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| author | Roger Frank <rfrank@pglaf.org> | 2025-10-15 02:18:19 -0700 |
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| committer | Roger Frank <rfrank@pglaf.org> | 2025-10-15 02:18:19 -0700 |
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+font-family: sans-serif; font-size: 90%;} + +div.mynote {margin: 1em 5%; padding: .5em 1em 1em;} +p.mynote, div.glossary p.mynote {margin: 1em 5%; text-indent: 0; +padding: 1em;} +div.mynote a, span.mynote a {text-decoration: none;} + +span.mynote {margin-left: 2em; padding: .25em .5em; font-size: 80%; +text-indent: 0em; white-space: nowrap;} + +div.endnote {padding: .5em 1em 1em; margin: 1em; border: 3px ridge #A9F; +font-family: sans-serif; font-size: 90%;} + +div.contents {font-family: sans-serif; font-size: 95%; margin-top: 4em; margin-bottom: 4em;} +div.contents p {margin-left: 1.5em; text-indent: -1.5em;} + +</style> +</head> + +<body> + + +<pre> + +Project Gutenberg's The Earliest Arithmetics in English, by Anonymous + +This eBook is for the use of anyone anywhere at no cost and with +almost no restrictions whatsoever. You may copy it, give it away or +re-use it under the terms of the Project Gutenberg License included +with this eBook or online at www.gutenberg.org + + +Title: The Earliest Arithmetics in English + +Author: Anonymous + +Editor: Robert Steele + +Release Date: June 1, 2008 [EBook #25664] + +Language: English + +Character set encoding: UTF-8 + +*** START OF THIS PROJECT GUTENBERG EBOOK THE EARLIEST ARITHMETICS IN ENGLISH *** + + + + +Produced by Louise Hope, David Starner and the Online +Distributed Proofreading Team at http://www.pgdp.net + + + + + + +</pre> + + +<div class = "mynote"> +<p><a name = "start" id = "start">This text</a> includes characters that +will only display in UTF-8 (Unicode) file encoding:</p> + +<p class = "inset"> +ȝ, ſ (yogh, long s)<br /> +ɳ, łł (n with curl, crossed l: see below)<br /> +φ (Greek phi, sometimes used in printed text for 0) +</p> + +<p>If any of these characters do not display properly, or if the +apostrophes and quotation marks in this paragraph appear as garbage, you +may have an incompatible browser or unavailable fonts. First, make sure +that the browser’s “character set” or “file encoding” is set to Unicode +(UTF-8). You may also need to change your browser’s default font.</p> + +<p>In <i>The Crafte of Nombrynge</i>, final <b>n</b> was sometimes +written with an extra curl as +<img src = "images/n_curl.gif" width = "17" height = "18" +alt = "n with curl" />. +It has been rendered as <b>ɳ</b> for visual effect; the character is not +intended to convey phonetic information. In the same selection, the +numeral “0” was sometimes printed as Greek φ (phi); this has been +retained for the e-text. Double <b>l</b> with a line +<img src = "images/l_pair.gif" width = "15" height = "19" +alt = "joined ll" /> +is shown as <b>łł</b>. The first few occurrences of <b>d</b> +(for “pence”) were printed with a curl as +<img src = "images/d_curl.gif" width = "15" height = "19" +alt = "d with curl" />. +The letter is shown with the same <b>d’</b> used in the +remainder of the text.</p> + +<p>The word “withdraw” or “w<i>i</i>t<i>h</i>draw” was inconsistently +hyphenated; it was left as printed, and line-end hyphens were retained. +All brackets [ ] are in the original.</p> + +<p>The diagrams in “Accomptynge by Counters” may not line up perfectly +in all browsers, but the contents should still be intelligible.</p> + +<p>The original text contained at least five types of marginal note. +Details are given at the <a href = "#endnote">end of the e-text</a>.</p> + +<p>Typographical errors are shown in the text with <ins class = +"correction" title = "like this">mouse-hover popups</ins>. Other +underlined words are cross-references to the <a class = "terms" href = +"#terms">Index of Technical Terms</a> and the <a class = "gloss" href = +"#glossary">Glossary</a>.</p> +</div> + +<div class = "titlepage"> + +<p class = "illustration"> +<img src = "images/title_main.png" width = "375" height = "219" alt = +"The Earliest Arithmetics in English / Early English Text Society. / +Extra Series, No. CXVIII. / 1922 (for 1916)." /> +</p> + +</div> + + +<hr class = "mid" /> +<div class = "contents"> +<h4><a name = "contents" id = "contents">Contents</a><br /> +<span class = "smaller">(added by transcriber)</span></h4> + +<table class = "toc" summary = "table of contents"> +<tr> +<td> +<a href = "#intro">Introduction</a> +</td> +<td class = "number">v</td> +</tr> +<tr> +<td> +<a href = "#crafte">The Crafte of Nombrynge</a> +</td> +<td class = "number">3</td> +</tr> +<tr> +<td> +<a href = "#art">The Art of Nombryng</a> +</td> +<td class = "number">33</td> +</tr> +<tr> +<td> +<a href = "#count">Accomptynge by Counters</a> +</td> +<td class = "number">52</td> +</tr> +<tr> +<td> +<p><a href = "#hand">The arte of nombrynge by the hande</a></p> +</td> +<td class = "number">66</td> +</tr> +<tr> +<td> +<p><a href = "#app1"><span class = "smallcaps">App. I.</span> A Treatise +on the Numeration of Algorism</a></p> +</td> +<td class = "number">70</td> +</tr> +<tr> +<td> +<p><a href = "#app2"><span class = "smallcaps">App. II.</span> Carmen de +Algorismo</a></p> +</td> +<td class = "number">72</td> +</tr> +<tr> +<td> +<a href = "#terms">Index of Technical Terms</a> +</td> +<td class = "number">81</td> +</tr> +<tr> +<td> +<a href = "#glossary">Glossary</a> +</td> +<td class = "number">83</td> +</tr> +</table> +</div> + +<hr class = "mid" /> + +<div class = "titlepage"> +<h1>The Earliest Arithmetics<br /> +in English</h1> + +<p> <br /> </p> + +<h2 class = "two">EDITED WITH INTRODUCTION</h2> + +<h2 class = "three">BY</h2> + +<h2 class = "one">ROBERT STEELE</h2> + +<p> <br /> </p> + +<p> <br /> </p> + +<h2 class = "two">LONDON:</h2> +<h2 class = "three">PUBLISHED FOR THE EARLY ENGLISH TEXT SOCIETY</h2> +<h2 class = "two">BY HUMPHREY MILFORD, OXFORD UNIVERSITY PRESS,</h2> +<h2 class = "three">AMEN CORNER, E.C. 4.<br /> +1922.</h2> + +</div> + +<div class = "intro"> + +<span class = "pagenum">v</span> +<a name = "page_v" id = "page_v"> </a> +<h3><a name = "intro" id = "intro">INTRODUCTION</a></h3> + + +<p><span class = "firstword">The</span> number of English arithmetics +before the sixteenth century is very small. This is hardly to be +wondered at, as no one requiring to use even the simplest operations of +the art up to the middle of the fifteenth century was likely to be +ignorant of Latin, in which language there were several treatises in a +considerable number of manuscripts, as shown by the quantity of them +still in existence. Until modern commerce was fairly well established, +few persons required more arithmetic than addition and subtraction, and +even in the thirteenth century, scientific treatises addressed to +advanced students contemplated the likelihood of their not being able to +do simple division. On the other hand, the study of astronomy +necessitated, from its earliest days as a science, considerable skill +and accuracy in computation, not only in the calculation of astronomical +tables but in their use, a knowledge of which latter was fairly +common from the thirteenth to the sixteenth centuries.</p> + +<p>The arithmetics in English known to me are:—</p> + +<div class = "inset hanging"> +<p>(1) Bodl. 790 G. VII. (2653) f. 146-154 (15th c.) <i>inc.</i> “Of +angrym ther be IX figures in numbray . . .” A mere +unfinished fragment, only getting as far as Duplation.</p> + +<p>(2) Camb. Univ. LI. IV. 14 (III.) f. 121-142 (15th c.) <i>inc.</i> +“Al maner of thyngis that prosedeth ffro the frist +begynnyng . . .”</p> + +<p>(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</p> + +<p>(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.</p> +</div> + +<p>The <span class = "smallcaps">Crafte of Nombrynge</span> is one of a +large number of scientific treatises, mostly in Latin, bound up together +as Egerton MS. 2622 in the British Museum Library. It measures +7” × 5”, 29-30 lines to the page, in a rough hand. The English +is N.E. Midland in dialect. It is a translation and amplification of one +of the numerous glosses on the <i>de algorismo</i> of Alexander de Villa +Dei (c. 1220), such as that of +<span class = "pagenum">vi</span> +<a name = "page_vi" id = "page_vi"> </a> +Thomas of Newmarket contained in the British Museum MS. Reg. 12, +E. 1. A fragment of another translation of the same gloss was +printed by Halliwell in his <i>Rara Mathematica</i> (1835) p. 29.<a +class = "tag" name = "tag_intro1" id = "tag_intro1" href = +"#note_intro1">1</a> 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.</p> + +<p>The <span class = "smallcaps">Art of Nombryng</span> is one of the +treatises bound up in the Bodleian MS. Ashmole 396. It measures +11½” × 17¾”, and is written with thirty-three lines to the +page in a fifteenth century hand. It is a translation, rather literal, +with amplifications of the <i>de arte numerandi</i> attributed to John +of Holywood (Sacrobosco) and the translator had obviously a poor MS. +before him. The <i>de arte numerandi</i> was printed in 1488, 1490 +(<i>s.n.</i>), 1501, 1503, 1510, 1517, 1521, 1522, 1523, 1582, and by +Halliwell separately and in his two editions of <i>Rara Mathematica</i>, +1839 and 1841, and reprinted by Curze in 1897.</p> + +<p>Both these tracts are here printed for the first time, but the first +having been circulated in proof a number of years ago, in an endeavour +to discover other manuscripts or parts of manuscripts of it, Dr. David +Eugene Smith, misunderstanding the position, printed some pages in a +curious transcript with four facsimiles in the <i>Archiv für die +Geschichte der Naturwissenschaften und der Technik</i>, 1909, and +invited the scientific world to take up the “not unpleasant task” of +editing it.</p> + +<p><span class = "smallcaps">Accomptynge by Counters</span> 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.</p> + +<p>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 +<i>Rara Mathemathica</i>, but I have added a number of stanzas from +<span class = "pagenum">vii</span> +<a name = "page_vii" id = "page_vii"> </a> +various manuscripts, selecting various readings on the principle that +the verses were made to scan, aided by the advice of my friend Mr. +Vernon Rendall, who is not responsible for the few doubtful lines I have +conserved. This poem is at the base of all other treatises on the +subject in medieval times, but I am unable to indicate its sources.</p> + + +<h4><a name = "intro_subject" id = "intro_subject"> +The Subject Matter.</a></h4> + +<p>Ancient and medieval writers observed a distinction between the +Science and the Art of Arithmetic. The classical treatises on the +subject, those of Euclid among the Greeks and Boethius among the Latins, +are devoted to the Science of Arithmetic, but it is obvious that coeval +with practical Astronomy the Art of Calculation must have existed and +have made considerable progress. If early treatises on this art existed +at all they must, almost of necessity, have been in Greek, which was the +language of science for the Romans as long as Latin civilisation +existed. But in their absence it is safe to say that no involved +operations were or could have been carried out by means of the +alphabetic notation of the Greeks and Romans. Specimen sums have indeed +been constructed by moderns which show its possibility, but it is absurd +to think that men of science, acquainted with Egyptian methods and in +possession of the abacus,<a class = "tag" name = "tag_intro2" id = +"tag_intro2" href = "#note_intro2">2</a> were unable to devise methods +for its use.</p> + + +<h4><a name = "intro_instruments" id = "intro_instruments"> +The Pre-Medieval Instruments Used in Calculation.</a></h4> + +<p>The following are known:—</p> + +<p>(1) A flat polished surface or tablets, strewn with sand, on which +figures were inscribed with a stylus.</p> + +<p>(2) A polished tablet divided longitudinally into nine columns (or +more) grouped in threes, with which counters were used, either plain or +marked with signs denoting the nine numerals, etc.</p> + +<p>(3) Tablets or boxes containing nine grooves or wires, in or on which +ran beads.</p> + +<p>(4) Tablets on which nine (or more) horizontal lines were marked, +each third being marked off.</p> + +<p>The only Greek counting board we have is of the fourth class and was +discovered at Salamis. It was engraved on a block of marble, and +measures 5 feet by 2½. Its chief part consists of eleven parallel lines, +the 3rd, 6th, and 9th being marked with a cross. Another section +consists of five parallel lines, and there are three +<span class = "pagenum">viii</span> +<a name = "page_viii" id = "page_viii"> </a> +rows of arithmetical symbols. This board could only have been used with +counters (<i>calculi</i>), preferably unmarked, as in our treatise of +<i>Accomptynge by Counters</i>.</p> + + +<h4><a name = "intro_roman" id = "intro_roman"> +Classical Roman Methods of Calculation.</a></h4> + +<p>We have proof of two methods of calculation in ancient Rome, one by +the first method, in which the surface of sand was divided into columns +by a stylus or the hand. Counters (<i>calculi</i>, or <i>lapilli</i>), +which were kept in boxes (<i>loculi</i>), were used in calculation, as +we learn from Horace’s schoolboys (Sat. 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, <i>in +fine</i>). The counters were made of various materials, ivory +principally, “Adeo nulla uncia nobis est eboris, etc.” (Juv. XI. 131), +sometimes of precious metals, “Pro calculis albis et nigris aureos +argenteosque habebat denarios” (Pet. Arb. Satyricon, 33).</p> + +<p>There are, however, still in existence four Roman counting boards of +a kind which does not appear to come into literature. A typical one +is of the third class. It consists of a number of transverse wires, +broken at the middle. On the left hand portion four beads are strung, on +the right one (or two). The left hand beads signify units, the right +hand one five units. Thus any number up to nine can be represented. This +instrument is in all essentials the same as the Swanpan or Abacus in use +throughout the Far East. The Russian stchota in use throughout Eastern +Europe is simpler still. The method of using this system is exactly the +same as that of <i>Accomptynge by Counters</i>, the right-hand five bead +replacing the counter between the lines.</p> + + +<h4><a name = "intro_abacus" id = "intro_abacus"> +The Boethian Abacus.</a></h4> + +<p>Between classical times and the tenth century we have little or no +guidance as to the art of calculation. Boethius (fifth century), at the +end of lib. II. of his <i>Geometria</i> gives us a figure of an +abacus of the second class with a set of counters arranged within it. It +has, however, been contended with great probability that the whole +passage is a tenth century interpolation. As no rules are given for its +use, the chief value of the figure is that it gives the signs of the +<span class = "pagenum">ix</span> +<a name = "page_ix" id = "page_ix"> </a> +nine numbers, known as the Boethian “apices” or “notae” (from whence our +word “notation”). To these we shall return later on.</p> + + +<h4><a name = "intro_abacists" id = "intro_abacists"> +The Abacists.</a></h4> + +<p>It would seem probable that writers on the calendar like Bede (<span +class = "smallroman">A.D.</span> 721) and Helpericus (<span class = +"smallroman">A.D.</span> 903) were able to perform simple calculations; +though we are unable to guess their methods, and for the most part they +were dependent on tables taken from Greek sources. We have no early +medieval treatises on arithmetic, till towards the end of the tenth +century we find a revival of the study of science, centring for us round +the name of Gerbert, who became Pope as Sylvester 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.</p> + +<p>The Abacists used a permanent board usually divided into twelve +columns; the columns were grouped in threes, each column being called an +“arcus,” and the value of a figure in it represented a tenth of what it +would have in the column to the left, as in our arithmetic of position. +With this board counters or jetons were used, either plain or, more +probably, marked with numerical signs, which with the early Abacists +were the “apices,” though counters from classical times were sometimes +marked on one side with the digital signs, on the other with Roman +numerals. Two ivory discs of this kind from the Hamilton collection may +be seen at the British Museum. Gerbert is said by Richer to have made +for the purpose of computation a thousand counters of horn; the usual +number of a set of counters in the sixteenth and seventeenth centuries +was a hundred.</p> + +<p>Treatises on the Abacus usually consist of chapters on Numeration +explaining the notation, and on the rules for Multiplication and +Division. Addition, as far as it required any rules, came naturally +under Multiplication, while Subtraction was involved in the process of +Division. These rules were all that were needed in Western Europe in +centuries when commerce hardly existed, and astronomy was unpractised, +and even they were only required in the preparation +<span class = "pagenum">x</span> +<a name = "page_x" id = "page_x"> </a> +of the calendar and the assignments of the royal exchequer. In England, +for example, when the hide developed from the normal holding of a +household into the unit of taxation, the calculation of the geldage in +each shire required a sum in division; as we know from the fact that one +of the Abacists proposes the sum: “If 200 marks are levied on the county +of Essex, which contains according to Hugh of Bocland 2500 hides, how +much does each hide pay?”<a class = "tag" name = "tag_intro3" id = +"tag_intro3" href = "#note_intro3">3</a> Exchequer methods up to the +sixteenth century were founded on the abacus, though when we have +details later on, a different and simpler form was used.</p> + +<p>The great difficulty of the early Abacists, owing to the absence of a +figure representing zero, was to place their results and operations in +the proper columns of the abacus, especially when doing a division sum. +The chief differences noticeable in their works are in the methods for +this rule. Division was either done directly or by means of differences +between the divisor and the next higher multiple of ten to the divisor. +Later Abacists made a distinction between “iron” and “golden” methods of +division. The following are examples taken from a twelfth century +treatise. In following the operations it must be remembered that a +figure asterisked represents a counter taken from the board. A zero +is obviously not needed, and the result may be written down in +words.</p> + +<h5>(<i>a</i>) <span class = "smallcaps">Multiplication.</span> 4600 +× 23.</h5> + +<table class = "grid center" summary = "example"> +<col class = "outline" /> +<col class = "outline" /> +<col class = "outline" /> +<col class = "outline" /> +<col class = "outline" /> +<col class = "outline" /> +<col /> +<tr class = "outline"> +<td colspan = "3">Thousands</td> +<td colspan = "3"> </td> +<td class = "edge"> </td> +</tr> +<tr class = "outline middle"> +<td>H<br />u<br />n<br />d<br />r<br />e<br />d<br />s</td> +<td>T<br />e<br />n<br />s</td> +<td>U<br />n<br />i<br />t<br />s</td> +<td>H<br />u<br />n<br />d<br />r<br />e<br />d<br />s</td> +<td>T<br />e<br />n<br />s</td> +<td>U<br />n<br />i<br />t<br />s</td> +<td class = "edge"> </td> +</tr> +<tr class = "outline"> +<td> </td> +<td> </td> +<td>4</td> +<td>6</td> +<td> </td> +<td> </td> +<td class = "edge"><b>Multiplicand.</b></td> +</tr> +<tr> +<td> </td> +<td> </td> +<td>1</td> +<td>8</td> +<td> </td> +<td> </td> +<td class = "edge">600 × 3.</td> +</tr> +<tr> +<td> </td> +<td>1</td> +<td>2</td> +<td> </td> +<td> </td> +<td> </td> +<td class = "edge">4000 × 3.</td> +</tr> +<tr> +<td> </td> +<td>1</td> +<td>2</td> +<td> </td> +<td> </td> +<td> </td> +<td class = "edge">600 × 20.</td> +</tr> +<tr> +<td> </td> +<td>8</td> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td class = "edge">4000 × 20.</td> +</tr> +<tr class = "outline"> +<td>1</td> +<td> </td> +<td>5</td> +<td>8</td> +<td> </td> +<td> </td> +<td class = "edge">Total product.</td> +</tr> +<tr class = "outline"> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td>2</td> +<td>3</td> +<td class = "edge"><b>Multiplier.</b></td> +</tr> +</table> + +<span class = "pagenum">xi</span> +<a name = "page_xi" id = "page_xi"> </a> + +<h5>(<i>b</i>) <span class = "smallcaps">Division: direct.</span> +100,000 ÷ 20,023. Here each counter in turn is a separate divisor.</h5> + +<table class = "grid center" summary = "example"> +<col class = "outline" /> +<col class = "outline" /> +<col class = "outline" /> +<col class = "outline" /> +<col class = "outline" /> +<col class = "outline" /> +<col /> +<tr class = "outline"> +<td>H.</td> +<td>T.</td> +<td>U.</td> +<td>H.</td> +<td>T.</td> +<td>U.</td> +<td class = "edge"> </td> +</tr> +<tr class = "outline"> +<td> </td> +<td>2</td> +<td> </td> +<td> </td> +<td>2</td> +<td>3</td> +<td class = "edge"><b>Divisors.</b></td> +</tr> +<tr> +<td> </td> +<td>2</td> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td class = "edge"><p>Place greatest divisor to right of +dividend.</p></td> +</tr> +<tr> +<td>1</td> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td class = "edge"><b>Dividend.</b></td> +</tr> +<tr> +<td> </td> +<td>2</td> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td class = "edge">Remainder.</td> +</tr> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td>1</td> +<td> </td> +<td> </td> +<td class = "edge"> </td> +</tr> +<tr> +<td> </td> +<td>1</td> +<td>9</td> +<td>9</td> +<td> </td> +<td> </td> +<td class = "edge">Another form of same.</td> +</tr> +<tr class = "underline"> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td>8</td> +<td> </td> +<td class = "edge"><p>Product of 1st Quotient and 20.</p></td> +</tr> +<tr> +<td> </td> +<td>1</td> +<td>9</td> +<td>9</td> +<td>2</td> +<td> </td> +<td class = "edge">Remainder.</td> +</tr> +<tr class = "underline"> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td>1</td> +<td>2</td> +<td class = "edge"><p>Product of 1st Quotient and 3.</p></td> +</tr> +<tr> +<td> </td> +<td>1</td> +<td>9</td> +<td>9</td> +<td> </td> +<td>8</td> +<td class = "edge"><b>Final remainder.</b></td> +</tr> +<tr class = "underline"> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td>4</td> +<td class = "edge">Quotient.</td> +</tr> +</table> + +<h5>(<i>c</i>) <span class = "smallcaps">Division by Differences.</span> +900 ÷ 8. Here we divide by (10-2).</h5> + +<table class = "grid center" summary = "example"> +<col class = "outline" /> +<col class = "outline" /> +<col class = "outline" /> +<col class = "outline" /> +<col class = "outline" /> +<col class = "outline" /> +<col /> +<tr class = "outline"> +<td> </td> +<td> </td> +<td> </td> +<td>H.</td> +<td>T.</td> +<td>U.</td> +<td class = "edge"> </td> +</tr> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td>2</td> +<td class = "edge">Difference.</td> +</tr> +<tr class = "underline"> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td>8</td> +<td class = "edge">Divisor.</td> +</tr> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td><a class = "tag" name = "tag_intro4" id = "tag_intro4" href = +"#note_intro4">4</a>9</td> +<td> </td> +<td> </td> +<td class = "edge"><b>Dividend.</b></td> +</tr> +<tr class = "underline"> +<td> </td> +<td> </td> +<td> </td> +<td><a class = "tag" href = "#note_intro4">4</a>1</td> +<td>8</td> +<td> </td> +<td class = "edge"><p>Product of difference by 1st +Quotient (9).</p></td> +</tr> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td>2</td> +<td> </td> +<td class = "edge"><p>Product of difference by 2nd +Quotient (1).</p></td> +</tr> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td><a class = "tag" href = "#note_intro4">4</a>1</td> +<td> </td> +<td> </td> +<td class = "edge">Sum of 8 and 2.</td> +</tr> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td>2</td> +<td> </td> +<td class = "edge"><p>Product of difference by 3rd +Quotient (1).</p></td> +</tr> +<tr class = "underline"> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td>4</td> +<td class = "edge"><p>Product of difference by 4th Quot. (2). +<b>Remainder.</b></p></td> +</tr> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td>2</td> +<td class = "edge">4th Quotient.</td> +</tr> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td>1</td> +<td> </td> +<td class = "edge">3rd Quotient.</td> +</tr> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td>1</td> +<td> </td> +<td class = "edge">2nd Quotient.</td> +</tr> +<tr class = "underline"> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td>9</td> +<td> </td> +<td class = "edge">1st Quotient.</td> +</tr> +<tr class = "underline"> +<td> </td> +<td> </td> +<td> </td> +<td>1</td> +<td>1</td> +<td>2</td> +<td class = "edge"><p><b>Quotient.</b> (<b>Total of all +four.</b>)</p></td> +</tr> +</table> + +<span class = "pagenum">xii</span> +<a name = "page_xii" id = "page_xii"> </a> + +<h5><span class = "smallcaps">Division.</span> 7800 ÷ 166.</h5> + +<table class = "grid center" summary = "example"> +<col class = "outline" /> +<col class = "outline" /> +<col class = "outline" /> +<col class = "outline" /> +<col class = "outline" /> +<col class = "outline" /> +<col /> +<tr class = "outline"> +<td colspan = "3">Thousands</td> +<td colspan = "3"> </td> +<td class = "edge"> </td> +</tr> +<tr class = "outline"> +<td>H.</td> +<td>T.</td> +<td>U.</td> +<td>H.</td> +<td>T.</td> +<td>U.</td> +<td class = "edge"> </td> +</tr> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td>3</td> +<td>4</td> +<td class = "edge"><p>Differences (making 200 trial divisor).</p></td> +</tr> +<tr class = "underline"> +<td> </td> +<td> </td> +<td> </td> +<td>1</td> +<td>6</td> +<td>6</td> +<td class = "edge">Divisors.</td> +</tr> +<tr> +<td> </td> +<td> </td> +<td><a class = "tag" href = "#note_intro4">4</a>7</td> +<td>8</td> +<td> </td> +<td> </td> +<td class = "edge"><b>Dividends.</b></td> +</tr> +<tr> +<td> </td> +<td> </td> +<td>1</td> +<td> </td> +<td> </td> +<td> </td> +<td class = "edge"><p>Remainder of greatest dividend.</p></td> +</tr> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td>1</td> +<td>2</td> +<td> </td> +<td class = "edge"><p>Product of 1st difference (4) by 1st +Quotient (3).</p></td> +</tr> +<tr class = "underline"> +<td> </td> +<td> </td> +<td> </td> +<td>9</td> +<td> </td> +<td> </td> +<td class = "edge"><p>Product of 2nd difference (3) by 1st +Quotient (3).</p></td> +</tr> +<tr> +<td> </td> +<td> </td> +<td><a class = "tag" href = "#note_intro4">4</a>2</td> +<td>8</td> +<td>2</td> +<td> </td> +<td class = "edge">New dividends.</td> +</tr> +<tr class = "underline"> +<td> </td> +<td> </td> +<td> </td> +<td>3</td> +<td>4</td> +<td> </td> +<td class = "edge"><p>Product of 1st and 2nd difference by 2nd +Quotient (1).</p></td> +</tr> +<tr> +<td> </td> +<td> </td> +<td><a class = "tag" href = "#note_intro4">4</a>1</td> +<td>1</td> +<td>6</td> +<td> </td> +<td class = "edge">New dividends.</td> +</tr> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td>2</td> +<td> </td> +<td class = "edge"><p>Product of 1st difference by 3rd +Quotient (5).</p></td> +</tr> +<tr class = "underline"> +<td> </td> +<td> </td> +<td> </td> +<td>1</td> +<td>5</td> +<td> </td> +<td class = "edge"><p>Product of 2nd difference by 3rd +Quotient (5).</p></td> +</tr> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td><a class = "tag" href = "#note_intro4">4</a>3</td> +<td>3</td> +<td> </td> +<td class = "edge">New dividends.</td> +</tr> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td>1</td> +<td> </td> +<td> </td> +<td class = "edge"><p>Remainder of greatest dividend.</p></td> +</tr> +<tr class = "underline"> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td>3</td> +<td>4</td> +<td class = "edge"><p>Product of 1st and 2nd difference by 4th +Quotient (1).</p></td> +</tr> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td>1</td> +<td>6</td> +<td>4</td> +<td class = "edge"><p><b>Remainder</b> (less than divisor).</p></td> +</tr> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td>1</td> +<td class = "edge">4th Quotient.</td> +</tr> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td>5</td> +<td class = "edge">3rd Quotient.</td> +</tr> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td>1</td> +<td> </td> +<td class = "edge">2nd Quotient.</td> +</tr> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td>3</td> +<td> </td> +<td class = "edge">1st Quotient.</td> +</tr> +<tr class = "outline"> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td>4</td> +<td>6</td> +<td class = "edge"><b>Quotient.</b></td> +</tr> +</table> + + +<span class = "pagenum">xiii</span> +<a name = "page_xiii" id = "page_xiii"> </a> + +<h5><span class = "smallcaps">Division.</span> 8000 ÷ 606.</h5> + +<table class = "grid center" summary = "example"> +<col class = "outline" /> +<col class = "outline" /> +<col class = "outline" /> +<col class = "outline" /> +<col class = "outline" /> +<col class = "outline" /> +<col /> +<tr class = "outline"> +<td colspan = "3">Thousands</td> +<td colspan = "3"> </td> +<td class = "edge"> </td> +</tr> +<tr class = "outline"> +<td>H.</td> +<td>T.</td> +<td>U.</td> +<td>H.</td> +<td>T.</td> +<td>U.</td> +<td class = "edge"> </td> +</tr> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td>9</td> +<td> </td> +<td class = "edge"><p>Difference (making 700 trial divisor).</p></td> +</tr> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td>4</td> +<td class = "edge">Difference.</td> +</tr> +<tr class = "underline"> +<td> </td> +<td> </td> +<td> </td> +<td>6</td> +<td> </td> +<td>6</td> +<td class = "edge">Divisors.</td> +</tr> +<tr> +<td> </td> +<td> </td> +<td><a class = "tag" href = "#note_intro4">4</a>8</td> +<td> </td> +<td> </td> +<td> </td> +<td class = "edge"><b>Dividend.</b></td> +</tr> +<tr> +<td> </td> +<td> </td> +<td>1</td> +<td> </td> +<td> </td> +<td> </td> +<td class = "edge">Remainder of dividend.</td> +</tr> +<tr class = "underline"> +<td> </td> +<td> </td> +<td> </td> +<td>9</td> +<td>4</td> +<td> </td> +<td class = "edge"><p>Product of difference 1 and 2 with 1st +Quotient (1).</p></td> +</tr> +<tr> +<td> </td> +<td> </td> +<td><a class = "tag" href = "#note_intro4">4</a>1</td> +<td>9</td> +<td>4</td> +<td> </td> +<td class = "edge">New dividends.</td> +</tr> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td>3</td> +<td> </td> +<td> </td> +<td class = "edge"><p>Remainder of greatest dividend.</p></td> +</tr> +<tr class = "underline"> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td>9</td> +<td>4</td> +<td class = "edge"><p>Product of difference 1 and 2 with 2nd +Quotient (1).</p></td> +</tr> +<tr> +<td> </td> +<td> </td> +<td><a class = "tag" href = "#note_intro4">4</a>1</td> +<td>3</td> +<td>3</td> +<td>4</td> +<td class = "edge">New dividends.</td> +</tr> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td>3</td> +<td> </td> +<td> </td> +<td class = "edge"><p>Remainder of greatest dividend.</p></td> +</tr> +<tr class = "underline"> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td>9</td> +<td>4</td> +<td class = "edge"><p>Product of difference 1 and 2 with 3rd +Quotient (1).</p></td> +</tr> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td>7</td> +<td>2</td> +<td>8</td> +<td class = "edge">New dividends.</td> +</tr> +<tr class = "underline"> +<td> </td> +<td> </td> +<td> </td> +<td>6</td> +<td> </td> +<td>6</td> +<td class = "edge"><p>Product of divisors by 4th +Quotient (1).</p></td> +</tr> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td>1</td> +<td>2</td> +<td>2</td> +<td class = "edge"><b>Remainder.</b></td> +</tr> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td>1</td> +<td class = "edge">4th Quotient.</td> +</tr> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td>1</td> +<td class = "edge">3rd Quotient.</td> +</tr> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td>1</td> +<td class = "edge">2nd Quotient.</td> +</tr> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td>1</td> +<td> </td> +<td class = "edge">1st Quotient.</td> +</tr> +<tr class = "outline"> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td>1</td> +<td>3</td> +<td class = "edge"><b>Quotient.</b></td> +</tr> +</table> + +<p>The chief Abacists are Gerbert (tenth century), Abbo, and Hermannus +Contractus (1054), who are credited with the revival of the art, +Bernelinus, Gerland, and Radulphus of Laon (twelfth century). We know as +English Abacists, Robert, bishop of Hereford, 1095, “abacum et lunarem +compotum et celestium cursum astrorum rimatus,” Turchillus Compotista +(Thurkil), and through him of Guilielmus R. . . . “the +best of living computers,” Gislebert, and Simonus de Rotellis (Simon of +the Rolls). They flourished most probably in the +<span class = "pagenum">xiv</span> +<a name = "page_xiv" id = "page_xiv"> </a> +first quarter of the twelfth century, as Thurkil’s treatise deals also +with fractions. Walcher of Durham, Thomas of York, and Samson of +Worcester are also known as Abacists.</p> + +<p>Finally, the term Abacists came to be applied to computers by manual +arithmetic. A 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.”</p> + +<p>In a composite treatise containing tracts written <span class = +"smallroman">A.D.</span> 1157 and 1208, on the calendar, the abacus, the +manual calendar and the manual abacus, we have a number of the methods +preserved. As an example we give the rule for multiplication (Claud. A. +IV., f. 54 vo). “Si numerus multiplicat alium numerum auferatur +differentia majoris a minore, et per residuum multiplicetur articulus, +et una differentia per aliam, et summa proveniet.” Example, +8 × 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>i.e.</i>, 20; but in such a case as 17 × 9 the difference +for each number must be taken from the higher article, <i>i.e.</i>, the +difference of 9 will be 11.</p> + + +<h4><a name = "intro_algorists" id = "intro_algorists"> +The Algorists.</a></h4> + +<p>Algorism (augrim, augrym, algram, agram, algorithm), owes its name to +the accident that the first arithmetical treatise translated from the +Arabic happened to be one written by Al-Khowarazmi in the early ninth +century, “de numeris Indorum,” beginning in its Latin form “Dixit +Algorismi. . . .” 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.</p> + +<p>Algorism is distinguished from Abacist computation by recognising +seven rules, Addition, Subtraction, Duplation, Mediation, +Multiplication, Division, and Extraction of Roots, to which were +afterwards +<span class = "pagenum">xv</span> +<a name = "page_xv" id = "page_xv"> </a> +added Numeration and Progression. It is further distinguished by the use +of the zero, which enabled the computer to dispense with the columns of +the Abacus. It obviously employs a board with fine sand or wax, and +later, as a substitute, paper or parchment; slate and pencil were also +used in the fourteenth century, how much earlier is unknown.<a class = +"tag" name = "tag_intro5" id = "tag_intro5" href = "#note_intro5">5</a> +Algorism quickly ousted the Abacus methods for all intricate +calculations, being simpler and more easily checked: in fact, the +astronomical revival of the twelfth and thirteenth centuries would have +been impossible without its aid.</p> + +<p>The number of Latin Algorisms still in manuscript is comparatively +large, but we are here only concerned with two—an Algorism in +prose attributed to Sacrobosco (John of Holywood) in the colophon of a +Paris manuscript, though this attribution is no longer regarded as +conclusive, and another in verse, most probably by Alexander de +Villedieu (Villa Dei). Alexander, who died in 1240, was teaching in +Paris in 1209. His verse treatise on the Calendar is dated 1200, and it +is to that period that his Algorism may be attributed; Sacrobosco died +in 1256 and quotes the verse Algorism. Several commentaries on +Alexander’s verse treatise were composed, from one of which our first +tractate was translated, and the text itself was from time to time +enlarged, sections on proofs and on mental arithmetic being added. We +have no indication of the source on which Alexander drew; it was most +likely one of the translations of Al-Khowarasmi, but he has also the +Abacists in mind, as shewn by preserving the use of differences in +multiplication. His treatise, first printed by Halliwell-Phillipps in +his <i>Rara Mathematica</i>, is adapted for use on a board covered with +sand, a 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.”</p> + + +<h4><a name = "intro_modern" id = "intro_modern">Modern +Arithmetic.</a></h4> + +<p>Modern Arithmetic begins with Leonardi Fibonacci’s treatise “de +Abaco,” written in 1202 and re-written in 1228. It is modern +<span class = "pagenum">xvi</span> +<a name = "page_xvi" id = "page_xvi"> </a> +rather in the range of its problems and the methods of attack than in +mere methods of calculation, which are of its period. Its sole interest +as regards the present work is that Leonardi makes use of the digital +signs described in Record’s treatise on <i>The arte of nombrynge by the +hand</i> in mental arithmetic, calling it “modus Indorum.” Leonardo also +introduces the method of proof by “casting out the nines.”</p> + + +<h4><a name = "intro_digital" id = "intro_digital">Digital +Arithmetic.</a></h4> + +<p>The method of indicating numbers by means of the fingers is of +considerable age. The British Museum possesses two ivory counters marked +on one side by carelessly scratched Roman numerals IIIV and VIIII, and +on the other by carefully engraved digital signs for 8 and 9. +Sixteen seems to have been the number of a complete set. These counters +were either used in games or for the counting board, and the Museum +ones, coming from the Hamilton collection, are undoubtedly not later +than the first century. Frohner has published in the <i>Zeitschrift des +Münchener Alterthumsvereins</i> a set, almost complete, of them with a +Byzantine treatise; a Latin treatise is printed among Bede’s works. +The use of this method is universal through the East, and a variety of +it is found among many of the native races in Africa. In medieval Europe +it was almost restricted to Italy and the Mediterranean basin, and in +the treatise already quoted (Sloane 3281) it is even called the Abacus, +perhaps a memory of Fibonacci’s work.</p> + +<p>Methods of calculation by means of these signs undoubtedly have +existed, but they were too involved and liable to error to be much +used.</p> + + +<h4><a name = "intro_arabic" id = "intro_arabic"> +The Use of “Arabic” Figures.</a></h4> + +<p>It may now be regarded as proved by Bubnov that our present numerals +are derived from Greek sources through the so-called Boethian “apices,” +which are first found in late tenth century manuscripts. That they were +not derived directly from the Arabic seems certain from the different +shapes of some of the numerals, especially the 0, which stands for 5 in +Arabic. Another Greek form existed, which was introduced into Europe by +John of Basingstoke in the thirteenth century, and is figured by Matthew +Paris (V. 285); but this form had no success. The date of the +introduction of the zero has been hotly debated, but it seems obvious +that the twelfth century Latin translators from the Arabic were +<span class = "pagenum">xvii</span> +<a name = "page_xvii" id = "page_xvii"> </a> +perfectly well acquainted with the system they met in their Arabic text, +while the earliest astronomical tables of the thirteenth century I have +seen use numbers of European and not Arabic origin. The fact that Latin +writers had a convenient way of writing hundreds and thousands without +any cyphers probably delayed the general use of the Arabic notation. Dr. +Hill has published a very complete survey of the various forms of +numerals in Europe. They began to be common at the middle of the +thirteenth century and a very interesting set of family notes concerning +births in a British Museum manuscript, Harl. 4350 shows their extension. +The first is dated <ins class = "correction" +title = "c printed directly above j">Mij<sup>c</sup></ins>. lviii., +the second Mij<sup>c</sup>. lxi., the third Mij<sup>c</sup>. 63, the +fourth 1264, and the fifth 1266. Another example is given in a set of +astronomical tables for 1269 in a manuscript of Roger Bacon’s works, +where the scribe began to write MCC6. and crossed out the figures, +substituting the “Arabic” form.</p> + + +<h4><a name = "intro_board" id = "intro_board"> +The Counting Board.</a></h4> + +<p>The treatise on pp. 52-65 is the only one in English known on the +subject. It describes a method of calculation which, with slight +modifications, is current in Russia, China, and Japan, to-day, though it +went out of use in Western Europe by the seventeenth century. In Germany +the method is called “Algorithmus Linealis,” and there are several +editions of a tract under this name (with a diagram of the counting +board), printed at Leipsic at the end of the fifteenth century and the +beginning of the sixteenth. They give the nine rules, but “Capitulum de +radicum extractione ad algoritmum integrorum reservato, cujus species +per ciffrales figuras ostenduntur ubi ad plenum de hac tractabitur.” The +invention of the art is there attributed to Appulegius the +philosopher.</p> + +<p>The advantage of the counting board, whether permanent or constructed +by chalking parallel lines on a table, as shown in some +sixteenth-century woodcuts, is that only five counters are needed to +indicate the number nine, counters on the lines representing units, and +those in the spaces above representing five times those on the line +below. The Russian abacus, the “tchatui” or “stchota” has ten beads on +the line; the Chinese and Japanese “Swanpan” economises by dividing the +line into two parts, the beads on one side representing five times the +value of those on the other. The “Swanpan” has usually many more lines +than the “stchota,” allowing for more extended calculations, see Tylor, +<i>Anthropology</i> (1892), p. 314.</p> + +<span class = "pagenum">xviii</span> +<a name = "page_xviii" id = "page_xviii"> </a> +<p>Record’s treatise also mentions another method of counter notation +(p. 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 +<i>Rara Mathematica</i> (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.”</p> + +<p>The Exchequer table is described in the <i>Dialogus de Scaccario</i> +(Oxford, 1902), p. 38.</p> + +<div class = "footnote"> + +<p> +<a name = "note_intro1" id = "note_intro1" href = "#tag_intro1">1.</a> +Halliwell printed the two sides of his leaf in the wrong order. This and +some obvious errors of transcription—‘ferye’ for ‘ferthe,’ ‘lest’ +for ‘left,’ etc., have not been corrected in the reprint on +pp. 70-71.</p> + +<p> +<a name = "note_intro2" id = "note_intro2" href = "#tag_intro2">2.</a> +For Egyptian use see Herodotus, ii. 36, Plato, <i>de Legibus</i>, +VII.</p> + +<p> +<a name = "note_intro3" id = "note_intro3" href = "#tag_intro3">3.</a> +See on this Dr. Poole, <i>The Exchequer in the Twelfth Century</i>, +Chap. III., and Haskins, <i>Eng. Hist. Review</i>, 27, 101. The hidage +of Essex in 1130 was 2364 hides.</p> + +<p> +<a name = "note_intro4" id = "note_intro4" href = "#tag_intro4">4.</a> +These figures are removed at the next step.</p> + +<p> +<a name = "note_intro5" id = "note_intro5" href = "#tag_intro5">5.</a> +Slates are mentioned by Chaucer, and soon after (1410) Prosdocimo de +Beldamandi speaks of the use of a “lapis” for making notes on by +calculators.</p> + +</div> + +</div> <!-- end div intro --> + +<div class = "titlepage"> + +<p> </p> + +<p class = "illustration"> +<img src = "images/title_inner.png" width = "324" height = "71" +alt = "The Earliest Arithmetics in English." +title = "The Earliest Arithmetics in English." /></p> + +<p> </p> + +</div> + +<div class = "maintext"> + +<div class = "craft"> + +<span class = "pagenum">3</span> +<a name = "page3" id = "page3"> </a> + +<p class = "illustration"> +<a name = "crafte" id = "crafte"> +<img src = "images/title_craft.png" width = "350" height = "36" +alt = "The Crafte of Nombrynge." +title = "The Crafte of Nombrynge." /></a></p> + +<hr class = "tiny" /> + +<p class = "subhead"><i>Egerton</i> 2622.</p> + +<span class = "linenum">leaf 136 <i>a</i>.</span> + +<div class = "verse"> +<p><span class = "dropcap">H</span>Ec algorism<i>us</i> ars +p<i>re</i>sens dicit<i>ur</i>; in qua</p> +<p>Talib<i>us</i> indor<i>um</i> fruim<i>ur</i> bis +qui<i>n</i>q<i>ue</i> figuris.</p> +</div> + +<p> +<span class = "sidenote">A derivation of Algorism.</span> +This boke is called þe boke of <a class = "terms" name = "algorym" id = +"algorym" href = "#terms_algorisme">algorym</a>, or Augrym aft<i>er</i> +<a class = "gloss" name = "lewder" id = "lewder" href = +"#gloss_lewder">lewd<i>er</i></a> vse. And þis boke tretys þe <a class = +"gloss" name = "craft" id = "craft" href = "#gloss_craft">Craft</a> of +Nombryng, þe quych crafte is called also Algorym. Ther was a kyng of +Inde, þe quich <a class = "gloss" name = "heyth" id = "heyth" href = +"#gloss_heyth">heyth</a> Algor, & he made þis craft. And +aft<i>er</i> his name he called hit algory<i>m</i>; +<span class = "sidenote">Another derivation of the word.</span> +or els anoþ<i>er</i> cause is quy it is called Algorym, for þe latyn +word of hit <a class = "gloss" name = "sc" id = "sc" href = +"#gloss_sc">s.</a> Algorism<i>us</i> com<i>es</i> of Algos, grece, +q<i>uid</i> e<i>st</i> ars, latine, craft oɳ englis, and rides, +q<i>uid</i> e<i>st</i> <i>nu</i>me<i>rus</i>, latine, +A nomb<i>ur</i> oɳ englys, inde d<i>icitu</i>r Algorism<i>us</i> +p<i>er</i> addic<i>i</i>one<i>m</i> hui<i>us</i> sillabe m<i>us</i> +& subtracc<i>i</i>onem d & e, q<i>ua</i>si ars num<i>er</i>andi. +¶ fforthermor<i>e</i> ȝe <a class = "gloss" name = "most1" id = +"most1" href = "#gloss_most">most</a> vnd<i>ir</i>stonde þ<i>a</i>t in +þis craft ben vsid teen figurys, as here ben<i>e</i> writen for +ensampul, φ 9 8 7 6 5 4 3 2 1. ¶ <a class = "gloss" name += "expone" id = "expone" href = "#gloss_expone">Expone</a> þe too +v<i>er</i>sus afor<i>e</i>: this p<i>re</i>sent craft ys called +Algorism<i>us</i>, in þe quych we vse teen signys of Inde. Questio. +¶ Why teɳ fyguris of Inde? Solucio. for as I haue sayd afore þai +wer<i>e</i> fonde fyrst in Inde of a kyng<i>e</i> of þat Cuntre, +þ<i>a</i>t was called Algor.</p> + +<p class = "headnote"><span class = "headnote"> +Notation and Numeration.</span></p> + +<span class = "sidenote">v<i>ersus</i> [in margin].</span> + +<div class = "verse"> +<p class = "pilcrow"> +¶ Prima sig<i>nifica</i>t unu<i>m</i>; duo ve<i>r</i>o +s<i>e</i>c<i>un</i>da:</p> +<p class = "pilcrow"> +¶ Tercia sig<i>nifica</i>t tria; sic procede sinistre.</p> +<p class = "pilcrow"> +¶ Don<i>e</i>c ad extrema<i>m</i> venias, que cifra voca<i>tur</i>.</p> +</div> + +<h5>¶ Cap<i>itulu</i>m primum de significac<i>i</i>o<i>n</i>e +figurar<i>um</i>.</h5> + +<p> +<span class = "sidenote original">Expo<i>sitio</i> v<i>ersus</i>.</span> +In þis verse is notifide þe significac<i>i</i>on of þese figur<i>is</i>. +And þus expone the verse. +<span class = "sidenote">The meaning and place of the figures.</span> +Þe first signifiyth on<i>e</i>, þe secu<i>n</i>de +<span class = "linenum">leaf 136 <i>b</i>.</span> +signi*fiyth tweyn<i>e</i>, þe thryd signifiyth thre, & the fourte +signifiyth 4. ¶ And so forthe towarde þe lyft syde of þe tabul or +of þe boke þ<i>a</i>t þe figures ben<i>e</i> writen<i>e</i> in, til þat +þ<i>o</i>u come to the last figure, þ<i>a</i>t is +<span class = "pagenum">4</span> +<a name = "page4" id = "page4"> </a> +called a <a class = "terms" name = "cifre" id = "cifre" href = +"#terms_cifre">cifre</a>. ¶ Questio. In quych syde sittes þe first +figur<i>e</i>? Soluc<i>io</i>, forsothe loke quich figure is first in þe +ryȝt side of þe bok or of þe tabul, & þ<i>a</i>t same is þe first +figur<i>e</i>, for þ<i>o</i>u schal write bakeward, as here, 3. 2. 6. 4. +1. 2. 5. +<span class = "sidenote">Which figure is read first.</span> +The fig<i>ur</i>e of 5. was first <a class = "gloss" name = "write1" id += "write1" href = "#gloss_write">write</a>, & he is þe first, for he +sittes oɳ þe riȝt syde. And the fig<i>ur</i>e of 3 is last. +¶ Neu<i>er</i>-þe-les wen he says ¶ P<i>ri</i>ma +sig<i>nifica</i>t vnu<i>m</i> &c., þat is to say, þe first betokenes +on<i>e</i>, þe secu<i>n</i>de. 2. & fore-þ<i>er</i>-mor<i>e</i>, he +vnd<i>ir</i>stondes <a class = "gloss" name = "noght1" id = "noght1" +href = "#gloss_noght">noȝt</a> of þe first fig<i>ur</i>e of eu<i>er</i>y +<a class = "gloss" name = "rew" id = "rew" href = "#gloss_rew">rew</a>. +¶ But he vnd<i>ir</i>stondes þe first figure þ<i>a</i>t is in þe +nomb<i>ur</i> of þe forsayd teen figuris, þe quych is on<i>e</i> of +þ<i>e</i>se. 1. And þe secu<i>n</i>de 2. & so forth.</p> + +<span class = "sidenote">v<i>ersus</i> [in margin].</span> + +<div class = "verse"> +<p class = "pilcrow"> +¶ Quelib<i>et</i> illar<i>um</i> si pr<i>im</i>o limite ponas,</p> +<p class = "pilcrow"> +¶ Simplicite<i>r</i> se significat: si v<i>er</i>o se<i>cun</i>do,</p> +<p>Se decies: sursu<i>m</i> <ins class = "correction" title = "italics as shown: error for ‘p{ro}’?"><i>pr</i>ocedas</ins> +m<i>u</i>ltiplicando.</p> +<p class = "pilcrow"> +¶ Na<i>m</i>q<i>ue</i> figura seque<i>n</i>s q<i>uam</i>uis signat +decies pl<i>us.</i></p> +<p class = "pilcrow"> +¶ Ipsa locata loco quam sign<i>ific</i>at p<i>ertin</i>ente.</p> +</div> + +<p><span class = "sidenote">Expo<i>sitio</i> [in margin].</span> +¶ Expone þis v<i>er</i>se þus. Eu<i>er</i>y of þese figuris +bitokens hym selfe & no mor<i>e</i>, yf he stonde in þe first place +of þe <a class = "gloss" name = "rewele" id = "rewele" href = +"#gloss_rewle">rewele</a> / this worde Simplicit<i>er</i> in þat verse +it is no more to say but þat, & no mor<i>e</i>. +<span class = "sidenote">An explanation of the principles of +notation.</span> +¶ If it stonde in the secu<i>n</i>de place of þe <a class = "gloss" +name = "rewle" id = "rewle" href = "#gloss_rewle">rewle</a>, he betokens +ten<i>e</i> tymes hym selfe, as þis figur<i>e</i> 2 here 20 tokens ten +tyme hym selfe, +<span class = "linenum">leaf 137 <i>a</i>.</span> +*þat is twenty, for he hym selfe betokenes twey<i>ne</i>, & ten +tymes <a class = "gloss" name = "twene1" id = "twene1" href = +"#gloss_twene">twene</a> is twenty. And for he stondis oɳ þe lyft side +& in þe secu<i>n</i>de place, he betokens ten tyme hy<i>m</i> selfe. +And so go forth. ¶ ffor eu<i>er</i>y <a class = "gloss" name = +"figure" id = "figure" href = "#gloss_figure">fig<i>ure</i></a>, & +he stonde aft<i>ur</i> a-noþ<i>er</i> toward the lyft side, he schal +betoken<i>e</i> ten tymes as mich mor<i>e</i> as he schul betoken & +he stode in þe place þ<i>ere</i> þat þe fig<i>ure</i> a-for<i>e</i> hym +stondes. +<span class = "sidenote">An example:</span> +loo an ensampull<i>e</i>. 9. 6. 3. 4. Þe fig<i>ure</i> of +4. þ<i>a</i>t hase þis schape <a href = "images/num4_full.png" target = +"_blank"><img src = "images/num4_full.png" width = "11" height = "14" +alt = "{4}" /></a>. betokens bot hymselfe, for he stondes in þe first +place. +<span class = "sidenote">units,</span> +The fig<i>ure</i> of 3. þat hase þis schape <a href = +"images/num3_full.png" target = "_blank"><img src = +"images/num3_full.png" width = "7" height = "15" alt = "{3}" /></a>. +betokens ten tymes mor<i>e</i> þen he schuld <a class = "gloss" name = +"amp" id = "amp" href = "#gloss_and">&</a> he stde þ<i>ere</i> +þ<i>a</i>t þe fig<i>ure</i> of 4. stondes, þ<i>a</i>t is thretty. +<span class = "sidenote">tens,</span> +The fig<i>ure</i> of 6, þ<i>a</i>t hase þis schape <a href = +"images/num6_full.png" target = "_blank"><img src = +"images/num6_full.png" width = "8" height = "14" alt = "{6}" /></a>, +betokens ten tymes mor<i>e</i> þan he schuld & he stode þ<i>ere</i> +as þe fig<i>ure</i> of <a href = "images/num3_full.png" target = +"_blank"><img src = "images/num3_full.png" width = "7" height = "15" alt += "{3}" /></a>. stondes, for þ<i>ere</i> he schuld tokyn<i>e</i> bot +sexty, & now he betokens ten tymes mor<i>e</i>, þat is sex hundryth. +<span class = "sidenote">hundreds,</span> +The fig<i>ure</i> of 9. þ<i>a</i>t hase þis schape <a href = +"images/num9_full.png" target = "_blank"><img src = +"images/num9_full.png" width = "8" height = "15" alt = "{9}" /></a>. +betokens ten tymes mor<i>e</i> þan<i>e</i> he schuld & he stode in +þe place þ<i>ere</i> þe fig<i>ure</i> of sex stondes, for þen he schuld +betoken to 9. hundryth, and in þe place þ<i>ere</i> he stondes now he +betokens 9. þousande. +<span class = "sidenote">thousands.</span> +Al þe <a class = "gloss" name = "hole" id = "hole" href = +"#gloss_hole">hole</a> nomb<i>ur</i> is 9 thousande sex hundryth & +four<i>e</i> & thretty. ¶ fforthermor<i>e</i>, when +<span class = "pagenum">5</span> +<a name = "page5" id = "page5"> </a> +þ<i>o</i>u schalt rede a nomb<i>ur</i> of fig<i>ure</i>, +<span class = "sidenote">How to read the number.</span> +þ<i>o</i>u schalt begyn<i>e</i> at þe last fig<i>ure</i> in the lyft +side, & rede so forth to þe riȝt side as her<i>e</i> 9. 6. +3. 4. Thou schal begyn to rede at þe fig<i>ure</i> of 9. & rede +forth þus. 9. +<span class = "linenum">leaf 137 <i>b</i>.</span> +*thousand sex hundryth thritty & foure. But when þ<i>o</i>u +schall<i>e</i> write, þ<i>o</i>u schalt be-gynne to write at þe ryȝt +side.</p> + +<div class = "verse"> +<p class = "pilcrow"> +¶ Nil cifra sig<i>nifica</i>t s<i>ed</i> dat signa<i>re</i> +sequenti.</p> +</div> + +<p><span class = "sidenote">The meaning and use of the cipher.</span> +Expone þis v<i>er</i>se. A cifre tokens <a class = "gloss" name = +"noght" id = "noght" href = "#gloss_noght">noȝt</a>, bot he makes þe +fig<i>ure</i> to betoken þat comes aft<i>ur</i> hym mor<i>e</i> þan he +schuld & he wer<i>e</i> away, as þus 1φ. her<i>e</i> þe +fig<i>ure</i> of on<i>e</i> tokens ten, & yf þe cifre wer<i>e</i> +away<a class = "tag" name = "tag_craft1" id = "tag_craft1" href = +"#note_craft1">1</a> & no fig<i>ure</i> by-for<i>e</i> hym he schuld +token bot on<i>e</i>, for þan he sch<i>ul</i>d stonde in þe first place. +¶ And þe cifre tokens nothyng hym selfe. for al þe nomb<i>ur</i> of +þe <a class = "gloss" name = "ylke" id = "ylke" href = +"#gloss_ylke">ylke</a> too fig<i>ure</i>s is bot ten. ¶ Questio. +Why says he þat a cifre makys a fig<i>ure</i> to <a class = "gloss" name += "signifyetyf" id = "signifyetyf" href = "#gloss_signifyetyf">signifye +(tyf)</a> mor<i>e</i> &c. ¶ I speke for þis worde <a class = +"terms" name = "significatyf" id = "significatyf" href = +"#terms_significatyf">significatyf</a>, ffor sothe it may happe +aft<i>ur</i> a cifre schuld come a-noþ<i>ur</i> cifre, as þus 2φφ. And +ȝet þe secunde cifre shuld token neu<i>er</i> þe mor<i>e</i> <a class = +"gloss" name = "excep" id = "excep" href = "#gloss_excep">excep</a> he +schuld kepe þe ord<i>er</i> of þe place. and a cifre is no fig<i>ure</i> +significatyf.</p> + +<div class = "verse"> +<p class = "pilcrow"> +¶ Q<i>ua</i>m p<i>re</i>cedentes plus ulti<i>m</i>a significabit /</p> +</div> + +<p><span class = "sidenote">The last figure means more than all the +others, since it is of the highest value.</span> +Expone þis v<i>er</i>se þus. Þe last figu<i>re</i> schal token +mor<i>e</i> þan all<i>e</i> þe oþ<i>er</i> afor<i>e</i>, <a class = +"gloss" name = "thought" id = "thought" href = +"#gloss_thought">thouȝt</a> þ<i>ere</i> wer<i>e</i> a hundryth thousant +figures afor<i>e</i>, as þus, 16798. Þe last fig<i>ure</i> þat is 1. +betokens ten thousant. And all<i>e</i> þe oþ<i>er</i> fig<i>ure</i>s +b<i>e</i>n bot betoken<i>e</i> bot sex thousant seuyn<i>e</i> +h<i>u</i>ndryth nynty & 8. ¶ And ten thousant is mor<i>e</i> +þen all<i>e</i> þat nomb<i>ur</i>, <i>er</i>go þe last figu<i>re</i> +tokens mor<i>e</i> þan all þe nomb<i>ur</i> afor<i>e</i>.</p> + +<p class = "headnote"><span class = "headnote"> +The Three Kinds of Numbers</span></p> + +<div class = "verse"> +<span class = "linenum">leaf 138 <i>a</i>.</span> +<p class = "pilcrow plus"> +* ¶ Post p<i>re</i>dicta scias breuit<i>er</i> q<i>uod</i> tres +num<i>er</i>or<i>um</i></p> +<p>Distincte species sunt; nam quidam digiti sunt;</p> +<p>Articuli quidam; quidam q<i>uoque</i> compositi sunt.</p> +</div> + +<h5>¶ Capit<i>ulu</i>m 2<sup>m</sup> de t<i>ri</i>plice divisione +nu<i>mer</i>or<i>um</i>.</h5> + +<p>¶ The auctor of þis tretis <a class = "terms" name = "departys" id = +"departys" href = "#terms_departys">dep<i>ar</i>tys</a> þis worde a +nomb<i>ur</i> into 3 p<i>ar</i>tes. Some nomb<i>ur</i> is called +digit<i>us</i> latine, a <a class = "terms" name = "digit" id = +"digit" href = "#terms_digit">digit</a> in englys. +<span class = "sidenote">Digits.</span> +So<i>m</i>me nomb<i>ur</i> is called articul<i>us</i> latine. An +<a class = "terms" name = "articul" id = "articul" href = +"#terms_article">Articul</a> in englys. +<span class = "sidenote">Articles.</span> +Some nomb<i>ur</i> is called a <a class = "terms" name = "composyt" id = +"composyt" href = "#terms_componede">composyt</a> in englys. +<span class = "sidenote">Composites.</span> +¶ Expone þis v<i>er</i>se. know þ<i>o</i>u aft<i>ur</i> þe forsayd +<a class = "gloss" name = "rewles" id = "rewles" href = +"#gloss_rewle">rewles</a> þ<i>a</i>t I sayd afore, þat þ<i>ere</i> ben +thre <a class = "gloss" name = "spices1" id = "spices1" href = +"#gloss_spices">spices</a> of nomb<i>ur</i>. Oon<i>e</i> is a digit, +Anoþ<i>er</i> is an Articul, & þe toþ<i>er</i> a Composyt. +v<i>er</i>sus.</p> + +<p class = "headnote"><span class = "headnote"> +Digits, Articles, and Composites.</span></p> + +<div class = "verse"> +<p class = "pilcrow"> +¶ Sunt digiti num<i>er</i>i qui cit<i>ra</i> denariu<i>m</i> +s<i>u</i>nt.</p> +</div> + +<p><span class = "sidenote">What are digits.</span> +¶ Her<i>e</i> he telles qwat is a digit, Expone v<i>er</i>su<i>s</i> +sic. Nomb<i>ur</i>s digitus ben<i>e</i> all<i>e</i> nomb<i>ur</i>s þat +ben w<i>i</i>t<i>h</i>-inne ten, as nyne, 8. 7. 6. 5. 4. 3. +2. 1.</p> + +<span class = "pagenum">6</span> +<a name = "page6" id = "page6"> </a> +<div class = "verse"> +<p class = "pilcrow"> +¶ Articupli decupli degito<i>rum</i>; compositi s<i>u</i>nt</p> +<p>Illi qui constant ex articulis degitisq<i>ue</i>.</p> +</div> + +<p>¶ Her<i>e</i> he telles what is a composyt and what is an<i>e</i> +articul. Expone sic v<i>er</i>sus. +<span class = "sidenote">What are articles.</span> +¶ Articulis ben<a class = "tag" name = "tag_craft2" id = +"tag_craft2" href = "#note_craft2">2</a> +all<i>e</i> þ<i>a</i>t may be deuidyt into nomb<i>urs</i> of ten & +nothyng<i>e</i> <a class = "gloss" name = "leue" id = "leue" href = +"#gloss_leue">leue</a> ou<i>er</i>, as twenty, thretty, fourty, +a hundryth, a thousand, & such oþ<i>er</i>, ffor twenty +may be dep<i>ar</i>tyt in-to 2 nomb<i>ur</i>s of ten, fforty in to +four<i>e</i> nomb<i>ur</i>s of ten, & so forth.</p> + +<p><span class = "linenum">leaf 138 <i>b</i>.</span> +<span class = "sidenote">What numbers are composites.</span> +*Compositys beɳ nomb<i>ur</i>s þat bene componyt of a digyt & of an +articull<i>e</i> as fouretene, fyftene, sextene, & such oþ<i>er</i>. +ffortene is co<i>m</i>ponyd of four<i>e</i> þat is a digit & of ten +þat is an articull<i>e</i>. ffiftene is componyd of 5 & ten, & +so of all oþ<i>er</i>, what þat þai ben. Short-lych eu<i>er</i>y +nomb<i>ur</i> þat be-gynnes w<i>i</i>t<i>h</i> a digit & endyth in a +articull<i>e</i> is a composyt, as fortene bygennyng<i>e</i> by +four<i>e</i> þat is a digit, & endes in ten.</p> + +<div class = "verse"> +<p class = "pilcrow"> +¶ Ergo, p<i>ro</i>posito nu<i>mer</i>o tibi scriber<i>e</i>, +p<i>ri</i>mo</p> +<p>Respicias quid sit nu<i>merus</i>; si digitus sit</p> +<p>P<i>ri</i>mo scribe loco digitu<i>m</i>, si compositus sit</p> +<p>P<i>ri</i>mo scribe loco digitu<i>m</i> post articulu<i>m</i>; +sic.</p> +</div> + +<p><span class = "sidenote">How to write a number,</span> +¶ here he telles how þ<i>o</i>u schalt <a class = "gloss" name = "wyrch" +id = "wyrch" href = "#gloss_worch">wyrch</a> whan þ<i>o</i>u schalt +write a nomb<i>ur</i>. Expone v<i>er</i>su<i>m</i> sic, & fac iuxta +expon<i>ent</i>is sentencia<i>m</i>; whan þ<i>o</i>u hast a +nomb<i>ur</i> to write, loke fyrst what man<i>er</i> nomb<i>ur</i> it ys +þ<i>a</i>t þ<i>o</i>u schalt write, whether it be a digit or a composit +or an Articul. +<span class = "sidenote">if it is a digit;</span> +¶ If he be a digit, write a digit, as yf it be seuen, write seuen +& write þ<i>a</i>t digit in þe first place toward þe ryght side. +<span class = "sidenote">if it is a composite.</span> +If it be a composyt, write þe digit of þe composit in þe first place +& write þe articul of þat digit in þe secunde place next toward þe +lyft side. As yf þ<i>o</i>u schal write sex & twenty. write þe digit +of þe nomb<i>ur</i> in þe first place þat is sex, and write þe articul +next aft<i>ur</i> þat is twenty, as þus 26. +<span class = "sidenote">How to read it.</span> +But whan þ<i>o</i>u schalt <a class = "gloss" name = "sowne" id = +"sowne" href = "#gloss_sowne">sowne</a> or speke +<span class = "linenum">leaf 139 <i>a</i>.</span> +*or rede an Composyt þou schalt first sowne þe articul & +aft<i>ur</i> þe digit, as þ<i>o</i>u seyst by þe comyn<i>e</i> speche, +Sex & twenty & nouȝt twenty & sex. v<i>er</i>sus.</p> + +<div class = "verse"> +<p class = "pilcrow"> +¶ Articul<i>us</i> si sit, in p<i>ri</i>mo limite cifram,</p> +<p>Articulu<i>m vero</i> reliq<i>ui</i>s insc<i>ri</i>be +figur<i>is</i>.</p> +</div> + +<p><span class = "sidenote">How to write Articles:</span> +¶ Here he tells how þ<i>o</i>u schal write when þe nombre þ<i>a</i>t +þ<i>o</i>u hase to write is an Articul. Expone v<i>er</i>sus sic & +fac s<i>ecundu</i>m sentenciam. Ife þe nomb<i>ur</i> þ<i>a</i>t +þ<i>o</i>u hast write be an Articul, write first a cifre & +aft<i>ur</i> þe cifer write an Articull<i>e</i> þus. 2φ. +<span class = "sidenote">tens,</span> +fforthermor<i>e</i> þ<i>o</i>u schalt vnd<i>ir</i>stonde yf þ<i>o</i>u +haue an Articul, loke how +<span class = "pagenum">7</span> +<a name = "page7" id = "page7"> </a> +mych he is, yf he be w<i>i</i>t<i>h</i>-ynne an hundryth, þ<i>o</i>u +schalt write bot on<i>e</i> cifre, afore, as her<i>e</i> .9φ. +<span class = "sidenote">hundreds,</span> +If þe articull<i>e</i> be by hym-silfe & be an hundrid euen<i>e</i>, +þen schal þ<i>o</i>u write .1. & 2 cifers afor<i>e</i>, þat he may +stonde in þe thryd place, for eu<i>er</i>y fig<i>ure</i> in þe thryd +place schal token a hundrid tymes hym selfe. +<span class = "sidenote">thousands, &c.</span> +If þe articul be a thousant or thousandes<a class = "tag" name = +"tag_craft3" id = "tag_craft3" href = "#note_craft3">3</a> +and he stonde by hy<i>m</i> selfe, write afor<i>e</i> 3 cifers & so +forþ of al oþ<i>er</i>.</p> + +<div class = "verse"> +<p class = "pilcrow"> +¶ Quolib<i>et</i> in nu<i>mer</i>o, si par sit p<i>ri</i>ma figura,</p> +<p>Par erit & to<i>tu</i>m, quicquid sibi +co<i>n</i>ti<i>nua</i>t<i>ur</i>;</p> +<p>Imp<i>ar</i> si fu<i>er</i>it, totu<i>m</i> tu<i>n</i>c fiet +<i>et</i> impar.</p> +</div> + +<p><span class = "sidenote">To tell an even number</span> +¶ Her<i>e</i> he teches a gen<i>er</i>all<i>e</i> rewle þ<i>a</i>t yf þe +first fig<i>ure</i> in þe <a class = "gloss" name = "rewle2" id = +"rewle2" href = "#gloss_rewle">rewle</a> of fig<i>ure</i>s token a +nomb<i>ur</i> þat is euen<i>e</i> al þ<i>a</i>t nomb<i>ur</i> of +fig<i>ur</i>ys in þat rewle schal be euen<i>e</i>, as her<i>e</i> +þ<i>o</i>u may see 6. 7. 3. 5. 4. Computa & p<i>ro</i>ba. +<span class = "sidenote">or an odd.</span> +¶ If þe first +<span class = "linenum">leaf 139 <i>b</i>.</span> +*fig<i>ur</i>e token an nomb<i>ur</i> þat is ode, all<i>e</i> þat +nomb<i>ur</i> in þat rewle schall<i>e</i> be ode, as her<i>e</i> 5 6 7 8 +6 7. Computa & p<i>ro</i>ba. v<i>er</i>sus.</p> + +<div class = "verse"> +<p class = "pilcrow"> +¶ Septe<i>m</i> su<i>n</i>t partes, no<i>n</i> pl<i>u</i>res, istius +artis;</p> +<p class = "pilcrow"> +¶ Adder<i>e</i>, subt<i>ra</i>her<i>e</i>, duplar<i>e</i>, +dimidiar<i>e</i>,</p> +<p>Sextaq<i>ue</i> diuider<i>e</i>, s<i>ed</i> qui<i>n</i>ta +m<i>u</i>ltiplicar<i>e</i>;</p> +<p>Radice<i>m</i> ext<i>ra</i>her<i>e</i> p<i>ar</i>s septi<i>m</i>a +dicitur esse.</p> +</div> + +<p class = "headnote"><span class = "headnote"> +The Seven Rules of Arithmetic.</span></p> + +<p><span class = "sidenote">The seven rules.</span> +¶ Her<i>e</i> telles þ<i>a</i>t þ<i>er</i> beɳ .7. spices or +p<i>ar</i>tes of þis craft. The first is called addicioñ, þe secunde is +called subtraccioñ. The thryd is called <a class = "terms" name = +"duplacion" id = "duplacion" href = "#terms_duplacion">duplacioñ</a>. +The 4. is called <a class = "terms" name = "dimydicion" id = +"dimydicion" href = "#terms_dimydicion">dimydicioñ</a>. The 5. is called +m<i>u</i>ltiplicacioñ. The 6 is called diuisioñ. The 7. is called +extraccioñ of þe <a class = "terms" name = "rote" id = "rote" href = +"#terms_rote">Rote</a>. What all þese spices ben<i>e</i> hit +schall<i>e</i> be tolde singillati<i>m</i> in <a class = "gloss" name = +"here" id = "here" href = "#gloss_here">her<i>e</i></a> <a class = +"gloss" name = "caputule" id = "caputule" href = +"#gloss_caputule">caputul<i>e</i></a>.</p> + +<div class = "verse"> +<p class = "pilcrow"> +¶ Subt<i>ra</i>his aut addis a dext<i>ri</i>s vel mediabis:</p> +</div> + +<p><span class = "sidenote">Add, subtract, or halve, from right to +left.</span> +Thou schal be-gynne in þe ryght side of þe boke or of a tabul. loke +wer<i>e</i> þ<i>o</i>u wul be-gynne to write latyn or englys in a boke, +& þ<i>a</i>t schall<i>e</i> be called þe lyft side of the boke, þat +þ<i>o</i>u writest toward þ<i>a</i>t side schal be called þe ryght side +of þe boke. V<i>er</i>sus.</p> + +<div class = "verse"> +<p>A leua dupla, diuide, m<i>u</i>ltiplica.</p> +</div> + +<p>Here he telles þe in quych side of þe boke or of þe tabul þ<i>o</i>u +schall<i>e</i> be-gyn<i>e</i> to wyrch duplacioñ, diuisioñ, and +m<i>u</i>ltiplicacioñ. +<span class = "sidenote">Multiply or divide from left to right.</span> +Thou schal begyn<i>e</i> to <a class = "gloss" name = "worch1" id = +"worch1" href = "#gloss_worch">worch</a> in þe lyft side of þe boke or +of þe tabul, but yn what wyse þ<i>o</i>u schal wyrch in hym <b>dicetur +<a class = "gloss" name = "singillatim" id = "singillatim" href = +"#gloss_singillatim">singillatim</a> in seque<i>n</i>tib<i>us</i> +capi<i>tulis</i> et de vtilitate cui<i>us</i>li<i>bet</i> art<i>is</i> +& sic Completur</b> +<span class = "linenum">leaf <ins class = "correction" title = "error for ‘140a’?">140</ins>.</span> +*<b>p<i>ro</i>hemi<i>um</i> & sequit<i>ur</i> tractat<i>us</i> & +p<i>ri</i>mo de arte addic<i>ion</i>is que p<i>ri</i>ma ars est in +ordine.</b></p> + +<span class = "pagenum">8</span> +<a name = "page8" id = "page8"> </a> + +<p class = "headnote"><span class = "headnote"> +The Craft of Addition.</span></p> + +<div class = "verse"> +<p><span class = "dropcap">A</span>dder<i>e</i> si nu<i>mer</i>o +num<i>e</i>ru<i>m</i> vis, ordine tali</p> +<p>Incipe; scribe duas p<i>rim</i>o series nu<i>mer</i>or<i>um</i></p> +<p>P<i>ri</i>ma<i>m</i> sub p<i>ri</i>ma recte pone<i>n</i>do +figura<i>m</i>,</p> +<p>Et sic de reliq<i>ui</i>s facias, si sint tibi plures.</p> +</div> + +<p><span class = "sidenote">Four things must be known:</span> +¶ Her<i>e</i> by-gynnes þe craft of Addicioñ. In þis craft þ<i>o</i>u +most knowe foure thyng<i>es</i>. ¶ Fyrst þ<i>ou</i> most know what +is addicioñ. Next þ<i>o</i>u most know how mony rewles of figurys þou +most haue. ¶ Next þ<i>o</i>u most know how mony diue<i>r</i>s casys +happes in þis craft of addicioñ. ¶ And next qwat is þe +p<i>ro</i>fet of þis craft. +<span class = "sidenote">what it is;</span> +¶ As for þe first þou most know þat addicioñ is a <a class = +"terms" name = "castyng" id = "castyng" href = "#terms_cast">castyng</a> +to-ged<i>ur</i> of twoo nomburys in-to on<i>e</i> nombr<i>e</i>. As yf I +aske qwat is <a class = "gloss" name = "twene" id = "twene" href = +"#gloss_twene">twene</a> & thre. Þ<i>o</i>u wyl <a class = "terms" +name = "cast" id = "cast" href = "#terms_cast">cast</a> þese twene +nomb<i>re</i>s to-ged<i>ur</i> & say þ<i>a</i>t it is fyue. +<span class = "sidenote">how many rows of figures;</span> +¶ As for þe secunde þou most know þ<i>a</i>t þou schall<i>e</i> haue +tweyne rewes of figures, on<i>e</i> vndur a-nother, as her<i>e</i> +þ<i>o</i>u mayst se. +<span class = "float"> +1234<br /> +2168.</span> +<span class = "sidenote">how many cases;</span> +¶ As for þe thryd þou most know þ<i>a</i>t ther<i>e</i> ben foure +diu<i>er</i>se cases. +<span class = "sidenote">what is its result.</span> +As for þe forthe þ<i>o</i>u most know þ<i>a</i>t þe p<i>ro</i>fet of þis +craft is to telle what is þe hole nomb<i>ur</i> þ<i>a</i>t comes of +diu<i>er</i>se nomburis. Now as to þe texte of oure verse, he teches +ther<i>e</i> how þ<i>o</i>u schal <a class = "gloss" name = "worch" id = +"worch" href = "#gloss_worch">worch</a> in þis craft. ¶ He says yf +þ<i>o</i>u wilt cast on<i>e</i> nomb<i>ur</i> to anoþ<i>er</i> +nomb<i>ur</i>, þou most by-gynne on þis wyse. +<span class = "sidenote">How to set down the sum.</span> +¶ ffyrst write +<span class = "linenum">leaf 140 <i>b</i>.</span> +*two rewes of figuris & nombris so þat þ<i>o</i>u write þe first +figur<i>e</i> of þe hyer nomb<i>ur</i> euen<i>e</i> vnd<i>ir</i> the +first fig<i>ure</i> of þe nether nomb<i>ur</i>, +<span class = "float"> +123<br /> +234.</span> +And þe secunde of þe nether nomb<i>ur</i> euen<i>e</i> vnd<i>ir</i> þe +secunde of þe hyer, & so forthe of eu<i>er</i>y fig<i>ur</i>e of +both þe rewes as þ<i>o</i>u mayst se.</p> + +<p class = "headnote"><span class = "headnote"> +The Cases of the Craft of Addition.</span></p> + +<div class = "verse"> +<p class = "pilcrow"> +¶ Inde duas adde p<i>ri</i>mas hac condic<i>i</i>one:</p> +<p>Si digitus crescat ex addic<i>i</i>one prior<i>um</i>;</p> +<p>P<i>ri</i>mo scribe loco digitu<i>m</i>, quicu<i>n</i>q<i>ue</i> sit +ille.</p> +</div> + +<p>¶ Here he teches what þ<i>o</i>u schalt do when þ<i>o</i>u hast write +too rewes of figuris <a class = "gloss" name = "on" id = "on" href = +"#gloss_on">on</a> vnder an-oþ<i>er</i>, as I sayd be-for<i>e</i>. +<span class = "sidenote">Add the first figures;</span> +¶ He says þ<i>o</i>u schalt take þe first fig<i>ur</i>e of þe heyer +nomb<i>re</i> & þe fyrst figur<i>e</i> of þe neþ<i>er</i> nombre, +& cast hem to-ged<i>er</i> vp-on þis condicioɳ. Thou schal loke +qweþ<i>er</i> þe nombe<i>r</i> þat comys þ<i>ere</i>-of be a digit or +no. +<span class = "sidenote">rub out the top figure;</span> +¶ If he be a digit þ<i>o</i>u schalt do away þe first fig<i>ur</i>e +of þe hyer nomb<i>re</i>, and write þ<i>ere</i> in his stede þat he +stode Inne þe digit, þ<i>a</i>t comes of þe ylke 2 fig<i>ur</i>es, & +so +<span class = "sidenote">write the result in its place.</span> +<a class = "gloss" name = "wrich" id = "wrich" href = +"#gloss_worch">wrich</a> forth oɳ oþ<i>er</i> figures yf þ<i>ere</i> be +ony moo, til þ<i>o</i>u come to þe ende toward þe lyft side. And +<a class = "gloss" name = "lede" id = "lede" href = "#gloss_lede">lede</a> +þe nether fig<i>ure</i> stonde still eu<i>er</i>-mor<i>e</i> til +þ<i>o</i>u haue <a class = "gloss" name = "ydo1" id = "ydo1" href = +"#gloss_ydo">ydo</a>. ffor þ<i>ere</i>-by þ<i>o</i>u schal <a class = +"gloss" name = "wyte" id = "wyte" href = "#gloss_wete">wyte</a> +wheþ<i>er</i> þ<i>o</i>u hast don<i>e</i> wel or no, as I schal tell þe +aft<i>er</i>ward in þe ende of þis Chapt<i>er</i>. ¶ And loke +<a class = "gloss" name = "allgate" id = "allgate" href = +"#gloss_allgate">allgate</a> +<span class = "linenum">leaf 141 <i>a</i>.</span> +þat þou be-gynne to worch in þis Craft of Addi*cioɳ in þe ryȝt side, +<span class = "pagenum">9</span> +<a name = "page9" id = "page9"> </a> +<span class = "sidenote">Here is an example.</span> +here is an ensampul of þis case. +<span class = "float"> +1234<br /> +2142.</span> +Caste 2 to four<i>e</i> & þat wel be sex, do away 4. & write in +þe same place þe fig<i>ur</i>e of sex. ¶ And lete þe fig<i>ur</i>e +of 2 in þe nether rewe stonde stil. When þ<i>o</i>u hast do so, cast 3 +& 4 to-ged<i>ur</i> and þat wel be seuen þ<i>a</i>t is a digit. Do +away þe 3, & set þ<i>ere</i> seueɳ, and lete þe neþ<i>er</i> +fig<i>ure</i> stonde still<i>e</i>, & so <a class = "gloss" name = +"worch_imp1" id = "worch_imp1" href = "#gloss_worch">worch</a> forth +bakward til þ<i>o</i>u hast <a class = "gloss" name = "ydo" id = "ydo" +href = "#gloss_ydo">ydo</a> all to-ged<i>er</i>.</p> + +<div class = "verse"> +<p>Et si composit<i>us</i>, in limite scribe seque<i>n</i>te</p> +<p>Articulum, p<i>ri</i>mo digitum; q<i>uia</i> sic iubet ordo.</p> +</div> + +<p>¶ Here is þe secunde case þ<i>a</i>t may happe in þis craft. And þe +case is þis, +<span class = "sidenote">Suppose it is a Composite, set down the digit, +and carry the tens.</span> +yf of þe casting of 2 nomburis to-ged<i>er</i>, as of þe fig<i>ur</i>e +of þe hyer rewe & of þe figure of þe neþ<i>er</i> rewe come a +Composyt, how schalt þ<i>ou</i> worch. Þ<i>us</i> þ<i>o</i>u schalt +worch. Thou shalt do away þe fig<i>ur</i>e of þe hyer nomb<i>er</i> þat +was cast to þe figure of þe neþ<i>er</i> nomber. ¶ And write +þ<i>ere</i> þe digit of þe Composyt. And set þe articul of þe composit +next aft<i>er</i> þe digit in þe same rewe, yf þ<i>ere</i> be no +<a class = "gloss" name = "mo" id = "mo" href = "#gloss_mo">mo</a> +fig<i>ur</i>es aft<i>er</i>. But yf þ<i>ere</i> be mo figuris +aft<i>er</i> þat digit. And þere he schall be rekend for hym selfe. And +when þ<i>o</i>u schalt adde þ<i>a</i>t ylke figure þ<i>a</i>t berys þe +articull<i>e</i> ou<i>er</i> his hed to þe figur<i>e</i> vnd<i>er</i> +hym, þ<i>o</i>u schalt cast þat articul to þe figure þ<i>a</i>t hase hym +ou<i>er</i> his hed, & þ<i>ere</i> þat Articul schal tokeɳ hym +selfe. +<span class = "sidenote">Here is an example.</span> +lo an Ensampull +<span class = "linenum">leaf 141 <i>b</i>.</span> +*of all. +<span class = "float"> +326<br /> +216.</span> +Cast 6 to 6, & þ<i>ere</i>-of wil arise twelue. do away þe hyer 6 +& write þ<i>ere</i> 2, þ<i>a</i>t is þe digit of þis composit. And +þe<i>n</i> write þe articull<i>e</i> þat is ten ou<i>er</i> þe figuris +hed of twene as þ<i>us</i>. +<span class = "float"> + 1<br /> +322<br /> +216.</span> +Now cast þe articull<i>e</i> þ<i>a</i>t standus vpon þe fig<i>ur</i>is +of twene hed to þe same fig<i>ur</i>e, & reken þat articul bot for +on<i>e</i>, and þan þ<i>ere</i> wil arise thre. Þan cast þat thre to þe +neþ<i>er</i> figure, þat is on<i>e</i>, & þat wul be four<i>e</i>. +do away þe fig<i>ur</i>e of 3, and write þ<i>ere</i> a fig<i>ur</i>e of +foure. and lete þe neþ<i>er</i> fig<i>ur</i>e stonde stil, & þan +worch forth. vn<i>de</i> <i>ver</i>sus.</p> + + +<div class = "verse"> +<p class = "pilcrow"> +¶ Articulus si sit, in p<i>ri</i>mo limite cifram,</p> +<p class = "pilcrow"> +¶ Articulu<i>m</i> v<i>er</i>o reliquis inscribe figuris,</p> +<p>Vel p<i>er</i> se scribas si nulla figura sequat<i>ur</i>.</p> +</div> + +<p>¶ Her<i>e</i> he puttes þe thryde case of þe craft of Addicioɳ. & +þe case is þis. +<span class = "sidenote">Suppose it is an Article, set down a cipher and +carry the tens.</span> +yf of Addiciouɳ of 2 figuris a-ryse an Articull<i>e</i>, how schal +þ<i>o</i>u do. thou most do away þe <a class = "gloss" name = "heer" id += "heer" href = "#gloss_heer">heer</a> fig<i>ur</i>e þ<i>a</i>t was +addid to þe neþ<i>er</i>, & write þ<i>ere</i> a cifre, and sett þe +<a class = "terms" name = "articuls" id = "articuls" href = +"#terms_article">articuls</a> on þe figuris hede, yf þ<i>a</i>t +þ<i>ere</i> come ony aft<i>er</i>. And wyrch þan as I haue tolde þe in +þe secunde case. An ensampull. +<span class = "float"> +25.<br /> +15</span> +Cast 5 to 5, þat wylle be ten. now do away þe hyer 5, & write +þ<i>ere</i> a cifer. And sette ten vpon þe figuris hed of 2. And reken +it but for on þus. lo +<span class = "sidenote">Here is an example.</span> +<span class = "pagenum">10</span> +<a name = "page10" id = "page10"> </a> +an Ensampull<i>e</i> +<span class = "float box"> +1 <br /> +2φ<br /> +15</span> +And +<span class = "linenum">leaf 142 <i>a</i>.</span> +*þan worch forth. But yf þ<i>ere</i> come no figure aft<i>er</i> þe +cifre, write þe articul next hym in þe same rewe as here +<span class = "float box"> +5<br /> +5</span> +cast 5 to 5, and it wel be ten. do away 5. þat is þe hier 5. and write +þ<i>ere</i> a cifre, & write aft<i>er</i> hym þe articul as þus +<span class = "float box"> +1φ<br /> + 5</span> +And þan þ<i>o</i>u hast done.</p> + +<div class = "verse"> +<p class = "pilcrow"> +¶ Si tibi cifra sup<i>er</i>ueniens occurrerit, illa<i>m</i></p> +<p>Dele sup<i>er</i>posita<i>m</i>; fac illic scribe figura<i>m</i>,</p> +<p>Postea procedas reliquas addendo figuras.</p> +</div> + +<p><span class = "sidenote">What to do when you have a cipher in the top +row.</span> +¶ Her<i>e</i> he putt<i>es</i> þe fourt case, & it is þis, þat +yf þ<i>ere</i> come a cifer in þe hier rewe, how þ<i>o</i>u schal do. +þus þ<i>o</i>u schalt do. do away þe cifer, & sett þ<i>ere</i> þe +digit þ<i>a</i>t comes of þe addiciou<i>n</i> as þus +<span class = "float"> +1φφ84.<br /> +17743</span> +<span class = "sidenote">An example of all the difficulties.</span> +In þis ensampul ben all<i>e</i> þe four<i>e</i> cases. Cast 3 to foure, +þ<i>a</i>t wol be seueɳ. do away 4. & write þ<i>ere</i> seueɳ; þan +cast 4 to þe figur<i>e</i> of 8. þ<i>a</i>t wel be 12. do away 8, & +sett þ<i>ere</i> 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<i>e</i> to a cifer, & hit wull<i>e</i> be but on, for +noȝt & on makes but on<i>e</i>. þan cast 7. þ<i>a</i>t stondes +vnd<i>er</i> þat on to hym, & þat wel be 8. do away þe cifer & +þat 1. & sette þ<i>ere</i> 8. þan go forthermor<i>e</i>. cast þe +oþ<i>er</i> 7 to þe cifer þ<i>a</i>t stondes ou<i>er</i> hy<i>m</i>. +þ<i>a</i>t wul be bot seuen, for þe cifer betokens noȝt. do away þe +cifer & sette þ<i>ere</i> seueɳ, +<span class = "linenum">leaf 142 <i>b</i>.</span> +*& þen go forþ<i>er</i>mor<i>e</i> & cast 1 to 1, & þat wel +be 2. do away þe hier 1, & sette þ<i>ere</i> 2. þan hast þ<i>o</i>u +do. And yf þ<i>o</i>u haue wel ydo þis nomber þat is sett +her<i>e</i>-aft<i>er</i> wel be þe nomber þat schall<i>e</i> aryse of +all<i>e</i> þe addicioɳ as her<i>e</i> 27827. ¶ Sequi<i>tu</i>r +alia sp<i>eci</i>es.</p> + +<p class = "headnote"><span class = "headnote"> +The Craft of Subtraction.</span></p> + +<div class = "verse"> +<p>A nu<i>mer</i>o num<i>er</i>u<i>m</i> si sit tibi demer<i>e</i> +cura</p> +<p>Scribe figurar<i>um</i> series, vt in addicione.</p> +</div> + +<p><span class = "sidenote">Four things to know about +subtraction:</span> +¶ This is þe Chapt<i>er</i> of subtraccioɳ, in the quych þou most +know foure nessessary thyng<i>es</i>. the first what is subtraccioɳ. þe +secunde is how mony nombers þou most haue to subt<i>ra</i>ccioɳ, the +thryd is how mony maners of cases þ<i>ere</i> may happe in þis craft of +subtraccioɳ. The fourte is qwat is þe p<i>ro</i>fet of þis craft. +¶ As for +<span class = "sidenote">the first;</span> +þe first, þ<i>o</i>u most know þ<i>a</i>t subtraccioɳ is drawyng<i>e</i> +of on<i>e</i> nowmb<i>er</i> oute of anoþ<i>er</i> nomber. +<span class = "sidenote">the second;</span> +As for þe secunde, þou most knowe þ<i>a</i>t þou most haue two rewes of +figuris on<i>e</i> vnd<i>er</i> anoþ<i>er</i>, as þ<i>o</i>u <a class = +"gloss" name = "addyst" id = "addyst" href = "#gloss_addyst">addyst</a> +in addicioɳ. +<span class = "sidenote">the third;</span> +As for þe thryd, þ<i>o</i>u moyst know þ<i>a</i>t four<i>e</i> +man<i>er</i> of diu<i>er</i>se casis mai happe in þis craft. +<span class = "sidenote">the fourth.</span> +¶ As for þe fourt, þou most know þ<i>a</i>t þe p<i>ro</i>fet of þis +craft is whenne þ<i>o</i>u hasse taken þe lasse nomber out of þe +mor<i>e</i> to telle what þ<i>ere</i> <a class = "gloss" name = "leues1" +id = "leues1" href = "#gloss_leue">leues</a> ou<i>er</i> +<span class = "pagenum">11</span> +<a name = "page11" id = "page11"> </a> +þ<i>a</i>t. & þ<i>o</i>u most be-gynne to wyrch in þ<i>is</i> craft +in þe ryght side of þe boke, as þ<i>o</i>u diddyst in addicioɳ. +V<i>er</i>sus.</p> + +<div class = "verse"> +<p class = "pilcrow"> +¶ Maiori nu<i>mer</i>o num<i>er</i>u<i>m</i> suppone minorem,</p> +<p class = "pilcrow"> +¶ Siue pari nu<i>mer</i>o supponat<i>ur</i> num<i>er</i>us par.</p> +</div> + +<p><span class = "linenum">leaf 143 <i>a</i>.</span> +* ¶ Her<i>e</i> he telles þat +<span class = "sidenote">Put the greater number above the less.</span> +þe hier nomber most be mor<i>e</i> þen þe neþ<i>er</i>, or els eueɳ as +mych. but he may not be lasse. And þe case is þis, þou schalt drawe þe +neþ<i>er</i> nomber out of þe hyer, & þou mayst not do þ<i>a</i>t yf +þe hier nomber wer<i>e</i> lasse þan þat. ffor þ<i>o</i>u mayst not draw +sex out of 2. But þ<i>o</i>u mast draw 2 out of sex. And þou maiste draw +twene out of twene, for þou schal leue noȝt of þe hier twene vn<i>de</i> +v<i>er</i>sus.</p> + +<p class = "headnote"><span class = "headnote"> +The Cases of the Craft of Subtraction.</span></p> + +<div class = "verse"> +<p class = "pilcrow"> +¶ Postea si possis a prima subt<i>ra</i>he p<i>ri</i>ma<i>m</i></p> +<p>Scribens quod remanet.</p> +</div> + +<p><span class = "sidenote">The first case of subtraction.</span> +Her<i>e</i> is þe first case put of subtraccioɳ, & he says þou +schalt begynne in þe ryght side, & draw þe first fig<i>ure</i> of þe +neþ<i>er</i> rewe out of þe first fig<i>ure</i> of þe hier rewe. qwether +þe hier fig<i>ur</i>e be mor<i>e</i> þen þe neþ<i>er</i>, or eueɳ as +mych. And þat is notified in þe vers when he says “Si possis.” Whan +þ<i>o</i>u has þus ydo, do away þe hiest fig<i>ur</i>e & sett +þ<i>ere</i> þat <a class = "gloss" name = "leues" id = "leues" href = +"#gloss_leue">leues</a> of þe subtraccioɳ, +<span class = "sidenote">Here is an example.</span> +lo an Ensampull<i>e</i> +<span class = "float box"> +234<br /> +122</span> +draw 2 out of 4. þan leues 2. do away 4 & write þ<i>ere</i> 2, & +latte þe neþ<i>er</i> figur<i>e</i> sto<i>n</i>de stille, & so go +<a class = "gloss" name = "forby" id = "forby" href = +"#gloss_forby">for-by</a> oþ<i>er</i> figuris till þ<i>o</i>u come to þe +ende, þan hast þ<i>o</i>u do.</p> + +<div class = "verse"> +<p class = "halfline"> +¶ Cifram si nil remanebit.</p> +</div> + +<p><span class = "sidenote">Put a cipher if nothing remains.</span> +¶ Her<i>e</i> he putt<i>es</i> þe secunde case, & hit is þis. +yf it happe þ<i>a</i>t qwen þ<i>o</i>u hast draw on neþ<i>er</i> +fig<i>ure</i> out of a hier, & þ<i>er</i>e leue noȝt aft<i>er</i> þe +subt<i>ra</i>ccioɳ, þus +<span class = "linenum">leaf 143 <i>b</i>.</span> +*þou schalt do. þ<i>o</i>u schall<i>e</i> do away þe hier fig<i>ur</i>e +& write þ<i>ere</i> a cifer, as +<span class = "sidenote">Here is an example.</span> +lo an Ensampull +<span class = "float box"> +24<br /> +24</span> +Take four<i>e</i> out of four<i>e</i> þan <a class = "gloss" name = +"leus" id = "leus" href = "#gloss_leue">leus</a> noȝt. +þ<i>er</i>efor<i>e</i> do away þe hier 4 & set þ<i>ere</i> a cifer, +þan take 2 out of 2, þan leues noȝt. do away þe hier 2, & set +þ<i>ere</i> a cifer, and so worch whar<i>e</i> so eu<i>er</i> þis +happe.</p> + +<div class = "verse"> +<p>Sed si no<i>n</i> possis a p<i>ri</i>ma dem<i>er</i>e +p<i>ri</i>ma<i>m</i></p> +<p>P<i>re</i>cedens vnu<i>m</i> de limite deme seque<i>n</i>te,</p> +<p>Quod demptu<i>m</i> p<i>ro</i> denario reputabis ab illo</p> +<p>Subt<i>ra</i>he to<i>ta</i>lem num<i>er</i>u<i>m</i> qu<i>em</i> +p<i>ro</i>posuisti</p> +<p>Quo facto sc<i>ri</i>be super quicquid remaneb<i>i</i>t.</p> +</div> + +<p><span class = "sidenote">Suppose you cannot take the lower figure +from the top one, borrow ten;</span> +Her<i>e</i> he puttes þe thryd case, þe quych is þis. yf it happe þat þe +neþ<i>er</i> fig<i>ur</i>e be mor<i>e</i> þen þe hier fig<i>ur</i>e þat +he schall<i>e</i> be draw out of. how schall<i>e</i> þou do. þus +þ<i>o</i>u schall<i>e</i> do. þou schall<i>e</i> <a class = "gloss" name += "borro" id = "borro" href = "#gloss_borro">borro</a> .1. oute of þe +next fig<i>ur</i>e þat comes aft<i>er</i> in þe same rewe, for þis case +may neu<i>er</i> happ but yf þ<i>ere</i> come figures aft<i>er</i>. þan +þ<i>o</i>u schalt sett +<span class = "pagenum">12</span> +<a name = "page12" id = "page12"> </a> +þat on ou<i>er</i> þe hier figur<i>es</i> hed, of the quych þou woldist +y-draw oute þe neyþ<i>er</i> fig<i>ur</i>e yf þ<i>o</i>u haddyst +<a class = "gloss" name = "ymyght" id = "ymyght" href = +"#gloss_ymyght">y-myȝt</a>. Whane þou hase þus ydo þou schall<i>e</i> +rekene þ<i>a</i>t .1. for ten. +<span class = "sidenote">take the lower number from ten;</span> +¶. And out of þat ten þ<i>o</i>u schal draw þe neyþermost fig<i>ur</i>e, +And all<i>e</i> þ<i>a</i>t leues þou schall<i>e</i> +<span class = "sidenote">add the answer to the top number.</span> +adde to þe figur<i>e</i> on whos hed þat .1. stode. And þen þ<i>o</i>u +schall<i>e</i> do away all<i>e</i> þat, & sett þ<i>ere</i> +all<i>e</i> that arisys of the addicioɳ of þe ylke 2 fig<i>ur</i>is. And +yf yt +<span class = "linenum">leaf 144 <i>a</i>.</span> +*happe þat þe fig<i>ur</i>e of þe quych þ<i>o</i>u schalt borro on be +hym self but 1. If þ<i>o</i>u schalt þat on<i>e</i> & sett it vppoɳ +þe oþ<i>er</i> figur<i>is</i> hed, and sett in þ<i>a</i>t 1. place a +cifer, yf þ<i>ere</i> come mony figur<i>es</i> aft<i>er</i>. +<span class = "sidenote">Example.</span> +lo an Ensampul. +<span class = "float box"> +2122<br /> +1134</span> +take 4 out of 2. it wyl not be, þerfor<i>e</i> borro on<i>e</i> of þe +next figur<i>e</i>, þ<i>a</i>t is 2. and sett þat ou<i>er</i> þe hed of +þe fyrst 2. & rekene it for ten. and þere þe secunde stondes write +1. for þ<i>o</i>u tokest on out of hy<i>m</i>. þan take þe neþ<i>er</i> +fig<i>ur</i>e, þat is 4, out of ten. And þen leues 6. cast to 6 þe +fig<i>ur</i>e of þat 2 þat stode vnd<i>er</i> þe hedde of 1. þat was +<a class = "gloss" name = "borwed" id = "borwed" href = +"#gloss_borro">borwed</a> & rekened for ten, and þat wylle be 8. do +away þ<i>a</i>t 6 & þat 2, & sette þ<i>ere</i> 8, & lette þe +neþ<i>er</i> fig<i>ur</i>e stonde stille. Whanne þ<i>o</i>u hast do þus, +go to þe next fig<i>ur</i>e þ<i>a</i>t is now bot 1. but first yt was 2, +& þ<i>ere</i>-of was <a class = "gloss" name = "borred" id = +"borred" href = "#gloss_borro">borred</a> 1. +<span class = "sidenote">How to ‘Pay back’ the borrowed ten.</span> +þan take out of þ<i>a</i>t þe fig<i>ur</i>e vnd<i>er</i> hym, þ<i>a</i>t +is 3. hit wel not be. þer-for<i>e</i> <a class = "gloss" name = "borowe" +id = "borowe" href = "#gloss_borro">borowe</a> of the next +fig<i>ur</i>e, þe quych is bot 1. Also take & sett hym ou<i>er</i> +þe hede of þe fig<i>ure</i> þat þou woldest haue y-draw oute of þe +nether figure, þe quych was 3. & þou myȝt not, & rekene +þ<i>a</i>t borwed 1 for ten & sett in þe same place, of þe quych +place þ<i>o</i>u tokest hy<i>m</i> of, a cifer, for he was bot 1. +Whanne þ<i>o</i>u hast þ<i>us</i> ydo, take out of þat 1. þ<i>a</i>t is +rekent for ten, þe neþ<i>er</i> figure of 3. And þ<i>ere</i> +leues 7. +<span class = "linenum">leaf 144 <i>b</i>.</span> +*cast þe ylke 7 to þe fig<i>ur</i>e þat had þe ylke ten vpon his hed, þe +quych fig<i>ur</i>e was 1, & þat wol be 8. þan do away +þ<i>a</i>t 1 and þ<i>a</i>t 7, & write þ<i>ere</i> 8. & þan +wyrch forth in oþ<i>er</i> figuris til þ<i>o</i>u come to þe ende, & +þan þ<i>o</i>u hast þe do. V<i>er</i>sus.</p> + +<div class = "verse"> +<p class = "pilcrow"> +¶ Facque nonenarios de cifris, cu<i>m</i> remeabis</p> +<p class = "pilcrow"> +¶ Occ<i>ur</i>rant si forte cifre; dum demps<i>er</i>is vnum</p> +<p class = "pilcrow"> +¶ Postea p<i>ro</i>cedas reliquas deme<i>n</i>do figuras.</p> +</div> + +<p><span class = "sidenote">A very hard case is put.</span> +¶ Her<i>e</i> he putt<i>es</i> þe fourte case, þe quych is þis, yf it +happe þat þe neþ<i>er</i> fig<i>ur</i>e, þe quych þ<i>o</i>u schalt draw +out of þe hier fig<i>ur</i>e be mor<i>e</i> pan þe hier figur +ou<i>er</i> hym, & þe next fig<i>ur</i>e of two or of thre or of +foure, or how mony þ<i>ere</i> be by cifers, how wold þ<i>o</i>u do. +Þ<i>o</i>u <a class = "gloss" name = "wost" id = "wost" href = +"#gloss_wete">wost</a> wel þ<i>o</i>u most nede borow, & þ<i>o</i>u +mayst not borow of þe cifers, for þai haue noȝt þat þai may <a class = +"gloss" name = "lene" id = "lene" href = "#gloss_lene">lene</a> or +spar<i>e</i>. Ergo<a class = "tag" name = "tag_craft4" id = "tag_craft4" +href = "#note_craft4">4</a> +how +<span class = "pagenum">13</span> +<a name = "page13" id = "page13"> </a> +woldest þ<i>o</i>u do. Certayɳ þus most þ<i>o</i>u do, þ<i>o</i>u most +borow on of þe next figure significatyf in þat rewe, for þis case may +not happe, but yf þ<i>ere</i> come figures significatyf aft<i>er</i> the +cifers. Whan þ<i>o</i>u hast borowede þ<i>a</i>t 1 of the next figure +significatyf, sett þ<i>a</i>t on ou<i>er</i> þe hede of þ<i>a</i>t +fig<i>ur</i>e of þe quych þ<i>o</i>u wold haue draw þe neþ<i>er</i> +figure out yf þ<i>o</i>u hadest myȝt, & reken it for ten as +þo<i>u</i> diddest i<i>n</i> þe oþ<i>er</i> case <a class = "gloss" name += "hereafore" id = "hereafore" href = +"#gloss_hereafore">her<i>e</i>-a-for<i>e</i></a>. Whaɳ þ<i>o</i>u hast +þus y-do loke how mony cifers þ<i>ere</i> wer<i>e</i> bye-twene þat +figur<i>e</i> significatyf, & þe fig<i>ur</i>e of þe quych +þ<i>o</i>u woldest haue y-draw the +<span class = "linenum">leaf 145 <i>a</i>.</span> +*neþ<i>er</i> figure, and of eu<i>er</i>y of þe ylke cifers make a +figur<i>e</i> of 9. +<span class = "sidenote">Here is an example.</span> +lo an Ensampull<i>e</i> after. +<span class = "float box"> +40002<br /> +10004</span> +Take 4 out of 2. it wel not be. borow 1 out of be next figure +significatyf, þe quych is 4, & þen leues 3. do away þ<i>a</i>t +figur<i>e</i> of 4 & write þ<i>ere</i> 3. & sett þ<i>a</i>t 1 +vppon þe fig<i>ur</i>e of 2 hede, & þan take 4 out of ten, & þan +þere leues 6. Cast 6 to the fig<i>ur</i>e of 2, þ<i>a</i>t wol be 8. do +away þat 6 & write þ<i>er</i>e 8. Whan þ<i>o</i>u hast þus y-do make +of eu<i>er</i>y 0 betweyn 3 & 8 a figure of 9, & þan worch forth +in goddes name. +<span class = "sidenote">Sic.</span> +<span class = "float box"> +39998<br /> +10004</span> +& yf þ<i>o</i>u hast wel y-do þ<i>o</i>u<a class = "tag" name = +"tag_craft5" id = "tag_craft5" href = "#note_craft5">5</a> +schalt haue þis nomb<i>er</i></p> + + +<p class = "headnote"><span class = "headnote"> +How to prove the Subtraction.</span></p> + +<div class = "verse"> +<p class = "pilcrow"> +¶ Si subt<i>ra</i>cc<i>i</i>o sit b<i>e</i>n<i>e</i> facta +p<i>ro</i>bar<i>e</i> valebis</p> +<p>Quas s<i>u</i>btraxisti p<i>ri</i>mas addendo figuras.</p> +</div> + +<p><span class = "sidenote">How to prove a subtraction sum.</span> +¶ Her<i>e</i> he teches þe Craft how þ<i>o</i>u schalt know, whan +þ<i>o</i>u hast subt<i>ra</i>yd, wheþ<i>er</i> þou hast wel ydo or no. +And þe Craft is þis, ryght as þ<i>o</i>u <a class = "gloss" name = +"subtrayd" id = "subtrayd" href = "#gloss_subtrahe">subtrayd</a> þe +neþ<i>er</i> figures fro þe hier figures, ryȝt so adde þe same +neþ<i>er</i> figures to þe hier figures. And yf þ<i>o</i>u haue well +<a class = "gloss" name = "ywroth" id = "ywroth" href = +"#gloss_worch">y-wroth</a> a-for<i>e</i> þou schalt haue þe hier nombre +þe same þ<i>o</i>u haddest <a class = "gloss" name = "or" id = "or" href += "#gloss_or">or</a> þou be-gan to worch. as for þis I bade þou schulde +kepe þe neþ<i>er</i> figures stylle. +<span class = "sidenote">Here is an example.</span> +lo an +<span class = "linenum">leaf 145 <i>b</i>.</span> +*Ensampull<i>e</i> of all<i>e</i> þe 4 cases toged<i>re</i>. worche +well<i>e</i> þis case +<span class = "float box"> +40003468<br /> +20004664</span> +And yf þou worch well<i>e</i> whan þou hast all<i>e</i> subtrayd þe +þ<i>a</i>t hier nombr<i>e</i> her<i>e</i>, þis schall<i>e</i> be þe +nombre here foloyng whan þ<i>o</i>u hast subtrayd. +<span class = "float box"> +39998804<br /> +20004664</span> +<span class = "sidenote">Our author makes a slip here (3 +for 1).</span> +And þou schalt know þ<i>us</i>. adde þe neþ<i>er</i> rowe of þe same +nombre to þe hier rewe as þus, cast 4 to 4. þat wol be 8. do away þe 4 +& write þ<i>ere</i> 8. by þe first case of addicioɳ. þan cast 6 to 0 +þat wol be 6. do away þe 0, & write þere 6. þan cast 6 to 8, +þ<i>a</i>t wel be 14. do away 8 & write þ<i>ere</i> a fig<i>ur</i>e +of 4, þat is þe digit, and write a fig<i>ur</i>e of 1. þ<i>a</i>t schall +be-token ten. þ<i>a</i>t is þe articul vpon þe hed of 8 next +aft<i>er</i>, þan reken þ<i>a</i>t 1. for 1. & cast it to 8. þat +schal be 9. cast to þat 9 þe neþ<i>er</i> fig<i>ur</i>e vnd<i>er</i> þat +þe quych is 4, & þat schall<i>e</i> be 13. do away þat 9 & sett +þ<i>er</i>e 3, & sett a figure of 1. þ<i>a</i>t schall be 10 vpon þe +next figur<i>is</i> hede þe +<span class = "pagenum">14</span> +<a name = "page14" id = "page14"> </a> +quych is 9. by þe secu<i>n</i>de case þ<i>a</i>t þ<i>o</i>u hadest in +addicioɳ. þan cast 1 to 9. & þat wol be 10. do away þe 9. & þat +1. And write þ<i>ere</i> a cifer. and write þe articull<i>e</i> þat is +1. betokenyng<i>e</i> 10. vpon þe hede of þe next figur<i>e</i> toward +þe lyft side, þe quych +<span class = "linenum">leaf 146 <i>a</i>.</span> +*is 9, & so do forth tyl þ<i>o</i>u come to þe last 9. +<span class = "sidenote">He works his proof through,</span> +take þe figur<i>e</i> of þat 1. þe quych þ<i>o</i>u schalt fynde +ou<i>er</i> þe hed of 9. & sett it ou<i>er</i> þe next figures hede +þat schal be 3. ¶ Also do away þe 9. & set þ<i>ere</i> a +cifer, & þen cast þat 1 þat stondes vpon þe hede of 3 to þe same 3, +& þ<i>a</i>t schall<i>e</i> make 4, þen caste to þe ylke 4 the +figur<i>e</i> in þe neyþ<i>er</i> rewe, þe quych is 2, and þat +schall<i>e</i> be 6. +<span class = "sidenote">and brings out a result.</span> +<span class = "float box"> +60003468<br /> +20004664</span> +And þen schal þ<i>o</i>u haue an Ensampull<i>e</i> aȝeyɳ, loke & se, +& but þ<i>o</i>u haue þis same þ<i>o</i>u hase <a class = "gloss" +name = "mysewroght" id = "mysewroght" href = +"#gloss_mysewroght">myse-wroȝt</a>.</p> + + +<p class = "headnote"><span class = "headnote"> +The Craft of Duplation.</span></p> + +<h5>Sequit<i>ur</i> de duplac<i>i</i>one</h5> + +<div class = "verse"> +<p><span class = "dropcap">S</span>i vis duplar<i>e</i> +num<i>er</i>u<i>m</i>, sic i<i>n</i>cipe p<i>rim</i>o</p> +<p>Scribe fig<i>ur</i>ar<i>um</i> serie<i>m</i> +q<i>ua</i>mcu<i>n</i>q<i>ue</i> vel<i>is</i> tu.</p> +</div> + +<p><span class = "sidenote">Four things must be known in +Duplation.</span> +¶ This is the Chaptur<i>e</i> of <a class = "terms" name = +"duplacion2" id = "duplacion2" href = "#terms_duplacion">duplacioɳ</a>, +in þe quych craft þ<i>o</i>u most haue & know 4 thing<i>es</i>. +¶ Þe first þ<i>a</i>t þ<i>o</i>u most know is what is duplacioɳ. þe +secu<i>n</i>de is how mony rewes of fig<i>ur</i>es þ<i>o</i>u most haue +to þis craft. ¶ þe thryde is how many cases may<a class = "tag" +name = "tag_craft6" id = "tag_craft6" href = "#note_craft6">6</a> +happe in þis craft. ¶ þe fourte is what is þe p<i>ro</i>fet of þe +craft. +<span class = "sidenote">Here they are.</span> +¶ As for þe first. duplacioɳ is a doublyng<i>e</i> of a nombre. +¶ As for þe secu<i>n</i>de þ<i>o</i>u most +<span class = "linenum">leaf 146 <i>b</i>.</span> +*haue on nombre or on rewe of figures, the quych called nu<i>merus</i> +dupland<i>us</i>. As for þe thrid þ<i>o</i>u most know þat 3 +diu<i>er</i>se cases may hap in þis craft. As for þe fourte. qwat is þe +p<i>ro</i>fet of þis craft, & þ<i>a</i>t is to know what <a class = +"gloss" name = "arisyght" id = "arisyght" href = +"#gloss_arisyght">a-risyȝt</a> of a nombre I-doublyde. +<span class = "sidenote">Mind where you begin.</span> +¶ fforþ<i>er</i>-mor<i>e</i>, þ<i>o</i>u most know & take gode +hede in quych side þ<i>o</i>u schall<i>e</i> be-gyn in þis craft, or +ellis þ<i>o</i>u mayst <a class = "gloss" name = "spyl" id = "spyl" href += "#gloss_spyl">spyl</a> all<i>e</i> þ<i>i</i> lab<i>er</i> þ<i>er</i>e +aboute. c<i>er</i>teyn þ<i>o</i>u schalt begyɳ in the lyft side in þis +Craft. thenke wel ou<i>er</i> þis verse. ¶ <a class = "tag" name = +"tag_craft7" id = "tag_craft7" href = "#note_craft7">7</a>A leua dupla, +diuide, m<i>u</i>ltiplica.<a class = "tag" href = +"#note_craft7">7</a></p> + +<p>The <a class = "gloss" name = "sentens" id = "sentens" href = +"#gloss_sentens">sentens</a> of þes verses afor<i>e</i>, as þ<i>o</i>u +may see if þ<i>o</i>u take hede. +<span class = "sidenote">Remember your rules.</span> +As þe text of þis verse, þat is to say, ¶ Si vis duplare. þis is þe +sentence. ¶ If þ<i>o</i>u <a class = "gloss" name = "wel" id = +"wel" href = "#gloss_wel">wel</a> double a nombre þus þ<i>o</i>u most +be-gynɳ. Write a rewe of figures of what nomb<i>re</i> þou welt. +v<i>er</i>sus.</p> + +<div class = "verse"> +<p>Postea p<i>ro</i>cedas p<i>ri</i>ma<i>m</i> duplando +figura<i>m</i></p> +<p>Inde q<i>uo</i>d excrescit scribas vbi iusserit ordo</p> +<p>Iuxta p<i>re</i>cepta tibi que dant<i>ur</i> in addic<i>i</i>one.</p> +</div> + +<p><span class = "sidenote">How to work a sum.</span> +¶ Her<i>e</i> he telles how þ<i>o</i>u schalt worch in þis Craft. he +says, fyrst, whan þ<i>o</i>u hast writen þe nombre þ<i>o</i>u schalt +be-gyn at þe first +<span class = "pagenum">15</span> +<a name = "page15" id = "page15"> </a> +figur<i>e</i> in the lyft side, & doubull<i>e</i> þat fig<i>ur</i>e, +& þe nombre þat comes þ<i>ere</i>-of þ<i>o</i>u schalt write as +þ<i>o</i>u diddyst in addicioɳ, as ¶ I schal telle þe in þe case. +v<i>er</i>sus.</p> + +<p class = "headnote"><span class = "headnote"> +The Cases of the Craft of Duplation.</span></p> + +<span class = "linenum">leaf 147 <i>a</i>.</span> +<div class = "verse"> +<p class = "pilcrow plus">* ¶ Nam si sit digitus in primo limite +scribas.</p> +</div> + +<p><span class = "sidenote">If the answer is a digit,</span> +¶ Her<i>e</i> is þe first case of þis craft, þe quych is þis. yf of +duplacioɳ of a figur<i>e</i> arise a digit. what schal þ<i>o</i>u do. +þus þ<i>o</i>u schal do. +<span class = "sidenote">write it in the place of the top figure.</span> +do away þe fig<i>ur</i>e þat was doublede, & sett þ<i>ere</i> þe +diget þat comes of þe duplacioɳ, as þus. 23. double 2, & þ<i>a</i>t +wel be 4. do away þe figur<i>e</i> of 2 & sett þ<i>ere</i> a +figur<i>e</i> of 4, & so <a class = "gloss" name = "worch_imp" id = +"worch_imp" href = "#gloss_worch">worch</a> forth till<i>e</i> +þ<i>o</i>u come to þe ende. v<i>er</i>sus.</p> + +<div class = "verse"> +<p class = "pilcrow"> +¶ Articul<i>us</i> si sit, in p<i>ri</i>mo limite cifram,</p> +<p class = "pilcrow"> +¶ Articulu<i>m</i> v<i>er</i>o reliquis inscribe figuris;</p> +<p class = "pilcrow"> +¶ Vel p<i>er</i> se scribas, si nulla figura sequat<i>ur</i>.</p> +</div> + +<p><span class = "sidenote">If it is an article,</span> +¶ Here is þe secunde case, þe quych is þis yf þ<i>ere</i> come an +articull<i>e</i> of þe duplacioɳ of a fig<i>ur</i>e þ<i>o</i>u schalt do +ryȝt as þ<i>o</i>u diddyst in addicioɳ, þat is to <a class = "gloss" +name = "wete" id = "wete" href = "#gloss_wete">wete</a> þat þ<i>o</i>u +schalt do away þe figur<i>e</i> þat is doublet & +<span class = "sidenote">put a cipher in the place, and ‘carry’ the +tens.</span> +sett þ<i>ere</i> a cifer, & write þe articull<i>e</i> ou<i>er</i> þe +next figur<i>is</i> hede, yf þ<i>ere</i> be any aft<i>er</i>-warde +toward þe lyft side as þus. 25. begyn at the lyft side, and +doubull<i>e</i> 2. þat wel be 4. do away þat 2 & sett þere 4. þan +doubul 5. þat wel be 10. do away 5, & sett þ<i>ere</i> a 0, & +sett 1 vpon þe next figur<i>is</i> hede þe quych is 4. & þen draw +downe 1 to 4 & þat woll<i>e</i> be 5, & þen do away þ<i>a</i>t 4 +& þat 1, & sett þ<i>ere</i> 5. for þat 1 schal be rekened in þe +drawyng<i>e</i> toged<i>re</i> for 1. wen +<span class = "linenum">leaf 147 <i>b</i>.</span> +*þou hast ydon þou schalt haue þis nomb<i>r</i>e 50. +<span class = "sidenote">If there is no figure to ‘carry’ them to, write +them down.</span> +yf þ<i>ere</i> come no figur<i>e</i> aft<i>er</i> þe fig<i>ur</i>e +þ<i>a</i>t is addit, of þe quych addicioɳ comes an articull<i>e</i>, +þ<i>o</i>u schalt do away þe figur<i>e</i> þ<i>a</i>t is dowblet & +sett þ<i>ere</i> 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<i>e</i> 5 & set þ<i>ere</i> a cifer, & sett þe articul +next aft<i>er</i> in þe same rewe toward þe lyft side, & þou schalt +haue þis nombre 1023. þen go forth & double þe oþ<i>er</i> nombers +þe quych is <a class = "gloss" name = "lyght" id = "lyght" href = +"#gloss_lyght">lyȝt</a> <a class = "gloss" name = "ynowght" id = +"ynowght" href = "#gloss_ynowght">y-nowȝt</a> to do. v<i>er</i>sus.</p> + +<div class = "verse"> +<p class = "pilcrow"> +¶ Compositus si sit, in limite sc<i>ri</i>be seq<i>uen</i>te</p> +<p>Articulu<i>m</i>, p<i>ri</i>mo digitu<i>m</i>; q<i>uia</i> sic iubet +ordo:</p> +<p>Et sic de reliq<i>ui</i>s facie<i>n</i>s, si sint tibi plures.</p> +</div> + +<p><span class = "sidenote">If it is a Composite,</span> +¶ Her<i>e</i> he putt<i>es</i> þe Thryd case, þe quych is þis, yf of +duplacioɳ of a fig<i>ur</i>e come a Composit. þ<i>o</i>u schalt do away +þe fig<i>u</i>re þ<i>a</i>t is doublet & set þ<i>ere</i> a digit of +þe Composit, +<span class = "sidenote">write down the digit, and ‘carry’ the +tens.</span> +& sett þe articull<i>e</i> ou<i>er</i> þe next figures hede, & +aft<i>er</i> draw hym downe w<i>i</i>t<i>h</i> þe figur<i>e</i> +ou<i>er</i> whos hede he stondes, & make þ<i>ere</i>-of an nombre as +þ<i>o</i>u hast done +<span class = "pagenum">16</span> +<a name = "page16" id = "page16"> </a> +afore, & yf þ<i>ere</i> come no fig<i>ur</i>e aft<i>er</i> þat digit +þat þ<i>o</i>u hast <a class = "gloss" name = "ywrite" id = "ywrite" +href = "#gloss_write">y-write</a>, þa<i>n</i> set þe articull<i>e</i> +next aft<i>er</i> hym in þe same rewe as þus, 67: double 6 þat wel be +12, do away 6 & write þ<i>ere</i> þe digit +<span class = "linenum">leaf 148 <i>a</i>.</span> +*of 12, þe quych is 2, +<span class = "sidenote">Here is an example.</span> +and set þe articull<i>e</i> next aft<i>er</i> toward þe lyft side in þe +same rewe, for þ<i>ere</i> comes no figur<i>e</i> aft<i>er</i>. þan +dowble þat oþ<i>er</i> figur<i>e</i>, þe quych is 7, þat wel be 14. +the quych is a Composit. þen do away 7 þat þ<i>o</i>u doublet & sett +þe þe diget of hy<i>m</i>, the quych is 4, sett þe articull<i>e</i> +ou<i>er</i> þe next figur<i>es</i> hed, þe quych is 2, & þen draw to +hym þat on, & make on nombre þe quych schall<i>e</i> be 3. And þen +yf þ<i>o</i>u haue wel y-do þ<i>o</i>u schall<i>e</i> haue þis nombre of +þe duplacioɳ, 134. v<i>er</i>sus.</p> + +<div class = "verse"> +<p class = "pilcrow"> +¶ Si super ext<i>re</i>ma<i>m</i> nota sit monade<i>m</i> dat +eid<i>em</i></p> +<p>Quod t<i>ibi</i> <i>con</i>tingat si p<i>ri</i>mo dimidiabis.</p> +</div> + +<p><span class = "sidenote">How to double the mark for one-half.</span> +¶ Her<i>e</i> he says, yf ou<i>er</i> þe fyrst fig<i>ur</i>e in þe +ryȝt side be such a merke as is her<i>e</i> made, <a class = "terms" +name = "sup_w" id = "sup_w" href = "#terms_sup_w"><sup>w</sup></a>, +þ<i>o</i>u schall<i>e</i> fyrst doubull<i>e</i> þe figur<i>e</i>, the +quych stondes vnd<i>er</i> þ<i>a</i>t merke, & þen þou schalt doubul +þat merke þe quych stond<i>es</i> for <a class = "gloss" name = +"haluendel" id = "haluendel" href = "#gloss_haluendel">haluendel</a> on. +for too <a class = "gloss" name = "haluedels" id = "haluedels" href = +"#gloss_haluendel">haluedels</a> makes on, & so þ<i>a</i>t wol be +on. cast þ<i>a</i>t on to þat duplacioɳ of þe figur<i>e</i> ou<i>er</i> +whos hed stode þat merke, & write it in þe same place þ<i>ere</i> +þat þe figur<i>e</i> þe quych was doublet stode, as þus 23<sup>w</sup>. +double 3, þat wol be 6; doubul þat halue on, & þat wol be on. cast +on to 6, þ<i>a</i>t wel be 7. do away 6 & þat 1, & sett +þ<i>ere</i> 7. þan hase þou do. as for þat figur<i>e</i>, þan go +<span class = "linenum">leaf 148 <i>b</i>.</span> +*to þe oþ<i>er</i> fig<i>ure</i> & worch forth. +<span class = "sidenote">This can only stand over the first +figure.</span> +& þ<i>o</i>u schall neu<i>er</i> haue such a merk but ou<i>er</i> þe +hed of þe furst figure in þe ryght side. And ȝet it schal not happe but +yf it were y-halued a-for<i>e</i>, þus þ<i>o</i>u schalt +vnd<i>er</i>stonde þe verse. ¶ Si sup<i>er</i> +ext<i>re</i>ma<i>m</i> &c. Et nota, talis fig<i>ur</i>a <sup>w</sup> +significans medietate<i>m</i>, unitat<i>is</i> veniat, <i>i.e.</i> +contingat u<i>e</i>l fiat sup<i>er</i> ext<i>re</i>ma<i>m</i>, +<i>i.e.</i> sup<i>er</i> p<i>ri</i>ma<i>m</i> figura<i>m</i> in +ext<i>re</i>mo sic v<i>er</i>sus dextram ars dat: <i>i.e.</i> reddit +monade<i>m</i>. <i>i.e.</i> vnitate<i>m</i> eide<i>m</i>. <i>i.e.</i> +eidem note & declina<i>tur</i> hec monos, d<i>i</i>s, di, dem, +&c. ¶ Quod <i>er</i>g<i>o</i> to<i>tum</i> ho<i>c</i> dabis +monade<i>m</i> note <i>con</i>ting<i>et</i>. <i>i.e.</i> eveniet tibi si +dimidiasti, <i>i.e.</i> accipisti u<i>e</i>l subtulisti medietatem +alicuius unius, in cuius principio sint figura nu<i>mer</i>u<i>m</i> +denotans i<i>m</i>pare<i>m</i> p<i>rim</i>o <i>i.e.</i> principiis.</p> + +<p class = "headnote"><span class = "headnote"> +The Craft of Mediation.</span></p> + +<h5>¶ Sequit<i>ur</i> de mediacione.</h5> + +<div class = "verse"> +<p><span class = "dropcap">I</span>ncipe sic, si vis alique<i>m</i> +nu<i>me</i>ru<i>m</i> mediar<i>e</i>:</p> +<p>Sc<i>ri</i>be figurar<i>um</i> seriem sola<i>m</i>, velut +an<i>te</i>.</p> +</div> + +<p><span class = "sidenote">The four things to be known in +mediation:</span> +¶ In þis Chapter is <a class = "gloss" name = "taght" id = "taght" +href = "#gloss_taght">taȝt</a> þe Craft of <a class = "terms" name = +"mediacioun" id = "mediacioun" href = +"#terms_mediacioun">mediaciouɳ</a>, in þe quych craft þ<i>o</i>u most +know 4 thynges. ffurst what is mediacioɳ. the secunde how mony rewes of +figur<i>es</i> þ<i>o</i>u most haue in þe wyrchyng<i>e</i> of þis craft. +þe thryde how mony diu<i>er</i>se cases may happ in þis craft.<a class = +"tag" name = "tag_craft8" id = "tag_craft8" href = "#note_craft8">8</a> +<span class = "sidenote">the first</span> +¶ As for þe furst, þ<i>o</i>u schalt vndurstonde þat mediacioɳ is a +<span class = "pagenum">17</span> +<a name = "page17" id = "page17"> </a> +takyng out of halfe a nomber out of a <a class = "gloss" name = "holle" +id = "holle" href = "#gloss_hole">holle</a> nomber, +<span class = "linenum">leaf 149 <i>a</i>.</span> +*as yf þ<i>o</i>u +<span class = "sidenote">the second;</span> +wolde take 3 out of 6. ¶ As for þe secunde, þ<i>o</i>u schalt know +þ<i>a</i>t þ<i>o</i>u most haue on<i>e</i> rewe of figures, & no +moo, as þ<i>o</i>u <a class = "gloss" name = "hayst" id = "hayst" href = +"#gloss_hayst">hayst</a> in þe +<span class = "sidenote">the third;</span> +craft of duplacioɳ. ¶ As for the thryd, þou most vnd<i>er</i>stonde +þat +<span class = "sidenote">the fourth.</span> +5 cases may happe in þis craft. ¶ As for þe fourte, þou +schall<i>e</i> know þat the p<i>ro</i>fet of þis craft is when +þ<i>o</i>u hast take away þe haluendel of a nomb<i>re</i> to telle qwat +þer<i>e</i> schall<i>e</i> leue. ¶ Incipe sic, &c. The sentence +of þis verse is þis. yf þ<i>o</i>u wold <a class = "terms" name = +"medye" id = "medye" href = "#terms_medye">medye</a>, þat is to say, +take halfe out of þe holle, or halfe out of halfe, þou most begynne +þ<i>us</i>. +<span class = "sidenote">Begin thus.</span> +Write on<i>e</i> rewe of figur<i>es</i> of what nombre þou wolte, as +þ<i>o</i>u dyddyst be-for<i>e</i> in þe Craft of duplacioɳ. +v<i>er</i>sus.</p> + +<div class = "verse"> +<p class = "pilcrow"> +¶ Postea p<i>ro</i>cedas medians, si p<i>ri</i>ma figura</p> +<p>Si par aut i<i>m</i>par videas.</p> +</div> + +<p>¶ Her<i>e</i> he says, when þ<i>o</i>u hast write a rewe of figures, +þ<i>o</i>u schalt +<span class = "sidenote">See if the number is even or odd.</span> +take hede wheþ<i>er</i> þe first figur<i>e</i> be eueɳ or odde in +nombre, & vnd<i>er</i>stonde þ<i>a</i>t he spekes of þe first figure +in þe ryȝt side. And i<i>n</i> the ryght side þ<i>o</i>u schall<i>e</i> +begynne in þis Craft.</p> + +<div class = "verse"> +<p class = "halfline"> +¶ Quia si fu<i>er</i>it par,</p> +<p>Dimidiab<i>is</i> eam, scribe<i>n</i>s quicq<i>ui</i>d remanebit:</p> +</div> + +<p><span class = "sidenote">If it is even, halve it, and write the +answer in its place.</span> +¶ Her<i>e</i> is the first case of þis craft, þe quych is þis, yf +þe first figur<i>e</i> be euen. þou schal take away fro þe figur<i>e</i> +euen halfe, & do away þat fig<i>ur</i>e and set þ<i>ere</i> þat +leues ou<i>er</i>, as þus, 4. take +<span class = "linenum">leaf 149 <i>b</i>.</span> +*halfe out of 4, & þan þ<i>ere</i> leues 2. do away 4 & sett +þ<i>ere</i> 2. þis is lyght y-nowȝt. v<i>er</i>sus.</p> + +<p class = "headnote"><span class = "headnote"> +The Mediation of an Odd Number.</span></p> + +<div class = "verse"> +<p class = "pilcrow"> +¶ Impar si fu<i>er</i>it vnu<i>m</i> demas mediar<i>e</i></p> +<p>Quod no<i>n</i> p<i>re</i>sumas, s<i>ed</i> quod sup<i>er</i>est +mediabis</p> +<p>Inde sup<i>er</i> tractu<i>m</i> fac demptu<i>m</i> quod no<i>ta</i>t +vnu<i>m</i>.</p> +</div> + +<p><span class = "sidenote">If it is odd, halve the even number less +than it.</span> +Her<i>e</i> is þe secunde case of þis craft, the quych is þis. yf þe +first figur<i>e</i> betoken<i>e</i> a nombre þat is odde, the quych odde +schal not be <a class = "terms" name = "mediete" id = "mediete" href = +"#terms_medye">mediete</a>, þen þ<i>o</i>u schalt medye þat nombre þat +leues, when the odde of þe same nomb<i>re</i> is take away, & write +þat þ<i>a</i>t leues as þ<i>o</i>u diddest in þe first case of þis +craft. Whaɳ þ<i>o</i>u <a class = "gloss" name = "hayst2" id = "hayst2" +href = "#gloss_hayst">hayst</a> write þat. for þ<i>a</i>t þat leues, +<span class = "sidenote">Then write the sign for one-half over +it.</span> +write such a merke as is her<i>e</i> <sup>w</sup> vpon his hede, þe +quych merke schal betokeɳ halfe of þe odde þat was take away. +<span class = "sidenote">Here is an example.</span> +lo an Ensampull. 245. the first figur<i>e</i> her<i>e</i> is +betokenyng<i>e</i> odde nombre, þe quych is 5, for 5 is odde; +þ<i>er</i>e-for<i>e</i> do away þat þ<i>a</i>t is odde, þe quych is 1. +þen leues 4. þen medye 4 & þen leues 2. do away 4. & sette +þ<i>ere</i> 2, & make such a merke <sup>w</sup> upon his hede, þat +is to say ou<i>er</i> his hede of 2 as þus. 242.<sup>w</sup> And þen +worch forth in þe oþ<i>er</i> figures tyll þ<i>o</i>u come to þe ende. +by þe furst case as þ<i>o</i>u schalt +<span class = "pagenum">18</span> +<a name = "page18" id = "page18"> </a> +vnd<i>er</i>stonde þat +<span class = "sidenote">Put the mark only over the first figure.</span> +þ<i>o</i>u schalt +<span class = "linenum">leaf 150 <i>a</i>.</span> +*neu<i>er</i> make such a merk but ou<i>er</i> þe first fig<i>ur</i>e +hed in þe riȝt side. Wheþ<i>er</i> þe other fig<i>ur</i>es þat comyɳ +aft<i>er</i> hym be eueɳ or odde. v<i>er</i>sus.</p> + +<p class = "headnote"><span class = "headnote"> +The Cases of the Craft of Mediation.</span></p> + +<div class = "verse"> +<p class = "pilcrow"> +¶ Si monos, dele; sit t<i>ibi</i> cifra post no<i>ta</i> supra.</p> +</div> + +<p><span class = "sidenote">If the first figure is one put a +cipher.</span> +¶ Here is þe thryde case, þe quych yf the first figur<i>e</i> be a +figur<i>e</i> of 1. þ<i>o</i>u schalt do away þat 1 & set +þ<i>ere</i> a cifer, & a merke ou<i>er</i> þe cifer as þus, 241. do +away 1, & sett þ<i>ere</i> a cifer w<i>i</i>t<i>h</i> a merke +ou<i>er</i> his hede, & þen hast þ<i>o</i>u ydo for þat 0. as þus +0<sup>w</sup> þen worch forth in þe oþer fig<i>ur</i>ys till þ<i>o</i>u +come to þe ende, for it is lyght as dyche water. vn<i>de</i> +v<i>er</i>sus.</p> + +<div class = "verse"> +<p class = "pilcrow"> +¶ Postea p<i>ro</i>cedas hac condic<i>i</i>one secu<i>n</i>da:</p> +<p>Imp<i>ar</i> si fu<i>er</i>it hinc vnu<i>m</i> deme +p<i>ri</i>ori,</p> +<p>Inscribens quinque, nam denos significabit</p> +<p>Monos p<i>re</i>d<i>ict</i>am.</p> +</div> + +<p><span class = "sidenote">What to do if any other figure is +odd.</span> +¶ Her<i>e</i> he putt<i>es</i> þe fourte case, þe quych is þis. yf +it happeɳ the secunde figur<i>e</i> betoken odde nombre, þou schal do +away on of þat odde nombre, þe quych is significatiue by þ<i>a</i>t +figure 1. þe quych 1 schall be rekende for 10. Whan þ<i>o</i>u hast take +away þ<i>a</i>t 1 out of þe nombre þ<i>a</i>t is signifiede by þat +figur<i>e</i>, þ<i>o</i>u schalt medie þ<i>a</i>t þat leues ou<i>er</i>, +& do away þat figur<i>e</i> þat is medied, & sette in his +<a class = "gloss" name = "styde" id = "styde" href = +"#gloss_styde">styde</a> halfe of þ<i>a</i>t nombre. +<span class = "sidenote">Write a figure of five over the next lower +number’s head.</span> +¶ Whan þ<i>o</i>u hase so done, þ<i>o</i>u schalt write +<span class = "linenum">leaf 150 <i>b</i>.</span> +*a figure of 5 ou<i>er</i> þe next figur<i>es</i> hede by-for<i>e</i> +toward þe ryȝt side, for þat 1, þe quych made odd nombr<i>e</i>, schall +stonde for ten, & 5 is halfe of 10; so þ<i>o</i>u most write 5 for +his haluendell<i>e</i>. +<span class = "sidenote">Example.</span> +lo an Ensampull<i>e</i>, 4678. begyɳ in þe ryȝt side as þ<i>o</i>u most +nedes. medie 8. þen þ<i>o</i>u schalt leue 4. do away þat 8 & sette +þ<i>ere</i> 4. þen out of 7. take away 1. þe quych makes odde, & +sett 5. vpon þe next figur<i>es</i> hede afor<i>e</i> toward þe ryȝt +side, þe quych is now 4. but afor<i>e</i> it was 8. for þat 1 schal be +rekenet for 10, of þe quych 10, 5 is halfe, as þou knowest wel. Whan +þ<i>o</i>u hast þus ydo, medye þ<i>a</i>t þe quych leues aft<i>er</i> þe +takying<i>e</i> away of þat þat is odde, þe quych leuyng<i>e</i> +schall<i>e</i> be 3; +<span class = "float"> + <span class = "smaller">5</span><br /> +4634.</span> +do away 6 & sette þ<i>er</i>e 3, & þou schalt haue such a nombre +aft<i>er</i> go forth to þe next fig<i>ur</i>e, & medy þat, & +worch forth, for it is lyȝt <a class = "gloss" name = "ynovght" id = +"ynovght" href = "#gloss_ynowght">ynovȝt</a> to þe <a class = "gloss" +name = "certayn" id = "certayn" href = +"#gloss_certayn">c<i>er</i>tayɳ</a>.</p> + +<div class = "verse"> +<p class = "halfline"> +¶ Si v<i>er</i>o s<i>e</i>c<i>un</i>da dat vnu<i>m</i>.</p> +<p>Illa deleta, sc<i>ri</i>bat<i>ur</i> cifra; p<i>ri</i>ori</p> +<p class = "pilcrow"> +¶ Tradendo quinque pro denario mediato;</p> +<p>Nec cifra sc<i>ri</i>batur, nisi dei<i>n</i>de fig<i>ur</i>a +seq<i>u</i>at<i>ur</i>:</p> +<p>Postea p<i>ro</i>cedas reliq<i>ua</i>s mediando figuras</p> +<p>Vt sup<i>ra</i> docui, si sint tibi mille figure.</p> +</div> + +<p><span class = "pagenum">19</span> +<a name = "page19" id = "page19"> </a> +¶ Her<i>e</i> he putt<i>es</i> þe 5 case, þe quych is +<span class = "linenum">leaf 151 <i>a</i>.</span> +*þis: +<span class = "sidenote">If the second figure is one, put a cipher, and +write five over the next figure.</span> +yf þe secunde figur<i>e</i> be of 1, as þis is here 12, þou schalt do +away þat 1 & sett þ<i>ere</i> a cifer. & sett 5 ou<i>er</i> þe +next fig<i>ur</i>e hede afor<i>e</i> toward þe riȝt side, as þou diddyst +afor<i>e</i>; & þat 5 schal be <a class = "gloss" name = "haldel" id += "haldel" href = "#gloss_haluendel">haldel</a> of þat 1, þe quych 1 is +rekent for 10. lo an Ensampull<i>e</i>, 214. medye 4. þ<i>a</i>t +schall<i>e</i> be 2. do away 4 & sett þ<i>ere</i> 2. þe<i>n</i> go +forth to þe next figur<i>e</i>. þe quych is bot 1. do away þat 1. & +sett þ<i>ere</i> a cifer. & set 5 vpon þe figur<i>es</i> hed +afor<i>e</i>, þe quych is nowe 2, & þen þou schalt haue þis +no<i>m</i>b<i>re</i> +<span class = "float"> + <span class = "smaller">5</span><br /> +202,</span> +þen worch forth to þe <a class = "gloss" name = "nex" id = "nex" href = +"#gloss_nex">nex</a> fig<i>ur</i>e. And also it is no <a class = "gloss" +name = "maystery" id = "maystery" href = +"#gloss_maystery">mayst<i>er</i>y</a> yf þ<i>ere</i> come no +figur<i>e</i> after þat on is medyet, þ<i>o</i>u schalt write no 0. ne +nowȝt ellis, but set 5 ou<i>er</i> þe next fig<i>ur</i>e afor<i>e</i> +toward þe ryȝt, as þus 14. +<span class = "sidenote">How to halve fourteen.</span> +medie 4 then leues 2, do away 4 & sett þ<i>ere</i> 2. þen medie 1. +þe q<i>ui</i>ch is rekende for ten, þe halue<i>n</i>del þ<i>ere</i>-of +wel be 5. sett þ<i>a</i>t 5 vpon þe hede of þ<i>a</i>t figur<i>e</i>, þe +quych is now 2, +<span class = "float"> + <span class = "smaller">5</span><br /> +2,</span> +& do away þ<i>a</i>t 1, & þou schalt haue þis nombre yf +þ<i>o</i>u worch wel, vn<i>de</i> v<i>er</i>sus.</p> + +<p class = "headnote"><span class = "headnote"> +How to prove the Mediation.</span></p> + +<div class = "verse"> +<p class = "pilcrow"> +¶ Si mediacio sit b<i>e</i>n<i>e</i> f<i>ac</i>ta p<i>ro</i>bar<i>e</i> +valeb<i>is</i></p> +<p class = "pilcrow"> +¶ Duplando num<i>er</i>u<i>m</i> que<i>m</i> p<i>ri</i>mo +di<i>m</i>ediasti</p> +</div> + +<p><span class = "sidenote">How to prove your mediation.</span> +¶ Her<i>e</i> he telles þe how þou schalt know wheþ<i>er</i> þou hase +wel ydo or no. doubul +<span class = "linenum">leaf 151 <i>b</i>.</span> +*þe nombre þe quych þ<i>o</i>u hase mediet, and yf þ<i>o</i>u haue wel +y-medyt after þe dupleacioɳ, þou schalt haue þe same nombre þat +þ<i>o</i>u haddyst in þe tabull<i>e</i> or þ<i>o</i>u began to medye, as +þus. +<span class = "sidenote">First example.</span> +¶ The furst ensampull<i>e</i> was þis. 4. þe quych <a class = +"terms" name = "imediet" id = "imediet" href = +"#terms_imediet">I-mediet</a> was <a class = "gloss" name = "laft" id = +"laft" href = "#gloss_leue">laft</a> 2, þe whych 2 was write in þe +place þ<i>a</i>t 4 was write afor<i>e</i>. Now doubull<i>e</i> þat 2, +& þ<i>o</i>u schal haue 4, as þ<i>o</i>u hadyst afor<i>e</i>. +<span class = "sidenote">The second.</span> +þe secunde Ensampull<i>e</i> was þis, 245. When þ<i>o</i>u haddyst +mediet all<i>e</i> þis nomb<i>re</i>, yf þou haue wel ydo þou schalt +haue of þ<i>a</i>t mediacioɳ þis nombre, 122<sup>w</sup>. Now +doubull<i>e</i> þis nombre, & begyn in þe lyft side; doubull<i>e</i> +1, þat schal be 2. do away þat 1 & sett þ<i>ere</i> 2. þen +doubull<i>e</i> þ<i>a</i>t oþ<i>er</i> 2 & sett þ<i>ere</i> 4, þen +doubull<i>e</i> þat oþ<i>er</i> 2, & þat wel be 4. þe<i>n</i> doubul +þat merke þat stondes for halue on. & þat schall<i>e</i> be 1. Cast +þat on to 4, & it schall<i>e</i> be 5. do away þat 2 & þat +merke, & sette þ<i>ere</i> 5, & þen þ<i>o</i>u schal haue þis +nombre 245. & þis wos þe same nombur þ<i>a</i>t þ<i>o</i>u haddyst +or þ<i>o</i>u began to medye, as þ<i>o</i>u mayst se yf þou take hede. +<span class = "sidenote">The third example.</span> +The nombre þe quych þou haddist for an Ensampul in þe 3 case of +mediacioɳ to be mediet was þis 241. whan þ<i>o</i>u haddist medied +all<i>e</i> þis nombur truly +<span class = "linenum">leaf 152 <i>a</i>.</span> +*by eu<i>er</i>y figur<i>e</i>, þou schall haue be þ<i>a</i>t mediacioɳ +þis nombur 120<sup>w</sup>. Now dowbul þis nomb<i>ur</i>, & begyn in +þe lyft side, as I tolde þe in þe Craft of duplacioɳ. þus +doubull<i>e</i> þe fig<i>ur</i>e of 1, þat wel be 2. do +<span class = "pagenum">20</span> +<a name = "page20" id = "page20"> </a> +away þat 1 & sett þ<i>ere</i> 2, þen doubul þe next figur<i>e</i> +afore, the quych is 2, & þat wel be 4; do away 2 & set +þ<i>ere</i> 4. þen doubul þe cifer, & þat wel be noȝt, for a 0 is +noȝt. And twyes noȝt is but noȝt. þ<i>ere</i>for<i>e</i> doubul the +merke aboue þe cifers hede, þe quych betokenes þe halue<i>n</i>del of 1, +& þat schal be 1. do away þe cifer & þe merke, & sett +þ<i>ere</i> 1, & þen þ<i>o</i>u schalt haue þis nombur 241. And þis +same nombur þ<i>o</i>u haddyst afore or þ<i>o</i>u began to medy, +<a class = "gloss" name = "and_yf" id = "and_yf" href = "#gloss_and">& +yf</a> þ<i>o</i>u take gode hede. +<span class = "sidenote">The fourth example.</span> +¶ The next ensampul þat had in þe 4 case of mediacioɳ was þis 4678. +Whan þ<i>o</i>u hast truly <a class = "terms" name = "ymedit" id = +"ymedit" href = "#terms_medye">ymedit</a> all<i>e</i> þis nombur fro þe +begynnyng<i>e</i> to þe endyng<i>e</i>, þ<i>o</i>u schalt haue of þe +mediacioɳ þis nombur +<span class = "float"> + <span class = "smaller">5</span><br /> +2334.</span> +Now doubul this nombur & begyn in þe lyft side, & +doubull<i>e</i> 2 þat schal be 4. do away 2 and sette þere 4; þen +doubul<i>e</i> 3, þ<i>a</i>t wol be 6; do away 3 & sett þ<i>ere</i> +6, þen doubul þat oþ<i>er</i> 3, & þat wel be 6; do away 3 & set +þ<i>ere</i> +<span class = "linenum">leaf 152 <i>b</i>.</span> +*6, þen doubul þe 4, þat welle be 8; þen doubul 5. þe quych stondes +ou<i>er</i> þe hed of 4, & þat wol be 10; cast 10 to 8, & +þ<i>a</i>t schal be 18; do away 4 & þat 5, & sett þ<i>ere</i> 8, +& sett that 1, þe quych is an articul of þe Composit þe quych is 18, +ou<i>er</i> þe next figur<i>es</i> hed toward þe lyft side, þe quych is +6. drav þ<i>a</i>t 1 to 6, þe quych 1 in þe dravyng schal be rekente bot +for 1, & þ<i>a</i>t 1 & þ<i>a</i>t 6 togedur wel be 7. do away +þat 6 & þat 1. the quych stondes ou<i>er</i> his hede, & sett +ther 7, & þen þou schalt haue þis nombur 4678. And þis same nombur +þ<i>o</i>u hadyst or þ<i>o</i>u began to medye, as þ<i>o</i>u mayst see +in þe secunde Ensampul þat þou had in þe 4 case of mediacioɳ, þat was +þis: +<span class = "sidenote">The fifth example.</span> +when þ<i>o</i>u had mediet truly all<i>e</i> the nombur, +a p<i>ri</i>ncipio usque ad fine<i>m</i>. þ<i>o</i>u schalt haue of +þat mediacioɳ þis nombur +<span class = "float"> + <span class = "smaller">5</span><br /> +102.</span> +Now doubul 1. þat wel be 2. do away 1 & sett þ<i>ere</i> 2. þen +doubul 0. þ<i>a</i>t will be noȝt. þ<i>ere</i>for<i>e</i> take þe 5, þe +quych stondes ou<i>er</i> þe next figur<i>es</i> hed, & doubul it, +& þat wol be 10. do away þe 0 þat stondes betwene þe two +fig<i>u</i>r<i>i</i>s, & sette þ<i>ere</i> in his stid 1, for +þ<i>a</i>t 1 now schal stonde in þe secunde place, wher<i>e</i> he schal +betoken 10; þen doubul 2, þat wol be 4. do away 2 & sett þere 4. +& +<span class = "linenum">leaf 153 <i>a</i>.</span> +*þou schal haue <a class = "gloss" name = "thus" id = "thus" href = +"#gloss_thus">þus</a> nombur 214. þis is þe same nu<i>m</i>bur þat +þ<i>o</i>u hadyst or þ<i>o</i>u began to medye, as þ<i>o</i>u may see. +And so do eu<i>er</i> mor<i>e</i>, yf þ<i>o</i>u wil knowe wheþ<i>er</i> +þou hase wel ymedyt or no. ¶. doubull<i>e</i> þe nu<i>m</i>bur þat +comes aft<i>er</i> þe mediaciouɳ, & þ<i>o</i>u schal haue þe same +nombur þ<i>a</i>t þ<i>o</i>u hadyst or þ<i>o</i>u began to medye, yf +þ<i>o</i>u haue welle ydo. or els doute þe noȝt, but yf þ<i>o</i>u haue +þe same, þ<i>o</i>u hase faylide in þ<i>i</i> Craft.</p> + + +<p class = "headnote"><span class = "headnote"> +The Craft of Multiplication.</span></p> + +<h5>Sequitur de multiplicatione.</h5> + +<span class = "pagenum">21</span> +<a name = "page21" id = "page21"> </a> + +<p class = "headnote"><span class = "headnote"> +To write down a Multiplication Sum.</span></p> + +<div class = "verse"> +<p><span class = "dropcap">S</span> i tu p<i>er</i> +num<i>er</i>u<i>m</i> num<i>er</i>u<i>m</i> vis +m<i>u</i>ltiplicar<i>e</i></p> +<p>Scribe duas q<i>ua</i>scu<i>nque</i> velis series +nu<i>me</i>ror<i>um</i></p> +<p>Ordo s<i>er</i>vet<i>ur</i> vt vltima m<i>u</i>ltiplicandi</p> +<p>Ponat<i>ur</i> sup<i>er</i> ant<i>er</i>iorem +multiplicant<i>is</i></p> +<p>A leua reliq<i>u</i>e sint scripte m<i>u</i>ltiplicantes.</p> +</div> + +<p><span class = "sidenote">Four things to be known of +Multiplication:</span> +¶ Her<i>e</i> be-gynnes þe Chapt<i>r</i>e of m<i>u</i>ltiplicatioɳ, +in þe quych þou most know 4 thynges. ¶ Ffirst, qwat is +m<i>u</i>ltiplicacioɳ. The secunde, how mony cases may hap in +multiplicacioɳ. The thryde, how mony rewes of figur<i>es</i> þ<i>ere</i> +most be. ¶ The 4. what is þe p<i>ro</i>fet of þis craft. +<span class = "sidenote">the first:</span> +¶ As for þe first, þ<i>o</i>u schal vnd<i>er</i>stonde þat +m<i>u</i>ltiplicacioɳ is a bryngyng<i>e</i> to-ged<i>er</i> of 2 +thyng<i>es</i> in on nombur, þe quych on nombur <a class = "gloss" name += "contynes" id = "contynes" href = +"#gloss_contynes"><i>con</i>tynes</a> so mony tymes on, howe +<span class = "linenum">leaf 153 <i>b</i>.</span> +*mony tymes þ<i>ere</i> ben vnytees in þe nowmb<i>re</i> of þat 2, as +twyes 4 is 8. now her<i>e</i> ben þe 2 nomb<i>er</i>s, of þe quych too +nowmbr<i>e</i>s on is betokened be an adu<i>er</i>be, þe quych is þe +worde twyes, & þis worde thryes, & þis worde four<i>e</i> +<a class = "gloss" name = "sythes" id = "sythes" href = +"#gloss_sythes">sythes</a>,<a class = "tag" name = "tag_craft9" id = +"tag_craft9" href = "#note_craft9">9</a> +& so furth of such other lyke wordes. ¶ And tweyn nombres schal +be tokenyde be a nowne, as þis worde four<i>e</i> showys þes tweyɳ +nombres <a class = "gloss" name = "ybroth" id = "ybroth" href = +"#gloss_ybroth">y-broth</a> in-to on hole nombur, þat is 8, for twyes 4 +is 8, as þ<i>o</i>u wost wel. ¶ And þes nomb<i>re</i> 8 conteynes +as oft tymes 4 as þ<i>ere</i> ben vnites in þ<i>a</i>t other +nomb<i>re</i>, þe quych is 2, for in 2 ben 2 vnites, & so oft tymes +4 ben in 8, as þ<i>o</i>u wottys wel. +<span class = "sidenote">the second:</span> +¶ ffor þe secu<i>n</i>de, þ<i>o</i>u most know þat þ<i>o</i>u most +haue too rewes of figures. +<span class = "sidenote">the third:</span> +¶ As for þe thryde, þ<i>o</i>u most know þ<i>a</i>t 8 man<i>er</i> +of diu<i>er</i>se case may happe in þis craft. +<span class = "sidenote">the fourth.</span> +The p<i>ro</i>fet of þis Craft is to telle when a nomb<i>re</i> is +m<i>u</i>ltiplyed be a noþ<i>er</i>, qwat co<i>m</i>mys +þ<i>ere</i> of. ¶ fforthermor<i>e</i>, as to þe sentence of +our<i>e</i> verse, yf þ<i>o</i>u wel m<i>u</i>ltiply a nombur be +a-noþ<i>er</i> nomb<i>ur</i>, þou schalt write +<span class = "linenum">leaf 154 <i>a</i>.</span> +*a rewe of figures of what nomb<i>ur</i>s so eu<i>er</i> þ<i>o</i>u +welt, +<span class = "sidenote">The multiplicand.</span> +& þat schal be called Num<i>erus</i> m<i>u</i>ltiplicand<i>us</i>, +Anglice, þe nomb<i>ur</i> the quych to be m<i>u</i>ltiplied. þen +þ<i>o</i>u schalt write a-nother rewe of figur<i>e</i>s, by þe quych +þ<i>o</i>u schalt m<i>u</i>ltiplie the nombre þat is to be +m<i>u</i>ltiplied, of þe quych nomb<i>ur</i> þe furst fig<i>ur</i>e +schal be write vnd<i>er</i> þe last figur<i>e</i> of þe nomb<i>ur</i>, +þe quych is to be m<i>u</i>ltiplied. +<span class = "sidenote">How to set down the sum.</span> +And so write forthe toward þe lyft side, as her<i>e</i> you may se, +<span class = "float box"> + 67324<br /> +1234</span> +And þis on<i>e</i> nomb<i>ur</i> schall<i>e</i> be called nu<i>meru</i>s +m<i>u</i>ltiplicans. An<i>gli</i>ce, þe nomb<i>ur</i> +m<i>u</i>ltipliyng<i>e</i>, for he schall<i>e</i> m<i>u</i>ltiply þe +hyer nounb<i>ur</i>, as þus on<i>e</i> tyme 6. And so forth, as I schal +telle the aft<i>er</i>warde. And þou schal begyn in þe lyft side. +<span class = "sidenote">Two sorts of Multiplication: mentally,</span> +¶ ffor-þ<i>ere</i>-more þou schalt vndurstonde þat þ<i>ere</i> is +two man<i>ur</i>s of m<i>u</i>ltiplicacioɳ; one ys of þe +wyrchyng<i>e</i> of þe boke only in þe mynde of a mon. fyrst he +<span class = "pagenum">22</span> +<a name = "page22" id = "page22"> </a> +teches of þe fyrst man<i>er</i> of duplacioɳ, þe quych is be +wyrchyng<i>e</i> of tabuls. +<span class = "sidenote">and on paper.</span> +Aft<i>er</i>warde he wol teche on þe secunde man<i>er</i>. vn<i>de</i> +v<i>er</i>sus.</p> + +<p class = "headnote"><span class = "headnote"> +To multiply one Digit by another.</span></p> + +<div class = "verse"> +<p>In digitu<i>m</i> cures digitu<i>m</i> si duc<i>er</i>e +ma<i>i</i>or</p> +<span class = "linenum">leaf 154 <i>b</i>.</span> +<p class = "pilcrow"> +* P<i>er</i> qua<i>n</i>tu<i>m</i> distat a denis respice debes</p> +<p class = "pilcrow"> +¶ Namq<i>ue</i> suo decuplo totiens deler<i>e</i> +mi<i>n</i>ore<i>m</i></p> +<p>Sitq<i>ue</i> tibi nu<i>meru</i>s veniens exinde patebit.</p> +</div> + +<p><span class = "sidenote">How to multiply two digits.</span> +¶ Her<i>e</i> he teches a rewle, how þ<i>o</i>u schalt fynde þe +nounb<i>r</i>e þat comes by þe m<i>u</i>ltiplicacioɳ of a digit be +anoþ<i>er</i>. loke how mony [vny]tes ben. bytwene þe mor<i>e</i> digit +and 10. And reken ten for on vnite. +<span class = "sidenote">Subtract the greater from ten;</span> +And so oft do away þe lasse nounbre out of his owne <a class = "terms" +name = "decuple" id = "decuple" href = "#terms_decuple">decuple</a>, þat +is to say, fro þat nounb<i>r</i>e þat is ten tymes so mych <ins class = +"correction" title = "error for ‘as’?">is</ins> þe nounb<i>re</i> +þ<i>a</i>t comes of þe m<i>u</i>ltiplicacioɳ. As yf þ<i>o</i>u wol +m<i>u</i>ltiply 2 be 4. loke how mony vnitees ben by-twene þe quych is +þe mor<i>e</i> nounb<i>re</i>, & be-twene ten. C<i>er</i>ten +þ<i>ere</i> wel be vj vnitees by-twene 4 & ten. yf þ<i>o</i>u reken +þ<i>ere</i> w<i>i</i>t<i>h</i> þe ten þe vnite, as þou may se. +<span class = "sidenote">take the less so many times from ten times +itself.</span> +so mony tymes take 2. out of his decuple, þe quych is 20. for 20 is þe +decuple of 2, 10 is þe decuple of 1, 30 is þe decuple of 3, 40 is þe +decuple of 4, And þe oþ<i>er</i> digetes til þ<i>o</i>u come to ten; +& whan þ<i>o</i>u +<span class = "sidenote">Example.</span> +hast y-take so mony tymes 2 out of twenty, þe quych is sex tymes, +þ<i>o</i>u schal leue 8 as þ<i>o</i>u wost wel, for 6 times 2 is twelue. +take [1]2 out of twenty, & þ<i>ere</i> schal leue 8. bot yf bothe þe +digett<i>es</i> +<span class = "linenum">leaf 155 <i>a</i>.</span> +*ben <a class = "gloss" name = "ylyech" id = "ylyech" href = +"#gloss_ylyech">y-lyech</a> mych as her<i>e</i>. 222 or too tymes +twenty, þen it is <a class = "gloss" name = "fors" id = "fors" href = +"#gloss_fors">no fors</a> quych of hem tweyn þ<i>o</i>u take out of here +decuple. <a class = "gloss" name = "als" id = "als" href = +"#gloss_als">als</a> mony +<span class = "sidenote">Better use this table, though.</span> +tymes as þ<i>a</i>t is fro 10. but neu<i>er</i>-þe-lesse, yf þ<i>o</i>u +haue <a class = "gloss" name = "hast" id = "hast" href = +"#gloss_hast">hast</a> to worch, þ<i>o</i>u schalt haue her<i>e</i> a +tabul of figures, wher<i>e</i>-by þ<i>o</i>u schalt se a-nonɳ ryght what +is þe nounbre þ<i>a</i>t comes of þe multiplicacioɳ of 2 digittes. þus +þ<i>o</i>u schalt worch in þis fig<i>ur</i>e.</p> + +<table class = "grid right" summary = "example"> +<tr> +<td class = "grid"> 1</td> +<td colspan = "9"></td> +</tr> +<tr> +<td class = "grid">2</td> +<td class = "grid">4</td> +<td colspan = "8"></td> +</tr> +<tr> +<td class = "grid">3</td> +<td class = "grid">6</td> +<td class = "grid">9</td> +<td colspan = "7"> </td> +</tr> +<tr> +<td class = "grid">4</td> +<td class = "grid">8</td> +<td class = "grid">12</td> +<td class = "grid">16</td> +<td colspan = "6"> </td> +</tr> +<tr> +<td class = "grid">5</td> +<td class = "grid">10</td> +<td class = "grid">15</td> +<td class = "grid">20</td> +<td class = "grid">25</td> +<td colspan = "5"> </td> +</tr> +<tr> +<td class = "grid">6</td> +<td class = "grid">12</td> +<td class = "grid">18</td> +<td class = "grid">24</td> +<td class = "grid">30</td> +<td class = "grid">36</td> +<td colspan = "4"> </td> +</tr> +<tr> +<td class = "grid">7</td> +<td class = "grid">14</td> +<td class = "grid">21</td> +<td class = "grid">28</td> +<td class = "grid">35</td> +<td class = "grid">42</td> +<td class = "grid">49</td> +<td colspan = "3"> </td> +</tr> +<tr> +<td class = "grid">8</td> +<td class = "grid">16</td> +<td class = "grid">24</td> +<td class = "grid">32</td> +<td class = "grid">40</td> +<td class = "grid">48</td> +<td class = "grid">56</td> +<td class = "grid">64</td> +<td colspan = "2"> </td> +</tr> +<tr> +<td class = "grid">9</td> +<td class = "grid">18</td> +<td class = "grid">27</td> +<td class = "grid">36</td> +<td class = "grid">45</td> +<td class = "grid">54</td> +<td class = "grid">63</td> +<td class = "grid">72</td> +<td class = "grid">81</td> +<td> </td> +</tr> +<tr> +<td class = "grid">1</td> +<td class = "grid">2</td> +<td class = "grid">3</td> +<td class = "grid">4</td> +<td class = "grid">5</td> +<td class = "grid">6</td> +<td class = "grid">7</td> +<td class = "grid">8</td> +<td class = "grid">9</td> +<td class = "grid"> </td> +</tr> +</table> + +<p><span class = "sidenote">How to use it.</span> +yf þe fig<i>ur</i>e, þe quych schall<i>e</i> be m<i>u</i>ltiplied, be +euen<i>e</i> as mych as þe diget be, þe quych þat oþ<i>er</i> +figur<i>e</i> schal be m<i>u</i>ltiplied, as two tymes twayɳ, or thre +tymes 3. or sych other. +<span class = "sidenote">The way to use the Multiplication table.</span> +loke qwer<i>e</i> þat fig<i>ur</i>e sittes in +<span class = "pagenum">23</span> +<a name = "page23" id = "page23"> </a> +þe lyft side of þe t<i>ri</i>angle, & loke qwer<i>e</i> þe diget +sittes in þe neþ<i>er</i> most rewe of þe triangle. & go fro hym +vpwarde in þe same rewe, þe quych rewe gose vpwarde til þ<i>o</i>u come +agaynes þe oþ<i>er</i> digette þat sittes in þe lyft side of þe +t<i>ri</i>angle. And þat nounbre, þe quych þou +<span class = "linenum">leaf 155 <i>b</i>.</span> +fyn*des þ<i>ere</i> is þe nounbre þat comes of the m<i>u</i>ltiplicacioɳ +of þe 2 digittes, as yf þou wold wete qwat is 2 tymes 2. loke +quer<i>e</i> sittes 2 in þe lyft side i<i>n</i> þe first rewe, he sittes +next 1 in þe lyft side al on hye, as þ<i>o</i>u may se; þe[<i>n</i>] +loke qwer<i>e</i> sittes 2 in þe lowyst rewe of þe t<i>ri</i>angle, +& go fro hym vpwarde in þe same rewe tyll<i>e</i> þou come <a class += "gloss" name = "aghenenes" id = "aghenenes" href = +"#gloss_aghenenes">a-ȝenenes</a> 2 in þe hyer place, & þer þou +schalt fynd ywrite 4, & þat is þe nounb<i>r</i>e þat comes of þe +multiplicacioɳ of two tymes tweyn is 4, as þow wotest well<i>e</i>. yf +þe diget. the quych is m<i>u</i>ltiplied, be mor<i>e</i> þan þe +oþ<i>er</i>, þou schalt loke qwer<i>e</i> þe mor<i>e</i> diget sittes in +þe lowest rewe of þe t<i>ri</i>angle, & go vpwarde in þe same rewe +tyl<a class = "tag" name = "tag_craft10" id = "tag_craft10" href = +"#note_craft10">10</a> +þ<i>o</i>u come <a class = "gloss" name = "anendes" id = "anendes" href += "#gloss_anendes">a-nendes</a> þe lasse diget in the lyft side. And +þ<i>ere</i> þ<i>o</i>u schalt fynde þe no<i>m</i>b<i>r</i>e þat comes of +þe m<i>u</i>ltiplicacioɳ; but þ<i>o</i>u schalt vnd<i>er</i>stonde þat +þis rewle, þe quych is in þis v<i>er</i>se. ¶ In digitu<i>m</i> +cures, &c., noþ<i>er</i> þis t<i>ri</i>angle schall<i>e</i> not +s<i>er</i>ue, bot to fynde þe nounbres þ<i>a</i>t comes of the +m<i>u</i>ltiplicacioɳ þat comes of 2 articuls or <i>com</i>posites, þe +nedes no craft but yf þou wolt m<i>u</i>ltiply in þi mynde. And +<span class = "linenum">leaf 156 <i>a</i>.</span> +*þere-to þou schalt haue a craft aft<i>er</i>warde, for þou schall wyrch +w<i>i</i>t<i>h</i> digettes in þe tables, as þou schalt know +aft<i>er</i>warde. v<i>er</i>sus.</p> + +<p class = "headnote"><span class = "headnote"> +To multiply one Composite by another.</span></p> + +<div class = "verse"> +<p class = "pilcrow"> +¶ Postea p<i>ro</i>cedas postrema<i>m</i> +m<i>u</i>ltiplica<i>n</i>do</p> +<p>[Recte multiplicans per cu<i>n</i>ctas i<i>n</i>feriores]</p> +<p>Condic<i>i</i>onem tamen t<i>a</i>li q<i>uod</i> +m<i>u</i>ltiplicant<i>es</i></p> +<p>Scribas in capite quicq<i>ui</i>d p<i>ro</i>cesserit inde</p> +<p>Sed postq<i>uam</i> fuit hec m<i>u</i>ltiplicate fig<i>ur</i>e</p> +<p>Anteriorent<i>ur</i> serei m<i>u</i>ltiplica<i>n</i>t<i>is</i></p> +<p>Et sic m<i>u</i>ltiplica velut isti m<i>u</i>ltiplicasti</p> +<p>Qui sequit<i>ur</i> nu<i>mer</i>u<i>m</i> sc<i>ri</i>ptu<i>m</i> +quiscu<i>n</i>q<i>ue</i> figur<i>is</i>.</p> +</div> + +<p><span class = "sidenote">How to multiply one number by +another.</span> +¶ Her<i>e</i> he teches how þ<i>o</i>u schalt wyrch in þis craft. +þou schalt m<i>ul</i>tiplye þe last figur<i>e</i> of þe nombre, and quen +þ<i>o</i>u hast so ydo þou schalt draw all<i>e</i> þe figures of þe +neþ<i>er</i> nounbre mor<i>e</i> <a class = "gloss" name = "taward" id = +"taward" href = "#gloss_taward">taward</a> þe ryȝt side, so qwe<i>n</i> +þ<i>o</i>u hast m<i>u</i>ltiplyed þe last figur<i>e</i> of þe heyer +nounbre by all<i>e</i> þe neþ<i>er</i> figures. +<span class = "sidenote">Multiply the ‘last’ figure of the higher by the +‘first’ of the lower number.</span> +And sette þe nounbir þat comes þer-of ou<i>er</i> þe last figur<i>e</i> +of þe neþ<i>er</i> nounb<i>re</i>, & þen þou schalt sette al þe +oþ<i>er</i> fig<i>ur</i>es of þe neþ<i>er</i> nounb<i>re</i> mor<i>e</i> +ner<i>e</i> to þe ryȝt side. ¶ And whan þou hast m<i>u</i>ltiplied +þat figur<i>e</i> þat schal be m<i>u</i>ltiplied þe next aft<i>er</i> +<span class = "pagenum">24</span> +<a name = "page24" id = "page24"> </a> +hym by al þe neþ<i>er</i> figures. And worch as þou dyddyst afor<i>e</i> +til +<span class = "linenum">leaf 156 <i>b</i>.</span> +*þou come to þe ende. And þou schalt vnd<i>er</i>stonde þat eu<i>er</i>y +figur<i>e</i> of þe hier nounb<i>re</i> schal be m<i>u</i>ltiplied be +all<i>e</i> þe figur<i>e</i>s of the neþ<i>er</i> nounbre, yf þe hier +nounb<i>re</i> be any figur<i>e</i> þen on<i>e</i>. +<span class = "sidenote">Set the answer over the first of the +lower:</span> +lo an Ensampul her<i>e</i> folowyng<i>e</i>. +<span class = "float box"> + 2465.<br /> +232</span> +þou schalt begyne to m<i>u</i>ltiplye in þe lyft side. M<i>u</i>ltiply 2 +be 2, and twyes 2 is 4. set 4 +<span class = "sidenote">then multiply the second of the lower, and so +on.</span> +ou<i>er</i> þe hed of þ<i>a</i>t 2, þen m<i>u</i>ltiplie þe same hier 2 +by 3 of þe nether nounbre, as thryes 2 þat schal be 6. set 6 ou<i>er</i> +þe hed of 3, þan m<i>u</i>ltiplie þe same hier 2 by þat 2 þe quych +stondes vnd<i>er</i> hym, þ<i>a</i>t wol be 4; do away þe hier 2 & +sette þ<i>ere</i> 4. +<span class = "sidenote">Then antery the lower number:</span> +¶ Now þ<i>o</i>u most <a class = "terms" name = "antery" id = +"antery" href = "#terms_antery">antery</a> þe nether nounbre, þat is to +say, þ<i>o</i>u most sett þe neþ<i>er</i> nounbre more towarde þe ryȝt +side, as þus. Take þe neþ<i>er</i> 2 toward þe ryȝt side, & sette it +eueɳ vnd<i>er</i> þe 4 of þe hyer nounb<i>r</i>e, & ant<i>er</i>y +all<i>e</i> þe figures þat comes aft<i>er</i> þat 2, as þus; sette 2 +vnd<i>er</i> þe 4. þen sett þe figur<i>e</i> of 3 þ<i>ere</i> þat þe +figure of 2 stode, þe quych is now vndur þ<i>a</i>t 4 in þe hier +nounbre; þen sett þe oþer figur<i>e</i> of 2, þe quych is þe last +fig<i>ur</i>e toward þe lyft side of þe neþ<i>er</i> nomb<i>er</i> +þ<i>ere</i> þe figur<i>e</i> of 3 stode. +<span class = "sidenote">as thus.</span> +þen þ<i>o</i>u schalt haue such a nombre. +<span class = "float box"> +464465<br /> + 232</span> +<span class = "linenum">leaf 157 <i>a</i>.</span> +* ¶ Now m<i>u</i>ltiply 4, þe quych comes next aft<i>er</i> 6, by þe +last 2 of þe neþ<i>er</i> nounbur toward þe lyft side. as 2 tymes 4, þat +wel be 8. sette þat 8 ou<i>er</i> þe figure the quych stondes +ou<i>er</i> þe hede of þat 2, þe quych is þe last figur<i>e</i> of þe +neþ<i>er</i> nounbre; þan multiplie þat same 4 by 3, þat comes in þe +neþ<i>er</i> rewe, þat wol be 12. sette þe digit of þe composyt +ou<i>er</i> þe figure þe quych stondes ou<i>er</i> þe hed of þat 3, +& sette þe articule of þis co<i>m</i>posit ou<i>er</i> al þe figures +þat stondes ou<i>er</i> þe neþ<i>er</i> 2 hede. +<span class = "sidenote">Now multiply by the last but one of the +higher:</span> +þen m<i>u</i>ltiplie þe same 4 by þe 2 in þe ryȝt side in þe +neþ<i>er</i> nounbur, þat wol be 8. do away 4. & sette þ<i>ere</i> +8. Eu<i>er</i> mor<i>e</i> qwen þ<i>o</i>u m<i>u</i>ltiplies þe hier +figur<i>e</i> by þat figur<i>e</i> þe quych stondes vnd<i>er</i> hym, +þou schalt do away þat hier figur<i>e</i>, & sett þer þat nounbre þe +quych comes of m<i>u</i>ltiplicacioɳ of ylke digittes. +<span class = "sidenote">as thus.</span> +Whan þou hast done as I haue byde þe, þ<i>o</i>u schalt haue suych an +ord<i>er</i> of figur<i>e</i> as is her<i>e</i>, +<span class = "float box"> + <span class = "smaller">1</span><br /> + <span class = "smaller">82</span><br /> +4648[65]<br /> + 232.</span> +þen take and ant<i>er</i>y þi neþ<i>er</i> figures. And sett þe fyrst +fig<i>ur</i>e of þe neþ<i>er</i> figures<a class = "tag" name = +"tag_craft11" id = "tag_craft11" href = "#note_craft11">11</a> +vndre be figur<i>e</i> of 6. ¶ And draw al þe oþ<i>er</i> figures +of þe same rewe to hym-warde, +<span class = "linenum">leaf 157 <i>b</i>.</span> +*as þ<i>o</i>u diddyst afore. þen m<i>u</i>ltiplye 6 be 2, & sett +þat þe quych comes ou<i>er</i> þ<i>ere</i>-of ou<i>er</i> al þe +oþ<i>er</i> figures hedes þat stondes ou<i>er</i> þat 2. þen +m<i>u</i>ltiply 6 be 3, & sett all<i>e</i> þat comes þ<i>ere</i>-of +vpon all<i>e</i> þe figur<i>e</i>s hedes þat standes ou<i>er</i> þat 3; +þa<i>n</i> m<i>u</i>ltiplye 6 be 2, þe quych +<span class = "pagenum">25</span> +<a name = "page25" id = "page25"> </a> +stondes vnd<i>er</i> þat 6, þen do away 6 & write þ<i>ere</i> þe +digitt of þe composit þat schal come þ<i>ere</i>of, & sette þe +articull ou<i>er</i> all<i>e</i> þe figures þat stondes ou<i>er</i> þe +hede of þat 3 as her<i>e</i>, +<span class = "float box"> + 11<br /> + 121<br /> + 828<br /> +464825<br /> + 232</span> +<span class = "sidenote">Antery the figures again, and multiply by +five:</span> +þen ant<i>er</i>y þi figures as þou diddyst afor<i>e</i>, and +m<i>u</i>ltipli 5 be 2, þat wol be 10; sett þe 0 ou<i>er</i> all þe +figures þ<i>a</i>t stonden ou<i>er</i> þat 2, & sett þ<i>a</i>t 1. +ou<i>er</i> the next figures hedes, all<i>e</i> on hye towarde þe lyft +side. þen m<i>u</i>ltiplye 5 be 3. þat wol be 15, write 5 ou<i>er</i> þe +figures hedes þat stonden ou<i>er</i> þ<i>a</i>t 3, & sett þat 1 +ou<i>er</i> þe next figur<i>e</i>s hedes toward þe lyft side. þen +m<i>u</i>ltiplye 5 be 2, þat wol be 10. do away þat 5 & sett +þ<i>ere</i> a 0, & sett þat 1 ou<i>er</i> þe figures hedes þat +stonden ou<i>er</i> 3. And þen +<span class = "linenum">leaf 158 <i>a</i>.</span> +þou schalt haue such a nounbre as here stondes aftur.* +<span class = "float box"> + 11<br /> + 1101<br /> + 1215<br /> + 82820<br /> +4648<br /> + 232</span> +¶ Now draw all<i>e</i> þese figures downe toged<i>er</i> as þus, 6.8.1. +& 1 draw to-gedur; þat wolle be 16, do away all<i>e</i> þese figures +saue 6. lat hym stonde, for <a class = "gloss" name = "thow" id = "thow" +href = "#gloss_thow">þow</a> þ<i>o</i>u take hym away þou most write þer +þe same aȝene. þ<i>ere</i>for<i>e</i> late hym stonde, & sett 1 +ou<i>er</i> þe figur<i>e</i> hede of 4 toward þe lyft side; þen draw on +to 4, þat woll<i>e</i> be 5. +<span class = "sidenote">Then add all the figures above the line:</span> +do away þat 4 & þat 1, & sette þ<i>ere</i> 5. þen draw 4221 +& 1 toged<i>ur</i>, þat wol be 10. do away all<i>e</i> þat, & +write þere þat 4 & þat 0, & sett þat 1 ou<i>er</i> þe next +figur<i>es</i> hede toward þe lyft side, þe quych is 6. þen draw þat 6 +& þat 1 togedur, & þat wolle be 7; do away 6 & sett +þ<i>ere</i> 7, þen draw 8810 & 1, & þat wel be 18; do away +all<i>e</i> þe figures þ<i>a</i>t stondes ou<i>er</i> þe hede of þat 8, +& lette 8 stonde stil, & write þat 1 ou<i>er</i> þe next +fig<i>u</i>r<i>is</i> hede, þe quych is a 0. þen do away þat 0, & +sett þ<i>ere</i> 1, þe quych stondes ou<i>er</i> þe 0. hede. þen draw 2, +5, & 1 toged<i>ur</i>, þat woll<i>e</i> be 8. þen do away +all<i>e</i> þat, & write þ<i>ere</i> 8. +<span class = "sidenote">and you will have the answer.</span> +¶ And þen þou schalt haue þis nounbre, 571880.</p> + +<p class = "headnote"><span class = "headnote"> +The Cases of this Craft.</span></p> + +<span class = "linenum">leaf 158 <i>b</i>.</span> +<div class = "verse"> +<p class = "pilcrow plus"> +* ¶ S<i>ed</i> cu<i>m</i> m<i>u</i>ltiplicabis, p<i>ri</i>mo sic +e<i>st</i> op<i>er</i>andu<i>m</i>,</p> +<p>Si dabit articulu<i>m</i> tibi m<i>u</i>ltiplicacio solu<i>m</i>;</p> +<p>P<i>ro</i>posita cifra su<i>m</i>ma<i>m</i> t<i>ra</i>nsferre +meme<i>n</i>to.</p> +</div> + +<p><span class = "sidenote">What to do if the first multiplication +results in an article.</span> +¶ Her<i>e</i> he puttes þe fyrst case of þis craft, þe quych is +þis: yf þ<i>ere</i> come an articulle of þe m<i>u</i>ltiplicacioɳ ysette +befor<i>e</i> the articull<i>e</i> in þe lyft side as þus +<span class = "float box"> + 51<br /> +23.</span> +multiplye 5 by 2, þat wol be 10; sette ou<i>er</i> þe hede of þat 2 a 0, +& sett þat on, þat is þe articul, in þe lyft side, þat is next hym, +þen þ<i>o</i>u schalt haue þis nounbre +<span class = "float box"> +1051.<br /> + 23</span> +¶ And þen worch forth as þou diddist afore. And þ<i>o</i>u schalt +vnd<i>er</i>stonde þat þ<i>o</i>u schalt write no 0. but whan þat place +where þou schal write þat 0 has no figure afore hy<i>m</i> noþ<i>er</i> +aft<i>er</i>. v<i>er</i>sus.</p> + +<span class = "pagenum">26</span> +<a name = "page26" id = "page26"> </a> +<div class = "verse"> +<p class = "pilcrow"> +¶ Si aut<i>em</i> digitus excreu<i>er</i>it +articul<i>us</i>q<i>ue</i>.</p> +<p>Articul<i>us</i><a class = "tag" name = "tag_craft12" id = +"tag_craft12" href = "#note_craft12">12</a> +sup<i>ra</i>p<i>osit</i>o digito salit vltra.</p> +</div> + +<p><span class = "sidenote">What to do if the result is a composite +number.</span> +¶ Her<i>e</i> is þe secunde case, þe quych is þis: yf hit happe þat +þ<i>ere</i> come a composyt, þou schalt write þe digitte ou<i>er</i> þe +hede of þe neþ<i>er</i> figur<i>e</i> by þe quych þ<i>o</i>u multipliest +þe hier figure; and sett þe articull<i>e</i> next hym toward þe lyft +side, as þou diddyst afore, as þ<i>us</i> +<span class = "float box"> + 83.<br /> +83</span> +Multiply 8 by 8, þat wol be 64. Write þe 4 ou<i>er</i> 8, þat is to say, +ou<i>er</i> þe hede of þe neþ<i>er</i> 8; & set 6, þe quych +<span class = "linenum">leaf 159 <i>a</i>.</span> +*is an articul, next aft<i>er</i>. +And þen þou schalt haue such a nounb<i>r</i>e as is her<i>e</i>, +<span class = "float box"> +6483<a class = "tag" name = "tag_craft13" id = "tag_craft13" href = +"#note_craft13">13</a>,<br /> + 83</span> +And þen worch forth.</p> + +<div class = "verse"> +<p class = "pilcrow"> +¶ Si digitus t<i>amen</i> ponas ip<i>su</i>m sup<i>er</i> +ip<i>s</i>am.</p> +</div> + +<p><span class = "sidenote">What if it be a digit.</span> +¶ Her<i>e</i> is þe thryde case, þe quych is þis: yf hit happe þat of þi +m<i>u</i>ltiplicaciouɳ come a digit, þ<i>o</i>u schalt write þe digit +ou<i>er</i> þe hede of þe neþ<i>er</i> figur<i>e</i>, by the quych þou +m<i>u</i>ltipliest þe hier<i>e</i> figur<i>e</i>, for þis nedes no +Ensampul.</p> + +<div class = "verse"> +<p class = "pilcrow"> +¶ Subdita m<i>u</i>ltiplica non hanc que [incidit] illi</p> +<p>Delet ea<i>m</i> penit<i>us</i> scribens quod p<i>ro</i>uenit +inde.</p> +</div> + +<p><span class = "sidenote">The fourth case of the craft.</span> +¶ Her<i>e</i> is þe 4 case, þe quych is: yf hit be happe þat þe +neþ<i>er</i> figur<i>e</i> schal multiplye þat figur<i>e</i>, þe quych +stondes ou<i>er</i> þat figures hede, þou schal do away þe hier +figur<i>e</i> & sett þ<i>er</i>e þat þ<i>a</i>t comys of þ<i>a</i>t +m<i>u</i>ltiplicacioɳ. As yf þ<i>er</i>e come of þat +m<i>u</i>ltiplicacioɳ an articuls þou schalt write þere þe hier +figur<i>e</i> stode a 0. ¶ And write þe articuls in þe lyft side, +yf þat hit be a digit write þ<i>er</i>e a digit. yf þat h<i>i</i>t be a +composit, write þe digit of þe composit. And þe articul in þe lyft side. +al þis is lyȝt y-nowȝt, þ<i>er</i>e-for<i>e</i> þer nedes no +Ensampul.</p> + +<div class = "verse"> +<p class = "pilcrow"> +¶ S<i>ed</i> si m<i>u</i>ltiplicat alia<i>m</i> ponas sup<i>er</i> +ip<i>s</i>am</p> +<p>Adiu<i>n</i>ges num<i>er</i>u<i>m</i> que<i>m</i> p<i>re</i>bet +duct<i>us</i> ear<i>um</i>.</p> +</div> + +<span class = "linenum">leaf 159 <i>b</i>.</span> + +<p><span class = "sidenote">The fifth case of the craft.</span> +¶ Her<i>e</i> is þe 5 case, þe quych is þis: yf *þe neþ<i>er</i> +figur<i>e</i> schul m<i>u</i>ltiplie þe hier, and þat hier figur<i>e</i> +is not <a class = "gloss" name = "recte1" id = "recte1" href = +"#gloss_recte">recte</a> ou<i>er</i> his hede. And þat neþ<i>er</i> +figur<i>e</i> hase oþ<i>er</i> figures, or on figure ou<i>er</i> his +hede by m<i>u</i>ltiplicacioɳ, þat hase be afor<i>e</i>, þou schalt +write þat nounbre, þe quych comes of þat, ou<i>er</i> all<i>e</i> þe +ylke figures hedes, as þus here: +<span class = "float box"> + 236<br /> +234</span> +Multiply 2 by 2, þat wol be 4; set 4 ou<i>er</i> þe hede of þat 2. þen<a +class = "tag" name = "tag_craft14" id = "tag_craft14" href = +"#note_craft14">14</a> +m<i>u</i>ltiplies þe hier 2 by þe neþ<i>er</i> 3, þat wol be 6. set +ou<i>er</i> his hede 6, multiplie þe hier 2 by þe neþ<i>er</i> 4, þat +wol be 8. do away þe hier 2, þe quych stondes ou<i>er</i> þe hede of þe +figur<i>e</i> of 4, +<span class = "pagenum">27</span> +<a name = "page27" id = "page27"> </a> +and set þ<i>er</i>e 8. And þou schalt haue þis nounb<i>re</i> here +<span class = "float box"> +46836<br /> +234</span> +And antery þi figur<i>e</i>s, þat is to say, set þi neþ<i>er</i> 4 +vnd<i>er</i> þe hier 3, and set þi 2 other figures ner<i>e</i> hym, so +þat þe neþ<i>er</i> 2 stonde vnd<i>ur</i> þe hier 6, þe quych 6 stondes +in þe lyft side. And þat 3 þat stondes vndur 8, as þus aftur ȝe +may se, +<span class = "float box"> +46836<br /> + 234</span> +Now worch forthermor<i>e</i>, And m<i>u</i>ltiplye þat hier 3 by 2, þat +wol be 6, set þ<i>a</i>t 6 þe quych stondes ou<i>er</i> þe hede of þat +2, And þen worch as I taȝt þe afore.</p> + +<span class = "linenum">leaf 160 <i>a</i>.</span> +<div class = "verse"> +<p class = "pilcrow plus"> +* ¶ Si sup<i>ra</i>posita cifra debet m<i>u</i>ltiplicar<i>e</i></p> +<p>Prorsus ea<i>m</i> deles & ibi scribi cifra debet.</p> +</div> + +<p><span class = "sidenote">The sixth case of the craft.</span> +¶ Her<i>e</i> is þe 6 case, þe quych is þis: yf hit happe þat þe +figur<i>e</i> by þe quych þou schal m<i>u</i>ltiplye þe hier +figur<i>e</i>, þe quych stondes ryght ou<i>er</i> hym by a 0, þou schalt +do away þat figur<i>e</i>, þe quych ou<i>er</i> þat cifre hede. +¶ And write þ<i>ere</i> þat nounbre þat comes of þe +m<i>u</i>ltiplicacioɳ as þus, 23. do away 2 and sett þ<i>er</i>e a 0. +vn<i>de</i> v<i>er</i>sus.</p> + +<div class = "verse"> +<p class = "pilcrow"> +¶ Si cifra m<i>u</i>ltiplicat alia<i>m</i> posita<i>m</i> sup<i>er</i> +ip<i>s</i>am</p> +<p>Sitq<i>ue</i> locus sup<i>ra</i> vacu<i>us</i> sup<i>er</i> hanc +cifra<i>m</i> fiet.</p> +</div> + +<p><span class = "sidenote">The seventh case of the craft.</span> +¶ Her<i>e</i> is þe 7 case, þe quych is þis: yf a 0 schal +m<i>u</i>ltiply a figur<i>e</i>, þe quych stondes not <a class = "gloss" +name = "recte" id = "recte" href = "#gloss_recte">recte</a> ou<i>er</i> +hym, And ou<i>er</i> þat 0 stonde no thyng, þou schalt write ou<i>er</i> +þat 0 anoþ<i>er</i> 0 as þus: +<span class = "float box"> + 24<br /> +03</span> +multiplye 2 be a 0, it wol be nothyng<i>e</i>. write þere a 0 +ou<i>er</i> þe hede of þe neþ<i>er</i> 0, And þen worch forth til þou +come to þe ende.</p> + + +<div class = "verse"> +<p class = "pilcrow"> +¶ Si sup<i>ra</i><a class = "tag" name = "tag_craft15" id = +"tag_craft15" href = "#note_craft15">15</a> fuerit cifra sem<i>per</i> +e<i>st</i> p<i>re</i>t<i>er</i>eunda.</p> +</div> + +<p><span class = "sidenote">The eighth case of the craft.</span> +¶ Her<i>e</i> is þe 8 case, þe quych is þis: yf þ<i>ere</i> be a 0 +or mony cifers in þe hier rewe, þ<i>o</i>u schalt not m<i>u</i>ltiplie +hem, bot let hem stonde. And antery þe figures beneþe to þe next +figur<i>e</i> sygnificatyf as þus: +<span class = "float box"> +00032.<br /> +22</span> +Ou<i>er</i>-lepe all<i>e</i> þese cifers & sett þat +<span class = "linenum">leaf 160 <i>b</i>.</span> +*neþ<i>er</i> 2 þat stondes toward þe ryght side, and sett hym +vnd<i>ur</i> þe 3, and sett þe oþ<i>er</i> nether 2 nere hym, so þat he +stonde vnd<i>ur</i> þe thrydde 0, þe quych stondes next 3. And þan +worch. vnd<i>e</i> v<i>er</i>sus.</p> + +<div class = "verse"> +<p class = "pilcrow"> +¶ Si dubites, an sit b<i>e</i>n<i>e</i> m<i>u</i>ltiplicac<i>i</i>o +facta,</p> +<p>Diuide totalem nu<i>mer</i>u<i>m</i> p<i>er</i> +multiplicante<i>m</i>.</p> +</div> + +<p><span class = "sidenote">How to prove the multiplication.</span> +¶ Her<i>e</i> he teches how þou schalt know wheþ<i>er</i> þou hase +wel I-do or no. And he says þat þou schalt deuide all<i>e</i> þe +nounb<i>r</i>e þat comes of þe m<i>u</i>ltiplicacioɳ by þe neþ<i>er</i> +figures. And þen þou schalt haue þe same nounbur þat þ<i>o</i>u hadyst +in þe begynnyng<i>e</i>. but ȝet þou hast not þe craft of dyuisioɳ, but +þ<i>o</i>u schalt haue hit aft<i>er</i>warde.</p> + +<span class = "pagenum">28</span> +<a name = "page28" id = "page28"> </a> +<div class = "verse"> +<p class = "pilcrow"> +¶ P<i>er</i> num<i>er</i>u<i>m</i> si vis nu<i>mer</i>u<i>m</i> +q<i>u</i>oq<i>ue</i> m<i>u</i>ltiplicar<i>e</i></p> +<p class = "pilcrow"> +¶ T<i>antu</i>m p<i>er</i> normas subtiles absq<i>ue</i> figuris</p> +<p>Has normas pot<i>er</i>is p<i>er</i> v<i>er</i>sus scir<i>e</i> +sequentes.</p> +</div> + +<p><span class = "sidenote">Mental multiplication.</span> +¶ Her<i>e</i> he teches þe to m<i>u</i>ltiplie <a class = "gloss" name = +"thowght" id = "thowght" href = "#gloss_thowght">be þowȝt</a> figures in +þi mynde. And þe sentence of þis v<i>er</i>se is þis: yf þo<i>u</i> wel +m<i>u</i>ltiplie on nounbre by anoþ<i>er</i> in þi mynde, þ<i>o</i>u +schal haue þ<i>er</i>eto rewles in þe v<i>er</i>ses þat schal come +aft<i>er</i>.</p> + +<div class = "verse"> +<p class = "pilcrow"> +¶ Si tu p<i>er</i> digitu<i>m</i> digitu<i>m</i> vis +m<i>u</i>ltiplicar<i>e</i></p> +<p>Re<i>gula</i> p<i>re</i>cedens dat qualit<i>er</i> est +op<i>er</i>andu<i>m</i>.</p> +</div> + +<p><span class = "sidenote">Digit by digit is easy.</span> +¶ Her<i>e</i> he teches a rewle as þou hast afor<i>e</i> to +m<i>u</i>ltiplie a digit be anoþ<i>er</i>, as yf þou wolde wete qwat is +sex tymes 6. þou +<span class = "linenum">leaf 161 <i>a</i>.</span> +*schalt wete by þe rewle þat I taȝt þe befor<i>e</i>, yf þou haue mynde +þ<i>er</i>of.</p> + +<div class = "verse"> +<p class = "pilcrow"> +¶ Articulu<i>m</i> si p<i>er</i> reliquu<i>m</i> reliquu<i>m</i> vis +m<i>u</i>lti<i>plica</i>r<i>e</i></p> +<p>In p<i>ro</i>p<i>r</i>iu<i>m</i> digitu<i>m</i> debet +vt<i>er</i>q<i>ue</i> resolui.</p> +<p class = "pilcrow"> +¶ Articul<i>us</i> digitos post se m<i>u</i>ltiplicantes</p> +<p>Ex digit<i>us</i> quociens retenerit +m<i>u</i>ltipli<i>ca</i>r<i>i</i></p> +<p>Articuli faciu<i>n</i>t tot centu<i>m</i> m<i>u</i>ltiplicati.</p> +</div> + +<p><span class = "sidenote">The first case of the craft.</span> +¶ Her<i>e</i> he teches þe furst rewle, þe quych is þis: yf þou wel +m<i>u</i>ltiplie an articul be anoþ<i>er</i>, so þat both þe articuls +bene w<i>i</i>t<i>h</i>-Inne an hundreth, þus þ<i>o</i>u schalt do. +<span class = "sidenote">Article by article;</span> +take þe digit of bothe the articuls, for eu<i>er</i>y articul hase a +digit, þen m<i>u</i>ltiplye þat on digit by þat oþ<i>er</i>, and loke +how mony vnytes ben in þe nounbre þat comes of þe m<i>u</i>ltiplicacioɳ +of þe 2 digittes, & so mony hundrythes ben in þe nounb<i>re</i> þat +schal come of þe m<i>u</i>ltiplicacioɳ of þe ylke 2 articuls as þus. +<span class = "sidenote">an example:</span> +yf þ<i>o</i>u wold wete qwat is ten tymes ten. take þe digit of ten, þe +quych is 1; take þe digit of þat oþ<i>er</i> ten, þe quych is on. +¶ Also m<i>u</i>ltiplie 1 be 1, as on tyme on þat is but 1. In on +is but on vnite as þou wost welle, þ<i>er</i>efor<i>e</i> ten tymes ten +is but a hundryth. +<span class = "sidenote">another example:</span> +¶ 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<i>u</i>ltiplie 3 be 2, þat is 6. Now in 6 ben 6 vnites, ¶ And so +mony hundrythes ben in 20 tymes 30*, +<span class = "linenum">leaf 161 <i>b</i>.</span> +þ<i>ere</i>for<i>e</i> 20 tymes 30 is 6 hundryth eueɳ. loke & se. +¶ But yf it be so þat on<i>e</i> articul be w<i>i</i>t<i>h</i>-Inne +an hundryth, <a class = "gloss" name = "or2" id = "or2" href = +"#gloss_or2">or</a> by-twene an hundryth and a thowsande, so þat it be +not a þowsande fully. þen loke how mony vnytes ben in þe nounbur þat +comys of þe m<i>u</i>ltiplicacioɳ <a class = "tag" name = "tag_craft16" +id = "tag_craft16" href = "#note_craft16">16</a>And so mony tymes<a +class = "tag" href = "#note_craft16">16</a> of 2 digitt<i>es</i> of ylke +articuls, so mony thowsant ben in þe nounbre, the qwych comes of þe +m<i>u</i>ltiplicacioɳ. And so mony tymes ten thowsand schal be in þe +nounbre þat comes of þe m<i>u</i>ltiplicacion of +<span class = "pagenum">29</span> +<a name = "page29" id = "page29"> </a> +2 articuls, as yf þ<i>o</i>u wold wete qwat is 4 hundryth tymes [two +hundryth]. Multiply 4 be 2,<a class = "tag" name = "tag_craft17" id = +"tag_craft17" href = "#note_craft17">17</a> +þat wol be 8. in 8 ben 8 vnites. +<span class = "headnote float"> +How to work subtly without Figures.</span> +<span class = "sidenote">Mental multiplication.</span> +¶ And so mony tymes ten thousand be in 4 hundryth tymes [2]<a class = +"tag" href = "#note_craft17">17</a> hundryth, þ<i>a</i>t is 80 thousand. +Take hede, I schall telle þe a +<span class = "sidenote">Another example.</span> +gen<i>e</i>rall<i>e</i> rewle whan þ<i>o</i>u hast 2 articuls, And þou +wold wete qwat comes of þe m<i>u</i>ltiplicacioɳ of hem 2. +m<i>u</i>ltiplie þe digit of þ<i>a</i>t on <a class = "terms" name = +"articuls2" id = "articuls2" href = "#terms_article">articuls</a>, and +kepe þat nounbre, þen loke how mony cifers schuld go befor<i>e</i> þat +on articuls, <a class = "gloss" name = "and" id = "and" href = +"#gloss_and">and</a> he wer<i>e</i> <a class = "gloss" name = "write" id += "write" href = "#gloss_write">write</a>. Als mony cifers schuld go +befor<i>e</i> þat other, & he wer<i>e</i> write of cifers. And haue +all<i>e</i> þe ylke cifers toged<i>ur</i> in þi mynde, +<span class = "linenum">leaf 162 <i>a</i>.</span> +*<a class = "gloss" name = "arowe" id = "arowe" href = +"#gloss_arowe">a-rowe</a> <a class = "gloss" name = "ychon" id = "ychon" +href = "#gloss_ychon">ychoɳ</a> aftur other, and in þe last plase set þe +nounbre þat comes of þe m<i>u</i>ltiplicacioɳ of þe 2 digittes. And loke +in þi mynde in what place he stondes, <a class = "gloss" name = "where" +id = "where" href = "#gloss_where">wher<i>e</i></a> in þe secunde, or in +þe thryd, or in þe 4, or wher<i>e</i> ellis, and loke qwat þe figures +by-token in þat place; & so mych is þe nounbre þat +<span class = "sidenote">Another example.</span> +comes of þe 2 articuls y-m<i>u</i>ltiplied to-ged<i>ur</i> as þus: yf +þ<i>o</i>u wold wete what is 20 thousant tymes 3 þowsande. +m<i>u</i>ltiply þe digit of þat articull<i>e</i> þe quych is 2 by þe +digitte of þat oþ<i>er</i> articul þe quych is 3, þat wol be 6. þen loke +how mony cifers schal go to 20 thousant as hit schuld be write in a +tabul. c<i>er</i>tainly 4 cifers schuld go to 20 þowsant. ffor þis +figure 2 in þe fyrst place betokenes twene. +<span class = "sidenote">Notation.</span> +¶ 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>i</i>t betokenes twenty þousant. þ<i>ere</i>for<i>e</i> he +most haue 4 cifers a-for<i>e</i> hym þat he may sto<i>n</i>de in þe 5 +place. kepe þese 4 cifers in thy mynde, þen loke how mony cifers goɳ to +3 thousant. Certayn to 3 thousante +<span class = "linenum">leaf 162 <i>b</i>.</span> +*goɳ 3 cifers afor<i>e</i>. Now cast ylke 4 cifers þat schuld go to +twenty thousant, And thes 3 cifers þat schuld go afor<i>e</i> 3 +thousant, & sette hem in rewe ychoɳ aft<i>er</i> oþ<i>er</i> in þi +mynde, as þai schuld stonde in a tabull<i>e</i>. And þen schal þou haue +7 cifers; þen sett þat 6 þe quych comes of þe m<i>u</i>ltiplicacioɳ of +þe 2 digitt<i>es</i> aft<i>u</i>r þe ylke cifers in þe 8 place as yf þat +hit stode in a tabul. And loke qwat a figur<i>e</i> of 6 schuld betoken +in þe 8 place. yf hit wer<i>e</i> in a tabul & 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. +<span class = "sidenote">Notation again.</span> +¶ 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<i>es</i>. ¶ In þe 8 place sexty þowsant +thousantes. þ<i>er</i>for<i>e</i> sett 6 in octauo loco, And he schal +betoken sexty þowsant +<span class = "pagenum">30</span> +<a name = "page30" id = "page30"> </a> +thousantes. +<span class = "sidenote">Mental multiplication.</span> +And so mych is twenty þowsant tymes 3 thousant, ¶ And þis rewle is +gen<i>er</i>all<i>e</i> for all<i>e</i> man<i>er</i> of articuls, +Whethir þai be hundryth or þowsant; but þ<i>o</i>u <a class = "gloss" +name = "most" id = "most" href = "#gloss_most">most</a> know well þe +craft of þe <a class = "gloss" name = "wryrchynge" id = "wryrchynge" +href = "#gloss_wryrchynge">wryrchyng<i>e</i></a> in þe tabull<i>e</i> +<span class = "linenum">leaf 163 <i>a</i>.</span> +*or þou know to do þus in þi mynde aftur þis rewle. Thou most þat þis +rewle <a class = "gloss" name = "holdythe" id = "holdythe" href = +"#gloss_holdythe">holdyþe</a> <a class = "gloss" name = "note" id = +"note" href = "#gloss_note">note</a> but wher<i>e</i> þ<i>ere</i> ben 2 +articuls and no mo of þe quych ayther of hem hase but on figur<i>e</i> +significatyf. As twenty tymes 3 thousant or 3 hundryth, and such +oþ<i>ur</i>.</p> + +<div class = "verse"> +<p class = "pilcrow"> +¶ Articulum digito si m<i>u</i>ltiplicare o<i>portet</i></p> +<p>Articuli digit[i sumi quo multiplicate]</p> +<p>Debem<i>us</i> reliquu<i>m</i> quod m<i>u</i>ltiplicat<i>ur</i> ab +ill<i>is</i></p> +<p>P<i>er</i> reliq<i>u</i>o decuplu<i>m</i> sic su<i>m</i>ma<i>m</i> +later<i>e</i> neq<i>ui</i>b<i>i</i>t.</p> +</div> + +<p><span class = "sidenote">The third case of the craft;</span> +¶ Her<i>e</i> he puttes þe thryde rewle, þe quych is þis. yf +þ<i>o</i>u wel m<i>u</i>ltiply in þi mynde, And þe Articul be a digitte, +þou schalt loke þat þe digitt be w<i>i</i>t<i>h</i>-Inne an hundryth, +þen þou schalt m<i>u</i>ltiply the digitt of þe Articulle by þe oþer +digitte. And eu<i>er</i>y vnite in þe nounbre þat schall<i>e</i> come +þ<i>ere</i>-of schal betoken ten. As þus: +<span class = "sidenote">an example.</span> +yf þat þ<i>o</i>u wold wete qwat is twyes 40. m<i>u</i>ltiplie þe +digitt<i>e</i> of 40, þe quych is 4, by þe oþ<i>er</i> diget, þe quych +is 2. And þat wolle be 8. And in þe nombre of 8 ben 8 vnites, & +eu<i>er</i>y of þe ylke vnites schuld stonde for 10. þ<i>ere</i>-fore +þ<i>ere</i> schal be 8 tymes 10, þat wol be 4 score. And so mony is +twyes 40. ¶ If þe articul be a hundryth or be 2 hundryth And a +þowsant, so þat hit be notte a thousant, +<span class = "linenum">leaf 163 <i>b</i>.</span> +*worch as þo<i>u</i> dyddyst afor<i>e</i>, saue þ<i>o</i>u schalt rekene +eu<i>er</i>y vnite for a hundryth.</p> + +<div class = "verse"> +<p class = "pilcrow"> +¶ In nu<i>mer</i>u<i>m</i> mixtu<i>m</i> digitu<i>m</i> si ducer<i>e</i> +cures</p> +<p>Articul<i>us</i> mixti sumat<i>ur</i> deinde resoluas</p> +<p>In digitu<i>m</i> post fac respectu de digitis</p> +<p>Articul<i>us</i>q<i>ue</i> docet excrescens in diriua<i>n</i>do</p> +<p>In digitu<i>m</i> mixti post ducas +m<i>u</i>ltiplica<i>n</i>te<i>m</i></p> +<p class = "pilcrow"> +¶ De digitis vt norma <a class = "tag" name = "tag_craft18" id = +"tag_craft18" href = "#note_craft18">18</a>[docet] de [hunc]</p> +<p>Multiplica si<i>mu</i>l et sic postea summa patebit.</p> +</div> + +<p><span class = "sidenote">The fourth case of the craft:</span> +Here he puttes þe 4 rewle, þe quych is þis: yf þou m<i>u</i>ltipliy on +composit be a digit as 6 tymes 24, <ins class = "correction" title = +"footnote tag positioned as shown"><a class = "tag" name = "tag_craft19" +id = "tag_craft19" href = "#note_craft19">19</a></ins>þen take þe diget +of þat composit, & m<i>u</i>ltiply þ<i>a</i>t digitt by þat +oþ<i>er</i> diget, and kepe þe nomb<i>ur</i> þat comes þ<i>ere</i>-of. +þen take þe digit of þat composit, & m<i>u</i>ltiply þat digit by +anoþ<i>er</i> diget, by þe quych þ<i>o</i>u hast m<i>u</i>ltiplyed þe +diget of þe articul, and loke qwat comes þ<i>ere</i>-of. +<span class = "sidenote">Composite by digit.</span> +þen take þ<i>o</i>u þat nounbur, & cast hit to þat other nounbur þat +þ<i>o</i>u secheste as þus yf þou wel +<span class = "pagenum">31</span> +<a name = "page31" id = "page31"> </a> +wete qwat comes of 6 tymes 4 & twenty. +<span class = "sidenote">Mental multiplication.</span> +multiply þat articull<i>e</i> of þe composit by þe digit, þe quych is 6, +as yn þe thryd rewle þ<i>o</i>u was tauȝt, And þat schal be 6 +scor<i>e</i>. þen m<i>u</i>ltiply þe diget of þe <i>com</i>posit, +<span class = "linenum">leaf 164 <i>a</i>.</span> +*þe quych is 4, and m<i>u</i>ltiply þat by þat other diget, þe quych is +6, as þou wast tauȝt in þe first rewle, yf þ<i>o</i>u haue mynde +þ<i>er</i>of, & þat wol be 4 & twenty. cast all ylke nounburs +to-ged<i>ir</i>, & hit schal be 144. And so mych is 6 tymes 4 & +twenty.</p> + +<p class = "headnote"><span class = "headnote"> +How to multiply without Figures.</span></p> + +<div class = "verse"> +<p class = "pilcrow"> +¶ Duct<i>us</i> in articulu<i>m</i> num<i>erus</i> si +<i>com</i>posit<i>us</i> sit</p> +<p>Articulu<i>m</i> puru<i>m</i> comites articulu<i>m</i> +q<i>u</i>o<i>que</i></p> +<p>Mixti pro digit<i>is</i> post fiat [et articulus vt]</p> +<p>Norma iubet [retinendo quod extra dicta ab illis]</p> +<p>Articuli digitu<i>m</i> post tu mixtu<i>m</i> digitu<i>m</i> duc</p> +<p>Re<i>gula</i> de digitis nec p<i>re</i>cipit +articul<i>us</i>q<i>ue</i></p> +<p>Ex quib<i>us</i> exc<i>re</i>scens su<i>m</i>me tu iunge +p<i>ri</i>ori</p> +<p>Sic ma<i>n</i>ifesta cito fiet t<i>ibi</i> su<i>m</i>ma petita.</p> +</div> + +<p><span class = "sidenote">The fifth case of the craft:</span> +¶ Her<i>e</i> he puttes þe 5 rewle, þe quych is þis: yf þ<i>o</i>u +wel m<i>u</i>ltiply an Articul be a composit, m<i>u</i>ltiplie þat +Articul by þe articul of þe composit, and worch as þou wos tauȝt in þe +secunde rewle, of þe quych rewle þe v<i>er</i>se begynnes þus. +<span class = "sidenote">Article by Composite.</span> +¶ Articulu<i>m</i> si p<i>er</i> Relicu<i>m</i> vis +m<i>u</i>ltiplicare. þen m<i>u</i>ltiply þe diget of þe composit by þat +oþ<i>ir</i> articul aft<i>ir</i> þe doctrine of þe 3 rewle. take +þ<i>er</i>of gode hede, I p<i>ra</i>y þe as þus. Yf þ<i>o</i>u wel +wete what is 24 tymes ten. +<span class = "sidenote">An example.</span> +Multiplie ten by 20, þat wel be 2 hundryth. þen m<i>u</i>ltiply þe diget +of þe 10, þe quych is 1, by þe diget of þe composit, þe quych is 4, +& þ<i>a</i>t +<span class = "linenum">leaf 164 <i>b</i>.</span> +*wol be 4. þen reken eu<i>er</i>y vnite þat is in 4 for 10, & þat +schal be 40. Cast 40 to 2 hundryth, & þat wol be 2 hundryth & +40. And so mych is 24 tymes ten.</p> + +<p class = "headnote"><span class = "headnote"> +How to work without Figures.</span></p> + +<div class = "verse"> +<p class = "pilcrow"> +¶ Compositu<i>m</i> num<i>er</i>u<i>m</i> mixto si[c] +m<i>u</i>ltiplicabis</p> +<p>Vndecies tredeci<i>m</i> sic e<i>st</i> ex hiis op<i>er</i>andum</p> +<p>In reliquu<i>m</i> p<i>rimu</i>m demu<i>m</i> duc post in +eund<i>em</i></p> +<p>Vnu<i>m</i> post den<i>u</i>m duc in t<i>ri</i>a dei<i>n</i>de +p<i>er</i> vnu<i>m</i></p> +<p>Multiplices<i>que</i> dem<i>u</i>m int<i>ra</i> o<i>mn</i>ia +m<i>u</i>ltiplicata</p> +<p>In su<i>m</i>ma decies q<i>ua</i>m si fu<i>er</i>it t<i>ibi</i> +doces</p> +<p>Multiplicandor<i>um</i> de normis sufficiunt h<i>ec</i>.</p> +</div> + +<p><span class = "sidenote">The sixth case of the craft:</span> +¶ Here he puttes þe 6 rewle, & þe last of all<i>e</i> +multiplicacioɳ, þe quych is þis: yf þ<i>o</i>u wel m<i>u</i>ltiplye a +<i>com</i>posit by a-noþ<i>er</i> composit, þou schalt do þus. +<span class = "sidenote">Composite by Composite.</span> +m<i>u</i>ltiplie þ<i>a</i>t on composit, qwych þ<i>o</i>u welt of the +twene, by þe articul of þe toþ<i>er</i> composit, as þ<i>o</i>u +wer<i>e</i> tauȝt in þe 5 rewle, þen m<i>u</i>ltiplie þ<i>a</i>t same +composit, þe quych þou hast m<i>u</i>ltiplied by þe oþ<i>er</i> articul, +by þe digit of þe oþ<i>er</i> composit, +<span class = "sidenote">Mental multiplication.</span> +as +<span class = "pagenum">32</span> +<a name = "page32" id = "page32"> </a> +þ<i>o</i>u was tauȝt in þe 4 rewle. +<span class = "sidenote">An example</span> +As þus, yf þou wold wete what is 11 tymes 13, as þ<i>o</i>u was tauȝt in +þe 5 rewle, & þat schal be an hundryth & ten, aft<i>er</i>warde +m<i>u</i>ltiply þat same co<i>m</i>posit þ<i>a</i>t þ<i>o</i>u hast +m<i>u</i>ltiplied, þe quych is a .11. And m<i>u</i>ltiplye hit be þe +digit of þe oþ<i>er</i> composit, þe quych is 3, for 3 is þe digit of +13, And þat wel be 30. þen take þe digit of þat composit, þe quych +composit þou m<i>u</i>ltiplied by þe digit of þ<i>a</i>t oþ<i>er</i> +<i>com</i>posit, +<span class = "linenum">leaf 165 <i>a</i>.</span> +*þe quych is a 11. +<span class = "sidenote">of the sixth case of the craft.</span> +¶ Also of the quych 11 on is þe digit. m<i>u</i>ltiplie þat digitt +by þe digett of þat oth<i>er</i> composit, þe quych diget is 3, as +þ<i>o</i>u was tauȝt in þe first rewle i<i>n</i> þe begynnyng<i>e</i> of +þis craft. þe quych rewle begynn<i>es</i> “In digitu<i>m</i> cures.” And +of all<i>e</i> þe m<i>u</i>ltiplicacioɳ of þe 2 digitt comys thre, for +onys 3 is but 3. Now cast all<i>e</i> þese nounbers toged<i>ur</i>, the +quych is þis, a 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<i>ur</i> towarde þe lyft side, And set aftur a +hundryth as her<i>e</i> an Ensampull<i>e</i>, 143.</p> + +<p class = "center">(Cetera desunt.)</p> + +<hr class = "mid" /> + +<div class = "footnote"> + +<p><a name = "note_craft1" id = "note_craft1" href = +"#tag_craft1">1.</a> +In MS, ‘awiy.’</p> + +<p><a name = "note_craft2" id = "note_craft2" href = +"#tag_craft2">2.</a> +‘ben’ repeated in MS.</p> + +<p><a name = "note_craft3" id = "note_craft3" href = +"#tag_craft3">3.</a> +In MS. ‘thausandes.’</p> + +<p><a name = "note_craft4" id = "note_craft4" href = +"#tag_craft4">4.</a> +Perhaps “So.”</p> + +<p><a name = "note_craft5" id = "note_craft5" href = +"#tag_craft5">5.</a> +‘hali’ marked for erasure in MS.</p> + +<p><a name = "note_craft6" id = "note_craft6" href = +"#tag_craft6">6.</a> +‘moy’ in MS.</p> + +<p><a name = "note_craft7" id = "note_craft7" href = +"#tag_craft7">7.</a> +<ins class = "correction" title = "open quote missing">‘S</ins>ubt<i>ra</i>has a<i>u</i>t addis a dext<i>ri</i>s +<i>ve</i>l medi<i>a</i>b<i>is</i>’ added on margin of MS.</p> + +<p><a name = "note_craft8" id = "note_craft8" href = +"#tag_craft8">8.</a> +After ‘craft’ insert ‘the .4. what is þe p<i>ro</i>fet of þis +craft.’</p> + +<p><a name = "note_craft9" id = "note_craft9" href = +"#tag_craft9">9.</a> +After ‘sythes’ insert ‘& þis wordes fyue sithe & sex +sythes.’</p> + +<p><a name = "note_craft10" id = "note_craft10" href = +"#tag_craft10">10.</a> +‘t’l’ marked for erasure before ‘tyl’ in MS.</p> + +<p><a name = "note_craft11" id = "note_craft11" href = +"#tag_craft11">11.</a> +Here ‘of þe same rew’ is marked for erasure in MS.</p> + +<p><a name = "note_craft12" id = "note_craft12" href = +"#tag_craft12">12.</a> +‘s<i>ed</i>’ deleted in MS.</p> + +<p><a name = "note_craft13" id = "note_craft13" href = +"#tag_craft13">13.</a> +6883 in MS.</p> + +<p><a name = "note_craft14" id = "note_craft14" href = +"#tag_craft14">14.</a> +‘þen’ overwritten on ‘þat’ marked for erasure.</p> + +<p><a name = "note_craft15" id = "note_craft15" href = +"#tag_craft15">15.</a> +‘Supra’ inserted in MS. in place of ‘cifra’ marked for erasure.</p> + +<p><a name = "note_craft16" id = "note_craft16" href = +"#tag_craft16">16–16.</a> +Marked for erasure in MS.</p> + +<p><a name = "note_craft17" id = "note_craft17" href = +"#tag_craft17">17.</a> +4 in MS.</p> + +<p><a name = "note_craft18" id = "note_craft18" href = +"#tag_craft18">18.</a> +docet. decet MS.</p> + +<p><a name = "note_craft19" id = "note_craft19" href = +"#tag_craft19">19.</a> +‘4 times 4’ in MS.</p> + +</div> + +</div> <!-- end div craft --> + +<div class = "art"> + +<span class = "pagenum">33</span> +<a name = "page33" id = "page33"> </a> + +<p class = "illustration"> +<a name = "art" id = "art"> +<img src = "images/title_art.png" width = "411" height = "108" +alt = "The Art of Nombryng. / a translation of / +John of Holywood’s De Arte Numerandi." +title = "The Art of Nombryng. /a translation of / +John of Holywood’s De Arte Numerandi." /></a></p> + +<hr class = "tiny" /> + +<p class = "subhead">[<i>Ashmole MS. 396, fol. 48.</i>]</p> + +<p class = "inset"><b><span class = "dropcap">B</span>oys seying in the +begynnyng of his <a class = "gloss" name = "arsemetrike" id = +"arsemetrike" href = +"#gloss_arsemetrike">Arsemetrik<i>e</i></a>:—All<i>e</i> +<span class = "linenum">Fol. 48.</span> +thynges that ben<i>e</i> fro the first begynnyng of thynges have +p<i>ro</i>ceded<i>e</i>, and come forth<i>e</i>, And by reso<i>u</i>n of +nombre ben formed<i>e</i>; And in wise as they ben<i>e</i>, So +oweth<i>e</i> they to be knowen<i>e</i>; wherfor in +vniu<i>er</i>sall<i>e</i> knowlechyng of thynges the Art of nombrynge is +best, and most operatyf<i>e</i>.</b></p> + +<p><span class = "dropcap">T</span>herfore <a class = "gloss" name = +"sithen" id = "sithen" href = "#gloss_sithen">sithen</a> the science of +the whiche at this tyme we +<span class = "sidenote">The name of the art.</span> +intenden<i>e</i> to write of standith<i>e</i> all<i>e</i> and about +nombre: ffirst we most se, what is the p<i>ro</i>pre name +therof<i>e</i>, and fro whens the name come: Afterward<i>e</i> what is +nombre, And how manye spices of nombre ther ben. The name is +cleped<i>e</i> Algorisme, +<span class = "sidenote">Derivation of Algorism.</span> +had<i>e</i> out of Algor<i>e</i>, other of Algos, in <a class = "gloss" +name = "grewe" id = "grewe" href = "#gloss_grewe">grewe</a>, That is +clepid<i>e</i> in englissh<i>e</i> art <a class = "gloss" name = "other" +id = "other" href = "#gloss_other">other</a> craft, And of +Rithm<i>us</i> that is called<i>e</i> nombre. So <a class = "terms" name += "algorisme" id = "algorisme" href = "#terms_algorisme">algorisme</a> +is cleped<i>e</i> the art of nombryng, +<span class = "sidenote">Another.</span> +other it is had of<i>e</i> en or in, and gogos that is +introduccio<i>u</i>n, and Rithm<i>us</i> nombre, that is to say +Interduccio<i>u</i>n of nombre. +<span class = "sidenote">Another.</span> +And thirdly it is had<i>e</i> of the name of a kyng that is +cleped<i>e</i> Algo and Rythm<i>us</i>; So called<i>e</i> +Algorism<i>us</i>. +<span class = "sidenote">Kinds of numbers.</span> +Sothely .2. maner<i>e</i> of nombres ben notified<i>e</i>; +Formall<i>e</i>,<a class = "tag" name = "tag_art1" id = "tag_art1" href += "#note_art1">1</a> +as nombr<i>e</i> i<i>s</i> vnitees gadred<i>e</i> to-gedres; +Materiall<i>e</i>,<a class = "tag" name = "tag_art2" id = "tag_art2" +href = "#note_art2">2</a> +as nombr<i>e</i> is a colleccio<i>u</i>n of vnitees. Other nombr<i>e</i> +is a multitude had<i>e</i> out of vnitees, vnitee is that thynge wher-by +eu<i>er</i>y thynge is called<i>e</i> oone, other <a class = "gloss" +name = "o1" id = "o1" href = "#gloss_oo">o</a> thynge. Of nombres, that +one is cleped<i>e</i> <a class = "terms" name = "digitalle" id = +"digitalle" href = "#terms_digit">digitall<i>e</i></a>, that +other<i>e</i> Article, Another a nombre <a class = "terms" name = +"componede" id = "componede" href = +"#terms_componede">componed<i>e</i></a> oþ<i>er</i> myxt. Another +digitall<i>e</i> is a nombre w<i>i</i>t<i>h</i>-in .10.; Article is +þ<i>a</i>t nombre that may be dyvyded<i>e</i> in .10. p<i>ar</i>ties +egally, And that there +<span class = "pagenum">34</span> +<a name = "page34" id = "page34"> </a> +leve no residue; Componed<i>e</i> or <a class = "terms" name = "medlede" +id = "medlede" href = "#terms_medlede">medled<i>e</i></a> is that nombre +that is come of a digite and of an <a class = "terms" name = "article" +id = "article" href = "#terms_article">article</a>. And +vndrestand<i>e</i> wele that all<i>e</i> nombres betwix .2. articles +next is a nombr<i>e</i> componed<i>e</i>. +<span class = "sidenote">The 9 rules of the Art.</span> +Of this art ben<i>e</i> .9. <a class = "gloss" name = "spices" id = +"spices" href = "#gloss_spices">spices</a>, that is forto sey, +num<i>er</i>acio<i>u</i>n, addicio<i>u</i>n, Subtraccio<i>u</i>n, +Mediac<i>i</i>o<i>u</i>n, Duplacio<i>u</i>n, Multipliacio<i>u</i>n, +Dyvysio<i>u</i>n, Progressio<i>u</i>n, And of Rootes the +extraccio<i>u</i>n, and that may be had<i>e</i> in .2. maners, that is +to sey in nombres quadrat, and in cubic<i>es</i>: Amonge the +which<i>e</i>, ffirst of Num<i>er</i>acio<i>u</i>n, and +aft<i>er</i>ward<i>e</i> of þe oþ<i>er</i>s by <a class = "gloss" name = +"ordure" id = "ordure" href = "#gloss_ordure">ordure</a>, y entende +to write.</p> + +<p class = "headnote"><span class = "headnote"> +Chapter I. Numeration.</span></p> + +<span class = "linenum">Fol. 48 <i>b</i>.</span> +<h5>*For-soth<i>e</i> num<i>er</i>acio<i>u</i>n is of eu<i>er</i>y +numbre by competent figures an artificiall<i>e</i> +rep<i>re</i>sentacio<i>u</i>n.</h5> + +<p><span class = "sidenote">Figures, differences, places, and +limits.</span> +Sothly figure, difference, places, and <a class = "gloss" name = "lynes" +id = "lynes" href = "#gloss_lymytes">lynes</a> supposen o thyng other +the same, But they ben sette here for dyue<i>r</i>s resons. ffigure is +cleped<i>e</i> for p<i>ro</i>traccio<i>u</i>n of figuracio<i>u</i>n; +Difference is called<i>e</i> for therby is shewed<i>e</i> eu<i>er</i>y +figure, how it hath<i>e</i> difference fro the figures before them: +place by cause of space, where-in <a class = "gloss" name = "me1" id = +"me1" href = "#gloss_me">me</a> writeth<i>e</i>: <a class = "gloss" name += "lynees" id = "lynees" href = "#gloss_lymytes">lynees</a>, for that is +ordeyned<i>e</i> for the p<i>re</i>sentacio<i>u</i>n of eu<i>er</i>y +figure. +<span class = "sidenote">The 9 figures.</span> +And vnderstonde that ther ben .9. <a class = "gloss" name = "lymytes" id += "lymytes" href = "#gloss_lymytes">lymytes</a> of figures that +rep<i>re</i>senten the .9. digit<i>es</i> that ben these. 0. 9. 8. 7. 6. +5. 4. 3. 2. 1. +<span class = "sidenote">The cipher.</span> +The .10. is cleped<i>e</i> theta, or a cercle, other a cifre, other a +figure of nought for nought it signyfieth<i>e</i>. Nathelesse she +holdyng that place giveth<i>e</i> others for to signyfie; for +with<i>e</i>-out cifre or cifres a pure article may not be writte. +<span class = "sidenote">The numeration</span> +And sithen that by these .9. figures significatif<i>es</i> +Ioyned<i>e</i> w<i>i</i>t<i>h</i> cifre or w<i>i</i>t<i>h</i> cifres +all<i>e</i> nombres ben and may be rep<i>re</i>sented<i>e</i>, It was, +<a class = "gloss" name = "nether" id = "nether" href = +"#gloss_nether">nether</a> is, no nede to fynde any more figures. +<span class = "sidenote">of digits,</span> +And note wele that eu<i>er</i>y digite shall<i>e</i> be writte +w<i>i</i>t<i>h</i> <a class = "gloss" name = "oo1" id = "oo1" href = +"#gloss_oo">oo</a> figure allone to it <a class = "gloss" name = +"aproprede" id = "aproprede" href = +"#gloss_aproprede">ap<i>ro</i>pred<i>e</i></a>. +<span class = "sidenote">of articles,</span> +And all<i>e</i> articles by a cifre, ffor eu<i>er</i>y article is +named<i>e</i> for oone of the digitis as <ins class = "correction" title += "punctuated as shown: error for ‘.10. of .1. 20. of .2.’?">.10. of 1.. +20. of. 2.</ins> and so of the others, &c. And all<i>e</i> nombres +digitall<i>e</i> owen to be sette in the first difference: All<i>e</i> +articles in the seconde. Also all<i>e</i> nombres fro .10. til an .100. +[which] is excluded<i>e</i>, with .2. figures mvst be writte; And yf it +be an article, by a cifre first put, and the figure y-writte +toward<i>e</i> the lift hond<i>e</i>, that signifieth<i>e</i> the digit +of the which<i>e</i> the article is named<i>e</i>; +<span class = "sidenote">of composites.</span> +And yf it be a nombre componed<i>e</i>, ffirst write the digit that is a +part of that componed<i>e</i>, and write to the lift side the article as +it is seid<i>e</i> be-fore. All<i>e</i> nombre that is fro an +hundred<i>e</i> tille a thousand<i>e</i> <a class = "gloss" name = +"exclusede" id = "exclusede" href = +"#gloss_exclusede">exclused<i>e</i></a>, owith<i>e</i> to be writ by .3. +figures; and all<i>e</i> nombre that is fro a thousand<i>e</i> +<span class = "pagenum">35</span> +<a name = "page35" id = "page35"> </a> +til .x. Mł. mvst be writ by .4. figures; And so forthe. +<span class = "sidenote">The value due to position.</span> +And vnderstond<i>e</i> wele that eu<i>er</i>y figure sette in the first +place signyfieth<i>e</i> his digit; In the second<i>e</i> place .10. +tymes his digit; In the .3. place an hundred<i>e</i> so moche; In the +.4. place a thousand<i>e</i> so moche; In the .5. place .x. +thousand<i>e</i> so moch<i>e</i>; In the .6. place an hundred<i>e</i> +thousand<i>e</i> so moch<i>e</i>; In the .7. place a thousand<i>e</i> +thousand<i>e</i>. And so infynytly mvltiplying by +<span class = "linenum">Fol. 49.</span> +*these .3. 10, 100, 1000. And vnderstand<i>e</i> wele that <a class = +"gloss" name = "competently" id = "competently" href = +"#gloss_competently">competently</a> me may sette vpon figure in the +place of a thousand<i>e</i>, a prik<i>e</i> to shewe how many +thousand<i>e</i> the last figure shall<i>e</i> rep<i>re</i>sent. +<span class = "sidenote">Numbers are written from right to left.</span> +We writen<i>e</i> in this art to the lift side-ward<i>e</i>, as +arabien<i>e</i> writen<i>e</i>, that weren fynders of this science, +other<i>e</i> for this reso<i>u</i>n, that for to kepe a custumable +ordr<i>e</i> in redyng, Sette we all<i>e</i>-wey the more nombre +before.</p> + +<p class = "headnote"><span class = "headnote"> +Chapter II. Addition.</span></p> + +<p><span class = "sidenote">Definition.</span> +<span class = "dropcap">A</span>ddicio<i>u</i>n is of nombre other of +nombres vnto nombre or to nombres aggregacio<i>u</i>n, that me may see +that that is come therof as <a class = "gloss" name = "excressent" id = +"excressent" href = "#gloss_excressent">exc<i>re</i>ssent</a>. In +addicio<i>u</i>n, 2. ordres of figures and .2. nombres ben necessary, +that is to sey, a nombre to be added<i>e</i> and the nombre wherto +the addic<i>i</i>oun shold<i>e</i> be made to. The nombre to be +added<i>e</i> is that þat shold<i>e</i> be added<i>e</i> therto, and +shall<i>e</i> be vnderwriten; the nombre vnto the which<i>e</i> +addicio<i>u</i>n shall<i>e</i> be made to is that nombre that +resceyueth<i>e</i> the addicion of þat other, and shall<i>e</i> be +writen above; +<span class = "sidenote">How the numbers should be written.</span> +and it is convenient that the lesse nombre be vnderwrit, and the more +added<i>e</i>, than the contrary. But whether it happ<i>e</i> one +<a class = "gloss" name = "other1" id = "other1" href = +"#gloss_other">other</a> other, the same comyth<i>e</i> of, Therfor, yf +þow wilt adde nombre to nombre, write the nombre wherto the +addicio<i>u</i>n shall<i>e</i> be made in the <a class = "gloss" name = +"omest" id = "omest" href = "#gloss_omest">omest</a> ordre by his +differences, so that the first of the lower ordre be vndre the first of +the <a class = "gloss" name = "omyst" id = "omyst" href = +"#gloss_omest">omyst</a> ordre, and so of others. +<span class = "sidenote">The method of working.</span> +That done, adde the first of the lower ordre to the first of the omyst +ordre. And of such<i>e</i> addicio<i>u</i>n, other þ<i>er</i>e +grow<i>i</i>t<i>h</i> therof a digit, An article, other a +composed<i>e</i>. +<span class = "sidenote">Begin at the right.</span> +If it be digit<i>us</i>, In the place of the omyst shalt thow write the +digit excrescyng, as thus:—</p> + +<table class = "grid outline float" summary = "example"> +<tr> +<td class = "words">The resultant</td> +<td>2</td> +</tr> +<tr> +<td class = "words">To whom it shal be added<i>e</i></td> +<td>1</td> +</tr> +<tr> +<td class = "words">The nombre to be added<i>e</i></td> +<td>1</td> +</tr> +</table> + +<p><span class = "sidenote">The Sum is a digit,</span> +If the article; in the place of the omyst put a-way by a cifre writte, +and the digit transferred<i>e</i>, of þe which<i>e</i> the article toke +his name, toward<i>e</i> the lift side, and be it added<i>e</i> to the +next figure folowyng, yf ther be any figure folowyng; or no, and yf it +be not, leve it [in the] void<i>e</i>, as thus:—</p> + +<span class = "pagenum">36</span> +<a name = "page36" id = "page36"> </a> +<span class = "sidenote">or an article,</span> + +<table class = "grid outline float" summary = "example"> +<tr> +<td class = "words">The resultant</td> +<td>10</td> +</tr> +<tr> +<td class = "words">To whom it shall<i>e</i> be added<i>e</i></td> +<td>7</td> +</tr> +<tr> +<td class = "words">The nombre to be added<i>e</i></td> +<td>3</td> +</tr> +</table> + +<table class = "grid outline floatleft" summary = "example"> +<tr> +<td class = "words">Resultans</td> +<td>2</td> +<td>7</td> +<td>8</td> +<td>2</td> +<td>7</td> +</tr> +<tr> +<td class = "words">Cui d<i>ebet</i> addi</td> +<td>1</td> +<td>0</td> +<td>0</td> +<td>8</td> +<td>4</td> +</tr> +<tr> +<td class = "words">Num<i>erus</i> addend<i>us</i></td> +<td>1</td> +<td>7</td> +<td>7</td> +<td>4</td> +<td>3</td> +</tr> +</table> + +<p class = "allclear"> +And yf it happe that the figure folowyng wherto the addicio<i>u</i>n +shall<i>e</i> be made by [the cifre of] an article, it sette a-side;</p> + +<table class = "grid outline float" summary = "example"> +<tr> +<td class = "words">The resultant</td> +<td>17</td> +</tr> +<tr> +<td class = "words">To whom it shall<i>e</i> be added<i>e</i></td> +<td>10</td> +</tr> +<tr> +<td class = "words">The nombre to be added<i>e</i></td> +<td>7</td> +</tr> +</table> + +<p class = "nospace"> +In his place write the +<span class = "linenum">Fol. 49 <i>b</i>.</span> +*[digit of the] Article as thus:—</p> + +<p class = "allclear"> +And yf it happe that a figure of .9. by the figure that me mvst adde +[one] to,</p> + +<table class = "grid outline float" summary = "example"> +<tr> +<td class = "words">The resultant</td> +<td>10</td> +</tr> +<tr> +<td class = "words">To whom it shall<i>e</i> be added<i>e</i></td> +<td>9</td> +</tr> +<tr> +<td class = "words">The nombre to be added<i>e</i></td> +<td>1</td> +</tr> +</table> + +<p class = "nospace"> +In the place of that 9. put a cifre <i>and</i> write þe article +toward<i>e</i> þe lift hond<i>e</i> as bifore, and thus:—</p> + +<p class = "allclear"> +<span class = "sidenote">or a composite.</span> +And yf<a class = "tag" name = "tag_art3" id = "tag_art3" href = +"#note_art3">3</a> +[therefrom grow a] nombre componed,<a class = "tag" name = "tag_art4" id += "tag_art4" href = "#note_art4">4</a> +[in the place of the nombre] put a-way<a class = "tag" name = "tag_art5" +id = "tag_art5" href = "#note_art5">5</a></p> + +<table class = "grid outline float" summary = "example"> +<tr> +<td class = "words">The resultant</td> +<td>12</td> +</tr> +<tr> +<td class = "words">To whom it shall<i>e</i> be added<i>e</i></td> +<td>8</td> +</tr> +<tr> +<td class = "words">The nombre to be added<i>e</i></td> +<td>4</td> +</tr> +</table> + +<p class = "nospace"> +[let] the digit [be]<a class = "tag" name = "tag_art6" id = "tag_art6" +href = "#note_art6">6</a> +writ þ<i>a</i>t is part of þ<i>a</i>t co<i>m</i>posid<i>e</i>, and þan +put to þe lift side the article as before, and þus:—</p> + +<p class = "allclear"> +<span class = "sidenote">The translator’s note.</span> +This done, adde the seconde to the second<i>e</i>, and write above +oþ<i>er</i> as before. Note wele þ<i>a</i>t in addic<i>i</i>ons and in +all<i>e</i> spices folowyng, whan he seith<i>e</i> one the other +shall<i>e</i> be writen aboue, and me most vse eu<i>er</i> figure, as +that eu<i>er</i>y figure were sette by half<i>e</i>, and by +hym-self<i>e</i>.</p> + +<p class = "headnote"><span class = "headnote"> +Chapter III. Subtraction.</span></p> + +<p><span class = "sidenote">Definition of Subtraction.</span> +<span class = "dropcap">S</span>ubtraccio<i>u</i>n is of .2. +p<i>ro</i>posed<i>e</i> nombres, the fyndyng of the excesse of the more +to the lasse: Other subtraccio<i>u</i>n is <a class = "gloss" name = +"ablacioun" id = "ablacioun" href = +"#gloss_ablacioun">ablacio<i>u</i>n</a> of o nombre fro a-nother, that +me may see a <a class = "gloss" name = "some1" id = "some1" href = +"#gloss_some">some</a> left. The lasse of the more, or even of even, may +be w<i>i</i>t<i>h</i>draw; The more fro the lesse may neu<i>er</i> be. +<span class = "sidenote">How it may be done.</span> +And sothly that nombre is more that hath<i>e</i> more figures, So that +the last be signyficatife<i>s</i>: And yf ther ben as many in that one +as in that other, me most deme it by the last, other by the next last. +<span class = "sidenote">What is required.</span> +More-ou<i>er</i> in w<i>i</i>t<i>h</i>-drawyng .2. nombres ben +necessary; A nombre to be w<i>i</i>t<i>h</i>draw, And a nombre that +me shall<i>e</i> w<i>i</i>t<i>h</i>-draw of. The nombre to be +w<i>i</i>t<i>h</i>-draw shall<i>e</i> be writ in the lower ordre by his +differences; +<span class = "sidenote">Write the greater number above.</span> +The +<span class = "pagenum">37</span> +<a name = "page37" id = "page37"> </a> +nombre fro the which<i>e</i> me shall<i>e</i> with<i>e</i>-draw in the +omyst ordre, so that the first be vnder the first, the second<i>e</i> +vnder the second<i>e</i>, And so of all<i>e</i> others. +<span class = "sidenote">Subtract the first figure if possible.</span> +With<i>e</i>-draw therfor the first of the lower<i>e</i> ordre fro the +first of the ordre above his hede, and that wolle be <a class = "gloss" +name = "other_or1" id = "other_or1" href = "#gloss_other">other</a> more +or lesse, oþ<i>er</i> egall<i>e</i>.</p> + +<table class = "grid outline float" summary = "example"> +<tr> +<td class = "words">The remanent</td> +<td>20</td> +</tr> +<tr> +<td class = "words">Wherof me shall<i>e</i> w<i>i</i>t<i>h</i>draw</td> +<td>22</td> +</tr> +<tr> +<td class = "words">The nombre to be w<i>i</i>t<i>h</i>draw</td> +<td>2</td> +</tr> +</table> + +<table class = "grid outline float" summary = "example"> +<tr> +<td class = "words">The remanent</td> +<td>2</td> +<td>2</td> +</tr> +<tr> +<td class = "words">Wherof me shall<i>e</i> w<i>i</i>t<i>h</i>-draw</td> +<td>2</td> +<td>8</td> +</tr> +<tr> +<td class = "words">Þe nombre to be w<i>i</i>t<i>h</i>draw</td> +<td></td> +<td>6</td> +</tr> +</table> + +<p class = "nospace"> +yf it be egall<i>e</i> or even the figure sette beside, put in his place +a cifre. And yf it be more put away þ<i>er</i>fro als many of vnitees +the lower figure conteyneth<i>e</i>, and writ the residue as thus</p> + +<span class = "linenum">Fol. 50.</span> + +<table class = "grid outline floatleft" summary = "example"> +<tr> +<td class = "words">*Remane<i>n</i>s</td> +<td>2</td> +<td>2</td> +<td>1</td> +<td>8</td> +<td>2</td> +<td>9</td> +<td>9</td> +<td>9</td> +<td>8</td> +</tr> +<tr> +<td class = "words"><p>A quo sit subtraccio</p></td> +<td>8</td> +<td>7</td> +<td>2</td> +<td>4</td> +<td>3</td> +<td>0</td> +<td>0</td> +<td>0</td> +<td>4</td> +</tr> +<tr> +<td class = "words"><p>Numerus subt<i>ra</i>hend<i>us</i></p></td> +<td>6</td> +<td>5</td> +<td><a class = "tag" name = "tag_art7" id = "tag_art7" href = +"#note_art7">7</a></td> +<td>[6]</td> +<td>.</td> +<td>.</td> +<td>.</td> +<td>.</td> +<td>6</td> +</tr> +</table> + +<p class = "rightclear"> +<span class = "sidenote">If it is not possible ‘borrow ten,’</span> +And yf it be lesse, by-cause the more may not be w<i>i</i>t<i>h</i>-draw +ther-fro, borow an vnyte of the next figure that is worth<i>e</i> 10. Of +that .10. and of the figure that ye wold<i>e</i> have +w<i>i</i>t<i>h</i>-draw fro +<span class = "sidenote">and then subtract.</span> +be-fore to-gedre Ioyned<i>e</i>,</p> + +<table class = "grid outline float" summary = "example"> +<tr> +<td class = "words">The remanent</td> +<td>1</td> +<td>8</td> +</tr> +<tr> +<td class = "words">Wherof me shall<i>e</i> w<i>i</i>t<i>h</i>-draw</td> +<td>2</td> +<td>4</td> +</tr> +<tr> +<td class = "words">The nombre to be w<i>i</i>t<i>h</i>-draw</td> +<td>0</td> +<td>6</td> +</tr> +</table> + +<p class = "nospace"> +w<i>i</i>t<i>h</i>-draw þe figure be-nethe, and put the residue in the +place of the figure put a-side as þ<i>us</i>:—</p> + +<p><span class = "sidenote">If the second figure is one.</span> +And yf the figure wherof me shal borow the vnyte be one, put it a-side, +and write a cifre in the place þ<i>er</i>of, lest the figures folowing +faile of thair<i>e</i> nombre, and þan worch<i>e</i> as it +shew<i>i</i>t<i>h</i> in this figure here:—</p> + +<table class = "grid outline float" summary = "example"> +<tr> +<td class = "words">The remanent</td> +<td>3</td> +<td>0</td> +<td>9<a class = "tag" name = "tag_art8" id = "tag_art8" href = +"#note_art8">8</a></td> +</tr> +<tr> +<td class = "words">Wherof me shal w<i>i</i>t<i>h</i>-draw</td> +<td>3</td> +<td>1</td> +<td>2</td> +</tr> +<tr> +<td class = "words">The nombre to be w<i>i</i>t<i>h</i>-draw</td> +<td>.</td> +<td>.</td> +<td>3</td> +</tr> +</table> + +<p><span class = "sidenote">If the second figure is a cipher.</span> +And yf the vnyte wherof me shal borow be a cifre, go ferther to the +figure signyficatif<i>e</i>, and ther borow one, and reto<i>ur</i>nyng +bak<i>e</i>, in the place of eu<i>er</i>y cifre þ<i>a</i>t ye +passid<i>e</i> ou<i>er</i>, sette figures of .9. as here it is +specified<i>e</i>:—</p> + +<table class = "grid outline float" summary = "example"> +<tr> +<td class = "words">The remenaunt</td> +<td>2</td> +<td>9</td> +<td>9</td> +<td>9</td> +<td>9</td> +</tr> +<tr> +<td class = "words">Wherof me shall<i>e</i> w<i>i</i>t<i>h</i>-draw</td> +<td>3</td> +<td>0</td> +<td>0</td> +<td>0</td> +<td>3</td> +</tr> +<tr> +<td class = "words">The nombre to be w<i>i</i>t<i>h</i>-draw</td> +<td></td> +<td></td> +<td></td> +<td></td> +<td>4</td> +</tr> +</table> + +<p>And whan me cometh<i>e</i> to the nombre wherof me intendith<i>e</i>, +there remayneth<i>e</i> all<i>e</i>-wayes .10. ffor þe which<i>e</i> +.10. &c. +<span class = "sidenote">A justification of the rule given.</span> +The reson why þat for eu<i>er</i>y cifre left behynde me setteth figures +ther of .9. this it is:—If fro the .3. place me borowed<i>e</i> an +vnyte, that vnyte by respect of the figure that he came fro +rep<i>re</i>sentith an .C., In the +<span class = "pagenum">38</span> +<a name = "page38" id = "page38"> </a> +place of that cifre [passed over] is left .9., [which is worth ninety], +and yit it remayneth<i>e</i> as .10., And the same reson<i>e</i> +wold<i>e</i> be yf me had<i>e</i> borowed<i>e</i> an vnyte fro the .4., +.5., .6., place, or ony other so vpward<i>e</i>. This done, withdraw the +second<i>e</i> of the lower ordre fro the figure above his hede of þe +omyst ordre, and wirch<i>e</i> as before. +<span class = "sidenote">Why it is better to work from right to +left.</span> +And note wele that in addicion or in subtracc<i>i</i>o<i>u</i>n me may +wele fro the lift side begynne and ryn to the right side, But it wol be +more p<i>ro</i>fitabler to be do, as it is taught. +<span class = "sidenote">How to prove subtraction,</span> +And yf thow wilt p<i>ro</i>ve yf thow have do wele or no, The figures +that thow hast withdraw, adde them <a class = "gloss" name = "ayene" id += "ayene" href = "#gloss_ayene">ayene</a> to the omyst figures, and they +wolle accorde w<i>i</i>t<i>h</i> the first that thow haddest yf thow +have labored wele; +<span class = "sidenote">and addition.</span> +and in like wise in addicio<i>u</i>n, whan thow hast added<i>e</i> +all<i>e</i> thy figures, w<i>i</i>t<i>h</i>draw them that thow first +<span class = "linenum">Fol. 50 <i>b</i>.</span> +*addest, and the same wolle reto<i>ur</i>ne. The subtraccio<i>u</i>n is +none other but a p<i>ro</i>uff<i>e</i> of the addicio<i>u</i>n, and the +contrarye in like wise.</p> + +<p class = "headnote"><span class = "headnote"> +Chapter IV. Mediation.</span></p> + +<p><span class = "sidenote">Definition of mediation.</span> +<span class = "dropcap">M</span><a class = "terms" name = "mediacioun2" +id = "mediacioun2" href = "#terms_mediacioun">ediacio<i>u</i>n</a> is +the fyndyng of the halfyng of eu<i>er</i>y nombre, that it may be +seyn<i>e</i> what and how moch<i>e</i> is eu<i>er</i>y half<i>e</i>. In +halfyng ay oo order of figures and oo nombre is necessary, that is to +sey the nombre to be halfed<i>e</i>. Therfor yf thow wilt half any +nombre, write that nombre by his differences, and +<span class = "sidenote">Where to begin.</span> +begynne at the right, that is to sey, fro the first figure to the right +side, so that it be signyficatif<i>e</i> other rep<i>re</i>sent vnyte or +eny other digitall<i>e</i> nombre. If it be vnyte write in his place a +cifre for the +<span class = "sidenote">If the first figure is unity.</span> +figures folowyng, [lest they signify less], and write that vnyte +w<i>i</i>t<i>h</i>out in the table, other resolue it in .60. <a class = +"gloss" name = "mynvtes" id = "mynvtes" href = +"#gloss_mynvtes">mynvt<i>es</i></a> and sette a-side half of <ins class += "correction" title = "error for the?">tho</ins> m<i>inutes</i> so, and +reserve the remen<i>au</i>nt w<i>i</i>t<i>h</i>out in the table, as thus +.30.; other sette w<i>i</i>t<i>h</i>out thus .<i>dī</i>: that +kepeth<i>e</i> none ordre of place, Nathelesse it hath<i>e</i> +signyficacio<i>u</i>n. And yf the other figure signyfie any other +digital nombre fro vnyte forth<i>e</i>, oþ<i>er</i> the nombre is +od<i>e</i> or even<i>e</i>.</p> + +<table class = "grid outline float" summary = "example"> +<tr> +<td class = "words">Halfed<i>e</i></td> +<td>2</td> +<td>2</td> +</tr> +<tr> +<td class = "words">to be halfed<i>e</i></td> +<td>4</td> +<td>4</td> +</tr> +</table> + +<table class = "grid outline float" summary = "example"> +<tr> +<td class = "words">halfed<i>e</i></td> +<td>2</td> +<td>3</td> +<td class = "edge">[di]</td> +</tr> +<tr> +<td class = "words">To be halfed<i>e</i></td> +<td>4</td> +<td>7</td> +</tr> +</table> + +<p class = "nospace"> +<span class = "sidenote">What to do if it is not unity.</span> +If it be even, write this half in this wise:—</p> + +<p>And if it be odde, Take the next even vndre hym conteyned<i>e</i>, +and put his half in the place of that odde, and of þe vnyte that +remayneth<i>e</i> to be halfed<i>e</i> do thus:—</p> + +<p><span class = "sidenote">Then halve the second figure.</span> +This done, the second<i>e</i> is to be halfed<i>e</i>, yf it be a cifre +put it be-side, and yf it be significatif<i>e</i>, <a class = "gloss" +name = "other_or" id = "other_or" href = "#gloss_other">other it is even +or od<i>e</i></a>: If it be even, write in the place of þe nombres +wiped<i>e</i> out the half<i>e</i>; yf it be od<i>e</i>, take the next +even vnder it <a class = "gloss" name = "contenythe" id = "contenythe" +href = "#gloss_contynes">co<i>n</i>tenyth<i>e</i></a>, and in the place +of the Impar sette a-side put half of the even: The +<span class = "pagenum">39</span> +<a name = "page39" id = "page39"> </a> +vnyte that remayneth<i>e</i> to be halfed<i>e</i>, respect had<i>e</i> +to them before, is worth<i>e</i> .10.</p> + +<table class = "grid outline float" summary = "example"> +<tr> +<td class = "words">Halfed<i>e</i></td> +<td> </td> +<td> </td> +<td> </td> +</tr> +<tr> +<td class = "words">to be halfed<i>e</i></td> +<td> </td> +<td> </td> +<td> </td> +</tr> +</table> + +<p class = "nospace"> +<span class = "sidenote">If it is odd, add 5 to the figure +before.</span> +Dyvide that .10. in .2., 5. is, and sette a-side that one, and adde that +other to the next figure p<i>re</i>cedent as here:—</p> + +<p>And yf þe addicio<i>u</i>n shold<i>e</i> be made to a cifre, sette it +a-side, and write in his place .5.</p> + +<table class = "grid outline float" summary = "example"> +<tr> +<td class = "words">doubled<i>e</i></td> +<td>2</td> +<td>6</td> +<td>8</td> +<td>9</td> +<td>0</td> +<td>10</td> +<td>17</td> +<td>4</td> +</tr> +<tr> +<td class = "words">to be doubled<i>e</i></td> +<td>1</td> +<td>3</td> +<td>4</td> +<td>4</td> +<td>5</td> +<td>5</td> +<td>8</td> +<td>7</td> +</tr> +</table> + +<p class = "nospace"> +And vnder this fo<i>ur</i>me me shall<i>e</i> write and worch<i>e</i>, +till<i>e</i> the totall<i>e</i> nombre be halfed<i>e</i>.</p> + +<p class = "headnote"><span class = "headnote"> +Chapter V. Duplation.</span></p> + +<p><span class = "sidenote">Definition of Duplation.</span> +<span class = "dropcap">D</span>uplicacio<i>u</i>n is ag<i>re</i>gacion +of nombre [to itself] þat me may se the nombre growen. In +doublyng<i>e</i> ay is but one ordre of figures necessarie. And me most +be-gynne w<i>i</i>t<i>h</i> the lift side, other of the more figure, And +after the nombre of the more figure <a class = "gloss" name = +"representithe" id = "representithe" href = +"#gloss_representithe">rep<i>re</i>sentith<i>e</i></a>. +<span class = "linenum">Fol. 51.</span> +*In the other .3. before we begynne all<i>e</i> way fro the right side +and fro the lasse nombre, +<span class = "sidenote">Where to begin.</span> +In this spice and in all<i>e</i> other folowyng we wolle begynne fro the +lift side, ffor and me bigon th<i>e</i> double fro the first, <a class = +"gloss" name = "omwhile1" id = "omwhile1" href = +"#gloss_omwhile">omwhile</a> me myght double oo thynge twyes. +<span class = "sidenote">Why.</span> +And how be it that me myght double fro the right, that wold<i>e</i> be +harder in techyng and in workyng. Therfor yf thow wolt double any +nombre, write that nombre by his differences, and double the last. And +of that doubly<i>n</i>g other growith<i>e</i> a nombre digital, article, +or componed<i>e</i>. [If it be a digit, write it in the place of the +first digit.]</p> + +<table class = "grid outline float" summary = "example"> +<tr> +<td class = "words">double</td> +<td>10</td> +</tr> +<tr> +<td class = "words">to be doubled<i>e</i></td> +<td>5</td> +</tr> +</table> + +<p class = "nospace"> +<span class = "sidenote">What to do with the result.</span> +If it be article, write in his place a cifre and transferre the article +toward<i>e</i> the lift, as thus:—</p> + +<p>And yf the nombre be componed<i>e</i>,</p> + +<table class = "grid outline float" summary = "example"> +<tr> +<td class = "words">doubled<i>e</i></td> +<td>16</td> +</tr> +<tr> +<td class = "words">to be doubled<i>e</i></td> +<td>8</td> +</tr> +</table> + +<p class = "nospace"> +write a digital that is part of his composicio<i>u</i>n, and sette the +article to the lift hand<i>e</i>, as thus:—</p> + +<p>That done, me most double the last save one, and what groweth<i>e</i> +þ<i>er</i>of me most worche as before. And yf a cifre be, touch<i>e</i> +it not. But yf any nombre shall<i>e</i> be added<i>e</i> to the +cifre,</p> + +<table class = "grid outline float" summary = "example"> +<tr> +<td class = "words">doubled<i>e</i></td> +<td>6</td> +<td>0</td> +<td>6</td> +</tr> +<tr> +<td class = "words">to be doubled<i>e</i></td> +<td>3</td> +<td>0</td> +<td>3</td> +</tr> +</table> + +<p class = "nospace"> +in þe place of þe figure wiped<i>e</i> out me most write the nombre to +be added<i>e</i>, as thus:—</p> + +<p>In the same wise me shall<i>e</i> wirch<i>e</i> of all<i>e</i> +others. +<span class = "sidenote">How to prove your answer.</span> +And this p<i>ro</i>bacio<i>u</i>n:</p> + +<table class = "grid outline float" summary = "example"> +<tr> +<td class = "words">Doubled<i>e</i></td> +<td>6</td> +<td>1</td> +<td>8</td> +</tr> +<tr> +<td class = "words">to be doubled<i>e</i></td> +<td>3</td> +<td>0</td> +<td>9</td> +</tr> +</table> + +<p class = "nospace"> +If thow truly double the halfis, and truly half the doubles, the same +nombre and figure shall<i>e</i> mete, such<i>e</i> as thow +labo<i>ur</i>ed<i>e</i> vpon<i>e</i> first, And of the contrarie.</p> + +<span class = "pagenum">40</span> +<a name = "page40" id = "page40"> </a> +<p class = "headnote"><span class = "headnote"> +Chapter VI. Multiplication.</span></p> + +<p><span class = "sidenote">Definition of Multiplication.</span> +<span class = "dropcap">M</span>ultiplicacio<i>u</i>n of nombre by +hym-self other by a-nother, w<i>i</i>t<i>h</i> p<i>ro</i>posid<i>e</i> +.2. nombres, [is] the fyndyng of the third<i>e</i>, That so oft +conteyneth<i>e</i> that other, as ther ben vnytes in the oþ<i>er</i>. In +multiplicacio<i>u</i>n .2. nombres pryncipally ben necessary, that is to +sey, the nombre multiplying and the nombre to be multiplied<i>e</i>, as +here;—twies fyve. +<span class = "sidenote">Multiplier.</span> +[The number multiplying] is designed<i>e</i> adu<i>er</i>bially. +<span class = "sidenote">Multiplicand.</span> +The nombre to be multiplied<i>e</i> resceyveth<i>e</i> a +no<i>m</i>i<i>n</i>all<i>e</i> appellacio<i>u</i>n, as twies .5. 5. is +the nombre multiplied<i>e</i>, and twies is the nombre to be <a class = +"gloss" name = "multipliede" id = "multipliede" href = +"#gloss_multipliede">multipliede</a>.</p> + +<table class = "grid outline" summary = "example"> +<tr> +<td class = "words">Resultans</td> +<td><a class = "tag" name = "tag_art9" id = "tag_art9" href = +"#note_art9">9</a></td> +<td>1</td> +<td>0</td> +<td class = "double">1</td> +<td>3</td> +<td>2</td> +<td>6</td> +<td>6</td> +<td>8</td> +<td>0</td> +<td>0</td> +<td>8</td> +</tr> +<tr> +<td class = "words">Multiplicand<i>us</i></td> +<td>.</td> +<td>.</td> +<td>5</td> +<td class = "double">.</td> +<td>.</td> +<td>4</td> +<td>.</td> +<td>3</td> +<td>4</td> +<td>0</td> +<td>0</td> +<td>4</td> +</tr> +<tr> +<td class = "words">Multiplicans</td> +<td>.</td> +<td>2</td> +<td>2</td> +<td class = "double">.</td> +<td>3</td> +<td>3</td> +<td>2</td> +<td>2</td> +<td>2</td> +<td>.</td> +<td>.</td> +<td>.</td> +</tr> +</table> + +<p><span class = "sidenote">Product.</span> +Also me may thervpon<i>e</i> to assigne the. 3. nombre, the +which<i>e</i> is +<span class = "linenum">Fol. 51 <i>b</i>.</span> +*cleped<i>e</i> p<i>ro</i>duct or p<i>ro</i>venient, of takyng out of +one fro another: as twyes .5 is .10., 5. the nombre to be +multiplied<i>e</i>, and .2. the multipliant, <ins class = "correction" +title = "error for ‘and .10.’?">and. 10.</ins> as before is come therof. +And vnderstonde wele, that of the multipliant may be made the nombre to +be multiplied<i>e</i>, and of the contrarie, remaynyng eu<i>er</i> the +same <a class = "gloss" name = "some" id = "some" href = +"#gloss_some">some</a>, and herof<i>e</i> cometh<i>e</i> the comen +speche, that seith<i>e</i> all nombre is converted<i>e</i> by +Multiplying in hym-self<i>e</i>. +<span class = "headnote float"> +The Cases of Multiplication.</span></p> + +<span class = "sidenote">There are 6 rules of Multiplication.</span> + +<table class = "grid outline float" summary = "example"> +<tr> +<td>1</td> +<td>2</td> +<td>3</td> +<td>4</td> +<td>5</td> +<td>6</td> +<td>7</td> +<td>8</td> +<td>9</td> +<td>10</td> +</tr> +<tr> +<td>2</td> +<td>4</td> +<td>6</td> +<td>8</td> +<td>10</td> +<td>10<a class = "tag" name = "tag_art10" id = "tag_art10" href = +"#note_art10">10</a></td> +<td>14</td> +<td>16</td> +<td>18</td> +<td>20</td> +</tr> +<tr> +<td>3</td> +<td>6</td> +<td>9</td> +<td>12</td> +<td>15</td> +<td>18</td> +<td>21</td> +<td>24</td> +<td>27</td> +<td>30</td> +</tr> +<tr> +<td>4</td> +<td>8</td> +<td>12</td> +<td>16</td> +<td>20</td> +<td>24</td> +<td>28</td> +<td>32</td> +<td>36</td> +<td>40</td> +</tr> +<tr> +<td>5</td> +<td>10</td> +<td>15</td> +<td>20</td> +<td>25</td> +<td>30</td> +<td>35</td> +<td>40</td> +<td>45</td> +<td>50</td> +</tr> +<tr> +<td>6</td> +<td>12</td> +<td>18</td> +<td>24</td> +<td>30</td> +<td>36</td> +<td>42</td> +<td>48</td> +<td>56</td> +<td>60</td> +</tr> +<tr> +<td>7</td> +<td>14</td> +<td>21</td> +<td>28</td> +<td>35</td> +<td>42</td> +<td>49</td> +<td>56</td> +<td>63</td> +<td>70</td> +</tr> +<tr> +<td>8</td> +<td>16</td> +<td>24</td> +<td>32</td> +<td>40</td> +<td>48</td> +<td>56</td> +<td>64</td> +<td>72</td> +<td>80</td> +</tr> +<tr> +<td>9</td> +<td>18</td> +<td>27</td> +<td>36</td> +<td>45</td> +<td>54</td> +<td>63</td> +<td>72</td> +<td>81</td> +<td>90</td> +</tr> +<tr> +<td>10</td> +<td>20</td> +<td>30</td> +<td>40</td> +<td>50</td> +<td>60</td> +<td>70</td> +<td>80</td> +<td>90</td> +<td>100</td> +</tr> +</table> + +<p class = "nospace"> +And ther ben .6 rules of Multiplicacio<i>u</i>n; +<span class = "sidenote">(1) Digit by digit.</span> +ffirst, yf a digit multiplie a digit, considr<i>e</i> how many of +vnytees ben betwix the digit by multiplying and his .10. beth<i>e</i> +to-gedre accompted<i>e</i>, and so oft w<i>i</i>t<i>h</i>-draw the digit +multiplying, vnder the article of his +deno<i>m</i>i<i>n</i>acio<i>u</i>n. Example of grace. If thow wolt wete +how moch<i>e</i> is .4. tymes .8., +<a class = "tag" name = "tag_art11" id = "tag_art11" href = +"#note_art11">11</a>se how many vnytees ben betwix .8.<a class = "tag" +name = "tag_art12" id = "tag_art12" href = "#note_art12">12</a> +and .10. to-geder rekened<i>e</i>, and it shew<i>i</i>t<i>h</i> that +.2.: withdraw ther-for the quat<i>e</i>rnary, of the article of his +deno<i>m</i>i<i>n</i>acion twies, of .40., And ther remayneth<i>e</i> +.32., that is, to <a class = "gloss" name = "some2" id = "some2" href = +"#gloss_some">some</a> of all<i>e</i> the multiplicacio<i>u</i>n. +<span class = "sidenote">See the table above.</span> +Wher-vpon for more evidence and declaracion the seid<i>e</i> table is +made. +<span class = "sidenote">(2) Digit by article.</span> +Whan a digit multiplieth<i>e</i> an article, thow most bryng the digit +into þe digit, of þe which<i>e</i> the article [has]<a class = "tag" +name = "tag_art13" id = "tag_art13" href = "#note_art13">13</a> +his name, and eu<i>er</i>y vnyte +<span class = "pagenum">41</span> +<a name = "page41" id = "page41"> </a> +shall<i>e</i> stond<i>e</i> for .10., and eu<i>er</i>y article an .100. +<span class = "sidenote">(3) Composite by digit.</span> +Whan the digit multiplieth<i>e</i> a nombre componed<i>e</i>, þ<i>o</i>u +most bryng the digit into aiþ<i>er</i> part of the nombre +componed<i>e</i>, so þ<i>a</i>t digit be had into digit by the first +rule, into an article by þe second<i>e</i> rule; and +aft<i>er</i>ward<i>e</i> Ioyne the p<i>ro</i>duccio<i>u</i>n, and +þ<i>er</i>e wol be the some totall<i>e</i>.</p> + +<table class = "grid outline" summary = "example"> +<tr> +<td class = "words">Resultans</td> +<td>1</td> +<td>2</td> +<td class = "double">6</td> +<td>7</td> +<td>3</td> +<td class = "double">6</td> +<td>1</td> +<td>2</td> +<td class = "double">0</td> +<td>1</td> +<td>2</td> +<td>0</td> +<td>8</td> +</tr> +<tr> +<td class = "words">Multiplicand<i>us</i></td> +<td> </td> +<td> </td> +<td class = "double">2</td> +<td> </td> +<td>3</td> +<td class = "double">2</td> +<td> </td> +<td> </td> +<td class = "double">6</td> +<td> </td> +<td> </td> +<td> </td> +<td>4</td> +</tr> +<tr> +<td class = "words">Multiplicans</td> +<td> </td> +<td>6</td> +<td class = "double">3</td> +<td>2</td> +<td>3</td> +<td class = "double"> </td> +<td> </td> +<td>2</td> +<td class = "double">0</td> +<td> </td> +<td>3</td> +<td>0</td> +<td>2</td> +</tr> +</table> + +<p><span class = "sidenote">(4) Article by article.</span> +Whan an article multiplieth<i>e</i> an article, the digit wherof he is +named<i>e</i> is to be brought Into the digit wherof the oþ<i>er</i> is +named<i>e</i>, and eu<i>er</i>y vnyte wol be worth<i>e</i> +<span class = "linenum">Fol. 52.</span> +*an .100., and eu<i>er</i>y article. a .1000. +<span class = "sidenote">(5) Composite by article.</span> +Whan an article multiplieth<i>e</i> a nombre componed<i>e</i>, thow most +bryng the digit of the article into aither part of the nombre +componed<i>e</i>; and Ioyne the p<i>ro</i>duccio<i>u</i>n, and +eu<i>er</i>y article wol be worth<i>e</i> .100., and eu<i>er</i>y vnyte +.10., and so woll<i>e</i> the some be open<i>e</i>. +<span class = "sidenote">(6) Composite by composite.</span> +Whan a nombre componed<i>e</i> multiplieth<i>e</i> a nombre +componed<i>e</i>, eu<i>er</i>y p<i>ar</i>t of the nombre multiplying is +to be had<i>e</i> into eu<i>er</i>y p<i>ar</i>t of the nombre to be +multiplied<i>e</i>, and so shall<i>e</i> the digit be had<i>e</i> twies, +onys in the digit, that other in the article. The article also twies, +ones in the digit, that other in the article. Therfor yf thow wilt any +nombre by hym-self other by any other multiplie, write the nombre to be +multiplied<i>e</i> in the ou<i>er</i> ordre by his differences, +<span class = "sidenote">How to set down your numbers.</span> +The nombre multiplying in the lower ordre by his differences, so that +the first of the lower ordre be vnder the last of the ou<i>er</i> ordre. +This done, of the multiplying, the last is to be had<i>e</i> into the +last of the nombre to be multiplied<i>e</i>. Wherof than wolle grow a +digit, an article, other a nombre componed<i>e</i>. +<span class = "sidenote">If the result is a digit,</span></p> + +<table class = "grid outline float" summary = "example"> +<tr> +<td class = "words">The resultant</td> +<td>6</td> +</tr> +<tr> +<td class = "words">To be multiplied<i>e</i></td> +<td>3</td> +</tr> +<tr> +<td class = "words">Þe nombre multipliyng</td> +<td>2</td> +</tr> +</table> + +<p class = "nospace"> +If it be a digit, even above the figure multiplying is hede write his +digit that come of, as it appereth<i>e</i> here:—</p> + +<p><span class = "sidenote">an article,</span> +And yf an article had be writ ou<i>er</i> the fig<i>ur</i>e multiplying +his hede, put a cifre þ<i>er</i> and transferre the article +toward<i>e</i> the lift hand<i>e</i>, as thus:—</p> + +<table class = "grid outline float" summary = "example"> +<tr> +<td class = "words">The resultant</td> +<td>1</td> +<td>0</td> +</tr> +<tr> +<td class = "words">to be multiplied<i>e</i></td> +<td></td> +<td>5</td> +</tr> +<tr> +<td class = "words">þe nombre m<i>u</i>ltipliyng</td> +<td></td> +<td>2</td> +</tr> +</table> + +<p><span class = "sidenote">or a composite.</span> +And yf a nombre componed<i>e</i> be writ ou<i>er</i> the figure +multyplying is hede, write the digit in the nombre componed<i>e</i> is +place, and sette the article to the lift hand<i>e</i>, as +thus:—</p> + +<span class = "pagenum">42</span> +<a name = "page42" id = "page42"> </a> + +<table class = "grid outline floatleft" summary = "example"> +<tr> +<td class = "words">Resultant</td> +<td>1</td> +<td>2</td> +</tr> +<tr> +<td class = "words">to be multiplied<i>e</i></td> +<td></td> +<td>4</td> +</tr> +<tr> +<td class = "words">the nombre multipliyng</td> +<td></td> +<td>3</td> +</tr> +</table> + +<p><span class = "sidenote">Multiply next by the last but one, and so +on.</span> +This done, <a class = "gloss" name = "me" id = "me" href = +"#gloss_me">me</a> most bryng the last save one of the multipliyng into +the last of þe nombre to be multiplied<i>e</i>, and se what +comyth<i>e</i> therof as before, and so do w<i>i</i>t<i>h</i> +all<i>e</i>, tille me come to the first of the nombre multiplying, that +must be brought into the last of the nombre to be multiplied<i>e</i>, +wherof growith<i>e</i> oþ<i>er</i> a digit, an article, +<span class = "linenum">Fol. 52 <i>b</i>.</span> +*other a nombre componed<i>e</i>.</p> + +<table class = "grid outline float" summary = "example"> +<tr> +<td class = "words">Resultant</td> +<td>6</td> +<td>6</td> +</tr> +<tr> +<td class = "words">to be multiplied<i>e</i></td> +<td></td> +<td>3</td> +</tr> +<tr> +<td class = "words">the nombre m<i>u</i>ltipliyng</td> +<td>2</td> +<td>2</td> +</tr> +</table> + +<table class = "grid outline float" summary = "example"> +<tr> +<td class = "words">The resultant</td> +<td>1</td> +<td>1</td> +<td>0</td> +</tr> +<tr> +<td class = "words">to be multiplied<i>e</i></td> +<td></td> +<td></td> +<td>5</td> +</tr> +<tr> +<td class = "words">þe nombre m<i>u</i>ltiplying</td> +<td></td> +<td>2</td> +<td>2</td> +</tr> +</table> + +<table class = "grid outline float" summary = "example"> +<tr> +<td class = "words">The resultant</td> +<td>1</td> +<td>3<a class = "tag" name = "tag_art15" id = "tag_art15" href = +"#note_art15">15</a></td> +<td>2</td> +</tr> +<tr> +<td class = "words">to be m<i>u</i>ltiplied<i>e</i></td> +<td></td> +<td></td> +<td>4</td> +</tr> +<tr> +<td class = "words">þe nombr<i>e</i> m<i>u</i>ltiplia<i>n</i>t</td> +<td></td> +<td>3</td> +<td>3</td> +</tr> +</table> + +<p class = "nospace"> +If it be a digit, In the place of the <a class = "gloss" name = "ouerer" +id = "ouerer" href = "#gloss_ouerer">ou<i>er</i>er</a>, sette a-side, as +here:</p> + +<p>If an article happe, there put a cifre in his place, and put hym to +the lift hand<i>e</i>, as here:</p> + +<p>If it be a nombre componed<i>e</i>, in the place of the ou<i>er</i>er +sette a-side, write a digit that<a class = "tag" name = "tag_art14" id = +"tag_art14" href = "#note_art14">14</a> +is a p<i>ar</i>t of the componed<i>e</i>, and sette on the left +hond<i>e</i> the article, as here:</p> + +<p><span class = "sidenote">Then antery the multiplier one place.</span> +That done, sette forward<i>e</i> the figures of the nombre multiplying +by <a class = "gloss" name = "oo" id = "oo" href = "#gloss_oo">oo</a> +difference, so that the first of the multipliant be vnder the last save +one of the nombre to be multiplied<i>e</i>, the other by <a class = +"gloss" name = "o" id = "o" href = "#gloss_oo">o</a> place sette +forward<i>e</i>. Than me shall<i>e</i> bryng<i>e</i> the last of the +m<i>u</i>ltipliant in hym to be multiplied<i>e</i>, vnder the +which<i>e</i> is the first multipliant. +<span class = "sidenote">Work as before.</span> +And than wolle growe oþ<i>er</i> a digit, an article, or a +componed<i>e</i> nombre. If it be a digit, adde hym even above his hede; +If it be an article, transferre hym to the lift side; And if it be a +nombre componed<i>e</i>, adde a digit to the figure above his hede, and +sette to the lift hand<i>e</i> the article. And all<i>e</i>-wayes +eu<i>er</i>y figure of the nombre multipliant is to be brought to the +last save one nombre to be multiplied<i>e</i>, til me come to the first +of the multipliant, where me shall<i>e</i> wirche as it is seid<i>e</i> +before of the first, and aft<i>er</i>ward<i>e</i> to put forward<i>e</i> +the figures by o difference and one till<i>e</i> they all<i>e</i> be +multiplied<i>e</i>. +<span class = "sidenote">How to deal with ciphers.</span> +And yf it happe that the first figure of þe multipliant be a cifre, and +<a class = "gloss" name = "boue" id = "boue" href = +"#gloss_boue">boue</a> it is sette the figure signyficatif<i>e</i>, +write a cifre in the place of the figur<i>e</i> sette a-side, as thus, +<i>et</i>c.:</p> + +<table class = "grid outline" summary = "example"> +<tr> +<td class = "words">The resultant</td> +<td>1</td> +<td>2</td> +<td>0</td> +</tr> +<tr> +<td class = "words">to be multiplied<i>e</i></td> +<td></td> +<td></td> +<td>6</td> +</tr> +<tr> +<td class = "words">the multipliant</td> +<td></td> +<td>2</td> +<td>0</td> +</tr> +</table> + +<span class = "pagenum">43</span> +<a name = "page43" id = "page43"> </a> + +<p><span class = "sidenote">How to deal with ciphers.</span> +And yf a cifre happe in the lower <a class = "gloss" name = "order" id = +"order" href = "#gloss_ordure">order</a> be-twix the first and the last, +and even above be sette the fig<i>ur</i>e signyficatif,</p> + +<table class = "grid outline float" summary = "example"> +<tr> +<td class = "words">The resultant</td> +<td>2</td> +<td>2</td> +<td>6</td> +<td>4</td> +<td>4</td> +</tr> +<tr> +<td class = "words">To be multiplied<i>e</i></td> +<td></td> +<td></td> +<td>2</td> +<td>2</td> +<td>2</td> +</tr> +<tr> +<td class = "words">The multipliant</td> +<td>1</td> +<td>0</td> +<td>2</td> +<td></td> +<td></td> +</tr> +</table> + +<p class = "nospace"> +leve it vntouched<i>e</i>, as here:—</p> + +<p>And yf the space above sette be void<i>e</i>, in that place write +thow a cifre. And yf the cifre happe betwix þe first and the last to be +m<i>u</i>ltiplied<i>e</i>, me most sette forward<i>e</i> the ordre of +the figures by thair<i>e</i> differences, for oft of <a class = "terms" +name = "duccioun" id = "duccioun" href = +"#terms_duccioun">duccio<i>u</i>n</a> of figur<i>e</i>s in cifres nought +is the resultant, as here,</p> + +<table class = "grid outline float" summary = "example"> +<tr> +<td class = "words">Resultant</td> +<td>8</td> +<td>0</td> +<td>0</td> +<td>8</td> +<td> </td> +</tr> +<tr> +<td class = "words">to be m<i>u</i>ltiplied<i>e</i></td> +<td>4</td> +<td>0</td> +<td>0</td> +<td>4</td> +<td> </td> +</tr> +<tr> +<td class = "words">the m<i>u</i>ltipliant</td> +<td>2</td> +<td>.</td> +<td>.</td> +<td>.</td> +<td> </td> +</tr> +</table> + +<p class = "nospace"> +<span class = "linenum">Fol. 53.</span> +*wherof it is evident and open, yf that the first figure of the nombre +be to be multiplied<i>e</i> be a cifre, vndir it shall<i>e</i> be none +sette as here:—</p> + +<table class = "grid outline floatleft" summary = "example"> +<tr> +<td class = "words">Resultant</td> +<td>3</td> +<td>2</td> +<td>0<a class = "tag" name = "tag_art16" id = "tag_art16" href = +"#note_art16">16</a></td> +</tr> +<tr> +<td class = "words">To be m<i>u</i>ltiplied<i>e</i></td> +<td></td> +<td>8</td> +<td>0</td> +</tr> +<tr> +<td class = "words">The m<i>u</i>ltipliant</td> +<td></td> +<td>4</td> +<td></td> +</tr> +</table> + +<p><span class = "sidenote">Leave room between the rows of +figures.</span> +Vnder[stand] also that in multiplicacio<i>u</i>n, divisio<i>u</i>n, and +of rootis the extraccio<i>u</i>n, competently me may leve a mydel space +betwix .2. ordres of figures, that me may write there what is come of +addyng other with<i>e</i>-drawyng, lest any thynge shold<i>e</i> be +<a class = "gloss" name = "ouerhippede" id = "ouerhippede" href = +"#gloss_ouerhippede">ou<i>er</i>-hipped<i>e</i></a> and sette out of +mynde.</p> + +<p class = "headnote"><span class = "headnote"> +Chapter VII. Division.</span></p> + +<p><span class = "sidenote">Definition of division.</span> +<span class = "dropcap">F</span>or to dyvyde oo nombre by a-nother, it +is of .2. nombres p<i>ro</i>posed<i>e</i>, It is forto depart the +<a class = "gloss" name = "moder" id = "moder" href = "#gloss_mo">moder</a> +nombre into as many p<i>ar</i>tis as ben of vnytees in the lasse nombre. +And note wele that in makyng<i>e</i> of dyvysio<i>u</i>n ther ben .3. +nombres necessary: +<span class = "sidenote">Dividend, Divisor, Quotient.</span> +that is to sey, the nombre to be dyvyded<i>e</i>; the nombre dyvydyng +and the nombre <a class = "gloss" name = "exeant" id = "exeant" href = +"#gloss_exeant">exeant</a>, <a class = "gloss" name = "other2" id = +"other2" href = "#gloss_other">other</a> how oft, or quocient. Ay +shall<i>e</i> the nombre that is to be dyvyded<i>e</i> be more, other at +the <a class = "gloss" name = "lest" id = "lest" href = +"#gloss_lest">lest</a> even<i>e</i> w<i>i</i>t<i>h</i> the nombre the +dyvysere, yf the nombre shall<i>e</i> be mad<i>e</i> by hole nombres. +<span class = "sidenote">How to set down your Sum.</span> +Therfor yf thow wolt any nombre dyvyde, write the nombre to be +dyvyded<i>e</i> in þe ou<i>er</i>er <a class = "gloss" name = "bordure" +id = "bordure" href = "#gloss_bordure">bordur<i>e</i></a> by his +differences, the dyviser<i>e</i> in the lower ordur<i>e</i> by his +differences, so that the last of the dyviser be vnder the last of the +nombre to be dyvyde, the next last vnder the next last, and so of the +others, yf it may competently be done; +<span class = "sidenote">An example.</span> +as here:—</p> + +<table class = "grid outline" summary = "example"> +<tr> +<td class = "words">The residue</td> +<td></td> +<td>2</td> +<td>7</td> +</tr> +<tr> +<td class = "words">The quotient</td> +<td></td> +<td></td> +<td>5</td> +</tr> +<tr> +<td class = "words">To be dyvyded<i>e</i></td> +<td>3</td> +<td>4</td> +<td>2</td> +</tr> +<tr> +<td class = "words">The dyvyser</td> +<td></td> +<td>6</td> +<td>3</td> +</tr> +</table> + +<span class = "pagenum">44</span> +<a name = "page44" id = "page44"> </a> +<span class = "sidenote">Examples.</span> + +<table class = "grid outline" summary = "example"> +<tr> +<td class = "words">Residuu<i>m</i></td> +<td> </td> +<td> </td> +<td class = "double">8</td> +<td> </td> +<td class = "double"> </td> +<td> </td> +<td>2</td> +<td class = "double">7</td> +<td> </td> +<td>2</td> +<td>6</td> +</tr> +<tr> +<td class = "words">Quociens</td> +<td> </td> +<td>2</td> +<td class = "double">1</td> +<td>2</td> +<td class = "double">2</td> +<td> </td> +<td> </td> +<td class = "double">5</td> +<td> </td> +<td> </td> +<td>9</td> +</tr> +<tr> +<td class = "words">Diuidend<i>us</i></td> +<td>6</td> +<td>8</td> +<td class = "double">0</td> +<td>6</td> +<td class = "double">6</td> +<td>3</td> +<td>4</td> +<td class = "double">2</td> +<td>3</td> +<td>3</td> +<td>2</td> +</tr> +<tr> +<td class = "words">Diuiser</td> +<td>3</td> +<td>2</td> +<td class = "double"> </td> +<td>3</td> +<td class = "double"> </td> +<td> </td> +<td>6</td> +<td class = "double">3</td> +<td> </td> +<td>3</td> +<td>4</td> +</tr> +</table> + +<p><span class = "sidenote">When the last of the divisor must not be set +below the last of the dividend.</span> +And ther ben .2. causes whan the last figure may not be sette vnder the +last, other that the last of the lower nombre may not be +w<i>i</i>t<i>h</i>-draw of the last of the ou<i>er</i>er nombre for it +is lasse than the lower, other <a class = "gloss" name = +"how_be_it_that" id = "how_be_it_that" href = +"#gloss_how_be_it_that">how be it, that</a> it myght be +w<i>i</i>t<i>h</i>-draw as for hym-self fro the ou<i>er</i>er the +remenaunt may not so oft of them above, other yf þe last of the lower be +even to the figure above his hede, and þe next last oþ<i>er</i> the +figure be-fore þ<i>a</i>t be more þan the figure above sette. +<span class = "linenum">Fol. 53<sup>2</sup>.</span> +*These so ordeyned<i>e</i>, me most wirch<i>e</i> from the last figure +of þe nombre of the dyvyser, and se how oft it may be +w<i>i</i>t<i>h</i>-draw of +<span class = "sidenote">How to begin.</span> +and fro the figure aboue his hede, namly so that the remen<i>au</i>nt +may be take of so oft, and to se the residue as here:—</p> + +<table class = "grid outline floatleft" summary = "example"> +<tr> +<td class = "words">The residue</td> +<td></td> +<td>2</td> +<td>6</td> +</tr> +<tr> +<td class = "words">The quocient</td> +<td></td> +<td></td> +<td>9</td> +</tr> +<tr> +<td class = "words">To be dyvyded<i>e</i></td> +<td>3</td> +<td>3</td> +<td>2</td> +</tr> +<tr> +<td class = "words">The dyvyser</td> +<td></td> +<td>3</td> +<td>4</td> +</tr> +</table> + +<p><span class = "sidenote">An example.</span> +And note wele that me may not with<i>e</i>-draw more than .9. tymes +nether lasse than ones. Therfor se how oft þe figures of the lower ordre +may be w<i>i</i>t<i>h</i>-draw fro the figures of the ou<i>er</i>er, and +the nombre that shew<i>i</i>t<i>h</i> þe q<i>u</i>ocient most be writ +ou<i>er</i> the hede of þat figure, vnder the which<i>e</i> the first +figure is, of the dyviser; +<span class = "sidenote">Where to set the <ins class = "correction" +title = "spelling (1922) unchanged">quotiente</ins></span> +And by that figure me most with<i>e</i>-draw all<i>e</i> oþ<i>er</i> +figures of the lower ordir and that of the figures aboue thair<i>e</i> +hedis. This so don<i>e</i>, me most sette forward<i>e</i> þe figures of +the diuiser by o difference toward<i>es</i> the right hond<i>e</i> and +worch<i>e</i> as before; and thus:— +<span class = "sidenote">Examples.</span></p> + +<table class = "grid outline" summary = "example"> +<tr> +<td class = "words">Residuu<i>m</i></td> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td class = "double"> </td> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td>.</td> +<td>1</td> +<td>2</td> +</tr> +<tr> +<td class = "words">quo<i>ciens</i></td> +<td> </td> +<td> </td> +<td> </td> +<td>6</td> +<td>5</td> +<td class = "double">4</td> +<td> </td> +<td> </td> +<td> </td> +<td>2</td> +<td>0</td> +<td>0</td> +<td>4</td> +</tr> +<tr> +<td class = "words">Diuidend<i>us</i></td> +<td>3</td> +<td>5</td> +<td>5</td> +<td>1</td> +<td>2</td> +<td class = "double">2</td> +<td>8</td> +<td>8</td> +<td>6</td> +<td>3</td> +<td>7</td> +<td>0</td> +<td>4</td> +</tr> +<tr> +<td class = "words">Diuisor</td> +<td> </td> +<td>5</td> +<td>4</td> +<td>3</td> +<td> </td> +<td class = "double"> </td> +<td>4</td> +<td>4</td> +<td>2</td> +<td>3</td> +<td> </td> +<td> </td> +<td> </td> +</tr> +</table> + +<table class = "grid outline" summary = "example"> +<tr> +<td class = "words">The quocient</td> +<td> </td> +<td> </td> +<td> </td> +<td>6</td> +<td>5</td> +<td>4</td> +</tr> +<tr> +<td class = "words">To be dyvyded<i>e</i></td> +<td>3</td> +<td>5</td> +<td>5</td> +<td>1</td> +<td>2</td> +<td>2</td> +</tr> +<tr> +<td class = "words">The dyvyser</td> +<td> </td> +<td>5</td> +<td>4</td> +<td>3</td> +<td> </td> +<td> </td> +</tr> +</table> + +<p><span class = "sidenote">A special case.</span> +And yf it happ<i>e</i> after þe settyng forward<i>e</i> of the +fig<i>ur</i>es þ<i>a</i>t þe last of the divisor may not so oft be +w<i>i</i>t<i>h</i>draw of the fig<i>ur</i>e above his hede, above þat +fig<i>ur</i>e vnder the which<i>e</i> the first of the diuiser is writ +me most sette a cifre in ordre of the nombre quocient, and sette the +fig<i>ur</i>es forward<i>e</i> as be-fore be o difference alone, and so +me shall<i>e</i> do in all<i>e</i> nombres to be dyvided<i>e</i>, for +where the dyviser may +<span class = "pagenum">45</span> +<a name = "page45" id = "page45"> </a> +not be w<i>i</i>t<i>h</i>-draw me most sette there a cifre, and sette +forward<i>e</i> the figures; as here:—</p> + +<table class = "grid outline floatleft" summary = "example"> +<tr> +<td class = "words">The residue</td> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td>1</td> +<td>2</td> +</tr> +<tr> +<td class = "words">The quocient</td> +<td> </td> +<td> </td> +<td> </td> +<td>2</td> +<td>0</td> +<td>0</td> +<td>4</td> +</tr> +<tr> +<td class = "words">To be dyvyded<i>e</i></td> +<td>8</td> +<td>8</td> +<td>6</td> +<td>3</td> +<td>7</td> +<td>0</td> +<td>4</td> +</tr> +<tr> +<td class = "words">The dyvyser</td> +<td>4</td> +<td>4</td> +<td>2</td> +<td>3</td> +<td> </td> +<td> </td> +<td> </td> +</tr> +</table> + +<p><span class = "sidenote">Another example.</span> +And me shall<i>e</i> not cesse fro such<i>e</i> settyng of +fig<i>ur</i>es forward<i>e</i>, nether of settyng<i>e</i> of þe quocient +into the dyviser, neþ<i>er</i> of subt<i>ra</i>ccio<i>u</i>n of the +dyvyser, till<i>e</i> the first of the dyvyser be +w<i>i</i>t<i>h</i>-draw fro þe first to be divided<i>e</i>. The +which<i>e</i> don<i>e</i>, or ought,<a class = "tag" name = "tag_art17" +id = "tag_art17" href = "#note_art17">17</a> oþ<i>er</i> nought +shall<i>e</i> remayne: and yf it be ought,<a class = "tag" href = +"#note_art17">17</a> +kepe it in the tables, And eu<i>er</i> <a class = "gloss" name = "vny" +id = "vny" href = "#gloss_vny">vny</a> it to þe diviser. And yf +þ<i>o</i>u wilt wete how many vnytees of þe divisio<i>u</i>n +<span class = "linenum">Fol. 53<sup>3</sup>.</span> +*wol growe to the nombre of the diviser<i>e</i>, +<span class = "sidenote">What the quotient shows.</span> +the nombre quocient wol shewe it: and whan such<i>e</i> divisio<i>u</i>n +is made, and þ<i>o</i>u <a class = "gloss" name = "lust" id = "lust" +href = "#gloss_lust">lust</a> p<i>ro</i>ve yf thow have wele done or +<span class = "sidenote">How to prove your division,</span> +no, Multiplie the quocient by the diviser, And the same fig<i>ur</i>es +wolle come ayene that thow haddest bifore and none other. And yf ought +be residue, than w<i>i</i>t<i>h</i> addicio<i>u</i>n therof +shall<i>e</i> come the same figures: And so multiplicacio<i>u</i>n +p<i>ro</i>vith<i>e</i> divisio<i>u</i>n, and dyvisio<i>u</i>n +multiplicacio<i>u</i>n: +<span class = "sidenote">or multiplication.</span> +as thus, yf multiplicacio<i>u</i>n be made, divide it by the +multipliant, and the nombre quocient wol shewe the nombre that was to be +multiplied<i>e</i>, <i>et</i>c.</p> + +<p class = "headnote"><span class = "headnote"> +Chapter VIII. Progression.</span></p> + +<p><span class = "sidenote">Definition <ins class = "correction" title = +"f illegible">of</ins> Progression.</span> +<span class = "dropcap">P</span>rogressio<i>u</i>n is of nombre after +<a class = "gloss" name = "egalle" id = "egalle" href = +"#gloss_egalle">egall<i>e</i></a> excesse fro oone or tweyn<i>e</i> +<a class = "gloss" name = "take" id = "take" href = "#gloss_take">take</a> +<a class = "gloss" name = "agregacioun" id = "agregacioun" href = +"#gloss_agregacioun">ag<i>r</i>egacio<i>u</i>n</a>. of +p<i>ro</i>gressio<i>u</i>n one is <a class = "terms" name = "naturelle" +id = "naturelle" href = "#terms_naturelle">naturell<i>e</i></a> or +co<i>n</i>tynuell<i>e</i>, þ<i>a</i>t oþ<i>er</i> broken and +discontynuell<i>e</i>. +<span class = "sidenote">Natural Progression.</span> +Naturell<i>e</i> it is, whan me begynneth<i>e</i> w<i>i</i>t<i>h</i> +one, and kepeth<i>e</i> ordure ou<i>er</i>lepyng one; as .1. 2. 3. 4. 5. +6., <i>et</i>c., so þ<i>a</i>t the nombre folowyng<i>e</i> +passith<i>e</i> the other be-fore in one. +<span class = "sidenote">Broken Progression.</span> +Broken it is, whan me lepith<i>e</i> fro o nombre till<i>e</i> another, +and kepith<i>e</i> not the contynuel ordir<i>e</i>; as 1. 3. 5. 7. 9, +<i>et</i>c. Ay me may begynne w<i>i</i>t<i>h</i> .2., as þus; .2. 4. 6. +8., <i>et</i>c., and the nombre folowyng passeth<i>e</i> the others +by-fore by .2. And note wele, that naturell<i>e</i> +p<i>ro</i>gressio<i>u</i>n ay begynneth<i>e</i> w<i>i</i>t<i>h</i> one, +and Int<i>er</i>cise or broken p<i>ro</i>gressio<i>u</i>n, <a class = +"gloss" name = "omwhile" id = "omwhile" href = +"#gloss_omwhile">omwhile</a> begynnyth<i>e</i> w<i>i</i>th one, omwhile +w<i>i</i>t<i>h</i> twayn<i>e</i>. Of p<i>ro</i>gressio<i>u</i>n naturell +.2. rules ther be <a class = "gloss" name = "yove" id = "yove" href = +"#gloss_yove">yove</a>, of the which<i>e</i> the first is this; +<span class = "sidenote">The 1st rule for Natural Progression.</span> +whan the p<i>ro</i>gressio<i>u</i>n naturell<i>e</i> endith<i>e</i> in +even nombre, by the half therof multiplie þe next totall<i>e</i> +ou<i>er</i>er<i>e</i> nombre; Example of grace: .1. 2. 3. 4. Multiplie +.5. by .2. and so .10. cometh<i>e</i> of, that is the totall<i>e</i> +nombre þ<i>er</i>of. +<span class = "sidenote">The second rule.</span> +The second<i>e</i> rule is such<i>e</i>, whan the +p<i>ro</i>gressio<i>u</i>n naturell<i>e</i> endith<i>e</i> in nombre +od<i>e</i>. Take the more porcio<i>u</i>n of the oddes, and multiplie +therby the totall<i>e</i> nombre. Example of grace 1. 2. 3. 4. 5., +multiplie +<span class = "pagenum">46</span> +<a name = "page46" id = "page46"> </a> +.5. by .3, and thryes .5. shall<i>e</i> be resultant. so the nombre +totall<i>e</i> is .15. +<span class = "sidenote">The first rule of Broken Progression.</span> +Of p<i>ro</i>gresio<i>u</i>n <a class = "terms" name = "intercise" id = +"intercise" href = "#terms_intercise">int<i>er</i>cise</a>, ther ben +also .2.<a class = "tag" name = "tag_art18" id = "tag_art18" href = +"#note_art18">18</a> +rules; and þe first is þis: Whan the Int<i>er</i>cise p<i>ro</i>gression +endith<i>e</i> in even nombre by half therof multiplie the next nombre +to þat half<i>e</i> as .2.<a class = "tag" href = "#note_art18">18</a> +4. 6. Multiplie .4. by .3. so þat is thryes .4., and .12. the nombre of +all<i>e</i> the p<i>ro</i>gressio<i>u</i>n, woll<i>e</i> folow. +<span class = "sidenote">The second rule.</span> +The second<i>e</i> rule is this: whan the p<i>ro</i>gressio<i>u</i>n +int<i>er</i>scise endith<i>e</i> in od<i>e</i>, take þe more +porcio<i>u</i>n of all<i>e</i> þe nombre, +<span class = "linenum">Fol. 53<sup>4</sup>.</span> +*and multiplie by hym-self<i>e</i>; as .1. 3. 5. Multiplie .3. by +hym-self<i>e</i>, and þe some of all<i>e</i> wolle be .9., +<i>et</i>c.</p> + +<p class = "headnote"><span class = "headnote"> +Chapter IX. Extraction of Roots.</span></p> + +<p><span class = "sidenote">The preamble of the extraction of +roots.</span> +<span class = "dropcap">H</span>ere folowith<i>e</i> the +extraccio<i>u</i>n of rotis, and first in nombre +q<i>ua</i>drat<i>es</i>. Wherfor me shall<i>e</i> se what is a <a class += "terms" name = "quadrat" id = "quadrat" href = "#terms_quadrat">nombre +quadrat</a>, and what is the rote of a nombre quadrat, and what it is to +draw out the rote of a nombre. And before other note this +divisio<i>u</i>n: +<span class = "sidenote">Linear, superficial, and solid numbers.</span> +Of nombres one is <a class = "terms" name = "lyneal" id = "lyneal" href += "#terms_lyneal">lyneal</a>, anoþ<i>er</i> <a class = "terms" name = +"superficial" id = "superficial" href = +"#terms_superficial">sup<i>er</i>ficiall<i>e</i></a>, anoþ<i>er</i> +quadrat, anoþ<i>er</i> cubik<i>e</i> or <a class = "gloss" name = +"hoole" id = "hoole" href = "#gloss_hole">hoole</a>. lyneal is that þat +is considred<i>e</i> after the p<i>ro</i>cesse, havyng<i>e</i> no +respect to the direccio<i>u</i>n of nombre in nombre, As a lyne +hath<i>e</i> but one dymensio<i>u</i>n that is to sey after the +length<i>e</i>. +<span class = "sidenote">Superficial numbers.</span> +Nombre sup<i>er</i>ficial is þ<i>a</i>t cometh<i>e</i> of ledyng<i>e</i> +of oo nombre into a-nother, wherfor it is called<i>e</i> +sup<i>er</i>ficial, for it hath<i>e</i> .2. nombres notyng or +mesuryng<i>e</i> hym, as a sup<i>er</i>ficiall<i>e</i> thyng<i>e</i> +hath<i>e</i> .2. dimensions, þ<i>a</i>t is to sey length<i>e</i> and +brede. +<span class = "sidenote">Square numbers.</span> +And for bycause a nombre may be had<i>e</i> in a-nother by .2. +man<i>er</i>s, þ<i>a</i>t is to sey other in hym-self<i>e</i>, +oþ<i>er</i> in anoþ<i>er</i>, Vnderstond<i>e</i> yf it be had in +hym-self, It is a quadrat. ffor dyvisio<i>u</i>n write by vnytes, +hath<i>e</i> .4. sides even as a quadrangill<i>e</i>. and yf the nombre +be had<i>e</i> in a-noþ<i>er</i>, the nombre is sup<i>er</i>ficiel and +not quadrat, as .2. had<i>e</i> in .3. maketh<i>e</i> .6. that is þe +first nombre sup<i>er</i>ficiell<i>e</i>; wherfor it is open þat +all<i>e</i> nombre quadrat is sup<i>er</i>ficiel, and not <a class = +"terms" name = "conuertide" id = "conuertide" href = +"#terms_conuertide">co<i>n</i>u<i>er</i>tid<i>e</i></a>. +<span class = "sidenote">The root of a square number.</span> +The rote of a nombre quadrat is þat nombre that is had of hym-self, as +twies .2. makith<i>e</i> 4. and .4. is the first nombre quadrat, and 2. +is his rote. 9. 8. 7. 6. 5. 4. 3. 2. 1. / The rote of the more quadrat +.3. 1. 4. 2. 6. +<span class = "sidenote">Notes of some examples of square roots here +interpolated.</span> +The most nombre quadrat 9. 8. 7. 5. 9. 3. 4. 7. 6. / the remenent +ou<i>er</i> the quadrat .6. 0. 8. 4. 5. / The first caas of nombre +quadrat .5. 4. 7. 5. 6. The rote .2. 3. 4. The second<i>e</i> caas .3. +8. 4. 5. The rote .6. 2. The third<i>e</i> caas .2. 8. 1. 9. The rote +.5. 3. The .4. caas .3. 2. 1. The rote .1. 7. / The 5. caas .9. 1. 2. 0. +4. / The rote 3. 0. 2. +<span class = "sidenote">Solid numbers.</span> +The <a class = "terms" name = "solide" id = "solide" href = +"#terms_solide">solid<i>e</i> nombre</a> or cubik<i>e</i> is þat +þ<i>a</i>t comytħe of double ledyng of nombre in nombre; +<span class = "sidenote">Three dimensions of solids.</span> +And it is cleped<i>e</i> a solid<i>e</i> body that hath<i>e</i> +þ<i>er</i>-in .3 +<span class = "pagenum">47</span> +<a name = "page47" id = "page47"> </a> +[dimensions] þat is to sey, length<i>e</i>, brede, and thiknesse. so +þ<i>a</i>t nombre hath<i>e</i> .3. nombres to be brought forth<i>e</i> +in hym. But nombre may be had<i>e</i> twies in nombre, for other it is +had<i>e</i> in hym-self<i>e</i>, oþ<i>er</i> in a-noþ<i>er</i>. +<span class = "sidenote">Cubic numbers.</span> +If a nombre be had<i>e</i> twies in hym-self, oþ<i>er</i> ones in his +quadrat, þ<i>a</i>t is the same, þ<i>a</i>t a <a class = "terms" name = +"cubike" id = "cubike" href = "#terms_cubike">cubik<i>e</i></a> +<span class = "linenum">Fol. 54.</span> +*is, And is the same that is solide. And yf a nombre twies be +had<i>e</i> in a-noþ<i>er</i>, the nombre is <a class = "gloss" name = +"clepede" id = "clepede" href = "#gloss_clepede">cleped<i>e</i></a> +solide and not cubik<i>e</i>, as twies .3. and þ<i>a</i>t .2. +makith<i>e</i> .12. +<span class = "sidenote">All cubics are solid numbers.</span> +Wherfor it is <a class = "gloss" name = "opyne" id = "opyne" href = +"#gloss_opyne">opyn<i>e</i></a> that all<i>e</i> cubik<i>e</i> nombre is +solid<i>e</i>, and not <a class = "terms" name = "conuertide2" id = +"conuertide2" href = +"#terms_conuertide"><i>con</i>u<i>er</i>tid<i>e</i></a>. Cubik<i>e</i> +is þ<i>a</i>t nombre þat comyth<i>e</i> of ledyng<i>e</i> of +hym-self<i>e</i> twyes, or ones in his quadrat. And here-by it is open +that o nombre is the <a class = "terms" name = "roote" id = "roote" href += "#terms_rote">roote</a> of a quadrat and of a cubik<i>e</i>. Natheles +the same nombre is not q<i>ua</i>drat and cubik<i>e</i>. +<span class = "sidenote">No number may be both linear and solid.</span> +Opyn<i>e</i> it is also that all<i>e</i> nombres may be a rote to a +q<i>ua</i>drat and cubik<i>e</i>, but not all<i>e</i> nombre quadrat or +cubik<i>e</i>. Therfor sithen þe ledyng<i>e</i> of vnyte in hym-self +ones or twies nought cometh<i>e</i> but vnytes, Seith<i>e</i> Boice in +Arsemetrik<i>e</i>, +<span class = "sidenote">Unity is not a number.</span> +that vnyte potencially is al nombre, and none in act. And +vndirstond<i>e</i> wele also that betwix euery .2. quadrat<i>es</i> ther +is a meene p<i>ro</i>porcionall<i>e</i>, +<span class = "sidenote">Examples of square roots.</span> +That is opened<i>e</i> thus; <a class = "terms" name = "lede_into" id = +"lede_into" href = "#terms_lede_into">lede the rote of o quadrat +into</a> the rote of the oþ<i>er</i> quadrat, and þan wolle þe meene +shew.</p> + +<table class = "grid outline" summary = "example"> +<tr> +<td class = "words">Residuu<i>m</i></td> +<td> </td> +<td> </td> +<td>0</td> +<td class = "double"> </td> +<td> </td> +<td> </td> +<td> </td> +<td class = "double">4</td> +<td> </td> +<td> </td> +<td>0</td> +<td> </td> +<td class = "double"> </td> +<td> </td> +<td> </td> +<td>0</td> +<td> </td> +</tr> +<tr> +<td class = "words">Quadrand<i>e</i></td> +<td>4</td> +<td>3</td> +<td>5</td> +<td class = "double">6</td> +<td>3</td> +<td>0</td> +<td>2</td> +<td class = "double">9</td> +<td>1</td> +<td>7</td> +<td>4</td> +<td>2</td> +<td class = "double">4</td> +<td>1</td> +<td>9 </td> +<td>3</td> +<td>6</td> +</tr> +<tr> +<td class = "words">Duplum</td> +<td>1</td> +<td>2</td> +<td> </td> +<td class = "double"> </td> +<td>1</td> +<td>0</td> +<td> </td> +<td class = "double"> </td> +<td>2</td> +<td> </td> +<td>6</td> +<td> </td> +<td class = "double"> </td> +<td> </td> +<td>[8]</td> +<td><a class = "tag" name = "tag_art19" id = "tag_art19" href = +"#note_art19">19</a></td> +<td> </td> +</tr> +<tr> +<td class = "words">Subduplu<i>m</i></td> +<td> </td> +<td>6</td> +<td> </td> +<td class = "double">6</td> +<td> </td> +<td>5</td> +<td> </td> +<td class = "double">5</td> +<td>1</td> +<td> </td> +<td>3</td> +<td> </td> +<td class = "double">2</td> +<td> </td> +<td>4 </td> +<td> </td> +<td>4</td> +</tr> +</table> + +<p><span class = "sidenote">A note on mean proportionals.</span> +Also betwix the next .2. cubikis, me may fynde a double meene, that is +to sey a more meene and a lesse. The more meene thus, as to +bryng<i>e</i> the rote of the lesse into a quadrat of the more. The +lesse thus, If the rote of the more be brought Into the quadrat of the +lesse.</p> + +<p class = "headnote"><span class = "headnote"> +Chapter X. Extraction of Square Root.</span></p> + +<p><span class = "dropcap">T</span>o<a class = "tag" name = "tag_art20" +id = "tag_art20" href = "#note_art20">20</a> draw a rote of the nombre +quadrat it is What-eu<i>er</i> nombre be p<i>ro</i>posed<i>e</i> to +fynde his rote and to se yf it be quadrat. +<span class = "sidenote">To find a square root.</span> +And yf it be not quadrat the rote of the most quadrat fynde out, vnder +the nombre p<i>ro</i>posed<i>e</i>. Therfor yf thow wilt the rote of any +quadrat nombre draw out, write the nombre by his differences, and +<a class = "gloss" name = "compt" id = "compt" href = +"#gloss_compt">compt</a> the nombre of the figures, and wete yf it be +od<i>e</i> or even. And yf +<span class = "sidenote">Begin with the last odd place.</span> +it be even, than most thow begynne worche vnder the last save one. And +yf it be od<i>e</i> w<i>i</i>t<i>h</i> the last; and forto sey it +shortly, al-weyes fro the last od<i>e</i> me shall<i>e</i> begynne. +Therfor vnder the last in an od place sette, +<span class = "sidenote">Find the nearest square root of that number, +subtract,</span> +me most fynd<i>e</i> a digit, the which<i>e</i> lad<i>e</i> in +hym-self<i>e</i> it puttith<i>e</i> away that, þat is ou<i>er</i> his +hede, oþ<i>er</i> as neigh<i>e</i> as me +<span class = "pagenum">48</span> +<a name = "page48" id = "page48"> </a> +may: suche a digit found<i>e</i> and w<i>i</i>t<i>h</i>draw fro his +ou<i>er</i>er, me most double that digit and sette the double vnder the +next figure toward<i>e</i> the right hond<i>e</i>, and his <a class = +"terms" name = "vnder_double" id = "vnder_double" href = +"#terms_vnder_double">vnder double</a> vnder hym. +<span class = "sidenote">double it,</span> +That done, than me most fy<i>n</i>d<i>e</i> a-noþ<i>er</i> digit vnder +the next figure bifore the doubled<i>e</i>, +<span class = "sidenote">and set the double one to the right.</span> +the which<i>e</i> +<span class = "linenum">Fol. 54 <i>b</i>.</span> +*brought in double setteth<i>e</i> a-way all<i>e</i> that is ou<i>er</i> +his hede as to <a class = "gloss" name = "rewarde" id = "rewarde" href = +"#gloss_rewarde">reward<i>e</i></a> of the doubled<i>e</i>: Than brought +into hym-self settith<i>e</i> all away in respect of hym-self, +<span class = "sidenote">Find the second figure by division.</span> +Other do it as nye as it may be do: other me may w<i>i</i>t<i>h</i>-draw +the digit +<a class = "tag" name = "tag_art21" id = "tag_art21" href = +"#note_art21">21</a>[last] found<i>e</i>, and lede hym in double or +double hym, and after in hym-self<i>e</i>; +<span class = "sidenote">Multiply the double by the second figure, and +add after it the square of the second figure, and subtract.</span> +Than Ioyne to-geder the p<i>ro</i>duccion<i>e</i> of them bothe, So that +the first figure of the last p<i>ro</i>duct be added<i>e</i> before the +first of the first p<i>ro</i>duct<i>es</i>, the second<i>e</i> of the +first, <i>et</i>c. and so forth<i>e</i>, <a class = "gloss" name = +"subtrahe" id = "subtrahe" href = "#gloss_subtrahe">subtrahe</a> fro the +totall<i>e</i> nombre in respect of þe digit.</p> + +<span class = "sidenote">Examples.</span> + +<table class = "grid outline" summary = "example"> +<tr> +<td class = "words">The residue</td> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td class = "double"> </td> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td class = "double"> </td> +<td> </td> +<td> </td> +<td> </td> +<td>5</td> +<td>4</td> +<td>3</td> +<td>2</td> +</tr> +<tr> +<td class = "words"><p>To be quadred<i>e</i></p></td> +<td>4</td> +<td>1</td> +<td>2</td> +<td>0</td> +<td class = "double">9</td> +<td>1</td> +<td>5</td> +<td>1</td> +<td>3</td> +<td class = "double">9</td> +<td>9</td> +<td>0</td> +<td>0</td> +<td>5</td> +<td>4</td> +<td>3</td> +<td>2</td> +</tr> +<tr> +<td class = "words">The double</td> +<td> </td> +<td>4</td> +<td>0</td> +<td> </td> +<td class = "double"> </td> +<td> </td> +<td>2</td> +<td> </td> +<td>4</td> +<td class = "double"> </td> +<td> </td> +<td>6</td> +<td> </td> +<td>0</td> +<td> </td> +<td> </td> +<td>0</td> +</tr> +<tr> +<td class = "words"><p>The vnder double</p></td> +<td>2</td> +<td> </td> +<td>0</td> +<td> </td> +<td class = "double">3</td> +<td>1</td> +<td> </td> +<td>2</td> +<td> </td> +<td class = "double">3</td> +<td>[3]</td> +<td> </td> +<td>[0]</td> +<td> </td> +<td>[0]</td> +<td> </td> +<td>0</td> +</tr> +</table> + +<p>And if it hap þ<i>a</i>t no digit may be found<i>e</i>, Than sette a +cifre vndre a cifre, and cesse not till<i>e</i> thow fynde a digit; and +whan thow hast founde it to double it, neþ<i>er</i> to sette the +doubled<i>e</i> forward<i>e</i> nether the vnder doubled<i>e</i>, +<span class = "sidenote">Special cases.</span> +Till thow fynde vndre the first figure a digit, the which<i>e</i> +lad<i>e</i> in all<i>e</i> double, settyng away all<i>e</i> that is +ou<i>er</i> hym in respect of the doubled<i>e</i>: Than lede hym into +hym-self<i>e</i>, and put a-way all<i>e</i> in regard<i>e</i> of hym, +other as nygh<i>e</i> as thow maist. +<span class = "sidenote">The residue.</span> +That done, other ought or nought wolle be the residue. If nought, than +it shewith<i>e</i> that a nombre componed<i>e</i> was the quadrat, and +his rote a digit last found<i>e</i> w<i>i</i>t<i>h</i> +vnder<i>e</i>-double other vndirdoubles, so that it be sette be-fore: +And yf ought<a class = "tag" name = "tag_art22" id = "tag_art22" href = +"#note_art22">22</a> +remayn<i>e</i>, that shew<i>i</i>t<i>h</i> that the nombre +p<i>ro</i>posed<i>e</i> was not quadrat,<a class = "tag" name = +"tag_art23" id = "tag_art23" href = "#note_art23">23</a> +but a digit [last found with the subduple or subduples +<span class = "pagenum">49</span> +<a name = "page49" id = "page49"> </a> +is]</p> + +<span class = "sidenote">This table is constructed for use in cube root +sums, giving the value of <ins class = "correction" title = "that is, ‘a(bˆ2).’">ab.<sup>2</sup></ins></span> + +<table class = "grid outline right" summary = "example"> +<tr> +<td class = "words">1</td> <!-- for extra spacing --> +<td>2</td> +<td>3</td> +<td>4</td> +<td>5</td> +<td>6</td> +<td>7 </td> +<td>8</td> +<td>9 </td> +</tr> +<tr> +<td class = "words">2</td> +<td>8</td> +<td>12</td> +<td>16</td> +<td>20</td> +<td>24</td> +<td>28 </td> +<td>32</td> +<td>36 </td> +</tr> +<tr> +<td class = "words">3</td> +<td>18</td> +<td>27</td> +<td>36</td> +<td>45</td> +<td>54</td> +<td>63 </td> +<td>72</td> +<td>81 </td> +</tr> +<tr> +<td class = "words">4</td> +<td>32</td> +<td>48</td> +<td>64</td> +<td>80</td> +<td>96</td> +<td>112<a class = "tag" name = "tag_art24" id = "tag_art24" href = +"#note_art24">24</a></td> +<td>128</td> +<td>144 </td> +</tr> +<tr> +<td class = "words">5</td> +<td>50</td> +<td>75</td> +<td>100</td> +<td>125</td> +<td>150</td> +<td>175 </td> +<td>200</td> +<td>225 </td> +</tr> +<tr> +<td class = "words">6</td> +<td>72</td> +<td>108</td> +<td>144</td> +<td>180</td> +<td>216</td> +<td>252 </td> +<td>288</td> +<td>324 </td> +</tr> +<tr> +<td class = "words">7</td> +<td>98</td> +<td>147</td> +<td>196</td> +<td>245</td> +<td>294</td> +<td>343 </td> +<td>393</td> +<td>441 </td> +</tr> +<tr> +<td class = "words">8</td> +<td>128</td> +<td>192</td> +<td>256</td> +<td>320</td> +<td>384</td> +<td>448 </td> +<td>512</td> +<td>576 </td> +</tr> +<tr> +<td class = "words">9</td> +<td>168</td> +<td>243</td> +<td>324</td> +<td>405</td> +<td>486</td> +<td>567 </td> +<td>648</td> +<td>729<a class = "tag" name = "tag_art25" id = "tag_art25" href = +"#note_art25">25</a></td> +</tr> +</table> + +<p>The rote of the most quadrat conteyned<i>e</i> vndre the nombre +p<i>ro</i>posed<i>e</i>. +<span class = "sidenote">How to prove the square root without or with a +remainder.</span> +Therfor yf thow wilt p<i>ro</i>ve yf thow have wele do or no, Multiplie +the digit last found<i>e</i> w<i>i</i>t<i>h</i> the vnder-double +oþ<i>er</i> vnder-doublis, and thow shalt fynde the same figures that +thow haddest before; And so that nought be the +<span class = "linenum">Fol. 55.</span> +*residue. And yf thow have any residue, than w<i>i</i>t<i>h</i> the +addicio<i>u</i>n þ<i>er</i>of that is res<i>er</i>ued<i>e</i> +w<i>i</i>t<i>h</i>-out in thy table, thow shalt fynd<i>e</i> thi first +figures as thow haddest them before, <i>et</i>c.</p> + +<p class = "headnote"><span class = "headnote"> +Chapter XI. Extraction of Cube Root.</span></p> + +<p><span class = "sidenote">Definition of a cubic number and a cube +root.</span> +<span class = "dropcap">H</span>eere folowith<i>e</i> the +extraccio<i>u</i>n of rotis in cubik<i>e</i> nombres; wher-for me most +se what is a nombre cubik<i>e</i>, and what is his roote, And what is +the extraccio<i>u</i>n of a rote. A nombre cubik<i>e</i> it is, as +it is before declared<i>e</i>, that cometh<i>e</i> of ledyng of any +nombre twies in hym-self<i>e</i>, other ones in his quadrat. The rote of +a nombre cubik<i>e</i> is the nombre that is twies had<i>e</i> in +hy<i>m</i>-self<i>e</i>, or ones in his quadrat. <a class = "gloss" name += "wherthurghe" id = "wherthurghe" href = +"#gloss_wherthurghe">Wher-thurgh<i>e</i></a> it is open, that +eu<i>er</i>y nombre quadrat or cubik<i>e</i> have the same rote, as it +is seid<i>e</i> before. And forto draw out the rote of a cubik<i>e</i>, +It is first to fynd<i>e</i> þe nombr<i>e</i> p<i>ro</i>posed<i>e</i> yf +it be a cubik<i>e</i>; And yf it be not, than thow most make +extraccio<i>u</i>n of his rote of the most cubik<i>e</i> vndre the +nombre p<i>ro</i>posid<i>e</i> his rote found<i>e</i>. Therfor +p<i>ro</i>posed<i>e</i> some nombre, whos cubical rote þ<i>o</i>u +woldest draw out; +<span class = "sidenote">Mark off the places in threes.</span> +First thow most compt the figures by fourthes, that is to sey in the +place of thousand<i>es</i>; +<span class = "sidenote">Find the first digit;</span> +And vnder the last thousand<i>e</i> place, thow most fynde a digit, the +which<i>e</i> lad<i>e</i> in hym-self cubikly puttith<i>e</i> a-way that +þat is ou<i>er</i> his hede as in respect of hym, other as nygh<i>e</i> +as thow maist. +<span class = "sidenote">treble it and place it under the next but one, +and multiply by the digit.</span> +That done, thow most <a class = "gloss" name = "trebille" id = +"trebille" href = "#gloss_trebille">trebill<i>e</i></a> the digit, and +that triplat is to be put vnder the .3. next figure toward<i>e</i> the +right hond<i>e</i>, And the <a class = "terms" name = "vnder_trebille" +id = "vnder_trebille" href = +"#terms_vnder_trebille">vnder-trebill<i>e</i></a> vnder the +trebill<i>e</i>; +<span class = "sidenote">Then find the second digit.</span> +Than me most fynd<i>e</i> a digit vndre the next figure bifore the +triplat, the which<i>e</i> w<i>i</i>t<i>h</i> his vnder-trebill<i>e</i> +had into a trebill<i>e</i>, aft<i>er</i>warde other vnder[trebille]<a +class = "tag" name = "tag_art26" id = "tag_art26" href = +"#note_art26">26</a> +had in his p<i>ro</i>duccio<i>u</i>n, putteth<i>e</i> a-way all<i>e</i> +that is ou<i>er</i> it in regard<i>e</i> of<a class = "tag" name = +"tag_art27" id = "tag_art27" href = "#note_art27">27</a> +[the triplat. Then lade in hymself puttithe away that þat is over his +hede as in respect of hym, other as nyghe as thou maist:] +<span class = "sidenote">Multiply the first triplate and the second +digit, twice by this digit.</span> +That done, thow most trebill<i>e</i> the digit ayene, and the triplat is +to be sette vnder the next .3. figure as before, And the +vnder-trebill<i>e</i> vnder the trebill<i>e</i>: and than most thow +sette forward<i>e</i> the first triplat w<i>i</i>t<i>h</i> his +vndre-trebill<i>e</i> by .2. differences. And than most thow fynde a +digit vnder the next figure before the triplat, the which<i>e</i> +with<i>e</i> his <a class = "terms" name = "vnder_triplat" id = +"vnder_triplat" href = "#terms_vnder_trebille">vnder-t<i>r</i>iplat</a> +had in his triplat afterward<i>e</i>, +<span class = "pagenum">50</span> +<a name = "page50" id = "page50"> </a> +<span class = "sidenote">Subtract.</span> +other vnder-treblis lad in p<i>ro</i>duct +<span class = "linenum">Fol. 55 <i>b</i>.</span> +*It sitteth<i>e</i> a-way ałł that is ou<i>er</i> his hede in respect of +the triplat than had in hym-self cubikly,<a class = "tag" name = +"tag_art28" id = "tag_art28" href = "#note_art28">28</a> +or as nygh<i>e</i> as ye may.</p> + +<span class = "sidenote">Examples.</span> + +<table class = "grid outline" summary = "example"> +<tr> +<td class = "words">Residuu<i>m</i></td> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td class = "double">5 </td> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td class = "double">4</td> +<td> </td> +<td>1</td> +<td>0</td> +<td>1</td> +<td>9</td> +<td> </td> +</tr> +<tr> +<td class = "words">Cubicandu<i>s</i></td> +<td>8</td> +<td>3</td> +<td>6</td> +<td>5</td> +<td>4</td> +<td>3</td> +<td class = "double">2 </td> +<td>3</td> +<td>0</td> +<td>0</td> +<td>7</td> +<td>6</td> +<td class = "double">7</td> +<td>1</td> +<td>1</td> +<td>6</td> +<td>6</td> +<td>7</td> +<td> </td> +</tr> +<tr> +<td class = "words">Triplum</td> +<td> </td> +<td> </td> +<td>6</td> +<td>0</td> +<td> </td> +<td> </td> +<td class = "double"> </td> +<td> </td> +<td> </td> +<td> </td> +<td>1</td> +<td>8</td> +<td class = "double"> </td> +<td> </td> +<td> </td> +<td> </td> +<td>4</td> +<td> </td> +<td> </td> +</tr> +<tr> +<td class = "words">Subt<i>r</i>iplu<i>m</i></td> +<td>2</td> +<td> </td> +<td> </td> +<td>0</td> +<td> </td> +<td> </td> +<td class = "double">[3]</td> +<td> </td> +<td> </td> +<td>6</td> +<td> </td> +<td> </td> +<td class = "double">7</td> +<td> </td> +<td>2</td> +<td> </td> +<td> </td> +<td>2</td> +<td> </td> +</tr> +</table> + +<p><span class = "sidenote">Continue this process till the first figure +is reached.</span> +Nother me shall<i>e</i> not cesse of the fyndyng<i>e</i> of that digit, +neither of his triplacio<i>u</i>n, neþ<i>er</i> of the triplat-is +<a class = "tag" name = "tag_art29" id = "tag_art29" href = +"#note_art29">29</a><a class = "terms" name = "anterioracioun" id = +"anterioracioun" href = +"#terms_anterioracioun">anteriorac<i>i</i>o<i>u</i>n</a>, that is to +sey, settyng forward<i>e</i> by .2. differences, Ne therof the +vndre-triple to be put vndre the triple, Nether of the +multiplicacio<i>u</i>n þ<i>er</i>of, Neither of the subtraccio<i>u</i>n, +till<i>e</i> it come to the first figure, vnder the which<i>e</i> is a +digitall<i>e</i> nombre to be found<i>e</i>, the which<i>e</i> +with<i>e</i> his vndre-treblis most be had<i>e</i> in tribles, +After-ward<i>e</i> w<i>i</i>t<i>h</i>out vnder-treblis to be had<i>e</i> +into <a class = "terms" name = "produccioun" id = "produccioun" href = +"#terms_produccioun">produccio<i>u</i>n</a>, settyng away all<i>e</i> +that is ou<i>er</i> the hed<i>e</i> of the triplat nombre, After had +into hymself<i>e</i> cubikly, +<span class = "sidenote">Examples.</span> +and sette all<i>e</i>-way that is ou<i>er</i> hym.</p> + +<table class = "grid outline float" summary = "example"> +<tr> +<td class = "words">To be <a class = "terms" name = "cubicede" id = +"cubicede" href = "#terms_cubicede">cubiced<i>e</i></a></td> +<td>1</td> +<td>7</td> +<td>2</td> +<td class = "double">8</td> +<td>3</td> +<td>2 </td> +<td>7</td> +<td>6</td> +<td>8</td> +</tr> +<tr> +<td class = "words">The triple</td> +<td> </td> +<td> </td> +<td>3</td> +<td class = "double">2</td> +<td> </td> +<td> </td> +<td> </td> +<td>9</td> +<td> </td> +</tr> +<tr> +<td class = "words">The vnder triple</td> +<td> </td> +<td> </td> +<td>1</td> +<td class = "double">2</td> +<td> </td> +<td>[3]</td> +<td> </td> +<td>3</td> +<td>3</td> +</tr> +</table> + +<p class = "nospace"> +Also note wele that the p<i>ro</i>ducc<i>i</i>on comyng<i>e</i> of the +ledyng of a digite found<i>e</i><a class = "tag" name = "tag_art30" id = +"tag_art30" href = "#note_art30">30</a> +me may adde to, and also w<i>i</i>t<i>h</i>-draw fro of the +totall<i>e</i> nombre sette above that digit so found<i>e</i>.<a class = +"tag" name = "tag_art31" id = "tag_art31" href = "#note_art31">31</a> +<span class = "sidenote">The residue.</span> +That done ought or nought most be the residue. If it be nought, It is +open that the nombre p<i>ro</i>posed<i>e</i> was a cubik<i>e</i> nombre, +And his rote a digit founde last w<i>i</i>t<i>h</i> the vnder-triples: +If the rote therof <a class = "gloss" name = "wex" id = "wex" href = +"#gloss_wex">wex</a> bad<i>e</i> in hym-self<i>e</i>, and +afterward<i>e</i> p<i>ro</i>duct they shall<i>e</i> make the first +fig<i>ur</i>es. And yf ought be in residue, kepe that +w<i>i</i>t<i>h</i>out in the table; and it is open<i>e</i> that the +nombre was not a cubik<i>e</i>. but a digit last founde +w<i>i</i>t<i>h</i> the vndirtriplis is rote of the most cubik<i>e</i> +vndre the nombre p<i>ro</i>posed<i>e</i> conteyned<i>e</i>, the +which<i>e</i> rote yf it be had<i>e</i> in hym-self<i>e</i>, +<span class = "sidenote">Special cases.</span> +And aft<i>er</i>ward<i>e</i> in a p<i>ro</i>duct of that shall<i>e</i> +growe the most cubik<i>e</i> vndre the nombre p<i>ro</i>posed<i>e</i> +conteyned<i>e</i>, And yf that be added<i>e</i> to a cubik<i>e</i> the +residue res<i>er</i>ued<i>e</i> in the table, woll<i>e</i> make the same +figures that ye had<i>e</i> first. +<span class = "sidenote">Special case.</span> +<span class = "linenum">Fol. 56.</span> +*And +<span class = "pagenum">51</span> +<a name = "page51" id = "page51"> </a> +yf no digit after the anterioracio<i>u</i>n<a class = "tag" name = +"tag_art32" id = "tag_art32" href = "#note_art32">32</a> +may not be found<i>e</i>, than put ther<i>e</i> a cifre vndre a cifre +vndir the third<i>e</i> figure, And put forward<i>e</i> þe +fig<i>ur</i>es. Note also wele that yf in the nombre +p<i>ro</i>posed<i>e</i> ther ben no place of thowsand<i>es</i>, me most +begynne vnder the first figure in the extraccio<i>u</i>n of the rote. +some vsen forto <a class = "gloss" name = "distingue" id = "distingue" +href = "#gloss_distingue">distingue</a> the nombre by threes, and ay +begynne forto wirch<i>e</i> vndre the first of the last <a class = +"terms" name = "ternary" id = "ternary" href = +"#terms_ternary">t<i>er</i>nary</a> other unco<i>m</i>plete nombre, the +which<i>e</i> maner of op<i>er</i>acio<i>u</i>n accordeth<i>e</i> +w<i>i</i>t<i>h</i> that before.</p> + +<span class = "sidenote">Examples.</span> + +<table class = "grid outline" summary = "example"> +<tr> +<td class = "words">The residue</td> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td class = "double">0</td> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td>1</td> +<td>1</td> +</tr> +<tr> +<td class = "words">The cubicand<i>us</i></td> +<td>8 </td> +<td>0</td> +<td>0</td> +<td>0</td> +<td>0</td> +<td>0</td> +<td class = "double">0</td> +<td>8</td> +<td>2</td> +<td>4</td> +<td>2</td> +<td>4</td> +<td>1</td> +<td>9</td> +</tr> +<tr> +<td class = "words">The triple</td> +<td> </td> +<td> </td> +<td><a class = "tag" name = "tag_art33" id = "tag_art33" href = +"#note_art33">33</a> </td> +<td>0</td> +<td>0</td> +<td> </td> +<td class = "double"> </td> +<td> </td> +<td> </td> +<td>6</td> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +</tr> +<tr> +<td class = "words">The vndert<i>r</i>iple</td> +<td>[2]</td> +<td> </td> +<td> </td> +<td>0</td> +<td>0</td> +<td> </td> +<td class = "double"> </td> +<td>2</td> +<td> </td> +<td> </td> +<td>6</td> +<td>2</td> +<td> </td> +<td></td> +</tr> +</table> + +<p class = "nospace"> +And this at this tyme suffiseth<i>e</i> in extraccio<i>u</i>n of nombres +quadrat or cubik<i>es</i> <i>et</i>c.</p> + +<p class = "headnote"><span class = "headnote"> +Table of Numbers, &c.</span></p> + +<span class = "sidenote">A table of numbers; probably from the +Abacus.</span> + +<table class = "gloss" summary = "table of numbers"> +<tr class = "gloss"> +<td>1</td> +<td>2</td> +<td>3</td> +<td>4</td> +<td>5</td> +</tr> +<tr class = "orig"> +<td>one.</td> +<td>x.</td> +<td>an. hundred<i>e</i> /</td> +<td>a thowsand<i>e</i> /</td> +<td>x. thowsand<i>e</i> /</td> +</tr> +</table> + +<table class = "gloss" summary = "table of numbers"> +<tr class = "gloss"> +<td>6</td> +<td>7</td> +</tr> +<tr class = "orig"> +<td>An hundred<i>e</i> thowsand<i>e</i> /</td> +<td>A thowsand<i>e</i> tymes a thowsand<i>e</i> /</td> +</tr> +</table> + +<p class = "wide">x. thousand<i>e</i> tymes a thousand<i>e</i> / An +hundred<i>e</i> thousand<i>e</i> tymes a thousand<i>e</i> +A thousand<i>e</i> thousand<i>e</i> tymes a thousand<i>e</i> / this +is the x place <i>et</i>c.</p> + +<p class = "center">[Ende.]</p> + +<hr class = "mid" /> + +<div class = "footnote"> + +<p><a name = "note_art1" id = "note_art1" href = "#tag_art1">1.</a> +MS. Materiall<i>e</i>.</p> + +<p><a name = "note_art2" id = "note_art2" href = "#tag_art2">2.</a> +MS. Formall<i>e</i>.</p> + +<p><a name = "note_art3" id = "note_art3" href = "#tag_art3">3.</a> +‘the’ in MS.</p> + +<p><a name = "note_art4" id = "note_art4" href = "#tag_art4">4.</a> +‘be’ in MS.</p> + +<p><a name = "note_art5" id = "note_art5" href = "#tag_art5">5.</a> +‘and’ in MS.</p> + +<p><a name = "note_art6" id = "note_art6" href = "#tag_art6">6.</a> +‘is’ in MS.</p> + +<p><a name = "note_art7" id = "note_art7" href = "#tag_art7">7.</a> +6 in MS.</p> + +<p><a name = "note_art8" id = "note_art8" href = "#tag_art8">8.</a> +0 in MS.</p> + +<p><a name = "note_art9" id = "note_art9" href = "#tag_art9">9.</a> +2 in MS.</p> + +<p><a name = "note_art10" id = "note_art10" href = "#tag_art10">10.</a> +<i>sic.</i></p> + +<p><a name = "note_art11" id = "note_art11" href = "#tag_art11">11.</a> +‘And’ inserted in MS.</p> + +<p><a name = "note_art12" id = "note_art12" href = "#tag_art12">12.</a> +‘4 the’ inserted in MS.</p> + +<p><a name = "note_art13" id = "note_art13" href = "#tag_art13">13.</a> +‘to’ in MS.</p> + +<p><a name = "note_art14" id = "note_art14" href = "#tag_art14">14.</a> +‘that’ repeated in MS.</p> + +<p><a name = "note_art15" id = "note_art15" href = "#tag_art15">15.</a> +‘1’ in MS.</p> + +<p><a name = "note_art16" id = "note_art16" href = "#tag_art16">16.</a> +Blank in MS.</p> + +<p><a name = "note_art17" id = "note_art17" href = "#tag_art17">17.</a> +‘nought’ in MS.</p> + +<p><a name = "note_art18" id = "note_art18" href = "#tag_art18">18.</a> +3 written for 2 in MS.</p> + +<p><a name = "note_art19" id = "note_art19" href = "#tag_art19">19.</a> +7 in MS.</p> + +<p><a name = "note_art20" id = "note_art20" href = "#tag_art20">20.</a> +runs on in MS.</p> + +<p><a name = "note_art21" id = "note_art21" href = "#tag_art21">21.</a> +‘so’ in MS.</p> + +<p><a name = "note_art22" id = "note_art22" href = "#tag_art22">22.</a> +‘nought’ in MS.</p> + +<p><a name = "note_art23" id = "note_art23" href = "#tag_art23">23.</a> +MS. adds here: ‘wher-vpon<i>e</i> se the table in the next side of the +next leef<i>e</i>.’</p> + +<p><a name = "note_art24" id = "note_art24" href = "#tag_art24">24.</a> +110 in MS.</p> + +<p><a name = "note_art25" id = "note_art25" href = "#tag_art25">25.</a> +0 in MS.</p> + +<p><a name = "note_art26" id = "note_art26" href = "#tag_art26">26.</a> +double in MS.</p> + +<p><a name = "note_art27" id = "note_art27" href = "#tag_art27">27.</a> +‘it hym-self<i>e</i>’ in MS.</p> + +<p><a name = "note_art28" id = "note_art28" href = "#tag_art28">28.</a> +MS. adds here: ‘it setteth<i>e</i> a-way all<i>e</i> his respect.’</p> + +<p><a name = "note_art29" id = "note_art29" href = "#tag_art29">29.</a> +‘aucterioracio<i>u</i>n’ in MS.</p> + +<p><a name = "note_art30" id = "note_art30" href = "#tag_art30">30.</a> +MS. adds here: ’w<i>i</i>t<i>h</i> an vndre-triple / other of an +vndre-triple in a triple or triplat is And after-ward<i>e</i> +w<i>i</i>t<i>h</i> out vndre-triple other vndre-triplis in the +p<i>ro</i>duct and ayene that p<i>ro</i>duct that cometh<i>e</i> of the +ledyng<i>e</i> of a digit found<i>e</i> in hym-self<i>e</i> +cubicall<i>e</i>’ /</p> + +<p><a name = "note_art31" id = "note_art31" href = "#tag_art31">31.</a> +MS. adds here: ‘as ther had be a divisio<i>u</i>n made as it is +opened<i>e</i> before.’</p> + +<p><a name = "note_art32" id = "note_art32" href = "#tag_art32">32.</a> +MS. anteriocacio<i>u</i>n.</p> + +<p><a name = "note_art33" id = "note_art33" href = "#tag_art33">33.</a> +4 in MS.</p> + +</div> + +</div> <!-- end div art --> + + +<div class = "count"> + +<span class = "pagenum">52</span> +<a name = "page52" id = "page52"> </a> + +<p class = "illustration"> +<a name = "count" id = "count"> +<img src = "images/title_count.png" width = "323" height = "36" +alt = "Accomptynge by counters." +title = "Accomptynge by counters." /></a></p> + +<p class = "mynote"> +The original text was printed as a single continuous paragraph, with +no break between speakers; all examples were shown inline. It has been +broken up for this e-text.</p> + +<span class = "linenum">116 <i>b</i>.</span> + +<h4>* ¶ The seconde dialoge of accomptynge by counters.</h4> + +<p class = "inset"><i>Mayster.</i></p> + +<p><span class = "dropcap">N</span>owe that you haue learned the commen +kyndes of Arithmetyke with the penne, you shall se the same art in +cou<i>n</i>ters: whiche feate doth not only serue for them that can not +write and rede, but also for them that can do bothe, but haue not at +some tymes theyr penne or tables redye with them. This sorte is in two +fourmes co<i>m</i>menly. The one by lynes, and the other without lynes: +in that <a class = "gloss" name = "yt" id = "yt" href = +"#gloss_yt">y<sup>t</sup></a> hath lynes, the lynes do stande for the +order of places: and in y<sup>t</sup> that hath no lynes, there must be +sette in theyr stede so many counters as shall nede, for eche lyne one, +and they shall supplye the stede of the lynes.</p> + +<p><i>S.</i> By examples I shuld better p<i>er</i>ceaue your +meanynge.</p> + +<p><i>M.</i> For example of the +<span class = "linenum">117 <i>a</i></span> +ly*nes:</p> + +<table class = "backline small float" summary = "counters example"> +<tr> +<td> </td> +<td>1</td> +<td>0 0 0 0 0</td> +<td> </td> +</tr> +<tr> +<td> </td> +<td>1</td> +<td>0 0 0 0</td> +<td> </td> +</tr> +<tr> +<td>X</td> +<td>1</td> +<td>0 0 0</td> +<td> </td> +</tr> +<tr> +<td> </td> +<td>1</td> +<td>0 0</td> +<td> </td> +</tr> +<tr> +<td> </td> +<td>1</td> +<td>0</td> +<td> </td> +</tr> +<tr> +<td> </td> +<td>1</td> +<td> </td> +<td> </td> +</tr> +</table> + +<p class = "nospace"> +Lo here you se .vi. lynes whiche stande for syxe places so that the +nethermost standeth for y<sup>e</sup> fyrst place, and the next aboue +it, for the second: and so vpward tyll you come to the hyghest, which is +the syxte lyne, and standeth for the syxte place. +<span class = "sidenote">Numeration.</span> +Now what is the valewe of euery place or lyne, you may perceaue by the +figures whiche I haue set on them, which is accordynge as you learned +before in the Numeration of figures by the penne: for the fyrste place +is the place of vnities or ones, and euery counter set in that lyne +betokeneth but one: <i>and</i> the seconde lyne is the place of 10, for +euery counter there, standeth for 10. The thyrd lyne the place of +hundredes: the fourth of thousandes: <i>and</i> so forth.</p> + +<p><i>S.</i> Syr I do perceaue that the same order is here of lynes, as +was in the other figures +<span class = "linenum">117 <i>b</i>.</span> +*by places, so that you shall not nede longer to stande about +Numeration, excepte there be any other difference.</p> + +<p><i>M.</i> Yf you do vndersta<i>n</i>de it, then how wyll you set +1543?</p> + +<table class = "backline small float" summary = "counters example"> +<tr> +<td> </td> +<td> </td> +<td> </td> +</tr> +<tr> +<td>X</td> +<td>1</td> +<td> </td> +</tr> +<tr> +<td> </td> +<td>5</td> +<td> </td> +</tr> +<tr> +<td> </td> +<td>4</td> +<td> </td> +</tr> +<tr> +<td> </td> +<td>3</td> +<td> </td> +</tr> +</table> + +<p><i>S.</i> Thus, as I suppose.</p> + +<p><i>M.</i> You haue set y<sup>e</sup> places truely, but your figures +be not mete for this vse: +<span class = "pagenum">53</span> +<a name = "page53" id = "page53"> </a> +for the metest figure in this behalfe, is the figure of a cou<i>n</i>ter +round, as you se here, where I haue expressed that same summe.</p> + +<table class = "backline float" summary = "counters example"> +<tr> +<td> </td> +<td> </td> +<td> </td> +</tr> +<tr> +<td>×</td> +<td>o</td> +<td> </td> +</tr> +<tr> +<td> </td> +<td> <span class = "five">°</span></td> +<td> </td> +</tr> +<tr> +<td> </td> +<td>o o o o</td> +<td> </td> +</tr> +<tr> +<td> </td> +<td>o o o</td> +<td> </td> +</tr> +</table> + +<p><i>S.</i> So that you haue not one figure for 2, nor 3, nor 4, and so +forth, but as many digettes as you haue, you set in the lowest lyne: and +for euery 10 you set one in the second line: and so of other. But I know +not by what reason you set that one counter for 500 betwene two +lynes.</p> + +<p><i>M.</i> you shall remember this, that when so euer you nede to set +downe 5, 50, or 500, or 5000, or so forth any other nomber, whose +numerator +<span class = "linenum">118 <i>a</i>.</span> +*is 5, you shall set one counter for it, in the next space aboue the +lyne that it hath his denomination of, as in this example of that 500, +bycause the numerator is 5, it must be set in a voyd space: and bycause +the denominator is hundred, I 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<i>m</i>bers aboue 5. And this is true bothe of digettes +and articles. And for example I wyll set downe this su<i>m</i>me +287965,</p> + +<table class = "backline float" summary = "counters example"> +<tr> +<td>X</td> +<td> </td> +</tr> +<tr> +<td> </td> +<td> o o</td> +</tr> +<tr> +<td> </td> +<td> o o<span class = "five">°</span>o</td> +</tr> +<tr> +<td>X</td> +<td> o o<span class = "five">°</span></td> +</tr> +<tr> +<td> </td> +<td>o o<span class = "five">°</span>o o</td> +</tr> +<tr> +<td> </td> +<td>o<span class = "five">°</span></td> +</tr> +<tr> +<td> </td> +<td> </td> +</tr> +</table> + +<p class = "nospace"> +which su<i>m</i>me yf you marke well, you nede none other +exa<i>m</i>ples for to lerne the numeration of +<span class = "linenum">118 <i>b</i>.</span> +*this forme. But this shal you marke, that as you dyd in the other kynde +of arithmetike, set a pricke in the places of thousa<i>n</i>des, in this +worke you shall sette a starre, as you se here.</p> + +<p class = "headnote"><span class = "headnote"> +Addition on the Counting Board.</span></p> + +<p><span class = "sidenote">Addition.</span> +<i>S.</i> Then I perceave numeration, but I praye you, howe shall I do +in this arte to adde two summes or more together?</p> + +<p><i>M.</i> The easyest way in this arte is, to adde but 2 +su<i>m</i>mes at ones together: how be it you may adde more, as I wyll +tell you anone. Therfore when you wyll adde two su<i>m</i>mes, you shall +fyrst set downe one of them, it <a class = "gloss" name = "forseth" id = +"forseth" href = "#gloss_forseth">forseth</a> not whiche, <i>and</i> +then by it drawe a lyne crosse the other lynes. And afterward set downe +the other su<i>m</i>me, so that that lyne may be betwene them, as yf you +wolde adde 2659 to 8342, you must set your su<i>m</i>mes as you se +here.</p> + +<table class = "backline float" summary = "counters example"> +<col /> +<col class = "rightline" /> +<col /> +<tr> +<td> </td> +<td> </td> +<td> </td> +</tr> +<tr> +<td>X</td> +<td>o<span class = "five">°</span>o o</td> +<td>o o</td> +</tr> +<tr> +<td> </td> +<td>o o o</td> +<td>o<span class = "five">°</span></td> +</tr> +<tr> +<td> </td> +<td>o o o o</td> +<td> <span class = "five">°</span></td> +</tr> +<tr> +<td> </td> +<td>o o</td> +<td>o<span class = "five">°</span>o o o</td> +</tr> +</table> + +<p class = "nospace"> +And then yf you lyst, you +<span class = "linenum">119 <i>a</i>.</span> +*may adde the one to the other in the same place, or els you may adde +them both together in a newe place: which waye, bycause it is moste +playnest, I wyll showe you fyrst. Therfore wyl I begynne at the +vnites, whiche in the fyrst su<i>m</i>me is but 2, <i>and</i> in +y<sup>e</sup> second su<i>m</i>me 9, that maketh 11, those do I take vp, +and for them I set 11 in the new roume, thus,</p> + +<span class = "pagenum">54</span> +<a name = "page54" id = "page54"> </a> + +<table class = "backline float" summary = "counters example"> +<col /> +<col class = "rightline" /> +<col class = "rightline" /> +<col /> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +</tr> +<tr> +<td>X</td> +<td>o<span class = "five">°</span>o o</td> +<td>o o</td> +<td> </td> +</tr> +<tr> +<td> </td> +<td>o o o</td> +<td>o<span class = "five">°</span></td> +<td> </td> +</tr> +<tr> +<td> </td> +<td>o o o o</td> +<td> <span class = "five">°</span></td> +<td>o</td> +</tr> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td>o</td> +</tr> +</table> + +<p class = "nospace"> +Then do I take vp all y<sup>e</sup> articles vnder a hundred, which in +the fyrst su<i>m</i>me are 40, and in the second summe 50, that maketh +90: or you may saye better, that in the fyrste summe there are 4 +articles of 10, and in the seconde summe 5, which make 9, but then take +hede that you sette them in theyr +<span class = "linenum">119 <i>b</i>.</span> +*ryght lynes as you se here.</p> + +<table class = "backline float" summary = "counters example"> +<col /> +<col class = "rightline" /> +<col class = "rightline" /> +<col /> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +</tr> +<tr> +<td>X</td> +<td>o<span class = "five">°</span>o o</td> +<td>o o</td> +<td> </td> +</tr> +<tr> +<td> </td> +<td>o o o</td> +<td>o<span class = "five">°</span></td> +<td> </td> +</tr> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td>o<span class = "five">°</span>o o o o</td> +</tr> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td>o</td> +</tr> +</table> + +<p class = "nospace"> +Where I haue taken awaye 40 fro<i>m</i> the fyrste su<i>m</i>me, and 50 +from y<sup>e</sup> second, and in theyr stede I haue set 90 in the +thyrde, whiche I haue set playnely y<sup>t</sup> you myght well perceaue +it: how be it seynge that 90 with the 10 that was in y<sup>e</sup> thyrd +roume all redy, doth make 100, I myghte better for those 6 +cou<i>n</i>ters set 1 in the thyrde lyne, thus:</p> + +<table class = "backline floatleft" summary = "counters example"> +<tr> +<td> </td> +<td> </td> +</tr> +<tr> +<td>X</td> +<td> </td> +</tr> +<tr> +<td> </td> +<td>o</td> +</tr> +<tr> +<td> </td> +<td> </td> +</tr> +<tr> +<td> </td> +<td>o</td> +</tr> +</table> + +<p class = "nospace"> +For it is all one summe as you may se, but it is beste, neuer to set 5 +cou<i>n</i>ters in any line, for that may be done with 1 cou<i>n</i>ter +in a hygher place.</p> + +<p><i>S.</i> I iudge that good reaso<i>n</i>, for many are vnnedefull, +where one wyll serue.</p> + +<p><i>M.</i> Well, then +<span class = "linenum">120 <i>a</i>.</span> +*wyll I adde forth of hundredes: I fynde 3 in the fyrste summe, and +6 in the seconde, whiche make 900, them do I take vp <i>and</i> set in +the thyrd roume where is one hundred all redy, to whiche I put 900, and +it wyll be 1000, therfore I set one cou<i>n</i>ter in the fourth lyne +for them all, as you se here.</p> + +<table class = "backline float" summary = "counters example"> +<col /> +<col class = "rightline" /> +<col class = "rightline" /> +<col /> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +</tr> +<tr> +<td>X</td> +<td>o<span class = "five">°</span>o o</td> +<td>o o</td> +<td>o</td> +</tr> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +</tr> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +</tr> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td>o</td> +</tr> +</table> + +<p class = "nospace"> +Then adde I y<sup>e</sup> thousandes together, whiche in the fyrst +su<i>m</i>me are 8000, <i>and</i> in y<sup>e</sup> second 2000, that +maketh 10000: them do I take vp fro<i>m</i> those two places, and for +them I set one counter in the fyfte lyne, and then appereth as +you se,</p> + +<table class = "backline floatleft" summary = "counters example"> +<tr> +<td> </td> +<td>o</td> +</tr> +<tr> +<td>X</td> +<td>o</td> +</tr> +<tr> +<td> </td> +<td> </td> +</tr> +<tr> +<td> </td> +<td> </td> +</tr> +<tr> +<td> </td> +<td>o</td> +</tr> +</table> + +<p class = "nospace"> +to be 11001, for so many doth amount of the addition of 8342 to +2659.</p> + +<p><span class = "linenum">120 <i>b</i>.</span> +*<i>S.</i> Syr, this I do perceave: but how shall I set one su<i>m</i>me +to an other, not chaungynge them to a thyrde place?</p> + +<p><i>M.</i> Marke well how I do it: I wyll adde together 65436, +and 3245, whiche fyrste I set downe thus.</p> + +<table class = "backline float" summary = "counters example"> +<col /> +<col class = "rightline" /> +<col /> +<tr> +<td> </td> +<td> </td> +<td> </td> +</tr> +<tr> +<td> </td> +<td> </td> +<td>o <span class = "five">°</span></td> +</tr> +<tr> +<td>X</td> +<td>o o o</td> +<td> <span class = "five">°</span></td> +</tr> +<tr> +<td> </td> +<td>o o</td> +<td>o o o o</td> +</tr> +<tr> +<td> </td> +<td>o o o o</td> +<td>o o o</td> +</tr> +<tr> +<td> </td> +<td> <span class = "five">°</span></td> +<td>o <span class = "five">°</span></td> +</tr> +</table> + +<p class = "nospace"> +Then do I begynne with the smalest, which in the fyrst summe is <ins +class = "correction" title = "invisible ‘5’ supplied by transcriber">5</ins>, that do I take vp, and wold put to the other 5 in +the seconde summe, sauynge that two counters can not be set in a voyd +place of 5, but for them bothe I must set 1 in the seconde lyne, which +is the place of 10, therfore I take vp the 5 of the fyrst su<i>m</i>me, +<i>and</i> the 5 of the seco<i>n</i>de, and for them I set 1 in the +seco<i>n</i>d lyne, +<span class = "linenum">121 <i>a</i>.</span> +*as you se here.</p> + +<table class = "backline float" summary = "counters example"> +<col /> +<col class = "rightline" /> +<col /> +<tr> +<td> </td> +<td> </td> +<td> </td> +</tr> +<tr> +<td> </td> +<td> </td> +<td>o<span class = "five">°</span></td> +</tr> +<tr> +<td>X</td> +<td>o o o</td> +<td> <span class = "five">°</span></td> +</tr> +<tr> +<td> </td> +<td>o o</td> +<td>o o o o</td> +</tr> +<tr> +<td> </td> +<td>o o o o</td> +<td>o o o o</td> +</tr> +<tr> +<td> </td> +<td> </td> +<td>o</td> +</tr> +</table> + +<p class = "nospace"> +Then do I lyke wayes take vp the 4 counters of the fyrste su<i>m</i>me +<i>and</i> +<span class = "pagenum">55</span> +<a name = "page55" id = "page55"> </a> +seconde lyne (which make 40) and adde them to the 4 counters of the same +lyne, in the second su<i>m</i>me, and it maketh 80, But as I sayde I +maye not conueniently set aboue 4 cou<i>n</i>ters in one lyne, therfore +to those 4 that I toke vp in the fyrst su<i>m</i>me, I take one +also of the seconde su<i>m</i>me, and then haue I taken vp 50, for +whiche 5 counters I sette downe one in the space ouer y<sup>e</sup> +second lyne, as here doth appere.</p> + +<table class = "backline float" summary = "counters example"> +<col /> +<col class = "rightline" /> +<col /> +<tr> +<td> </td> +<td> </td> +<td> </td> +</tr> +<tr> +<td> </td> +<td> </td> +<td>o<span class = "five">°</span></td> +</tr> +<tr> +<td>X</td> +<td>o o o</td> +<td> <span class = "five">°</span></td> +</tr> +<tr> +<td> </td> +<td>o o</td> +<td>o o o o</td> +</tr> +<tr> +<td> </td> +<td> </td> +<td>o<span class = "five">°</span>o o</td> +</tr> +<tr> +<td> </td> +<td> </td> +<td>o</td> +</tr> +</table> + +<p class = "nospace"> +<span class = "linenum">121 <i>b</i></span> +*and then is there 80, as well <a class = "gloss" name = "wt" id = "wt" +href = "#gloss_wt">w<sup>t</sup></a> those 4 counters, as yf I had set +downe y<sup>e</sup> other 4 also. Now do I take the 200 in the fyrste +su<i>m</i>me, and adde them to the 400 in the seconde summe, and it +maketh 600, therfore I take vp the 2 counters in the fyrste summe, and 3 +of them in the seconde summe, and for them 5 I set 1 in y<sup>e</sup> +space aboue, thus.</p> + +<table class = "backline float" summary = "counters example"> +<col /> +<col class = "rightline" /> +<col /> +<tr> +<td> </td> +<td> </td> +<td> </td> +</tr> +<tr> +<td> </td> +<td> </td> +<td>o<span class = "five">°</span></td> +</tr> +<tr> +<td>X</td> +<td>o o o</td> +<td> <span class = "five">°</span></td> +</tr> +<tr> +<td> </td> +<td> </td> +<td>o<span class = "five">°</span></td> +</tr> +<tr> +<td> </td> +<td> </td> +<td>o<span class = "five">°</span>o o</td> +</tr> +<tr> +<td> </td> +<td> </td> +<td>o</td> +</tr> +</table> + +<p class = "nospace"> +Then I take y<sup>e</sup> 3000 in y<sup>e</sup> fyrste su<i>m</i>me, +vnto whiche there are none in the second summe agreynge, therfore I do +onely remoue those 3 counters from the fyrste summe into the seconde, as +here doth appere.</p> + +<table class = "backline floatleft" summary = "counters example"> +<col class = "rightline" /> +<col /> +<tr> +<td> </td> +<td> </td> +</tr> +<tr> +<td> </td> +<td> o<span class = "five">°</span></td> +</tr> +<tr> +<td>X</td> +<td> o<span class = "five">°</span>o o</td> +</tr> +<tr> +<td> </td> +<td>o <span class = "five">°</span></td> +</tr> +<tr> +<td> </td> +<td> o<span class = "five">°</span>o o</td> +</tr> +<tr> +<td> </td> +<td> o</td> +</tr> +</table> + +<p class = "nospace"> +<span class = "linenum">122 <i>a</i></span> +*And so you see the hole su<i>m</i>me, that amou<i>n</i>teth of the +addytio<i>n</i> of 65436 with 3245 to be 6868[1]. And yf you haue marked +these two exa<i>m</i>ples well, you nede no farther enstructio<i>n</i> +in Addition of 2 only summes: but yf you haue more then two summes to +adde, you may adde them thus.</p> + +<table class = "backline float" summary = "counters example"> +<col /> +<col class = "rightline" /> +<col class = "rightline" /> +<col /> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +</tr> +<tr> +<td>X</td> +<td>o o</td> +<td>o o o o</td> +<td>o<span class = "five">°</span></td> +</tr> +<tr> +<td> </td> +<td>o<span class = "five">°</span></td> +<td>o o</td> +<td>o<span class = "five">°</span>o o o</td> +</tr> +<tr> +<td> </td> +<td>o<span class = "five">°</span>o</td> +<td>o<span class = "five">°</span>o o</td> +<td>o<span class = "five">°</span></td> +</tr> +<tr> +<td> </td> +<td>o<span class = "five">°</span>o o o</td> +<td>o<span class = "five">°</span></td> +<td> <span class = "five">°</span></td> +</tr> +</table> + +<p class = "nospace"> +Fyrst adde two of them, and then adde the thyrde, and y<sup>e</sup> +fourth, or more yf there be so many: as yf I wolde adde 2679 with 4286 +and 1391. Fyrste I adde the two fyrste summes thus. +<span class = "linenum">122 <i>b</i>.</span> +*And then I adde the thyrde thereto thus.</p> + +<table class = "backline float" summary = "counters example"> +<col /> +<col class = "rightline" /> +<col class = "rightline" /> +<col /> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +</tr> +<tr> +<td>X</td> +<td>o</td> +<td>o<span class = "five">°</span></td> +<td>o o o<span class = "five">°</span></td> +</tr> +<tr> +<td> </td> +<td>o o o</td> +<td>o<span class = "five">°</span>o o o</td> +<td>o o o</td> +</tr> +<tr> +<td> </td> +<td>o<span class = "five">°</span>o o o</td> +<td> <span class = "five">°</span>o</td> +<td> <span class = "five">°</span></td> +</tr> +<tr> +<td> </td> +<td>o</td> +<td> <span class = "five">°</span></td> +<td>o<span class = "five">°</span></td> +</tr> +</table> + +<p class = "nospace"> +And so of more yf you haue them.</p> + +<p class = "headnote"><span class = "headnote"> +Subtraction on the Counting Board.</span></p> + +<p><i>S.</i> Nowe I thynke beste that you passe forth to Subtraction, +except there be any wayes to examyn this maner of Addition, then I +thynke that were good to be knowen nexte.</p> + +<p><i>M.</i> There is the same profe here that is +<span class = "sidenote">Subtraction.</span> +in the other Addition by the penne, I 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 +<span class = "linenum">116 <i>a</i> <ins class = "correction" title = +"editor’s ‘sic’ for jump back in line numbering">(<i>sic</i>)</ins>.</span> +*to set the lesser no<i>m</i>ber fyrste, thus.</p> + +<table class = "backline float" summary = "counters example"> +<col /> +<col class = "rightline" /> +<col /> +<tr> +<td> </td> +<td> </td> +<td> </td> +</tr> +<tr> +<td>X</td> +<td>o o</td> +<td>o o<span class = "five">°</span>o</td> +</tr> +<tr> +<td> </td> +<td>o<span class = "five">°</span>o o</td> +<td>o<span class = "five">°</span>o</td> +</tr> +<tr> +<td> </td> +<td>o<span class = "five">°</span>o o o</td> +<td>o o o o</td> +</tr> +<tr> +<td> </td> +<td>o o</td> +<td>o<span class = "five">°</span></td> +</tr> +</table> + +<p class = "nospace"> +Then shall I begynne to subtracte the greatest nombres fyrste (contrary +to the vse of the penne) +<span class = "pagenum">56</span> +<a name = "page56" id = "page56"> </a> +y<sup>t</sup> is the thousandes in this exa<i>m</i>ple: therfore I fynd +amongest the thousandes 2, for which I withdrawe so many fro<i>m</i> the +seconde summe (where are 8) and so remayneth there 6, as this +exa<i>m</i>ple showeth.</p> + +<table class = "backline float" summary = "counters example"> +<col /> +<col class = "rightline" /> +<col /> +<tr> +<td> </td> +<td> </td> +<td> </td> +</tr> +<tr> +<td> </td> +<td> </td> +<td>o<span class = "five">°</span></td> +</tr> +<tr> +<td>+</td> +<td>o<span class = "five">°</span>o o</td> +<td>o<span class = "five">°</span>o</td> +</tr> +<tr> +<td> </td> +<td>o<span class = "five">°</span>o o o</td> +<td>o o o o</td> +</tr> +<tr> +<td> </td> +<td>o o</td> +<td>o<span class = "five">°</span></td> +</tr> +</table> + +<p class = "nospace"> +Then do I lyke wayes with the hundredes, of whiche in the fyrste summe +<span class = "linenum">116 <i>b</i>.</span> +*I fynde 8, and is the seconde summe but 7, out of whiche I can not take +8, therfore thus muste I do: I muste loke how moche my summe +dyffereth from 10, whiche I fynde here to be 2, then must I bate for my +su<i>m</i>me of 800, one thousande, and set downe the excesse of +hundredes, that is to saye 2, for so moche 100[0] is more then I shuld +take vp. Therfore fro<i>m</i> the fyrste su<i>m</i>me I take that 800, +and from the second su<i>m</i>me where are 6000, I take vp one +thousande, and leue 5000; but then set I downe the 200 unto the 700 +y<sup>t</sup> are there all redye, and make them 900 thus.</p> + +<table class = "backline float" summary = "counters example"> +<col /> +<col class = "rightline" /> +<col /> +<tr> +<td> </td> +<td> </td> +<td> </td> +</tr> +<tr> +<td>+</td> +<td> </td> +<td> <span class = "five">°</span></td> +</tr> +<tr> +<td> </td> +<td> </td> +<td>o<span class = "five">°</span>o o o</td> +</tr> +<tr> +<td> </td> +<td>o o<span class = "five">°</span>o o</td> +<td>o o o o</td> +</tr> +<tr> +<td> </td> +<td>o o</td> +<td>o<span class = "five">°</span></td> +</tr> +</table> + +<p class = "nospace"> +Then come I to the articles of te<i>n</i>nes where in the fyrste +su<i>m</i>me I fynde 90, +<span class = "linenum">117 <i>a</i>.</span> +*and in the seconde su<i>m</i>me but only 40: Now consyderyng that 90 +can not be bated from 40, I loke how moche y<sup>t</sup> 90 doth +dyffer from the next summe aboue it, that is 100 (or elles whiche is all +to one effecte, I loke how moch 9 doth dyffer fro<i>m</i> 10) +<i>and</i> I fynd it to be 1, then in the stede of that 90, I do +take from the second summe 100: but consyderynge that it is 10 to moche, +I set downe 1 in y<sup>e</sup> nexte lyne beneth for it, as you se +here.</p> + +<table class = "backline float" summary = "counters example"> +<col /> +<col class = "rightline" /> +<col /> +<tr> +<td> </td> +<td> </td> +<td> </td> +</tr> +<tr> +<td>+</td> +<td> </td> +<td> <span class = "five">°</span></td> +</tr> +<tr> +<td> </td> +<td> </td> +<td>o<span class = "five">°</span>o o</td> +</tr> +<tr> +<td> </td> +<td> </td> +<td> <span class = "five">°</span></td> +</tr> +<tr> +<td> </td> +<td>o o</td> +<td>o<span class = "five">°</span></td> +</tr> +</table> + +<p class = "nospace"> +Sauynge that here I haue set one counter in y<sup>e</sup> space in stede +of 5 in y<sup>e</sup> nexte lyne. And thus haue I subtracted all saue +two, which I must bate from the 6 in the second summe, and there wyll +remayne 4, thus.</p> + +<table class = "backline float" summary = "counters example"> +<col class = "rightline" /> +<col /> +<tr> +<td> </td> +<td> </td> +</tr> +<tr> +<td>=</td> +<td> <span class = "five">°</span></td> +</tr> +<tr> +<td> </td> +<td>o<span class = "five">°</span>o o</td> +</tr> +<tr> +<td> </td> +<td> <span class = "five">°</span></td> +</tr> +<tr> +<td> </td> +<td>o o o o</td> +</tr> +</table> + +<p class = "nospace"> +So y<sup>t</sup> yf I subtracte 2892 fro<i>m</i> 8746, the <a class = +"gloss" name = "remayner" id = "remayner" href = +"#gloss_remayner">remayner</a> wyll be 5854, +<span class = "linenum">117 <i>b</i>.</span> +*And that this is truely wrought, you maye proue by Addition: for yf you +adde to this remayner the same su<i>m</i>me that you dyd subtracte, then +wyll the formar su<i>m</i>me 8746 amount agayne.</p> + +<p><i>S.</i> That wyll I proue: and fyrst I set the su<i>m</i>me that +was subtracted, which was 2892, <i>and</i> the<i>n</i> the remayner +5854, thus.</p> + +<table class = "backline float" summary = "counters example"> +<col /> +<col class = "rightline" /> +<col /> +<tr> +<td> </td> +<td> </td> +<td> </td> +</tr> +<tr> +<td>||</td> +<td>o o</td> +<td> <span class = "five">°</span></td> +</tr> +<tr> +<td> </td> +<td>o<span class = "five">°</span>o o</td> +<td>o o<span class = "five">°</span>o</td> +</tr> +<tr> +<td> </td> +<td>o o<span class = "five">°</span>o o</td> +<td> <span class = "five">°</span></td> +</tr> +<tr> +<td> </td> +<td>o o</td> +<td>o o o o</td> +</tr> +</table> + +<p class = "nospace"> +Then do I adde fyrst y<sup>e</sup> 2 to 4, whiche maketh 6, so take I vp +5 of those counters, and in theyr stede I sette 1 in the space, as here +appereth.</p> + +<table class = "backline float" summary = "counters example"> +<col /> +<col class = "rightline" /> +<col /> +<tr> +<td> </td> +<td> </td> +<td> </td> +</tr> +<tr> +<td>||</td> +<td>o o</td> +<td> <span class = "five">°</span></td> +</tr> +<tr> +<td> </td> +<td>o<span class = "five">°</span>o o</td> +<td>o<span class = "five">°</span>o o</td> +</tr> +<tr> +<td> </td> +<td>o<span class = "five">°</span>o o o</td> +<td> <span class = "five">°</span></td> +</tr> +<tr> +<td> </td> +<td> </td> +<td>o<span class = "five">°</span></td> +</tr> +</table> + +<p class = "nospace"> +<span class = "linenum">118 <i>a</i>.</span> +*Then do I adde the 90 nexte aboue to the 50, and it maketh 140, +therfore I take vp those 6 counters, and for them I sette 1 to the +hundredes in y<sup>e</sup> thyrde lyne, <i>and</i> 4 in y<sup>e</sup> +<span class = "pagenum">57</span> +<a name = "page57" id = "page57"> </a> +second lyne, thus.</p> + +<table class = "backline float" summary = "counters example"> +<col /> +<col class = "rightline" /> +<col /> +<tr> +<td> </td> +<td> </td> +<td> </td> +</tr> +<tr> +<td>||</td> +<td>o o</td> +<td> <span class = "five">°</span></td> +</tr> +<tr> +<td> </td> +<td>o<span class = "five">°</span>o o</td> +<td>o<span class = "five">°</span>o o o</td> +</tr> +<tr> +<td> </td> +<td> </td> +<td>o o o o</td> +</tr> +<tr> +<td> </td> +<td> </td> +<td> <span class = "five">°</span>o</td> +</tr> +</table> + +<p class = "nospace"> +Then do I come to the hundredes, of whiche I fynde 8 in the fyrst summe, +and 9 in y<sup>e</sup> second, that maketh 1700, therfore I take vp +those 9 counters, and in theyr stede I sette 1 in the .iiii. lyne, and 1 +in the space nexte beneth, and 2 in the thyrde lyne, as you se here.</p> + +<table class = "backline float" summary = "counters example"> +<col /> +<col class = "rightline" /> +<col /> +<tr> +<td> </td> +<td> </td> +<td> </td> +</tr> +<tr> +<td>||</td> +<td>o o</td> +<td>o<span class = "five">°</span></td> +</tr> +<tr> +<td> </td> +<td> </td> +<td>o<span class = "five">°</span>o</td> +</tr> +<tr> +<td> </td> +<td> </td> +<td>o o o o</td> +</tr> +<tr> +<td> </td> +<td> </td> +<td>o<span class = "five">°</span></td> +</tr> +</table> + +<p class = "nospace"> +Then is there lefte in the fyrste summe but only 2000, whiche I shall +take vp from thence, and set +<span class = "linenum">118 <i>b</i>.</span> +*in the same lyne in y<sup>e</sup> second su<i>m</i>me, to y<sup>e</sup> +one y<sup>t</sup> is there all redy: <i>and</i> then wyll the hole +su<i>m</i>me appere (as you may wel se) to be 8746, which was +y<sup>e</sup> fyrst grosse summe, <i>and</i> therfore I do perceaue, +that I hadde well subtracted before.</p> + +<table class = "backline float" summary = "counters example"> +<col class = "rightline" /> +<col /> +<tr> +<td> </td> +<td> </td> +</tr> +<tr> +<td>X</td> +<td>o o<span class = "five">°</span>o</td> +</tr> +<tr> +<td> </td> +<td>o o<span class = "five">°</span></td> +</tr> +<tr> +<td> </td> +<td>o o o o</td> +</tr> +<tr> +<td> </td> +<td> <span class = "five">°</span>o</td> +</tr> +</table> + +<p class = "nospace"> +And thus you may se how Subtraction maye be tryed by Addition.</p> + +<p><i>S.</i> I perceaue the same order here w<sup>t</sup> +cou<i>n</i>ters, y<sup>t</sup> I lerned before in figures.</p> + +<p><i>M.</i> Then let me se howe can you trye Addition by +Subtraction.</p> + +<p><i>S.</i> Fyrste I wyl set forth this exa<i>m</i>ple of +Additio<i>n</i> where I haue added 2189 to 4988, and the hole +su<i>m</i>me appereth to be 7177,</p> + +<table class = "backline float" summary = "counters example"> +<col /> +<col class = "rightline" /> +<col class = "rightline" /> +<col /> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +</tr> +<tr> +<td>||</td> +<td>o o</td> +<td>o o o o</td> +<td>o<span class = "five">°</span>o</td> +</tr> +<tr> +<td> </td> +<td>o</td> +<td>o<span class = "five">°</span>o o o</td> +<td>o</td> +</tr> +<tr> +<td> </td> +<td>o<span class = "five">°</span>o o</td> +<td>o<span class = "five">°</span>o o</td> +<td>o<span class = "five">°</span>o</td> +</tr> +<tr> +<td> </td> +<td>o<span class = "five">°</span>o o o</td> +<td>o<span class = "five">°</span>o o</td> +<td>o<span class = "five">°</span>o</td> +</tr> +</table> + +<p class = "nospace"> +<span class = "linenum">119 <i>a</i>.</span> +*Nowe to trye whether that su<i>m</i>me be well added or no, I wyll +subtract one of the fyrst two su<i>m</i>mes from the thyrd, and yf I +haue well done y<sup>e</sup> remayner wyll be lyke that other +su<i>m</i>me. As for example: I wyll subtracte the fyrste summe +from the thyrde, whiche I set thus in theyr order.</p> + +<table class = "backline float" summary = "counters example"> +<col /> +<col class = "rightline" /> +<col /> +<tr> +<td> </td> +<td> </td> +<td> </td> +</tr> +<tr> +<td>||</td> +<td>o o</td> +<td>o<span class = "five">°</span>o</td> +</tr> +<tr> +<td> </td> +<td>o</td> +<td>o</td> +</tr> +<tr> +<td> </td> +<td>o<span class = "five">°</span>o o</td> +<td>o<span class = "five">°</span>o</td> +</tr> +<tr> +<td> </td> +<td>o<span class = "five">°</span>o o o</td> +<td>o<span class = "five">°</span>o</td> +</tr> +</table> + +<p class = "nospace"> +Then do I subtract 2000 of the fyrste summe fro<i>m</i> y<sup>e</sup> +second su<i>m</i>me, and then remayneth there 5000 thus.</p> + +<table class = "backline float" summary = "counters example"> +<col /> +<col class = "rightline" /> +<col /> +<tr> +<td> </td> +<td> </td> +<td> </td> +</tr> +<tr> +<td>X</td> +<td> </td> +<td> <span class = "five">°</span></td> +</tr> +<tr> +<td> </td> +<td>o</td> +<td>o</td> +</tr> +<tr> +<td> </td> +<td>o<span class = "five">°</span>o o</td> +<td>o<span class = "five">°</span>o</td> +</tr> +<tr> +<td> </td> +<td>o<span class = "five">°</span>o o o</td> +<td>o<span class = "five">°</span>o</td> +</tr> +</table> + +<p class = "nospace"> +Then in the thyrd lyne, I subtract y<sup>e</sup> 100 of the fyrste +summe, fro<i>m</i> the second su<i>m</i>me, where is onely 100 also, and +then in y<sup>e</sup> thyrde lyne <a class = "gloss" name = "resteth1" +id = "resteth1" href = "#gloss_resteth">resteth</a> nothyng. Then in the +second lyne with his space ouer hym, I fynde 80, which I shuld +subtract +<span class = "linenum">119 <i>b</i>.</span> +*from the other su<i>m</i>me, then seyng there are but only 70 I must +take it out of some hygher summe, which is here only 5000, therfore I +take vp 5000, and seyng that it is to moch by 4920, I 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.</p> + +<table class = "backline float" summary = "counters example"> +<col /> +<col class = "rightline" /> +<col /> +<tr> +<td> </td> +<td> </td> +<td> </td> +</tr> +<tr> +<td>||</td> +<td> </td> +<td>o o o o</td> +</tr> +<tr> +<td> </td> +<td> </td> +<td>o<span class = "five">°</span>o o o</td> +</tr> +<tr> +<td> </td> +<td> </td> +<td>o<span class = "five">°</span>o o o</td> +</tr> +<tr> +<td> </td> +<td>o<span class = "five">°</span>o o o</td> +<td>o<span class = "five">°</span>o</td> +</tr> +</table> + +<p class = "nospace"> +Yet remayneth there in the fyrst su<i>m</i>me 9, to be bated from the +second summe, where in that place of vnities dothe appere only 7, then I +muste bate a hygher su<i>m</i>me, that is to saye 10, but seynge that 10 +is more then 9 (which I shulde abate) by 1, therfore shall I take vp one +counter from the seconde lyne, <i>and</i> set downe the same in the +fyrst +<span class = "linenum">120 <i>a</i>.</span> +*or +<span class = "pagenum">58</span> +<a name = "page58" id = "page58"> </a> +lowest lyne, as you se here.</p> + +<table class = "backline float" summary = "counters example"> +<col class = "rightline" /> +<col /> +<tr> +<td> </td> +<td> </td> +</tr> +<tr> +<td>||</td> +<td>o o o o</td> +</tr> +<tr> +<td> </td> +<td>o<span class = "five">°</span>o o o</td> +</tr> +<tr> +<td> </td> +<td>o<span class = "five">°</span>o o</td> +</tr> +<tr> +<td> </td> +<td>o<span class = "five">°</span>o o</td> +</tr> +</table> + +<p class = "nospace"> +And so haue I ended this worke, <i>and</i> the su<i>m</i>me appereth to +be y<sup>e</sup> same, whiche was y<sup>e</sup> seconde summe of my +addition, and therfore I perceaue, I haue wel done.</p> + +<p><i>M.</i> To stande longer about this, it is but folye: excepte that +this you maye also vnderstande, that many do begynne to subtracte with +counters, not at the hyghest su<i>m</i>me, as I haue taught you, but at +the nethermoste, as they do vse to adde: and when the summe to be +abatyd, in any lyne appeareth greater then the other, then do they +borowe one of the next hygher roume, as for example: yf they shuld abate +1846 from 2378, they set y<sup>e</sup> summes thus.</p> + +<table class = "backline float" summary = "counters example"> +<col /> +<col class = "rightline" /> +<col /> +<tr> +<td> </td> +<td> </td> +<td> </td> +</tr> +<tr> +<td>||</td> +<td>o</td> +<td>o o</td> +</tr> +<tr> +<td> </td> +<td>o<span class = "five">°</span>o o</td> +<td>o o o</td> +</tr> +<tr> +<td> </td> +<td>o o o o</td> +<td>o<span class = "five">°</span>o</td> +</tr> +<tr> +<td> </td> +<td>o<span class = "five">°</span></td> +<td>o<span class = "five">°</span>o o</td> +</tr> +</table> + +<p class = "nospace"> +<span class = "linenum">120 <i>b</i>.</span> +*And fyrste they take 6 whiche is in the lower lyne, and his space from +8 in the same roumes, in y<sup>e</sup> second su<i>m</i>me, and yet +there remayneth 2 counters in the lowest lyne. Then in the second lyne +must 4 be subtracte from 7, and so remayneth there 3. Then 8 in the +thyrde lyne and his space, from 3 of the second summe can not be, +therfore do they bate it from a hygher roume, that is, from 1000, and +bycause that 1000 is to moch by 200, therfore must I sette downe 200 in +the thyrde lyne, after I haue taken vp 1000 from the fourth lyne: then +is there yet 1000 in the fourth lyne of the fyrst summe, whiche yf I +withdrawe from the seconde summe, then doth all y<sup>e</sup> figures +stande in this order.</p> + +<table class = "backline float" summary = "counters example"> +<col class = "rightline" /> +<col /> +<tr> +<td> </td> +<td> </td> +</tr> +<tr> +<td>||</td> +<td></td> +</tr> +<tr> +<td> </td> +<td> <span class = "five">°</span></td> +</tr> +<tr> +<td> </td> +<td>o o o</td> +</tr> +<tr> +<td> </td> +<td>o o</td> +</tr> +</table> + +<p class = "nospace"> +So that (as you se) it differeth not greatly whether you begynne +subtractio<i>n</i> at the hygher lynes, or at +<span class = "linenum">121 <i>a</i>.</span> +*the lower. How be it, as some menne lyke the one waye beste, so some +lyke the other: therfore you now knowyng bothe, may vse whiche you +lyst.</p> + +<p class = "headnote"><span class = "headnote"> +Multiplication by Counters.</span></p> + +<p><span class = "sidenote">Multiplication.</span> +But nowe touchynge Multiplicatio<i>n</i>: you shall set your +no<i>m</i>bers in two roumes, as you dyd in those two other kyndes, but +so that the multiplier be set in the fyrste roume. Then shall you begyn +with the hyghest no<i>m</i>bers of y<sup>e</sup> seconde roume, and +multiply them fyrst after this sort. Take that ouermost lyne in your +fyrst workynge, as yf it were the lowest lyne, setting on it some +mouable marke, as you lyste, and loke how many counters be in hym, take +them vp, and for them set downe the hole multyplyer, so many tymes as +you toke vp counters, reckenyng, I saye that lyne for the vnites: +<i>and</i> when you haue so done with the hygheest no<i>m</i>ber then +come to the nexte lyne beneth, <i>and</i> do euen so with it, and so +with y<sup>e</sup> next, tyll you haue done all. And yf there be any +nomber in a space, then for it +<span class = "linenum">121 <i>b</i>.</span> +*shall you take y<sup>e</sup> multiplyer 5 tymes, and then must you +recken that lyne for the vnites whiche is nexte beneth that space: or +els +<span class = "pagenum">59</span> +<a name = "page59" id = "page59"> </a> +after a shorter way, you shall take only halfe the multyplyer, but then +shall you take the lyne nexte aboue that space, for the lyne of vnites: +but in suche workynge, yf chau<i>n</i>ce your multyplyer be an odde +nomber, so that you can not take the halfe of it iustly, then muste you +take the greater halfe, and set downe that, as if that it were the iuste +halfe, and farther you shall set one cou<i>n</i>ter in the space beneth +that line, which you recken for the lyne of vnities, or els only remoue +forward the same that is to be multyplyed.</p> + +<p><i>S.</i> Yf you set forth an example hereto I thynke I shal perceaue +you.</p> + +<p><i>M.</i> Take this exa<i>m</i>ple: I wold multiply 1542 by 365, +therfore I set y<sup>e</sup> nombers thus.</p> + +<table class = "backline float" summary = "counters example"> +<col /> +<col class = "rightline" /> +<col /> +<tr> +<td> </td> +<td> </td> +<td> </td> +</tr> +<tr> +<td>||</td> +<td> </td> +<td>o</td> +</tr> +<tr> +<td> </td> +<td>o o o</td> +<td> <span class = "five">°</span></td> +</tr> +<tr> +<td> </td> +<td>o<span class = "five">°</span></td> +<td>o o o o</td> +</tr> +<tr> +<td> </td> +<td> <span class = "five">°</span></td> +<td>o o</td> +</tr> +</table> + +<p class = "nospace"> +<span class = "linenum">122 <i>a</i>.</span> +*Then fyrste I begynne at the 1000 in y<sup>e</sup> hyghest roume, as yf +it were y<sup>e</sup> fyrst place, & I take it vp, settynge downe +for it so often (that is ones) the multyplyer, which is 365, thus, as +you se here:</p> + +<table class = "backline float" summary = "counters example"> +<col /> +<col class = "rightline" /> +<col class = "rightline" /> +<col /> +<col /> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td class = "plain"> </td> +</tr> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td>o o o</td> +<td class = "plain"> </td> +</tr> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td>o<span class = "five">°</span></td> +<td class = "plain"> </td> +</tr> +<tr> +<td>X</td> +<td> </td> +<td> <span class = "five">°</span></td> +<td> </td> +<td class = "plain"> +<img src = "images/finger_left.gif" width = "30" height = "13" +alt = "<--" /></td> +</tr> +<tr> +<td> </td> +<td>o o o</td> +<td> </td> +<td> <span class = "five">°</span></td> +<td class = "plain"> </td> +</tr> +<tr> +<td> </td> +<td>o<span class = "five">°</span></td> +<td>o o o o</td> +<td> </td> +<td class = "plain"> </td> +</tr> +<tr> +<td> </td> +<td> <span class = "five">°</span></td> +<td>o o</td> +<td> </td> +<td class = "plain"> </td> +</tr> +</table> + +<p class = "nospace"> +where for the one counter taken vp from the fourth lyne, I haue +sette downe other 6, whiche make y<sup>e</sup> su<i>m</i>me of the +multyplyer, reckenynge that fourth lyne, as yf it were the fyrste: +whiche thyng I haue marked by the hand set at the begynnyng of +y<sup>e</sup> same,</p> + +<p><i>S.</i> I perceaue this well: for in dede, this summe that you haue +set downe is 365000, for so moche doth amount +<span class = "linenum">122 <i>b</i>.</span> +*of 1000, multiplyed by 365.</p> + +<p><i>M.</i> Well the<i>n</i> to go forth, in the nexte space I fynde +one counter which I remoue forward but take not vp, but do (as in such +case I must) set downe the greater halfe of my multiplier (seyng it is +an odde no<i>m</i>ber) which is 182, <i>and</i> here I do styll let that +fourth place stand, as yf it were y<sup>e</sup> fyrst:</p> + +<table class = "backline float" summary = "counters example"> +<col /> +<col class = "rightline" /> +<col class = "rightline" /> +<col class = "rightline" /> +<col /> +<col /> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td>o o o</td> +<td>o</td> +<td class = "plain"> </td> +</tr> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td>o<span class = "five">°</span></td> +<td>o<span class = "five">°</span>o o</td> +<td class = "plain"> </td> +</tr> +<tr> +<td>||</td> +<td> </td> +<td> </td> +<td> <span class = "five">°</span></td> +<td>o<span class = "five">°</span>o</td> +<td class = "plain"> +<img src = "images/finger_left.gif" width = "30" height = "13" +alt = "<--" /></td> +</tr> +<tr> +<td> </td> +<td>o o o</td> +<td> </td> +<td> </td> +<td> <span class = "five">°</span></td> +<td class = "plain"> </td> +</tr> +<tr> +<td> </td> +<td>o<span class = "five">°</span></td> +<td>o o o o</td> +<td> </td> +<td> </td> +<td class = "plain"> </td> +</tr> +<tr> +<td> </td> +<td> <span class = "five">°</span></td> +<td>o o</td> +<td> </td> +<td> </td> +<td class = "plain"> </td> +</tr> +</table> + +<p class = "nospace"> +as in this fourme you se, where I haue set this multiplycatio<i>n</i> +with y<sup>e</sup> other: but for the ease of your +vndersta<i>n</i>dynge, I haue set a lytell lyne betwene them: now +shulde they both in one su<i>m</i>me stand thus.</p> + +<table class = "backline float" summary = "counters example"> +<col /> +<col class = "rightline" /> +<col class = "rightline" /> +<col /> +<col /> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td>o o o o o</td> +<td class = "plain"> </td> +</tr> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td>o o o o</td> +<td class = "plain"> </td> +</tr> +<tr> +<td>||</td> +<td> </td> +<td> </td> +<td>o<span class = "five">°</span>o</td> +<td class = "plain"> +<img src = "images/finger_left.gif" width = "30" height = "13" +alt = "<--" /></td> +</tr> +<tr> +<td> </td> +<td>o o o</td> +<td> </td> +<td> <span class = "five">°</span></td> +<td class = "plain"> </td> +</tr> +<tr> +<td> </td> +<td>o<span class = "five">°</span></td> +<td>o o o o</td> +<td> </td> +<td class = "plain"> </td> +</tr> +<tr> +<td> </td> +<td> <span class = "five">°</span></td> +<td>o o</td> +<td> </td> +<td class = "plain"> </td> +</tr> +</table> + +<p class = "nospace"> +<span class = "linenum">123 <i>a</i>.</span> +*Howe be it an other fourme to multyplye suche cou<i>n</i>ters i<i>n</i> +space is this: Fyrst to remoue the fynger to the lyne nexte benethe +y<sup>e</sup> space, <i>and</i> then to take vp y<sup>e</sup> +cou<i>n</i>ter, <i>and</i> to set downe y<sup>e</sup> multiplyer .v. +tymes, as here you se.</p> + +<!-- 5th column typo 35500 for 36500 --> + +<table class = "backline" summary = "counters example"> +<col /> +<col class = "rightline" /> +<col class = "rightline" /> +<col class = "rightline" /> +<col class = "rightline" /> +<col class = "rightline" /> +<col class = "rightline" /> +<col class = "rightline" /> +<col class = "rightline" /> +<col /> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td>o o o</td> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +</tr> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td>o<span class = "five">°</span></td> +<td>o o o</td> +<td>o o o</td> +<td>o o o</td> +<td>o o o</td> +<td>o o o</td> +<td> </td> +</tr> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td> <span class = "five">°</span></td> +<td>o<span class = "five">°</span></td> +<td><ins class = "correction" title = "shown as printed"> <span +class = "five">°</span></ins></td> +<td>o<span class = "five">°</span></td> +<td>o<span class = "five">°</span></td> +<td>o<span class = "five">°</span></td> +<td> </td> +</tr> +<tr> +<td> +<span class = "finger left"> +<img src = "images/finger.gif" width = "30" height = "13" +alt = "-->" /></span> +X</td> +<td>o o o</td> +<td> </td> +<td> </td> +<td> <span class = "five">°</span></td> +<td> <span class = "five">°</span></td> +<td> <span class = "five">°</span></td> +<td> <span class = "five">°</span></td> +<td> <span class = "five">°</span></td> +<td> </td> +</tr> +<tr> +<td> </td> +<td>o<span class = "five">°</span></td> +<td>o o o o</td> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +</tr> +<tr> +<td> </td> +<td> <span class = "five">°</span></td> +<td>o o</td> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +</tr> +</table> + +<p class = "nospace"> +Which su<i>m</i>mes yf you do adde together into one su<i>m</i>me, you +shal p<i>er</i>ceaue that it wyll be y<sup>e</sup> +<span class = "pagenum">60</span> +<a name = "page60" id = "page60"> </a> +same y<sup>t</sup> appeareth of y<sup>e</sup> other worki<i>n</i>g +before, so that +<span class = "linenum">123 <i>b</i>.</span> +*bothe sortes are to one entent, but as the other is much shorter, so +this is playner to reason, for suche as haue had small exercyse in this +arte. Not withstandynge you maye adde them in your mynde before you +sette them downe, as in this exa<i>m</i>ple, you myghte haue sayde 5 +tymes 300 is 1500, <i>and</i> 5 tymes 60 is 300, also 5 tymes 5 is 25, +whiche all put together do make 1825, which you maye at one tyme set +downe yf you lyste. But nowe to go forth, I must remoue the hand to +the nexte counters, whiche are in the second lyne, and there must I take +vp those 4 counters, settynge downe for them my multiplyer 4 tymes, +whiche thynge other I maye do at 4 tymes seuerally, or elles I may +gather that hole summe in my mynde fyrste, and then set it downe: as to +saye 4 tymes 300 is 1200: 4 tymes 60 are 240: and 4 tymes 5 make 20: +y<sup>t</sup> is in all 1460, y<sup>t</sup> shall I set downe also: as +here you se.</p> + +<table class = "backline float" summary = "counters example"> +<col /> +<col class = "rightline" /> +<col class = "rightline" /> +<col class = "rightline" /> +<col /> +<col /> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td> <span class = "five">°</span></td> +<td> </td> +<td class = "plain"> </td> +</tr> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td>o o o o</td> +<td>o</td> +<td class = "plain"> </td> +</tr> +<tr> +<td>X</td> +<td> </td> +<td> </td> +<td>o<span class = "five">°</span>o</td> +<td>o o o o</td> +<td class = "plain"> </td> +</tr> +<tr> +<td> </td> +<td>o o o</td> +<td> </td> +<td> <span class = "five">°</span></td> +<td>o<span class = "five">°</span></td> +<td class = "plain"> </td> +</tr> +<tr> +<td> </td> +<td>o<span class = "five">°</span></td> +<td> </td> +<td> </td> +<td> </td> +<td class = "plain"><img src = "images/finger_left.gif" width = "30" +height = "13" +alt = "<--" /></td> +</tr> +<tr> +<td> </td> +<td> <span class = "five">°</span></td> +<td>o o</td> +<td> </td> +<td> </td> +<td class = "plain"> </td> +</tr> +</table> + +<p class = "nospace"> +<span class = "linenum">124 <i>a</i>.</span> +*whiche yf I ioyne in one summe with the formar nombers, it wyll appeare +thus.</p> + +<table class = "backline floatleft" summary = "counters example"> +<col /> +<col class = "rightline" /> +<col class = "rightline" /> +<col /> +<tr> +<td class = "plain"> </td> +<td> </td> +<td> </td> +<td> <span class = "five">°</span></td> +</tr> +<tr> +<td class = "plain"> </td> +<td> </td> +<td> </td> +<td>o<span class = "five">°</span></td> +</tr> +<tr> +<td class = "plain"> </td> +<td> </td> +<td> </td> +<td>o o</td> +</tr> +<tr> +<td class = "plain"> +<img src = "images/finger.gif" width = "30" height = "13" +alt = "<--" /></td> +<td>o o o</td> +<td> </td> +<td>o</td> +</tr> +<tr> +<td class = "plain"> </td> +<td>o<span class = "five">°</span></td> +<td> </td> +<td> </td> +</tr> +<tr> +<td class = "plain"> </td> +<td> <span class = "five">°</span></td> +<td>o o</td> +<td> </td> +</tr> +</table> + +<p class = "nospace"> +Then to ende this multiplycation, I remoue the fynger to the lowest +lyne, where are onely 2, them do I take vp, and in theyr stede do I set +downe twyse 365, that is 730, for which I set +<span class = "linenum">124 <i>b</i>.</span> +*one in the space aboue the thyrd lyne for 500, and 2 more in the thyrd +lyne with that one that is there all redye, and the reste in theyr +order, <i>and</i> so haue I ended the hole summe thus.</p> + +<table class = "backline float" summary = "counters example"> +<col class = "rightline" /> +<col class = "rightline" /> +<col /> +<tr> +<td> </td> +<td> </td> +<td> <span class = "five">°</span></td> +</tr> +<tr> +<td> </td> +<td> </td> +<td>o<span class = "five">°</span></td> +</tr> +<tr> +<td> </td> +<td> </td> +<td>o o</td> +</tr> +<tr> +<td>o o o</td> +<td> </td> +<td>o<span class = "five">°</span>o o</td> +</tr> +<tr> +<td>o<span class = "five">°</span></td> +<td> </td> +<td>o o o</td> +</tr> +<tr> +<td> <span class = "five">°</span></td> +<td> </td> +<td> </td> +</tr> +</table> + +<p class = "nospace"> +Wherby you se, that 1542 (which is the nomber of yeares syth Ch[r]ystes +incarnation) beyng multyplyed by 365 <ins class = "correction" title = +"open parenthesis missing">(which</ins> is the nomber of dayes in one +yeare) dothe amounte vnto 562830, which declareth y<sup>e</sup> +no<i>m</i>ber of daies sith Chrystes incarnatio<i>n</i> vnto the ende of +1542<a class = "tag" name = "tag_count1" id = "tag_count1" href = +"#note_count1">1</a> +yeares. (besyde 385 dayes and 12 houres for lepe yeares).</p> + +<p><i>S.</i> Now wyll I proue by an other exa<i>m</i>ple, as this: 40 +labourers (after 6 d. y<sup>e</sup> day for eche man) haue wrought +28 dayes, I wold +<span class = "linenum">125 <i>a</i>.</span> +*know what theyr wages doth amou<i>n</i>t vnto: In this case muste I +worke doublely: fyrst I must multyplye the nomber of the labourers by +y<sup>e</sup> wages of a man for one day, so wyll y<sup>e</sup> charge +of one daye amount: then secondarely shall I multyply that charge of one +daye, by the hole nomber of dayes, <i>and</i> so wyll the hole summe +appeare: fyrst therefore I shall set the su<i>m</i>mes thus.</p> + +<table class = "backline floatleft" summary = "counters example"> +<col class = "rightline" /> +<col /> +<tr> +<td> </td> +<td> </td> +</tr> +<tr> +<td> </td> +<td> </td> +</tr> +<tr> +<td> </td> +<td> </td> +</tr> +<tr> +<td> </td> +<td>o o o o</td> +</tr> +<tr> +<td>o<span class = "five">°</span></td> +<td> </td> +</tr> +</table> + +<span class = "pagenum">61</span> +<a name = "page61" id = "page61"> </a> +<p class = "nospace"> +Where in the fyrste space is the multyplyer (y<sup>t</sup> is one dayes +wages for one man) <i>and</i> in the second space is set the nomber of +the worke men to be multyplyed: the<i>n</i> saye I, 6 tymes 4 +(reckenynge that second lyne as the lyne of vnites) maketh 24, for +whiche summe I shulde set 2 counters in the thyrde lyne, and 4 in the +seconde, therfore do I set 2 in the thyrde lyne, and let the 4 stand +styll in the seconde lyne, thus.* +<span class = "linenum">125 <i>b</i>.</span></p> + +<table class = "backline float" summary = "counters example"> +<col class = "rightline" /> +<col /> +<tr> +<td> </td> +<td> </td> +</tr> +<tr> +<td> </td> +<td> </td> +</tr> +<tr> +<td> </td> +<td>o o</td> +</tr> +<tr> +<td> </td> +<td>o o o o</td> +</tr> +<tr> +<td> </td> +<td> </td> +</tr> +</table> + +<p class = "nospace"> +So <a class = "gloss" name = "apwereth" id = "apwereth" href = +"#gloss_apwereth">apwereth</a> the hole dayes wages to be 240d’. that is +20 s. Then do I multiply agayn the same summe by the no<i>m</i>ber +of dayes and fyrste I sette the nombers, thus.</p> + +<table class = "backline floatleft" summary = "counters example"> +<col class = "rightline" /> +<col /> +<tr> +<td> </td> +<td> </td> +</tr> +<tr> +<td> </td> +<td> </td> +</tr> +<tr> +<td> </td> +<td>o o</td> +</tr> +<tr> +<td>o o</td> +<td>o o o o</td> +</tr> +<tr> +<td>o<span class = "five">°</span>o o</td> +<td> </td> +</tr> +</table> + +<p class = "nospace"> +The<i>n</i> 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<sup>t</sup> is twyse, +then wyll y<sup>e</sup> su<i>m</i>me stande thus.</p> + +<table class = "backline float" summary = "counters example"> +<col class = "rightline" /> +<col /> +<tr> +<td> </td> +<td> </td> +</tr> +<tr> +<td> </td> +<td> <span class = "five">°</span></td> +</tr> +<tr> +<td> </td> +<td>o<span class = "five">°</span></td> +</tr> +<tr> +<td> </td> +<td>o o o o</td> +</tr> +<tr> +<td> </td> +<td> </td> +</tr> +</table> + +<p class = "nospace"> +Then come I to y<sup>e</sup> seconde lyne, and take vp those 4 +cou<i>n</i>ters, settynge for them the multiplyer foure tymes, so wyll +the hole summe appeare thus.* +<span class = "linenum">126 <i>a</i>.</span></p> + +<table class = "backline floatleft" summary = "counters example"> +<col class = "rightline" /> +<col /> +<tr> +<td> </td> +<td> </td> +</tr> +<tr> +<td> </td> +<td>o<span class = "five">°</span></td> +</tr> +<tr> +<td> </td> +<td>o<span class = "five">°</span>o</td> +</tr> +<tr> +<td> </td> +<td>o o</td> +</tr> +<tr> +<td> </td> +<td> </td> +</tr> +</table> + +<p class = "nospace"> +So is the hole wages of 40 workeme<i>n</i>, for 28 dayes (after 6d’. +eche daye for a man) 6720d’. that is 560 s. or 28 l’i.</p> + +<p class = "headnote"><span class = "headnote"> +Division on the Counting Board.</span></p> + +<p><span class = "sidenote">Diuision.</span> +<i>M.</i> Now if you wold proue Multiplycatio<i>n</i>, the surest way is +by Dyuision: therfore wyll I ouer passe it tyll I haue taught you +y<sup>e</sup> arte of Diuision, whiche you shall worke thus. Fyrste +sette downe the Diuisor for feare of forgettynge, and then set the +nomber that shalbe deuided, at y<sup>e</sup> ryghte syde, so farre from +the diuisor, that the quotient may be set betwene them: as for +exa<i>m</i>ple: Yf 225 shepe cost 45 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<i>m</i>ber of +<span class = "linenum">126 <i>b</i>.</span> +*shepe, whiche is 225, these two nombers therfore I sette thus.</p> + +<table class = "backline float" summary = "counters example"> +<col class = "rightline" /> +<col class = "rightline" /> +<col /> +<tr> +<td> </td> +<td> </td> +<td> </td> +</tr> +<tr> +<td> </td> +<td> </td> +<td> </td> +</tr> +<tr> +<td>o o</td> +<td> </td> +<td>o<span class = "five">°</span>o o o</td> +</tr> +<tr> +<td>o o</td> +<td> </td> +<td> </td> +</tr> +<tr> +<td> <span class = "five">°</span></td> +<td> </td> +<td> </td> +</tr> +</table> + +<p class = "nospace"> +Then begynne I at the hyghest lyne of the diuident, and seke how often I +may haue the diuisor therin, and that maye I do 4 tymes, then say I, 4 +tymes 2 are 8, whyche yf I take from 9, there resteth but 1, thus</p> + +<table class = "backline float" summary = "counters example"> +<col class = "rightline" /> +<col class = "rightline" /> +<col /> +<tr> +<td> </td> +<td> </td> +<td> </td> +</tr> +<tr> +<td> </td> +<td> </td> +<td> </td> +</tr> +<tr> +<td>o o</td> +<td> </td> +<td>o</td> +</tr> +<tr> +<td>o o</td> +<td> </td> +<td> </td> +</tr> +<tr> +<td> <span class = "five">°</span></td> +<td>o o o o</td> +<td> </td> +</tr> +</table> + +<p class = "nospace"> +And bycause I founde the diuisor 4 tymes in the diuidente, I haue +set (as you se) 4 in the myddle roume, which +<span class = "linenum">127 <i>a</i>.</span> +*is the place of the quotient: but now must I take the reste of the +diuisor as often out of the remayner: therfore come +<span class = "pagenum">62</span> +<a name = "page62" id = "page62"> </a> +I to the seconde lyne of the diuisor, sayeng 2 foure tymes make 8, +take 8 from 10, <i>and</i> there resteth 2, thus.</p> + +<table class = "backline float" summary = "counters example"> +<col /> +<col class = "rightline" /> +<col class = "rightline" /> +<col /> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +</tr> +<tr> +<td>||</td> +<td> </td> +<td> </td> +<td> </td> +</tr> +<tr> +<td> </td> +<td>o o</td> +<td> </td> +<td> </td> +</tr> +<tr> +<td> </td> +<td>o o</td> +<td> </td> +<td>o o</td> +</tr> +<tr> +<td> </td> +<td> <span class = "five">°</span></td> +<td>o o o o</td> +<td> </td> +</tr> +</table> + +<p class = "nospace"> +Then come I to the lowest nomber, which is 5, and multyply it 4 tymes, +so is it 20, that take I from 20, and there remayneth nothynge, so that +I se my quotient to be 4, whiche are in valewe shyllynges, for so was +the diuident: and therby I knowe, that yf 225 shepe dyd coste +45 l’i. euery shepe coste 4 s.</p> + +<p><i>S.</i> This can I do, as you shall perceaue by this +exa<i>m</i>ple: Yf 160 sowldyars do spende euery moneth 68 l’i. +what spendeth eche man? Fyrst +<span class = "linenum">127 <i>b</i>.</span> +*bycause I can not diuide the 68 by 160, therfore I wyll turne the +pou<i>n</i>des into pennes by multiplicacio<i>n</i>, so shall there be +16320 d’. Nowe muste I diuide this su<i>m</i>me by the nomber of +sowldyars, therfore I set the<i>m</i> i<i>n</i> order, thus.</p> + +<table class = "backline float" summary = "counters example"> +<col /> +<col class = "rightline" /> +<col class = "rightline" /> +<col /> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td>o</td> +</tr> +<tr> +<td>||</td> +<td> </td> +<td> </td> +<td>o<span class = "five">°</span></td> +</tr> +<tr> +<td> </td> +<td>o</td> +<td> </td> +<td>o o o</td> +</tr> +<tr> +<td> </td> +<td>o<span class = "five">°</span></td> +<td> </td> +<td>o o</td> +</tr> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +</tr> +</table> + +<p class = "nospace"> +Then begyn I at the hyghest place of the diuidente, sekynge my diuisor +there, whiche I fynde ones, Therfore set I 1 in the nether lyne.</p> + +<p><i>M.</i> Not in the nether line of the hole summe, but in the nether +lyne of that worke, whiche is the thyrde lyne.</p> + +<p><i>S.</i> So standeth it with reason.</p> + +<p><i>M.</i> Then thus do they stande.* +<span class = "linenum">128 <i>a</i>.</span></p> + +<table class = "backline float" summary = "counters example"> +<col /> +<col class = "rightline" /> +<col class = "rightline" /> +<col /> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +</tr> +<tr> +<td>||</td> +<td> </td> +<td> </td> +<td> </td> +</tr> +<tr> +<td> </td> +<td>o</td> +<td>o</td> +<td>o o o</td> +</tr> +<tr> +<td> </td> +<td>o<span class = "five">°</span></td> +<td> </td> +<td>o o</td> +</tr> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +</tr> +</table> + +<p class = "nospace"> +Then seke I agayne in the reste, how often I may fynde my diuisor, and I +se that in the 300 I myghte fynde 100 thre tymes, but then the 60 wyll +not be so often founde in 20, therfore I take 2 for my quotient: then +take I 100 twyse from 300, and there resteth 100, out of whiche with the +20 (that maketh 120) I may take 60 also twyse, and then standeth +the nombers thus,</p> + +<table class = "backline float" summary = "counters example"> +<col /> +<col class = "rightline" /> +<col class = "rightline" /> +<col /> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +</tr> +<tr> +<td>||</td> +<td> </td> +<td> </td> +<td> </td> +</tr> +<tr> +<td> </td> +<td>o</td> +<td>o</td> +<td> </td> +</tr> +<tr> +<td> </td> +<td>o<span class = "five">°</span></td> +<td> </td> +<td> </td> +</tr> +<tr> +<td> </td> +<td> </td> +<td>o o</td> +<td> </td> +</tr> +</table> + +<p class = "nospace"> +<span class = "linenum">128 <i>b</i>.</span> +*where I haue sette the quotient 2 in the lowest lyne: So is euery +sowldyars portion 102 d’. that is 8 s. 6 d’.</p> + +<p><i>M.</i> But yet bycause you shall perceaue iustly the reason of +Diuision, it shall be good that you do set your diuisor styll agaynst +those nombres fro<i>m</i> whiche you do take it: as by this example I +wyll declare. Yf y<sup>e</sup> purchace of 200 acres of ground dyd coste +290 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.</p> + +<table class = "backline float" summary = "counters example"> +<col /> +<col class = "rightline" /> +<col class = "rightline" /> +<col /> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +</tr> +<tr> +<td> </td> +<td>o o</td> +<td> </td> +<td>o<span class = "five">°</span></td> +</tr> +<tr> +<td>X</td> +<td> </td> +<td> </td> +<td>o<span class = "five">°</span>o o o</td> +</tr> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td>o <span class = "five">°</span></td> +</tr> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +</tr> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +</tr> +</table> + +<p class = "nospace"> +Fyrst set the diuident on the ryghte hande as it oughte, and then +<span class = "linenum">129 <i>a</i>.</span> +*the diuisor on the lefte hande agaynst those nombers, fro<i>m</i> which +I entende to take hym fyrst as here you se, wher I haue set the diuisor +two lynes hygher the<i>n</i> is theyr owne place.</p> + +<p><i>S.</i> This is lyke the order of diuision by the penne. +<span class = "pagenum">63</span> +<a name = "page63" id = "page63"> </a></p> + +<p><i>M.</i> Truth you say, and nowe must I set y<sup>e</sup> quotient +of this worke in the thyrde lyne, for that is the lyne of vnities in +respecte to the diuisor in this worke. Then I seke howe often the +diuisor maye be founde in the diuident, <i>and</i> that I fynde 3 tymes, +then set I 3 in the thyrde lyne for the quotient, and take awaye that +60000 fro<i>m</i> the diuident, and farther I do set the diuisor one +line lower, as yow se here.</p> + +<table class = "backline float" summary = "counters example"> +<col /> +<col class = "rightline" /> +<col class = "rightline" /> +<col /> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +</tr> +<tr> +<td>||</td> +<td>o o</td> +<td> </td> +<td>o<span class = "five">°</span>o o o</td> +</tr> +<tr> +<td> </td> +<td> </td> +<td>o o o</td> +<td> <span class = "five">°</span>o</td> +</tr> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +</tr> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +</tr> +</table> + +<p class = "nospace"> +<span class = "linenum">129 <i>b</i>.</span> +*And then seke I how often the diuisor wyll be taken from the nomber +agaynste it, whiche wyll be 4 tymes and 1 remaynynge.</p> + +<p><i>S.</i> But what yf it chaunce that when the diuisor is so remoued, +it can not be ones taken out of the diuident agaynste it?</p> + +<p><i>M.</i> Then must the diuisor be set in an other line lower.</p> + +<p><i>S.</i> So was it in diuision by the penne, and therfore was there +a cypher set in the quotient: but howe shall that be noted here?</p> + +<p><i>M.</i> Here nedeth no token, for the lynes do represente the +places: onely loke that you set your quotient in that place which +standeth for vnities in respecte of the diuisor: but now to returne to +the example, I fynde the diuisor 4 tymes in the diuidente, and 1 +remaynynge, for 4 tymes 2 make 8, which I take from 9, and there +<a class = "gloss" name = "resteth" id = "resteth" href = +"#gloss_resteth">resteth</a> 1, as this figure sheweth:</p> + +<table class = "backline float" summary = "counters example"> +<col /> +<col class = "rightline" /> +<col class = "rightline" /> +<col /> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +</tr> +<tr> +<td>||</td> +<td>o o</td> +<td> </td> +<td>o</td> +</tr> +<tr> +<td> </td> +<td> </td> +<td>o o o</td> +<td>o<span class = "five">°</span></td> +</tr> +<tr> +<td> </td> +<td> </td> +<td>o o o o</td> +<td> </td> +</tr> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +</tr> +</table> + +<p class = "nospace"> +and in the myddle space for the quotient I set 4 in the seconde lyne, +whiche is in this worke the place of vnities.* +<span class = "linenum">130 <i>a</i>.</span> +Then remoue I y<sup>e</sup> diuisor to the next lower line, and seke how +often I may haue it in the dyuident, which I may do here 8 tymes iust, +and nothynge remayne, as in this fourme,</p> + +<table class = "backline float" summary = "counters example"> +<col /> +<col class = "rightline" /> +<col class = "rightline" /> +<col /> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +</tr> +<tr> +<td>||</td> +<td> </td> +<td> </td> +<td> </td> +</tr> +<tr> +<td> </td> +<td>o o</td> +<td>o o o</td> +<td> </td> +</tr> +<tr> +<td> </td> +<td> </td> +<td>o o o o</td> +<td> </td> +</tr> +<tr> +<td> </td> +<td> </td> +<td>o<span class = "five">°</span>o o</td> +<td> </td> +</tr> +</table> + +<p class = "nospace"> +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.</p> + +<p><i>S.</i> Now resteth the profes of Multiplycatio<i>n</i>, and also +of Diuisio<i>n</i>.</p> + +<p><i>M.</i> Ther best profes are eche +<span class = "linenum">130 <i>b</i>.</span> +*one by the other, for Multyplication is proued by Diuision, and +Diuision by Multiplycation, as in the worke by the penne you +learned.</p> + +<p><i>S.</i> Yf that be all, you shall not nede to repete agayne that, +y<sup>t</sup> was sufficye<i>n</i>tly taughte all redye: and excepte you +wyll teache me any other feate, here maye you make an ende of this arte +I suppose.</p> + +<p><i>M.</i> So wyll I do as touchynge hole nomber, and as for broken +nomber, I wyll not trouble your wytte with it, tyll you haue +practised this so well, y<sup>t</sup> you be full perfecte, so that you +nede not to doubte in any poynte that I haue taught you, and thenne maye +I boldly enstructe you in y<sup>e</sup> arte of fractions or broken +no<i>m</i>ber, wherin I +<span class = "pagenum">64</span> +<a name = "page64" id = "page64"> </a> +wyll also showe you the reasons of all that you haue nowe learned. But +yet before I make an ende, I wyll showe you the order of +co<i>m</i>men castyng, wher in are bothe pennes, shyllynges, and +poundes, procedynge by no grounded reason, but onely by a receaued +<span class = "linenum">131 <i>a</i>.</span> +*fourme, and that dyuersly of dyuers men: for marchau<i>n</i>tes vse one +fourme, and auditors an other:</p> + +<p class = "headnote"><span class = "headnote"> +Merchants’ Casting Counters.</span></p> + +<p><span class = "sidenote">Merchants’ casting.</span> +But fyrste for marchauntes fourme marke this example here,</p> + +<table class = "noline float" summary = "merchant counters example"> +<tr> +<td>o</td> +<td>o o o o</td> +</tr> +<tr> +<td> </td> +<td> o</td> +</tr> +<tr> +<td>o</td> +<td>o o o</td> +</tr> +<tr> +<td> </td> +<td> o</td> +</tr> +<tr> +<td>o</td> +<td>o o o o</td> +</tr> +<tr> +<td> </td> +<td> o</td> +</tr> +<tr> +<td> </td> +<td>o o o o o</td> +</tr> +</table> + +<p class = "nospace"> +in which I haue expressed this summe 198 l’i.<a class = "tag" name = +"tag_count2" id = "tag_count2" href = "#note_count2">2</a> 19 s. +11 d’. So that you maye se that the lowest lyne serueth for +pe<i>n</i>nes, the next aboue for shyllynges, the thyrde for poundes, +and the fourth for scores of pou<i>n</i>des. And farther you maye se, +that the space betwene pennes and shyllynges may receaue but one counter +(as all other spaces lyke wayes do) and that one standeth in that place +for 6 d’. Lyke wayes betwene the shyllynges <i>and</i> the +pou<i>n</i>des, one cou<i>n</i>ter standeth for 10 s. And betwene +the poundes and 20 l’i. one counter standeth for 10 pou<i>n</i>des. +But besyde those you maye see at the left syde of shyllynges, that one +counter standeth alone, <i>and</i> betokeneth 5 s. +<span class = "linenum">131 <i>b</i>.</span> +*So agaynste the poundes, that one cou<i>n</i>ter standeth for +5 l’i. And agaynst the 20 poundes, the one counter standeth for 5 +score pou<i>n</i>des, that is 100 l’i. so that euery syde counter +is 5 tymes so moch as one of them agaynst whiche he standeth. +<span class = "sidenote">Auditors’ casting.</span> +Now for the accompt of auditors take this example.</p> + +<table class = "noline float" summary = "merchant counters example"> +<tr> +<td> o</td> +<td>o o</td> +<td>o o</td> +<td> o</td> +</tr> +<tr> +<td>o o o</td> +<td>o o o</td> +<td>o o o</td> +<td>o o o</td> +</tr> +<tr> +<td>o</td> +<td></td> +<td>o</td> +<td>o o</td> +</tr> +</table> + +<p class = "nospace"> +where I haue expressed y<sup>e</sup> same su<i>m</i>me 198 l’i. +19 s. 11 d’. But here you se the pe<i>n</i>nes stande toward +y<sup>e</sup> ryght hande, and the other encreasynge orderly towarde the +lefte hande. Agayne you maye se, that auditours wyll make 2 lynes (yea +and more) for pennes, shyllynges, <i>and</i> all other valewes, yf theyr +summes extende therto. Also you se, that they set one counter at the +ryght ende of eche rowe, whiche so set there standeth for 5 of that +roume: and on +<span class = "linenum">132 <i>a</i>.</span> +*the lefte corner of the rowe it sta<i>n</i>deth for 10, of +y<sup>e</sup> same row. But now yf you wold adde other subtracte after +any of both those sortes, yf you marke y<sup>e</sup> order of +y<sup>t</sup> other feate which I taught you, you may easely do the same +here without moch teachynge: for in Additio<i>n</i> you must fyrst set +downe one su<i>m</i>me and to the same set the other orderly, and lyke +maner yf you haue many: but in Subtraction you must sette downe fyrst +the greatest summe, and from it must you abate that other euery +denominatio<i>n</i> from his dewe place.</p> + +<p><i>S.</i> I do not doubte but with a +<span class = "pagenum">65</span> +<a name = "page65" id = "page65"> </a> +lytell practise I shall attayne these bothe: but how shall I multiply +and diuide after these fourmes?</p> + +<p><i>M.</i> You can not duely do none of both by these sortes, therfore +in suche case, you must resort to your other artes.</p> + +<p><i>S.</i> Syr, yet I se not by these sortes how to expresse +hu<i>n</i>dreddes, yf they excede one hundred, nother yet +thousandes.</p> + +<p><i>M.</i> They that vse such accomptes that it excede 200 +<span class = "linenum">132 <i>b</i>.</span> +*in one summe, they sette no 5 at the lefte hande of the scores of +poundes, but they set all the hundredes in an other farther rowe +<i>and</i> 500 at the lefte hand therof, and the thousandes they set in +a farther rowe yet, <i>and</i> at the lefte syde therof they sette the +5000, and in the space ouer they sette the 10000, and in a hygher rowe +20000, whiche all I haue expressed in this exa<i>m</i>ple,</p> + +<table class = "noline float" summary = "merchant counters example"> +<tr> +<td></td> +<td>o o o o</td> +</tr> +<tr> +<td></td> +<td> o</td> +</tr> +<tr> +<td>o</td> +<td>o o</td> +</tr> +<tr> +<td>o</td> +<td>o o o</td> +</tr> +<tr> +<td></td> +<td>o o o</td> +</tr> +<tr> +<td>o</td> +<td>o o o o</td> +</tr> +<tr> +<td></td> +<td> o</td> +</tr> +<tr> +<td></td> +<td>o o</td> +</tr> +<tr> +<td></td> +<td> o</td> +</tr> +<tr> +<td></td> +<td>o o o</td> +</tr> +<tr> +<td></td> +<td> o o</td> +</tr> +<tr> +<td></td> +<td> o</td> +</tr> +</table> + +<p class = "nospace"> +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: <i>and</i> more +there can not be, for 4 farthynges +<span class = "linenum">133 <i>a</i>.</span> +*do make 1 d’. which must be set in his dewe place.</p> + +<p class = "headnote"><span class = "headnote"> +Auditors’ Casting Counters.</span></p> + +<p>And yf you desyre y<sup>e</sup> same summe after audytors maner, lo +here it is.</p> + +<table class = "noline" summary = "merchant counters example"> +<tr> +<td></td> +<td> o o</td> +<td> o</td> +<td></td> +<td> o</td> +<td>o</td> +<td> o</td> +</tr> +<tr> +<td>o o o</td> +<td>o o</td> +<td>o o o</td> +<td>o o o</td> +<td>o o o</td> +<td>o o</td> +<td>o o o</td> +<td></td> +</tr> +<tr> +<td>o</td><td></td><td></td><td></td> +<td>o</td><td></td><td></td><td>o o</td> +</tr> +<tr> +<td></td><td></td><td></td><td></td> +<td></td><td></td><td></td><td>o</td> +</tr> +</table> + +<p class = "nospace"> +But in this thyng, you shall take this for suffycyent, and the reste you +shall obserue as you maye se by the working of eche sorte: for the +dyuers wittes of men haue inuented dyuers and sundry wayes almost +vnnumerable. But one feate I shall teache you, whiche not only for the +straungenes and secretnes is moche pleasaunt, but also for the good +co<i>m</i>moditie of it ryghte worthy to be well marked. This feate hath +ben vsed aboue 2000 yeares at the leaste, and yet was it neuer +come<i>n</i>ly knowen, especyally in Englysshe it was neuer taughte yet. +This is the arte of nombrynge on the hand, with diuers gestures of the +fyngers, expressynge any summe conceaued in the +<span class = "linenum">133 <i>b</i>.</span> +*mynde. And fyrst to begynne, yf you wyll expresse any summe vnder 100, +you shall expresse it with your lefte hande: and from 100 vnto 10000, +you shall expresse it with your ryght hande, as here orderly by this +table folowynge you may perceaue.</p> + +<h5>¶ Here foloweth the table<br /> +of the arte of the<br /> +hande</h5> + +<p> </p> + +<span class = "pagenum">66</span> +<a name = "page66" id = "page66"> </a> + +<p class = "illustration"> +<a name = "hand" id = "hand"> +<img src = "images/title_hand.png" width = "472" height = "35" +alt = "The arte of nombrynge by the hande." +title = "The arte of nombrynge by the hande." /></a></p> + +<span class = "linenum"> +<img src = "images/num_134.png" width = "32" height = "17" +alt = "page number ‘134’ from original illustration" +title = "page number ‘134’ from original illustration" /></span> + +<p class = "illustration"> +<img src = "images/hand_count.png" width = "410" height = "612" +alt = "hand numbering as described in text" /></p> + +<p><span class = "sidenote">1</span> +<span class = "linenum">134 <i>b</i>.</span> +*In which as you may se 1 is expressed by y<sup>e</sup> lyttle fynger of +y<sup>e</sup> lefte hande closely and harde croked.</p> + +<p><span class = "sidenote">2</span> +<a class = "tag" name = "tag_count3" id = "tag_count3" href = +"#note_count3">3</a>2 is declared by lyke bowynge of the weddynge fynger +(whiche is the nexte to the lyttell fynger) together with the lytell +fynger.</p> + +<p><span class = "sidenote">3</span> +3 is signified by the myddle fynger bowed in lyke maner, with those +other two.</p> + +<p><span class = "sidenote"><ins class = "correction" title = "missing sidenote added by transcriber"><i> 4 </i></ins></span> +4 is declared by the bowyng of the myddle fynger and the rynge +<span class = "pagenum">67</span> +<a name = "page67" id = "page67"> </a> +fynger, or weddynge fynger, with the other all stretched forth.</p> + +<p><span class = "sidenote">5</span> +5 is represented by the myddle fynger onely bowed.</p> + +<p><span class = "sidenote">6</span> +And 6 by the weddynge fynger only crooked: and this you may marke in +these a certayne order. But now 7, 8, and 9, are expressed +w<i>i</i>t<i>h</i> the bowynge of the same fyngers as are 1, 2, and 3, +but after an other fourme.</p> + +<p><span class = "sidenote">7</span> +For 7 is declared by the bowynge of the lytell fynger, as is 1, saue +that for 1 the fynger is <ins class = "correction" title = "in at least one printing, text reads ‘elapsed’">clasped</ins> in, harde <i>and</i> +<span class = "linenum">135 <i>a</i>.</span> +*rounde, but for to expresse 7, you shall bowe the myddle ioynte of the +lytell fynger only, and holde the other ioyntes streyght.</p> + +<p><i>S.</i> Yf you wyll geue me leue to expresse it after my rude +maner, thus I vnderstand your meanyng: that 1 is expressed by crookynge +in the lyttell fynger lyke the head of a bysshoppes <a class = "gloss" +name = "bagle" id = "bagle" href = "#gloss_bagle">bagle</a>: and 7 is +declared by the same fynger bowed lyke a gybbet.</p> + +<p><i>M.</i> So I perceaue, you vnderstande it.</p> + +<p><span class = "sidenote">8</span> +Then to expresse 8, you shall bowe after the same maner both the lyttell +fynger and the rynge fynger.</p> + +<p><span class = "sidenote">9</span> +And yf you bowe lyke wayes with them the myddle fynger, then doth it +betoken 9.</p> + +<p><span class = "sidenote">10</span> +Now to expresse 10, you shall bowe your fore fynger rounde, and set the +ende of it on the hyghest ioynte of the thombe.</p> + +<p><span class = "sidenote">20</span> +And for to expresse 20, you must set your fyngers streyght, and the ende +of your thombe to the partitio<i>n</i> of the +<span class = "linenum">135 <i>b</i>.</span> +*fore moste and myddle fynger.</p> + +<p><span class = "sidenote">30</span> +30 is represented by the ioynynge together of y<sup>e</sup> headdes of +the foremost fynger and the thombe.</p> + +<p><span class = "sidenote">40</span> +40 is declared by settynge of the thombe crossewayes on the foremost +fynger.</p> + +<p><span class = "sidenote">50</span> +50 is signified by ryght stretchyng forth of the fyngers ioyntly, and +applyenge of the thombes ende to the partition of the myddle fynger +<i>and</i> the rynge fynger, or weddynge fynger.</p> + +<p><span class = "sidenote">60</span> +60 is formed by bendynge of the thombe croked and crossynge it with the +fore fynger.</p> + +<p><span class = "sidenote">70</span> +70 is expressed by the bowynge of the foremost fynger, and settynge the +ende of the thombe between the 2 foremost or hyghest ioyntes of it.</p> + +<p><span class = "sidenote">80</span> +80 is expressed by settynge of the foremost fynger crossewayes on the +thombe, so that 80 dyffereth thus fro<i>m</i> 40, that for 80 the +forefynger is set crosse on the thombe, and for 40 the thombe is set +crosse ouer y<sup>e</sup> forefinger.</p> + +<p><span class = "sidenote">90</span> +<span class = "linenum">136 <i>a</i>.</span> +*90 is signified, by bendynge the fore fynger, and settyng the ende of +it in the innermost ioynte of y<sup>e</sup> thombe, that is euen at the +foote of it. And thus are all the no<i>m</i>bers ended vnder 100.</p> + +<p><i>S.</i> In dede these be all the nombers fro<i>m</i> 1 to 10, +<i>and</i> then all the +tenthes within 100, +<span class = "sidenote">11, 12, 13,<br /> +21, 22, 23</span> +but this <ins class = "correction" title = "spelling unchanged">teacyed</ins> me not how to expresse 11, 12, 13, <i>et</i>c. +21, 22, 23, <i>et</i>c. and such lyke.</p> + +<p><i>M.</i> You can lytell vnderstande, yf you can not do that without +teachynge: what is +<span class = "pagenum">68</span> +<a name = "page68" id = "page68"> </a> +11? is it not 10 and 1? then expresse 10 as you were taught, and 1 also, +and that is 11: and for 12 expresse 10 and 2: for 23 set 20 and 3: and +so for 68 you muste make 60 and there to 8: and so of all other +sortes.</p> + +<p><span class = "sidenote">100</span> +But now yf you wolde represente 100 other any nomber aboue it, you muste +do that with the ryghte hande, after this maner.</p> + +<p>You must expresse 100 in the ryght hand, with the lytell fynger so +bowed as you dyd expresse 1 in the left hand.</p> + +<p><span class = "sidenote">200</span> +<span class = "linenum">136 <i>b</i>.</span> +*And as you expressed 2 in the lefte hande, the same fasshyon in the +ryght hande doth declare 200.</p> + +<p><span class = "sidenote">300</span> +The fourme of 3 in the ryght hand standeth for 300.</p> + +<p><span class = "sidenote">400</span> +The fourme of 4, for 400.</p> + +<p><span class = "sidenote">500</span> +Lykewayes the fourme of 5, for 500.</p> + +<p><span class = "sidenote">600</span> +The fourme of 6, for 600. And to be shorte: loke how you did expresse +single vnities and tenthes in the lefte hande, so must you expresse +vnities <i>and</i> tenthes of hundredes, in the ryghte hande.</p> + +<p><span class = "sidenote">900</span> +<i>S.</i> I vnderstande you thus: that yf I wold represent 900, +I must so fourme the fyngers of my ryghte hande, as I shuld do in +my left hand to expresse 9, +<span class = "sidenote">1000</span> +And as in my lefte hand I expressed 10, so in my ryght hande must I +expresse 1000.</p> + +<p>And so the fourme of euery tenthe in the lefte hande serueth to +expresse lyke no<i>m</i>ber of thousa<i>n</i>des, +<span class = "sidenote">4000</span> +so y<sup>e</sup> fourme of 40 standeth for 4000.</p> + +<p><span class = "sidenote">8000</span> +The fourme of 80 for 8000.</p> + +<span class = "sidenote">9000</span> +<span class = "linenum">137 <i>a</i>.</span> +<p class = "center">*And the fourme of 90 (whiche is<br /> +the greatest) for 9000, and aboue that<br /> +I can not expresse any nomber. <i>M.</i><br /> +No not with one fynger: how be it,<br /> +w<i>i</i>t<i>h</i> dyuers fyngers you maye expresse<br /> +9999, and all at one tyme, and that lac<br /> +keth but 1 of 10000. So that vnder<br /> +10000 you may by your fyngers ex-<br /> +presse any summe. And this shal suf-<br /> +fyce for Numeration on the fyngers.<br /> +And as for Addition, Subtraction,<br /> +Multiplicatio<i>n</i>, and Diuision (which<br /> +yet were neuer taught by any man as<br /> +farre as I do knowe) I wyll enstruct<br /> +you after the treatyse of fractions.<br /> +And now for this tyme fare well,<br /> +<span class = "pagenum">69</span> +<a name = "page69" id = "page69"> </a> +and loke that you cease not to<br /> +practyse that you haue lear<br /> +ned. <i>S.</i> Syr, with moste<br /> +harty mynde I thanke<br /> +you, bothe for your<br /> +good learnyng, <i>and</i><br /> +also your good<br /> +cou<i>ns</i>el, which<br /> +(god wyllyng) I truste to folow.</p> + +<p class = "center">Finis.</p> + +<hr class = "mid" /> + +<div class = "footnote"> + +<p><a name = "note_count1" id = "note_count1" href = +"#tag_count1">1.</a> +1342 in original.</p> + +<p><a name = "note_count2" id = "note_count2" href = +"#tag_count2">2.</a> +168 in original.</p> + +<p><a name = "note_count3" id = "note_count3" href = +"#tag_count3">3.</a> +Bracket ([) denotes new paragraph in original.</p> +<p class = "mynote"> +For this e-text, the brackets have been omitted in favor of restoring +the paragraph breaks. Numbers 200 and up were printed as separate +paragraphs and are unchanged. Sidenote 4 was missing and has been +supplied by the transcriber; the pairs 5, 6 and 9, 10 (originally on one +line) have been separated.</p> + +</div> + +</div> <!-- end div count --> + +<div class = "app"> + +<span class = "pagenum">70</span> +<a name = "page70" id = "page70"> </a> + +<h3><a name = "app1" id = "app1">APPENDIX I.</a></h3> + +<hr class = "tiny" /> + +<p class = "illustration"> +<img src = "images/title_app1.png" width = "441" height = "71" +alt = "A Treatise on the Numeration of Algorism." +title = "A Treatise on the Numeration of Algorism." /></p> + + +<p class = "subhead">[<i>From a MS. of the 14th Century.</i>]</p> + +<p>To alle suche even nombrys the most have cifrys as to ten. twenty. +thirtty. an hundred. an thousand and suche other. but ye schal +vnderstonde that a cifre tokeneth nothinge but he maketh other the more +significatyf that comith after hym. Also ye schal vnderstonde that in +nombrys composyt and in alle other nombrys that ben of diverse figurys +ye schal begynne in the ritht syde and to rekene backwarde and so he +schal be wryte as thus—1000. the sifre in the ritht side was first +wryte and yit he tokeneth nothinge to the secunde no the thridde but +thei maken that figure of 1 the more signyficatyf that comith after hem +by as moche as he born oute of his first place where he schuld yf he +stode ther tokene but one. And there he stondith nowe in the <a class = +"gloss" name = "ferye" id = "ferye" href = "#gloss_ferye">ferye</a> +place he tokeneth a thousand as by this rewle. In the first place he +tokeneth but hymself. In the secunde place he tokeneth ten times +hymself. In the thridde place he tokeneth an hundred tymes himself. In +the ferye he tokeneth a thousand tymes himself. In the <a class = +"gloss" name = "fyftye" id = "fyftye" href = "#gloss_fyftye">fyftye</a> +place he tokeneth ten thousand tymes himself. In the sexte place he +tokeneth an hundred thousand tymes hymself. In the seveth place he +tokeneth ten hundred thousand tymes hymself, &c. And ye schal +vnderstond that this worde nombre is partyd into thre partyes. Somme is +callyd nombre of digitys for alle ben digitys that ben withine ten as +ix, viii, vii, vi, v, iv, iii, ii, i. Articules ben alle thei that mow +be devyded into nombrys of ten as xx, xxx, xl, and suche other. +Composittys be alle nombrys that ben componyd of a digyt and of an +articule as fourtene fyftene thrittene and suche other. Fourtene is +componyd of four that is a digyt +<span class = "pagenum">71</span> +<a name = "page71" id = "page71"> </a> +and of ten that is an articule. Fyftene is componyd of fyve that is a +digyt and of ten that is an articule and so of others +. . . . . . But as to this rewle. In the firste +place he tokeneth but himself that is to say he tokeneth but that and no +more. If that he stonde in the secunde place he tokeneth ten tymes +himself as this figure 2 here 21. this is oon and twenty. This figure 2 +stondith in the secunde place and therfor he tokeneth ten tymes himself +and ten tymes 2 is twenty and so <a class = "gloss" name = "forye" id = +"forye" href = "#gloss_forye">forye</a> of every figure and he stonde +after another toward the <a class = "gloss" name = "lest2" id = "lest2" +href = "#gloss_lest2">lest</a> syde he schal tokene ten tymes as moche +more as he schuld token and he stode in that place ther that the figure +afore him stondeth: lo an example as thus 9634. This figure of foure +that hath this schape 4 tokeneth but himself for he stondeth in the +first place. The figure of thre that hath this schape 3 tokeneth ten +tyme himself for he stondeth in the secunde place and that is thritti. +The figure of sexe that hath this schape 6 tokeneth ten tyme more than +he schuld and he stode in the place yer the figure of thre stondeth for +ther he schuld tokene but sexty. And now he tokeneth ten tymes that is +sexe hundrid. The figure of nyne that hath this schape 9 tokeneth ten +tymes more than he schulde and he stode in the place ther the figure of +6 stondeth inne for thanne he schuld tokene but nyne hundryd. And in the +place that he stondeth inne nowe he tokeneth nine thousand. Alle the +hole nombre of these foure figurys. Nine thousand sexe hundrid and foure +and thritti.</p> + + + + +<span class = "pagenum">72</span> +<a name = "page72" id = "page72"> </a> + +<h3><a name = "app2" id = "app2">APPENDIX II.</a></h3> + +<hr class = "tiny" /> + +<p class = "illustration"> +<img src = "images/title_app2.png" width = "288" height = "35" +alt = "Carmen de Algorismo." +title = "Carmen de Algorismo." /></p> + +<p class = "subhead">[<i>From a B.M. MS., </i>8<i> C. </i>iv.<i>, with +additions from </i>12<i> E. </i>1<i> & Eg. </i>2622<i>.</i>]</p> + +<div class = "verse"> +<p>Hec algorismus ars presens dicitur<a class = "tag" name = "tag_app_1" +id = "tag_app_1" href = "#note_app_1">1</a>; in qua</p> +<p>Talibus Indorum<a class = "tag" name = "tag_app_2" id = "tag_app_2" +href = "#note_app_2">2</a> fruimur his quinque figuris.</p> +<p><span class = "gap">0. 9. 8. 7. 6. 5. 4. 3. 2. 1.</span></p> +<p>Prima significat unum: duo vero secunda:</p> +<span class = "linenum">4</span> +<p>Tercia significat tria: sic procede sinistre</p> +<p>Donec ad extremam venies, qua cifra vocatur;</p> +<p><a class = "tag" name = "tag_app_3" id = "tag_app_3" href = +"#note_app_3">3</a>[Que nil significat; dat significare sequenti.]</p> +<p>Quelibet illarum si primo limite ponas,</p> +<span class = "linenum">8</span> +<p>Simpliciter se significat: si vero secundo,</p> +<p>Se decies: sursum procedas multiplicando.<a class = "tag" name = +"tag_app_4" id = "tag_app_4" href = "#note_app_4">4</a></p> +<p>[Namque figura sequens quevis signat decies plus,</p> +<span class = "linenum">12</span> +<p>Ipsa locata loco quam significet pereunte:</p> +<p>Nam precedentes plus ultima significabit.]</p> + +<p class = "stanza"> +<a class = "tag" name = "tag_app_5" id = "tag_app_5" href = +"#note_app_5">5</a>Post predicta scias quod tres breuiter numerorum</p> +<p>Distincte species sunt; nam quidam digiti sunt;</p> +<span class = "linenum">16</span> +<p>Articuli quidam; quidam quoque compositi sunt.</p> +<p>[Sunt digiti numeri qui citra denarium sunt;</p> +<p>Articuli decupli degitorum; compositi sunt</p> +<p>Illi qui constant ex articulis digitisque.]</p> +<span class = "linenum">20</span> +<p>Ergo, proposito numero tibi scribere, primo</p> +<p>Respicias quis sit numerus; quia si digitus sit,</p> +<p><a class = "tag" href = "#note_app_5">5</a>[Una figura satis sibi; +sed si compositus sit,]</p> +<p>Primo scribe loco digitum post articulum fac</p> +<span class = "linenum">24</span> +<p>Articulus si sit, cifram post articulum sit,</p> +<p>[Articulum vero reliquenti in scribe figure.]</p> +<span class = "pagenum">73</span> +<a name = "page73" id = "page73"> </a> +<p>Quolibet in numero, si par sit prima figura,</p> +<p>Par erit et totum, quicquid sibi continetur;</p> +<span class = "linenum">28</span> +<p>Impar si fuerit, totum sibi fiet et impar.</p> + +<p class = "stanza"> +Septem<a class = "tag" name = "tag_app_6" id = "tag_app_6" href = +"#note_app_6">6</a> sunt partes, non plures, istius artis;</p> +<p>Addere, subtrahere, duplare, dimidiare;</p> +<p>Sexta est diuidere, set quinta est multiplicare;</p> +<span class = "linenum">32</span> +<p>Radicem extrahere pars septima dicitur esse.</p> +<p>Subtrahis aut addis a dextris vel mediabis;</p> +<p>A leua dupla, diuide, multiplicaque;</p> +<p>Extrahe radicem semper sub parte sinistra.</p> + +<span class = "linenum">36</span> +<p class = "stanza"> +<span class = "sidenote">Addition.</span> +Addere si numero numerum vis, ordine tali</p> +<p>Incipe; scribe duas primo series numerorum</p> +<p>Prima sub prima recte ponendo figuram,</p> +<p>Et sic de reliquis facias, si sint tibi plures.</p> +<span class = "linenum">40</span> +<p>Inde duas adde primas hac condicione;</p> +<p>Si digitus crescat ex addicione priorum,</p> +<p>Primo scribe loco digitum, quicunque sit ille;</p> +<p>Si sit compositus, in limite scribe sequenti</p> +<span class = "linenum">44</span> +<p>Articulum, primo digitum; quia sic iubet ordo.</p> +<p>Articulus si sit, in primo limite cifram,</p> +<p>Articulum vero reliquis inscribe figuris;</p> +<p>Vel per se scribas si nulla figura sequatur.</p> +<span class = "linenum">48</span> +<p>Si tibi cifra superueniens occurrerit, illam</p> +<p>Deme suppositam; post illic scribe figuram:</p> +<p>Postea procedas reliquas addendo figuras.</p> + +<p class = "stanza"> +<span class = "sidenote">Subtraction.</span> +A numero numerum si sit tibi demere cura,</p> +<span class = "linenum">52</span> +<p>Scribe figurarum series, vt in addicione;</p> +<p>Maiori numero numerum suppone minorem,</p> +<p>Siue pari numero supponatur numerus par.</p> +<p>Postea si possis a prima subtrahe primam,</p> +<span class = "linenum">56</span> +<p>Scribens quod remanet, cifram si nil remanebit.</p> +<p>Set si non possis a prima demere primam;</p> +<p>Procedens, vnum de limite deme sequenti;</p> +<span class = "pagenum">74</span> +<a name = "page74" id = "page74"> </a> +<p>Et demptum pro denario reputabis ab illo,</p> +<span class = "linenum">60</span> +<p>Subtrahe totaliter numerum quem proposuisti.</p> +<p>Quo facto, scribe supra quicquit remanebit,</p> +<p>Facque novenarios de cifris, cum remanebis,</p> +<p>Occurrant si forte cifre, dum demseris vnum;</p> +<span class = "linenum">64</span> +<p>Postea procedas reliquas demendo figuras.</p> + +<p class = "stanza"> +<span class = "sidenote">Proof.</span> +<a class = "tag" name = "tag_app_7" id = "tag_app_7" href = +"#note_app_7">7</a>[Si subtracio sit bene facta probare valebis,</p> +<p>Quas subtraxisti primas addendo figuras.</p> +<p>Nam, subtractio si bene sit, primas retinebis,</p> +<span class = "linenum">68</span> +<p>Et subtractio facta tibi probat additionem.]</p> + +<p class = "stanza"> +<span class = "sidenote">Duplation.</span> +Si vis duplare numerum, sic incipe; solam</p> +<p>Scribe figurarum seriem, quamcumque voles que</p> +<p>Postea procedas primam duplando figuram;</p> +<span class = "linenum">72</span> +<p>Inde quod excrescet, scribens, vbi iusserit ordo,</p> +<p>Juxta precepta que dantur in addicione.</p> +<p>Nam si sit digitus, in primo limite scribe;</p> +<p>Articulus si sit, in primo limite cifram,</p> +<span class = "linenum">76</span> +<p>Articulum vero reliquis inscribe figuris;</p> +<p>Vel per se scribas, si nulla figura sequatur:</p> +<p>Compositus si sit, in limite scribe sequenti</p> +<p>Articulum primo, digitum; quia sic jubet ordo:</p> +<span class = "linenum">80</span> +<p>Et sic de reliquis facias, si sint tibi plures.</p> +<p><a class = "tag" name = "tag_app_8" id = "tag_app_8" href = +"#note_app_8">8</a>[Si super extremam nota sit, monadem dat eidem,</p> +<p>Quod tibi contingit, si primo dimidiabis.]</p> + +<p class = "stanza"> +<span class = "sidenote">Mediation.</span> +Incipe sic, si vis aliquem numerum mediare:</p> +<span class = "linenum">84</span> +<p>Scribe figurarum seriem solam, velud ante;</p> +<p>Postea procedens medias, et prima figura</p> +<p>Si par aut impar videas; quia si fuerit par,</p> +<p>Dimidiabis eam, scribens quicquit remanebit;</p> +<span class = "linenum">88</span> +<p>Impar si fuerit, vnum demas, mediare,</p> +<p>Nonne presumas, sed quod superest mediabis;</p> +<p>Inde super tractum, fac demptum quod notat unum;</p> +<p>Si monos, dele; sit ibi cifra post nota supra.</p> +<span class = "linenum">92</span> +<p>Postea procedas hac condicione secunda:<a class = "tag" name = +"tag_app_9" id = "tag_app_9" href = "#note_app_9">9</a></p> +<p>Impar<a class = "tag" name = "tag_app_10" id = "tag_app_10" href = +"#note_app_10">10</a> si fuerit hic vnum deme priori,</p> +<p>Inscribens quinque, nam denos significabit</p> +<p>Monos prædictam: si vero secunda dat vnam,</p> +<span class = "linenum">96</span> +<p>Illa deleta, scribatur cifra; priori</p> +<span class = "pagenum">75</span> +<a name = "page75" id = "page75"> </a> +<p>Tradendo quinque pro denario mediato;</p> +<p>Nec cifra scribatur, nisi inde figura sequatur:</p> +<p>Postea <ins class = "correction" title = "text reads ‘procdeas’">procedas</ins> reliquas mediando figuras,</p> +<span class = "linenum">100</span> +<p>Quin supra docui, si sint tibi mille figure.</p> +<p><a class = "tag" name = "tag_app_11" id = "tag_app_11" href = +"#note_app_11">11</a>[Si mediatio sit bene facta probare valebis,</p> +<p>Duplando numerum quem primo dimidiasti.]</p> + +<p class = "stanza"> +<span class = "sidenote">Multiplication.</span> +Si tu per numerum numerum vis multiplicare,</p> +<span class = "linenum">104</span> +<p>Scribe duas, quascunque volis, series numerorum;</p> +<p>Ordo tamen seruetur vt vltima multiplicandi</p> +<p>Ponatur super anteriorem multiplicantis;</p> +<p><a class = "tag" name = "tag_app_12" id = "tag_app_12" href = +"#note_app_12">12</a>[A leua relique sint scripte multiplicantes.]</p> +<span class = "linenum">108</span> +<p>In digitum cures digitum si ducere, major</p> +<p>Per quantes distat a denis respice, debes</p> +<p>Namque suo decuplo tociens delere minorem;</p> +<p>Sicque tibi numerus veniens exinde patebit.</p> +<span class = "linenum">112</span> +<p>Postea procedas postremam multiplicando,</p> +<p>Juste multiplicans per cunctas inferiores,</p> +<p>Condicione tamen tali; quod multiplicantis</p> +<p>Scribas in capite, quicquid processerit inde;</p> +<span class = "linenum">116</span> +<p>Set postquam fuerit hec multiplicata, figure</p> +<p>Anteriorentur seriei multiplicantis;</p> +<p>Et sic multiplica, velut istam multiplicasti,</p> +<p>Qui sequitur numerum scriptum quicunque figuris.</p> +<span class = "linenum">120</span> +<p>Set cum multiplicas, primo sic est operandum,</p> +<p>Si dabit articulum tibi multiplicacio solum;</p> +<p>Proposita cifra, summam transferre memento.</p> +<p>Sin autem digitus excrescerit articulusque,</p> +<span class = "linenum">124</span> +<p>Articulus supraposito digito salit ultra;</p> +<p>Si digitus tamen, ponas illum super ipsam,</p> +<p>Subdita multiplicans hanc que super incidit illi</p> +<p>Delet eam penitus, scribens quod provenit inde;</p> +<span class = "linenum">128</span> +<p>Sed si multiplices illam posite super ipsam,</p> +<p>Adiungens numerum quem prebet ductus earum;</p> +<p>Si supraimpositam cifra debet multiplicare,</p> +<p>Prorsus eam delet, scribi que loco cifra debet,</p> +<span class = "linenum">132</span> +<p><a class = "tag" href = "#note_app_12">12</a>[Si cifra multiplicat +aliam positam super ipsam,</p> +<p>Sitque locus supra vacuus super hanc cifra fiet;]</p> +<span class = "pagenum">76</span> +<a name = "page76" id = "page76"> </a> +<p>Si supra fuerit cifra semper pretereunda est;</p> +<p>Si dubites, an sit bene multiplicando secunda,</p> +<span class = "linenum">136</span> +<p>Diuide totalem numerum per multiplicantem,</p> +<p>Et reddet numerus emergens inde priorem.</p> + + + +<span class = "sidenote">Mental Multiplication.</span> + +<p><a class = "tag" name = "tag_app_13" id = "tag_app_13" href = +"#note_app_13">13</a>[Per numerum si vis numerum quoque multiplicare</p> +<p>Tantum per normas subtiles absque figuris</p> +<span class = "linenum">140</span> +<p>Has normas poteris per versus scire sequentes.</p> +<p>Si tu per digitum digitum quilibet multiplicabis</p> +<p>Regula precedens dat qualiter est operandum</p> +<p>Articulum si per reliquum vis multiplicare</p> +<span class = "linenum">144</span> +<p>In proprium digitum debebit uterque resolvi</p> +<p>Articulus digitos post per se multiplicantes</p> +<p>Ex digitis quociens teneret multiplicatum</p> +<p>Articuli faciunt tot centum multiplicati.</p> +<span class = "linenum">148</span> +<p>Articulum digito si multiplicamus oportet</p> +<p>Articulum digitum sumi quo multiplicare</p> +<p>Debemus reliquum quod multiplicaris ab illis</p> +<p>Per reliquo decuplum sic omne latere nequibit</p> +<span class = "linenum">152</span> +<p>In numerum mixtum digitum si ducere cures</p> +<p>Articulus mixti sumatur deinde resolvas</p> +<p>In digitum post hec fac ita de digitis nec</p> +<p>Articulusque docet excrescens in detinendo</p> +<span class = "linenum">156</span> +<p>In digitum mixti post ducas multiplicantem</p> +<p>De digitis ut norma docet sit juncta secundo</p> +<p>Multiplica summam et postea summa patebit</p> +<p>Junctus in articulum purum articulumque</p> +<span class = "linenum">160</span> +<p><a class = "tag" name = "tag_app_14" id = "tag_app_14" href = +"#note_app_14">14</a>[Articulum purum comittes articulum que]</p> +<p>Mixti pro digitis post fiat et articulus vt</p> +<p>Norma jubet retinendo quod egreditur ab illis</p> +<p>Articuli digitum post <ins class = "correction" title = "text reads ‘iu’">in</ins> digitum mixti duc</p> +<span class = "linenum">164</span> +<p>Regula de digitis ut percipit articulusque</p> +<p>Ex quibus excrescens summe tu junge priori</p> +<p>Sic manifesta cito fiet tibi summa petita.</p> +<p>Compositum numerum mixto sic multiplicabis</p> +<span class = "linenum">168</span> +<p>Vndecies tredecem sic est ex hiis operandum</p> +<p>In reliquum primum demum duc post in eundem</p> +<p>Unum post deinde duc in tercia deinde per unum</p> +<p>Multiplices tercia demum tunc omnia multiplicata</p> +<span class = "linenum">172</span> +<p>In summa duces quam que fuerit te dices</p> +<span class = "pagenum">77</span> +<a name = "page77" id = "page77"> </a> +<p>Hic ut hic mixtus intentus est operandum</p> +<p>Multiplicandorum de normis sufficiunt hec.]</p> + + +<span class = "sidenote">Division.</span> + +<p>Si vis dividere numerum, sic incipe primo;</p> +<span class = "linenum">176</span> +<p>Scribe duas, quascunque voles, series numerorum;</p> +<p>Majori numero numerum suppone minorem,</p> +<p><a class = "tag" name = "tag_app_15" id = "tag_app_15" href = +"#note_app_15">15</a>[Nam docet ut major teneat bis terve minorem;]</p> + +<p>Et sub supprima supprimam pone figuram,</p> +<span class = "linenum">180</span> +<p>Sic reliquis reliquas a dextra parte locabis;</p> +<p>Postea de prima primam sub parte sinistra</p> +<p>Subtrahe, si possis, quociens potes adminus istud,</p> +<p>Scribens quod remanet sub tali conditione;</p> +<span class = "linenum">184</span> +<p>Ut totiens demas demendas a remanente,</p> +<p>Que serie recte ponentur in anteriori,</p> +<p>Unica si, tantum sit ibi decet operari;</p> +<p>Set si non possis a prima demere primam,</p> +<span class = "linenum">188</span> +<p>Procedas, et eam numero suppone sequenti;</p> +<p>Hanc uno retrahendo gradu quo comites retrahantur,</p> +<p>Et, quotiens poteris, ab eadem deme priorem,</p> +<p>Ut totiens demas demendas a remanenti,</p> +<span class = "linenum">192</span> +<p>Nec plus quam novies quicquam tibi demere debes,</p> +<p>Nascitur hinc numerus quociens supraque sequentem</p> +<p>Hunc primo scribas, retrahas exinde figuras,</p> +<p>Dum fuerit major supra positus inferiori,</p> +<span class = "linenum">196</span> +<p>Et rursum fiat divisio more priori;</p> +<p>Et numerum quotiens supra scribas pereunti,</p> +<p>Si fiat saliens retrahendo, cifra locetur,</p> +<p>Et pereat numero quotiens, proponas eidem</p> +<span class = "linenum">200</span> +<p>Cifram, ne numerum pereat vis, dum locus illic</p> +<p>Restat, et expletis divisio non valet ultra:</p> +<p>Dum fuerit numerus numerorum inferiore seorsum</p> +<p>Illum servabis; hinc multiplicando probabis,</p> + +<span class = "sidenote">Proof.</span> + +<span class = "linenum">204</span> +<p>Si bene fecisti, divisor multiplicetur</p> +<p>Per numerum quotiens; cum multiplicaveris, adde</p> +<p>Totali summæ, quod servatum fuit ante,</p> +<p>Reddeturque tibi numerus quem proposuisti;</p> +<span class = "linenum">208</span> +<p>Et si nil remanet, hunc multiplicando reddet,</p> + +<span class = "sidenote">Square Numbers.</span> + +<p>Cum ducis numerum per se, qui provenit inde</p> +<p>Sit tibi quadratus, ductus radix erit hujus,</p> +<p>Nec numeros omnes quadratos dicere debes,</p> +<span class = "linenum">212</span> +<p>Est autem omnis numerus radix alicujus.</p> +<span class = "pagenum">78</span> +<a name = "page78" id = "page78"> </a> +<p>Quando voles numeri radicem querere, scribi</p> +<p>Debet; inde notes si sit locus ulterius impar,</p> +<p>Estque figura loco talis scribenda sub illo,</p> +<span class = "linenum">216</span> +<p>Que, per se dicta, numerum tibi destruat illum,</p> +<p>Vel quantum poterit ex inde delebis eandem;</p> +<p>Vel retrahendo duples retrahens duplando sub ista</p> +<p>Que primo sequitur, duplicatur per duplacationem,</p> +<span class = "linenum">220</span> +<p>Post per se minuens pro posse quod est minuendum.</p> +<p><a class = "tag" name = "tag_app_16" id = "tag_app_16" href = +"#note_app_16">16</a>Post his propones digitum, qui, more priori</p> +<p>Per precedentes, post per se multiplicatus,</p> +<p>Destruat in quantum poterit numerum remanentem,</p> +<span class = "linenum">224</span> +<p>Et sic procedens retrahens duplando figuram,</p> +<p>Preponendo novam donec totum peragatur,</p> +<p>Subdupla propriis servare docetque duplatis;</p> +<p>Si det compositum numerum duplacio, debet</p> +<span class = "linenum">228</span> +<p>Inscribi digitus a parte dextra parte propinqua,</p> +<p>Articulusque loco quo non duplicata resessit;</p> +<p>Si dabit articulum, sit cifra loco pereunte</p> +<p>Articulusque locum tenet unum, de duplicata resessit;</p> +<span class = "linenum">232</span> +<p>Si donet digitum, sub prima pone sequente,</p> +<p>Si supraposita fuerit duplicata figura</p> +<p>Major proponi debet tantummodo cifra,</p> +<p>Has retrahens solito propones more figuram,</p> +<span class = "linenum">236</span> +<p>Usque sub extrema ita fac retrahendo figuras,</p> +<p>Si totum deles numerum quem proposuisti,</p> +<p>Quadratus fuerit, de dupla quod duplicasti,</p> +<p>Sicque tibi radix illius certa patebit,</p> +<span class = "linenum">240</span> +<p>Si de duplatis fit juncta supprima figura;</p> +<p>Radicem per se multiplices habeasque</p> +<p>Primo propositum, bene te fecisse probasti;</p> +<p>Non est quadratus, si quis restat, sed habentur</p> +<span class = "linenum">244</span> +<p>Radix quadrati qui stat major sub eadem;</p> +<p>Vel quicquid remanet tabula servare memento;</p> +<p>Hoc casu radix per se quoque multiplicetur,</p> +<p>Vel sic quadratus sub primo major habetur,</p> +<span class = "linenum">248</span> +<p>Hinc addas remanens, et prius debes haberi;</p> +<p>Si locus extremus fuerit par, scribe figuram</p> +<p>Sub pereunte loco per quam debes operari,</p> +<p>Que quantum poterit supprimas destruat ambas,</p> +<span class = "pagenum">79</span> +<a name = "page79" id = "page79"> </a> +<span class = "linenum">252</span> +<p>Vel penitus legem teneas operando priorem,</p> +<p>Si suppositum digitus suo fine repertus,</p> +<p>Omnino delet illic scribi cifra debet,</p> +<p>A leva si qua sit ei sociata figura;</p> +<span class = "linenum">256</span> +<p>Si cifre remanent in fine pares decet harum</p> +<p>Radices, numero mediam proponere partem,</p> +<p>Tali quesita radix patet arte reperta.</p> +<p>Per numerum recte si nosti multiplicare</p> +<span class = "linenum">260</span> +<p>Ejus quadratum, numerus qui pervenit inde</p> +<p>Dicetur cubicus; primus radix erit ejus;</p> +<p>Nec numeros omnes cubicatos dicere debes,</p> +<p>Est autem omnis numerus radix alicujus;</p> + +<span class = "sidenote">Cube Root.</span> + +<span class = "linenum">264</span> +<p>Si curas cubici radicem quærere, primo</p> +<p>Inscriptum numerum distinguere per loca debes;</p> +<p>Que tibi mille notant a mille notante suprema</p> +<p>Initiam, summa operandi parte sinistra,</p> +<span class = "linenum">268</span> +<p>Illic sub scribas digitum, qui multiplicatus</p> +<p>In semet cubice suprapositum sibi perdat,</p> +<p>Et si quid fuerit adjunctum parte sinistra</p> +<p>Si non omnino, quantum poteris minuendo,</p> +<span class = "linenum">272</span> +<p>Hinc triplans retrahe saltum, faciendo sub illa</p> +<p>Que manet a digito deleto terna, figuram</p> +<p>Illi propones quo sub triplo asocietur,</p> +<p>Ut cum subtriplo per eam tripla multiplicatur;</p> +<span class = "linenum">276</span> +<p>Hinc per eam solam productum multiplicabis,</p> +<p>Postea totalem numerum, qui provenit inde</p> +<p>A suprapositis respectu tolle triplate</p> +<p>Addita supprimo cubice tunc multiplicetur,</p> +<span class = "linenum">280</span> +<p>Respectu cujus, numerus qui progredietur</p> +<p>Ex cubito ductu, supra omnes adimetur;</p> +<p>Tunc ipsam delens triples saltum faciendo,</p> +<p>Semper sub ternas, retrahens alias triplicatas</p> +<span class = "linenum">284</span> +<p>Ex hinc triplatis aliam propone figuram,</p> +<p>Que per triplatas ducatur more priori;</p> +<p>Primo sub triplis sibi junctis, postea per se,</p> +<p>In numerum ducta, productum de triplicatis:</p> +<span class = "linenum">288</span> +<p>Utque prius dixi numerus qui provenit inde</p> +<p>A suprapositis has respiciendo trahatur,</p> +<p>Huic cubice ductum sub primo multiplicabis,</p> +<p>Respectumque sui, removebis de remanenti,</p> +<span class = "linenum">292</span> +<p>Et sic procedas retrahendo triplando figuram.</p> +<span class = "pagenum">80</span> +<a name = "page80" id = "page80"> </a> +<p>Et proponendo nonam, donec totum peragatur,</p> +<p>Subtripla sub propriis servare decet triplicatis;</p> +<p>Si nil in fine remanet, numerus datus ante</p> +<span class = "linenum">296</span> +<p>Est cubicus; cubicam radicem sub tripla prebent,</p> +<p>Cum digito juncto quem supprimo posuisti,</p> +<p>Hec cubice ducta, numerum reddant tibi primum.</p> +<p>Si quid erit remanens non est cubicus, sed habetur</p> +<span class = "linenum">300</span> +<p>Major sub primo qui stat radix cubicam,</p> +<p>Servari debet quicquid radice remansit,</p> +<p>Extracto numero, decet hec addi cubicato.</p> +<p>Quo facto, numerus reddi debet tibi primus.</p> +<span class = "linenum">304</span> +<p>Nam debes per se radicem multiplicare</p> +<p>Ex hinc in numerum duces, qui provenit inde</p> +<p>Sub primo cubicus major sic invenietur;</p> +<p>Illi jungatur remanens, et primus habetur,</p> +<span class = "linenum">308</span> +<p>Si per triplatum numerum nequeas operari;</p> +<p>Cifram propones, nil vero per hanc operare</p> +<p>Set retrahens illam cum saltu deinde triplata,</p> +<p>Propones illi digitum sub lege priori,</p> +<span class = "linenum">312</span> +<p>Cumque cifram retrahas saliendo, non triplicabis,</p> +<p>Namque nihil cifre triplacio dicitur esse;</p> +<p>At tu cum cifram protraxeris aut triplicata,</p> +<p>Hanc cum subtriplo semper servare memento:</p> +<span class = "linenum">316</span> +<p>Si det compositum, digiti triplacio debet</p> +<p>Illius scribi, digitus saliendo sub ipsam;</p> +<p>Digito deleto, que terna dicitur esse;</p> +<p>Jungitur articulus cum triplata pereunte,</p> +<span class = "linenum">320</span> +<p>Set facit hunc scribi per se triplacio prima,</p> +<p>Que si det digitum per se scribi facit illum;</p> +<p>Consumpto numero, si sole fuit tibi cifre</p> +<p>Triplato, propone cifram saltum faciendo,</p> +<span class = "linenum">324</span> +<p>Cumque cifram retrahe triplam, scribendo figuram,</p> +<p>Preponas cifre, sic procedens operare,</p> +<p>Si tres vel duo serie in sint, pone sub yma,</p> +<p>A dextris digitum servando prius documentum.</p> +<span class = "linenum">328</span> +<p>Si sit continua progressio terminus nuper</p> +<p>Per majus medium totalem multiplicato;</p> +<p>Si par, per medium tunc multiplicato sequentem.</p> +<p>Set si continua non sit progressio finis:</p> +<span class = "linenum">332</span> +<p>Impar, tunc majus medium si multiplicabis,</p> +<span class = "linenum">333</span> +<p>Si par per medium sibi multiplicato propinquum.</p> + +</div> <!-- end div verse --> + +<hr class = "tiny" /> + +<div class = "footnote"> + +<p><a name = "note_app_1" id = "note_app_1" href = "#tag_app_1">1.</a> +“Hec præsens ars dicitur algorismus ab Algore rege ejus inventore, vel +dicitur ab <i>algos</i> quod est ars, et <i>rodos</i> quod est numerus; +quæ est ars numerorum vel numerandi, ad quam artem bene sciendum +inveniebantur apud Indos bis quinque (id est decem) figuræ.” +—<i>Comment. Thomæ de Novo-Mercatu.</i> MS. Bib. Reg. Mus. Brit. +12 E. 1.</p> + +<p><a name = "note_app_2" id = "note_app_2" href = "#tag_app_2">2.</a> +“Hæ necessariæ figuræ sunt Indorum characteros.” <i>MS. de +numeratione.</i> Bib. Sloan. Mus. Brit. 513, fol. 58. “Cum vidissem +Yndos constituisse <span class = "smallroman">IX</span> literas in +universo numero suo propter dispositionem suam quam posuerunt, volui +patefacere de opere quod sit per eas aliquidque esset levius +discentibus, si Deus voluerit. Si autem Indi hoc voluerunt et intentio +illorum nihil novem literis fuit, causa que mihi potuit. Deus direxit me +ad hoc. Si vero alia dicam preter eam quam ego exposui, hoc fecerunt per +hoc quod ego exposui, eadem tam certissime et absque ulla dubitatione +poterit inveniri. Levitasque patebit aspicientibus et discentibus.” MS. +U.L.C., Ii. vi. 5, f. 102.</p> + +<p><a name = "note_app_3" id = "note_app_3" href = "#tag_app_3">3.</a> +From Eg. 2622.</p> + +<p><a name = "note_app_4" id = "note_app_4" href = "#tag_app_4">4.</a> +8 C. iv. inserts Nullum cipa significat: dat significare sequenti.</p> + +<p><a name = "note_app_5" id = "note_app_5" href = "#tag_app_5">5.</a> +From 12 E. 1.</p> + +<p><a name = "note_app_6" id = "note_app_6" href = +"#tag_app_5">6.</a></p> +<div class = "verse"> +<p>En argorisme devon prendre</p> +<p>Vii especes . . . .</p> +<p>Adision subtracion</p> +<p>Doubloison mediacion</p> +<p>Monteploie et division</p> +<p>Et de radix eustracion</p> +<p>A chez vii especes savoir</p> +<p>Doit chascun en memoire avoir</p> +<p>Letres qui figures sont dites</p> +<p>Et qui excellens sont ecrites.—MS. <i>Seld. Arch.</i> +B. 26.</p> +</div> + +<p><a name = "note_app_7" id = "note_app_7" href = "#tag_app_7">7.</a> +From 12 E. 1.</p> + +<p><a name = "note_app_8" id = "note_app_8" href = "#tag_app_8">8.</a> +From 12 E. 1.</p> + +<p><a name = "note_app_9" id = "note_app_9" href = "#tag_app_9">9.</a> +8 C. iv. inserts Atque figura prior nuper fuerit mediando.</p> + +<p><a name = "note_app_10" id = "note_app_10" href = +"#tag_app_10">10.</a> +<i>I.e.</i> figura secundo loco posita.</p> + +<p><a name = "note_app_11" id = "note_app_11" href = +"#tag_app_11">11.</a> +So 12 E. 1; 8 C. iv. inserts—</p> +<div class = "verse"> +<p>Si super extremam nota sit monades dat eidem</p> +<p>Quod contingat cum primo dimiabis</p> +<p>Atque figura prior nuper fuerit mediando.</p> +</div> + +<p><a name = "note_app_12" id = "note_app_12" href = +"#tag_app_12">12.</a> +12 E. 1 inserts.</p> + +<p><a name = "note_app_13" id = "note_app_13" href = +"#tag_app_13">13.</a> +12 E. 1 inserts to l. 174.</p> + +<p><a name = "note_app_14" id = "note_app_14" href = +"#tag_app_14">14.</a> +12 E. 1 omits, Eg. 2622 inserts.</p> + +<p><a name = "note_app_15" id = "note_app_15" href = +"#tag_app_15">15.</a> +12 E. 1 inserts.</p> + +<p><a name = "note_app_16" id = "note_app_16" href = +"#tag_app_16">16.</a> +8 C. iv. inserts—</p> +<div class = "verse"> +<p>Hinc illam dele duplans sub ei psalliendo</p> +<p>Que sequitur retrahens quicquid fuerit duplicatum.</p> +</div> + +</div> + +</div> <!-- end div app --> + +</div> <!-- end div maintext --> + +<hr class = "mid" /> + +<div class = "index"> + +<span class = "pagenum">81</span> +<a name = "page81" id = "page81"> </a> + +<h3><a name = "terms" id = "terms">INDEX OF TECHNICAL TERMS</a><a class += "tag" name = "tag_terms1" id = "tag_terms1" href = +"#note_terms1">1</a></h3> + +<p><a name = "terms_algorisme" id = +"terms_algorisme"><b>algorisme</b></a>, +<a href = "#algorisme">33/12</a>; +<b>algorym</b>, <b>augrym</b>, <a href = "#algorym">3/3</a>; +the art of computing, using the so-called Arabic numerals.</p> + +<p class = "inset"> +The word in its various forms is derived from the Arabic +<i>al-Khowarazmi</i> (i.e. the native of Khwarazm (Khiva)). This was the +surname of Ja’far Mohammad ben Musa, who wrote a treatise early in the +9th century (see p. xiv).</p> + +<p class = "inset"> +The form <i>algorithm</i> is also found, being suggested by a supposed +derivation from the Greek <span class = "greek" title = +"arithmos">ἀριθμός</span> (number).</p> + +<p><a name = "terms_antery" id = "terms_antery"><b>antery</b></a>, +<a href = "#antery">24/11</a>; +to move figures to the right of the position in which they are first +written. This operation is performed repeatedly upon the multiplier in +multiplication, and upon certain figures which arise in the process of +root extraction.</p> + +<p><a name = "terms_anterioracioun" id = +"terms_anterioracioun"><b>anterioracioun</b></a>, +<a href = "#anterioracioun">50/5</a>; +the operation of moving figures to the right.</p> + +<p><a name = "terms_article" id = "terms_article"><b>article</b></a>, +<a href = "#article">34/23</a>; +<b>articul</b>, <a href = "#articul">5/31</a>; +<b>articuls</b>, <a href = "#articuls">9/36</a>, +<a href = "#articuls2">29/7,8</a>; +a number divisible by ten without remainder.</p> + +<p><a name = "terms_cast" id = "terms_cast"><b>cast</b></a>, +<a href = "#cast">8/12</a>; +to add one number to another.</p> + +<p class = "inset"> +‘Addition is a <i>casting</i> together of two numbers into one number,’ +<a href = "#castyng">8/10</a>.</p> + +<p><a name = "terms_cifre" id = "terms_cifre"><b>cifre</b></a>, +<a href = "#cifre">4/1</a>; +the name of the figure 0. The word is derived from the Arabic +<i>sifr</i> = empty, nothing. Hence <i>zero</i>.</p> + +<p class = "inset"> +A cipher is the symbol of the absence of number or of zero quantity. It +may be used alone or in conjunction with digits or other ciphers, and in +the latter case, according to the position which it occupies relative to +the other figures, indicates the absence of units, or tens, or hundreds, +etc. The great superiority of the Arabic to all other systems of +notation resides in the employment of this symbol. When the cipher is +not used, the place value of digits has to be indicated by writing them +in assigned rows or columns. Ciphers, however, may be interpolated +amongst the significant figures used, and as they sufficiently indicate +the positions of the empty rows or columns, the latter need not be +indicated in any other way. The practical performance of calculations is +thus enormously facilitated (see <a href = +"#intro_modern">p. xvi</a>).</p> + +<p><a name = "terms_componede" id = +"terms_componede"><b>componede</b></a>, <a href = +"#componede">33/24</a>; +<b>composyt</b>, <a href = "#composyt">5/35</a>; +with reference to numbers, one compounded of a multiple of ten and a +digit.</p> + +<p><a name = "terms_conuertide" id = +"terms_conuertide"><b>conuertide</b></a> = conversely, +<a href = "#conuertide">46/29</a>, +<a href = "#conuertide2">47/9</a>.</p> + +<p><a name = "terms_cubicede" id = "terms_cubicede"><b>cubicede</b></a>, +<a href = "#cubicede">50/13</a>; +<b>to be c.</b>, to have its cube root found.</p> + +<span class = "pagenum">82</span> +<a name = "page82" id = "page82"> </a> + +<p><a name = "terms_cubike" id = "terms_cubike"><b>cubike +nombre</b></a>, +<a href = "#cubike">47/8</a>; +a number formed by multiplying a given number twice by itself, +<i>e.g.</i> 27 = 3 × 3 × 3. Now called simply a +cube.</p> + +<p><a name = "terms_decuple" id = "terms_decuple"><b>decuple</b></a>, +<a href = "#decuple">22/12</a>; +the product of a number by ten. Tenfold.</p> + +<p><a name = "terms_departys" id = "terms_departys"><b>departys</b></a> += divides, +<a href = "#departys">5/29</a>.</p> + +<p><a name = "terms_digit" id = "terms_digit"><b>digit</b></a>, +<a href = "#digit">5/30</a>; +<b>digitalle</b>, <a href = "#digitalle">33/24</a>; +a number less than ten, represented by one of the nine Arabic +numerals.</p> + +<p><a name = "terms_dimydicion" id = +"terms_dimydicion"><b>dimydicion</b></a>, +<a href = "#dimydicion">7/23</a>; +the operation of dividing a number by two. Halving.</p> + +<p><a name = "terms_duccioun" id = "terms_duccioun"><b>duccioun</b></a>, +multiplication, <a href = "#duccioun">43/9</a>.</p> + +<p><a name = "terms_duplacion" id = +"terms_duplacion"><b>duplacion</b></a>, +<a href = "#duplacion">7/23</a>, +<a href = "#duplacion2">14/15</a>; the operation of multiplying a number +by two. Doubling.</p> + +<p><a name = "terms_imediet" id = "terms_imediet"><b>i-mediet</b></a> = +halved, <a href = "#imediet">19/23</a>.</p> + +<p><a name = "terms_intercise" id = +"terms_intercise"><b>intercise</b></a> = broken, <a href = +"#intercise">46/2</a>; +intercise Progression is the name given to either of the Progressions 1, +3, 5, 7, etc.; 2, 4, 6, 8, etc., in which the common difference +is 2.</p> + +<p><a name = "terms_lede_into" id = "terms_lede_into"><b>lede +into</b></a>, multiply by, <a href = "#lede_into">47/18</a>.</p> + +<p><a name = "terms_lyneal" id = "terms_lyneal"><b>lyneal +nombre</b></a>, <a href = "#lyneal">46/14</a>; +a number such as that which expresses the measure of the length of a +line, and therefore is not <i>necessarily</i> the product of two or more +numbers (<i>vide</i> Superficial, Solid). This appears to be the meaning +of the phrase as used in <i>The Art of Nombryng</i>. It is possible that +the numbers so designated are the prime numbers, that is, numbers not +divisible by any other number except themselves and unity, but it is not +clear that this limitation is intended.</p> + +<p><a name = "terms_mediacioun" id = +"terms_mediacioun"><b>mediacioun</b></a>, +<a href = "#mediacioun">16/36</a>, +<a href = "#mediacioun2">38/16</a>; dividing by two (see also +<b>dimydicion</b>).</p> + +<p><a name = "terms_medlede" id = "terms_medlede"><b>medlede +nombre</b></a>, +<a href = "#medlede">34/1</a>; +a number formed of a multiple of ten and a digit (<i>vide</i> +componede, composyt).</p> + +<p><a name = "terms_medye" id = "terms_medye"><b>medye</b></a>, +<a href = "#medye">17/8</a>, to halve; +<b>mediete</b>, halved, <a href = "#mediete">17/30</a>; +<b>ymedit</b>, <a href = "#ymedit">20/9</a>.</p> + +<p><a name = "terms_naturelle" id = "terms_naturelle"><b>naturelle +progressioun</b></a>, +<a href = "#naturelle">45/22</a>; +the series of numbers 1, 2, 3, etc.</p> + +<p><a name = "terms_produccioun" id = +"terms_produccioun"><b>produccioun</b></a>, multiplication, +<a href = "#produccioun">50/11</a>.</p> + +<p><a name = "terms_quadrat" id = "terms_quadrat"><b>quadrat +nombre</b></a>, +<a href = "#quadrat">46/12</a>; +a number formed by multiplying a given number by itself, <i>e.g.</i> 9 = +3 × 3, a square.</p> + +<p><a name = "terms_rote" id = "terms_rote"><b>rote</b></a>, +<a href = "#rote">7/25</a>; +<b>roote</b>, <a href = "#roote">47/11</a>; +root. The roots of squares and cubes are the numbers from which the +squares and cubes are derived by multiplication into themselves.</p> + +<p><a name = "terms_significatyf" id = +"terms_significatyf"><b>significatyf</b></a>, significant, +<a href = "#significatyf">5/14</a>; +The significant figures of a number are, strictly speaking, those other +than zero, <i>e.g.</i> in 3 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.</p> + +<p><a name = "terms_solide" id = "terms_solide"><b>solide +nombre</b></a>, <a href = "#solide">46/37</a>; +a number which is the product of three other numbers, <i>e.g.</i> +66 = 11 × 2 × 3.</p> + +<p><a name = "terms_superficial" id = "terms_superficial"><b>superficial +nombre</b></a>, <a href = "#superficial">46/18</a>; +a number which is the product of two other numbers, <i>e.g.</i> 6 = +2 × 3.</p> + +<p><a name = "terms_ternary" id = "terms_ternary"><b>ternary</b></a>, +consisting of three digits, <a href = "#ternary">51/7</a>.</p> + +<p><a name = "terms_vnder_double" id = "terms_vnder_double"><b>vnder +double</b></a>, a digit which has been doubled, +<a href = "#vnder_double">48/3</a>.</p> + +<p><a name = "terms_vnder_trebille" id = +"terms_vnder_trebille"><b>vnder-trebille</b></a>, a digit which has been +trebled, +<a href = "#vnder_trebille">49/28</a>; +<b>vnder-triplat</b>, <a href = "#vnder_triplat">49/39</a>.</p> + +<p><a name = "terms_sup_w" id = "terms_sup_w"><b>w</b></a>, +a symbol used to denote half a unit, <a href = "#sup_w">17/33</a>.</p> + +</div> <!-- end div index --> + +<p class = "footnote"> +<a name = "note_terms1" id = "note_terms1" href = "#tag_terms1">1.</a> +This Index has been kindly prepared by Professor J. B. Dale, of +King’s College, University of London, and the best thanks of the Society +are due to him for his valuable contribution.</p> + +<hr class = "mid" /> + +<div class = "glossary"> + +<span class = "pagenum">83</span> +<a name = "page83" id = "page83"> </a> + +<h3><a name = "glossary" id = "glossary">GLOSSARY</a></h3> + +<p class = "mynote"> +Words whose first appearance is earlier than the page cited in the +Glossary are identified in supplementary notes, and both occurrences are +marked in the main text.</p> + +<p><a name = "gloss_ablacioun" id = +"gloss_ablacioun"><b>ablacioun</b></a>, taking away, <a href = +"#ablacioun">36/21</a></p> + +<p><a name = "gloss_addyst" id = "gloss_addyst"><b>addyst</b></a>, +haddest, <a href = "#addyst">10/37</a></p> + +<p><a name = "gloss_agregacioun" id = +"gloss_agregacioun"><b>agregacioun</b></a>, addition, <a href = +"#agregacioun">45/22</a>. (First example in N.E.D., 1547.)</p> + +<p><a name = "gloss_aghenenes" id = +"gloss_aghenenes"><b>a-ȝenenes</b></a>, against, <a href = +"#aghenenes">23/10</a></p> + +<p><a name = "gloss_allgate" id = "gloss_allgate"><b>allgate</b></a>, +always, <a href = "#allgate">8/39</a></p> + +<p><a name = "gloss_als" id = "gloss_als"><b>als</b></a>, as, <a href = +"#als">22/24</a></p> + +<p><a name = "gloss_and" id = "gloss_and"><b>and</b></a>, if, <a href = +"#and">29/8</a>;</p> +<p class = "inset"> +<b>&</b>, <a href = "#amp">4/27</a>;</p> +<p class = "inset"> +<b>& yf</b>, <a href = "#and_yf">20/7</a></p> + +<p><a name = "gloss_anendes" id = "gloss_anendes"><b>a-nendes</b></a>, +towards, <a href = "#anendes">23/15</a></p> + +<p><a name = "gloss_aproprede" id = +"gloss_aproprede"><b>aproprede</b></a>, appropriated, <a href = +"#aproprede">34/27</a></p> + +<p><a name = "gloss_apwereth" id = "gloss_apwereth"><b>apwereth</b></a>, +appears, <a href = "#apwereth">61/8</a></p> + +<p><a name = "gloss_arisyght" id = "gloss_arisyght"><b>a-risyȝt</b></a>, +arises, <a href = "#arisyght">14/24</a></p> + +<p><a name = "gloss_arowe" id = "gloss_arowe"><b>a-rowe</b></a>, in a +row, <a href = "#arowe">29/10</a></p> + +<p><a name = "gloss_arsemetrike" id = +"gloss_arsemetrike"><b>arsemetrike</b></a>, arithmetic, <a href = +"#arsemetrike">33/1</a></p> + +<p><a name = "gloss_ayene" id = "gloss_ayene"><b>ayene</b></a>, again, +<a href = "#ayene">45/15</a></p> + + +<p><a name = "gloss_bagle" id = "gloss_bagle"><b>bagle</b></a>, crozier, +<a href = "#bagle">67/12</a></p> + +<p><a name = "gloss_bordure" id = "gloss_bordure"><b>bordure</b></a> = +ordure, row, <a href = "#bordure">43/30</a></p> + +<p><a name = "gloss_borro" id = "gloss_borro"><b>borro</b></a>, +<i>inf.</i> borrow, <a href = "#borro">11/38</a>;</p> +<p class = "inset"> +<i>imp. s.</i> <b>borowe</b>, <a href = "#borowe">12/20</a>;</p> +<p class = "inset"> +<i>pp.</i> <b>borwed</b>, <a href = "#borwed">12/15</a>;</p> +<p class = "inset"> +<b>borred</b>, <a href = "#borred">12/19</a></p> + +<p><a name = "gloss_boue" id = "gloss_boue"><b>boue</b></a>, above, +<a href = "#boue">42/34</a></p> + + +<p><a name = "gloss_caputule" id = "gloss_caputule"><b>caputule</b></a>, +chapter, <a href = "#caputule">7/26</a></p> + +<p><a name = "gloss_certayn" id = "gloss_certayn"><b>certayn</b></a>, +assuredly, <a href = "#certayn">18/34</a></p> + +<p><a name = "gloss_clepede" id = "gloss_clepede"><b>clepede</b></a>, +called, <a href = "#clepede">47/7</a></p> + +<p><a name = "gloss_competently" id = +"gloss_competently"><b>competently</b></a>, conveniently, <a href = +"#competently">35/8</a></p> + +<p><a name = "gloss_compt" id = "gloss_compt"><b>compt</b></a>, count, +<a href = "#compt">47/29</a></p> + +<p><a name = "gloss_contynes" id = "gloss_contynes"><b>contynes</b></a>, +contains, <a href = "#contynes">21/12</a>;</p> +<p class = "inset"> +<i>pp.</i> <b>contenythe</b>, <a href = "#contenythe">38/39</a></p> + +<p><a name = "gloss_craft" id = "gloss_craft"><b>craft</b></a>, art, +<a href = "#craft">3/4</a></p> + + +<p><a name = "gloss_distingue" id = +"gloss_distingue"><b>distingue</b></a>, divide, <a href = +"#distingue">51/5</a></p> + + +<p><a name = "gloss_egalle" id = "gloss_egalle"><b>egalle</b></a>, +equal, <a href = "#egalle">45/21</a></p> + +<p><a name = "gloss_excep" id = "gloss_excep"><b>excep</b></a>, except, +<a href = "#excep">5/16</a></p> + +<p><a name = "gloss_exclusede" id = +"gloss_exclusede"><b>exclusede</b></a>, excluded, <a href = +"#exclusede">34/37</a></p> + +<p><a name = "gloss_excressent" id = +"gloss_excressent"><b>excressent</b></a>, resulting, <a href = +"#excressent">35/16</a></p> + +<p><a name = "gloss_exeant" id = "gloss_exeant"><b>exeant</b></a>, +resulting, <a href = "#exeant">43/26</a></p> + +<p><a name = "gloss_expone" id = "gloss_expone"><b>expone</b></a>, +expound, <a href = "#expone">3/23</a></p> + + +<p><a name = "gloss_ferye" id = "gloss_ferye"><b>ferye</b></a> = ferþe, +fourth, <a href = "#ferye">70/12</a></p> + +<p><a name = "gloss_figure" id = "gloss_figure"><b>figure</b></a> = +figures, <a href = "#figure">5/1</a></p> + +<p><a name = "gloss_forby" id = "gloss_forby"><b>for-by</b></a>, past, +<a href = "#forby">12/11</a></p> + +<p><a name = "gloss_fors" id = "gloss_fors"><b>fors; no f.</b></a>, no +matter, <a href = "#fors">22/24</a></p> + +<p><a name = "gloss_forseth" id = "gloss_forseth"><b>forseth</b></a>, +matters, <a href = "#forseth">53/30</a></p> + +<p><a name = "gloss_forye" id = "gloss_forye"><b>forye</b></a> = forþe, +forth, <a href = "#forye">71/8</a></p> + +<p><a name = "gloss_fyftye" id = "gloss_fyftye"><b>fyftye</b></a> = +fyftþe, fifth, <a href = "#fyftye">70/16</a></p> + + +<p><a name = "gloss_grewe" id = "gloss_grewe"><b>grewe</b></a>, Greek, +<a href = "#grewe">33/13</a></p> + + +<p><a name = "gloss_haluendel" id = +"gloss_haluendel"><b>haluendel</b></a>, half, <a href = +"#haluendel">16/16</a>;</p> +<p class = "inset"> +<b>haldel</b>, <a href = "#haldel">19/4</a>;</p> +<p class = "inset"> +<i>pl.</i> <b>haluedels</b>, <a href = "#haluedels">16/16</a></p> + +<p><a name = "gloss_hayst" id = "gloss_hayst"><b>hayst</b></a>, hast, +<a href = "#hayst">17/3</a>, 32</p> + +<p><a name = "gloss_hast" id = "gloss_hast"><b>hast</b></a>, haste, +<a href = "#hast">22/25</a></p> + +<p><a name = "gloss_heer" id = "gloss_heer"><b>heer</b></a>, higher, +<a href = "#heer">9/35</a></p> + +<p><a name = "gloss_here" id = "gloss_here"><b>here</b></a>, their, +<a href = "#here">7/26</a></p> + +<p><a name = "gloss_hereafore" id = +"gloss_hereafore"><b>here-a-fore</b></a>, heretofore, <a href = +"#hereafore">13/7</a></p> + +<p><a name = "gloss_heyth" id = "gloss_heyth"><b>heyth</b></a>, was +called, <a href = "#heyth">3/5</a></p> + +<p><a name = "gloss_hole" id = "gloss_hole"><b>hole</b></a>, whole, +<a href = "#hole">4/39</a>;</p> +<p class = "inset"> +<b>holle</b>, <a href = "#holle">17/1</a>;</p> +<p class = "inset"> +<b>hoole</b>, of three dimensions, <a href = "#hoole">46/15</a></p> + +<p><a name = "gloss_holdythe" id = "gloss_holdythe"><b>holdyþe</b></a>, +holds good, <a href = "#holdythe">30/5</a></p> + +<p><a name = "gloss_how_be_it_that" id = "gloss_how_be_it_that"><b>how +be it that</b></a>, although, <a href = "#how_be_it_that">44/4</a></p> + + +<p><a name = "gloss_lede" id = "gloss_lede"><b>lede</b></a> = lete, let, +<a href = "#lede">8/37</a></p> + +<p><a name = "gloss_lene" id = "gloss_lene"><b>lene</b></a>, lend, +<a href = "#lene">12/39</a></p> + +<p><a name = "gloss_lest" id = "gloss_lest"><b>lest</b></a>, least, +<a href = "#lest">43/27</a></p> + +<p><a name = "gloss_lest2" id = "gloss_lest2"><b>lest</b></a> = left, +<a href = "#lest2">71/9</a></p> + +<p><a name = "gloss_leue" id = "gloss_leue"><b>leue</b></a>, leave, +<a href = "#leue">6/5</a>;</p> +<p class = "inset"> +<i>pr. 3 s.</i> <b>leues</b>, remains, <a href = "#leues">11/19</a>; +<span class = "mynote">First used in <a href = +"#leues1">10/40</a></span></p> +<p class = "inset"> +<b>leus</b>, <a href = "#leus">11/28</a>;</p> +<p class = "inset"> +<i>pp.</i> <b>laft</b>, left, <a href = "#laft">19/24</a></p> + +<p><a name = "gloss_lewder" id = "gloss_lewder"><b>lewder</b></a>, more +ignorant, <a href = "#lewder">3/3</a></p> + +<p><a name = "gloss_lust" id = "gloss_lust"><b>lust</b></a>, desirest +to, <a href = "#lust">45/13</a></p> + +<p><a name = "gloss_lyght" id = "gloss_lyght"><b>lyȝt</b></a>, easy, +<a href = "#lyght">15/31</a></p> + +<p><a name = "gloss_lymytes" id = "gloss_lymytes"><b>lymytes</b></a>, +limits, <a href = "#lymytes">34/18</a>;</p> +<p class = "inset"> +<b>lynes</b>, <a href = "#lynes">34/12</a>;</p> +<p class = "inset"> +<b>lynees</b>, <a href = "#lynees">34/17</a>;</p> +<p class = "inset"> +Lat. limes, <i>pl.</i> limites.</p> + + +<p><a name = "gloss_maystery" id = "gloss_maystery"><b>maystery</b></a>, +achievement;</p> +<p class = "inset"> +<b>no m.</b>, no achievement, i.e. easy, <a href = +"#maystery">19/10</a></p> + +<p><a name = "gloss_me" id = "gloss_me"><b>me</b></a>, <i>indef. +pron.</i> one, <a href = "#me">42/1</a> +<span class = "mynote">First used in <a href = +"#me1">34/16</a></span></p> + +<p><a name = "gloss_mo" id = "gloss_mo"><b>mo</b></a>, more, <a href = +"#mo">9/16</a> +<span class = "pagenum">84</span> +<a name = "page84" id = "page84"> </a></p> +<p class = "inset"> +<b>moder</b> = more (Lat. majorem), <a href = "#moder">43/22</a></p> + +<p><a name = "gloss_most" id = "gloss_most"><b>most</b></a>, must, +<a href = "#most">30/3</a> +<span class = "mynote">First used in <a href = +"#most1">3/12</a></span></p> + +<p><a name = "gloss_multipliede" id = +"gloss_multipliede"><b>multipliede</b></a>, <b>to be m.</b> = +multiplying, <a href = "#multipliede">40/9</a></p> + +<p><a name = "gloss_mynvtes" id = "gloss_mynvtes"><b>mynvtes</b></a>, +the sixty parts into which a unit is divided, <a href = +"#mynvtes">38/25</a></p> + +<p><a name = "gloss_mysewroght" id = +"gloss_mysewroght"><b>myse-wroȝt</b></a>, mis-wrought, <a href = +"#mysewroght">14/11</a></p> + + +<p><a name = "gloss_nether" id = "gloss_nether"><b>nether</b></a>, nor, +<a href = "#nether">34/25</a></p> + +<p><a name = "gloss_nex" id = "gloss_nex"><b>nex</b></a>, next, +<a href = "#nex">19/9</a></p> + +<p><a name = "gloss_noght" id = "gloss_noght"><b>noȝt</b></a>, nought, +<a href = "#noght">5/7</a> +<span class = "mynote">First used in <a href = +"#noght1">4/8</a></span></p> + +<p><a name = "gloss_note" id = "gloss_note"><b>note</b></a>, not, +<a href = "#note">30/5</a></p> + + +<p><a name = "gloss_oo" id = "gloss_oo"><b>oo</b></a>, one, <a href = +"#oo">42/20</a>; <b>o</b>, <a href = "#o">42/21</a> +<span class = "mynote">First used in <a href = +"#oo1">34/27</a> (oo); <a href = +"#o1">33/22</a> (o)</span></p> + +<p><a name = "gloss_omest" id = "gloss_omest"><b>omest</b></a>, +uppermost, higher, <a href = "#omest">35/26</a>;</p> +<p class = "inset"> +<b>omyst</b>, <a href = "#omyst">35/28</a></p> + +<p><a name = "gloss_omwhile" id = "gloss_omwhile"><b>omwhile</b></a>, +sometimes, <a href = "#omwhile">45/31</a> +<span class = "mynote">First used in <a href = +"#omwhile1">39/17</a></span></p> + +<p><a name = "gloss_on" id = "gloss_on"><b>on</b></a>, one, <a href = +"#on">8/29</a></p> + +<p><a name = "gloss_opyne" id = "gloss_opyne"><b>opyne</b></a>, plain, +<a href = "#opyne">47/8</a></p> + +<p><a name = "gloss_or" id = "gloss_or"><b>or</b></a>, before, <a href = +"#or">13/25</a></p> + +<p><a name = "gloss_or2" id = "gloss_or2"><b>or</b></a> = þe +oþ<i>er</i>, the other, <a href = "#or">28/34</a></p> + +<p><a name = "gloss_ordure" id = "gloss_ordure"><b>ordure</b></a>, +order, <a href = "#ordure">34/9</a>;</p> +<p class = "inset"> +row, <a href = "#order">43/1</a> +<span class = "mynote">Word form is “order”</span></p> + +<p><a name = "gloss_other" id = "gloss_other"><b>other</b></a>, or, +<a href = "#other">33/13</a>, <a href = "#other2">43/26</a>; +<span class = "mynote">Note also “one other other” in <a href = +"#other1">35/24</a></span></p> +<p class = "inset"> +<b>other . . . or</b>, either . . . or, <a href = "#other_or">38/37</a> +<span class = "mynote">First used in <a href = +"#other_or1">37/5</a></span></p> + +<p><a name = "gloss_ouerer" id = "gloss_ouerer"><b>ouerer</b></a>, +upper, <a href = "#ouerer">42/15</a></p> + +<p><a name = "gloss_ouerhippede" id = +"gloss_ouerhippede"><b>ouer-hippede</b></a>, passed over, <a href = +"#ouerhippede">43/19</a></p> + + +<p><a name = "gloss_recte" id = "gloss_recte"><b>recte</b></a>, +directly, <a href = "#recte">27/20</a> +<span class = "mynote">First used in <a href = +"#recte1">26/31</a></span></p> + +<p><a name = "gloss_remayner" id = "gloss_remayner"><b>remayner</b></a>, +remainder, <a href = "#remayner">56/28</a></p> + +<p><a name = "gloss_representithe" id = +"gloss_representithe"><b>representithe</b></a>, represented, <a href = +"#representithe">39/14</a></p> + +<p><a name = "gloss_resteth" id = "gloss_resteth"><b>resteth</b></a>, +remains, <a href = "#resteth">63/29</a> +<span class = "mynote">First used in <a href = +"#resteth1">57/29</a></span></p> + +<p><a name = "gloss_rewarde" id = "gloss_rewarde"><b>rewarde</b></a>, +regard, <a href = "#rewarde">48/6</a></p> + +<p><a name = "gloss_rew" id = "gloss_rew"><b>rew</b></a>, row, <a href = +"#rew">4/8</a></p> + +<p><a name = "gloss_rewle" id = "gloss_rewle"><b>rewle</b></a>, row, +<a href = "#rewle">4/20</a>, <a href = "#rewle2">7/12</a>;</p> +<p class = "inset"> +<b>rewele</b>, <a href = "#rewele">4/18</a>;</p> +<p class = "inset"> +<b>rewles</b>, rules, <a href = "#rewles">5/33</a></p> + + +<p><a name = "gloss_sc" id = "gloss_sc"><b>s.</b></a> = scilicet, +<a href = "#sc">3/8</a></p> + +<p><a name = "gloss_sentens" id = "gloss_sentens"><b>sentens</b></a>, +meaning, <a href = "#sentens">14/29</a></p> + +<p><a name = "gloss_signifyetyf" id = +"gloss_signifyetyf"><b>signifye(tyf)</b></a>, <a href = +"#signifyetyf">5/13</a>. The last three letters are added above the +line, evidently because of the word ‘significatyf’ in l. 14. But +the ‘Solucio,’ which contained the word, has been omitted.</p> + +<p><a name = "gloss_sithen" id = "gloss_sithen"><b>sithen</b></a>, +since, <a href = "#sithen">33/8</a></p> + +<p><a name = "gloss_some" id = "gloss_some"><b>some</b></a>, sum, +result, <a href = "#some">40/17</a>, 32 +<span class = "mynote">First used in <a href = +"#some1">36/21</a></span></p> + +<p><a name = "gloss_sowne" id = "gloss_sowne"><b>sowne</b></a>, +pronounce, <a href = "#sowne">6/29</a></p> + +<p><a name = "gloss_singillatim" id = +"gloss_singillatim"><b>singillatim</b></a>, singly, <a href = +"#singillatim">7/25</a></p> + +<p><a name = "gloss_spices" id = "gloss_spices"><b>spices</b></a>, +species, kinds, <a href = "#spices">34/4</a><span class = "mynote">First +used in <a href = "#spices1">5/34</a></span></p> + +<p><a name = "gloss_spyl" id = "gloss_spyl"><b>spyl</b></a>, waste, +<a href = "#spyl">14/26</a></p> + +<p><a name = "gloss_styde" id = "gloss_styde"><b>styde</b></a>, stead, +<a href = "#styde">18/20</a></p> + +<p><a name = "gloss_subtrahe" id = "gloss_subtrahe"><b>subtrahe</b></a>, +subtract, <a href = "#subtrahe">48/12</a>;</p> +<p class = "inset"> +<i>pp.</i> <b>subtrayd</b>, <a href = "#subtrayd">13/21</a></p> + +<p><a name = "gloss_sythes" id = "gloss_sythes"><b>sythes</b></a>, +times, <a href = "#sythes">21/16</a></p> + + +<p><a name = "gloss_taght" id = "gloss_taght"><b>taȝt</b></a>, taught, +<a href = "#taght">16/36</a></p> + +<p><a name = "gloss_take" id = "gloss_take"><b>take</b></a>, <i>pp.</i> +taken;</p> +<p class = "inset"> +<b>t. fro</b>, starting from, <a href = "#take">45/22</a></p> + +<p><a name = "gloss_taward" id = "gloss_taward"><b>taward</b></a>, +toward, <a href = "#taward">23/34</a></p> + +<p><a name = "gloss_thought" id = "gloss_thought"><b>thouȝt</b></a>, +though, <a href = "#thought">5/20</a></p> + +<p><a name = "gloss_trebille" id = "gloss_trebille"><b>trebille</b></a>, +multiply by three, <a href = "#trebille">49/26</a></p> + +<p><a name = "gloss_twene" id = "gloss_twene"><b>twene</b></a>, two, +<a href = "#twene">8/11</a> +<span class = "mynote">First used in <a href = +"#twene1">4/23</a></span></p> + +<p><a name = "gloss_thow" id = "gloss_thow"><b>þow</b></a>, though, +<a href = "#thow">25/15</a></p> + +<p><a name = "gloss_thowght" id = "gloss_thowght"><b>þowȝt</b></a>, +thought;</p> +<p class = "inset"> +<b>be þ.</b>, mentally, <a href = "#thowght">28/4</a></p> + +<p><a name = "gloss_thus" id = "gloss_thus"><b>þus</b></a> = þis, this, +<a href = "#thus">20/33</a></p> + + +<p><a name = "gloss_vny" id = "gloss_vny"><b>vny</b></a>, unite, +<a href = "#vny">45/10</a></p> + + +<p><a name = "gloss_wel" id = "gloss_wel"><b>wel</b></a>, wilt, +<a href = "#wel">14/31</a></p> + +<p><a name = "gloss_wete" id = "gloss_wete"><b>wete</b></a>, wit, +<a href = "#wete">15/16</a>;</p> +<p class = "inset"> +<b>wyte</b>, know, <a href = "#wyte">8/38</a>;</p> +<p class = "inset"> +<i>pr. 2 s.</i> <b>wost</b>, <a href = "#wost">12/38</a></p> + +<p><a name = "gloss_wex" id = "gloss_wex"><b>wex</b></a>, become, +<a href = "#wex">50/18</a></p> + +<p><a name = "gloss_where" id = "gloss_where"><b>where</b></a>, whether, +<a href = "#where">29/12</a></p> + +<p><a name = "gloss_wherthurghe" id = +"gloss_wherthurghe"><b>wher-thurghe</b></a>, whence, <a href = +"#wherthurghe">49/15</a></p> + +<p><a name = "gloss_worch" id = "gloss_worch"><b>worch</b></a>, work, +<a href = "#worch">8/19</a>; +<span class = "mynote">First used in <a href = +"#worch1">7/35</a></span></p> +<p class = "inset"> +<b>wrich</b>, <a href = "#wrich">8/35</a>;</p> +<p class = "inset"> +<b>wyrch</b>, <a href = "#wyrch">6/19</a>;</p> +<p class = "inset"> +<i>imp. s.</i> <b>worch</b>, <a href = "#worch">15/9</a>; +<span class = "mynote">First used in <a href = +"#worch1">9/6</a></span></p> +<p class = "inset"> +<i>pp.</i> <b>y-wroth</b>, <a href = "#ywroth">13/24</a></p> + +<p><a name = "gloss_write" id = "gloss_write"><b>write</b></a>, written, +<a href = "#write">29/19</a>; +<span class = "mynote">First used in <a href = +"#write1">4/5</a></span></p> +<p class = "inset"> +<b>y-write</b>, <a href = "#ywrite">16/1</a></p> + +<p><a name = "gloss_wryrchynge" id = +"gloss_wryrchynge"><b>wryrchynge</b></a> = wyrchynge, working, <a href = +"#wryrchynge">30/4</a></p> + +<p><a name = "gloss_wt" id = "gloss_wt"><b>w<sup>t</sup></b></a>, with, +<a href = "#wt">55/8</a></p> + + +<p><a name = "gloss_ybroth" id = "gloss_ybroth"><b>y-broth</b></a>, +brought, <a href = "#ybroth">21/18</a></p> + +<p><a name = "gloss_ychon" id = "gloss_ychon"><b>ychon</b></a>, each +one, <a href = "#ychon">29/10</a></p> + +<p><a name = "gloss_ydo" id = "gloss_ydo"><b>ydo</b></a>, done, added, +<a href = "#ydo">9/6</a> +<span class = "mynote">First used in <a href = +"#ydo1">8/37</a></span></p> + +<p><a name = "gloss_ylke" id = "gloss_ylke"><b>ylke</b></a>, same, +<a href = "#ylke">5/12</a></p> + +<p><a name = "gloss_ylyech" id = "gloss_ylyech"><b>y-lyech</b></a>, +alike, <a href = "#ylyech">22/23</a></p> + +<p><a name = "gloss_ymyght" id = "gloss_ymyght"><b>y-myȝt</b></a>, been +able, <a href = "#ymyght">12/2</a></p> + +<p><a name = "gloss_ynowght" id = "gloss_ynowght"><b>y-nowȝt</b></a>, +enough, <a href = "#ynowght">15/31</a>;</p> +<p class = "inset"> +<b>ynovȝt</b>, <a href = "#ynovght">18/34</a></p> + +<p><a name = "gloss_yove" id = "gloss_yove"><b>yove</b></a>, given, +<a href = "#yove">45/33</a></p> + +<p><a name = "gloss_yt" id = "gloss_yt"><b>y<sup>t</sup></b></a>, that, +<a href = "#yt">52/8</a></p> + +<p><b>y-write</b>, <i>v.</i> <a href = +"#gloss_write"><b>write.</b></a></p> + +<p><b>y-wroth</b>, <i>v.</i> <a href = +"#gloss_worch"><b>worch.</b></a></p> + +</div> <!-- end div glossary --> + +<div class = "endnote"> +<h4><a name = "endnote" id = "endnote">MARGINAL NOTES</a></h4> + +<p><b>Headnotes</b> have been moved to the beginning of the appropriate +paragraph. Headnotes were omitted from the two Appendixes, as sidenotes +give the same information.</p> + +<p><b>Line Numbers</b> are cited in the Index and Glossary. They have +been omitted from the e-text except in the one verse selection +(App. II, <i>Carmen de Algorismo</i>). Instead, the Index and +Glossary are linked directly to each word.</p> + +<p><b>Numbered Notes</b>:</p> + +<p class = "inset"> +Numbered sidenotes show page or leaf numbers from the original MSS. In +the e-text, sidenote numbers have been replaced with simple +asterisks.</p> + +<p class = "inset"> +Footnotes give textual information such as variant readings. They +have been numbered sequentially within each title.</p> + +<p><b>Sidenotes</b> giving a running synopsis of the text have been kept +as close as possible to their original format and location.</p> + +</div> + + + + + + + + +<pre> + + + + + +End of Project Gutenberg's The Earliest Arithmetics in English, by Anonymous + +*** END OF THIS PROJECT GUTENBERG EBOOK THE EARLIEST ARITHMETICS IN ENGLISH *** + +***** This file should be named 25664-h.htm or 25664-h.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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