;</p>
+<p><a class = "tag" name = "tag_app_3" id = "tag_app_3" href =
+"#note_app_3">3</a>[Que nil significat; dat significare sequenti.]</p>
+<p>Quelibet illarum si primo limite ponas,</p>
+<span class = "linenum">8</span>
+<p>Simpliciter se significat: si vero secundo,</p>
+<p>Se decies: sursum procedas multiplicando.<a class = "tag" name =
+"tag_app_4" id = "tag_app_4" href = "#note_app_4">4</a></p>
+<p>[Namque figura sequens quevis signat decies plus,</p>
+<span class = "linenum">12</span>
+<p>Ipsa locata loco quam significet pereunte:</p>
+<p>Nam precedentes plus ultima significabit.]</p>
+
+<p class = "stanza">
+<a class = "tag" name = "tag_app_5" id = "tag_app_5" href =
+"#note_app_5">5</a>Post predicta scias quod tres breuiter numerorum</p>
+<p>Distincte species sunt; nam quidam digiti sunt;</p>
+<span class = "linenum">16</span>
+<p>Articuli quidam; quidam quoque compositi sunt.</p>
+<p>[Sunt digiti numeri qui citra denarium sunt;</p>
+<p>Articuli decupli degitorum; compositi sunt</p>
+<p>Illi qui constant ex articulis digitisque.]</p>
+<span class = "linenum">20</span>
+<p>Ergo, proposito numero tibi scribere, primo</p>
+<p>Respicias quis sit numerus; quia si digitus sit,</p>
+<p><a class = "tag" href = "#note_app_5">5</a>[Una figura satis sibi;
+sed si compositus sit,]</p>
+<p>Primo scribe loco digitum post articulum fac</p>
+<span class = "linenum">24</span>
+<p>Articulus si sit, cifram post articulum sit,</p>
+<p>[Articulum vero reliquenti in scribe figure.]</p>
+<span class = "pagenum">73</span>
+<a name = "page73" id = "page73"> </a>
+<p>Quolibet in numero, si par sit prima figura,</p>
+<p>Par erit et totum, quicquid sibi continetur;</p>
+<span class = "linenum">28</span>
+<p>Impar si fuerit, totum sibi fiet et impar.</p>
+
+<p class = "stanza">
+Septem<a class = "tag" name = "tag_app_6" id = "tag_app_6" href =
+"#note_app_6">6</a> sunt partes, non plures, istius artis;</p>
+<p>Addere, subtrahere, duplare, dimidiare;</p>
+<p>Sexta est diuidere, set quinta est multiplicare;</p>
+<span class = "linenum">32</span>
+<p>Radicem extrahere pars septima dicitur esse.</p>
+<p>Subtrahis aut addis a dextris vel mediabis;</p>
+<p>A leua dupla, diuide, multiplicaque;</p>
+<p>Extrahe radicem semper sub parte sinistra.</p>
+
+<span class = "linenum">36</span>
+<p class = "stanza">
+<span class = "sidenote">Addition.</span>
+Addere si numero numerum vis, ordine tali</p>
+<p>Incipe; scribe duas primo series numerorum</p>
+<p>Prima sub prima recte ponendo figuram,</p>
+<p>Et sic de reliquis facias, si sint tibi plures.</p>
+<span class = "linenum">40</span>
+<p>Inde duas adde primas hac condicione;</p>
+<p>Si digitus crescat ex addicione priorum,</p>
+<p>Primo scribe loco digitum, quicunque sit ille;</p>
+<p>Si sit compositus, in limite scribe sequenti</p>
+<span class = "linenum">44</span>
+<p>Articulum, primo digitum; quia sic iubet ordo.</p>
+<p>Articulus si sit, in primo limite cifram,</p>
+<p>Articulum vero reliquis inscribe figuris;</p>
+<p>Vel per se scribas si nulla figura sequatur.</p>
+<span class = "linenum">48</span>
+<p>Si tibi cifra superueniens occurrerit, illam</p>
+<p>Deme suppositam; post illic scribe figuram:</p>
+<p>Postea procedas reliquas addendo figuras.</p>
+
+<p class = "stanza">
+<span class = "sidenote">Subtraction.</span>
+A numero numerum si sit tibi demere cura,</p>
+<span class = "linenum">52</span>
+<p>Scribe figurarum series, vt in addicione;</p>
+<p>Maiori numero numerum suppone minorem,</p>
+<p>Siue pari numero supponatur numerus par.</p>
+<p>Postea si possis a prima subtrahe primam,</p>
+<span class = "linenum">56</span>
+<p>Scribens quod remanet, cifram si nil remanebit.</p>
+<p>Set si non possis a prima demere primam;</p>
+<p>Procedens, vnum de limite deme sequenti;</p>
+<span class = "pagenum">74</span>
+<a name = "page74" id = "page74"> </a>
+<p>Et demptum pro denario reputabis ab illo,</p>
+<span class = "linenum">60</span>
+<p>Subtrahe totaliter numerum quem proposuisti.</p>
+<p>Quo facto, scribe supra quicquit remanebit,</p>
+<p>Facque novenarios de cifris, cum remanebis,</p>
+<p>Occurrant si forte cifre, dum demseris vnum;</p>
+<span class = "linenum">64</span>
+<p>Postea procedas reliquas demendo figuras.</p>
+
+<p class = "stanza">
+<span class = "sidenote">Proof.</span>
+<a class = "tag" name = "tag_app_7" id = "tag_app_7" href =
+"#note_app_7">7</a>[Si subtracio sit bene facta probare valebis,</p>
+<p>Quas subtraxisti primas addendo figuras.</p>
+<p>Nam, subtractio si bene sit, primas retinebis,</p>
+<span class = "linenum">68</span>
+<p>Et subtractio facta tibi probat additionem.]</p>
+
+<p class = "stanza">
+<span class = "sidenote">Duplation.</span>
+Si vis duplare numerum, sic incipe; solam</p>
+<p>Scribe figurarum seriem, quamcumque voles que</p>
+<p>Postea procedas primam duplando figuram;</p>
+<span class = "linenum">72</span>
+<p>Inde quod excrescet, scribens, vbi iusserit ordo,</p>
+<p>Juxta precepta que dantur in addicione.</p>
+<p>Nam si sit digitus, in primo limite scribe;</p>
+<p>Articulus si sit, in primo limite cifram,</p>
+<span class = "linenum">76</span>
+<p>Articulum vero reliquis inscribe figuris;</p>
+<p>Vel per se scribas, si nulla figura sequatur:</p>
+<p>Compositus si sit, in limite scribe sequenti</p>
+<p>Articulum primo, digitum; quia sic jubet ordo:</p>
+<span class = "linenum">80</span>
+<p>Et sic de reliquis facias, si sint tibi plures.</p>
+<p><a class = "tag" name = "tag_app_8" id = "tag_app_8" href =
+"#note_app_8">8</a>[Si super extremam nota sit, monadem dat eidem,</p>
+<p>Quod tibi contingit, si primo dimidiabis.]</p>
+
+<p class = "stanza">
+<span class = "sidenote">Mediation.</span>
+Incipe sic, si vis aliquem numerum mediare:</p>
+<span class = "linenum">84</span>
+<p>Scribe figurarum seriem solam, velud ante;</p>
+<p>Postea procedens medias, et prima figura</p>
+<p>Si par aut impar videas; quia si fuerit par,</p>
+<p>Dimidiabis eam, scribens quicquit remanebit;</p>
+<span class = "linenum">88</span>
+<p>Impar si fuerit, vnum demas, mediare,</p>
+<p>Nonne presumas, sed quod superest mediabis;</p>
+<p>Inde super tractum, fac demptum quod notat unum;</p>
+<p>Si monos, dele; sit ibi cifra post nota supra.</p>
+<span class = "linenum">92</span>
+<p>Postea procedas hac condicione secunda:<a class = "tag" name =
+"tag_app_9" id = "tag_app_9" href = "#note_app_9">9</a></p>
+<p>Impar<a class = "tag" name = "tag_app_10" id = "tag_app_10" href =
+"#note_app_10">10</a> si fuerit hic vnum deme priori,</p>
+<p>Inscribens quinque, nam denos significabit</p>
+<p>Monos prædictam: si vero secunda dat vnam,</p>
+<span class = "linenum">96</span>
+<p>Illa deleta, scribatur cifra; priori</p>
+<span class = "pagenum">75</span>
+<a name = "page75" id = "page75"> </a>
+<p>Tradendo quinque pro denario mediato;</p>
+<p>Nec cifra scribatur, nisi inde figura sequatur:</p>
+<p>Postea <ins class = "correction" title = "text reads ‘procdeas’">procedas</ins> reliquas mediando figuras,</p>
+<span class = "linenum">100</span>
+<p>Quin supra docui, si sint tibi mille figure.</p>
+<p><a class = "tag" name = "tag_app_11" id = "tag_app_11" href =
+"#note_app_11">11</a>[Si mediatio sit bene facta probare valebis,</p>
+<p>Duplando numerum quem primo dimidiasti.]</p>
+
+<p class = "stanza">
+<span class = "sidenote">Multiplication.</span>
+Si tu per numerum numerum vis multiplicare,</p>
+<span class = "linenum">104</span>
+<p>Scribe duas, quascunque volis, series numerorum;</p>
+<p>Ordo tamen seruetur vt vltima multiplicandi</p>
+<p>Ponatur super anteriorem multiplicantis;</p>
+<p><a class = "tag" name = "tag_app_12" id = "tag_app_12" href =
+"#note_app_12">12</a>[A leua relique sint scripte multiplicantes.]</p>
+<span class = "linenum">108</span>
+<p>In digitum cures digitum si ducere, major</p>
+<p>Per quantes distat a denis respice, debes</p>
+<p>Namque suo decuplo tociens delere minorem;</p>
+<p>Sicque tibi numerus veniens exinde patebit.</p>
+<span class = "linenum">112</span>
+<p>Postea procedas postremam multiplicando,</p>
+<p>Juste multiplicans per cunctas inferiores,</p>
+<p>Condicione tamen tali; quod multiplicantis</p>
+<p>Scribas in capite, quicquid processerit inde;</p>
+<span class = "linenum">116</span>
+<p>Set postquam fuerit hec multiplicata, figure</p>
+<p>Anteriorentur seriei multiplicantis;</p>
+<p>Et sic multiplica, velut istam multiplicasti,</p>
+<p>Qui sequitur numerum scriptum quicunque figuris.</p>
+<span class = "linenum">120</span>
+<p>Set cum multiplicas, primo sic est operandum,</p>
+<p>Si dabit articulum tibi multiplicacio solum;</p>
+<p>Proposita cifra, summam transferre memento.</p>
+<p>Sin autem digitus excrescerit articulusque,</p>
+<span class = "linenum">124</span>
+<p>Articulus supraposito digito salit ultra;</p>
+<p>Si digitus tamen, ponas illum super ipsam,</p>
+<p>Subdita multiplicans hanc que super incidit illi</p>
+<p>Delet eam penitus, scribens quod provenit inde;</p>
+<span class = "linenum">128</span>
+<p>Sed si multiplices illam posite super ipsam,</p>
+<p>Adiungens numerum quem prebet ductus earum;</p>
+<p>Si supraimpositam cifra debet multiplicare,</p>
+<p>Prorsus eam delet, scribi que loco cifra debet,</p>
+<span class = "linenum">132</span>
+<p><a class = "tag" href = "#note_app_12">12</a>[Si cifra multiplicat
+aliam positam super ipsam,</p>
+<p>Sitque locus supra vacuus super hanc cifra fiet;]</p>
+<span class = "pagenum">76</span>
+<a name = "page76" id = "page76"> </a>
+<p>Si supra fuerit cifra semper pretereunda est;</p>
+<p>Si dubites, an sit bene multiplicando secunda,</p>
+<span class = "linenum">136</span>
+<p>Diuide totalem numerum per multiplicantem,</p>
+<p>Et reddet numerus emergens inde priorem.</p>
+
+
+
+<span class = "sidenote">Mental Multiplication.</span>
+
+<p><a class = "tag" name = "tag_app_13" id = "tag_app_13" href =
+"#note_app_13">13</a>[Per numerum si vis numerum quoque multiplicare</p>
+<p>Tantum per normas subtiles absque figuris</p>
+<span class = "linenum">140</span>
+<p>Has normas poteris per versus scire sequentes.</p>
+<p>Si tu per digitum digitum quilibet multiplicabis</p>
+<p>Regula precedens dat qualiter est operandum</p>
+<p>Articulum si per reliquum vis multiplicare</p>
+<span class = "linenum">144</span>
+<p>In proprium digitum debebit uterque resolvi</p>
+<p>Articulus digitos post per se multiplicantes</p>
+<p>Ex digitis quociens teneret multiplicatum</p>
+<p>Articuli faciunt tot centum multiplicati.</p>
+<span class = "linenum">148</span>
+<p>Articulum digito si multiplicamus oportet</p>
+<p>Articulum digitum sumi quo multiplicare</p>
+<p>Debemus reliquum quod multiplicaris ab illis</p>
+<p>Per reliquo decuplum sic omne latere nequibit</p>
+<span class = "linenum">152</span>
+<p>In numerum mixtum digitum si ducere cures</p>
+<p>Articulus mixti sumatur deinde resolvas</p>
+<p>In digitum post hec fac ita de digitis nec</p>
+<p>Articulusque docet excrescens in detinendo</p>
+<span class = "linenum">156</span>
+<p>In digitum mixti post ducas multiplicantem</p>
+<p>De digitis ut norma docet sit juncta secundo</p>
+<p>Multiplica summam et postea summa patebit</p>
+<p>Junctus in articulum purum articulumque</p>
+<span class = "linenum">160</span>
+<p><a class = "tag" name = "tag_app_14" id = "tag_app_14" href =
+"#note_app_14">14</a>[Articulum purum comittes articulum que]</p>
+<p>Mixti pro digitis post fiat et articulus vt</p>
+<p>Norma jubet retinendo quod egreditur ab illis</p>
+<p>Articuli digitum post <ins class = "correction" title = "text reads ‘iu’">in</ins> digitum mixti duc</p>
+<span class = "linenum">164</span>
+<p>Regula de digitis ut percipit articulusque</p>
+<p>Ex quibus excrescens summe tu junge priori</p>
+<p>Sic manifesta cito fiet tibi summa petita.</p>
+<p>Compositum numerum mixto sic multiplicabis</p>
+<span class = "linenum">168</span>
+<p>Vndecies tredecem sic est ex hiis operandum</p>
+<p>In reliquum primum demum duc post in eundem</p>
+<p>Unum post deinde duc in tercia deinde per unum</p>
+<p>Multiplices tercia demum tunc omnia multiplicata</p>
+<span class = "linenum">172</span>
+<p>In summa duces quam que fuerit te dices</p>
+<span class = "pagenum">77</span>
+<a name = "page77" id = "page77"> </a>
+<p>Hic ut hic mixtus intentus est operandum</p>
+<p>Multiplicandorum de normis sufficiunt hec.]</p>
+
+
+<span class = "sidenote">Division.</span>
+
+<p>Si vis dividere numerum, sic incipe primo;</p>
+<span class = "linenum">176</span>
+<p>Scribe duas, quascunque voles, series numerorum;</p>
+<p>Majori numero numerum suppone minorem,</p>
+<p><a class = "tag" name = "tag_app_15" id = "tag_app_15" href =
+"#note_app_15">15</a>[Nam docet ut major teneat bis terve minorem;]</p>
+
+<p>Et sub supprima supprimam pone figuram,</p>
+<span class = "linenum">180</span>
+<p>Sic reliquis reliquas a dextra parte locabis;</p>
+<p>Postea de prima primam sub parte sinistra</p>
+<p>Subtrahe, si possis, quociens potes adminus istud,</p>
+<p>Scribens quod remanet sub tali conditione;</p>
+<span class = "linenum">184</span>
+<p>Ut totiens demas demendas a remanente,</p>
+<p>Que serie recte ponentur in anteriori,</p>
+<p>Unica si, tantum sit ibi decet operari;</p>
+<p>Set si non possis a prima demere primam,</p>
+<span class = "linenum">188</span>
+<p>Procedas, et eam numero suppone sequenti;</p>
+<p>Hanc uno retrahendo gradu quo comites retrahantur,</p>
+<p>Et, quotiens poteris, ab eadem deme priorem,</p>
+<p>Ut totiens demas demendas a remanenti,</p>
+<span class = "linenum">192</span>
+<p>Nec plus quam novies quicquam tibi demere debes,</p>
+<p>Nascitur hinc numerus quociens supraque sequentem</p>
+<p>Hunc primo scribas, retrahas exinde figuras,</p>
+<p>Dum fuerit major supra positus inferiori,</p>
+<span class = "linenum">196</span>
+<p>Et rursum fiat divisio more priori;</p>
+<p>Et numerum quotiens supra scribas pereunti,</p>
+<p>Si fiat saliens retrahendo, cifra locetur,</p>
+<p>Et pereat numero quotiens, proponas eidem</p>
+<span class = "linenum">200</span>
+<p>Cifram, ne numerum pereat vis, dum locus illic</p>
+<p>Restat, et expletis divisio non valet ultra:</p>
+<p>Dum fuerit numerus numerorum inferiore seorsum</p>
+<p>Illum servabis; hinc multiplicando probabis,</p>
+
+<span class = "sidenote">Proof.</span>
+
+<span class = "linenum">204</span>
+<p>Si bene fecisti, divisor multiplicetur</p>
+<p>Per numerum quotiens; cum multiplicaveris, adde</p>
+<p>Totali summæ, quod servatum fuit ante,</p>
+<p>Reddeturque tibi numerus quem proposuisti;</p>
+<span class = "linenum">208</span>
+<p>Et si nil remanet, hunc multiplicando reddet,</p>
+
+<span class = "sidenote">Square Numbers.</span>
+
+<p>Cum ducis numerum per se, qui provenit inde</p>
+<p>Sit tibi quadratus, ductus radix erit hujus,</p>
+<p>Nec numeros omnes quadratos dicere debes,</p>
+<span class = "linenum">212</span>
+<p>Est autem omnis numerus radix alicujus.</p>
+<span class = "pagenum">78</span>
+<a name = "page78" id = "page78"> </a>
+<p>Quando voles numeri radicem querere, scribi</p>
+<p>Debet; inde notes si sit locus ulterius impar,</p>
+<p>Estque figura loco talis scribenda sub illo,</p>
+<span class = "linenum">216</span>
+<p>Que, per se dicta, numerum tibi destruat illum,</p>
+<p>Vel quantum poterit ex inde delebis eandem;</p>
+<p>Vel retrahendo duples retrahens duplando sub ista</p>
+<p>Que primo sequitur, duplicatur per duplacationem,</p>
+<span class = "linenum">220</span>
+<p>Post per se minuens pro posse quod est minuendum.</p>
+<p><a class = "tag" name = "tag_app_16" id = "tag_app_16" href =
+"#note_app_16">16</a>Post his propones digitum, qui, more priori</p>
+<p>Per precedentes, post per se multiplicatus,</p>
+<p>Destruat in quantum poterit numerum remanentem,</p>
+<span class = "linenum">224</span>
+<p>Et sic procedens retrahens duplando figuram,</p>
+<p>Preponendo novam donec totum peragatur,</p>
+<p>Subdupla propriis servare docetque duplatis;</p>
+<p>Si det compositum numerum duplacio, debet</p>
+<span class = "linenum">228</span>
+<p>Inscribi digitus a parte dextra parte propinqua,</p>
+<p>Articulusque loco quo non duplicata resessit;</p>
+<p>Si dabit articulum, sit cifra loco pereunte</p>
+<p>Articulusque locum tenet unum, de duplicata resessit;</p>
+<span class = "linenum">232</span>
+<p>Si donet digitum, sub prima pone sequente,</p>
+<p>Si supraposita fuerit duplicata figura</p>
+<p>Major proponi debet tantummodo cifra,</p>
+<p>Has retrahens solito propones more figuram,</p>
+<span class = "linenum">236</span>
+<p>Usque sub extrema ita fac retrahendo figuras,</p>
+<p>Si totum deles numerum quem proposuisti,</p>
+<p>Quadratus fuerit, de dupla quod duplicasti,</p>
+<p>Sicque tibi radix illius certa patebit,</p>
+<span class = "linenum">240</span>
+<p>Si de duplatis fit juncta supprima figura;</p>
+<p>Radicem per se multiplices habeasque</p>
+<p>Primo propositum, bene te fecisse probasti;</p>
+<p>Non est quadratus, si quis restat, sed habentur</p>
+<span class = "linenum">244</span>
+<p>Radix quadrati qui stat major sub eadem;</p>
+<p>Vel quicquid remanet tabula servare memento;</p>
+<p>Hoc casu radix per se quoque multiplicetur,</p>
+<p>Vel sic quadratus sub primo major habetur,</p>
+<span class = "linenum">248</span>
+<p>Hinc addas remanens, et prius debes haberi;</p>
+<p>Si locus extremus fuerit par, scribe figuram</p>
+<p>Sub pereunte loco per quam debes operari,</p>
+<p>Que quantum poterit supprimas destruat ambas,</p>
+<span class = "pagenum">79</span>
+<a name = "page79" id = "page79"> </a>
+<span class = "linenum">252</span>
+<p>Vel penitus legem teneas operando priorem,</p>
+<p>Si suppositum digitus suo fine repertus,</p>
+<p>Omnino delet illic scribi cifra debet,</p>
+<p>A leva si qua sit ei sociata figura;</p>
+<span class = "linenum">256</span>
+<p>Si cifre remanent in fine pares decet harum</p>
+<p>Radices, numero mediam proponere partem,</p>
+<p>Tali quesita radix patet arte reperta.</p>
+<p>Per numerum recte si nosti multiplicare</p>
+<span class = "linenum">260</span>
+<p>Ejus quadratum, numerus qui pervenit inde</p>
+<p>Dicetur cubicus; primus radix erit ejus;</p>
+<p>Nec numeros omnes cubicatos dicere debes,</p>
+<p>Est autem omnis numerus radix alicujus;</p>
+
+<span class = "sidenote">Cube Root.</span>
+
+<span class = "linenum">264</span>
+<p>Si curas cubici radicem quærere, primo</p>
+<p>Inscriptum numerum distinguere per loca debes;</p>
+<p>Que tibi mille notant a mille notante suprema</p>
+<p>Initiam, summa operandi parte sinistra,</p>
+<span class = "linenum">268</span>
+<p>Illic sub scribas digitum, qui multiplicatus</p>
+<p>In semet cubice suprapositum sibi perdat,</p>
+<p>Et si quid fuerit adjunctum parte sinistra</p>
+<p>Si non omnino, quantum poteris minuendo,</p>
+<span class = "linenum">272</span>
+<p>Hinc triplans retrahe saltum, faciendo sub illa</p>
+<p>Que manet a digito deleto terna, figuram</p>
+<p>Illi propones quo sub triplo asocietur,</p>
+<p>Ut cum subtriplo per eam tripla multiplicatur;</p>
+<span class = "linenum">276</span>
+<p>Hinc per eam solam productum multiplicabis,</p>
+<p>Postea totalem numerum, qui provenit inde</p>
+<p>A suprapositis respectu tolle triplate</p>
+<p>Addita supprimo cubice tunc multiplicetur,</p>
+<span class = "linenum">280</span>
+<p>Respectu cujus, numerus qui progredietur</p>
+<p>Ex cubito ductu, supra omnes adimetur;</p>
+<p>Tunc ipsam delens triples saltum faciendo,</p>
+<p>Semper sub ternas, retrahens alias triplicatas</p>
+<span class = "linenum">284</span>
+<p>Ex hinc triplatis aliam propone figuram,</p>
+<p>Que per triplatas ducatur more priori;</p>
+<p>Primo sub triplis sibi junctis, postea per&nbsp;se,</p>
+<p>In numerum ducta, productum de triplicatis:</p>
+<span class = "linenum">288</span>
+<p>Utque prius dixi numerus qui provenit inde</p>
+<p>A suprapositis has respiciendo trahatur,</p>
+<p>Huic cubice ductum sub primo multiplicabis,</p>
+<p>Respectumque sui, removebis de remanenti,</p>
+<span class = "linenum">292</span>
+<p>Et sic procedas retrahendo triplando figuram.</p>
+<span class = "pagenum">80</span>
+<a name = "page80" id = "page80"> </a>
+<p>Et proponendo nonam, donec totum peragatur,</p>
+<p>Subtripla sub propriis servare decet triplicatis;</p>
+<p>Si nil in fine remanet, numerus datus ante</p>
+<span class = "linenum">296</span>
+<p>Est cubicus; cubicam radicem sub tripla prebent,</p>
+<p>Cum digito juncto quem supprimo posuisti,</p>
+<p>Hec cubice ducta, numerum reddant tibi primum.</p>
+<p>Si quid erit remanens non est cubicus, sed habetur</p>
+<span class = "linenum">300</span>
+<p>Major sub primo qui stat radix cubicam,</p>
+<p>Servari debet quicquid radice remansit,</p>
+<p>Extracto numero, decet hec addi cubicato.</p>
+<p>Quo facto, numerus reddi debet tibi primus.</p>
+<span class = "linenum">304</span>
+<p>Nam debes per se radicem multiplicare</p>
+<p>Ex hinc in numerum duces, qui provenit inde</p>
+<p>Sub primo cubicus major sic invenietur;</p>
+<p>Illi jungatur remanens, et primus habetur,</p>
+<span class = "linenum">308</span>
+<p>Si per triplatum numerum nequeas operari;</p>
+<p>Cifram propones, nil vero per hanc operare</p>
+<p>Set retrahens illam cum saltu deinde triplata,</p>
+<p>Propones illi digitum sub lege priori,</p>
+<span class = "linenum">312</span>
+<p>Cumque cifram retrahas saliendo, non triplicabis,</p>
+<p>Namque nihil cifre triplacio dicitur esse;</p>
+<p>At tu cum cifram protraxeris aut triplicata,</p>
+<p>Hanc cum subtriplo semper servare memento:</p>
+<span class = "linenum">316</span>
+<p>Si det compositum, digiti triplacio debet</p>
+<p>Illius scribi, digitus saliendo sub ipsam;</p>
+<p>Digito deleto, que terna dicitur esse;</p>
+<p>Jungitur articulus cum triplata pereunte,</p>
+<span class = "linenum">320</span>
+<p>Set facit hunc scribi per se triplacio prima,</p>
+<p>Que si det digitum per se scribi facit illum;</p>
+<p>Consumpto numero, si sole fuit tibi cifre</p>
+<p>Triplato, propone cifram saltum faciendo,</p>
+<span class = "linenum">324</span>
+<p>Cumque cifram retrahe triplam, scribendo figuram,</p>
+<p>Preponas cifre, sic procedens operare,</p>
+<p>Si tres vel duo serie in sint, pone sub yma,</p>
+<p>A dextris digitum servando prius documentum.</p>
+<span class = "linenum">328</span>
+<p>Si sit continua progressio terminus nuper</p>
+<p>Per majus medium totalem multiplicato;</p>
+<p>Si par, per medium tunc multiplicato sequentem.</p>
+<p>Set si continua non sit progressio finis:</p>
+<span class = "linenum">332</span>
+<p>Impar, tunc majus medium si multiplicabis,</p>
+<span class = "linenum">333</span>
+<p>Si par per medium sibi multiplicato propinquum.</p>
+
+</div> <!-- end div verse -->
+
+<hr class = "tiny" />
+
+<div class = "footnote">
+
+<p><a name = "note_app_1" id = "note_app_1" href = "#tag_app_1">1.</a>
+“Hec præsens ars dicitur algorismus ab Algore rege ejus inventore, vel
+dicitur ab <i>algos</i> quod est ars, et <i>rodos</i> quod est numerus;
+quæ est ars numerorum vel numerandi, ad quam artem bene sciendum
+inveniebantur apud Indos bis quinque (id est decem) figuræ.”
+&mdash;<i>Comment. Thomæ de Novo-Mercatu.</i> MS. Bib. Reg. Mus. Brit.
+12 E.&nbsp;1.</p>
+
+<p><a name = "note_app_2" id = "note_app_2" href = "#tag_app_2">2.</a>
+“Hæ necessariæ figuræ sunt Indorum characteros.” <i>MS. de
+numeratione.</i> Bib. Sloan. Mus. Brit. 513, fol. 58. “Cum vidissem
+Yndos constituisse <span class = "smallroman">IX</span> literas in
+universo numero suo propter dispositionem suam quam posuerunt, volui
+patefacere de opere quod sit per eas aliquidque esset levius
+discentibus, si Deus voluerit. Si autem Indi hoc voluerunt et intentio
+illorum nihil novem literis fuit, causa que mihi potuit. Deus direxit me
+ad hoc. Si vero alia dicam preter eam quam ego exposui, hoc fecerunt per
+hoc quod ego exposui, eadem tam certissime et absque ulla dubitatione
+poterit inveniri. Levitasque patebit aspicientibus et discentibus.” MS.
+U.L.C., Ii. vi. 5, f.&nbsp;102.</p>
+
+<p><a name = "note_app_3" id = "note_app_3" href = "#tag_app_3">3.</a>
+From Eg. 2622.</p>
+
+<p><a name = "note_app_4" id = "note_app_4" href = "#tag_app_4">4.</a>
+8 C. iv. inserts Nullum cipa significat: dat significare sequenti.</p>
+
+<p><a name = "note_app_5" id = "note_app_5" href = "#tag_app_5">5.</a>
+From 12 E. 1.</p>
+
+<p><a name = "note_app_6" id = "note_app_6" href =
+"#tag_app_5">6.</a></p>
+<div class = "verse">
+<p>En argorisme devon prendre</p>
+<p>Vii especes . . . .</p>
+<p>Adision subtracion</p>
+<p>Doubloison mediacion</p>
+<p>Monteploie et division</p>
+<p>Et de radix eustracion</p>
+<p>A chez vii especes savoir</p>
+<p>Doit chascun en memoire avoir</p>
+<p>Letres qui figures sont dites</p>
+<p>Et qui excellens sont ecrites.&mdash;MS. <i>Seld. Arch.</i>
+B.&nbsp;26.</p>
+</div>
+
+<p><a name = "note_app_7" id = "note_app_7" href = "#tag_app_7">7.</a>
+From 12 E. 1.</p>
+
+<p><a name = "note_app_8" id = "note_app_8" href = "#tag_app_8">8.</a>
+From 12 E. 1.</p>
+
+<p><a name = "note_app_9" id = "note_app_9" href = "#tag_app_9">9.</a>
+8 C. iv. inserts Atque figura prior nuper fuerit mediando.</p>
+
+<p><a name = "note_app_10" id = "note_app_10" href =
+"#tag_app_10">10.</a>
+<i>I.e.</i> figura secundo loco posita.</p>
+
+<p><a name = "note_app_11" id = "note_app_11" href =
+"#tag_app_11">11.</a>
+So 12 E. 1; 8 C. iv. inserts&mdash;</p>
+<div class = "verse">
+<p>Si super extremam nota sit monades dat eidem</p>
+<p>Quod contingat cum primo dimiabis</p>
+<p>Atque figura prior nuper fuerit mediando.</p>
+</div>
+
+<p><a name = "note_app_12" id = "note_app_12" href =
+"#tag_app_12">12.</a>
+12 E. 1 inserts.</p>
+
+<p><a name = "note_app_13" id = "note_app_13" href =
+"#tag_app_13">13.</a>
+12 E. 1 inserts to l. 174.</p>
+
+<p><a name = "note_app_14" id = "note_app_14" href =
+"#tag_app_14">14.</a>
+12 E. 1 omits, Eg. 2622 inserts.</p>
+
+<p><a name = "note_app_15" id = "note_app_15" href =
+"#tag_app_15">15.</a>
+12 E. 1 inserts.</p>
+
+<p><a name = "note_app_16" id = "note_app_16" href =
+"#tag_app_16">16.</a>
+8 C. iv. inserts&mdash;</p>
+<div class = "verse">
+<p>Hinc illam dele duplans sub ei psalliendo</p>
+<p>Que sequitur retrahens quicquid fuerit duplicatum.</p>
+</div>
+
+</div>
+
+</div> <!-- end div app -->
+
+</div> <!-- end div maintext -->
+
+<hr class = "mid" />
+
+<div class = "index">
+
+<span class = "pagenum">81</span>
+<a name = "page81" id = "page81"> </a>
+
+<h3><a name = "terms" id = "terms">INDEX OF TECHNICAL TERMS</a><a class
+= "tag" name = "tag_terms1" id = "tag_terms1" href =
+"#note_terms1">1</a></h3>
+
+<p><a name = "terms_algorisme" id =
+"terms_algorisme"><b>algorisme</b></a>,
+<a href = "#algorisme">33/12</a>;
+<b>algorym</b>, <b>augrym</b>, <a href = "#algorym">3/3</a>;
+the art of computing, using the so-called Arabic numerals.</p>
+
+<p class = "inset">
+The word in its various forms is derived from the Arabic
+<i>al-Khowarazmi</i> (i.e. the native of Khwarazm (Khiva)). This was the
+surname of Ja’far Mohammad ben Musa, who wrote a treatise early in the
+9th century (see p.&nbsp;xiv).</p>
+
+<p class = "inset">
+The form <i>algorithm</i> is also found, being suggested by a supposed
+derivation from the Greek <span class = "greek" title =
+"arithmos">ἀριθμός</span> (number).</p>
+
+<p><a name = "terms_antery" id = "terms_antery"><b>antery</b></a>,
+<a href = "#antery">24/11</a>;
+to move figures to the right of the position in which they are first
+written. This operation is performed repeatedly upon the multiplier in
+multiplication, and upon certain figures which arise in the process of
+root extraction.</p>
+
+<p><a name = "terms_anterioracioun" id =
+"terms_anterioracioun"><b>anterioracioun</b></a>,
+<a href = "#anterioracioun">50/5</a>;
+the operation of moving figures to the right.</p>
+
+<p><a name = "terms_article" id = "terms_article"><b>article</b></a>,
+<a href = "#article">34/23</a>;
+<b>articul</b>, <a href = "#articul">5/31</a>;
+<b>articuls</b>, <a href = "#articuls">9/36</a>,
+<a href = "#articuls2">29/7,8</a>;
+a&nbsp;number divisible by ten without remainder.</p>
+
+<p><a name = "terms_cast" id = "terms_cast"><b>cast</b></a>,
+<a href = "#cast">8/12</a>;
+to add one number to another.</p>
+
+<p class = "inset">
+‘Addition is a <i>casting</i> together of two numbers into one number,’
+<a href = "#castyng">8/10</a>.</p>
+
+<p><a name = "terms_cifre" id = "terms_cifre"><b>cifre</b></a>,
+<a href = "#cifre">4/1</a>;
+the name of the figure 0. The word is derived from the Arabic
+<i>sifr</i> = empty, nothing. Hence <i>zero</i>.</p>
+
+<p class = "inset">
+A cipher is the symbol of the absence of number or of zero quantity. It
+may be used alone or in conjunction with digits or other ciphers, and in
+the latter case, according to the position which it occupies relative to
+the other figures, indicates the absence of units, or tens, or hundreds,
+etc. The great superiority of the Arabic to all other systems of
+notation resides in the employment of this symbol. When the cipher is
+not used, the place value of digits has to be indicated by writing them
+in assigned rows or columns. Ciphers, however, may be interpolated
+amongst the significant figures used, and as they sufficiently indicate
+the positions of the empty rows or columns, the latter need not be
+indicated in any other way. The practical performance of calculations is
+thus enormously facilitated (see <a href =
+"#intro_modern">p.&nbsp;xvi</a>).</p>
+
+<p><a name = "terms_componede" id =
+"terms_componede"><b>componede</b></a>, <a href =
+"#componede">33/24</a>;
+<b>composyt</b>, <a href = "#composyt">5/35</a>;
+with reference to numbers, one compounded of a multiple of ten and a
+digit.</p>
+
+<p><a name = "terms_conuertide" id =
+"terms_conuertide"><b>conuertide</b></a> = conversely,
+<a href = "#conuertide">46/29</a>,
+<a href = "#conuertide2">47/9</a>.</p>
+
+<p><a name = "terms_cubicede" id = "terms_cubicede"><b>cubicede</b></a>,
+<a href = "#cubicede">50/13</a>;
+<b>to be c.</b>, to have its cube root found.</p>
+
+<span class = "pagenum">82</span>
+<a name = "page82" id = "page82"> </a>
+
+<p><a name = "terms_cubike" id = "terms_cubike"><b>cubike
+nombre</b></a>,
+<a href = "#cubike">47/8</a>;
+a&nbsp;number formed by multiplying a given number twice by itself,
+<i>e.g.</i> 27 = 3&nbsp;×&nbsp;3&nbsp;×&nbsp;3. Now called simply a
+cube.</p>
+
+<p><a name = "terms_decuple" id = "terms_decuple"><b>decuple</b></a>,
+<a href = "#decuple">22/12</a>;
+the product of a number by ten. Tenfold.</p>
+
+<p><a name = "terms_departys" id = "terms_departys"><b>departys</b></a>
+= divides,
+<a href = "#departys">5/29</a>.</p>
+
+<p><a name = "terms_digit" id = "terms_digit"><b>digit</b></a>,
+<a href = "#digit">5/30</a>;
+<b>digitalle</b>, <a href = "#digitalle">33/24</a>;
+a&nbsp;number less than ten, represented by one of the nine Arabic
+numerals.</p>
+
+<p><a name = "terms_dimydicion" id =
+"terms_dimydicion"><b>dimydicion</b></a>,
+<a href = "#dimydicion">7/23</a>;
+the operation of dividing a number by two. Halving.</p>
+
+<p><a name = "terms_duccioun" id = "terms_duccioun"><b>duccioun</b></a>,
+multiplication, <a href = "#duccioun">43/9</a>.</p>
+
+<p><a name = "terms_duplacion" id =
+"terms_duplacion"><b>duplacion</b></a>,
+<a href = "#duplacion">7/23</a>,
+<a href = "#duplacion2">14/15</a>; the operation of multiplying a number
+by two. Doubling.</p>
+
+<p><a name = "terms_imediet" id = "terms_imediet"><b>i-mediet</b></a> =
+halved, <a href = "#imediet">19/23</a>.</p>
+
+<p><a name = "terms_intercise" id =
+"terms_intercise"><b>intercise</b></a> = broken, <a href =
+"#intercise">46/2</a>;
+intercise Progression is the name given to either of the Progressions 1,
+3, 5, 7, etc.; 2, 4, 6, 8, etc., in which the common difference
+is&nbsp;2.</p>
+
+<p><a name = "terms_lede_into" id = "terms_lede_into"><b>lede
+into</b></a>, multiply by, <a href = "#lede_into">47/18</a>.</p>
+
+<p><a name = "terms_lyneal" id = "terms_lyneal"><b>lyneal
+nombre</b></a>, <a href = "#lyneal">46/14</a>;
+a number such as that which expresses the measure of the length of a
+line, and therefore is not <i>necessarily</i> the product of two or more
+numbers (<i>vide</i> Superficial, Solid). This appears to be the meaning
+of the phrase as used in <i>The Art of Nombryng</i>. It is possible that
+the numbers so designated are the prime numbers, that is, numbers not
+divisible by any other number except themselves and unity, but it is not
+clear that this limitation is intended.</p>
+
+<p><a name = "terms_mediacioun" id =
+"terms_mediacioun"><b>mediacioun</b></a>,
+<a href = "#mediacioun">16/36</a>,
+<a href = "#mediacioun2">38/16</a>; dividing by two (see also
+<b>dimydicion</b>).</p>
+
+<p><a name = "terms_medlede" id = "terms_medlede"><b>medlede
+nombre</b></a>,
+<a href = "#medlede">34/1</a>;
+a&nbsp;number formed of a multiple of ten and a digit (<i>vide</i>
+componede, composyt).</p>
+
+<p><a name = "terms_medye" id = "terms_medye"><b>medye</b></a>,
+<a href = "#medye">17/8</a>, to halve;
+<b>mediete</b>, halved, <a href = "#mediete">17/30</a>;
+<b>ymedit</b>, <a href = "#ymedit">20/9</a>.</p>
+
+<p><a name = "terms_naturelle" id = "terms_naturelle"><b>naturelle
+progressioun</b></a>,
+<a href = "#naturelle">45/22</a>;
+the series of numbers 1, 2, 3, etc.</p>
+
+<p><a name = "terms_produccioun" id =
+"terms_produccioun"><b>produccioun</b></a>, multiplication,
+<a href = "#produccioun">50/11</a>.</p>
+
+<p><a name = "terms_quadrat" id = "terms_quadrat"><b>quadrat
+nombre</b></a>,
+<a href = "#quadrat">46/12</a>;
+a number formed by multiplying a given number by itself, <i>e.g.</i> 9 =
+3&nbsp;×&nbsp;3, a&nbsp;square.</p>
+
+<p><a name = "terms_rote" id = "terms_rote"><b>rote</b></a>,
+<a href = "#rote">7/25</a>;
+<b>roote</b>, <a href = "#roote">47/11</a>;
+root. The roots of squares and cubes are the numbers from which the
+squares and cubes are derived by multiplication into themselves.</p>
+
+<p><a name = "terms_significatyf" id =
+"terms_significatyf"><b>significatyf</b></a>, significant,
+<a href = "#significatyf">5/14</a>;
+The significant figures of a number are, strictly speaking, those other
+than zero, <i>e.g.</i> in 3&nbsp;6 5 0 4 0&nbsp;0, the significant
+figures are 3, 6, 5,&nbsp;4. Modern usage, however, regards all figures
+between the two extreme significant figures as significant, even when
+some are zero. Thus, in the above example, 3&nbsp;6 5 0&nbsp;4 are
+considered significant.</p>
+
+<p><a name = "terms_solide" id = "terms_solide"><b>solide
+nombre</b></a>, <a href = "#solide">46/37</a>;
+a&nbsp;number which is the product of three other numbers, <i>e.g.</i>
+66 = 11&nbsp;×&nbsp;2&nbsp;×&nbsp;3.</p>
+
+<p><a name = "terms_superficial" id = "terms_superficial"><b>superficial
+nombre</b></a>, <a href = "#superficial">46/18</a>;
+a&nbsp;number which is the product of two other numbers, <i>e.g.</i> 6 =
+2&nbsp;×&nbsp;3.</p>
+
+<p><a name = "terms_ternary" id = "terms_ternary"><b>ternary</b></a>,
+consisting of three digits, <a href = "#ternary">51/7</a>.</p>
+
+<p><a name = "terms_vnder_double" id = "terms_vnder_double"><b>vnder
+double</b></a>, a digit which has been doubled,
+<a href = "#vnder_double">48/3</a>.</p>
+
+<p><a name = "terms_vnder_trebille" id =
+"terms_vnder_trebille"><b>vnder-trebille</b></a>, a digit which has been
+trebled,
+<a href = "#vnder_trebille">49/28</a>;
+<b>vnder-triplat</b>, <a href = "#vnder_triplat">49/39</a>.</p>
+
+<p><a name = "terms_sup_w" id = "terms_sup_w"><b>w</b></a>,
+a symbol used to denote half a unit, <a href = "#sup_w">17/33</a>.</p>
+
+</div> <!-- end div index -->
+
+<p class = "footnote">
+<a name = "note_terms1" id = "note_terms1" href = "#tag_terms1">1.</a>
+This Index has been kindly prepared by Professor J.&nbsp;B. Dale, of
+King’s College, University of London, and the best thanks of the Society
+are due to him for his valuable contribution.</p>
+
+<hr class = "mid" />
+
+<div class = "glossary">
+
+<span class = "pagenum">83</span>
+<a name = "page83" id = "page83"> </a>
+
+<h3><a name = "glossary" id = "glossary">GLOSSARY</a></h3>
+
+<p class = "mynote">
+Words whose first appearance is earlier than the page cited in the
+Glossary are identified in supplementary notes, and both occurrences are
+marked in the main text.</p>
+
+<p><a name = "gloss_ablacioun" id =
+"gloss_ablacioun"><b>ablacioun</b></a>, taking away, <a href =
+"#ablacioun">36/21</a></p>
+
+<p><a name = "gloss_addyst" id = "gloss_addyst"><b>addyst</b></a>,
+haddest, <a href = "#addyst">10/37</a></p>
+
+<p><a name = "gloss_agregacioun" id =
+"gloss_agregacioun"><b>agregacioun</b></a>, addition, <a href =
+"#agregacioun">45/22</a>. (First example in N.E.D., 1547.)</p>
+
+<p><a name = "gloss_aghenenes" id =
+"gloss_aghenenes"><b>a-ȝenenes</b></a>, against, <a href =
+"#aghenenes">23/10</a></p>
+
+<p><a name = "gloss_allgate" id = "gloss_allgate"><b>allgate</b></a>,
+always, <a href = "#allgate">8/39</a></p>
+
+<p><a name = "gloss_als" id = "gloss_als"><b>als</b></a>, as, <a href =
+"#als">22/24</a></p>
+
+<p><a name = "gloss_and" id = "gloss_and"><b>and</b></a>, if, <a href =
+"#and">29/8</a>;</p>
+<p class = "inset">
+<b>&amp;</b>, <a href = "#amp">4/27</a>;</p>
+<p class = "inset">
+<b>&amp; yf</b>, <a href = "#and_yf">20/7</a></p>
+
+<p><a name = "gloss_anendes" id = "gloss_anendes"><b>a-nendes</b></a>,
+towards, <a href = "#anendes">23/15</a></p>
+
+<p><a name = "gloss_aproprede" id =
+"gloss_aproprede"><b>aproprede</b></a>, appropriated, <a href =
+"#aproprede">34/27</a></p>
+
+<p><a name = "gloss_apwereth" id = "gloss_apwereth"><b>apwereth</b></a>,
+appears, <a href = "#apwereth">61/8</a></p>
+
+<p><a name = "gloss_arisyght" id = "gloss_arisyght"><b>a-risyȝt</b></a>,
+arises, <a href = "#arisyght">14/24</a></p>
+
+<p><a name = "gloss_arowe" id = "gloss_arowe"><b>a-rowe</b></a>, in a
+row, <a href = "#arowe">29/10</a></p>
+
+<p><a name = "gloss_arsemetrike" id =
+"gloss_arsemetrike"><b>arsemetrike</b></a>, arithmetic, <a href =
+"#arsemetrike">33/1</a></p>
+
+<p><a name = "gloss_ayene" id = "gloss_ayene"><b>ayene</b></a>, again,
+<a href = "#ayene">45/15</a></p>
+
+
+<p><a name = "gloss_bagle" id = "gloss_bagle"><b>bagle</b></a>, crozier,
+<a href = "#bagle">67/12</a></p>
+
+<p><a name = "gloss_bordure" id = "gloss_bordure"><b>bordure</b></a> =
+ordure, row, <a href = "#bordure">43/30</a></p>
+
+<p><a name = "gloss_borro" id = "gloss_borro"><b>borro</b></a>,
+<i>inf.</i> borrow, <a href = "#borro">11/38</a>;</p>
+<p class = "inset">
+<i>imp. s.</i> <b>borowe</b>, <a href = "#borowe">12/20</a>;</p>
+<p class = "inset">
+<i>pp.</i> <b>borwed</b>, <a href = "#borwed">12/15</a>;</p>
+<p class = "inset">
+<b>borred</b>, <a href = "#borred">12/19</a></p>
+
+<p><a name = "gloss_boue" id = "gloss_boue"><b>boue</b></a>, above,
+<a href = "#boue">42/34</a></p>
+
+
+<p><a name = "gloss_caputule" id = "gloss_caputule"><b>caputule</b></a>,
+chapter, <a href = "#caputule">7/26</a></p>
+
+<p><a name = "gloss_certayn" id = "gloss_certayn"><b>certayn</b></a>,
+assuredly, <a href = "#certayn">18/34</a></p>
+
+<p><a name = "gloss_clepede" id = "gloss_clepede"><b>clepede</b></a>,
+called, <a href = "#clepede">47/7</a></p>
+
+<p><a name = "gloss_competently" id =
+"gloss_competently"><b>competently</b></a>, conveniently, <a href =
+"#competently">35/8</a></p>
+
+<p><a name = "gloss_compt" id = "gloss_compt"><b>compt</b></a>, count,
+<a href = "#compt">47/29</a></p>
+
+<p><a name = "gloss_contynes" id = "gloss_contynes"><b>contynes</b></a>,
+contains, <a href = "#contynes">21/12</a>;</p>
+<p class = "inset">
+<i>pp.</i> <b>contenythe</b>, <a href = "#contenythe">38/39</a></p>
+
+<p><a name = "gloss_craft" id = "gloss_craft"><b>craft</b></a>, art,
+<a href = "#craft">3/4</a></p>
+
+
+<p><a name = "gloss_distingue" id =
+"gloss_distingue"><b>distingue</b></a>, divide, <a href =
+"#distingue">51/5</a></p>
+
+
+<p><a name = "gloss_egalle" id = "gloss_egalle"><b>egalle</b></a>,
+equal, <a href = "#egalle">45/21</a></p>
+
+<p><a name = "gloss_excep" id = "gloss_excep"><b>excep</b></a>, except,
+<a href = "#excep">5/16</a></p>
+
+<p><a name = "gloss_exclusede" id =
+"gloss_exclusede"><b>exclusede</b></a>, excluded, <a href =
+"#exclusede">34/37</a></p>
+
+<p><a name = "gloss_excressent" id =
+"gloss_excressent"><b>excressent</b></a>, resulting, <a href =
+"#excressent">35/16</a></p>
+
+<p><a name = "gloss_exeant" id = "gloss_exeant"><b>exeant</b></a>,
+resulting, <a href = "#exeant">43/26</a></p>
+
+<p><a name = "gloss_expone" id = "gloss_expone"><b>expone</b></a>,
+expound, <a href = "#expone">3/23</a></p>
+
+
+<p><a name = "gloss_ferye" id = "gloss_ferye"><b>ferye</b></a> = ferþe,
+fourth, <a href = "#ferye">70/12</a></p>
+
+<p><a name = "gloss_figure" id = "gloss_figure"><b>figure</b></a> =
+figures, <a href = "#figure">5/1</a></p>
+
+<p><a name = "gloss_forby" id = "gloss_forby"><b>for-by</b></a>, past,
+<a href = "#forby">12/11</a></p>
+
+<p><a name = "gloss_fors" id = "gloss_fors"><b>fors; no f.</b></a>, no
+matter, <a href = "#fors">22/24</a></p>
+
+<p><a name = "gloss_forseth" id = "gloss_forseth"><b>forseth</b></a>,
+matters, <a href = "#forseth">53/30</a></p>
+
+<p><a name = "gloss_forye" id = "gloss_forye"><b>forye</b></a> = forþe,
+forth, <a href = "#forye">71/8</a></p>
+
+<p><a name = "gloss_fyftye" id = "gloss_fyftye"><b>fyftye</b></a> =
+fyftþe, fifth, <a href = "#fyftye">70/16</a></p>
+
+
+<p><a name = "gloss_grewe" id = "gloss_grewe"><b>grewe</b></a>, Greek,
+<a href = "#grewe">33/13</a></p>
+
+
+<p><a name = "gloss_haluendel" id =
+"gloss_haluendel"><b>haluendel</b></a>, half, <a href =
+"#haluendel">16/16</a>;</p>
+<p class = "inset">
+<b>haldel</b>, <a href = "#haldel">19/4</a>;</p>
+<p class = "inset">
+<i>pl.</i> <b>haluedels</b>, <a href = "#haluedels">16/16</a></p>
+
+<p><a name = "gloss_hayst" id = "gloss_hayst"><b>hayst</b></a>, hast,
+<a href = "#hayst">17/3</a>, 32</p>
+
+<p><a name = "gloss_hast" id = "gloss_hast"><b>hast</b></a>, haste,
+<a href = "#hast">22/25</a></p>
+
+<p><a name = "gloss_heer" id = "gloss_heer"><b>heer</b></a>, higher,
+<a href = "#heer">9/35</a></p>
+
+<p><a name = "gloss_here" id = "gloss_here"><b>here</b></a>, their,
+<a href = "#here">7/26</a></p>
+
+<p><a name = "gloss_hereafore" id =
+"gloss_hereafore"><b>here-a-fore</b></a>, heretofore, <a href =
+"#hereafore">13/7</a></p>
+
+<p><a name = "gloss_heyth" id = "gloss_heyth"><b>heyth</b></a>, was
+called, <a href = "#heyth">3/5</a></p>
+
+<p><a name = "gloss_hole" id = "gloss_hole"><b>hole</b></a>, whole,
+<a href = "#hole">4/39</a>;</p>
+<p class = "inset">
+<b>holle</b>, <a href = "#holle">17/1</a>;</p>
+<p class = "inset">
+<b>hoole</b>, of three dimensions, <a href = "#hoole">46/15</a></p>
+
+<p><a name = "gloss_holdythe" id = "gloss_holdythe"><b>holdyþe</b></a>,
+holds good, <a href = "#holdythe">30/5</a></p>
+
+<p><a name = "gloss_how_be_it_that" id = "gloss_how_be_it_that"><b>how
+be it that</b></a>, although, <a href = "#how_be_it_that">44/4</a></p>
+
+
+<p><a name = "gloss_lede" id = "gloss_lede"><b>lede</b></a> = lete, let,
+<a href = "#lede">8/37</a></p>
+
+<p><a name = "gloss_lene" id = "gloss_lene"><b>lene</b></a>, lend,
+<a href = "#lene">12/39</a></p>
+
+<p><a name = "gloss_lest" id = "gloss_lest"><b>lest</b></a>, least,
+<a href = "#lest">43/27</a></p>
+
+<p><a name = "gloss_lest2" id = "gloss_lest2"><b>lest</b></a> = left,
+<a href = "#lest2">71/9</a></p>
+
+<p><a name = "gloss_leue" id = "gloss_leue"><b>leue</b></a>, leave,
+<a href = "#leue">6/5</a>;</p>
+<p class = "inset">
+<i>pr. 3 s.</i> <b>leues</b>, remains, <a href = "#leues">11/19</a>;
+<span class = "mynote">First used in <a href =
+"#leues1">10/40</a></span></p>
+<p class = "inset">
+<b>leus</b>, <a href = "#leus">11/28</a>;</p>
+<p class = "inset">
+<i>pp.</i> <b>laft</b>, left, <a href = "#laft">19/24</a></p>
+
+<p><a name = "gloss_lewder" id = "gloss_lewder"><b>lewder</b></a>, more
+ignorant, <a href = "#lewder">3/3</a></p>
+
+<p><a name = "gloss_lust" id = "gloss_lust"><b>lust</b></a>, desirest
+to, <a href = "#lust">45/13</a></p>
+
+<p><a name = "gloss_lyght" id = "gloss_lyght"><b>lyȝt</b></a>, easy,
+<a href = "#lyght">15/31</a></p>
+
+<p><a name = "gloss_lymytes" id = "gloss_lymytes"><b>lymytes</b></a>,
+limits, <a href = "#lymytes">34/18</a>;</p>
+<p class = "inset">
+<b>lynes</b>, <a href = "#lynes">34/12</a>;</p>
+<p class = "inset">
+<b>lynees</b>, <a href = "#lynees">34/17</a>;</p>
+<p class = "inset">
+Lat. limes, <i>pl.</i> limites.</p>
+
+
+<p><a name = "gloss_maystery" id = "gloss_maystery"><b>maystery</b></a>,
+achievement;</p>
+<p class = "inset">
+<b>no m.</b>, no achievement, i.e. easy, <a href =
+"#maystery">19/10</a></p>
+
+<p><a name = "gloss_me" id = "gloss_me"><b>me</b></a>, <i>indef.
+pron.</i> one, <a href = "#me">42/1</a>
+<span class = "mynote">First used in <a href =
+"#me1">34/16</a></span></p>
+
+<p><a name = "gloss_mo" id = "gloss_mo"><b>mo</b></a>, more, <a href =
+"#mo">9/16</a>
+<span class = "pagenum">84</span>
+<a name = "page84" id = "page84"> </a></p>
+<p class = "inset">
+<b>moder</b> = more (Lat. majorem), <a href = "#moder">43/22</a></p>
+
+<p><a name = "gloss_most" id = "gloss_most"><b>most</b></a>, must,
+<a href = "#most">30/3</a>
+<span class = "mynote">First used in <a href =
+"#most1">3/12</a></span></p>
+
+<p><a name = "gloss_multipliede" id =
+"gloss_multipliede"><b>multipliede</b></a>, <b>to be m.</b> =
+multiplying, <a href = "#multipliede">40/9</a></p>
+
+<p><a name = "gloss_mynvtes" id = "gloss_mynvtes"><b>mynvtes</b></a>,
+the sixty parts into which a unit is divided, <a href =
+"#mynvtes">38/25</a></p>
+
+<p><a name = "gloss_mysewroght" id =
+"gloss_mysewroght"><b>myse-wroȝt</b></a>, mis-wrought, <a href =
+"#mysewroght">14/11</a></p>
+
+
+<p><a name = "gloss_nether" id = "gloss_nether"><b>nether</b></a>, nor,
+<a href = "#nether">34/25</a></p>
+
+<p><a name = "gloss_nex" id = "gloss_nex"><b>nex</b></a>, next,
+<a href = "#nex">19/9</a></p>
+
+<p><a name = "gloss_noght" id = "gloss_noght"><b>noȝt</b></a>, nought,
+<a href = "#noght">5/7</a>
+<span class = "mynote">First used in <a href =
+"#noght1">4/8</a></span></p>
+
+<p><a name = "gloss_note" id = "gloss_note"><b>note</b></a>, not,
+<a href = "#note">30/5</a></p>
+
+
+<p><a name = "gloss_oo" id = "gloss_oo"><b>oo</b></a>, one, <a href =
+"#oo">42/20</a>; <b>o</b>, <a href = "#o">42/21</a>
+<span class = "mynote">First used in <a href =
+"#oo1">34/27</a>&nbsp;(oo); <a href =
+"#o1">33/22</a>&nbsp;(o)</span></p>
+
+<p><a name = "gloss_omest" id = "gloss_omest"><b>omest</b></a>,
+uppermost, higher, <a href = "#omest">35/26</a>;</p>
+<p class = "inset">
+<b>omyst</b>, <a href = "#omyst">35/28</a></p>
+
+<p><a name = "gloss_omwhile" id = "gloss_omwhile"><b>omwhile</b></a>,
+sometimes, <a href = "#omwhile">45/31</a>
+<span class = "mynote">First used in <a href =
+"#omwhile1">39/17</a></span></p>
+
+<p><a name = "gloss_on" id = "gloss_on"><b>on</b></a>, one, <a href =
+"#on">8/29</a></p>
+
+<p><a name = "gloss_opyne" id = "gloss_opyne"><b>opyne</b></a>, plain,
+<a href = "#opyne">47/8</a></p>
+
+<p><a name = "gloss_or" id = "gloss_or"><b>or</b></a>, before, <a href =
+"#or">13/25</a></p>
+
+<p><a name = "gloss_or2" id = "gloss_or2"><b>or</b></a> = þe
+oþ<i>er</i>, the other, <a href = "#or">28/34</a></p>
+
+<p><a name = "gloss_ordure" id = "gloss_ordure"><b>ordure</b></a>,
+order, <a href = "#ordure">34/9</a>;</p>
+<p class = "inset">
+row, <a href = "#order">43/1</a>
+<span class = "mynote">Word form is “order”</span></p>
+
+<p><a name = "gloss_other" id = "gloss_other"><b>other</b></a>, or,
+<a href = "#other">33/13</a>, <a href = "#other2">43/26</a>;
+<span class = "mynote">Note also “one other other” in <a href =
+"#other1">35/24</a></span></p>
+<p class = "inset">
+<b>other . . . or</b>, either . . . or, <a href = "#other_or">38/37</a>
+<span class = "mynote">First used in <a href =
+"#other_or1">37/5</a></span></p>
+
+<p><a name = "gloss_ouerer" id = "gloss_ouerer"><b>ouerer</b></a>,
+upper, <a href = "#ouerer">42/15</a></p>
+
+<p><a name = "gloss_ouerhippede" id =
+"gloss_ouerhippede"><b>ouer-hippede</b></a>, passed over, <a href =
+"#ouerhippede">43/19</a></p>
+
+
+<p><a name = "gloss_recte" id = "gloss_recte"><b>recte</b></a>,
+directly, <a href = "#recte">27/20</a>
+<span class = "mynote">First used in <a href =
+"#recte1">26/31</a></span></p>
+
+<p><a name = "gloss_remayner" id = "gloss_remayner"><b>remayner</b></a>,
+remainder, <a href = "#remayner">56/28</a></p>
+
+<p><a name = "gloss_representithe" id =
+"gloss_representithe"><b>representithe</b></a>, represented, <a href =
+"#representithe">39/14</a></p>
+
+<p><a name = "gloss_resteth" id = "gloss_resteth"><b>resteth</b></a>,
+remains, <a href = "#resteth">63/29</a>
+<span class = "mynote">First used in <a href =
+"#resteth1">57/29</a></span></p>
+
+<p><a name = "gloss_rewarde" id = "gloss_rewarde"><b>rewarde</b></a>,
+regard, <a href = "#rewarde">48/6</a></p>
+
+<p><a name = "gloss_rew" id = "gloss_rew"><b>rew</b></a>, row, <a href =
+"#rew">4/8</a></p>
+
+<p><a name = "gloss_rewle" id = "gloss_rewle"><b>rewle</b></a>, row,
+<a href = "#rewle">4/20</a>, <a href = "#rewle2">7/12</a>;</p>
+<p class = "inset">
+<b>rewele</b>, <a href = "#rewele">4/18</a>;</p>
+<p class = "inset">
+<b>rewles</b>, rules, <a href = "#rewles">5/33</a></p>
+
+
+<p><a name = "gloss_sc" id = "gloss_sc"><b>s.</b></a> = scilicet,
+<a href = "#sc">3/8</a></p>
+
+<p><a name = "gloss_sentens" id = "gloss_sentens"><b>sentens</b></a>,
+meaning, <a href = "#sentens">14/29</a></p>
+
+<p><a name = "gloss_signifyetyf" id =
+"gloss_signifyetyf"><b>signifye(tyf)</b></a>, <a href =
+"#signifyetyf">5/13</a>. The last three letters are added above the
+line, evidently because of the word ‘significatyf’ in l.&nbsp;14. But
+the ‘Solucio,’ which contained the word, has been omitted.</p>
+
+<p><a name = "gloss_sithen" id = "gloss_sithen"><b>sithen</b></a>,
+since, <a href = "#sithen">33/8</a></p>
+
+<p><a name = "gloss_some" id = "gloss_some"><b>some</b></a>, sum,
+result, <a href = "#some">40/17</a>, 32
+<span class = "mynote">First used in <a href =
+"#some1">36/21</a></span></p>
+
+<p><a name = "gloss_sowne" id = "gloss_sowne"><b>sowne</b></a>,
+pronounce, <a href = "#sowne">6/29</a></p>
+
+<p><a name = "gloss_singillatim" id =
+"gloss_singillatim"><b>singillatim</b></a>, singly, <a href =
+"#singillatim">7/25</a></p>
+
+<p><a name = "gloss_spices" id = "gloss_spices"><b>spices</b></a>,
+species, kinds, <a href = "#spices">34/4</a><span class = "mynote">First
+used in <a href = "#spices1">5/34</a></span></p>
+
+<p><a name = "gloss_spyl" id = "gloss_spyl"><b>spyl</b></a>, waste,
+<a href = "#spyl">14/26</a></p>
+
+<p><a name = "gloss_styde" id = "gloss_styde"><b>styde</b></a>, stead,
+<a href = "#styde">18/20</a></p>
+
+<p><a name = "gloss_subtrahe" id = "gloss_subtrahe"><b>subtrahe</b></a>,
+subtract, <a href = "#subtrahe">48/12</a>;</p>
+<p class = "inset">
+<i>pp.</i> <b>subtrayd</b>, <a href = "#subtrayd">13/21</a></p>
+
+<p><a name = "gloss_sythes" id = "gloss_sythes"><b>sythes</b></a>,
+times, <a href = "#sythes">21/16</a></p>
+
+
+<p><a name = "gloss_taght" id = "gloss_taght"><b>taȝt</b></a>, taught,
+<a href = "#taght">16/36</a></p>
+
+<p><a name = "gloss_take" id = "gloss_take"><b>take</b></a>, <i>pp.</i>
+taken;</p>
+<p class = "inset">
+<b>t. fro</b>, starting from, <a href = "#take">45/22</a></p>
+
+<p><a name = "gloss_taward" id = "gloss_taward"><b>taward</b></a>,
+toward, <a href = "#taward">23/34</a></p>
+
+<p><a name = "gloss_thought" id = "gloss_thought"><b>thouȝt</b></a>,
+though, <a href = "#thought">5/20</a></p>
+
+<p><a name = "gloss_trebille" id = "gloss_trebille"><b>trebille</b></a>,
+multiply by three, <a href = "#trebille">49/26</a></p>
+
+<p><a name = "gloss_twene" id = "gloss_twene"><b>twene</b></a>, two,
+<a href = "#twene">8/11</a>
+<span class = "mynote">First used in <a href =
+"#twene1">4/23</a></span></p>
+
+<p><a name = "gloss_thow" id = "gloss_thow"><b>þow</b></a>, though,
+<a href = "#thow">25/15</a></p>
+
+<p><a name = "gloss_thowght" id = "gloss_thowght"><b>þowȝt</b></a>,
+thought;</p>
+<p class = "inset">
+<b>be þ.</b>, mentally, <a href = "#thowght">28/4</a></p>
+
+<p><a name = "gloss_thus" id = "gloss_thus"><b>þus</b></a> = þis, this,
+<a href = "#thus">20/33</a></p>
+
+
+<p><a name = "gloss_vny" id = "gloss_vny"><b>vny</b></a>, unite,
+<a href = "#vny">45/10</a></p>
+
+
+<p><a name = "gloss_wel" id = "gloss_wel"><b>wel</b></a>, wilt,
+<a href = "#wel">14/31</a></p>
+
+<p><a name = "gloss_wete" id = "gloss_wete"><b>wete</b></a>, wit,
+<a href = "#wete">15/16</a>;</p>
+<p class = "inset">
+<b>wyte</b>, know, <a href = "#wyte">8/38</a>;</p>
+<p class = "inset">
+<i>pr. 2 s.</i> <b>wost</b>, <a href = "#wost">12/38</a></p>
+
+<p><a name = "gloss_wex" id = "gloss_wex"><b>wex</b></a>, become,
+<a href = "#wex">50/18</a></p>
+
+<p><a name = "gloss_where" id = "gloss_where"><b>where</b></a>, whether,
+<a href = "#where">29/12</a></p>
+
+<p><a name = "gloss_wherthurghe" id =
+"gloss_wherthurghe"><b>wher-thurghe</b></a>, whence, <a href =
+"#wherthurghe">49/15</a></p>
+
+<p><a name = "gloss_worch" id = "gloss_worch"><b>worch</b></a>, work,
+<a href = "#worch">8/19</a>;
+<span class = "mynote">First used in <a href =
+"#worch1">7/35</a></span></p>
+<p class = "inset">
+<b>wrich</b>, <a href = "#wrich">8/35</a>;</p>
+<p class = "inset">
+<b>wyrch</b>, <a href = "#wyrch">6/19</a>;</p>
+<p class = "inset">
+<i>imp. s.</i> <b>worch</b>, <a href = "#worch">15/9</a>;
+<span class = "mynote">First used in <a href =
+"#worch1">9/6</a></span></p>
+<p class = "inset">
+<i>pp.</i> <b>y-wroth</b>, <a href = "#ywroth">13/24</a></p>
+
+<p><a name = "gloss_write" id = "gloss_write"><b>write</b></a>, written,
+<a href = "#write">29/19</a>;
+<span class = "mynote">First used in <a href =
+"#write1">4/5</a></span></p>
+<p class = "inset">
+<b>y-write</b>, <a href = "#ywrite">16/1</a></p>
+
+<p><a name = "gloss_wryrchynge" id =
+"gloss_wryrchynge"><b>wryrchynge</b></a> = wyrchynge, working, <a href =
+"#wryrchynge">30/4</a></p>
+
+<p><a name = "gloss_wt" id = "gloss_wt"><b>w<sup>t</sup></b></a>, with,
+<a href = "#wt">55/8</a></p>
+
+
+<p><a name = "gloss_ybroth" id = "gloss_ybroth"><b>y-broth</b></a>,
+brought, <a href = "#ybroth">21/18</a></p>
+
+<p><a name = "gloss_ychon" id = "gloss_ychon"><b>ychon</b></a>, each
+one, <a href = "#ychon">29/10</a></p>
+
+<p><a name = "gloss_ydo" id = "gloss_ydo"><b>ydo</b></a>, done, added,
+<a href = "#ydo">9/6</a>
+<span class = "mynote">First used in <a href =
+"#ydo1">8/37</a></span></p>
+
+<p><a name = "gloss_ylke" id = "gloss_ylke"><b>ylke</b></a>, same,
+<a href = "#ylke">5/12</a></p>
+
+<p><a name = "gloss_ylyech" id = "gloss_ylyech"><b>y-lyech</b></a>,
+alike, <a href = "#ylyech">22/23</a></p>
+
+<p><a name = "gloss_ymyght" id = "gloss_ymyght"><b>y-myȝt</b></a>, been
+able, <a href = "#ymyght">12/2</a></p>
+
+<p><a name = "gloss_ynowght" id = "gloss_ynowght"><b>y-nowȝt</b></a>,
+enough, <a href = "#ynowght">15/31</a>;</p>
+<p class = "inset">
+<b>ynovȝt</b>, <a href = "#ynovght">18/34</a></p>
+
+<p><a name = "gloss_yove" id = "gloss_yove"><b>yove</b></a>, given,
+<a href = "#yove">45/33</a></p>
+
+<p><a name = "gloss_yt" id = "gloss_yt"><b>y<sup>t</sup></b></a>, that,
+<a href = "#yt">52/8</a></p>
+
+<p><b>y-write</b>, <i>v.</i> <a href =
+"#gloss_write"><b>write.</b></a></p>
+
+<p><b>y-wroth</b>, <i>v.</i> <a href =
+"#gloss_worch"><b>worch.</b></a></p>
+
+</div> <!-- end div glossary -->
+
+<div class = "endnote">
+<h4><a name = "endnote" id = "endnote">MARGINAL NOTES</a></h4>
+
+<p><b>Headnotes</b> have been moved to the beginning of the appropriate
+paragraph. Headnotes were omitted from the two Appendixes, as sidenotes
+give the same information.</p>
+
+<p><b>Line Numbers</b> are cited in the Index and Glossary. They have
+been omitted from the e-text except in the one verse selection
+(App.&nbsp;II, <i>Carmen de Algorismo</i>). Instead, the Index and
+Glossary are linked directly to each word.</p>
+
+<p><b>Numbered Notes</b>:</p>
+
+<p class = "inset">
+Numbered sidenotes show page or leaf numbers from the original MSS. In
+the e-text, sidenote numbers have been replaced with simple
+asterisks.</p>
+
+<p class = "inset">
+Footnotes give textual information such as variant readings. They
+have been numbered sequentially within each title.</p>
+
+<p><b>Sidenotes</b> giving a running synopsis of the text have been kept
+as close as possible to their original format and location.</p>
+
+</div>
+
+
+
+
+
+
+
+
+<pre>
+
+
+
+
+
+End of Project Gutenberg's The Earliest Arithmetics in English, by Anonymous
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+Project Gutenberg (https://www.gutenberg.org) public repository for
+eBook #25664 (https://www.gutenberg.org/ebooks/25664)