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diff --git a/25664-0.txt b/25664-0.txt new file mode 100644 index 0000000..b0f4b1d --- /dev/null +++ b/25664-0.txt @@ -0,0 +1,6592 @@ +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 + +*** END OF THIS PROJECT GUTENBERG EBOOK THE EARLIEST ARITHMETICS IN ENGLISH *** + +***** This file should be named 25664-0.txt or 25664-0.zip ***** +This and all associated files of various formats will be found in: + http://www.gutenberg.org/2/5/6/6/25664/ + +Produced by Louise Hope, David Starner and the Online +Distributed Proofreading Team at http://www.pgdp.net + + +Updated editions will replace the previous one--the old editions +will be renamed. + +Creating the works from public domain print editions means that no +one owns a United States copyright in these works, so the Foundation +(and you!) can copy and distribute it in the United States without +permission and without paying copyright royalties. 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