,transcriber's ,note3 ,this e-text includes characters that will only display in ,,utf-#h 7,unicode7 text readers3 1 7yogh1 long s7 1 7n with curl1 crossed l3 see below7 7,greek phi3 see below7 7small raised 8w07 ,if any of these characters do not display properly1 or if the apostrophes and quotation marks in this paragraph appear as garbage1 make sure your text reader's 8character set0 or 8file encoding0 is set to ,unicode 7,,utf-#h74 ,you may also need to change the default font4 ,in ,the ,crafte of ,nombrynge1 final 8n0 was sometimes written with an extra curl4 ,it has been rendered as for visual effect2 the character is not intended to convey phonetic information4 ,in the same selection1 the numeral 8#j0 was sometimes printed as ,greek 7phi72 this has been retained for the e-text4 ,double 8l0 with a line is shown as 4 ,the first few occurrences of 8d0 7for 8pence07 were printed with a decorative curl4 ,the letter is shown with the same 8d'0 used in the remainder of the text4 ,the word 8withdraw0 or 8withdraw0 was inconsistently hyphenated2 it was left as printed1 and line-end hyphens were retained4 ,superscripts are shown with carets as e4 ,except for ,illustration markers and similar1 all brackets are in the original4 ,individual letters were italicized to show expanded abbreviations2 these are shown in braces4 ,other italicized words are shown conventionally with lines1 boldface with marks4 ,when a footnote called for added text1 the addition is shown in the body text with double brackets4 ,the original text contained at least five types of marginal note4 ,details are given at the end of the e-text1 followed by a listing of typographical errors4 99 99 99 99 99 99 99 99 99 99 99 99 99 99 ,the ,earliest ,arithmetics in ,english ,early ,english ,text ,society4 ,extra ,series1 ,no4 ,,cxviii4 #aibb 7for #aiaf74 ,,the ,,earliest ,,arithmetics ,,in ,,english ,edited ,with ,introduction by ,,robert ,,steele ,london3 ,published for the ,early ,english ,text ,society ,by ,humphrey ,milford1 ,oxford ,university ,press1 ,amen ,corner1 ,e4,c4 #d4 #aibb4 ,titles 7list added by transcriber73 ,the ,crafte of ,nombrynge ,the ,art of ,nombryng ,accomptynge by ,counters ,the arte of nombrynge by the hande ,,app4 ,i4 ,a ,treatise on the ,numeration of ,algorism ,,app4 ,,ii4 ,carmen de ,algorismo ,,introduction ,the number of ,english arithmetics before the sixteenth century is very small4 ,this is hardly to be wondered at1 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 ,latin1 in which language there were several treatises in a considerable number of manuscripts1 as shown by the quantity of them still in existence4 ,until modern commerce was fairly well established1 few persons required more arithmetic than addition and subtraction1 and even in the thirteenth century1 scientific treatises addressed to advanced students contemplated the likelihood of their not being able to do simple division4 ,on the other hand1 the study of astronomy necessitated1 from its earliest days as a science1 considerable skill and accuracy in computation1 not only in the calculation of astronomical tables but in their use1 a knowledge of which latter was fairly common from the thirteenth to the sixteenth centuries4 ,the arithmetics in ,english known to me are3-- 7#a7 ,bodl4 #gij ,g4 ,,vii4 7#bfec7 f4 #adf-aed 7#aeth c47 inc4 8,of angrym ther be ,,ix figures in numbray 4 4 40 ,a mere unfinished fragment1 only getting as far as ,duplation4 7#b7 ,camb4 ,univ4 ,,li4 ,,iv4 #ad 7,,iii47 f4 #aba-adb 7#aeth c47 inc4 8,al maner of thyngis that prosedeth ffro the frist begynnyng 4 4 40 7#c7 ,fragmentary passages or diagrams in ,sloane #bac f4 #abj-c 7a fourteenth-century counting board71 ,egerton #bheb f4 #e-ac1 ,harl4 #bah f4 #adg and 7#d7 ,the two ,,mss4 here printed2 ,eg4 #bfbb f4 #acf and ,ashmole #cif f4 #dh4 ,all of these1 as the language shows1 are of the fifteenth century4 ,the ,,crafte ,,of ,,nombrynge is one of a large number of scientific treatises1 mostly in ,latin1 bound up together as ,egerton ,,ms4 #bfbb in the ,british ,museum ,library4 ,it measures #g0 #e01 #bi-cj lines to the page1 in a rough hand4 ,the ,english is ,n4,e4 ,midland in dialect4 ,it is a translation and amplification of one of the numerous glosses on the de algorismo of ,alexander de ,villa ,dei 7c4 #abbj71 such as that of ,thomas of ,newmarket contained in the ,british ,museum ,,ms4 ,reg4 #ab1 ,e4 #a4 ,a fragment of another translation of the same gloss was printed by ,halliwell in his ,rara ,mathematica 7#ahce7 p4 #bi4a99 ,it corresponds1 as far as p4 #ga1 l4 #b1 roughly to p4 #c of our version1 and from thence to the end p4 #b1 ll4 #af-dj4 ,footnote #a993 ,halliwell printed the two sides of his leaf in the wrong order4 ,this and some obvious errors of transcription-- #8ferye' for #8ferthe1' #8lest' for #8left1' etc41 have not been corrected in the reprint on pp4 #gj-ga4 ,the ,,art ,,of ,,nombryng is one of the treatises bound up in the ,bodleian ,,ms4 ,ashmole #cif4 ,it measures #aa0 #ag01 and is written with thirty-three lines to the page in a fifteenth century hand4 ,it is a translation1 rather literal1 with amplifications of the de arte numerandi attributed to ,john of ,holywood 7,sacrobosco7 and the translator had obviously a poor ,,ms4 before him4 ,the de arte numerandi was printed in #adhh1 #adij 7s4n471 #aeja1 #aejc1 #aeaj1 #aeag1 #aeba1 #aebb1 #aebc1 #aehb1 and by ,halliwell separately and in his two editions of ,rara ,mathematica1 #ahci and #ahda1 and reprinted by ,curze in #ahig4 ,both these tracts are here printed for the first time1 but the first having been circulated in proof a number of years ago1 in an endeavour to discover other manuscripts or parts of manuscripts of it1 ,dr4 ,david ,eugene ,smith1 misunderstanding the position1 printed some pages in a curious transcript with four facsimiles in the ,archiv fr die ,geschichte der ,naturwissenschaften und der ,technik1 #aiji1 and invited the scientific world to take up the 8not unpleasant task0 of editing it4 ,,accomptynge ,,by ,,counters is reprinted from the #aedc edition of ,robert ,record's ,arithmetic1 printed by ,r4 ,wolfe4 ,it has been reprinted within the last few years by ,mr4 ,f4 ,p4 ,barnard1 in his work on ,casting ,counters4 ,it is the earliest ,english treatise we have on this variety of the ,abacus 7there are ,latin ones of the end of the fifteenth century71 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 methods4 ,at the end of the treatise there follows a note on merchants' and auditors' ways of setting down sums1 and lastly1 a system of digital numeration which seems of great antiquity and almost world-wide extension4 ,after the fragment already referred to1 ,i print as an appendix the #8,carmen de ,algorismo' of ,alexander de ,villa ,dei in an enlarged and corrected form4 ,it was printed for the first time by ,halliwell in ,rara ,mathemathica1 but ,i have added a number of stanzas from various manuscripts1 selecting various readings on the principle that the verses were made to scan1 aided by the advice of my friend ,mr4 ,vernon ,rendall1 who is not responsible for the few doubtful lines ,i have conserved4 ,this poem is at the base of all other treatises on the subject in medieval times1 but ,i am unable to indicate its sources4 ,,the ,,subject ,,matter4 ,ancient and medieval writers observed a distinction between the ,science and the ,art of ,arithmetic4 ,the classical treatises on the subject1 those of ,euclid among the ,greeks and ,boethius among the ,latins1 are devoted to the ,science of ,arithmetic1 but it is obvious that coeval with practical ,astronomy the ,art of ,calculation must have existed and have made considerable progress4 ,if early treatises on this art existed at all they must1 almost of necessity1 have been in ,greek1 which was the language of science for the ,romans as long as ,latin civilisation existed4 ,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 ,romans4 ,specimen sums have indeed been constructed by moderns which show its possibility1 but it is absurd to think that men of science1 acquainted with ,egyptian methods and in possession of the abacus1#b99 were unable to devise methods for its use4 ,footnote #b993 ,for ,egyptian use see ,herodotus1 ii4 #cf1 ,plato1 de ,legibus1 ,,vii4 ,,the ,,pre-,,medieval ,,instruments ,,used ,,in ,,calculation4 ,the following are known3-- 7#a7 ,a flat polished surface or tablets1 strewn with sand1 on which figures were inscribed with a stylus4 7#b7 ,a polished tablet divided longitudinally into nine columns 7or more7 grouped in threes1 with which counters were used1 either plain or marked with signs denoting the nine numerals1 etc4 7#c7 ,tablets or boxes containing nine grooves or wires1 in or on which ran beads4 7#d7 ,tablets on which nine 7or more7 horizontal lines were marked1 each third being marked off4 ,the only ,greek counting board we have is of the fourth class and was discovered at ,salamis4 ,it was engraved on a block of marble1 and measures #e feet by #b4 ,its chief part consists of eleven parallel lines1 the #crd1 #fth1 and #ith being marked with a cross4 ,another section consists of five parallel lines1 and there are three rows of arithmetical symbols4 ,this board could only have been used with counters 7calculi71 preferably unmarked1 as in our treatise of ,accomptynge by ,counters4 ,,classical ,,roman ,,methods ,,of ,,calculation4 ,we have proof of two methods of calculation in ancient ,rome1 one by the first method1 in which the surface of sand was divided into columns by a stylus or the hand4 ,counters 7calculi1 or lapilli71 which were kept in boxes 7loculi71 were used in calculation1 as we learn from ,horace's schoolboys 7,sat4 #a4 vi4 #gd74 ,for the sand see ,persius ,i4 #aca1 8,nec qui abaco numeros et secto in pulvere metas scit risisse10 ,apul4 ,apolog4 #af 7pulvisculo71 ,mart4 ,capella1 lib4 vii4 #c1 #d1 etc4 ,cicero says of an expert calculator 8eruditum attigisse pulverem10 7de nat4 ,deorum1 ii4 #ah74 ,tertullian calls a teacher of arithmetic 8primus numerorum arenarius0 7de ,pallio1 in fine74 ,the counters were made of various materials1 ivory principally1 8,adeo nulla uncia nobis est eboris1 etc40 7,juv4 ,,xi4 #aca71 sometimes of precious metals1 8,pro calculis albis et nigris aureos argenteosque habebat denarios0 7,pet4 ,arb4 ,satyricon1 #cc74 ,there are1 however1 still in existence four ,roman counting boards of a kind which does not appear to come into literature4 ,a typical one is of the third class4 ,it consists of a number of transverse wires1 broken at the middle4 ,on the left hand portion four beads are strung1 on the right one 7or two74 ,the left hand beads signify units1 the right hand one five units4 ,thus any number up to nine can be represented4 ,this instrument is in all essentials the same as the ,swanpan or ,abacus in use throughout the ,far ,east4 ,the ,russian stchota in use throughout ,eastern ,europe is simpler still4 ,the method of using this system is exactly the same as that of ,accomptynge by ,counters1 the right-hand five bead replacing the counter between the lines4 ,,the ,,boethian ,,abacus4 ,between classical times and the tenth century we have little or no guidance as to the art of calculation4 ,boethius 7fifth century71 at the end of lib4 ,,ii4 of his ,geometria gives us a figure of an abacus of the second class with a set of counters arranged within it4 ,it has1 however1 been contended with great probability that the whole passage is a tenth century interpolation4 ,as no rules are given for its use1 the chief value of the figure is that it gives the signs of the nine numbers1 known as the ,boethian 8apices0 or 8notae0 7from whence our word 8notation074 ,to these we shall return later on4 ,,the ,,abacists4 ,it would seem probable that writers on the calendar like ,bede 7,a4,d4 #gba7 and ,helpericus 7,a4,d4 #ijc7 were able to perform simple calculations2 though we are unable to guess their methods1 and for the most part they were dependent on tables taken from ,greek sources4 ,we have no early medieval treatises on arithmetic1 till towards the end of the tenth century we find a revival of the study of science1 centring for us round the name of ,gerbert1 who became ,pope as ,sylvester ,,ii4 in #iii4 ,his treatise on the use of the ,abacus was written 7c4 #ihj7 to a friend ,constantine1 and was first printed among the works of ,bede in the ,basle 7#aefc7 edition of his works1 ,i4 #aei1 in a somewhat enlarged form4 ,another tenth century treatise is that of ,abbo of ,fleury 7c4 #ihh71 preserved in several manuscripts4 ,very few treatises on the use of the ,abacus can be certainly ascribed to the eleventh century1 but from the beginning of the twelfth century their numbers increase rapidly1 to judge by those that have been preserved4 ,the ,abacists used a permanent board usually divided into twelve columns2 the columns were grouped in threes1 each column being called an 8arcus10 and the value of a figure in it represented a tenth of what it would have in the column to the left1 as in our arithmetic of position4 ,with this board counters or jetons were used1 either plain or1 more probably1 marked with numerical signs1 which with the early ,abacists were the 8apices10 though counters from classical times were sometimes marked on one side with the digital signs1 on the other with ,roman numerals4 ,two ivory discs of this kind from the ,hamilton collection may be seen at the ,british ,museum4 ,gerbert is said by ,richer to have made for the purpose of computation a thousand counters of horn2 the usual number of a set of counters in the sixteenth and seventeenth centuries was a hundred4 ,treatises on the ,abacus usually consist of chapters on ,numeration explaining the notation1 and on the rules for ,multiplication and ,division4 ,addition1 as far as it required any rules1 came naturally under ,multiplication1 while ,subtraction was involved in the process of ,division4 ,these rules were all that were needed in ,western ,europe in centuries when commerce hardly existed1 and astronomy was unpractised1 and even they were only required in the preparation of the calendar and the assignments of the royal exchequer4 ,in ,england1 for example1 when the hide developed from the normal holding of a household into the unit of taxation1 the calculation of the geldage in each shire required a sum in division2 as we know from the fact that one of the ,abacists proposes the sum3 8,if #bjj marks are levied on the county of ,essex1 which contains according to ,hugh of ,bocland #bejj hides1 how much does each hide pay80#c99 ,exchequer methods up to the sixteenth century were founded on the abacus1 though when we have details later on1 a different and simpler form was used4 ,footnote #c993 ,see on this ,dr4 ,poole1 ,the ,exchequer in the ,twelfth ,century1 ,chap4 ,,iii41 and ,haskins1 ,eng4 ,hist4 ,review1 #bg1 #aja4 ,the hidage of ,essex in #aacj was #bcfd hides4 ,the great difficulty of the early ,abacists1 owing to the absence of a figure representing zero1 was to place their results and operations in the proper columns of the abacus1 especially when doing a division sum4 ,the chief differences noticeable in their works are in the methods for this rule4 ,division was either done directly or by means of differences between the divisor and the next higher multiple of ten to the divisor4 ,later ,abacists made a distinction between 8iron0 and 8golden0 methods of division4 ,the following are examples taken from a twelfth century treatise4 ,in following the operations it must be remembered that a figure asterisked represents a counter taken from the board4 ,a zero is obviously not needed1 and the result may be written down in words4 7a7 ,,multiplication4 #dfjj #bc4 ---------------------- ,thousands ------------------ ,h ,t ,u ,h ,t ,u u e n u e n n n i n n i d s t d s t r s r s e e d d s s ------------------ #d #f ,multiplicand4 ------------------ #a #h #fjj #c4 #a #b #djjj #c4 #a #b #fjj #bj4 #h #djjj #bj4 ------------------ #a #e #h ,total product4 ------------------ #b #c ,multiplier4 ------------------ 7b7 ,,division3 ,,direct4 #ajj1jjj #bj1jbc4 ,here each counter in turn is a separate divisor4 ---------------------- ,thousands ------------------ ,h4 ,t4 ,u4 ,h4 ,t4 ,u4 ------------------ #b #b #c ,divisors4 ------------------ #b ,place greatest divisor to right of dividend4 #a ,dividend4 #b ,remainder4 #a #a #i #i ,another form of same4 #h ,product of #ast ,quotient and #bj4 ------------------ #a #i #i #b ,remainder4 #a #b ,product of #ast ,quotient and #c4 ------------------ #a #i #i #h ,final remainder4 #d ,quotient4 ------------------ 7c7 ,,division ,,by ,,differences4 #ijj #h4 ,here we divide by 7#aj-b74 -------------------- ,h4 ,t4 ,u4 -------------------- #b ,difference4 #h ,divisor4 -------------------- #d99i ,dividend4 #d99a #h ,product of difference by #ast ,quotient 7#i74 #b ,product of difference by #bnd ,quotient 7#a74 -------------------- #d99a ,sum of #h and #b4 #b ,product of difference by #crd ,quotient 7#a74 #d ,product of difference by #dth ,quot4 7#b74 ,remainder4 -------------------- #b #dth ,quotient4 #a #crd ,quotient4 #a #bnd ,quotient4 #i #ast ,quotient4 -------------------- #a #a #b ,quotient4 7,total of all four47 -------------------- ,footnote #d993 ,these figures are removed at the next step4 ,,division4 #ghjj #aff4 ------------------------------ ,thousands -------------------------- ,h4 ,t4 ,u4 ,h4 ,t4 ,u4 -------------------------- #c #d ,differences 7making #bjj trial divisor74 #a #f #f ,divisors4 -------------------------- #d99g #h ,dividends4 #a ,remainder of greatest dividend4 #a #b ,product of #ast difference 7#d7 by #ast ,quotient 7#c74 #i ,product of #bnd difference 7#c7 by #ast ,quotient 7#c74 -------------------------- #d99b #h #b ,new dividends4 #c #d ,product of #ast and #bnd difference by #bnd ,quotient 7#a74 -------------------------- #d99a #a #f ,new dividends4 #b ,product of #ast difference by #crd ,quotient 7#e74 #a #e ,product of #bnd difference by #crd ,quotient 7#e74 -------------------------- #d99c #c ,new dividends4 #a ,remainder of greatest dividend4 #c #d ,product of #ast and #bnd difference by #dth ,quotient 7#a74 -------------------------- #a #f #d ,remainder 7less than divisor74 #a #dth ,quotient4 #e #crd ,quotient4 #a #bnd ,quotient4 #c #ast ,quotient4 -------------------------- #d #f ,quotient4 -------------------------- ,footnote #d993 ,these figures are removed at the next step4 ,,division4 #hjjj #fjf4 ------------------------ ,thousands -------------------- ,h4 ,t4 ,u4 ,h4 ,t4 ,u4 -------------------- #i ,difference 7making #gjj trial divisor74 #d ,difference4 #f #f ,divisors4 -------------------- #d99h ,dividend4 #a ,remainder of dividend4 #i #d ,product of difference #a and #b with #ast ,quotient 7#a74 -------------------- #d99a #i #d ,new dividends4 #c ,remainder of greatest dividend4 #i #d ,product of difference #a and #b with #bnd ,quotient 7#a74 -------------------- #d99a #c #c #d ,new dividends4 #c ,remainder of greatest dividend4 #i #d ,product of difference #a and #b with #crd ,quotient 7#a74 -------------------- #g #b #h ,new dividends4 #f #f ,product of divisors by #dth ,quotient 7#a74 -------------------- #a #b #b ,remainder4 #a #dth ,quotient4 #a #crd ,quotient4 #a #bnd ,quotient4 #a #ast ,quotient4 -------------------- #a #c ,quotient4 -------------------- ,footnote #d993 ,these figures are removed at the next step4 ,the chief ,abacists are ,gerbert 7tenth century71 ,abbo1 and ,hermannus ,contractus 7#ajed71 who are credited with the revival of the art1 ,bernelinus1 ,gerland1 and ,radulphus of ,laon 7twelfth century74 ,we know as ,english ,abacists1 ,robert1 bishop of ,hereford1 #ajie1 8abacum et lunarem compotum et celestium cursum astrorum rimatus10 ,turchillus ,compotista 7,thurkil71 and through him of ,guilielmus ,r4 4 4 4 8the best of living computers10 ,gislebert1 and ,simonus de ,rotellis 7,simon of the ,rolls74 ,they flourished most probably in the first quarter of the twelfth century1 as ,thurkil's treatise deals also with fractions4 ,walcher of ,durham1 ,thomas of ,york1 and ,samson of ,worcester are also known as ,abacists4 ,finally1 the term ,abacists came to be applied to computers by manual arithmetic4 ,a ,,ms4 ,algorithm of the thirteenth century 7,sl4 #cbha1 f4 #f1 b71 contains the following passage3 8,est et alius modus secundum operatores sive practicos1 quorum unus appellatur ,abacus2 et modus ejus est in computando per digitos et junctura manuum1 et iste utitur ultra ,alpes40 ,in a composite treatise containing tracts written ,a4,d4 #aaeg and #abjh1 on the calendar1 the abacus1 the manual calendar and the manual abacus1 we have a number of the methods preserved4 ,as an example we give the rule for multiplication 7,claud4 ,a4 ,,iv41 f4 #ed vo74 8,si numerus multiplicat alium numerum auferatur differentia majoris a minore1 et per residuum multiplicetur articulus1 et una differentia per aliam1 et summa proveniet40 ,example1 #h #g4 ,the difference of #h is #b1 of #g is #c1 the next article being #aj2 #g - #b is #e4 #e #aj #ej2 #b #c #f4 #ej #f #ef answer4 ,the rule will hold in such cases as #ag #ae where the article next higher is the same for both1 i4e41 #bj2 but in such a case as #ag #i the difference for each number must be taken from the higher article1 i4e41 the difference of #i will be #aa4 ,,the ,,algorists4 ,algorism 7augrim1 augrym1 algram1 agram1 algorithm71 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 century1 8de numeris ,indorum10 beginning in its ,latin form 8,dixit ,algorismi4 4 4 40 ,the translation1 of which only one ,,ms4 is known1 was made about #aabj by ,adelard of ,bath1 who also wrote on the ,abacus and translated with a commentary ,euclid from the ,arabic4 ,it is probable that another version was made by ,gerard of ,cremona 7#aaad-aahg72 the number of important works that were not translated more than once from the ,arabic decreases every year with our knowledge of medieval texts4 ,a few lines of this translation1 as copied by ,halliwell1 are given on p4 #gb1 note #b4 ,another translation still seems to have been made by ,johannes ,hispalensis4 ,algorism is distinguished from ,abacist computation by recognising seven rules1 ,addition1 ,subtraction1 ,duplation1 ,mediation1 ,multiplication1 ,division1 and ,extraction of ,roots1 to which were afterwards added ,numeration and ,progression4 ,it is further distinguished by the use of the zero1 which enabled the computer to dispense with the columns of the ,abacus4 ,it obviously employs a board with fine sand or wax1 and later1 as a substitute1 paper or parchment2 slate and pencil were also used in the fourteenth century1 how much earlier is unknown4#e99 ,algorism quickly ousted the ,abacus methods for all intricate calculations1 being simpler and more easily checked3 in fact1 the astronomical revival of the twelfth and thirteenth centuries would have been impossible without its aid4 ,footnote #e993 ,slates are mentioned by ,chaucer1 and soon after 7#adaj7 ,prosdocimo de ,beldamandi speaks of the use of a 8lapis0 for making notes on by calculators4 ,the number of ,latin ,algorisms still in manuscript is comparatively large1 but we are here only concerned with two--an ,algorism in prose attributed to ,sacrobosco 7,john of ,holywood7 in the colophon of a ,paris manuscript1 though this attribution is no longer regarded as conclusive1 and another in verse1 most probably by ,alexander de ,villedieu 7,villa ,dei74 ,alexander1 who died in #abdj1 was teaching in ,paris in #abji4 ,his verse treatise on the ,calendar is dated #abjj1 and it is to that period that his ,algorism may be attributed2 ,sacrobosco died in #abef and quotes the verse ,algorism4 ,several commentaries on ,alexander's verse treatise were composed1 from one of which our first tractate was translated1 and the text itself was from time to time enlarged1 sections on proofs and on mental arithmetic being added4 ,we have no indication of the source on which ,alexander drew2 it was most likely one of the translations of ,al-,khowarasmi1 but he has also the ,abacists in mind1 as shewn by preserving the use of differences in multiplication4 ,his treatise1 first printed by ,halliwell-,phillipps in his ,rara ,mathematica1 is adapted for use on a board covered with sand1 a method almost universal in the thirteenth century1 as some passages in the algorism of that period already quoted show3 8,est et alius modus qui utitur apud ,indos1 et doctor hujusmodi ipsos erat quidem nomine ,algus4 ,et modus suus erat in computando per quasdam figuras scribendo in pulvere4 4 4 40 8,si voluerimus depingere in pulvere predictos digitos secundum consuetudinem algorismi 4 4 40 8et sciendum est quod in nullo loco minutorum sive secundorum 4 4 4 in pulvere debent scribi plusquam sexaginta40 ,,modern ,,arithmetic4 ,modern ,arithmetic begins with ,leonardi ,fibonacci's treatise 8de ,abaco10 written in #abjb and re-written in #abbh4 ,it is modern rather in the range of its problems and the methods of attack than in mere methods of calculation1 which are of its period4 ,its sole interest as regards the present work is that ,leonardi makes use of the digital signs described in ,record's treatise on ,the arte of nombrynge by the hand in mental arithmetic1 calling it 8modus ,indorum40 ,leonardo also introduces the method of proof by 8casting out the nines40 ,,digital ,,arithmetic4 ,the method of indicating numbers by means of the fingers is of considerable age4 ,the ,british ,museum possesses two ivory counters marked on one side by carelessly scratched ,roman numerals ,,iiiv and ,,viiii1 and on the other by carefully engraved digital signs for #h and #i4 ,sixteen seems to have been the number of a complete set4 ,these counters were either used in games or for the counting board1 and the ,museum ones1 coming from the ,hamilton collection1 are undoubtedly not later than the first century4 ,frohner has published in the ,zeitschrift des ,mnchener ,alterthumsvereins a set1 almost complete1 of them with a ,byzantine treatise2 a ,latin treatise is printed among ,bede's works4 ,the use of this method is universal through the ,east1 and a variety of it is found among many of the native races in ,africa4 ,in medieval ,europe it was almost restricted to ,italy and the ,mediterranean basin1 and in the treatise already quoted 7,sloane #cbha7 it is even called the ,abacus1 perhaps a memory of ,fibonacci's work4 ,methods of calculation by means of these signs undoubtedly have existed1 but they were too involved and liable to error to be much used4 ,,the ,,use ,,of 8,,arabic0 ,,figures4 ,it may now be regarded as proved by ,bubnov that our present numerals are derived from ,greek sources through the so-called ,boethian 8apices10 which are first found in late tenth century manuscripts4 ,that they were not derived directly from the ,arabic seems certain from the different shapes of some of the numerals1 especially the #j1 which stands for #e in ,arabic4 ,another ,greek form existed1 which was introduced into ,europe by ,john of ,basingstoke in the thirteenth century1 and is figured by ,matthew ,paris 7,v4 #bhe72 but this form had no success4 ,the date of the introduction of the zero has been hotly debated1 but it seems obvious that the twelfth century ,latin translators from the ,arabic were perfectly well acquainted with the system they met in their ,arabic text1 while the earliest astronomical tables of the thirteenth century ,i have seen use numbers of ,european and not ,arabic origin4 ,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 notation4 ,dr4 ,hill has published a very complete survey of the various forms of numerals in ,europe4 ,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 manuscript1 ,harl4 #dcej shows their extension4 ,the first is dated ,mijc4 lviii41 the second ,mijc4 lxi41 the third ,mijc4 #fc1 the fourth #abfd1 and the fifth #abff4 ,another example is given in a set of astronomical tables for #abfi in a manuscript of ,roger ,bacon's works1 where the scribe began to write ,,mcc#f4 and crossed out the figures1 substituting the 8,arabic0 form4 ,,the ,,counting ,,board4 ,the treatise on pp4 #eb-fe is the only one in ,english known on the subject4 ,it describes a method of calculation which1 with slight modifications1 is current in ,russia1 ,china1 and ,japan1 to-day1 though it went out of use in ,western ,europe by the seventeenth century4 ,in ,germany the method is called 8,algorithmus ,linealis10 and there are several editions of a tract under this name 7with a diagram of the counting board71 printed at ,leipsic at the end of the fifteenth century and the beginning of the sixteenth4 ,they give the nine rules1 but 8,capitulum de radicum extractione ad algoritmum integrorum reservato1 cujus species per ciffrales figuras ostenduntur ubi ad plenum de hac tractabitur40 ,the invention of the art is there attributed to ,appulegius the philosopher4 ,the advantage of the counting board1 whether permanent or constructed by chalking parallel lines on a table1 as shown in some sixteenth-century woodcuts1 is that only five counters are needed to indicate the number nine1 counters on the lines representing units1 and those in the spaces above representing five times those on the line below4 ,the ,russian abacus1 the 8tchatui0 or 8stchota0 has ten beads on the line2 the ,chinese and ,japanese 8,swanpan0 economises by dividing the line into two parts1 the beads on one side representing five times the value of those on the other4 ,the 8,swanpan0 has usually many more lines than the 8stchota10 allowing for more extended calculations1 see ,tylor1 ,anthropology 7#ahib71 p4 #cad4 ,record's treatise also mentions another method of counter notation 7p4 #fd7 8merchants' casting0 and 8auditors' casting40 ,these were adapted for the usual ,english method of reckoning numbers up to #bjj by scores4 ,this method seems to have been used in the ,exchequer4 ,a counting board for merchants' use is printed by ,halliwell in ,rara ,mathematica 7p4 #gb7 from ,sloane ,,ms4 #bac1 and two others are figured in ,egerton #bfbb f4 #hb and f4 #hc4 ,the latter is said to be 8novus modus computandi secundum inventionem ,magistri ,thome ,thorleby10 and is in principle1 the same as the 8,swanpan40 ,the ,exchequer table is described in the ,dialogus de ,scaccario 7,oxford1 #aijb71 p4 #ch4 ,the ,earliest ,arithmetics in ,english4 ,the ,crafte of ,nombrynge ,egerton #bfbb4 99leaf #acf;a ,hec algorismus ars presens dicitur2 in qua ,talibus indorum fruimur bis quinque figuris4 ,sidenote3 ,a derivation of ,algorism4 ,another derivation of the word4 ,this boke is called e boke of algorym1 or ,augrym after lewder vse4 ,and is boke tretys e ,craft of ,nombryng1 e quych crafte is called also ,algorym4 ,ther was a kyng of ,inde1 e quich heyth ,algor1 he made is craft4 ,and after his name he called hit algorym2 or els anoer cause is quy it is called ,algorym1 for e latyn word of hit s4 ,algorismus comes of ,algos1 grece1 quid est ars1 latine1 craft o englis1 and rides1 quid est numerus1 latine1 ,a nombur o englys1 inde dicitur ,algorismus per addicionem huius sillabe mus subtraccionem d e1 quasi ars numerandi4 fforthermore e most vndirstonde at in is craft ben vsid teen figurys1 as here bene writen for ensampul1 #i #h #g #f #e #d #c #b #a4 ,expone e too versus afore3 this present craft ys called ,algorismus1 in e quych we vse teen signys of ,inde4 ,questio4 ,why te fyguris of ,inde8 ,solucio4 for as ,i haue sayd afore ai were fonde fyrst in ,inde of a kynge of at ,cuntre1 at was called ,algor4 ,headnote3 ,notation and ,numeration4 ,sidenote3 versus in margin4 ,prima significat unum2 duo vero secunda3 ,tercia significat tria2 sic procede sinistre4 ,donec ad extremam venias1 que cifra vocatur4 ,capitulum primum de significacione figurarum4 ,sidenote3 ,expositio versus4 ,sidenote3 ,the meaning and place of the figures4 ,which figure is read first4 ,in is verse is notifide e significacion of ese figuris4 ,and us expone the verse4 e first signifiyth one1 e secunde 99leaf #acf;b signi99fiyth tweyne1 e thryd signifiyth thre1 the fourte signifiyth #d4 ,and so forthe towarde e lyft syde of e tabul or of e boke at e figures bene writene in1 til at ou come to the last figure1 at is called a cifre4 ,questio4 ,in quych syde sittes e first figure8 ,solucio1 forsothe loke quich figure is first in e ryt side of e bok or of e tabul1 at same is e first figure1 for ou schal write bakeward1 as here1 #c4 #b4 #f4 #d4 #a4 #b4 #e4 ,the figure of #e4 was first write1 he is e first1 for he sittes o e rit syde4 ,and the figure of #c is last4 ,neuer-e-les wen he says ,prima significat vnum c41 at is to say1 e first betokenes one1 e secunde4 #b4 fore-er-more1 he vndirstondes not of e first figure of euery rew4 ,but he vndirstondes e first figure at is in e nombur of e forsayd teen figuris1 e quych is one of ese4 #a4 ,and e secunde #b4 so forth4 ,sidenote3 versus in margin4 ,quelibet illarum si primo limite ponas1 ,simpliciter se significat3 si vero secundo1 ,se decies3 sursum procedas multiplicando4 ,namque figura sequens quamuis signat decies plus4 ,ipsa locata loco quam significat pertinente4 ,transcriber's ,note3 ,in the following section1 numerals shown in marks were printed in a different font1 possibly as facsimiles of the original ,,ms form4 ,sidenote3 ,expositio in margin4 ,sidenote3 ,an explanation of the principles of notation4 ,an example3 units1 tens1 hundreds1 thousands4 ,how to read the number4 ,expone is verse us4 ,euery of ese figuris bitokens hym selfe no more1 yf he stonde in e first place of e rewele _/ this worde ,simpliciter in at verse it is no more to say but at1 no more4 ,if it stonde in the secunde place of e rewle1 he betokens tene tymes hym selfe1 as is figure #b here #bj tokens ten tyme hym selfe1 99leaf #acg;a at is twenty1 for he hym selfe betokenes tweyne1 ten tymes twene is twenty4 ,and for he stondis o e lyft side in e secunde place1 he betokens ten tyme hym selfe4 ,and so go forth4 ffor euery figure1 he stonde aftur a-noer toward the lyft side1 he schal betokene ten tymes as mich more as he schul betoken he stode in e place ere at e figure a-fore hym stondes4 loo an ensampulle4 #i4 #f4 #c4 #d4 e figure of #d4 at hase is schape #d4 betokens bot hymselfe1 for he stondes in e first place4 ,the figure of #c4 at hase is schape #c4 betokens ten tymes more en he schuld he stode ere at e figure of #d4 stondes1 at is thretty4 ,the figure of #f1 at hase is schape #f1 betokens ten tymes more an he schuld he stode ere as e figure of #c4 stondes1 for ere he schuld tokyne bot sexty1 now he betokens ten tymes more1 at is sex hundryth4 ,the figure of #i4 at hase is schape #i4 betokens ten tymes more ane he schuld he stode in e place ere e figure of sex stondes1 for en he schuld betoken to #i4 hundryth1 and in e place ere he stondes now he betokens #i4 ousande4 ,al e hole nombur is #i thousande sex hundryth foure thretty4 fforthermore1 when ou schalt rede a nombur of figure1 ou schalt begyne at e last figure in the lyft side1 rede so forth to e rit side as here #i4 #f4 #c4 #d4 ,thou schal begyn to rede at e figure of #i4 rede forth us4 #i4 99leaf #acg;b thousand sex hundryth thritty foure4 ,but when ou schalle write1 ou schalt be-gynne to write at e ryt side4 ,nil cifra significat sed dat signare sequenti4 ,sidenote3 ,the meaning and use of the cipher4 ,expone is verse4 ,a cifre tokens not1 bot he makes e figure to betoken at comes aftur hym more an he schuld he were away1 as us #a4 here e figure of one tokens ten1 yf e cifre were away#a no figure by-fore hym he schuld token bot one1 for an he schuld stonde in e first place4 ,and e cifre tokens nothyng hym selfe4 for al e nombur of e ylke too figures is bot ten4 ,questio4 ,why says he at a cifre makys a figure to signifye 7tyf7 more c4 ,i speke for is worde significatyf1 ffor sothe it may happe aftur a cifre schuld come a-nour cifre1 as us #b4 ,and et e secunde cifre shuld token neuer e more excep he schuld kepe e order of e place4 and a cifre is no figure significatyf4 ,quam precedentes plus ultima significabit _/ ,sidenote3 ,the last figure means more than all the others1 since it is of the highest value4 ,expone is verse us4 e last figure schal token more an alle e oer afore1 thout ere were a hundryth thousant figures afore1 as us1 #afgih4 e last figure at is #a4 betokens ten thousant4 ,and alle e oer figures ben bot betokene bot sex thousant seuyne hundryth nynty #h4 ,and ten thousant is more en alle at nombur1 ergo e last figure tokens more an all e nombur afore4 ,headnote3 ,the ,three ,kinds of ,numbers 99leaf #ach;a ,post predicta scias breuiter quod tres numerorum ,distincte species sunt2 nam quidam digiti sunt2 ,articuli quidam2 quidam quoque compositi sunt4 ,capitulum #bm de triplice divisione numerorum4 ,sidenote3 ,digits4 ,articles4 ,composites4 ,the auctor of is tretis departys is worde a nombur into #c partes4 ,some nombur is called digitus latine1 a digit in englys4 ,somme nombur is called articulus latine4 ,an ,articul in englys4 ,some nombur is called a composyt in englys4 ,expone is verse4 know ou aftur e forsayd rewles at ,i sayd afore1 at ere ben thre spices of nombur4 ,oone is a digit1 ,anoer is an ,articul1 e toer a ,composyt4 versus4 ,headnote3 ,digits1 ,articles1 and ,composites4 ,sunt digiti numeri qui citra denarium sunt4 ,sidenote3 ,what are digits4 ,here he telles qwat is a digit1 ,expone versus sic4 ,nomburs digitus bene alle nomburs at ben with-inne ten1 as nyne1 #h4 #g4 #f4 #e4 #d4 #c4 #b4 #a4 ,articupli decupli degitorum2 compositi sunt ,illi qui constant ex articulis degitisque4 ,sidenote3 ,what are articles4 ,here he telles what is a composyt and what is ane articul4 ,expone sic versus4 ,articulis ben#b alle at may be deuidyt into nomburs of ten nothynge leue ouer1 as twenty1 thretty1 fourty1 a hundryth1 a thousand1 such oer1 ffor twenty may be departyt in-to #b nomburs of ten1 fforty in to foure nomburs of ten1 so forth4 ,sidenote3 ,what numbers are composites4 99leaf #ach;b ,compositys be nomburs at bene componyt of a digyt of an articulle as fouretene1 fyftene1 sextene1 such oer4 ffortene is componyd of foure at is a digit of ten at is an articulle4 ffiftene is componyd of #e ten1 so of all oer1 what at ai ben4 ,short-lych euery nombur at be-gynnes with a digit endyth in a articulle is a composyt1 as fortene bygennynge by foure at is a digit1 endes in ten4 ,ergo1 proposito numero tibi scribere1 primo ,respicias quid sit numerus2 si digitus sit ,primo scribe loco digitum1 si compositus sit ,primo scribe loco digitum post articulum2 sic4 ,sidenote3 ,how to write a number1 if it is a digit2 if it is a composite4 ,how to read it4 here he telles how ou schalt wyrch whan ou schalt write a nombur4 ,expone versum sic1 fac iuxta exponentis sentenciam2 whan ou hast a nombur to write1 loke fyrst what maner nombur it ys at ou schalt write1 whether it be a digit or a composit or an ,articul4 ,if he be a digit1 write a digit1 as yf it be seuen1 write seuen write at digit in e first place toward e ryght side4 ,if it be a composyt1 write e digit of e composit in e first place write e articul of at digit in e secunde place next toward e lyft side4 ,as yf ou schal write sex twenty4 write e digit of e nombur in e first place at is sex1 and write e articul next aftur at is twenty1 as us #bf4 ,but whan ou schalt sowne or speke 99leaf #aci;a or rede an ,composyt ou schalt first sowne e articul aftur e digit1 as ou seyst by e comyne speche1 ,sex twenty nout twenty sex4 versus4 ,articulus si sit1 in primo limite cifram1 ,articulum vero reliquis inscribe figuris4 ,sidenote3 ,how to write ,articles3 tens1 hundreds1 thousands1 c4 ,here he tells how ou schal write when e nombre at ou hase to write is an ,articul4 ,expone versus sic fac secundum sentenciam4 ,ife e nombur at ou hast write be an ,articul1 write first a cifre aftur e cifer write an ,articulle us4 #b4 fforthermore ou schalt vndirstonde yf ou haue an ,articul1 loke how mych he is1 yf he be with-ynne an hundryth1 ou schalt write bot one cifre1 afore1 as here 4#i4 ,if e articulle be by hym-silfe be an hundrid euene1 en schal ou write 4#a4 #b cifers afore1 at he may stonde in e thryd place1 for euery figure in e thryd place schal token a hundrid tymes hym selfe4 ,if e articul be a thousant or thousandes#c and he stonde by hym selfe1 write afore #c cifers so for of al oer4 ,quolibet in numero1 si par sit prima figura1 ,par erit totum1 quicquid sibi continuatur2 ,impar si fuerit1 totum tunc fiet et impar4 ,sidenote3 ,to tell an even number or an odd4 ,here he teches a generalle rewle at yf e first figure in e rewle of figures token a nombur at is euene al at nombur of figurys in at rewle schal be euene1 as here ou may see #f4 #g4 #c4 #e4 #d4 ,computa proba4 ,if e first 99leaf #aci;b figure token an nombur at is ode1 alle at nombur in at rewle schalle be ode1 as here #e #f #g #h #f #g4 ,computa proba4 versus4 ,septem sunt partes1 non plures1 istius artis2 ,addere1 subtrahere1 duplare1 dimidiare1 ,sextaque diuidere1 sed quinta multiplicare2 ,radicem extrahere pars septima dicitur esse4 ,headnote3 ,the ,seven ,rules of ,arithmetic4 ,sidenote3 ,the seven rules4 ,here telles at er be 4#g4 spices or partes of is craft4 ,the first is called addicio1 e secunde is called subtraccio4 ,the thryd is called duplacio4 ,the #d4 is called dimydicio4 ,the #e4 is called multiplicacio4 ,the #f is called diuisio4 ,the #g4 is called extraccio of e ,rote4 ,what all ese spices bene hit schalle be tolde singillatim in here caputule4 ,subtrahis aut addis a dextris vel mediabis3 ,sidenote3 ,add1 subtract1 or halve1 from right to left4 ,thou schal be-gynne in e ryght side of e boke or of a tabul4 loke were ou wul be-gynne to write latyn or englys in a boke1 at schalle be called e lyft side of the boke1 at ou writest toward at side schal be called e ryght side of e boke4 ,versus4 ,a leua dupla1 diuide1 multiplica4 ,sidenote3 ,multiply or divide from left to right4 ,here he telles e in quych side of e boke or of e tabul ou schalle be-gyne to wyrch duplacio1 diuisio1 and multiplicacio4 ,thou schal begyne to worch in e lyft side of e boke or of e tabul1 but yn what wyse ou schal wyrch in hym dicetur singillatim in sequentibus capitulis et de vtilitate cuiuslibet artis sic ,completur 99leaf #adj4 prohemium sequitur tractatus primo de arte addicionis que prima ars est in ordine4 ,headnote3 ,the ,craft of ,addition4 ,addere si numero numerum vis1 ordine tali ,incipe2 scribe duas primo series numerorum ,primam sub prima recte ponendo figuram1 ,et sic de reliquis facias1 si sint tibi plures4 ,sidenote3 ,four things must be known3 what it is2 how many rows of figures2 how many cases2 what is its result4 ,how to set down the sum4 ,here by-gynnes e craft of ,addicio4 ,in is craft ou most knowe foure thynges4 ,fyrst ou most know what is addicio4 ,next ou most know how mony rewles of figurys ou most haue4 ,next ou most know how mony diuers casys happes in is craft of addicio4 ,and next qwat is e profet of is craft4 ,as for e first ou most know at addicio is a castyng to-gedur of twoo nomburys in-to one nombre4 ,as yf ,i aske qwat is twene thre4 ou wyl cast ese twene nombres to-gedur say at it is fyue4 ,as for e secunde ou most know at ou schalle haue tweyne rewes of figures1 one vndur a-nother1 as here ou mayst se4 #abcd #bafh4 ,as for e thryd ou most know at there ben foure diuerse cases4 ,as for e forthe ou most know at e profet of is craft is to telle what is e hole nombur at comes of diuerse nomburis4 ,now as to e texte of oure verse1 he teches there how ou schal worch in is craft4 ,he says yf ou wilt cast one nombur to anoer nombur1 ou most by-gynne on is wyse4 ffyrst write 99leaf #adj;b two rewes of figuris nombris so at ou write e first figure of e hyer nombur euene vndir the first figure of e nether nombur1 ,and e secunde of e nether nombur euene vndir e secunde of e hyer1 so forthe of euery figure of both e rewes as ou mayst se4 #abc #bcd4 ,headnote3 ,the ,cases of the ,craft of ,addition4 ,inde duas adde primas hac condicione3 ,si digitus crescat ex addicione priorum2 ,primo scribe loco digitum1 quicunque sit ille4 ,sidenote3 ,add the first figures2 rub out the top figure2 write the result in its place4 ,here is an example4 ,here he teches what ou schalt do when ou hast write too rewes of figuris on vnder an-oer1 as ,i sayd be-fore4 ,he says ou schalt take e first figure of e heyer nombre e fyrst figure of e neer nombre1 cast hem to-geder vp-on is condicio4 ,thou schal loke qweer e nomber at comys ere-of be a digit or no4 ,if he be a digit ou schalt do away e first figure of e hyer nombre1 and write ere in his stede at he stode ,inne e digit1 at comes of e ylke #b figures1 so wrich forth o oer figures yf ere be ony moo1 til ou come to e ende toward e lyft side4 ,and lede e nether figure stonde still euer-more til ou haue ydo4 ffor ere-by ou schal wyte wheer ou hast done wel or no1 as ,i schal tell e afterward in e ende of is ,chapter4 ,and loke allgate at ou be-gynne to worch in is ,craft of 99leaf #ada;a ,addi99cio in e ryt side1 here is an ensampul of is case4 #abcd #badb4 ,caste #b to foure at wel be sex1 do away #d4 write in e same place e figure of sex4 ,and lete e figure of #b in e nether rewe stonde stil4 ,when ou hast do so1 cast #c #d to-gedur and at wel be seuen at is a digit4 ,do away e #c1 set ere seue1 and lete e neer figure stonde stille1 so worch forth bakward til ou hast ydo all to-geder4 ,et si compositus1 in limite scribe sequente ,articulum1 primo digitum2 quia sic iubet ordo4 ,sidenote3 ,suppose it is a ,composite1 set down the digit1 and carry the tens4 ,here is an example4 ,here is e secunde case at may happe in is craft4 ,and e case is is1 yf of e casting of #b nomburis to-geder1 as of e figure of e hyer rewe of e figure of e neer rewe come a ,composyt1 how schalt ou worch4 us ou schalt worch4 ,thou shalt do away e figure of e hyer nomber at was cast to e figure of e neer nomber4 ,and write ere e digit of e ,composyt4 ,and set e articul of e composit next after e digit in e same rewe1 yf ere be no mo figures after4 ,but yf ere be mo figuris after at digit4 ,and ere he schall be rekend for hym selfe4 ,and when ou schalt adde at ylke figure at berys e articulle ouer his hed to e figure vnder hym1 ou schalt cast at articul to e figure at hase hym ouer his hed1 ere at ,articul schal toke hym selfe4 lo an ,ensampull 99leaf #ada;b of all4 #cbf #baf4 ,cast #f to #f1 ere-of wil arise twelue4 do away e hyer #f write ere #b1 at is e digit of is composit4 ,and en write e articulle at is ten ouer e figuris hed of twene as us4 #a #cbb #baf4 ,now cast e articulle at standus vpon e figuris of twene hed to e same figure1 reken at articul bot for one1 and an ere wil arise thre4 an cast at thre to e neer figure1 at is one1 at wul be foure4 do away e figure of #c1 and write ere a figure of foure4 and lete e neer figure stonde stil1 an worch forth4 vnde versus4 ,articulus si sit1 in primo limite cifram1 ,articulum vero reliquis inscribe figuris1 ,vel per se scribas si nulla figura sequatur4 ,sidenote3 ,suppose it is an ,article1 set down a cipher and carry the tens4 ,here is an example4 ,here he puttes e thryde case of e craft of ,addicio4 e case is is4 yf of ,addiciou of #b figuris a-ryse an ,articulle1 how schal ou do4 thou most do away e heer figure at was addid to e neer1 write ere a cifre1 and sett e articuls on e figuris hede1 yf at ere come ony after4 ,and wyrch an as ,i haue tolde e in e secunde case4 ,an ensampull4 #be4 #ae ,cast #e to #e1 at wylle be ten4 now do away e hyer #e1 write ere a cifer4 ,and sette ten vpon e figuris hed of #b4 ,and reken it but for on us4 lo an ,ensampulle ---- #a #b #ae ---- ,and 99leaf #adb;a an worch forth4 ,but yf ere come no figure after e cifre1 write e articul next hym in e same rewe as here --- #e #e --- cast #e to #e1 and it wel be ten4 do away #e4 at is e hier #e4 and write ere a cifre1 write after hym e articul as us ---- #a #e ---- ,and an ou hast done4 ,si tibi cifra superueniens occurrerit1 illam ,dele superpositam2 fac illic scribe figuram1 ,postea procedas reliquas addendo figuras4 ,sidenote3 ,what to do when you have a cipher in the top row4 ,an example of all the difficulties4 ,here he puttes e fourt case1 it is is1 at yf ere come a cifer in e hier rewe1 how ou schal do4 us ou schalt do4 do away e cifer1 sett ere e digit at comes of e addicioun as us #ahd4 #aggdc ,in is ensampul ben alle e foure cases4 ,cast #c to foure1 at wol be seue4 do away #d4 write ere seue2 an cast #d to e figure of #h4 at wel be #ab4 do away #h1 sett ere #b4 at is a digit1 and sette e articul of e composit1 at is ten1 vpon e cifers hed1 reken it for hym selfe at is on4 an cast one to a cifer1 hit wulle be but on1 for not on makes but one4 an cast #g4 at stondes vnder at on to hym1 at wel be #h4 do away e cifer at #a4 sette ere #h4 an go forthermore4 cast e oer #g to e cifer at stondes ouer hym4 at wul be bot seuen1 for e cifer betokens not4 do away e cifer sette ere seue1 99leaf #adb;b en go forermore cast #a to #a1 at wel be #b4 do away e hier #a1 sette ere #b4 an hast ou do4 ,and yf ou haue wel ydo is nomber at is sett here-after wel be e nomber at schalle aryse of alle e addicio as here #bghbg4 ,sequitur alia species4 ,headnote3 ,the ,craft of ,subtraction4 ,a numero numerum si sit tibi demere cura ,scribe figurarum series1 vt in addicione4 ,sidenote3 ,four things to know about subtraction3 the first2 the second2 the third2 the fourth4 ,this is e ,chapter of subtraccio1 in the quych ou most know foure nessessary thynges4 the first what is subtraccio4 e secunde is how mony nombers ou most haue to subtraccio1 the thryd is how mony maners of cases ere may happe in is craft of subtraccio4 ,the fourte is qwat is e profet of is craft4 ,as for e first1 ou most know at subtraccio is drawynge of one nowmber oute of anoer nomber4 ,as for e secunde1 ou most knowe at ou most haue two rewes of figuris one vnder anoer1 as ou addyst in addicio4 ,as for e thryd1 ou moyst know at foure maner of diuerse casis mai happe in is craft4 ,as for e fourt1 ou most know at e profet of is craft is whenne ou hasse taken e lasse nomber out of e more to telle what ere leues ouer at4 ou most be-gynne to wyrch in is craft in e ryght side of e boke1 as ou diddyst in addicio4 ,versus4 ,maiori numero numerum suppone minorem1 ,siue pari numero supponatur numerus par4 ,sidenote3 ,put the greater number above the less4 99leaf #adc;a ,here he telles at e hier nomber most be more en e neer1 or els eue as mych4 but he may not be lasse4 ,and e case is is1 ou schalt drawe e neer nomber out of e hyer1 ou mayst not do at yf e hier nomber were lasse an at4 ffor ou mayst not draw sex out of #b4 ,but ou mast draw #b out of sex4 ,and ou maiste draw twene out of twene1 for ou schal leue not of e hier twene vnde versus4 ,headnote3 ,the ,cases of the ,craft of ,subtraction4 ,postea si possis a prima subtrahe primam ,scribens quod remanet4 ,sidenote3 ,the first case of subtraction4 ,here is an example4 ,here is e first case put of subtraccio1 he says ou schalt begynne in e ryght side1 draw e first figure of e neer rewe out of e first figure of e hier rewe4 qwether e hier figure be more en e neer1 or eue as mych4 ,and at is notified in e vers when he says 8,si possis40 ,whan ou has us ydo1 do away e hiest figure sett ere at leues of e subtraccio1 lo an ,ensampulle ----- #bcd #abb ----- draw #b out of #d4 an leues #b4 do away #d write ere #b1 latte e neer figure stonde stille1 so go for-by oer figuris till ou come to e ende1 an hast ou do4 ,cifram si nil remanebit4 ,sidenote3 ,put a cipher if nothing remains4 ,here is an example4 ,here he puttes e secunde case1 hit is is4 yf it happe at qwen ou hast draw on neer figure out of a hier1 ere leue not after e subtraccio1 us 99leaf #adc;b ou schalt do4 ou schalle do away e hier figure write ere a cifer1 as lo an ,ensampull ---- #bd #bd ---- ,take foure out of foure an leus not4 erefore do away e hier #d set ere a cifer1 an take #b out of #b1 an leues not4 do away e hier #b1 set ere a cifer1 and so worch whare so euer is happe4 ,sed si non possis a prima demere primam ,precedens vnum de limite deme sequente1 ,quod demptum pro denario reputabis ab illo ,subtrahe totalem numerum quem proposuisti ,quo facto scribe super quicquid remanebit4 ,sidenote3 ,suppose you cannot take the lower figure from the top one1 borrow ten2 take the lower number from ten2 add the answer to the top number4 ,how to #8,pay back' the borrowed ten4 ,example4 ,here he puttes e thryd case1 e quych is is4 yf it happe at e neer figure be more en e hier figure at he schalle be draw out of4 how schalle ou do4 us ou schalle do4 ou schalle borro 4#a4 oute of e next figure at comes after in e same rewe1 for is case may neuer happ but yf ere come figures after4 an ou schalt sett at on ouer e hier figures hed1 of the quych ou woldist y-draw oute e neyer figure yf ou haddyst y-myt4 ,whane ou hase us ydo ou schalle rekene at 4#a4 for ten4 4 ,and out of at ten ou schal draw e neyermost figure1 ,and alle at leues ou schalle adde to e figure on whos hed at 4#a4 stode4 ,and en ou schalle do away alle at1 sett ere alle that arisys of the addicio of e ylke #b figuris4 ,and yf yt 99leaf #add;a happe at e figure of e quych ou schalt borro on be hym self but #a4 ,if ou schalt at one sett it vppo e oer figuris hed1 and sett in at #a4 place a cifer1 yf ere come mony figures after4 lo an ,ensampul4 ------ #babb #aacd ------ take #d out of #b4 it wyl not be1 erfore borro one of e next figure1 at is #b4 and sett at ouer e hed of e fyrst #b4 rekene it for ten4 and ere e secunde stondes write #a4 for ou tokest on out of hym4 an take e neer figure1 at is #d1 out of ten4 ,and en leues #f4 cast to #f e figure of at #b at stode vnder e hedde of #a4 at was borwed rekened for ten1 and at wylle be #h4 do away at #f at #b1 sette ere #h1 lette e neer figure stonde stille4 ,whanne ou hast do us1 go to e next figure at is now bot #a4 but first yt was #b1 ere-of was borred #a4 an take out of at e figure vnder hym1 at is #c4 hit wel not be4 er-fore borowe of the next figure1 e quych is bot #a4 ,also take sett hym ouer e hede of e figure at ou woldest haue y-draw oute of e nether figure1 e quych was #c4 ou myt not1 rekene at borwed #a for ten sett in e same place1 of e quych place ou tokest hym of1 a cifer1 for he was bot #a4 ,whanne ou hast us ydo1 take out of at #a4 at is rekent for ten1 e neer figure of #c4 ,and ere leues #g4 99leaf #add;b cast e ylke #g to e figure at had e ylke ten vpon his hed1 e quych figure was #a1 at wol be #h4 an do away at #a and at #g1 write ere #h4 an wyrch forth in oer figuris til ou come to e ende1 an ou hast e do4 ,versus4 ,facque nonenarios de cifris1 cum remeabis ,occurrant si forte cifre2 dum dempseris vnum ,postea procedas reliquas demendo figuras4 ,sidenote3 ,a very hard case is put4 ,here is an example4 ,here he puttes e fourte case1 e quych is is1 yf it happe at e neer figure1 e quych ou schalt draw out of e hier figure be more pan e hier figur ouer hym1 e next figure of two or of thre or of foure1 or how mony ere be by cifers1 how wold ou do4 ou wost wel ou most nede borow1 ou mayst not borow of e cifers1 for ai haue not at ai may lene or spare4 ,ergo#d how woldest ou do4 ,certay us most ou do1 ou most borow on of e next figure significatyf in at rewe1 for is case may not happe1 but yf ere come figures significatyf after the cifers4 ,whan ou hast borowede at #a of the next figure significatyf1 sett at on ouer e hede of at figure of e quych ou wold haue draw e neer figure out yf ou hadest myt1 reken it for ten as ou diddest in e oer case here-a-fore4 ,wha ou hast us y-do loke how mony cifers ere were bye-twene at figure significatyf1 e figure of e quych ou woldest haue y-draw the 99leaf #ade;a neer figure1 and of euery of e ylke cifers make a figure of #i4 lo an ,ensampulle after4 ----- #djjjb #ajjjd ----- ,take #d out of #b4 it wel not be4 borow #a out of be next figure significatyf1 e quych is #d1 en leues #c4 do away at figure of #d write ere #c4 sett at #a vppon e figure of #b hede1 an take #d out of ten1 an ere leues #f4 ,cast #f to the figure of #b1 at wol be #h4 do away at #f write ere #h4 ,whan ou hast us y-do make of euery #j betweyn #c #h a figure of #i1 an worch forth in goddes name4 yf ou hast wel y-do ou#e schalt haue is nomber ----- #ciiih ,sic4 #ajjjd ----- ,headnote3 ,how to prove the ,subtraction4 ,si subtraccio sit bene facta probare valebis ,quas subtraxisti primas addendo figuras4 ,sidenote3 ,how to prove a subtraction sum4 ,here is an example4 ,he works his proof through1 and brings out a result4 ,here he teches e ,craft how ou schalt know1 whan ou hast subtrayd1 wheer ou hast wel ydo or no4 ,and e ,craft is is1 ryght as ou subtrayd e neer figures fro e hier figures1 ryt so adde e same neer figures to e hier figures4 ,and yf ou haue well y-wroth a-fore ou schalt haue e hier nombre e same ou haddest or ou be-gan to worch4 as for is ,i bade ou schulde kepe e neer figures stylle4 lo an 99leaf #ade;b ,ensampulle of alle e #d cases togedre4 worche welle is case -------- #djjjcdfh4 #bjjjdffd -------- ,and yf ou worch welle whan ou hast alle subtrayd e at hier nombre here1 is schalle be e nombre here foloyng whan ou hast subtrayd4 -------- #ciiihhjd4 ,sidenote3 ,our author makes a slip here 7#c for #a74 #bjjjdffd -------- ,and ou schalt know us4 adde e neer rowe of e same nombre to e hier rewe as us1 cast #d to #d4 at wol be #h4 do away e #d write ere #h4 by e first case of addicio4 an cast #f to #j at wol be #f4 do away e #j1 write ere #f4 an cast #f to #h1 at wel be #ad4 do away #h write ere a figure of #d1 at is e digit1 and write a figure of #a4 at schall be-token ten4 at is e articul vpon e hed of #h next after1 an reken at #a4 for #a4 cast it to #h4 at schal be #i4 cast to at #i e neer figure vnder at e quych is #d1 at schalle be #ac4 do away at #i sett ere #c1 sett a figure of #a4 at schall be #aj vpon e next figuris hede e quych is #i4 by e secunde case at ou hadest in addicio4 an cast #a to #i4 at wol be #aj4 do away e #i4 at #a4 ,and write ere a cifer4 and write e articulle at is #a4 betokenynge #aj4 vpon e hede of e next figure toward e lyft side1 e quych 99leaf #adf;a is #i1 so do forth tyl ou come to e last #i4 take e figure of at #a4 e quych ou schalt fynde ouer e hed of #i4 sett it ouer e next figures hede at schal be #c4 ,also do away e #i4 set ere a cifer1 en cast at #a at stondes vpon e hede of #c to e same #c1 at schalle make #d1 en caste to e ylke #d the figure in e neyer rewe1 e quych is #b1 and at schalle be #f4 ,and en schal ou haue an ,ensampulle aey1 loke se1 but ou haue is same ou hase myse-wrot4 -------- #fjjjcdfh #bjjjdffd -------- ,sequitur de duplacione ,headnote3 ,the ,craft of ,duplation4 ,si vis duplare numerum1 sic incipe primo ,scribe figurarum seriem quamcunque velis tu4 ,sidenote3 ,four things must be known in ,duplation4 ,here they are4 ,mind where you begin4 ,remember your rules4 ,this is the ,chapture of duplacio1 in e quych craft ou most haue know #d thinges4 e first at ou most know is what is duplacio4 e secunde is how mony rewes of figures ou most haue to is craft4 e thryde is how many cases may#f happe in is craft4 e fourte is what is e profet of e craft4 ,as for e first4 duplacio is a doublynge of a nombre4 ,as for e secunde ou most 99leaf #adf;b haue on nombre or on rewe of figures1 the quych called numerus duplandus4 ,as for e thrid ou most know at #c diuerse cases may hap in is craft4 ,as for e fourte4 qwat is e profet of is craft1 at is to know what a-risyt of a nombre ,i-doublyde4 fforer-more1 ou most know take gode hede in quych side ou schalle be-gyn in is craft1 or ellis ou mayst spyl alle i laber ere aboute4 certeyn ou schalt begy in the lyft side in is ,craft4 thenke wel ouer is verse4 #g;,a leua dupla1 diuide1 multiplica4#g ,subtrahas aut addis a dextris vel mediabis ,the sentens of es verses afore1 as ou may see if ou take hede4 ,as e text of is verse1 at is to say1 ,si vis duplare4 is is e sentence4 ,if ou wel double a nombre us ou most be-gyn4 ,write a rewe of figures of what nombre ou welt4 versus4 ,postea procedas primam duplando figuram ,inde quod excrescit scribas vbi iusserit ordo ,iuxta precepta tibi que dantur in addicione4 ,sidenote3 ,how to work a sum4 ,here he telles how ou schalt worch in is ,craft4 he says1 fyrst1 whan ou hast writen e nombre ou schalt be-gyn at e first figure in the lyft side1 doubulle at figure1 e nombre at comes ere-of ou schalt write as ou diddyst in addicio1 as ,i schal telle e in e case4 versus4 ,headnote3 ,the ,cases of the ,craft of ,duplation4 99leaf #adg;a ,nam si sit digitus in primo limite scribas4 ,sidenote3 ,if the answer is a digit1 write it in the place of the top figure4 ,here is e first case of is craft1 e quych is is4 yf of duplacio of a figure arise a digit4 what schal ou do4 us ou schal do4 do away e figure at was doublede1 sett ere e diget at comes of e duplacio1 as us4 #bc4 double #b1 at wel be #d4 do away e figure of #b sett ere a figure of #d1 so worch forth tille ou come to e ende4 versus4 ,articulus si sit1 in primo limite cifram1 ,articulum vero reliquis inscribe figuris2 ,vel per se scribas1 si nulla figura sequatur4 ,sidenote3 ,if it is an article1 put a cipher in the place1 and #8carry' the tens4 ,if there is no figure to #8carry' them to1 write them down4 ,here is e secunde case1 e quych is is yf ere come an articulle of e duplacio of a figure ou schalt do ryt as ou diddyst in addicio1 at is to wete at ou schalt do away e figure at is doublet sett ere a cifer1 write e articulle ouer e next figuris hede1 yf ere be any after-warde toward e lyft side as us4 #be4 begyn at the lyft side1 and doubulle #b4 at wel be #d4 do away at #b sett ere #d4 an doubul #e4 at wel be #aj4 do away #e1 sett ere a #j1 sett #a vpon e next figuris hede e quych is #d4 en draw downe #a to #d at wolle be #e1 en do away at #d at #a1 sett ere #e4 for at #a schal be rekened in e drawynge togedre for #a4 wen 99leaf #adg;b ou hast ydon ou schalt haue is nombre #ej4 yf ere come no figure after e figure at is addit1 of e quych addicio comes an articulle1 ou schalt do away e figure at is dowblet sett ere a #j4 write e articul next by in e same rewe toward e lyft syde as us1 #ebc4 double #e at woll be ten4 do away e figure #e set ere a cifer1 sett e articul next after in e same rewe toward e lyft side1 ou schalt haue is nombre #ajbc4 en go forth double e oer nombers e quych is lyt y-nowt to do4 versus4 ,compositus si sit1 in limite scribe sequente ,articulum1 primo digitum2 quia sic iubet ordo3 ,et sic de reliquis faciens1 si sint tibi plures4 ,sidenote3 ,if it is a ,composite1 write down the digit1 and #8carry' the tens4 ,here is an example4 ,here he puttes e ,thryd case1 e quych is is1 yf of duplacio of a figure come a ,composit4 ou schalt do away e figure at is doublet set ere a digit of e ,composit1 sett e articulle ouer e next figures hede1 after draw hym downe with e figure ouer whos hede he stondes1 make ere-of an nombre as ou hast done afore1 yf ere come no figure after at digit at ou hast y-write1 an set e articulle next after hym in e same rewe as us1 #fg3 double #f at wel be #ab1 do away #f write ere e digit 99leaf #adh;a of #ab1 e quych is #b1 and set e articulle next after toward e lyft side in e same rewe1 for ere comes no figure after4 an dowble at oer figure1 e quych is #g1 at wel be #ad4 the quych is a ,composit4 en do away #g at ou doublet sett e e diget of hym1 the quych is #d1 sett e articulle ouer e next figures hed1 e quych is #b1 en draw to hym at on1 make on nombre e quych schalle be #c4 ,and en yf ou haue wel y-do ou schalle haue is nombre of e duplacio1 #acd4 versus4 ,si super extremam nota sit monadem dat eidem ,quod tibi contingat si primo dimidiabis4 ,sidenote3 ,how to double the mark for one-half4 ,this can only stand over the first figure4 ,here he says1 yf ouer e fyrst figure in e ryt side be such a merke as is here made1 1 ou schalle fyrst doubulle e figure1 the quych stondes vnder at merke1 en ou schalt doubul at merke e quych stondes for haluendel on4 for too haluedels makes on1 so at wol be on4 cast at on to at duplacio of e figure ouer whos hed stode at merke1 write it in e same place ere at e figure e quych was doublet stode1 as us #bc4 double #c1 at wol be #f2 doubul at halue on1 at wol be on4 cast on to #f1 at wel be #g4 do away #f at #a1 sett ere #g4 an hase ou do4 as for at figure1 an go 99leaf #adh;b to e oer figure worch forth4 ou schall neuer haue such a merk but ouer e hed of e furst figure in e ryght side4 ,and et it schal not happe but yf it were y-halued a-fore1 us ou schalt vnderstonde e verse4 ,si super extremam c4 ,et nota1 talis figura significans medietatem1 unitatis veniat1 i4e4 contingat uel fiat super extremam1 i4e4 super primam figuram in extremo sic versus dextram ars dat3 i4e4 reddit monadem4 i4e4 vnitatem eidem4 i4e4 eidem note declinatur hec monos1 dis1 di1 dem1 c4 ,quod ergo totum hoc dabis monadem note continget4 i4e4 eveniet tibi si dimidiasti1 i4e4 accipisti uel subtulisti medietatem alicuius unius1 in cuius principio sint figura numerum denotans imparem primo i4e4 principiis4 ,headnote3 ,the ,craft of ,mediation4 ,sequitur de mediacione4 ,incipe sic1 si vis aliquem numerum mediare3 ,scribe figurarum seriem solam1 velut ante4 ,sidenote3 ,the four things to be known in mediation3 the first the second2 the third2 the fourth4 ,begin thus4 ,in is ,chapter is tat e ,craft of mediaciou1 in e quych craft ou most know #d thynges4 ffurst what is mediacio4 the secunde how mony rewes of figures ou most haue in e wyrchynge of is craft4 e thryde how mony diuerse cases may happ in is craft4#h the 4#d4 what is e profet of is craft4 ,as for e furst1 ou schalt vndurstonde at mediacio is a takyng out of halfe a nomber out of a holle nomber1 99leaf #adi;a as yf ou wolde take #c out of #f4 ,as for e secunde1 ou schalt know at ou most haue one rewe of figures1 no moo1 as ou hayst in e craft of duplacio4 ,as for the thryd1 ou most vnderstonde at #e cases may happe in is craft4 ,as for e fourte1 ou schalle know at the profet of is craft is when ou hast take away e haluendel of a nombre to telle qwat ere schalle leue4 ,incipe sic1 c4 ,the sentence of is verse is is4 yf ou wold medye1 at is to say1 take halfe out of e holle1 or halfe out of halfe1 ou most begynne us4 ,write one rewe of figures of what nombre ou wolte1 as ou dyddyst be-fore in e ,craft of duplacio4 versus4 ,postea procedas medians1 si prima figura ,si par aut impar videas4 ,sidenote3 ,see if the number is even or odd4 ,here he says1 when ou hast write a rewe of figures1 ou schalt take hede wheer e first figure be eue or odde in nombre1 vnderstonde at he spekes of e first figure in e ryt side4 ,and in the ryght side ou schalle begynne in is ,craft4 ,quia si fuerit par1 ,dimidiabis eam1 scribens quicquid remanebit3 ,sidenote3 ,if it is even1 halve it1 and write the answer in its place4 ,here is the first case of is craft1 e quych is is1 yf e first figure be euen4 ou schal take away fro e figure euen halfe1 do away at figure and set ere at leues ouer1 as us1 #d4 take 99leaf #adi;b halfe out of #d1 an ere leues #b4 do away #d sett ere #b4 is is lyght y-nowt4 versus4 ,headnote3 ,the ,mediation of an ,odd ,number4 ,impar si fuerit vnum demas mediare ,quod non presumas1 sed quod superest mediabis ,inde super tractum fac demptum quod notat vnum4 ,sidenote3 ,if it is odd1 halve the even number less than it4 ,here is an example4 ,then write the sign for one-half over it4 ,put the mark only over the first figure4 ,here is e secunde case of is craft1 the quych is is4 yf e first figure betokene a nombre at is odde1 the quych odde schal not be mediete1 en ou schalt medye at nombre at leues1 when the odde of e same nombre is take away1 write at at leues as ou diddest in e first case of is craft4 ,wha ou hayst write at4 for at at leues1 write such a merke as is here vpon his hede1 e quych merke schal betoke halfe of e odde at was take away4 lo an ,ensampull4 #bde4 the first figure here is betokenynge odde nombre1 e quych is #e1 for #e is odde2 ere-fore do away at at is odde1 e quych is #a4 en leues #d4 en medye #d en leues #b4 do away #d4 sette ere #b1 make such a merke upon his hede1 at is to say ouer his hede of #b as us4 #bdb4 ,and en worch forth in e oer figures tyll ou come to e ende4 by e furst case as ou schalt vnderstonde at ou schalt 99leaf #aej;a neuer make such a merk but ouer e first figure hed in e rit side4 ,wheer e other figures at comy after hym be eue or odde4 versus4 ,headnote3 ,the ,cases of the ,craft of ,mediation4 ,si monos1 dele2 sit tibi cifra post nota supra4 ,sidenote3 ,if the first figure is one put a cipher4 ,here is e thryde case1 e quych yf the first figure be a figure of #a4 ou schalt do away at #a set ere a cifer1 a merke ouer e cifer as us1 #bda4 do away #a1 sett ere a cifer with a merke ouer his hede1 en hast ou ydo for at #j4 as us #j en worch forth in e oer figurys till ou come to e ende1 for it is lyght as dyche water4 vnde versus4 ,postea procedas hac condicione secunda3 ,impar si fuerit hinc vnum deme priori1 ,inscribens quinque1 nam denos significabit ,monos predictam4 ,sidenote3 ,what to do if any other figure is odd4 ,write a figure of five over the next lower number's head4 ,example4 ,here he puttes e fourte case1 e quych is is4 yf it happe the secunde figure betoken odde nombre1 ou schal do away on of at odde nombre1 e quych is significatiue by at figure #a4 e quych #a schall be rekende for #aj4 ,whan ou hast take away at #a out of e nombre at is signifiede by at figure1 ou schalt medie at at leues ouer1 do away at figure at is medied1 sette in his styde halfe of at nombre4 ,whan ou hase so done1 ou schalt write 99leaf #aej;b a figure of #e ouer e next figures hede by-fore toward e ryt side1 for at #a1 e quych made odd nombre1 schall stonde for ten1 #e is halfe of #aj2 so ou most write #e for his haluendelle4 lo an ,ensampulle1 #dfgh4 begy in e ryt side as ou most nedes4 medie #h4 en ou schalt leue #d4 do away at #h sette ere #d4 en out of #g4 take away #a4 e quych makes odde1 sett #e4 vpon e next figures hede afore toward e ryt side1 e quych is now #d4 but afore it was #h4 for at #a schal be rekenet for #aj1 of e quych #aj1 #e is halfe1 as ou knowest wel4 ,whan ou hast us ydo1 medye at e quych leues after e takyinge away of at at is odde1 e quych leuynge schalle be #c2 do away #f sette ere #c1 ou schalt haue such a nombre #e #dfcd4 after go forth to e next figure1 medy at1 worch forth1 for it is lyt ynovt to e certay4 ,si vero secunda dat vnum4 ,illa deleta1 scribatur cifra2 priori ,tradendo quinque pro denario mediato2 ,nec cifra scribatur1 nisi deinde figura sequatur3 ,postea procedas reliquas mediando figuras ,vt supra docui1 si sint tibi mille figure4 ,sidenote3 ,if the second figure is one1 put a cipher1 and write five over the next figure4 ,how to halve fourteen4 ,here he puttes e #e case1 e quych is 99leaf #aea;a is3 yf e secunde figure be of #a1 as is is here #ab1 ou schalt do away at #a sett ere a cifer4 sett #e ouer e next figure hede afore toward e rit side1 as ou diddyst afore2 at #e schal be haldel of at #a1 e quych #a is rekent for #aj4 lo an ,ensampulle1 #bad4 medye #d4 at schalle be #b4 do away #d sett ere #b4 en go forth to e next figure4 e quych is bot #a4 do away at #a4 sett ere a cifer4 set #e vpon e figures hed afore1 e quych is nowe #b1 en ou schalt haue is nombre #e #bjb1 en worch forth to e nex figure4 ,and also it is no maystery yf ere come no figure after at on is medyet1 ou schalt write no #j4 ne nowt ellis1 but set #e ouer e next figure afore toward e ryt1 as us #ad4 medie #d then leues #b1 do away #d sett ere #b4 en medie #a4 e quich is rekende for ten1 e haluendel ere-of wel be #e4 sett at #e vpon e hede of at figure1 e quych is now #b1 do away at #a1 ou schalt haue is nombre yf ou worch wel1 #e #b4 vnde versus4 ,headnote3 ,how to prove the ,mediation4 ,si mediacio sit bene facta probare valebis ,duplando numerum quem primo dimediasti ,sidenote3 ,how to prove your mediation4 ,first example4 ,the second4 ,the third example4 ,the fourth example4 ,the fifth example4 ,here he telles e how ou schalt know wheer ou hase wel ydo or no4 doubul 99leaf #aea;b e nombre e quych ou hase mediet1 and yf ou haue wel y-medyt after e dupleacio1 ou schalt haue e same nombre at ou haddyst in e tabulle or ou began to medye1 as us4 ,the furst ensampulle was is4 #d4 e quych ,i-mediet was laft #b1 e whych #b was write in e place at #d was write afore4 ,now doubulle at #b1 ou schal haue #d1 as ou hadyst afore4 e secunde ,ensampulle was is1 #bde4 ,when ou haddyst mediet alle is nombre1 yf ou haue wel ydo ou schalt haue of at mediacio is nombre1 #abb4 ,now doubulle is nombre1 begyn in e lyft side2 doubulle #a1 at schal be #b4 do away at #a sett ere #b4 en doubulle at oer #b sett ere #d1 en doubulle at oer #b1 at wel be #d4 en doubul at merke at stondes for halue on4 at schalle be #a4 ,cast at on to #d1 it schalle be #e4 do away at #b at merke1 sette ere #e1 en ou schal haue is nombre #bde4 is wos e same nombur at ou haddyst or ou began to medye1 as ou mayst se yf ou take hede4 ,the nombre e quych ou haddist for an ,ensampul in e #c case of mediacio to be mediet was is #bda4 whan ou haddist medied alle is nombur truly 99leaf #aeb;a by euery figure1 ou schall haue be at mediacio is nombur #abj4 ,now dowbul is nombur1 begyn in e lyft side1 as ,i tolde e in e ,craft of duplacio4 us doubulle e figure of #a1 at wel be #b4 do away at #a sett ere #b1 en doubul e next figure afore1 the quych is #b1 at wel be #d2 do away #b set ere #d4 en doubul e cifer1 at wel be not1 for a #j is not4 ,and twyes not is but not4 erefore doubul the merke aboue e cifers hede1 e quych betokenes e haluendel of #a1 at schal be #a4 do away e cifer e merke1 sett ere #a1 en ou schalt haue is nombur #bda4 ,and is same nombur ou haddyst afore or ou began to medy1 yf ou take gode hede4 ,the next ensampul at had in e #d case of mediacio was is #dfgh4 ,whan ou hast truly ymedit alle is nombur fro e begynnynge to e endynge1 ou schalt haue of e mediacio is nombur #e #bccd4 ,now doubul this nombur begyn in e lyft side1 doubulle #b at schal be #d4 do away #b and sette ere #d2 en doubule #c1 at wol be #f2 do away #c sett ere #f1 en doubul at oer #c1 at wel be #f2 do away #c set ere 99leaf #aeb;b #f1 en doubul e #d1 at welle be #h2 en doubul #e4 e quych stondes ouer e hed of #d1 at wol be #aj2 cast #aj to #h1 at schal be #ah2 do away #d at #e1 sett ere #h1 sett that #a1 e quych is an articul of e ,composit e quych is #ah1 ouer e next figures hed toward e lyft side1 e quych is #f4 drav at #a to #f1 e quych #a in e dravyng schal be rekente bot for #a1 at #a at #f togedur wel be #g4 do away at #f at #a4 the quych stondes ouer his hede1 sett ther #g1 en ou schalt haue is nombur #dfgh4 ,and is same nombur ou hadyst or ou began to medye1 as ou mayst see in e secunde ,ensampul at ou had in e #d case of mediacio1 at was is3 when ou had mediet truly alle the nombur1 a principio usque ad finem4 ou schalt haue of at mediacio is nombur #e #ajb4 ,now doubul #a4 at wel be #b4 do away #a sett ere #b4 en doubul #j4 at will be not4 erefore take e #e1 e quych stondes ouer e next figures hed1 doubul it1 at wol be #aj4 do away e #j at stondes betwene e two figuris1 sette ere in his stid #a1 for at #a now schal stonde in e secunde place1 where he schal betoken #aj2 en doubul #b1 at wol be #d4 do away #b sett ere #d4 99leaf #aec;a ou schal haue us nombur #bad4 is is e same numbur at ou hadyst or ou began to medye1 as ou may see4 ,and so do euer more1 yf ou wil knowe wheer ou hase wel ymedyt or no4 4 doubulle e numbur at comes after e mediaciou1 ou schal haue e same nombur at ou hadyst or ou began to medye1 yf ou haue welle ydo4 or els doute e not1 but yf ou haue e same1 ou hase faylide in i ,craft4 ,sequitur de multiplicatione4 ,headnote3 ,the ,craft of ,multiplication4 ,headnote3 ,to write down a ,multiplication ,sum4 ,si tu per numerum numerum vis multiplicare ,scribe duas quascunque velis series numerorum ,ordo servetur vt vltima multiplicandi ,ponatur super anteriorem multiplicantis ,a leua relique sint scripte multiplicantes4 ,sidenote3 ,four things to be known of ,multiplication3 the first3 the second3 the third3 the fourth4 ,how to set down the sum4 ,two sorts of ,multiplication3 mentally1 and on paper4 ,here be-gynnes e ,chaptre of multiplicatio1 in e quych ou most know #d thynges4 ,ffirst1 qwat is multiplicacio4 ,the secunde1 how mony cases may hap in multiplicacio4 ,the thryde1 how mony rewes of figures ere most be4 ,the #d4 what is e profet of is craft4 ,as for e first1 ou schal vnderstonde at multiplicacio is a bryngynge to-geder of #b thynges in on nombur1 e quych on nombur contynes so mony tymes on1 howe 99leaf #aec;b mony tymes ere ben vnytees in e nowmbre of at #b1 as twyes #d is #h4 now here ben e #b nombers1 of e quych too nowmbres on is betokened be an aduerbe1 e quych is e worde twyes1 is worde thryes1 is worde foure sythes1#i is wordes fyue sithe sex sythes4 so furth of such other lyke wordes4 ,and tweyn nombres schal be tokenyde be a nowne1 as is worde foure showys es twey nombres y-broth in-to on hole nombur1 at is #h1 for twyes #d is #h1 as ou wost wel4 ,and es nombre #h conteynes as oft tymes #d as ere ben vnites in at other nombre1 e quych is #b1 for in #b ben #b vnites1 so oft tymes #d ben in #h1 as ou wottys wel4 ffor e secunde1 ou most know at ou most haue too rewes of figures4 ,as for e thryde1 ou most know at #h maner of diuerse case may happe in is craft4 ,the profet of is ,craft is to telle when a nombre is multiplyed be a noer1 qwat commys ere of4 fforthermore1 as to e sentence of oure verse1 yf ou wel multiply a nombur be a-noer nombur1 ou schalt write 99leaf #aed;a a rewe of figures of what nomburs so euer ou welt1 at schal be called ,numerus multiplicandus1 ,anglice1 e nombur the quych to be multiplied4 en ou schalt write a-nother rewe of figures1 by e quych ou schalt multiplie the nombre at is to be multiplied1 of e quych nombur e furst figure schal be write vnder e last figure of e nombur1 e quych is to be multiplied4 ,and so write forthe toward e lyft side1 as here you may se1 ---------- #fgcbd #abcd ---------- ,and is one nombur schalle be called numerus multiplicans4 ,anglice1 e nombur multipliynge1 for he schalle multiply e hyer nounbur1 as us one tyme #f4 ,and so forth1 as ,i schal telle the afterwarde4 ,and ou schal begyn in e lyft side4 ffor-ere-more ou schalt vndurstonde at ere is two manurs of multiplicacio2 one ys of e wyrchynge of e boke only in e mynde of a mon4 fyrst he teches of e fyrst maner of duplacio1 e quych is be wyrchynge of tabuls4 ,afterwarde he wol teche on e secunde maner4 vnde versus4 ,headnote3 ,to multiply one ,digit by another4 ,in digitum cures digitum si ducere maior 99leaf #aed;b4 ,per quantum distat a denis respice debes ,namque suo decuplo totiens delere minorem ,sitque tibi numerus veniens exinde patebit4 ,sidenote3 ,how to multiply two digits4 ,subtract the greater from ten2 take the less so many times from ten times itself4 ,example4 ,here he teches a rewle1 how ou schalt fynde e nounbre at comes by e multiplicacio of a digit be anoer4 loke how mony vnytes ben4 bytwene e more digit and #aj4 ,and reken ten for on vnite4 ,and so oft do away e lasse nounbre out of his owne decuple1 at is to say1 fro at nounbre at is ten tymes so mych is e nounbre at comes of e multiplicacio4 ,as yf ou wol multiply #b be #d4 loke how mony vnitees ben by-twene e quych is e more nounbre1 be-twene ten4 ,certen ere wel be vj vnitees by-twene #d ten4 yf ou reken ere with e ten e vnite1 as ou may se4 so mony tymes take #b4 out of his decuple1 e quych is #bj4 for #bj is e decuple of #b1 #aj is e decuple of #a1 #cj is e decuple of #c1 #dj is e decuple of #d1 ,and e oer digetes til ou come to ten2 whan ou hast y-take so mony tymes #b out of twenty1 e quych is sex tymes1 ou schal leue #h as ou wost wel1 for #f times #b is twelue4 take #ab out of twenty1 ere schal leue #h4 bot yf bothe e digettes 99leaf #aee;a ben y-lyech mych as here4 #bbb or too tymes twenty1 en it is no fors quych of hem tweyn ou take out of here decuple4 als mony tymes as at is fro #aj4 but neuer-e-lesse1 yf ou haue hast to worch1 ou schalt haue here a tabul of figures1 where-by ou schalt se a-non ryght what is e nounbre at comes of e multiplicacio of #b digittes4 us ou schalt worch in is figure4 ,sidenote3 ,better use this table1 though4 ,how to use it4 ,the way to use the ,multiplication table4 #a ----- #b #d -------- #c #f #i ----------- #d #habaf -------------- #eajaebjbe ----------------- #fabahbdcjcf -------------------- #gadbabhcedbdi ----------------------- #hafbdcbdjdheffd -------------------------- #iahbgcfdeedfcgbha ---------------------------- #a #b #c #d #e #f #g #h #i ---------------------------- yf e figure1 e quych schalle be multiplied1 be euene as mych as e diget be1 e quych at oer figure schal be multiplied1 as two tymes tway1 or thre tymes #c4 or sych other4 loke qwere at figure sittes in e lyft side of e triangle1 loke qwere e diget sittes in e neer most rewe of e triangle4 go fro hym vpwarde in e same rewe1 e quych rewe gose vpwarde til ou come agaynes e oer digette at sittes in e lyft side of e triangle4 ,and at nounbre1 e quych ou 99leaf #aee;b fyn99des ere is e nounbre at comes of the multiplicacio of e #b digittes1 as yf ou wold wete qwat is #b tymes #b4 loke quere sittes #b in e lyft side in e first rewe1 he sittes next #a in e lyft side al on hye1 as ou may se2 en loke qwere sittes #b in e lowyst rewe of e triangle1 go fro hym vpwarde in e same rewe tylle ou come a-enenes #b in e hyer place1 er ou schalt fynd ywrite #d1 at is e nounbre at comes of e multiplicacio of two tymes tweyn is #d1 as ow wotest welle4 yf e diget4 the quych is multiplied1 be more an e oer1 ou schalt loke qwere e more diget sittes in e lowest rewe of e triangle1 go vpwarde in e same rewe tyl#aj ou come a-nendes e lasse diget in the lyft side4 ,and ere ou schalt fynde e nombre at comes of e multiplicacio2 but ou schalt vnderstonde at is rewle1 e quych is in is verse4 ,in digitum cures1 c41 noer is triangle schalle not serue1 bot to fynde e nounbres at comes of the multiplicacio at comes of #b articuls or composites1 e nedes no craft but yf ou wolt multiply in i mynde4 ,and 99leaf #aef;a ere-to ou schalt haue a craft afterwarde1 for ou schall wyrch with digettes in e tables1 as ou schalt know afterwarde4 versus4 ,headnote3 ,to multiply one ,composite by another4 ,postea procedas postremam multiplicando ,recte multiplicans per cunctas inferiores ,condicionem tamen tali quod multiplicantes ,scribas in capite quicquid processerit inde ,sed postquam fuit hec multiplicate figure ,anteriorentur serei multiplicantis ,et sic multiplica velut isti multiplicasti ,qui sequitur numerum scriptum quiscunque figuris4 ,sidenote3 ,how to multiply one number by another4 ,multiply the #8last' figure of the higher by the #8first' of the lower number4 ,set the answer over the first of the lower3 then multiply the second of the lower1 and so on4 ,then antery the lower number3 as thus4 ,now multiply by the last but one of the higher3 as thus4 ,antery the figures again1 and multiply by five3 ,then add all the figures above the line3 and you will have the answer4 ,here he teches how ou schalt wyrch in is craft4 ou schalt multiplye e last figure of e nombre1 and quen ou hast so ydo ou schalt draw alle e figures of e neer nounbre more taward e ryt side1 so qwen ou hast multiplyed e last figure of e heyer nounbre by alle e neer figures4 ,and sette e nounbir at comes er-of ouer e last figure of e neer nounbre1 en ou schalt sette al e oer figures of e neer nounbre more nere to e ryt side4 ,and whan ou hast multiplied at figure at schal be multiplied e next after hym by al e neer figures4 ,and worch as ou dyddyst afore til 99leaf #aef;b ou come to e ende4 ,and ou schalt vnderstonde at euery figure of e hier nounbre schal be multiplied be alle e figures of the neer nounbre1 yf e hier nounbre be any figure en one4 lo an ,ensampul here folowynge4 ------ #bdfe4 #bcb ------ ou schalt begyne to multiplye in e lyft side4 ,multiply #b be #b1 and twyes #b is #d4 set #d ouer e hed of at #b1 en multiplie e same hier #b by #c of e nether nounbre1 as thryes #b at schal be #f4 set #f ouer e hed of #c1 an multiplie e same hier #b by at #b e quych stondes vnder hym1 at wol be #d2 do away e hier #b sette ere #d4 ,now ou most antery e nether nounbre1 at is to say1 ou most sett e neer nounbre more towarde e ryt side1 as us4 ,take e neer #b toward e ryt side1 sette it eue vnder e #d of e hyer nounbre1 antery alle e figures at comes after at #b1 as us2 sette #b vnder e #d4 en sett e figure of #c ere at e figure of #b stode1 e quych is now vndur at #d in e hier nounbre2 en sett e oer figure of #b1 e quych is e last figure toward e lyft side of e neer nomber ere e figure of #c stode4 en ou schalt haue such a nombre4 ------ #dfddfe #bcb ------ 99leaf #aeg;a ,now multiply #d1 e quych comes next after #f1 by e last #b of e neer nounbur toward e lyft side4 as #b tymes #d1 at wel be #h4 sette at #h ouer e figure the quych stondes ouer e hede of at #b1 e quych is e last figure of e neer nounbre2 an multiplie at same #d by #c1 at comes in e neer rewe1 at wol be #ab4 sette e digit of e composyt ouer e figure e quych stondes ouer e hed of at #c1 sette e articule of is composit ouer al e figures at stondes ouer e neer #b hede4 en multiplie e same #d by e #b in e ryt side in e neer nounbur1 at wol be #h4 do away #d4 sette ere #h4 ,euer more qwen ou multiplies e hier figure by at figure e quych stondes vnder hym1 ou schalt do away at hier figure1 sett er at nounbre e quych comes of multiplicacio of ylke digittes4 ,whan ou hast done as ,i haue byde e1 ou schalt haue suych an order of figure as is here1 -------- #a 4 #hb #dfdhfe #bcb -------- en take and antery i neer figures4 ,and sett e fyrst figure of e neer figures#aa vndre be figure of #f4 ,and draw al e oer figures of e same rewe to hym-warde1 99leaf #aeg;b as ou diddyst afore4 en multiplye #f be #b1 sett at e quych comes ouer ere-of ouer al e oer figures hedes at stondes ouer at #b4 en multiply #f be #c1 sett alle at comes ere-of vpon alle e figures hedes at standes ouer at #c2 an multiplye #f be #b1 e quych stondes vnder at #f1 en do away #f write ere e digitt of e composit at schal come ereof1 sette e articull ouer alle e figures at stondes ouer e hede of at #c as here1 ------ #aa #aba #hbh #dfdhbe #bcb ------ en antery i figures as ou diddyst afore1 and multipli #e be #b1 at wol be #aj2 sett e #j ouer all e figures at stonden ouer at #b1 sett at #a4 ouer the next figures hedes1 alle on hye towarde e lyft side4 en multiplye #e be #c4 at wol be #ae1 write #e ouer e figures hedes at stonden ouer at #c1 sett at #a ouer e next figures hedes toward e lyft side4 en multiplye #e be #b1 at wol be #aj4 do away at #e sett ere a #j1 sett at #a ouer e figures hedes at stonden ouer #c4 ,and en ou schalt haue such a nounbre as here stondes aftur499leaf #aeh;a ------ #aa #aaja #abae #hbhbj #dfdh #bcb ------ ,now draw alle ese figures downe togeder as us1 #f4h4a4 #a draw to-gedur2 at wolle be #af1 do away alle ese figures saue #f4 lat hym stonde1 for ow ou take hym away ou most write er e same aene4 erefore late hym stonde1 sett #a ouer e figure hede of #d toward e lyft side2 en draw on to #d1 at wolle be #e4 do away at #d at #a1 sette ere #e4 en draw #dbba #a togedur1 at wol be #aj4 do away alle at1 write ere at #d at #j1 sett at #a ouer e next figures hede toward e lyft side1 e quych is #f4 en draw at #f at #a togedur1 at wolle be #g2 do away #f sett ere #g1 en draw #hhaj #a1 at wel be #ah2 do away alle e figures at stondes ouer e hede of at #h1 lette #h stonde stil1 write at #a ouer e next figuris hede1 e quych is a #j4 en do away at #j1 sett ere #a1 e quych stondes ouer e #j4 hede4 en draw #b1 #e1 #a togedur1 at wolle be #h4 en do away alle at1 write ere #h4 ,and en ou schalt haue is nounbre1 #egahhj4 ,headnote3 ,the ,cases of this ,craft4 99leaf #aeh;b ,sed cum multiplicabis1 primo sic est operandum1 ,si dabit articulum tibi multiplicacio solum2 ,proposita cifra summam transferre memento4 ,sidenote3 ,what to do if the first multiplication results in an article4 ,here he puttes e fyrst case of is craft1 e quych is is3 yf ere come an articulle of e multiplicacio ysette before the articulle in e lyft side as us --- #ea4 #bc --- multiplye #e by #b1 at wol be #aj2 sette ouer e hede of at #b a #j1 sett at on1 at is e articul1 in e lyft side1 at is next hym1 en ou schalt haue is nounbre ---- #ajea4 #bc ---- ,and en worch forth as ou diddist afore4 ,and ou schalt vnderstonde at ou schalt write no #j4 but whan at place where ou schal write at #j has no figure afore hym noer after4 versus4 ,si autem digitus excreuerit articulusque4 ,articulus#ab supraposito digito salit vltra4 ,sidenote3 ,what to do if the result is a composite number4 ,here is e secunde case1 e quych is is3 yf hit happe at ere come a composyt1 ou schalt write e digitte ouer e hede of e neer figure by e quych ou multipliest e hier figure2 and sett e articulle next hym toward e lyft side1 as ou diddyst afore1 as us --- #hc4 #hc --- ,multiply #h by #h1 at wol be #fd4 ,write e #d ouer #h1 at is to say1 ouer e hede of e neer #h2 set #f1 e quych 99leaf #aei;a is an articul1 next after4 ,and en ou schalt haue such a nounbre as is here1 ----------- #fdhcac1 #hc ----------- ,and en worch forth4 ,si digitus tamen ponas ipsum super ipsam4 ,sidenote3 ,what if it be a digit4 ,here is e thryde case1 e quych is is3 yf hit happe at of i multiplicaciou come a digit1 ou schalt write e digit ouer e hede of e neer figure1 by the quych ou multipliest e hiere figure1 for is nedes no ,ensampul4 ,subdita multiplica non hanc que incidit illi ,delet eam penitus scribens quod prouenit inde4 ,sidenote3 ,the fourth case of the craft4 ,here is e #d case1 e quych is3 yf hit be happe at e neer figure schal multiplye at figure1 e quych stondes ouer at figures hede1 ou schal do away e hier figure sett ere at at comys of at multiplicacio4 ,as yf ere come of at multiplicacio an articuls ou schalt write ere e hier figure stode a #j4 ,and write e articuls in e lyft side1 yf at hit be a digit write ere a digit4 yf at hit be a composit1 write e digit of e composit4 ,and e articul in e lyft side4 al is is lyt y-nowt1 ere-fore er nedes no ,ensampul4 ,sed si multiplicat aliam ponas super ipsam ,adiunges numerum quem prebet ductus earum4 ,sidenote3 ,the fifth case of the craft4 ,here is e #e case1 e quych is is3 yf 99leaf #aei;b e neer figure schul multiplie e hier1 and at hier figure is not recte ouer his hede4 ,and at neer figure hase oer figures1 or on figure ouer his hede by multiplicacio1 at hase be afore1 ou schalt write at nounbre1 e quych comes of at1 ouer alle e ylke figures hedes1 as us here3 ----- #bcf #bcd ----- ,multiply #b by #b1 at wol be #d2 set #d ouer e hede of at #b4 en#ad multiplies e hier #b by e neer #c1 at wol be #f4 set ouer his hede #f1 multiplie e hier #b by e neer #d1 at wol be #h4 do away e hier #b1 e quych stondes ouer e hede of e figure of #d1 and set ere #h4 ,and ou schalt haue is nounbre here ------- #dfhcf #bcd ------- ,and antery i figures1 at is to say1 set i neer #d vnder e hier #c1 and set i #b other figures nere hym1 so at e neer #b stonde vndur e hier #f1 e quych #f stondes in e lyft side4 ,and at #c at stondes vndur #h1 as us aftur e may se1 ------- #dfhcf #bcd ------- ,now worch forthermore1 ,and multiplye at hier #c by #b1 at wol be #f1 set at #f e quych stondes ouer e hede of at #b1 ,and en worch as ,i tat e afore4 99leaf #afj;a ,si supraposita cifra debet multiplicare ,prorsus eam deles ibi scribi cifra debet4 ,sidenote3 ,the sixth case of the craft4 ,here is e #f case1 e quych is is3 yf hit happe at e figure by e quych ou schal multiplye e hier figure1 e quych stondes ryght ouer hym by a #j1 ou schalt do away at figure1 e quych ouer at cifre hede4 ,and write ere at nounbre at comes of e multiplicacio as us1 #bc4 do away #b and sett ere a #j4 vnde versus4 ,si cifra multiplicat aliam positam super ipsam ,sitque locus supra vacuus super hanc cifram fiet4 ,sidenote3 ,the seventh case of the craft4 ,here is e #g case1 e quych is is3 yf a #j schal multiply a figure1 e quych stondes not recte ouer hym1 ,and ouer at #j stonde no thyng1 ou schalt write ouer at #j anoer #j as us3 ---- #bd #jc ---- multiplye #b be a #j1 it wol be nothynge4 write ere a #j ouer e hede of e neer #j1 ,and en worch forth til ou come to e ende4 ,si supra#ae fuerit cifra semper est pretereunda4 ,sidenote3 ,the eighth case of the craft4 ,here is e #h case1 e quych is is3 yf ere be a #j or mony cifers in e hier rewe1 ou schalt not multiplie hem1 bot let hem stonde4 ,and antery e figures benee to e next figure sygnificatyf as us3 ----- #jjjcb4 #bb ----- ,ouer-lepe alle ese cifers sett at 99leaf #afj;b neer #b at stondes toward e ryght side1 and sett hym vndur e #c1 and sett e oer nether #b nere hym1 so at he stonde vndur e thrydde #j1 e quych stondes next #c4 ,and an worch4 vnde versus4 ,si dubites1 an sit bene multiplicacio facta1 ,diuide totalem numerum per multiplicantem4 ,sidenote3 ,how to prove the multiplication4 ,here he teches how ou schalt know wheer ou hase wel ,i-do or no4 ,and he says at ou schalt deuide alle e nounbre at comes of e multiplicacio by e neer figures4 ,and en ou schalt haue e same nounbur at ou hadyst in e begynnynge4 but et ou hast not e craft of dyuisio1 but ou schalt haue hit afterwarde4 ,per numerum si vis numerum quoque multiplicare ,tantum per normas subtiles absque figuris ,has normas poteris per versus scire sequentes4 ,sidenote3 ,mental multiplication4 ,here he teches e to multiplie be owt figures in i mynde4 ,and e sentence of is verse is is3 yf ou wel multiplie on nounbre by anoer in i mynde1 ou schal haue ereto rewles in e verses at schal come after4 ,si tu per digitum digitum vis multiplicare ,regula precedens dat qualiter est operandum4 ,sidenote3 ,digit by digit is easy4 ,here he teches a rewle as ou hast afore to multiplie a digit be anoer1 as yf ou wolde wete qwat is sex tymes #f4 ou 99leaf #afa;a schalt wete by e rewle at ,i tat e before1 yf ou haue mynde erof4 ,articulum si per reliquum reliquum vis multiplicare ,in proprium digitum debet vterque resolui4 ,articulus digitos post se multiplicantes ,ex digitus quociens retenerit multiplicari ,articuli faciunt tot centum multiplicati4 ,sidenote3 ,the first case of the craft4 ,article by article2 an example3 another example3 ,headnote3 ,how to work subtly without ,figures4 ,sidenote3 ,mental multiplication4 ,another example4 ,another example4 ,notation4 ,notation again4 ,mental multiplication4 ,here he teches e furst rewle1 e quych is is3 yf ou wel multiplie an articul be anoer1 so at both e articuls bene with-,inne an hundreth1 us ou schalt do4 take e digit of bothe the articuls1 for euery articul hase a digit1 en multiplye at on digit by at oer1 and loke how mony vnytes ben in e nounbre at comes of e multiplicacio of e #b digittes1 so mony hundrythes ben in e nounbre at schal come of e multiplicacio of e ylke #b articuls as us4 yf ou wold wete qwat is ten tymes ten4 take e digit of ten1 e quych is #a2 take e digit of at oer ten1 e quych is on4 ,also multiplie #a be #a1 as on tyme on at is but #a4 ,in on is but on vnite as ou wost welle1 erefore ten tymes ten is but a hundryth4 ,also yf ou wold wete what is twenty tymes #cj4 take e digit of twenty1 at is #b2 take e digitt of thrytty1 at is #c4 multiplie #c be #b1 at is #f4 ,now in #f ben #f vnites1 ,and so mony hundrythes ben in #bj tymes #cj99leaf #afa;b1 erefore #bj tymes #cj is #f hundryth eue4 loke se4 ,but yf it be so at one articul be with-,inne an hundryth1 or by-twene an hundryth and a thowsande1 so at it be not a owsande fully4 en loke how mony vnytes ben in e nounbur at comys of e multiplicacio #af;,and so mony tymes#af of #b digittes of ylke articuls1 so mony thowsant ben in e nounbre1 the qwych comes of e multiplicacio4 ,and so mony tymes ten thowsand schal be in e nounbre at comes of e multiplicacion of #b articuls1 as yf ou wold wete qwat is #d hundryth tymes two hundryth4 ,multiply #d be #b1ag at wol be #h4 in #h ben #h vnites4 ,and so mony tymes ten thousand be in #d hundryth tymes #bag hundryth1 at is #hj thousand4 ,take hede1 ,i schall telle e a generalle rewle whan ou hast #b articuls1 ,and ou wold wete qwat comes of e multiplicacio of hem #b4 multiplie e digit of at on articuls1 and kepe at nounbre1 en loke how mony cifers schuld go before at on articuls1 and he were write4 ,als mony cifers schuld go before at other1 he were write of cifers4 ,and haue alle e ylke cifers togedur in i mynde1 99leaf #afb;a a-rowe ycho aftur other1 and in e last plase set e nounbre at comes of e multiplicacio of e #b digittes4 ,and loke in i mynde in what place he stondes1 where in e secunde1 or in e thryd1 or in e #d1 or where ellis1 and loke qwat e figures by-token in at place2 so mych is e nounbre at comes of e #b articuls y-multiplied to-gedur as us3 yf ou wold wete what is #bj thousant tymes #c owsande4 multiply e digit of at articulle e quych is #b by e digitte of at oer articul e quych is #c1 at wol be #f4 en loke how mony cifers schal go to #bj thousant as hit schuld be write in a tabul4 certainly #d cifers schuld go to #bj owsant4 ffor is figure #b in e fyrst place betokenes twene4 ,in e secunde place hit betokenes twenty4 ,in e #c4 place hit betokenes #b hundryth4 44 ,in e #d place #b thousant4 ,in e #e place hit betokenes twenty ousant4 erefore he most haue #d cifers a-fore hym at he may stonde in e #e place4 kepe ese #d cifers in thy mynde1 en loke how mony cifers go to #c thousant4 ,certayn to #c thousante 99leaf #afb;b go #c cifers afore4 ,now cast ylke #d cifers at schuld go to twenty thousant1 ,and thes #c cifers at schuld go afore #c thousant1 sette hem in rewe ycho after oer in i mynde1 as ai schuld stonde in a tabulle4 ,and en schal ou haue #g cifers2 en sett at #f e quych comes of e multiplicacio of e #b digittes aftur e ylke cifers in e #h place as yf at hit stode in a tabul4 ,and loke qwat a figure of #f schuld betoken in e #h place4 yf hit were in a tabul so mych it is4 yf at figure of #f stonde in e fyrst place he schuld betoken but #f4 ,in e #b place he schuld betoken sexty4 ,in the #c place he schuld betoke sex hundryth4 ,in e #d place sex thousant4 ,in e #e place sexty owsant4 ,in e sext place sex hundryth owsant4 ,in e #g place sex owsant thousantes4 ,in e #h place sexty owsant thousantes4 erfore sett #f in octauo loco1 ,and he schal betoken sexty owsant thousantes4 ,and so mych is twenty owsant tymes #c thousant1 ,and is rewle is generalle for alle maner of articuls1 ,whethir ai be hundryth or owsant2 but ou most know well e craft of e wryrchynge in e tabulle 99leaf #afc;a or ou know to do us in i mynde aftur is rewle4 ,thou most at is rewle holdye note but where ere ben #b articuls and no mo of e quych ayther of hem hase but on figure significatyf4 ,as twenty tymes #c thousant or #c hundryth1 and such our4 ,articulum digito si multiplicare oportet ,articuli digiti sumi quo multiplicate ,debemus reliquum quod multiplicatur ab illis ,per reliquo decuplum sic summam latere nequibit4 ,sidenote3 ,the third case of the craft2 an example4 ,here he puttes e thryde rewle1 e quych is is4 yf ou wel multiply in i mynde1 ,and e ,articul be a digitte1 ou schalt loke at e digitt be with-,inne an hundryth1 en ou schalt multiply the digitt of e ,articulle by e oer digitte4 ,and euery vnite in e nounbre at schalle come ere-of schal betoken ten4 ,as us3 yf at ou wold wete qwat is twyes #dj4 multiplie e digitte of #dj1 e quych is #d1 by e oer diget1 e quych is #b4 ,and at wolle be #h4 ,and in e nombre of #h ben #h vnites1 euery of e ylke vnites schuld stonde for #aj4 ere-fore ere schal be #h tymes #aj1 at wol be #d score4 ,and so mony is twyes #dj4 ,if e articul be a hundryth or be #b hundryth ,and a owsant1 so at hit be notte a thousant1 99leaf #afc;b worch as ou dyddyst afore1 saue ou schalt rekene euery vnite for a hundryth4 ,in numerum mixtum digitum si ducere cures ,articulus mixti sumatur deinde resoluas ,in digitum post fac respectu de digitis ,articulusque docet excrescens in diriuando ,in digitum mixti post ducas multiplicantem ,de digitis vt norma #ah;docet de hunc ,multiplica simul et sic postea summa patebit4 ,sidenote3 ,the fourth case of the craft3 ,composite by digit4 ,mental multiplication4 ,here he puttes e #d rewle1 e quych is is3 yf ou multipliy on composit be a digit as #f tymes #bd1 #ai;en take e diget of at composit1 multiply at digitt by at oer diget1 and kepe e nombur at comes ere-of4 en take e digit of at composit1 multiply at digit by anoer diget1 by e quych ou hast multiplyed e diget of e articul1 and loke qwat comes ere-of4 en take ou at nounbur1 cast hit to at other nounbur at ou secheste as us yf ou wel wete qwat comes of #f tymes #d twenty4 multiply at articulle of e composit by e digit1 e quych is #f1 as yn e thryd rewle ou was taut1 ,and at schal be #f score4 en multiply e diget of e composit1 99leaf #afd;a e quych is #d1 and multiply at by at other diget1 e quych is #f1 as ou wast taut in e first rewle1 yf ou haue mynde erof1 at wol be #d twenty4 cast all ylke nounburs to-gedir1 hit schal be #add4 ,and so mych is #f tymes #d twenty4 ,headnote3 ,how to multiply without ,figures4 ,ductus in articulum numerus si compositus sit ,articulum purum comites articulum quoque ,mixti pro digitis post fiat et articulus vt ,norma iubet retinendo quod extra dicta ab illis ,articuli digitum post tu mixtum digitum duc ,regula de digitis nec precipit articulusque ,ex quibus excrescens summe tu iunge priori ,sic manifesta cito fiet tibi summa petita4 ,sidenote3 ,the fifth case of the craft3 ,article by ,composite4 ,an example4 ,here he puttes e #e rewle1 e quych is is3 yf ou wel multiply an ,articul be a composit1 multiplie at ,articul by e articul of e composit1 and worch as ou wos taut in e secunde rewle1 of e quych rewle e verse begynnes us4 ,articulum si per ,relicum vis multiplicare4 en multiply e diget of e composit by at oir articul aftir e doctrine of e #c rewle4 take erof gode hede1 ,i pray e as us4 ,yf ou wel wete what is #bd tymes ten4 ,multiplie ten by #bj1 at wel be #b hundryth4 en multiply e diget of e #aj1 e quych is #a1 by e diget of e composit1 e quych is #d1 at 99leaf #afd;b wol be #d4 en reken euery vnite at is in #d for #aj1 at schal be #dj4 ,cast #dj to #b hundryth1 at wol be #b hundryth #dj4 ,and so mych is #bd tymes ten4 ,headnote3 ,how to work without ,figures4 ,compositum numerum mixto sic multiplicabis ,vndecies tredecim sic est ex hiis operandum ,in reliquum primum demum duc post in eundem ,vnum post denum duc in tria deinde per vnum ,multiplicesque demum intra omnia multiplicata ,in summa decies quam si fuerit tibi doces ,multiplicandorum de normis sufficiunt hec4 ,sidenote3 ,the sixth case of the craft3 ,composite by ,composite4 ,mental multiplication4 ,an example of the sixth case of the craft4 ,here he puttes e #f rewle1 e last of alle multiplicacio1 e quych is is3 yf ou wel multiplye a composit by a-noer composit1 ou schalt do us4 multiplie at on composit1 qwych ou welt of the twene1 by e articul of e toer composit1 as ou were taut in e #e rewle1 en multiplie at same composit1 e quych ou hast multiplied by e oer articul1 by e digit of e oer composit1 as ou was taut in e #d rewle4 ,as us1 yf ou wold wete what is #aa tymes #ac1 as ou was taut in e #e rewle1 at schal be an hundryth ten1 afterwarde multiply at same composit at ou hast multiplied1 e quych is a 4#aa4 ,and multiplye hit be e digit of e oer composit1 e quych is #c1 for #c is e digit of #ac1 ,and at wel be #cj4 en take e digit of at composit1 e quych composit ou multiplied by e digit of at oer composit1 99leaf #afe;a e quych is a #aa4 ,also of the quych #aa on is e digit4 multiplie at digitt by e digett of at other composit1 e quych diget is #c1 as ou was taut in e first rewle in e begynnynge of is craft4 e quych rewle begynnes 8,in digitum cures40 ,and of alle e multiplicacio of e #b digitt comys thre1 for onys #c is but #c4 ,now cast alle ese nounbers togedur1 the quych is is1 a hundryth ten #cj #c4 ,and al at wel be #adc4 ,write #c first in e ryght side4 ,and cast #aj to #cj1 at wol be #dj4 set #dj next aftur towarde e lyft side1 ,and set aftur a hundryth as here an ,ensampulle1 #adc4 7,cetera desunt47 ,,footnotes 7,the ,crafte of ,nombrynge73 #a3 ,in ,,ms1 #8awiy4' #b3 #8ben' repeated in ,,ms4 #c3 ,in ,,ms4 #8thausandes4' #d3 ,perhaps 8,so40 #e3 #8hali' marked for erasure in ,,ms4 #f3 #8moy' in ,,ms4 #g3 #8,subtrahas aut addis a dextris vel mediabis' added on margin of ,,ms4 #h3 ,after #8craft' insert #8the 4#d4 what is e profet of is craft4' #i3 ,after #8sythes' insert #8 is wordes fyue sithe sex sythes4' #aj3 #8t'l' marked for erasure before #8tyl' in ,,ms4 #aa3 ,here #8of e same rew' is marked for erasure in ,,ms4 #ab3 #8sed' deleted in ,,ms4 #ac3 #fhhc in ,,ms4 #ad3 #8en' overwritten on #8at' marked for erasure4 #ae3 #8,supra' inserted in ,,ms4 in place of #8cifra' marked for erasure4 #af--af3 ,marked for erasure in ,,ms4 #ag3 #d in ,,ms4 #ah3 docet4 decet ,,ms4 #ai3 #8#d times #d' in ,,ms4 ,the ,art of ,nombryng4 ,a ,,translation ,,of ,john of ,holywood's ,de ,arte ,numerandi4 ,ashmole ,,ms4 #cif1 fol4 #dh4 ,boys seying in the begynnyng of his ,arsemetrike3--,alle 99,fol4 #dh4 thynges that bene fro the first begynnyng of thynges have procedede1 and come forthe1 ,and by resoun of nombre ben formede2 ,and in wise as they bene1 ,so owethe they to be knowene2 wherfor in vniuersalle knowlechyng of thynges the ,art of nombrynge is best1 and most operatyfe4 ,sidenote3 ,the name of the art4 ,derivation of ,algorism4 ,another4 ,another4 ,kinds of numbers4 ,the #i rules of the ,art4 ,therfore sithen the science of the whiche at this tyme we intendene to write of standithe alle and about nombre3 ffirst we most se1 what is the propre name therofe1 and fro whens the name come3 ,afterwarde what is nombre1 ,and how manye spices of nombre ther ben4 ,the name is clepede ,algorisme1 hade out of ,algore1 other of ,algos1 in grewe1 ,that is clepide in englisshe art other craft1 ,and of ,rithmus that is callede nombre4 ,so algorisme is clepede the art of nombryng1 other it is had ofe en or in1 and gogos that is introduccioun1 and ,rithmus nombre1 that is to say ,interduccioun of nombre4 ,and thirdly it is hade of the name of a kyng that is clepede ,algo and ,rythmus2 ,so callede ,algorismus4 ,sothely 4#b4 manere of nombres ben notifiede2 ,formalle1#a as nombre is vnitees gadrede to-gedres2 ,materialle1#b as nombre is a colleccioun of vnitees4 ,other nombre is a multitude hade out of vnitees1 vnitee is that thynge wher-by euery thynge is callede oone1 other o thynge4 ,of nombres1 that one is clepede digitalle1 that othere ,article1 ,another a nombre componede oer myxt4 ,another digitalle is a nombre with-in 4#aj42 ,article is at nombre that may be dyvydede in 4#aj4 parties egally1 ,and that there leve no residue2 ,componede or medlede is that nombre that is come of a digite and of an article4 ,and vndrestande wele that alle nombres betwix 4#b4 articles next is a nombre componede4 ,of this art bene 4#i4 spices1 that is forto sey1 numeracioun1 addicioun1 ,subtraccioun1 ,mediacioun1 ,duplacioun1 ,multipliacioun1 ,dyvysioun1 ,progressioun1 ,and of ,rootes the extraccioun1 and that may be hade in 4#b4 maners1 that is to sey in nombres quadrat1 and in cubices3 ,amonge the whiche1 ffirst of ,numeracioun1 and afterwarde of e oers by ordure1 y entende to write4 ,headnote3 ,chapter ,i4 ,numeration4 99,fol4 #dh;b ,for-sothe numeracioun is of euery numbre by competent figures an artificialle representacioun4 ,sidenote3 ,figures1 differences1 places1 and limits4 ,the #i figures4 ,the cipher4 ,the numeration of digits1 of articles1 of composites4 ,the value due to position4 ,numbers are written from right to left4 ,sothly figure1 difference1 places1 and lynes supposen o thyng other the same1 ,but they ben sette here for dyuers resons4 ffigure is clepede for protraccioun of figuracioun2 ,difference is callede for therby is shewede euery figure1 how it hathe difference fro the figures before them3 place by cause of space1 where-in me writethe3 lynees1 for that is ordeynede for the presentacioun of euery figure4 ,and vnderstonde that ther ben 4#i4 lymytes of figures that representen the 4#i4 digites that ben these4 #j4 #i4 #h4 #g4 #f4 #e4 #d4 #c4 #b4 #a4 ,the 4#aj4 is clepede theta1 or a cercle1 other a cifre1 other a figure of nought for nought it signyfiethe4 ,nathelesse she holdyng that place givethe others for to signyfie2 for withe-out cifre or cifres a pure article may not be writte4 ,and sithen that by these 4#i4 figures significatifes ,ioynede with cifre or with cifres alle nombres ben and may be representede1 ,it was1 nether is1 no nede to fynde any more figures4 ,and note wele that euery digite shalle be writte with oo figure allone to it aproprede4 ,and alle articles by a cifre1 ffor euery article is namede for oone of the digitis as 4#aj4 of #a44 #bj4 of4 #b4 and so of the others1 c4 ,and alle nombres digitalle owen to be sette in the first difference3 ,alle articles in the seconde4 ,also alle nombres fro 4#aj4 til an 4#ajj4 which is excludede1 with 4#b4 figures mvst be writte2 ,and yf it be an article1 by a cifre first put1 and the figure y-writte towarde the lift honde1 that signifiethe the digit of the whiche the article is namede2 ,and yf it be a nombre componede1 ffirst write the digit that is a part of that componede1 and write to the lift side the article as it is seide be-fore4 ,alle nombre that is fro an hundrede tille a thousande exclusede1 owithe to be writ by 4#c4 figures2 and alle nombre that is fro a thousande til 4x4 ,m4 mvst be writ by 4#d4 figures2 ,and so forthe4 ,and vnderstonde wele that euery figure sette in the first place signyfiethe his digit2 ,in the seconde place 4#aj4 tymes his digit2 ,in the 4#c4 place an hundrede so moche2 ,in the 4#d4 place a thousande so moche2 ,in the 4#e4 place 4x4 thousande so moche2 ,in the 4#f4 place an hundrede thousande so moche2 ,in the 4#g4 place a thousande thousande4 ,and so infynytly mvltiplying by 99,fol4 #di4 these 4#c4 #aj1 #ajj1 #ajjj4 ,and vnderstande wele that competently me may sette vpon figure in the place of a thousande1 a prike to shewe how many thousande the last figure shalle represent4 ,we writene in this art to the lift side-warde1 as arabiene writene1 that weren fynders of this science1 othere for this resoun1 that for to kepe a custumable ordre in redyng1 ,sette we alle-wey the more nombre before4 ,headnote3 ,chapter ,,ii4 ,addition4 ,sidenote3 ,definition4 ,how the numbers should be written4 ,the method of working4 ,begin at the right4 ,the ,sum is a digit1 or an article1 or a composite4 ,addicioun is of nombre other of nombres vnto nombre or to nombres aggregacioun1 that me may see that that is come therof as excressent4 ,in addicioun1 #b4 ordres of figures and 4#b4 nombres ben necessary1 that is to sey1 a nombre to be addede and the nombre wherto the addicioun sholde be made to4 ,the nombre to be addede is that at sholde be addede therto1 and shalle be vnderwriten2 the nombre vnto the whiche addicioun shalle be made to is that nombre that resceyuethe the addicion of at other1 and shalle be writen above2 and it is convenient that the lesse nombre be vnderwrit1 and the more addede1 than the contrary4 ,but whether it happe one other other1 the same comythe of1 ,therfor1 yf ow wilt adde nombre to nombre1 write the nombre wherto the addicioun shalle be made in the omest ordre by his differences1 so that the first of the lower ordre be vndre the first of the omyst ordre1 and so of others4 ,that done1 adde the first of the lower ordre to the first of the omyst ordre4 ,and of suche addicioun1 other ere growith therof a digit1 ,an article1 other a composede4 ,if it be digitus1 ,in the place of the omyst shalt thow write the digit excrescyng1 as thus3-- ------------------------------- ,the resultant #b ------------------------------- ,to whom it shal be addede #a ------------------------------- ,the nombre to be addede #a ------------------------------- ,if the article2 in the place of the omyst put a-way by a cifre writte1 and the digit transferrede1 of e whiche the article toke his name1 towarde the lift side1 and be it addede to the next figure folowyng1 yf ther be any figure folowyng2 or no1 and yf it be not1 leve it in the voide1 as thus3-- ------------------------------------- ,the resultant #aj ------------------------------------- ,to whom it shalle be addede #g ------------------------------------- ,the nombre to be addede #c ------------------------------------- ------------------------------------- ,resultans #b #g #h #b #g ------------------------------------- ,cui debet addi #a #j #j #h #d ------------------------------------- ,numerus addendus #a #g #g #d #c ------------------------------------- ,and yf it happe that the figure folowyng wherto the addicioun shalle be made by the cifre of an article1 it sette a-side2 ,in his place write the 99,fol4 #di;b digit of the ,article as thus3-- ------------------------------------- ,the resultant #ag ------------------------------------- ,to whom it shalle be addede #aj ------------------------------------- ,the nombre to be addede #g ------------------------------------- ,and yf it happe that a figure of 4#i4 by the figure that me mvst adde one to1 ,in the place of that #i4 put a cifre and write e article towarde e lift honde as bifore1 and thus3-- ------------------------------------- ,the resultant #aj ------------------------------------- ,to whom it shalle be addede #i ------------------------------------- ,the nombre to be addede #a ------------------------------------- ,and yf#c therefrom grow a nombre componed1#d in the place of the nombre put a-way#elet the digit be#fwrit at is part of at composide1 and an put to e lift side the article as before1 and us3-- ------------------------------------- ,the resultant #ab ------------------------------------- ,to whom it shalle be addede #h ------------------------------------- ,the nombre to be addede #d ------------------------------------- ,this done1 adde the seconde to the seconde1 and write above oer as before4 ,sidenote3 ,the translator's note4 ,note wele at in addicions and in alle spices folowyng1 whan he seithe one the other shalle be writen aboue1 and me most vse euer figure1 as that euery figure were sette by halfe1 and by hym-selfe4 ,headnote3 ,chapter ,,iii4 ,subtraction4 ,sidenote3 ,definition of ,subtraction4 ,how it may be done4 ,what is required4 ,write the greater number above4 ,subtract the first figure if possible4 ,if it is not possible #8borrow ten1' and then subtract4 ,subtraccioun is of 4#b4 proposede nombres1 the fyndyng of the excesse of the more to the lasse3 ,other subtraccioun is ablacioun of o nombre fro a-nother1 that me may see a some left4 ,the lasse of the more1 or even of even1 may be withdraw2 ,the more fro the lesse may neuer be4 ,and sothly that nombre is more that hathe more figures1 ,so that the last be signyficatifes3 ,and yf ther ben as many in that one as in that other1 me most deme it by the last1 other by the next last4 ,more-ouer in with-drawyng 4#b4 nombres ben necessary2 ,a nombre to be withdraw1 ,and a nombre that me shalle with-draw of4 ,the nombre to be with-draw shalle be writ in the lower ordre by his differences2 ,the nombre fro the whiche me shalle withe-draw in the omyst ordre1 so that the first be vnder the first1 the seconde vnder the seconde1 ,and so of alle others4 ,withe-draw therfor the first of the lowere ordre fro the first of the ordre above his hede1 and that wolle be other more or lesse1 oer egalle4 ------------------------------------- ,the remanent #bj ------------------------------------- ,wherof me shalle withdraw #bb ------------------------------------- ,the nombre to be withdraw #b ------------------------------------- yf it be egalle or even the figure sette beside1 put in his place a cifre4 ,and yf it be more put away erfro als many of vnitees the lower figure conteynethe1 and writ the residue as thus ---------------------------------------- ,the remanent #b #b ---------------------------------------- ,wherof me shalle with-draw #b #h ---------------------------------------- e nombre to be withdraw #f ---------------------------------------- 99,fol4 #ej4 ---------------------------------------- --------------- ,remanens #b #b #a #h #b #i #i #i #h ---------------------------------------- --------------- ,a quo sit subtraccio #h #g #b #d #c #j #j #j #d ---------------------------------------- --------------- ,numerus subtrahendus #f #e #gf 4 4 4 4 #f ---------------------------------------- --------------- ,and yf it be lesse1 by-cause the more may not be with-draw ther-fro1 borow an vnyte of the next figure that is worthe #aj4 ,of that 4#aj4 and of the figure that ye wolde have with-draw fro be-fore to-gedre ,ioynede1 with-draw e figure be-nethe1 and put the residue in the place of the figure put a-side as us3-- ---------------------------------------- ,the remanent #a #h ---------------------------------------- ,wherof me shalle with-draw #b #d ---------------------------------------- ,the nombre to be with-draw #j #f ---------------------------------------- ,sidenote3 ,if the second figure is one4 ,and yf the figure wherof me shal borow the vnyte be one1 put it a-side1 and write a cifre in the place erof1 lest the figures folowing faile of thaire nombre1 and an worche as it shewith in this figure here3-- ---------------------------------------- ---- ,the remanent #c #j #ih ---------------------------------------- ---- ,wherof me shal with-draw #c #a #b ---------------------------------------- ---- ,the nombre to be with-draw 4 4 #c ---------------------------------------- ---- ,sidenote3 ,if the second figure is a cipher4 ,and yf the vnyte wherof me shal borow be a cifre1 go ferther to the figure signyficatife1 and ther borow one1 and retournyng bake1 in the place of euery cifre at ye passide ouer1 sette figures of 4#i4 as here it is specifiede3-- ---------------------------------------- --------- ,the remenaunt #b #i #i #i #i ---------------------------------------- --------- ,wherof me shalle with-draw #c #j #j #j #c ---------------------------------------- --------- ,the nombre to be with-draw #d ---------------------------------------- --------- ,sidenote3 ,a justification of the rule given4 ,why it is better to work from right to left4 ,how to prove subtraction1 and addition4 ,and whan me comethe to the nombre wherof me intendithe1 there remaynethe alle-wayes 4#aj4 ffor e whiche 4#aj4 c4 ,the reson why at for euery cifre left behynde me setteth figures ther of 4#i4 this it is3--,if fro the 4#c4 place me borowede an vnyte1 that vnyte by respect of the figure that he came fro representith an 4,c41 ,in the place of that cifre passed over is left 4#i41 which is worth ninety1 and yit it remaynethe as 4#aj41 ,and the same resone wolde be yf me hade borowede an vnyte fro the 4#d41 4#e41 4#f41 place1 or ony other so vpwarde4 ,this done1 withdraw the seconde of the lower ordre fro the figure above his hede of e omyst ordre1 and wirche as before4 ,and note wele that in addicion or in subtraccioun me may wele fro the lift side begynne and ryn to the right side1 ,but it wol be more profitabler to be do1 as it is taught4 ,and yf thow wilt prove yf thow have do wele or no1 ,the figures that thow hast withdraw1 adde them ayene to the omyst figures1 and they wolle accorde with the first that thow haddest yf thow have labored wele2 and in like wise in addicioun1 whan thow hast addede alle thy figures1 withdraw them that thow first 99,fol4 #ej;b addest1 and the same wolle retourne4 ,the subtraccioun is none other but a prouffe of the addicioun1 and the contrarye in like wise4 ,headnote3 ,chapter ,,iv4 ,mediation4 ,sidenote3 ,definition of mediation4 ,where to begin4 ,if the first figure is unity4 ,what to do if it is not unity4 ,mediacioun is the fyndyng of the halfyng of euery nombre1 that it may be seyne what and how moche is euery halfe4 ,in halfyng ay oo order of figures and oo nombre is necessary1 that is to sey the nombre to be halfede4 ,therfor yf thow wilt half any nombre1 write that nombre by his differences1 and begynne at the right1 that is to sey1 fro the first figure to the right side1 so that it be signyficatife other represent vnyte or eny other digitalle nombre4 ,if it be vnyte write in his place a cifre for the figures folowyng1 lest they signify less1 and write that vnyte without in the table1 other resolue it in 4#fj4 mynvtes and sette a-side half of tho minutes so1 and reserve the remenaunt without in the table1 as thus 4#cj42 other sette without thus 4d3 that kepethe none ordre of place1 ,nathelesse it hathe signyficacioun4 ,and yf the other figure signyfie any other digital nombre fro vnyte forthe1 oer the nombre is ode or evene4 ,if it be even1 write this half in this wise3-- ----------------------- ,halfede #b #b ----------------------- to be halfede #d #d ----------------------- ,and if it be odde1 ,take the next even vndre hym conteynede1 and put his half in the place of that odde1 and of e vnyte that remaynethe to be halfede do thus3-- ----------------------- halfede #b #c di ----------------------- ,to be halfede #d #g ----------------------- ,sidenote3 ,then halve the second figure4 ,if it is odd1 add #e to the figure before4 ,this done1 the seconde is to be halfede1 yf it be a cifre put it be-side1 and yf it be significatife1 other it is even or ode3 ,if it be even1 write in the place of e nombres wipede out the halfe2 yf it be ode1 take the next even vnder it contenythe1 and in the place of the ,impar sette a-side put half of the even3 ,the vnyte that remaynethe to be halfede1 respect hade to them before1 is worthe 4#aj4 ,dyvide that 4#aj4 in 4#b41 #e4 is1 and sette a-side that one1 and adde that other to the next figure precedent as here3-- -------------------------- ,halfede -------------------------- to be halfede -------------------------- ,and yf e addicioun sholde be made to a cifre1 sette it a-side1 and write in his place 4#e4 ,and vnder this fourme me shalle write and worche1 tille the totalle nombre be halfede4 ---------------------------------------- ---- doublede #b #f #h #i #j #aj #ag #d ---------------------------------------- ---- to be doublede #a #c #d #d #e #e #h #g ---------------------------------------- ---- ,headnote3 ,chapter ,v4 ,duplation4 ,sidenote3 ,definition of ,duplation4 ,where to begin4 ,why4 ,what to do with the result4 ,duplicacioun is agregacion of nombre to itself at me may se the nombre growen4 ,in doublynge ay is but one ordre of figures necessarie4 ,and me most be-gynne with the lift side1 other of the more figure1 ,and after the nombre of the more figure representithe4 99,fol4 #ea4 ,in the other 4#c4 before we begynne alle way fro the right side and fro the lasse nombre1 ,in this spice and in alle other folowyng we wolle begynne fro the lift side1 ffor and me bigon the double fro the first1 omwhile me myght double oo thynge twyes4 ,and how be it that me myght double fro the right1 that wolde be harder in techyng and in workyng4 ,therfor yf thow wolt double any nombre1 write that nombre by his differences1 and double the last4 ,and of that doublyng other growithe a nombre digital1 article1 or componede4 ,if it be a digit1 write it in the place of the first digit4 ,if it be article1 write in his place a cifre and transferre the article towarde the lift1 as thus3-- ---------------------- double #aj ---------------------- to be doublede #e ---------------------- ,and yf the nombre be componede1 write a digital that is part of his composicioun1 and sette the article to the lift hande1 as thus3-- ---------------------- doublede #af ---------------------- to be doublede #h ---------------------- ,that done1 me most double the last save one1 and what growethe erof me most worche as before4 ,and yf a cifre be1 touche it not4 ,but yf any nombre shalle be addede to the cifre1 in e place of e figure wipede out me most write the nombre to be addede1 as thus3-- --------------------------- doublede #f #j #f --------------------------- to be doublede #c #j #c --------------------------- ,sidenote3 ,how to prove your answer4 ,in the same wise me shalle wirche of alle others4 ,and this probacioun3 ,if thow truly double the halfis1 and truly half the doubles1 the same nombre and figure shalle mete1 suche as thow labourede vpone first1 ,and of the contrarie4 --------------------------- ,doublede #f #a #h --------------------------- to be doublede #c #j #i --------------------------- ,headnote3 ,chapter ,,vi4 ,multiplication4 ,sidenote3 ,definition of ,multiplication4 ,multiplier4 ,multiplicand4 ,product4 ,multiplicacioun of nombre by hym-self other by a-nother1 with proposide 4#b4 nombres1 is the fyndyng of the thirde1 ,that so oft conteynethe that other1 as ther ben vnytes in the oer4 ,in multiplicacioun 4#b4 nombres pryncipally ben necessary1 that is to sey1 the nombre multiplying and the nombre to be multipliede1 as here2--twies fyve4 ,the number multiplying is designede aduerbially4 ,the nombre to be multipliede resceyvethe a nominalle appellacioun1 as twies 4#e4 #e4 is the nombre multipliede1 and twies is the nombre to be multipliede4 ---------------------------------------- --------------- ,resultans #i #a #j #a #c #b #f #f #h #j #j #h ---------------------------------------- --------------- ,multiplicandus 4 4 #e 4 4 #d 4 #c #d #j #j #d ---------------------------------------- --------------- ,multiplicans 4 #b #b 4 #c #c #b #b #b 4 4 4 ---------------------------------------- --------------- ,also me may thervpone to assigne the4 #c4 nombre1 the whiche is 99,fol4 #ea;b clepede product or provenient1 of takyng out of one fro another3 as twyes 4#e is 4#aj41 #e4 the nombre to be multipliede1 and 4#b4 the multipliant1 and4 #aj4 as before is come therof4 ,and vnderstonde wele1 that of the multipliant may be made the nombre to be multipliede1 and of the contrarie1 remaynyng euer the same some1 and herofe comethe the comen speche1 that seithe all nombre is convertede by ,multiplying in hym-selfe4 ---------------------------------------- ----- #a #b #c #d #e #f #g #h #i #aj ---------------------------------------- ----- #b #d #f #h #aj #ajaj #ad #af #ah #bj ---------------------------------------- ----- #c #f #i #ab #ae #ah #ba #bd #bg #cj ---------------------------------------- ----- #d #h #ab #af #bj #bd #bh #cb #cf #dj ---------------------------------------- ----- #e #aj #ae #bj #be #cj #ce #dj #de #ej ---------------------------------------- ----- #f #ab #ah #bd #cj #cf #db #dh #ef #fj ---------------------------------------- ----- #g #ad #ba #bh #ce #db #di #ef #fc #gj ---------------------------------------- ----- #h #af #bd #cb #dj #dh #ef #fd #gb #hj ---------------------------------------- ----- #i #ah #bg #cf #de #ed #fc #gb #ha #ij ---------------------------------------- ----- #aj #bj #cj #dj #ej #fj #gj #hj #ij #ajj ---------------------------------------- ----- ,headnote3 ,the ,cases of ,multiplication4 ,sidenote3 ,there are #f rules of ,multiplication4 7#a7 ,digit by digit4 ,see the table above4 7#b7 ,digit by article4 7#c7 ,composite by digit4 ,and ther ben 4#f rules of ,multiplicacioun2 ffirst1 yf a digit multiplie a digit1 considre how many of vnytees ben betwix the digit by multiplying and his 4#aj4 bethe to-gedre accomptede1 and so oft with-draw the digit multiplying1 vnder the article of his denominacioun4 ,example of grace4 ,if thow wolt wete how moche is 4#d4 tymes 4#h41 #aase how many vnytees ben betwix 4#h4ab and 4#aj4 to-geder rekenede1 and it shewith that 4#b43 withdraw ther-for the quaternary1 of the article of his denominacion twies1 of 4#dj41 ,and ther remaynethe 4#cb41 that is1 to some of alle the multiplicacioun4 ,wher-vpon for more evidence and declaracion the seide table is made4 ,whan a digit multipliethe an article1 thow most bryng the digit into e digit1 of e whiche the article has#ac his name1 and euery vnyte shalle stonde for 4#aj41 and euery article an 4#ajj4 ,whan the digit multipliethe a nombre componede1 ou most bryng the digit into aier part of the nombre componede1 so at digit be had into digit by the first rule1 into an article by e seconde rule2 and afterwarde ,ioyne the produccioun1 and ere wol be the some totalle4 ---------------------------------------- ------------ ,resultans #a #b #f #g #c #f #a #b #j #a #b #j #h ---------------------------------------- ------------ ,multiplicandus #b #c #b #f #d ---------------------------------------- ------------ ,multiplicans #f #c #b #c #b #j #c #j #b ---------------------------------------- ------------ ,sidenote3 7#d7 ,article by article4 7#e7 ,composite by article4 7#f7 ,composite by composite4 ,how to set down your numbers4 ,if the result is a digit1 an article1 or a composite4 ,multiply next by the last but one1 and so on4 ,whan an article multipliethe an article1 the digit wherof he is namede is to be brought ,into the digit wherof the oer is namede1 and euery vnyte wol be worthe 99,fol4 #eb4 an 4#ajj41 and euery article4 a 4#ajjj4 ,whan an article multipliethe a nombre componede1 thow most bryng the digit of the article into aither part of the nombre componede2 and ,ioyne the produccioun1 and euery article wol be worthe 4#ajj41 and euery vnyte 4#aj41 and so wolle the some be opene4 ,whan a nombre componede multipliethe a nombre componede1 euery part of the nombre multiplying is to be hade into euery part of the nombre to be multipliede1 and so shalle the digit be hade twies1 onys in the digit1 that other in the article4 ,the article also twies1 ones in the digit1 that other in the article4 ,therfor yf thow wilt any nombre by hym-self other by any other multiplie1 write the nombre to be multipliede in the ouer ordre by his differences1 ,the nombre multiplying in the lower ordre by his differences1 so that the first of the lower ordre be vnder the last of the ouer ordre4 ,this done1 of the multiplying1 the last is to be hade into the last of the nombre to be multipliede4 ,wherof than wolle grow a digit1 an article1 other a nombre componede4 ,if it be a digit1 even above the figure multiplying is hede write his digit that come of1 as it apperethe here3-- -------------------------- ,the resultant #f -------------------------- ,to be multipliede #c -------------------------- e nombre multipliyng #b -------------------------- ,and yf an article had be writ ouer the figure multiplying his hede1 put a cifre er and transferre the article towarde the lift hande1 as thus3-- ------------------------------- ,the resultant #a #j ------------------------------- to be multipliede #e ------------------------------- e nombre multipliyng #b ------------------------------- ,and yf a nombre componede be writ ouer the figure multyplying is hede1 write the digit in the nombre componede is place1 and sette the article to the lift hande1 as thus3-- ------------------------------ ,resultant #a #b ------------------------------ to be multipliede #d ------------------------------ the nombre multipliyng #c ------------------------------ ,this done1 me most bryng the last save one of the multipliyng into the last of e nombre to be multipliede1 and se what comythe therof as before1 and so do with alle1 tille me come to the first of the nombre multiplying1 that must be brought into the last of the nombre to be multipliede1 wherof growithe oer a digit1 an article1 99,fol4 #eb;b other a nombre componede4 ,if it be a digit1 ,in the place of the ouerer1 sette a-side1 as here3 -------------------------------- ,resultant #f #f -------------------------------- to be multipliede #c -------------------------------- the nombre multipliyng #b #b -------------------------------- ,if an article happe1 there put a cifre in his place1 and put hym to the lift hande1 as here3 ---------------------------------- ,the resultant #a #a #j ---------------------------------- to be multipliede #e ---------------------------------- e nombre multiplying #b #b ---------------------------------- ,if it be a nombre componede1 in the place of the ouerer sette a-side1 write a digit that#ad is a part of the componede1 and sette on the left honde the article1 as here3 ---------------------------------------- -- ,the resultant #a #cae #b ---------------------------------------- -- to be multipliede #d ---------------------------------------- -- e nombre multipliant #c #c ---------------------------------------- -- ,sidenote3 ,then antery the multiplier one place4 ,work as before4 ,how to deal with ciphers4 ,that done1 sette forwarde the figures of the nombre multiplying by oo difference1 so that the first of the multipliant be vnder the last save one of the nombre to be multipliede1 the other by o place sette forwarde4 ,than me shalle brynge the last of the multipliant in hym to be multipliede1 vnder the whiche is the first multipliant4 ,and than wolle growe oer a digit1 an article1 or a componede nombre4 ,if it be a digit1 adde hym even above his hede2 ,if it be an article1 transferre hym to the lift side2 ,and if it be a nombre componede1 adde a digit to the figure above his hede1 and sette to the lift hande the article4 ,and alle-wayes euery figure of the nombre multipliant is to be brought to the last save one nombre to be multipliede1 til me come to the first of the multipliant1 where me shalle wirche as it is seide before of the first1 and afterwarde to put forwarde the figures by o difference and one tille they alle be multipliede4 ,and yf it happe that the first figure of e multipliant be a cifre1 and boue it is sette the figure signyficatife1 write a cifre in the place of the figure sette a-side1 as thus1 etc43 ------------------------------ ,the resultant #a #b #j ------------------------------ to be multipliede #f ------------------------------ the multipliant #b #j ------------------------------ ,sidenote3 ,how to deal with ciphers4 ,and yf a cifre happe in the lower order be-twix the first and the last1 and even above be sette the figure signyficatif1 leve it vntouchede1 as here3-- ------------------------------------ ,the resultant #b #b #f #d #d ------------------------------------ ,to be multipliede #b #b #b ------------------------------------ ,the multipliant #a #j #b ------------------------------------ ,and yf the space above sette be voide1 in that place write thow a cifre4 ,and yf the cifre happe betwix e first and the last to be multipliede1 me most sette forwarde the ordre of the figures by thaire differences1 for oft of duccioun of figures in cifres nought is the resultant1 as here1 -------------------------------------- ,resultant #h #j #j #h -------------------------------------- to be multipliede #d #j #j #d -------------------------------------- the multipliant #b 4 4 4 -------------------------------------- 99,fol4 #ec4 wherof it is evident and open1 yf that the first figure of the nombre be to be multipliede be a cifre1 vndir it shalle be none sette as here3-- ------------------------------------- ,resultant #c #b #jaf ------------------------------------- ,to be multipliede #h #j ------------------------------------- ,the multipliant #d ------------------------------------- ,sidenote3 ,leave room between the rows of figures4 ,vnderstand also that in multiplicacioun1 divisioun1 and of rootis the extraccioun1 competently me may leve a mydel space betwix 4#b4 ordres of figures1 that me may write there what is come of addyng other withe-drawyng1 lest any thynge sholde be ouer-hippede and sette out of mynde4 ,headnote3 ,chapter ,,vii4 ,division4 ,sidenote3 ,definition of division4 ,dividend1 ,divisor1 ,quotient4 ,how to set down your ,sum4 ,an example4 ,examples4 ,for to dyvyde oo nombre by a-nother1 it is of 4#b4 nombres proposede1 ,it is forto depart the moder nombre into as many partis as ben of vnytees in the lasse nombre4 ,and note wele that in makynge of dyvysioun ther ben 4#c4 nombres necessary3 that is to sey1 the nombre to be dyvydede2 the nombre dyvydyng and the nombre exeant1 other how oft1 or quocient4 ,ay shalle the nombre that is to be dyvydede be more1 other at the lest evene with the nombre the dyvysere1 yf the nombre shalle be made by hole nombres4 ,therfor yf thow wolt any nombre dyvyde1 write the nombre to be dyvydede in e ouerer bordure by his differences1 the dyvisere in the lower ordure by his differences1 so that the last of the dyviser be vnder the last of the nombre to be dyvyde1 the next last vnder the next last1 and so of the others1 yf it may competently be done2 as here3-- --------------------------- ,the residue #b #g --------------------------- ,the quotient #e --------------------------- ,to be dyvydede #c #d #b --------------------------- ,the dyvyser #f #c --------------------------- ---------------------------------------- -------- ,residuum #h #b #g #b #f ---------------------------------------- ------- ,quociens #b #a #b #b #e #i ---------------------------------------- ------- ,diuidendus #f #h #j #f #f #c #d #b #c #c #b ---------------------------------------- ------- ,diuiser #c #b #c #f #c #c #d ---------------------------------------- ------- ,sidenote3 ,when the last of the divisor must not be set below the last of the dividend4 ,how to begin4 ,and ther ben 4#b4 causes whan the last figure may not be sette vnder the last1 other that the last of the lower nombre may not be with-draw of the last of the ouerer nombre for it is lasse than the lower1 other how be it1 that it myght be with-draw as for hym-self fro the ouerer the remenaunt may not so oft of them above1 other yf e last of the lower be even to the figure above his hede1 and e next last oer the figure be-fore at be more an the figure above sette4 99,fol4 #ecb4 ,these so ordeynede1 me most wirche from the last figure of e nombre of the dyvyser1 and se how oft it may be with-draw of and fro the figure aboue his hede1 namly so that the remenaunt may be take of so oft1 and to se the residue as here3-- ,sidenote3 ,an example4 --------------------------- ,the residue #b #f --------------------------- ,the quocient #i --------------------------- ,to be dyvydede #c #c #b --------------------------- ,the dyvyser #c #d --------------------------- ,sidenote3 ,where to set the quotiente4 ,examples4 ,and note wele that me may not withe-draw more than 4#i4 tymes nether lasse than ones4 ,therfor se how oft e figures of the lower ordre may be with-draw fro the figures of the ouerer1 and the nombre that shewith e quocient most be writ ouer the hede of at figure1 vnder the whiche the first figure is1 of the dyviser2 ,and by that figure me most withe-draw alle oer figures of the lower ordir and that of the figures aboue thaire hedis4 ,this so done1 me most sette forwarde e figures of the diuiser by o difference towardes the right honde and worche as before2 and thus3-- ---------------------------------------- ------------- ,residuum 4 #a #b ---------------------------------------- ------------- quociens #f #e #d #b #j #j #d ---------------------------------------- ------------- ,diuidendus #c #e #e #a #b #b #h #h #f #c #g #j #d ---------------------------------------- ------------- ,diuisor #e #d #c #d #d #b #c ---------------------------------------- ------------- ------------------------------------ ,the quocient #f #e #d ------------------------------------ ,to be dyvydede #c #e #e #a #b #b ------------------------------------ ,the dyvyser #e #d #c ------------------------------------ ,sidenote3 ,a special case4 ,and yf it happe after e settyng forwarde of the figures at e last of the divisor may not so oft be withdraw of the figure above his hede1 above at figure vnder the whiche the first of the diuiser is writ me most sette a cifre in ordre of the nombre quocient1 and sette the figures forwarde as be-fore be o difference alone1 and so me shalle do in alle nombres to be dyvidede1 for where the dyviser may not be with-draw me most sette there a cifre1 and sette forwarde the figures2 as here3-- --------------------------------------- ,the residue #a #b --------------------------------------- ,the quocient #b #j #j #d --------------------------------------- ,to be dyvydede #h #h #f #c #g #j #d --------------------------------------- ,the dyvyser #d #d #b #c --------------------------------------- ,sidenote3 ,another example4 ,what the quotient shows4 ,how to prove your division1 or multiplication4 ,and me shalle not cesse fro suche settyng of figures forwarde1 nether of settynge of e quocient into the dyviser1 neer of subtraccioun of the dyvyser1 tille the first of the dyvyser be with-draw fro e first to be dividede4 ,the whiche done1 or ought1#ag oer nought shalle remayne3 and yf it be ought1#ag kepe it in the tables1 ,and euer vny it to e diviser4 ,and yf ou wilt wete how many vnytees of e divisioun 99,fol4 #ecc4 wol growe to the nombre of the divisere1 the nombre quocient wol shewe it3 and whan suche divisioun is made1 and ou lust prove yf thow have wele done or no1 ,multiplie the quocient by the diviser1 ,and the same figures wolle come ayene that thow haddest bifore and none other4 ,and yf ought be residue1 than with addicioun therof shalle come the same figures3 ,and so multiplicacioun provithe divisioun1 and dyvisioun multiplicacioun3 as thus1 yf multiplicacioun be made1 divide it by the multipliant1 and the nombre quocient wol shewe the nombre that was to be multipliede1 etc4 ,headnote3 ,chapter ,,viii4 ,progression4 ,sidenote3 ,definition of ,progression4 ,natural ,progression4 ,broken ,progression4 ,the #ast rule for ,natural ,progression4 ,the second rule4 ,the first rule of ,broken ,progression4 ,the second rule4 ,progressioun is of nombre after egalle excesse fro oone or tweyne take agregacioun4 of progressioun one is naturelle or contynuelle1 at oer broken and discontynuelle4 ,naturelle it is1 whan me begynnethe with one1 and kepethe ordure ouerlepyng one2 as 4#a4 #b4 #c4 #d4 #e4 #f41 etc41 so at the nombre folowynge passithe the other be-fore in one4 ,broken it is1 whan me lepithe fro o nombre tille another1 and kepithe not the contynuel ordire2 as #a4 #c4 #e4 #g4 #i1 etc4 ,ay me may begynne with 4#b41 as us2 4#b4 #d4 #f4 #h41 etc41 and the nombre folowyng passethe the others by-fore by 4#b4 ,and note wele1 that naturelle progressioun ay begynnethe with one1 and ,intercise or broken progressioun1 omwhile begynnythe with one1 omwhile with twayne4 ,of progressioun naturell 4#b4 rules ther be yove1 of the whiche the first is this2 whan the progressioun naturelle endithe in even nombre1 by the half therof multiplie e next totalle ouerere nombre2 ,example of grace3 4#a4 #b4 #c4 #d4 ,multiplie 4#e4 by 4#b4 and so 4#aj4 comethe of1 that is the totalle nombre erof4 ,the seconde rule is suche1 whan the progressioun naturelle endithe in nombre ode4 ,take the more porcioun of the oddes1 and multiplie therby the totalle nombre4 ,example of grace #a4 #b4 #c4 #d4 #e41 multiplie 4#e4 by 4#c1 and thryes 4#e4 shalle be resultant4 so the nombre totalle is 4#ae4 ,of progresioun intercise1 ther ben also 4#b4ah rules2 and e first is is3 ,whan the ,intercise progression endithe in even nombre by half therof multiplie the next nombre to at halfe as 4#b4ah #d4 #f4 ,multiplie 4#d4 by 4#c4 so at is thryes 4#d41 and 4#ab4 the nombre of alle the progressioun1 wolle folow4 ,the seconde rule is this3 whan the progressioun interscise endithe in ode1 take e more porcioun of alle e nombre1 99,fol4 #ecd4 and multiplie by hym-selfe2 as 4#a4 #c4 #e4 ,multiplie 4#c4 by hym-selfe1 and e some of alle wolle be 4#i41 etc4 ,headnote3 ,chapter ,,ix4 ,extraction of ,roots4 ,sidenote3 ,the preamble of the extraction of roots4 ,linear1 superficial1 and solid numbers4 ,superficial numbers4 ,square numbers4 ,the root of a square number4 ,notes of some examples of square roots here interpolated4 ,solid numbers4 ,three dimensions of solids4 ,cubic numbers4 ,all cubics are solid numbers4 ,no number may be both linear and solid4 ,unity is not a number4 ,here folowithe the extraccioun of rotis1 and first in nombre quadrates4 ,wherfor me shalle se what is a nombre quadrat1 and what is the rote of a nombre quadrat1 and what it is to draw out the rote of a nombre4 ,and before other note this divisioun3 ,of nombres one is lyneal1 anoer superficialle1 anoer quadrat1 anoer cubike or hoole4 lyneal is that at is considrede after the processe1 havynge no respect to the direccioun of nombre in nombre1 ,as a lyne hathe but one dymensioun that is to sey after the lengthe4 ,nombre superficial is at comethe of ledynge of oo nombre into a-nother1 wherfor it is callede superficial1 for it hathe 4#b4 nombres notyng or mesurynge hym1 as a superficialle thynge hathe 4#b4 dimensions1 at is to sey lengthe and brede4 ,and for bycause a nombre may be hade in a-nother by 4#b4 maners1 at is to sey other in hym-selfe1 oer in anoer1 ,vnderstonde yf it be had in hym-self1 ,it is a quadrat4 ffor dyvisioun write by vnytes1 hathe 4#d4 sides even as a quadrangille4 and yf the nombre be hade in a-noer1 the nombre is superficiel and not quadrat1 as 4#b4 hade in 4#c4 makethe 4#f4 that is e first nombre superficielle2 wherfor it is open at alle nombre quadrat is superficiel1 and not conuertide4 ,the rote of a nombre quadrat is at nombre that is had of hym-self1 as twies 4#b4 makithe #d4 and 4#d4 is the first nombre quadrat1 and #b4 is his rote4 #i4 #h4 #g4 #f4 #e4 #d4 #c4 #b4 #a4 _/ ,the rote of the more quadrat 4#c4 #a4 #d4 #b4 #f4 ,the most nombre quadrat #i4 #h4 #g4 #e4 #i4 #c4 #d4 #g4 #f4 _/ the remenent ouer the quadrat 4#f4 #j4 #h4 #d4 #e4 _/ ,the first caas of nombre quadrat 4#e4 #d4 #g4 #e4 #f4 ,the rote 4#b4 #c4 #d4 ,the seconde caas 4#c4 #h4 #d4 #e4 ,the rote 4#f4 #b4 ,the thirde caas 4#b4 #h4 #a4 #i4 ,the rote 4#e4 #c4 ,the 4#d4 caas 4#c4 #b4 #a4 ,the rote 4#a4 #g4 _/ ,the #e4 caas 4#i4 #a4 #b4 #j4 #d4 _/ ,the rote #c4 #j4 #b4 ,the solide nombre or cubike is at at comyte of double ledyng of nombre in nombre2 ,and it is clepede a solide body that hathe er-in 4#c dimensions at is to sey1 lengthe1 brede1 and thiknesse4 so at nombre hathe 4#c4 nombres to be brought forthe in hym4 ,but nombre may be hade twies in nombre1 for other it is hade in hym-selfe1 oer in a-noer4 ,if a nombre be hade twies in hym-self1 oer ones in his quadrat1 at is the same1 at a cubike 99,fol4 #ed4 is1 ,and is the same that is solide4 ,and yf a nombre twies be hade in a-noer1 the nombre is clepede solide and not cubike1 as twies 4#c4 and at 4#b4 makithe 4#ab4 ,wherfor it is opyne that alle cubike nombre is solide1 and not conuertide4 ,cubike is at nombre at comythe of ledynge of hym-selfe twyes1 or ones in his quadrat4 ,and here-by it is open that o nombre is the roote of a quadrat and of a cubike4 ,natheles the same nombre is not quadrat and cubike4 ,opyne it is also that alle nombres may be a rote to a quadrat and cubike1 but not alle nombre quadrat or cubike4 ,therfor sithen e ledynge of vnyte in hym-self ones or twies nought comethe but vnytes1 ,seithe ,boice in ,arsemetrike1 that vnyte potencially is al nombre1 and none in act4 ,and vndirstonde wele also that betwix euery 4#b4 quadrates ther is a meene proporcionalle1 ,that is openede thus2 lede the rote of o quadrat into the rote of the oer quadrat1 and an wolle e meene shew4 ,sidenote3 ,examples of square roots4 ------------------------------------- ,residuum #j #d #j #j ------------------------------------- ,quadrande #dcefcjbiagdbda #i #c #f ------------------------------------- ,duplum #ab #aj #b #f #hai ------------------------------------- ,subduplum #f #f #e #ea #c #b #d #d ------------------------------------- ,sidenote3 ,a note on mean proportionals4 ,also betwix the next 4#b4 cubikis1 me may fynde a double meene1 that is to sey a more meene and a lesse4 ,the more meene thus1 as to brynge the rote of the lesse into a quadrat of the more4 ,the lesse thus1 ,if the rote of the more be brought ,into the quadrat of the lesse4 ,headnote3 ,chapter ,x4 ,extraction of ,square ,root4 ,sidenote3 ,to find a square root4 ,begin with the last odd place4 ,find the nearest square root of that number1 subtract1 double it1 and set the double one to the right4 ,find the second figure by division4 ,multiply the double by the second figure1 and add after it the square of the second figure1 and subtract4 #bj,to draw a rote of the nombre quadrat it is ,what-euer nombre be proposede to fynde his rote and to se yf it be quadrat4 ,and yf it be not quadrat the rote of the most quadrat fynde out1 vnder the nombre proposede4 ,therfor yf thow wilt the rote of any quadrat nombre draw out1 write the nombre by his differences1 and compt the nombre of the figures1 and wete yf it be ode or even4 ,and yf it be even1 than most thow begynne worche vnder the last save one4 ,and yf it be ode with the last2 and forto sey it shortly1 al-weyes fro the last ode me shalle begynne4 ,therfor vnder the last in an od place sette1 me most fynde a digit1 the whiche lade in hym-selfe it puttithe away that1 at is ouer his hede1 oer as neighe as me may3 suche a digit founde and withdraw fro his ouerer1 me most double that digit and sette the double vnder the next figure towarde the right honde1 and his vnder double vnder hym4 ,that done1 than me most fynde a-noer digit vnder the next figure bifore the doublede1 the whiche 99,fol4 #ed;b brought in double settethe a-way alle that is ouer his hede as to rewarde of the doublede3 ,than brought into hym-self settithe all away in respect of hym-self1 ,other do it as nye as it may be do3 other me may with-draw the digit #balast founde1 and lede hym in double or double hym1 and after in hym-selfe2 ,than ,ioyne to-geder the produccione of them bothe1 ,so that the first figure of the last product be addede before the first of the first productes1 the seconde of the first1 etc4 and so forthe1 subtrahe fro the totalle nombre in respect of e digit4 ,sidenote3 ,examples4 ---------------------------------------- - ,the residue #e #d #cb ---------------------------------------- - ,to be quadrede #dabjiaeaci #i #j #j #e #d #cb ---------------------------------------- - ,the double #dj #b #d #f #j #j ---------------------------------------- - ,the vnder double #b #j #ca #b #cc #j #j #j ---------------------------------------- - ,sidenote3 ,special cases4 ,the residue4 ,and if it hap at no digit may be founde1 ,than sette a cifre vndre a cifre1 and cesse not tille thow fynde a digit2 and whan thow hast founde it to double it1 neer to sette the doublede forwarde nether the vnder doublede1 ,till thow fynde vndre the first figure a digit1 the whiche lade in alle double1 settyng away alle that is ouer hym in respect of the doublede3 ,than lede hym into hym-selfe1 and put a-way alle in regarde of hym1 other as nyghe as thow maist4 ,that done1 other ought or nought wolle be the residue4 ,if nought1 than it shewithe that a nombre componede was the quadrat1 and his rote a digit last founde with vndere-double other vndirdoubles1 so that it be sette be-fore3 ,and yf ought#bb remayne1 that shewith that the nombre proposede was not quadrat1#bc wher-vpone se the table in the next side of the next leefe4 but a digit last found with the subduple or subduples is ,sidenote3 ,this table is constructed for use in cube root sums1 giving the value of ab4#b ---------------------------------------- ----------- #a #b #c #d #e #f #g #h #i ---------------------------------------- ----------- #b #h #ab #af #bj #bd #bh #cb #cf ---------------------------------------- ----------- #c #ah #bg #cf #de #ed #fc #gb #ha ---------------------------------------- ----------- #d #cb #dh #fd #hj #if #aabbd #abh #add ---------------------------------------- ----------- #e #ej #ge #ajj #abe #aej #age #bjj #bbe ---------------------------------------- ----------- #f #gb #ajh #add #ahj #baf #beb #bhh #cbd ---------------------------------------- ----------- #g #ih #adg #aif #bde #bid #cdc #cic #dda ---------------------------------------- ----------- #h #abh #aib #bef #cbj #chd #ddh #eab #egf ---------------------------------------- ----------- #i #afh #bdc #cbd #dje #dhf #efg #fdh #gbibe ---------------------------------------- ----------- ,sidenote3 ,how to prove the square root without or with a remainder4 ,the rote of the most quadrat conteynede vndre the nombre proposede4 ,therfor yf thow wilt prove yf thow have wele do or no1 ,multiplie the digit last founde with the vnder-double oer vnder-doublis1 and thow shalt fynde the same figures that thow haddest before2 ,and so that nought be the 99,fol4 #ee4 residue4 ,and yf thow have any residue1 than with the addicioun erof that is reseruede with-out in thy table1 thow shalt fynde thi first figures as thow haddest them before1 etc4 ,headnote3 ,chapter ,,xi4 ,extraction of ,cube ,root4 ,sidenote3 ,definition of a cubic number and a cube root4 ,mark off the places in threes4 ,find the first digit2 treble it and place it under the next but one1 and multiply by the digit4 ,then find the second digit4 ,multiply the first triplate and the second digit1 twice by this digit4 ,subtract4 ,examples4 ,heere folowithe the extraccioun of rotis in cubike nombres2 wher-for me most se what is a nombre cubike1 and what is his roote1 ,and what is the extraccioun of a rote4 ,a nombre cubike it is1 as it is before declarede1 that comethe of ledyng of any nombre twies in hym-selfe1 other ones in his quadrat4 ,the rote of a nombre cubike is the nombre that is twies hade in hym-selfe1 or ones in his quadrat4 ,wher-thurghe it is open1 that euery nombre quadrat or cubike have the same rote1 as it is seide before4 ,and forto draw out the rote of a cubike1 ,it is first to fynde e nombre proposede yf it be a cubike2 ,and yf it be not1 than thow most make extraccioun of his rote of the most cubike vndre the nombre proposide his rote founde4 ,therfor proposede some nombre1 whos cubical rote ou woldest draw out2 ,first thow most compt the figures by fourthes1 that is to sey in the place of thousandes2 ,and vnder the last thousande place1 thow most fynde a digit1 the whiche lade in hym-self cubikly puttithe a-way that at is ouer his hede as in respect of hym1 other as nyghe as thow maist4 ,that done1 thow most trebille the digit1 and that triplat is to be put vnder the 4#c4 next figure towarde the right honde1 ,and the vnder-trebille vnder the trebille2 ,than me most fynde a digit vndre the next figure bifore the triplat1 the whiche with his vnder-trebille had into a trebille1 afterwarde other vndertrebille#bf had in his produccioun1 puttethe a-way alle that is ouer it in regarde of#bg the triplat4 ,then lade in hymself puttithe away that at is over his hede as in respect of hym1 other as nyghe as thou maist3 ,that done1 thow most trebille the digit ayene1 and the triplat is to be sette vnder the next 4#c4 figure as before1 ,and the vnder-trebille vnder the trebille3 and than most thow sette forwarde the first triplat with his vndre-trebille by 4#b4 differences4 ,and than most thow fynde a digit vnder the next figure before the triplat1 the whiche withe his vnder-triplat had in his triplat afterwarde1 other vnder-treblis lad in product 99,fol4 #ee;b ,it sittethe a-way a that is ouer his hede in respect of the triplat than had in hym-self cubikly1#bh it settethe a-way alle his respect or as nyghe as ye may4 ---------------------------------------- ---- ,residuum #e #d #aja #i ---------------------------------------- ---- ,cubicandus #hcfedc #b #cjjgf #g #a #aff #g ---------------------------------------- ---- ,triplum #fj #ah #d ---------------------------------------- ----- ,subtriplum #b #j #c #f #g #b #b ---------------------------------------- ---- ,sidenote3 ,continue this process till the first figure is reached4 ,examples4 ,the residue4 ,special cases4 ,special case4 ,nother me shalle not cesse of the fyndynge of that digit1 neither of his triplacioun1 neer of the triplat-is #bi;anterioracioun1 that is to sey1 settyng forwarde by 4#b4 differences1 ,ne therof the vndre-triple to be put vndre the triple1 ,nether of the multiplicacioun erof1 ,neither of the subtraccioun1 tille it come to the first figure1 vnder the whiche is a digitalle nombre to be founde1 the whiche withe his vndre-treblis most be hade in tribles1 ,after-warde without vnder-treblis to be hade into produccioun1 settyng away alle that is ouer the hede of the triplat nombre1 ,after had into hymselfe cubikly1 and sette alle-way that is ouer hym4 ---------------------------------------- ----- ,to be cubicede #a #g #b #h #c #b #g #f #h ---------------------------------------- ----- ,the triple #c #b #i ---------------------------------------- ----- ,the vnder triple #a #b #c #c #c ---------------------------------------- ----- ,also note wele that the produccion comynge of the ledyng of a digite founde#cj with an vndre-triple _/ other of an vndre-triple in a triple or triplat is ,and after-warde with out vndre-triple other vndre-triplis in the product and ayene that product that comethe of the ledynge of a digit founde in hym-selfe cubicalle me may adde to1 and also with-draw fro of the totalle nombre sette above that digit so founde4#ca as ther had be a divisioun made as it is openede before ,that done ought or nought most be the residue4 ,if it be nought1 ,it is open that the nombre proposede was a cubike nombre1 ,and his rote a digit founde last with the vnder-triples3 ,if the rote therof wex bade in hym-selfe1 and afterwarde product they shalle make the first figures4 ,and yf ought be in residue1 kepe that without in the table2 and it is opene that the nombre was not a cubike4 but a digit last founde with the vndirtriplis is rote of the most cubike vndre the nombre proposede conteynede1 the whiche rote yf it be hade in hym-selfe1 ,and afterwarde in a product of that shalle growe the most cubike vndre the nombre proposede conteynede1 ,and yf that be addede to a cubike the residue reseruede in the table1 wolle make the same figures that ye hade first4 99,fol4 #ef4 ,and yf no digit after the anterioracioun#cb may not be founde1 than put there a cifre vndre a cifre vndir the thirde figure1 ,and put forwarde e figures4 ,note also wele that yf in the nombre proposede ther ben no place of thowsandes1 me most begynne vnder the first figure in the extraccioun of the rote4 some vsen forto distingue the nombre by threes1 and ay begynne forto wirche vndre the first of the last ternary other uncomplete nombre1 the whiche maner of operacioun accordethe with that before4 ,and this at this tyme suffisethe in extraccioun of nombres quadrat or cubikes etc4 ,sidenote3 ,examples4 ---------------------------------------- ------------ ,the residue #j #a #a ---------------------------------------- ------------ ,the cubicandus #h #j #j #j #j #j #j #h #b #d #b #d #a #i ---------------------------------------- ------------ ,the triple #cc #j #j #f ---------------------------------------- ------------ ,the vndertriple #b #j #j #b #f #b ---------------------------------------- ------------ ,headnote3 ,table of ,numbers1 c4 ,sidenote3 ,a table of numbers2 probably from the ,abacus4 #a #b #c #d #e #f one4 x4 an4 hundrede_/ a thowsande_/ x4 thowsande_/ ,an hundrede #g thowsande_/ ,a thowsande tymes a thowsande_/ x4 thousande tymes a thousande_/ ,an hundrede thousande tymes a thousande ,a thousande thousande tymes a thousande_/ this is the x place etc4 ,ende4 ,,footnotes 7,the ,art of ,nombryng73 #a3 ,,ms4 ,materialle4 #b3 ,,ms4 ,formalle4 #c3 #8the' in ,,ms4 #d3 #8be' in ,,ms4 #e3 #8and' in ,,ms4 #f3 #8is' in ,,ms4 #g3 #f in ,,ms4 #h3 #j in ,,ms4 #i3 #b in ,,ms4 #aj3 sic4 #aa3 #8,and' inserted in ,,ms4 #ab3 #8#d the' inserted in ,,ms4 #ac3 #8to' in ,,ms4 #ad3 #8that' repeated in ,,ms4 #ae3 #8#a' in ,,ms4 #af3 ,blank in ,,ms4 #ag3 #8nought' in ,,ms4 #ah3 #c written for #b in ,,ms4 #ai3 #g in ,,ms4 #bj3 runs on in ,,ms4 #ba3 #8so' in ,,ms4 #bb3 #8nought' in ,,ms4 #bc3 ,,ms4 adds here3 #8wher-vpone se the table in the next side of the next leefe4' #bd3 #aaj in ,,ms4 #be3 #j in ,,ms4 #bf3 double in ,,ms4 #bg3 #8it hym-selfe' in ,,ms4 #bh3 ,,ms4 adds here3 #8it settethe a-way alle his respect4' #bi3 #8aucterioracioun' in ,,ms4 #cj3 ,,ms4 adds here3 'with an vndre-triple _/ other of an vndre-triple in a triple or triplat is ,and after-warde with out vndre-triple other vndre-triplis in the product and ayene that product that comethe of the ledynge of a digit founde in hym-selfe cubicalle' _/ #ca3 ,,ms4 adds here3 #8as ther had be a divisioun made as it is openede before4' #cb3 ,,ms4 anteriocacioun4 #cc3 #d in ,,ms4 ,accomptynge by counters4 ,transcriber's ,note3 ,the original text was printed as a single continuous paragraph1 with no break between speakers2 all examples were shown inline4 ,it has been broken up for this e-text4 99#aaf;b ,the seconde dialoge of accomptynge by counters4 ,mayster4 ,nowe that you haue learned the commen kyndes of ,arithmetyke with the penne1 you shall se the same art in counters3 whiche feate doth not only serue for them that can not write and rede1 but also for them that can do bothe1 but haue not at some tymes theyr penne or tables redye with them4 ,this sorte is in two fourmes commenly4 ,the one by lynes1 and the other without lynes3 in that yt hath lynes1 the lynes do stande for the order of places3 and in yt that hath no lynes1 there must be sette in theyr stede so many counters as shall nede1 for eche lyne one1 and they shall supplye the stede of the lynes4 ,s4 ,by examples ,i shuld better perceaue your meanynge4 ,m4 ,for example of the 99#aag;a4 ly99nes3 ----#a-j-j-j-j-j-- ----#a-j-j-j-j---- -,x--#a-j-j-j------ ----#a-j-j-------- ----#a-j---------- ----#a------------ ,sidenote3 ,numeration4 ,lo here you se 4vi4 lynes whiche stande for syxe places so that the nethermost standeth for ye fyrst place1 and the next aboue it1 for the second3 and so vpward tyll you come to the hyghest1 which is the syxte lyne1 and standeth for the syxte place4 ,now what is the valewe of euery place or lyne1 you may perceaue by the figures whiche ,i haue set on them1 which is accordynge as you learned before in the ,numeration of figures by the penne3 for the fyrste place is the place of vnities or ones1 and euery counter set in that lyne betokeneth but one3 and the seconde lyne is the place of #aj1 for euery counter there1 standeth for #aj4 ,the thyrd lyne the place of hundredes3 the fourth of thousandes3 and so forth4 ,s4 ,syr ,i do perceaue that the same order is here of lynes1 as was in the other figures 99#aag;b by places1 so that you shall not nede longer to stande about ,numeration1 excepte there be any other difference4 ,m4 ,yf you do vnderstande it1 then how wyll you set #aedc8 ,s4 ,thus1 as ,i suppose4 ------- -,x--#a-- ----#e-- ----#d-- ----#c-- ,m4 ,you haue set ye places truely1 but your figures be not mete for this vse3 for the metest figure in this behalfe1 is the figure of a counter round1 as you se here1 where ,i haue expressed that same summe4 ------------- -,x--o-------- o ------------- ----o-o-o-o-- ----o-o-o---- ,s4 ,so that you haue not one figure for #b1 nor #c1 nor #d1 and so forth1 but as many digettes as you haue1 you set in the lowest lyne3 and for euery #aj you set one in the second line3 and so of other4 ,but ,i know not by what reason you set that one counter for #ejj betwene two lynes4 ,m4 you shall remember this1 that when so euer you nede to set downe #e1 #ej1 or #ejj1 or #ejjj1 or so forth any other nomber1 whose numerator 99#aah;a is #e1 you shall set one counter for it1 in the next space aboue the lyne that it hath his denomination of1 as in this example of that #ejj1 bycause the numerator is #e1 it must be set in a voyd space3 and bycause the denominator is hundred1 ,i knowe that his place is the voyde space next aboue hundredes1 that is to say1 aboue the thyrd lyne4 ,and farther you shall marke1 that in all workynge by this sorte1 yf you shall sette downe any summe betwene #d and #aj1 for the fyrste parte of that nomber you shall set downe #e1 then so many counters more1 as there reste nombers aboue #e4 ,and this is true bothe of digettes and articles4 ,and for example ,i wyll set downe this summe #bhgife1 -,x----------- ------o-o---- o ------o-o-o-- o -,x----o-o---- o ----o-o-o-o-- o ----o-------- o ------------- which summe yf you marke well1 you nede none other examples for to lerne the numeration of 99#aah;b this forme4 ,but this shal you marke1 that as you dyd in the other kynde of arithmetike1 set a pricke in the places of thousandes1 in this worke you shall sette a starre1 as you se here4 ,headnote3 ,addition on the ,counting ,board4 ,sidenote3 ,addition4 ,s4 ,then ,i perceave numeration1 but ,i praye you1 howe shall ,i do in this arte to adde two summes or more together8 ,m4 ,the easyest way in this arte is1 to adde but #b summes at ones together3 how be it you may adde more1 as ,i wyll tell you anone4 ,therfore when you wyll adde two summes1 you shall fyrst set downe one of them1 it forseth not whiche1 and then by it drawe a lyne crosse the other lynes4 ,and afterward set downe the other summe1 so that that lyne may be betwene them1 as yf you wolde adde #bfei to #hcdb1 you must set your summes as you se ------------------------ o -,x--o-o-o------o-o------ o ----o-o-o------o-------- o ----o-o-o-o------------- o ----o-o--------o-o-o-o-- here4 ,and then yf you lyst1 you 99#aai;a may adde the one to the other in the same place1 or els you may adde them both together in a newe place3 which waye1 bycause it is moste playnest1 ,i wyll showe you fyrst4 ,therfore wyl ,i begynne at the vnites1 whiche in the fyrst summe is but #b1 and in ye second summe #i1 that maketh #aa1 those do ,i take vp1 and for them ,i set #aa in the new roume1 thus1 --------------------------- o -,x--o-o-o------o-o--------- o ----o-o-o------o----------- o ----o-o-o-o----------o----- ---------------------o----- ,then do ,i take vp all ye articles vnder a hundred1 which in the fyrst summe are #dj1 and in the second summe #ej1 that maketh #ij3 or you may saye better1 that in the fyrste summe there are #d articles of #aj1 and in the seconde summe #e1 which make #i1 but then take hede that you sette them in theyr 99#aai;b ryght lynes as you se here4 ---------------------------------- o -,x--o-o-o----o-o------------------ o ----o-o-o----o-------------------- o -----------------------o-o-o-o-o-- -----------------------o---------- ,where ,i haue taken awaye #dj from the fyrste summe1 and #ej from ye second1 and in theyr stede ,i haue set #ij in the thyrde1 whiche ,i haue set playnely yt you myght well perceaue it3 how be it seynge that #ij with the #aj that was in ye thyrd roume all redy1 doth make #ajj1 ,i myghte better for those #f counters set #a in the thyrde lyne1 thus3 ---------- -,x-------- ----o----- ---------- ----o----- ,for it is all one summe as you may se1 but it is beste1 neuer to set #e counters in any line1 for that may be done with #a counter in a hygher place4 ,s4 ,i iudge that good reason1 for many are vnnedefull1 where one wyll serue4 ,m4 ,well1 then 99#abj;a wyll ,i adde forth of hundredes3 ,i fynde #c in the fyrste summe1 and #f in the seconde1 whiche make #ijj1 them do ,i take vp and set in the thyrd roume where is one hundred all redy1 to whiche ,i put #ijj1 and it wyll be #ajjj1 therfore ,i set one counter in the fourth lyne for them all1 as you se here4 -------------------------- o -,x--o-o-o----o-o----o----- -------------------------- -------------------------- --------------------o----- ,then adde ,i ye thousandes together1 whiche in the fyrst summe are #hjjj1 and in ye second #bjjj1 that maketh #ajjjj3 them do ,i take vp from those two places1 and for them ,i set one counter in the fyfte lyne1 and then appereth as you se1 to be #aajja1 for so many doth amount of the addition of #hcdb to #bfei4 ----o----- -,x--o----- ---------- ---------- ----o----- 99#abj;b ,s4 ,syr1 this ,i do perceave3 but how shall ,i set one summe to an other1 not chaungynge them to a thyrde place8 ,m4 ,marke well how ,i do it3 ,i wyll adde together #fedcf1 and #cbde1 whiche fyrste ,i set downe thus4 --------------------------- o ---------------o----------- o -,x--o-o-o------------------ ----o-o--------o-o-o-o----- ----o-o-o-o----o-o-o------- o o ---------------o----------- ,then do ,i begynne with the smalest1 which in the fyrst summe is #e1 that do ,i take vp1 and wold put to the other #e in the seconde summe1 sauynge that two counters can not be set in a voyd place of #e1 but for them bothe ,i must set #a in the seconde lyne1 which is the place of #aj1 therfore ,i take vp the #e of the fyrst summe1 and the #e of the seconde1 and for them ,i set #a in the second lyne1 99#aba;a as you se here4 --------------------------- o ---------------o----------- o -,x--o-o-o------------------ ----o-o--------o-o-o-o----- ----o-o-o-o----o-o-o-o----- ---------------o----------- ,then do ,i lyke wayes take vp the #d counters of the fyrste summe and seconde lyne 7which make #dj7 and adde them to the #d counters of the same lyne1 in the second summe1 and it maketh #hj1 ,but as ,i sayde ,i maye not conueniently set aboue #d counters in one lyne1 therfore to those #d that ,i toke vp in the fyrst summe1 ,i take one also of the seconde summe1 and then haue ,i taken vp #ej1 for whiche #e counters ,i sette downe one in the space ouer ye second lyne1 as here doth appere4 ------------------------- o -------------o----------- o -,x--o-o-o---------------- ----o-o------o-o-o-o----- o -------------o-o-o------- -------------o----------- 99#aba;b4 and then is there #hj1 as well wt those #d counters1 as yf ,i had set downe ye other #d also4 ,now do ,i take the #bjj in the fyrste summe1 and adde them to the #djj in the seconde summe1 and it maketh #fjj1 therfore ,i take vp the #b counters in the fyrste summe1 and #c of them in the seconde summe1 and for them #e ,i set #a in ye space aboue1 thus4 ----------------------- o -------------o--------- o -,x--o-o-o-------------- o -------------o--------- o -------------o-o-o----- -------------o--------- ,then ,i take ye #cjjj in ye fyrste summe1 vnto whiche there are none in the second summe agreynge1 therfore ,i do onely remoue those #c counters from the fyrste summe into the seconde1 as here doth appere4 ----------------- o -------o--------- o -,x-----o-o-o----- o -----o----------- o -------o-o-o----- -------o--------- 99#abb;a4 ,and so you see the hole summe1 that amounteth of the addytion of #fedcf with #cbde to be #fhfha4 ,and yf you haue marked these two examples well1 you nede no farther enstruction in ,addition of #b only summes3 but yf you haue more then two summes to adde1 you may adde them thus4 ,fyrst adde two of them1 and then adde the thyrde1 and ye fourth1 or more yf there be so many3 as yf ,i wolde adde #bfgi with #dbhf and #acia4 ,fyrste ,i adde the two fyrste summes thus4 -------------------------------------- o -,x--o-o--------o-o-o-o----o----------- o o ----o----------o-o--------o-o-o-o----- o o o ----o-o--------o-o-o------o----------- o o o ----o-o-o-o----o---------------------- 99#abb;b4 ,and then ,i adde the thyrde thereto thus4 ,and so of more yf you haue them4 ------------------------------------ o o -,x--o----------o----------o-o-o----- o ----o-o-o------o-o-o-o----o-o-o----- o o o ----o-o-o-o------o------------------ o o ----o---------------------o--------- ,headnote3 ,subtraction on the ,counting ,board4 ,sidenote3 ,subtraction4 ,s4 ,nowe ,i thynke beste that you passe forth to ,subtraction1 except there be any wayes to examyn this maner of ,addition1 then ,i thynke that were good to be knowen nexte4 ,m4 ,there is the same profe here that is in the other ,addition by the penne1 ,i meane ,subtraction1 for that onely is a sure waye3 but consyderynge that ,subtraction must be fyrste knowen1 ,i wyl fyrste teache you the arte of ,subtraction1 and that by this example3 ,i wolde subtracte #bhib out of #hgdf4 ,these summes must ,i set downe as ,i dyd in ,addition3 but here it is best 99#aaf;a 7sic74 to set the lesser nomber fyrste1 thus4 --------------------------- o -,x--o-o--------o-o-o------- o o ----o-o-o------o-o--------- o ----o-o-o-o----o-o-o-o----- o ----o-o--------o----------- ,then shall ,i begynne to subtracte the greatest nombres fyrste 7contrary to the vse of the penne7 yt is the thousandes in this example3 therfore ,i fynd amongest the thousandes #b1 for which ,i withdrawe so many from the seconde summe 7where are #h7 and so remayneth there #f1 as this example showeth4 --------------------------- o --------------o----------- o o ----o-o-o------o-o--------- o ----o-o-o-o----o-o-o-o----- o ----o-o--------o----------- ,then do ,i lyke wayes with the hundredes1 of whiche in the fyrste summe 99#aaf;b ,i fynde #h1 and is the seconde summe but #g1 out of whiche ,i can not take #h1 therfore thus muste ,i do3 ,i muste loke how moche my summe dyffereth from #aj1 whiche ,i fynde here to be #b1 then must ,i bate for my summe of #hjj1 one thousande1 and set downe the excesse of hundredes1 that is to saye #b1 for so moche #ajjj is more then ,i shuld take vp4 ,therfore from the fyrste summe ,i take that #hjj1 and from the second summe where are #fjjj1 ,i take vp one thousande1 and leue #ejjj2 but then set ,i downe the #bjj unto the #gjj yt are there all redye1 and make them #ijj thus4 --------------------------- o -------------------------- o ---------------o-o-o-o----- o ----o-o-o-o----o-o-o-o----- o ----o-o--------o----------- ,then come ,i to the articles of tennes where in the fyrste summe ,i fynde #ij1 99#aag;a and in the seconde summe but only #dj3 ,now consyderyng that #ij can not be bated from #dj1 ,i loke how moche yt #ij doth dyffer from the next summe aboue it1 that is #ajj 7or elles whiche is all to one effecte1 ,i loke how moch #i doth dyffer from #aj7 and ,i fynd it to be #a1 then in the stede of that #ij1 ,i do take from the second summe #ajj3 but consyderynge that it is #aj to moche1 ,i set downe #a in ye nexte lyne beneth for it1 as you se here4 --------------------- o -------------------- o -----------o-o-o----- o --------------------- o ----o-o----o--------- ,sauynge that here ,i haue set one counter in ye space in stede of #e in ye nexte lyne4 ,and thus haue ,i subtracted all saue two1 which ,i must bate from the #f in the second summe1 and there wyll remayne #d1 thus4 ------------------ o ----------------- o ------o-o-o------- o ------------------ ------o-o-o-o----- ,so yt yf ,i subtracte #bhib from #hgdf1 the remayner wyll be #ehed1 99#aag;b ,and that this is truely wrought1 you maye proue by ,addition3 for yf you adde to this remayner the same summe that you dyd subtracte1 then wyll the formar summe #hgdf amount agayne4 ,s4 ,that wyll ,i proue3 and fyrst ,i set the summe that was subtracted1 which was #bhib1 and then the remayner #ehed1 thus4 ---------------------------- o ---o-o-------------------- o o -----o-o-o------o-o-o------- o o -----o-o-o-o---------------- -----o-o--------o-o-o-o----- ,then do ,i adde fyrst ye #b to #d1 whiche maketh #f1 so take ,i vp #e of those counters1 and in theyr stede ,i sette #a in the space1 as here appereth4 -------------------------- o ---o-o------------------ o o -----o-o-o------o-o-o----- o o -----o-o-o-o-------------- o ----------------o--------- 99#aah;a ,then do ,i adde the #ij nexte aboue to the #ej1 and it maketh #adj1 therfore ,i take vp those #f counters1 and for them ,i sette #a to the hundredes in ye thyrde lyne1 and #d in ye second lyne1 thus4 -------------------------- o ---o-o------------------ o o -----o-o-o----o-o-o-o----- --------------o-o-o-o----- o ----------------o--------- ,then do ,i come to the hundredes1 of whiche ,i fynde #h in the fyrst summe1 and #i in ye second1 that maketh #agjj1 therfore ,i take vp those #i counters1 and in theyr stede ,i sette #a in the 4iiii4 lyne1 and #a in the space nexte beneth1 and #b in the thyrde lyne1 as you se here4 ------------------------ o ---o-o----o----------- o ------------o-o--------- ------------o-o-o-o----- o ------------o----------- ,then is there lefte in the fyrste summe but only #bjjj1 whiche ,i shall take vp from thence1 and set 99#aah;b in the same lyne in ye second summe1 to ye one yt is there all redy3 and then wyll the hole summe appere 7as you may wel se7 to be #hgdf1 which was ye fyrst grosse summe1 and therfore ,i do perceaue1 that ,i hadde well subtracted before4 ,and thus you may se how ,subtraction maye be tryed by ,addition4 ------------------ o -,x----o-o-o------- o ------o-o--------- ------o-o-o-o----- o --------o--------- ,s4 ,i perceaue the same order here wt counters1 yt ,i lerned before in figures4 ,m4 ,then let me se howe can you trye ,addition by ,subtraction4 ,s4 ,fyrste ,i wyl set forth this example of ,addition where ,i haue added #bahi to #dihh1 and the hole summe appereth to be #gagg1 ----------------------------------- o ---o-o--------o-o-o-o----o-o----- o -----o----------o-o-o-o----o------- o o o -----o-o-o------o-o-o------o-o----- o o o -----o-o-o-o----o-o-o------o-o----- 99#aai;a ,nowe to trye whether that summe be well added or no1 ,i wyll subtract one of the fyrst two summes from the thyrd1 and yf ,i haue well done ye remayner wyll be lyke that other summe4 ,as for example3 ,i wyll subtracte the fyrste summe from the thyrde1 whiche ,i set thus in theyr order4 ------------------------ o ---o-o--------o-o----- -----o----------o------- o o -----o-o-o------o-o----- o o -----o-o-o-o----o-o----- ,then do ,i subtract #bjjj of the fyrste summe from ye second summe1 and then remayneth there #ejjj thus4 ----------------------- o -,x---------------------- ----o----------o------- o o ----o-o-o------o-o----- o o ----o-o-o-o----o-o----- ,then in the thyrd lyne1 ,i subtract ye #ajj of the fyrste summe1 from the second summe1 where is onely #ajj also1 and then in ye thyrde lyne resteth nothyng4 ,then in the second lyne with his space ouer hym1 ,i fynde #hj1 which ,i shuld subtract 99#aai;b from the other summe1 then seyng there are but only #gj ,i must take it out of some hygher summe1 which is here only #ejjj1 therfore ,i take vp #ejjj1 and seyng that it is to moch by #dibj1 ,i sette downe so many in the seconde roume1 whiche with the #gj beynge there all redy do make #diij1 then the summes doth stande thus4 ---------------------------- --------------o-o-o-o----- o ----------------o-o-o-o----- o ----------------o-o-o-o----- o o -----o-o-o-o----o-o--------- ,yet remayneth there in the fyrst summe #i1 to be bated from the second summe1 where in that place of vnities dothe appere only #g1 then ,i muste bate a hygher summe1 that is to saye #aj1 but seynge that #aj is more then #i 7which ,i shulde abate7 by #a1 therfore shall ,i take vp one counter from the seconde lyne1 and set downe the same in the fyrst 99#abj;a or lowest lyne1 as you se here4 ------------------- -----o-o-o-o----- o -------o-o-o-o----- o -------o-o-o------- o -------o-o-o------- ,and so haue ,i ended this worke1 and the summe appereth to be ye same1 whiche was ye seconde summe of my addition1 and therfore ,i perceaue1 ,i haue wel done4 ,m4 ,to stande longer about this1 it is but folye3 excepte that this you maye also vnderstande1 that many do begynne to subtracte with counters1 not at the hyghest summe1 as ,i haue taught you1 but at the nethermoste1 as they do vse to adde3 and when the summe to be abatyd1 in any lyne appeareth greater then the other1 then do they borowe one of the next hygher roume1 as for example3 yf they shuld abate #ahdf from #bcgh1 they set ye summes thus4 -------------------------- ---o----------o-o------- o -----o-o-o------o-o-o----- o -----o-o-o-o----o-o------- o o -----o----------o-o-o----- 99#abj;b ,and fyrste they take #f whiche is in the lower lyne1 and his space from #h in the same roumes1 in ye second summe1 and yet there remayneth #b counters in the lowest lyne4 ,then in the second lyne must #d be subtracte from #g1 and so remayneth there #c4 ,then #h in the thyrde lyne and his space1 from #c of the second summe can not be1 therfore do they bate it from a hygher roume1 that is1 from #ajjj1 and bycause that #ajjj is to moch by #bjj1 therfore must ,i sette downe #bjj in the thyrde lyne1 after ,i haue taken vp #ajjj from the fourth lyne3 then is there yet #ajjj in the fourth lyne of the fyrst summe1 whiche yf ,i withdrawe from the seconde summe1 then doth all ye figures stande in this order4 ----------------- --------------- o ----------------- -------o-o-o----- -------o-o------- ,so that 7as you se7 it differeth not greatly whether you begynne subtraction at the hygher lynes1 or at 99#aba;a the lower4 ,how be it1 as some menne lyke the one waye beste1 so some lyke the other3 therfore you now knowyng bothe1 may vse whiche you lyst4 ,headnote3 ,multiplication by ,counters4 ,sidenote3 ,multiplication4 ,but nowe touchynge ,multiplication3 you shall set your nombers in two roumes1 as you dyd in those two other kyndes1 but so that the multiplier be set in the fyrste roume4 ,then shall you begyn with the hyghest nombers of ye seconde roume1 and multiply them fyrst after this sort4 ,take that ouermost lyne in your fyrst workynge1 as yf it were the lowest lyne1 setting on it some mouable marke1 as you lyste1 and loke how many counters be in hym1 take them vp1 and for them set downe the hole multyplyer1 so many tymes as you toke vp counters1 reckenyng1 ,i saye that lyne for the vnites3 and when you haue so done with the hygheest nomber then come to the nexte lyne beneth1 and do euen so with it1 and so with ye next1 tyll you haue done all4 ,and yf there be any nomber in a space1 then for it 99#aba;b shall you take ye multiplyer #e tymes1 and then must you recken that lyne for the vnites whiche is nexte beneth that space3 or els after a shorter way1 you shall take only halfe the multyplyer1 but then shall you take the lyne nexte aboue that space1 for the lyne of vnites3 but in suche workynge1 yf chaunce your multyplyer be an odde nomber1 so that you can not take the halfe of it iustly1 then muste you take the greater halfe1 and set downe that1 as if that it were the iuste halfe1 and farther you shall set one counter in the space beneth that line1 which you recken for the lyne of vnities1 or els only remoue forward the same that is to be multyplyed4 ,s4 ,yf you set forth an example hereto ,i thynke ,i shal perceaue you4 ,m4 ,take this example3 ,i wold multiply #aedb by #cfe1 therfore ,i set ye nombers thus4 -------------------------- ------------o----------- o -----o-o-o---------------- o -----o--------o-o-o-o----- o --------------o-o--------- 99#abb;a ,then fyrste ,i begynne at the #ajjj in ye hyghest roume1 as yf it were ye fyrst place1 ,i take it vp1 settynge downe for it so often 7that is ones7 the multyplyer1 which is #cfe1 thus1 as you se here3 ---------------------------------- ------------------------o-o-o----- o ------------------------o--------- o -,x-------------------------------- - o ----o-o-o------------------------- o ----o--------o-o-o-o-------------- o -------------o-o------------------ where for the one counter taken vp from the fourth lyne1 ,i haue sette downe other #f1 whiche make ye summe of the multyplyer1 reckenynge that fourth lyne1 as yf it were the fyrste3 whiche thyng ,i haue marked by the hand set at the begynnyng of ye same1 ,s4 ,i perceaue this well3 for in dede1 this summe that you haue set downe is #cfejjj1 for so moche doth amount 99#abb;b of #ajjj1 multiplyed by #cfe4 ,m4 ,well then to go forth1 in the nexte space ,i fynde one counter which ,i remoue forward but take not vp1 but do 7as in such case ,i must7 set downe the greater halfe of my multiplier 7seyng it is an odde nomber7 which is #ahb1 and here ,i do styll let that fourth place stand1 as yf it were ye fyrst3 -------------------------o-o-o----o----- ---- o o -------------------------o--------o-o-o- ---- o --------------------------------o-o----- -- - o -----o-o-o------------------------------ ---- o -----o--------o-o-o-o------------------- ---- o --------------o-o----------------------- ---- as in this fourme you se1 where ,i haue set this multiplycation with ye other3 but for the ease of your vnderstandynge1 ,i haue set a lytell lyne betwene them3 now shulde they both in one summe stand thus4 -------------------------o-o-o-o-o----- -------------------------o-o-o-o------- o -----------------------o-o----------- - o -----o-o-o----------------------------- o -----o--------o-o-o-o------------------ o --------------o-o---------------------- 99#abc;a ,howe be it an other fourme to multyplye suche counters in space is this3 ,fyrst to remoue the fynger to the lyne nexte benethe ye space1 and then to take vp ye counter1 and to set downe ye multiplyer 4v4 tymes1 as here you se4 -------------------o-o-o---------------- ---------------- o -------------------o-----o-o-o-o-o-o-o-o -o-o-o-o-o-o-o-- o o o o o o -------------------------o-----------o-- ---o-----o------ o o o o o --,x-o-o-o------------------------------ ------------------ o ---o------o-o-o-o----------------------- ---------------- o ----------o-o--------------------------- ---------------- ,which summes yf you do adde together into one summe1 you shal perceaue that it wyll be ye same yt appeareth of ye other working before1 so that 99#abc;b bothe sortes are to one entent1 but as the other is much shorter1 so this is playner to reason1 for suche as haue had small exercyse in this arte4 ,not withstandynge you maye adde them in your mynde before you sette them downe1 as in this example1 you myghte haue sayde #e tymes #cjj is #aejj1 and #e tymes #fj is #cjj1 also #e tymes #e is #be1 whiche all put together do make #ahbe1 which you maye at one tyme set downe yf you lyste4 ,but nowe to go forth1 ,i must remoue the hand to the nexte counters1 whiche are in the second lyne1 and there must ,i take vp those #d counters1 settynge downe for them my multiplyer #d tymes1 whiche thynge other ,i maye do at #d tymes seuerally1 or elles ,i may gather that hole summe in my mynde fyrste1 and then set it downe3 as to saye #d tymes #cjj is #abjj3 #d tymes #fj are #bdj3 and #d tymes #e make #bj3 yt is in all #adfj1 yt shall ,i set downe also3 as here you se4 o ---------------------------------------- --- --------------------o-o-o-o----o-------- --- o -,x------------------o-o--------o-o-o-o- ---- o o ----o-o-o----------------------o-------- --- o - ----o----------------------------------- --- o -------------o-o------------------------ --- 99#abd;a whiche yf ,i ioyne in one summe with the formar nombers1 it wyll appeare thus4 o -------------------------- o ------------------o------- ------------------o-o----- --o-o-o----------o-------- o - --o----------------------- o -----------o-o------------ ,then to ende this multiplycation1 ,i remoue the fynger to the lowest lyne1 where are onely #b1 them do ,i take vp1 and in theyr stede do ,i set downe twyse #cfe1 that is #gcj1 for which ,i set 99#abd;b one in the space aboue the thyrd lyne for #ejj1 and #b more in the thyrd lyne with that one that is there all redye1 and the reste in theyr order1 and so haue ,i ended the hole summe thus4 o -------------------------- o ----------------o--------- ----------------o-o------- o --o-o-o---------o-o-o----- o --o-------------o-o-o----- o -------------------------- ,wherby you se1 that #aedb 7which is the nomber of yeares syth ,chrystes incarnation7 beyng multyplyed by #cfe 7which is the nomber of dayes in one yeare7 dothe amounte vnto #efbhcj1 which declareth ye nomber of daies sith ,chrystes incarnation vnto the ende of #aedba yeares4 7besyde #che dayes and #ab houres for lepe yeares74 ,s4 ,now wyll ,i proue by an other example1 as this3 #dj labourers 7after #f d4 ye day for eche man7 haue wrought #bh dayes1 ,i wold 99#abe;a know what theyr wages doth amount vnto3 ,in this case muste ,i worke doublely3 fyrst ,i must multyplye the nomber of the labourers by ye wages of a man for one day1 so wyll ye charge of one daye amount3 then secondarely shall ,i multyply that charge of one daye1 by the hole nomber of dayes1 and so wyll the hole summe appeare3 fyrst therefore ,i shall set the summes thus4 -------------------- -------------------- -------------------- --------o-o-o-o----- o --o----------------- ,where in the fyrste space is the multyplyer 7yt is one dayes wages for one man7 and in the second space is set the nomber of the worke men to be multyplyed3 then saye ,i1 #f tymes #d 7reckenynge that second lyne as the lyne of vnites7 maketh #bd1 for whiche summe ,i shulde set #b counters in the thyrde lyne1 and #d in the seconde1 therfore do ,i set #b in the thyrde lyne1 and let the #d stand styll in the seconde lyne1 thus499#abe;b ------------------- ------------------- -------o-o--------- -------o-o-o-o----- ------------------- ,so apwereth the hole dayes wages to be #bdj;d'4 that is #bj s4 ,then do ,i multiply agayn the same summe by the nomber of dayes and fyrste ,i sette the nombers1 thus4 ----------------------- ----------------------- -----------o-o--------- --o-o------o-o-o-o----- o --o-o-o--------------- ,then bycause there are counters in dyuers lynes1 ,i shall begynne with the hyghest1 and take them vp1 settynge for them the multyplyer so many tymes1 as ,i toke vp counters1 yt is twyse1 then wyll ye summe stande thus4 ------------------- o ------------------- o -------o----------- -------o-o-o-o----- ------------------- ,then come ,i to ye seconde lyne1 and take vp those #d counters1 settynge for them the multiplyer foure tymes1 so wyll the hole summe appeare thus499#abf;a --------------- o -------o------- o -------o-o----- -------o-o----- --------------- ,so is the hole wages of #dj workemen1 for #bh dayes 7after #f;d'4 eche daye for a man7 #fgbj;d'4 that is #efj s4 or #bh l'i4 ,headnote3 ,division on the ,counting ,board4 ,sidenote3 ,diuision4 ,m4 ,now if you wold proue ,multiplycation1 the surest way is by ,dyuision3 therfore wyll ,i ouer passe it tyll ,i haue taught you ye arte of ,diuision1 whiche you shall worke thus4 ,fyrste sette downe the ,diuisor for feare of forgettynge1 and then set the nomber that shalbe deuided1 at ye ryghte syde1 so farre from the diuisor1 that the quotient may be set betwene them3 as for example3 ,yf #bbe shepe cost #de l'i4 what dyd euery shepe cost8 ,to knowe this1 ,i shulde diuide the hole summe1 that is #de l'i4 by #bbe1 but that can not be1 therfore must ,i fyrste reduce that #de l'i4 into a lesser denomination1 as into shyllynges3 then ,i multiply #de by #bj1 and it is #ijj1 that summe shall ,i diuide by the nomber of 99#abf;b shepe1 whiche is #bbe1 these two nombers therfore ,i sette thus4 -------------------------- -------------------------- o --o-o---------o-o-o-o----- --o-o--------------------- o -------------------------- ,then begynne ,i at the hyghest lyne of the diuident1 and seke how often ,i may haue the diuisor therin1 and that maye ,i do #d tymes1 then say ,i1 #d tymes #b are #h1 whyche yf ,i take from #i1 there resteth but #a1 thus -------------------------- -------------------------- --o-o---------------o----- --o-o--------------------- o ---------o-o-o-o---------- ,and bycause ,i founde the diuisor #d tymes in the diuidente1 ,i haue set 7as you se7 #d in the myddle roume1 which 99#abg;a is the place of the quotient3 but now must ,i take the reste of the diuisor as often out of the remayner3 therfore come ,i to the seconde lyne of the diuisor1 sayeng #b foure tymes make #h1 take #h from #aj1 and there resteth #b1 thus4 ------------------------------- ----------------------------- -----o-o----------------------- -----o-o---------------o-o----- o ------------o-o-o-o------------ ,then come ,i to the lowest nomber1 which is #e1 and multyply it #d tymes1 so is it #bj1 that take ,i from #bj1 and there remayneth nothynge1 so that ,i se my quotient to be #d1 whiche are in valewe shyllynges1 for so was the diuident3 and therby ,i knowe1 that yf #bbe shepe dyd coste #de l'i4 euery shepe coste #d s4 ,s4 ,this can ,i do1 as you shall perceaue by this example3 ,yf #afj sowldyars do spende euery moneth #fh l'i4 what spendeth eche man8 ,fyrst 99#abg;b bycause ,i can not diuide the #fh by #afj1 therfore ,i wyll turne the poundes into pennes by multiplicacion1 so shall there be #afcbj d'4 ,nowe muste ,i diuide this summe by the nomber of sowldyars1 therfore ,i set them in order1 thus4 ----------------o--------- o --------------o--------- -----o----------o-o-o----- o -----o----------o-o------- -------------------------- ,then begyn ,i at the hyghest place of the diuidente1 sekynge my diuisor there1 whiche ,i fynde ones1 ,therfore set ,i #a in the nether lyne4 ,m4 ,not in the nether line of the hole summe1 but in the nether lyne of that worke1 whiche is the thyrde lyne4 ,s4 ,so standeth it with reason4 ,m4 ,then thus do they stande499#abh;a -------------------------- ------------------------ -----o-----o----o-o-o----- o -----o----------o-o------- -------------------------- ,then seke ,i agayne in the reste1 how often ,i may fynde my diuisor1 and ,i se that in the #cjj ,i myghte fynde #ajj thre tymes1 but then the #fj wyll not be so often founde in #bj1 therfore ,i take #b for my quotient3 then take ,i #ajj twyse from #cjj1 and there resteth #ajj1 out of whiche with the #bj 7that maketh #abj7 ,i may take #fj also twyse1 and then standeth the nombers thus1 --------------------- ------------------- -----o-----o--------- o -----o--------------- -----------o-o------- 99#abh;b where ,i haue sette the quotient #b in the lowest lyne3 ,so is euery sowldyars portion #ajb d'4 that is #h s4 #f d'4 ,m4 ,but yet bycause you shall perceaue iustly the reason of ,diuision1 it shall be good that you do set your diuisor styll agaynst those nombres from whiche you do take it3 as by this example ,i wyll declare4 ,yf ye purchace of #bjj acres of ground dyd coste #bij l'i4 what dyd one acre coste8 ,fyrst wyl ,i turne the poundes into pennes1 so wyll there be #fifjj d' ,then in settynge downe these nombers ,i shall do thus4 ---------------------------- o ----o-o---------o----------- o -,x--------------o-o-o-o----- o ----------------o----------- ---------------------------- ---------------------------- ,fyrst set the diuident on the ryghte hande as it oughte1 and then 99#abi;a the diuisor on the lefte hande agaynst those nombers1 from which ,i entende to take hym fyrst as here you se1 wher ,i haue set the diuisor two lynes hygher then is theyr owne place4 ,s4 ,this is lyke the order of diuision by the penne4 ,m4 ,truth you say1 and nowe must ,i set ye quotient of this worke in the thyrde lyne1 for that is the lyne of vnities in respecte to the diuisor in this worke4 ,then ,i seke howe often the diuisor maye be founde in the diuident1 and that ,i fynde #c tymes1 then set ,i #c in the thyrde lyne for the quotient1 and take awaye that #fjjjj from the diuident1 and farther ,i do set the diuisor one line lower1 as yow se here4 --------------------------------- o ---o-o-------------o-o-o-o----- o ------------o-o-o------o--------- --------------------------------- --------------------------------- 99#abi;b ,and then seke ,i how often the diuisor wyll be taken from the nomber agaynste it1 whiche wyll be #d tymes and #a remaynynge4 ,s4 ,but what yf it chaunce that when the diuisor is so remoued1 it can not be ones taken out of the diuident agaynste it8 ,m4 ,then must the diuisor be set in an other line lower4 ,s4 ,so was it in diuision by the penne1 and therfore was there a cypher set in the quotient3 but howe shall that be noted here8 ,m4 ,here nedeth no token1 for the lynes do represente the places3 onely loke that you set your quotient in that place which standeth for vnities in respecte of the diuisor3 but now to returne to the example1 ,i fynde the diuisor #d tymes in the diuidente1 and #a remaynynge1 for #d tymes #b make #h1 which ,i take from #i1 and there resteth #a1 as this figure sheweth3 ------------------------------ ---o-o---------------o------ o ------------o-o-o------o------ ------------o-o-o-o----------- ------------------------------ and in the myddle space for the quotient ,i set #d in the seconde lyne1 whiche is in this worke the place of vnities499#acj;a ,then remoue ,i ye diuisor to the next lower line1 and seke how often ,i may haue it in the dyuident1 which ,i may do here #h tymes iust1 and nothynge remayne1 as in this fourme1 -------------------------- ---o-o------------------ ------------o-o-o--------- ------------o-o-o-o------- o ------------o-o-o--------- where you may se that the hole quotient is #cdh d'1 that is #bi s4 wherby ,i knowe that so moche coste the purchace of one aker4 ,s4 ,now resteth the profes of ,multiplycation1 and also of ,diuision4 ,m4 ,ther best profes are eche 99#acj;b one by the other1 for ,multyplication is proued by ,diuision1 and ,diuision by ,multiplycation1 as in the worke by the penne you learned4 ,s4 ,yf that be all1 you shall not nede to repete agayne that1 yt was sufficyently taughte all redye3 and excepte you wyll teache me any other feate1 here maye you make an ende of this arte ,i suppose4 ,m4 ,so wyll ,i do as touchynge hole nomber1 and as for broken nomber1 ,i wyll not trouble your wytte with it1 tyll you haue practised this so well1 yt you be full perfecte1 so that you nede not to doubte in any poynte that ,i haue taught you1 and thenne maye ,i boldly enstructe you in ye arte of fractions or broken nomber1 wherin ,i wyll also showe you the reasons of all that you haue nowe learned4 ,but yet before ,i make an ende1 ,i wyll showe you the order of commen castyng1 wher in are bothe pennes1 shyllynges1 and poundes1 procedynge by no grounded reason1 but onely by a receaued 99#aca;a fourme1 and that dyuersly of dyuers men3 for marchauntes vse one fourme1 and auditors an other3 ,headnote3 ,merchants' ,casting ,counters4 ,sidenote3 ,merchants' casting4 ,but fyrste for marchauntes fourme marke this example here1 o o o o o o o o o o o o o o o o o o o o o o in which ,i haue expressed this summe #aih l'i4#b #ai s4 #aa d'4 ,so that you maye se that the lowest lyne serueth for pennes1 the next aboue for shyllynges1 the thyrde for poundes1 and the fourth for scores of poundes4 ,and farther you maye se1 that the space betwene pennes and shyllynges may receaue but one counter 7as all other spaces lyke wayes do7 and that one standeth in that place for #f d'4 ,lyke wayes betwene the shyllynges and the poundes1 one counter standeth for #aj s4 ,and betwene the poundes and #bj l'i4 one counter standeth for #aj poundes4 ,but besyde those you maye see at the left syde of shyllynges1 that one counter standeth alone1 and betokeneth #e s4 99#aca;b ,so agaynste the poundes1 that one counter standeth for #e l'i4 ,and agaynst the #bj poundes1 the one counter standeth for #e score poundes1 that is #ajj l'i4 so that euery syde counter is #e tymes so moch as one of them agaynst whiche he standeth4 ,sidenote3 ,auditors' casting4 ,now for the accompt of auditors take this example4 o o o o o o o o o o o o o o o o o o o o o o where ,i haue expressed ye same summe #aih l'i4 #ai s4 #aa d'4 ,but here you se the pennes stande toward ye ryght hande1 and the other encreasynge orderly towarde the lefte hande4 ,agayne you maye se1 that auditours wyll make #b lynes 7yea and more7 for pennes1 shyllynges1 and all other valewes1 yf theyr summes extende therto4 ,also you se1 that they set one counter at the ryght ende of eche rowe1 whiche so set there standeth for #e of that roume3 and on 99#acb;a the lefte corner of the rowe it standeth for #aj1 of ye same row4 ,but now yf you wold adde other subtracte after any of both those sortes1 yf you marke ye order of yt other feate which ,i taught you1 you may easely do the same here without moch teachynge3 for in ,addition you must fyrst set downe one summe and to the same set the other orderly1 and lyke maner yf you haue many3 but in ,subtraction you must sette downe fyrst the greatest summe1 and from it must you abate that other euery denomination from his dewe place4 ,s4 ,i do not doubte but with a lytell practise ,i shall attayne these bothe3 but how shall ,i multiply and diuide after these fourmes8 ,m4 ,you can not duely do none of both by these sortes1 therfore in suche case1 you must resort to your other artes4 ,s4 ,syr1 yet ,i se not by these sortes how to expresse hundreddes1 yf they excede one hundred1 nother yet thousandes4 ,m4 ,they that vse such accomptes that it excede #bjj 99#acb;b in one summe1 they sette no #e at the lefte hande of the scores of poundes1 but they set all the hundredes in an other farther rowe and #ejj at the lefte hand therof1 and the thousandes they set in a farther rowe yet1 and at the lefte syde therof they sette the #ejjj1 and in the space ouer they sette the #ajjjj1 and in a hygher rowe #bjjjj1 whiche all ,i haue expressed in this example1 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o which is #ighfi l'i4 #ab s4 #i d' ob4 q4 for ,i had not told you before where1 nother how you shuld set downe farthynges1 which 7as you se here7 must be set in a voyde space sydelynge beneth the pennes3 for q one counter3 for ob4 #b counters3 for ob4 q4 #c counters3 and more there can not be1 for #d farthynges 99#acc;a do make #a d'4 which must be set in his dewe place4 ,headnote3 ,auditors' ,casting ,counters4 ,and yf you desyre ye same summe after audytors maner1 lo here it is4 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o ,but in this thyng1 you shall take this for suffycyent1 and the reste you shall obserue as you maye se by the working of eche sorte3 for the dyuers wittes of men haue inuented dyuers and sundry wayes almost vnnumerable4 ,but one feate ,i shall teache you1 whiche not only for the straungenes and secretnes is moche pleasaunt1 but also for the good commoditie of it ryghte worthy to be well marked4 ,this feate hath ben vsed aboue #bjjj yeares at the leaste1 and yet was it neuer comenly knowen1 especyally in ,englysshe it was neuer taughte yet4 ,this is the arte of nombrynge on the hand1 with diuers gestures of the fyngers1 expressynge any summe conceaued in the 99#acc;b mynde4 ,and fyrst to begynne1 yf you wyll expresse any summe vnder #ajj1 you shall expresse it with your lefte hande3 and from #ajj vnto #ajjjj1 you shall expresse it with your ryght hande1 as here orderly by this table folowynge you may perceaue4 ,here foloweth the table of the arte of the hande ,the arte of nombrynge by the hande4 ,transcriber's ,note3 ,footnote #c reads3 8,bracket 77 denotes new paragraph in original40 ,for this e-text1 the brackets have been omitted in favor of restoring the paragraph breaks4 ,changes of speaker 7,m1 ,s7 are also marked by paragraphs1 as in the previous selection4 ,the illustration includes the printed page number #acd2 there is therefore no sidenote 99#acd;a4 ,the sidenote for 8#d0 is missing4 ,illustration3 7,numbers as described in text7 ,sidenote3 #a 99#acd;b ,in which as you may se #a is expressed by ye lyttle fynger of ye lefte hande closely and harde croked4 ,sidenote3 #b #cb is declared by lyke bowynge of the weddynge fynger 7whiche is the nexte to the lyttell fynger7 together with the lytell fynger4 ,sidenote3 #c #c is signified by the myddle fynger bowed in lyke maner1 with those other two4 #d is declared by the bowyng of the myddle fynger and the rynge fynger1 or weddynge fynger1 with the other all stretched forth4 ,sidenote3 #e1 #f #e is represented by the myddle fynger onely bowed4 ,and #f by the weddynge fynger only crooked3 and this you may marke in these a certayne order4 ,but now #g1 #h1 and #i1 are expressed with the bowynge of the same fyngers as are #a1 #b1 and #c1 but after an other fourme4 ,sidenote3 #g ,for #g is declared by the bowynge of the lytell fynger1 as is #a1 saue that for #a the fynger is clasped in1 harde and 99#ace;a rounde1 but for to expresse #g1 you shall bowe the myddle ioynte of the lytell fynger only1 and holde the other ioyntes streyght4 ,s4 ,yf you wyll geue me leue to expresse it after my rude maner1 thus ,i vnderstand your meanyng3 that #a is expressed by crookynge in the lyttell fynger lyke the head of a bysshoppes bagle3 and #g is declared by the same fynger bowed lyke a gybbet4 ,m4 ,so ,i perceaue1 you vnderstande it4 ,sidenote3 #h ,then to expresse #h1 you shall bowe after the same maner both the lyttell fynger and the rynge fynger4 ,sidenote3 #i1 #aj ,and yf you bowe lyke wayes with them the myddle fynger1 then doth it betoken #i4 ,now to expresse #aj1 you shall bowe your fore fynger rounde1 and set the ende of it on the hyghest ioynte of the thombe4 ,sidenote3 #bj ,and for to expresse #bj1 you must set your fyngers streyght1 and the ende of your thombe to the partition of the 99#ace;b fore moste and myddle fynger4 ,sidenote3 #cj #cj is represented by the ioynynge together of ye headdes of the foremost fynger and the thombe4 ,sidenote3 #dj #dj is declared by settynge of the thombe crossewayes on the foremost fynger4 ,sidenote3 #ej #ej is signified by ryght stretchyng forth of the fyngers ioyntly1 and applyenge of the thombes ende to the partition of the myddle fynger and the rynge fynger1 or weddynge fynger4 ,sidenote3 #fj #fj is formed by bendynge of the thombe croked and crossynge it with the fore fynger4 ,sidenote3 #gj #gj is expressed by the bowynge of the foremost fynger1 and settynge the ende of the thombe between the #b foremost or hyghest ioyntes of it4 ,sidenote3 #hj #hj is expressed by settynge of the foremost fynger crossewayes on the thombe1 so that #hj dyffereth thus from #dj1 that for #hj the forefynger is set crosse on the thombe1 and for #dj the thombe is set crosse ouer ye forefinger4 ,sidenote3 #ij 99#acf;a #ij is signified1 by bendynge the fore fynger1 and settyng the ende of it in the innermost ioynte of ye thombe1 that is euen at the foote of it4 ,and thus are all the nombers ended vnder #ajj4 ,sidenote3 #aa1 #ab1 #ac1 #ba1 #bb1 #bc ,s4 ,in dede these be all the nombers from #a to #aj1 and then all the tenthes within #ajj1 but this teacyed me not how to expresse #aa1 #ab1 #ac1 etc4 #ba1 #bb1 #bc1 etc4 and such lyke4 ,m4 ,you can lytell vnderstande1 yf you can not do that without teachynge3 what is #aa8 is it not #aj and #a8 then expresse #aj as you were taught1 and #a also1 and that is #aa3 and for #ab expresse #aj and #b3 for #bc set #bj and #c3 and so for #fh you muste make #fj and there to #h3 and so of all other sortes4 ,sidenote3 #ajj ,but now yf you wolde represente #ajj other any nomber aboue it1 you muste do that with the ryghte hande1 after this maner4 ,you must expresse #ajj in the ryght hand1 with the lytell fynger so bowed as you dyd expresse #a in the left hand4 ,sidenote3 #bjj 99#acf;b ,and as you expressed #b in the lefte hande1 the same fasshyon in the ryght hande doth declare #bjj4 ,sidenote3 #cjj ,the fourme of #c in the ryght hand standeth for #cjj4 ,sidenote3 #djj ,the fourme of #d1 for #djj4 ,sidenote3 #ejj ,lykewayes the fourme of #e1 for #ejj4 ,sidenote3 #fjj ,the fourme of #f1 for #fjj4 ,and to be shorte3 loke how you did expresse single vnities and tenthes in the lefte hande1 so must you expresse vnities and tenthes of hundredes1 in the ryghte hande4 ,sidenote3 #ijj ,s4 ,i vnderstande you thus3 that yf ,i wold represent #ijj1 ,i must so fourme the fyngers of my ryghte hande1 as ,i shuld do in my left hand to expresse #i1 ,sidenote3 #ajjj ,and as in my lefte hand ,i expressed #aj1 so in my ryght hande must ,i expresse #ajjj4 ,and so the fourme of euery tenthe in the lefte hande serueth to expresse lyke nomber of thousandes1 ,sidenote3 #djjj so ye fourme of #dj standeth for #djjj4 ,sidenote3 #hjjj ,the fourme of #hj for #hjjj4 ,sidenote3 #ijjj 99#acg;a ,and the fourme of #ij 7whiche is the greatest7 for #ijjj1 and aboue that ,i can not expresse any nomber4 ,m4 ,no not with one fynger3 how be it1 with dyuers fyngers you maye expresse #iiii1 and all at one tyme1 and that lac keth but #a of #ajjjj4 ,so that vnder #ajjjj you may by your fyngers ex- presse any summe4 ,and this shal suf- fyce for ,numeration on the fyngers4 ,and as for ,addition1 ,subtraction1 ,multiplication1 and ,diuision 7which yet were neuer taught by any man as farre as ,i do knowe7 ,i wyll enstruct you after the treatyse of fractions4 ,and now for this tyme fare well1 and loke that you cease not to practyse that you haue lear ned4 ,s4 ,syr1 with moste harty mynde ,i thanke you1 bothe for your good learnyng1 and also your good counsel1 which 7god wyllyng7 ,i truste to folow4 ,finis4 ,,footnotes 7,accomptynge by counters and ,the arte of nombrynge by the hande73 #a3 #acdb in original4 #b3 #afh in original4 #c3 ,bracket 77 denotes new paragraph in original4 ,,appendix ,i4 ,a ,treatise on the ,numeration of ,algorism4 ,from a ,,ms4 of the #adth ,century4 ,to alle suche even nombrys the most have cifrys as to ten4 twenty4 thirtty4 an hundred4 an thousand and suche other4 but ye schal vnderstonde that a cifre tokeneth nothinge but he maketh other the more significatyf that comith after hym4 ,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--#ajjj4 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 #a 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 one4 ,and there he stondith nowe in the ferye place he tokeneth a thousand as by this rewle4 ,in the first place he tokeneth but hymself4 ,in the secunde place he tokeneth ten times hymself4 ,in the thridde place he tokeneth an hundred tymes himself4 ,in the ferye he tokeneth a thousand tymes himself4 ,in the fyftye place he tokeneth ten thousand tymes himself4 ,in the sexte place he tokeneth an hundred thousand tymes hymself4 ,in the seveth place he tokeneth ten hundred thousand tymes hymself1 c4 ,and ye schal vnderstond that this worde nombre is partyd into thre partyes4 ,somme is callyd nombre of digitys for alle ben digitys that ben withine ten as ix1 viii1 vii1 vi1 v1 iv1 iii1 ii1 i4 ,articules ben alle thei that mow be devyded into nombrys of ten as xx1 xxx1 xl1 and suche other4 ,composittys be alle nombrys that ben componyd of a digyt and of an articule as fourtene fyftene thrittene and suche other4 ,fourtene is componyd of four that is a digyt and of ten that is an articule4 ,fyftene is componyd of fyve that is a digyt and of ten that is an articule and so of others 4 4 4 4 4 4 ,but as to this rewle4 ,in the firste place he tokeneth but himself that is to say he tokeneth but that and no more4 ,if that he stonde in the secunde place he tokeneth ten tymes himself as this figure #b here #ba4 this is oon and twenty4 ,this figure #b stondith in the secunde place and therfor he tokeneth ten tymes himself and ten tymes #b is twenty and so forye of every figure and he stonde after another toward the lest syde he schal tokene ten tymes as moche more as he schuld token and he stode in that place ther that the figure afore him stondeth3 lo an example as thus #ifcd4 ,this figure of foure that hath this schape #d tokeneth but himself for he stondeth in the first place4 ,the figure of thre that hath this schape #c tokeneth ten tyme himself for he stondeth in the secunde place and that is thritti4 ,the figure of sexe that hath this schape #f 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 sexty4 ,and now he tokeneth ten tymes that is sexe hundrid4 ,the figure of nyne that hath this schape #i tokeneth ten tymes more than he schulde and he stode in the place ther the figure of #f stondeth inne for thanne he schuld tokene but nyne hundryd4 ,and in the place that he stondeth inne nowe he tokeneth nine thousand4 ,alle the hole nombre of these foure figurys4 ,nine thousand sexe hundrid and foure and thritti4 ,,appendix ,,ii4 ,carmen de ,algorismo4 ,from a ,b4,m4 ,,ms41 #h ,c4 iv41 with additions from #ab ,e4 #a ,eg4 #bfbb4 ,hec algorismus ars presens dicitur#a2 in qua ,talibus ,indorum#b fruimur his quinque figuris4 #j4 #i4 #h4 #g4 #f4 #e4 #d4 #c4 #b4 #a4 ,prima significat unum3 duo vero secunda3 ,tercia significat tria3 sic procede sinistre #d ,donec ad extremam venies1 qua cifra vocatur2 #c,que nil significat2 dat significare sequenti4 ,quelibet illarum si primo limite ponas1 ,simpliciter se significat3 si vero secundo1 #h ,se decies3 sursum procedas multiplicando4#d ,namque figura sequens quevis signat decies plus1 ,ipsa locata loco quam significet pereunte3 #ab ,nam precedentes plus ultima significabit4 #e,post predicta scias quod tres breuiter numerorum ,distincte species sunt2 nam quidam digiti sunt2 ,articuli quidam2 quidam quoque compositi sunt4 #af ,sunt digiti numeri qui citra denarium sunt2 ,articuli decupli degitorum2 compositi sunt ,illi qui constant ex articulis digitisque4 ,ergo1 proposito numero tibi scribere1 primo #bj ,respicias quis sit numerus2 quia si digitus sit1 #e,una figura satis sibi2 sed si compositus sit1 ,primo scribe loco digitum post articulum fac ,articulus si sit1 cifram post articulum sit1 #bd ,articulum vero reliquenti in scribe figure4 ,quolibet in numero1 si par sit prima figura1 ,par erit et totum1 quicquid sibi continetur2 ,impar si fuerit1 totum sibi fiet et impar4 #bh ,septem#f sunt partes1 non plures1 istius artis2 ,addere1 subtrahere1 duplare1 dimidiare2 ,sexta est diuidere1 set quinta est multiplicare2 ,radicem extrahere pars septima dicitur esse4 #cb ,subtrahis aut addis a dextris vel mediabis2 ,a leua dupla1 diuide1 multiplicaque2 ,extrahe radicem semper sub parte sinistra4 ,sidenote3 ,addition4 ,addere si numero numerum vis1 ordine tali #cf ,incipe2 scribe duas primo series numerorum ,prima sub prima recte ponendo figuram1 ,et sic de reliquis facias1 si sint tibi plures4 ,inde duas adde primas hac condicione2 #dj ,si digitus crescat ex addicione priorum1 ,primo scribe loco digitum1 quicunque sit ille2 ,si sit compositus1 in limite scribe sequenti ,articulum1 primo digitum2 quia sic iubet ordo4 #dd ,articulus si sit1 in primo limite cifram1 ,articulum vero reliquis inscribe figuris2 ,vel per se scribas si nulla figura sequatur4 ,si tibi cifra superueniens occurrerit1 illam #dh ,deme suppositam2 post illic scribe figuram3 ,postea procedas reliquas addendo figuras4 ,sidenote3 ,subtraction4 ,a numero numerum si sit tibi demere cura1 ,scribe figurarum series1 vt in addicione2 #eb ,maiori numero numerum suppone minorem1 ,siue pari numero supponatur numerus par4 ,postea si possis a prima subtrahe primam1 ,scribens quod remanet1 cifram si nil remanebit4 #ef ,set si non possis a prima demere primam2 ,procedens1 vnum de limite deme sequenti2 ,et demptum pro denario reputabis ab illo1 ,subtrahe totaliter numerum quem proposuisti4 #fj ,quo facto1 scribe supra quicquit remanebit1 ,facque novenarios de cifris1 cum remanebis1 ,occurrant si forte cifre1 dum demseris vnum2 ,postea procedas reliquas demendo figuras4 #fd ,sidenote3 ,proof4 #g,si subtracio sit bene facta probare valebis1 ,quas subtraxisti primas addendo figuras4 ,nam1 subtractio si bene sit1 primas retinebis1 ,et subtractio facta tibi probat additionem4 #fh ,sidenote3 ,duplation4 ,si vis duplare numerum1 sic incipe2 solam ,scribe figurarum seriem1 quamcumque voles que ,postea procedas primam duplando figuram2 ,inde quod excrescet1 scribens1 vbi iusserit ordo1 #gb ,juxta precepta que dantur in addicione4 ,nam si sit digitus1 in primo limite scribe2 ,articulus si sit1 in primo limite cifram1 ,articulum vero reliquis inscribe figuris2 #gf ,vel per se scribas1 si nulla figura sequatur3 ,compositus si sit1 in limite scribe sequenti ,articulum primo1 digitum2 quia sic jubet ordo3 ,et sic de reliquis facias1 si sint tibi plures4 #hj #h,si super extremam nota sit1 monadem dat eidem1 ,quod tibi contingit1 si primo dimidiabis4 ,sidenote3 ,mediation4 ,incipe sic1 si vis aliquem numerum mediare3 ,scribe figurarum seriem solam1 velud ante2 #hd ,postea procedens medias1 et prima figura ,si par aut impar videas2 quia si fuerit par1 ,dimidiabis eam1 scribens quicquit remanebit2 ,impar si fuerit1 vnum demas1 mediare1 #hh ,nonne presumas1 sed quod superest mediabis2 ,inde super tractum1 fac demptum quod notat unum2 ,si monos1 dele2 sit ibi cifra post nota supra4 ,postea procedas hac condicione secunda3#i #ib ,impar#aj si fuerit hic vnum deme priori1 ,inscribens quinque1 nam denos significabit ,monos prdictam3 si vero secunda dat vnam1 ,illa deleta1 scribatur cifra2 priori #if ,tradendo quinque pro denario mediato2 ,nec cifra scribatur1 nisi inde figura sequatur3 ,postea procedas reliquas mediando figuras1 ,quin supra docui1 si sint tibi mille figure4 #ajj #aa,si mediatio sit bene facta probare valebis1 ,duplando numerum quem primo dimidiasti4 ,si super extremam nota sit monades dat eidem ,quod contingat cum primo dimiabis ,atque figura prior nuper fuerit mediando4 ,sidenote3 ,multiplication4 ,si tu per numerum numerum vis multiplicare1 ,scribe duas1 quascunque volis1 series numerorum2 #ajd ,ordo tamen seruetur vt vltima multiplicandi ,ponatur super anteriorem multiplicantis2 #ab;,a leua relique sint scripte multiplicantes4 ,in digitum cures digitum si ducere1 major #ajh ,per quantes distat a denis respice1 debes ,namque suo decuplo tociens delere minorem2 ,sicque tibi numerus veniens exinde patebit4 ,postea procedas postremam multiplicando1 #aab ,juste multiplicans per cunctas inferiores1 ,condicione tamen tali2 quod multiplicantis ,scribas in capite1 quicquid processerit inde2 ,set postquam fuerit hec multiplicata1 figure #aaf ,anteriorentur seriei multiplicantis2 ,et sic multiplica1 velut istam multiplicasti1 ,qui sequitur numerum scriptum quicunque figuris4 ,set cum multiplicas1 primo sic est operandum1 #abj ,si dabit articulum tibi multiplicacio solum2 ,proposita cifra1 summam transferre memento4 ,sin autem digitus excrescerit articulusque1 ,articulus supraposito digito salit ultra2 #abd ,si digitus tamen1 ponas illum super ipsam1 ,subdita multiplicans hanc que super incidit illi ,delet eam penitus1 scribens quod provenit inde2 ,sed si multiplices illam posite super ipsam1 #abh ,adiungens numerum quem prebet ductus earum2 ,si supraimpositam cifra debet multiplicare1 ,prorsus eam delet1 scribi que loco cifra debet1 #ab,si cifra multiplicat aliam positam super ipsam1 #acb ,sitque locus supra vacuus super hanc cifra fiet2 ,si supra fuerit cifra semper pretereunda est2 ,si dubites1 an sit bene multiplicando secunda1 ,diuide totalem numerum per multiplicantem1 #acf ,et reddet numerus emergens inde priorem4 ,sidenote3 ,mental ,multiplication4 #ac,per numerum si vis numerum quoque multiplicare ,tantum per normas subtiles absque figuris ,has normas poteris per versus scire sequentes4 #adj ,si tu per digitum digitum quilibet multiplicabis ,regula precedens dat qualiter est operandum ,articulum si per reliquum vis multiplicare ,in proprium digitum debebit uterque resolvi #add ,articulus digitos post per se multiplicantes ,ex digitis quociens teneret multiplicatum ,articuli faciunt tot centum multiplicati4 ,articulum digito si multiplicamus oportet #adh ,articulum digitum sumi quo multiplicare ,debemus reliquum quod multiplicaris ab illis ,per reliquo decuplum sic omne latere nequibit ,in numerum mixtum digitum si ducere cures #aeb ,articulus mixti sumatur deinde resolvas ,in digitum post hec fac ita de digitis nec ,articulusque docet excrescens in detinendo ,in digitum mixti post ducas multiplicantem #aef ,de digitis ut norma docet sit juncta secundo ,multiplica summam et postea summa patebit ,junctus in articulum purum articulumque #ad;,articulum purum comittes articulum que #afj ,mixti pro digitis post fiat et articulus vt ,norma jubet retinendo quod egreditur ab illis ,articuli digitum post in digitum mixti duc ,regula de digitis ut percipit articulusque #afd ,ex quibus excrescens summe tu junge priori ,sic manifesta cito fiet tibi summa petita4 ,compositum numerum mixto sic multiplicabis ,vndecies tredecem sic est ex hiis operandum #afh ,in reliquum primum demum duc post in eundem ,unum post deinde duc in tercia deinde per unum ,multiplices tercia demum tunc omnia multiplicata ,in summa duces quam que fuerit te dices #agb ,hic ut hic mixtus intentus est operandum ,multiplicandorum de normis sufficiunt hec4 ,sidenote3 ,division4 ,si vis dividere numerum1 sic incipe primo2 ,scribe duas1 quascunque voles1 series numerorum2 #agf ,majori numero numerum suppone minorem1 #ae,nam docet ut major teneat bis terve minorem2 ,et sub supprima supprimam pone figuram1 ,sic reliquis reliquas a dextra parte locabis2 #ahj ,postea de prima primam sub parte sinistra ,subtrahe1 si possis1 quociens potes adminus istud1 ,scribens quod remanet sub tali conditione2 ,ut totiens demas demendas a remanente1 #ahd ,que serie recte ponentur in anteriori1 ,unica si1 tantum sit ibi decet operari2 ,set si non possis a prima demere primam1 ,procedas1 et eam numero suppone sequenti2 #ahh ,hanc uno retrahendo gradu quo comites retrahantur1 ,et1 quotiens poteris1 ab eadem deme priorem1 ,ut totiens demas demendas a remanenti1 ,nec plus quam novies quicquam tibi demere debes1 #aib ,nascitur hinc numerus quociens supraque sequentem ,hunc primo scribas1 retrahas exinde figuras1 ,dum fuerit major supra positus inferiori1 ,et rursum fiat divisio more priori2 #aif ,et numerum quotiens supra scribas pereunti1 ,si fiat saliens retrahendo1 cifra locetur1 ,et pereat numero quotiens1 proponas eidem ,cifram1 ne numerum pereat vis1 dum locus illic #bjj ,restat1 et expletis divisio non valet ultra3 ,dum fuerit numerus numerorum inferiore seorsum ,illum servabis2 hinc multiplicando probabis1 ,sidenote3 ,proof4 ,si bene fecisti1 divisor multiplicetur #bjd ,per numerum quotiens2 cum multiplicaveris1 adde ,totali summ1 quod servatum fuit ante1 ,reddeturque tibi numerus quem proposuisti2 ,et si nil remanet1 hunc multiplicando reddet1 #bjh ,sidenote3 ,square ,numbers4 ,cum ducis numerum per se1 qui provenit inde ,sit tibi quadratus1 ductus radix erit hujus1 ,nec numeros omnes quadratos dicere debes1 ,est autem omnis numerus radix alicujus4 #bab ,quando voles numeri radicem querere1 scribi ,debet2 inde notes si sit locus ulterius impar1 ,estque figura loco talis scribenda sub illo1 ,que1 per se dicta1 numerum tibi destruat illum1 #baf ,vel quantum poterit ex inde delebis eandem2 ,vel retrahendo duples retrahens duplando sub ista ,que primo sequitur1 duplicatur per duplacationem1 ,post per se minuens pro posse quod est minuendum4 #bbj #af,post his propones digitum1 qui1 more priori ,per precedentes1 post per se multiplicatus1 ,destruat in quantum poterit numerum remanentem1 ,et sic procedens retrahens duplando figuram1 #bbd ,preponendo novam donec totum peragatur1 ,subdupla propriis servare docetque duplatis2 ,si det compositum numerum duplacio1 debet ,inscribi digitus a parte dextra parte propinqua1 #bbh ,articulusque loco quo non duplicata resessit2 ,si dabit articulum1 sit cifra loco pereunte ,articulusque locum tenet unum1 de duplicata resessit2 ,si donet digitum1 sub prima pone sequente1 #bcb ,si supraposita fuerit duplicata figura ,major proponi debet tantummodo cifra1 ,has retrahens solito propones more figuram1 ,usque sub extrema ita fac retrahendo figuras1 #bcf ,si totum deles numerum quem proposuisti1 ,quadratus fuerit1 de dupla quod duplicasti1 ,sicque tibi radix illius certa patebit1 ,si de duplatis fit juncta supprima figura2 #bdj ,radicem per se multiplices habeasque ,primo propositum1 bene te fecisse probasti2 ,non est quadratus1 si quis restat1 sed habentur ,radix quadrati qui stat major sub eadem2 #bdd ,vel quicquid remanet tabula servare memento2 ,hoc casu radix per se quoque multiplicetur1 ,vel sic quadratus sub primo major habetur1 ,hinc addas remanens1 et prius debes haberi2 #bdh ,si locus extremus fuerit par1 scribe figuram ,sub pereunte loco per quam debes operari1 ,que quantum poterit supprimas destruat ambas1 ,vel penitus legem teneas operando priorem1 #beb ,si suppositum digitus suo fine repertus1 ,omnino delet illic scribi cifra debet1 ,a leva si qua sit ei sociata figura2 ,si cifre remanent in fine pares decet harum #bef ,radices1 numero mediam proponere partem1 ,tali quesita radix patet arte reperta4 ,per numerum recte si nosti multiplicare ,ejus quadratum1 numerus qui pervenit inde #bfj ,dicetur cubicus2 primus radix erit ejus2 ,nec numeros omnes cubicatos dicere debes1 ,est autem omnis numerus radix alicujus2 ,sidenote3 ,cube ,root4 ,si curas cubici radicem qurere1 primo #bfd ,inscriptum numerum distinguere per loca debes2 ,que tibi mille notant a mille notante suprema ,initiam1 summa operandi parte sinistra1 ,illic sub scribas digitum1 qui multiplicatus #bfh ,in semet cubice suprapositum sibi perdat1 ,et si quid fuerit adjunctum parte sinistra ,si non omnino1 quantum poteris minuendo1 ,hinc triplans retrahe saltum1 faciendo sub illa #bgb ,que manet a digito deleto terna1 figuram ,illi propones quo sub triplo asocietur1 ,ut cum subtriplo per eam tripla multiplicatur2 ,hinc per eam solam productum multiplicabis1 #bgf ,postea totalem numerum1 qui provenit inde ,a suprapositis respectu tolle triplate ,addita supprimo cubice tunc multiplicetur1 ,respectu cujus1 numerus qui progredietur #bhj ,ex cubito ductu1 supra omnes adimetur2 ,tunc ipsam delens triples saltum faciendo1 ,semper sub ternas1 retrahens alias triplicatas ,ex hinc triplatis aliam propone figuram1 #bhd ,que per triplatas ducatur more priori2 ,primo sub triplis sibi junctis1 postea per se1 ,in numerum ducta1 productum de triplicatis3 ,utque prius dixi numerus qui provenit inde #bhh ,a suprapositis has respiciendo trahatur1 ,huic cubice ductum sub primo multiplicabis1 ,respectumque sui1 removebis de remanenti1 ,et sic procedas retrahendo triplando figuram4 #bib ,et proponendo nonam1 donec totum peragatur1 ,subtripla sub propriis servare decet triplicatis2 ,si nil in fine remanet1 numerus datus ante ,est cubicus2 cubicam radicem sub tripla prebent1 #bif ,cum digito juncto quem supprimo posuisti1 ,hec cubice ducta1 numerum reddant tibi primum4 ,si quid erit remanens non est cubicus1 sed habetur ,major sub primo qui stat radix cubicam1 #cjj ,servari debet quicquid radice remansit1 ,extracto numero1 decet hec addi cubicato4 ,quo facto1 numerus reddi debet tibi primus4 ,nam debes per se radicem multiplicare #cjd ,ex hinc in numerum duces1 qui provenit inde ,sub primo cubicus major sic invenietur2 ,illi jungatur remanens1 et primus habetur1 ,si per triplatum numerum nequeas operari2 #cjh ,cifram propones1 nil vero per hanc operare ,set retrahens illam cum saltu deinde triplata1 ,propones illi digitum sub lege priori1 ,cumque cifram retrahas saliendo1 non triplicabis1 #cab ,namque nihil cifre triplacio dicitur esse2 ,at tu cum cifram protraxeris aut triplicata1 ,hanc cum subtriplo semper servare memento3 ,si det compositum1 digiti triplacio debet #caf ,illius scribi1 digitus saliendo sub ipsam2 ,digito deleto1 que terna dicitur esse2 ,jungitur articulus cum triplata pereunte1 ,set facit hunc scribi per se triplacio prima1 #cbj ,que si det digitum per se scribi facit illum2 ,consumpto numero1 si sole fuit tibi cifre ,triplato1 propone cifram saltum faciendo1 ,cumque cifram retrahe triplam1 scribendo figuram1 #cbd ,preponas cifre1 sic procedens operare1 ,si tres vel duo serie in sint1 pone sub yma1 ,a dextris digitum servando prius documentum4 ,si sit continua progressio terminus nuper #cbh ,per majus medium totalem multiplicato2 ,si par1 per medium tunc multiplicato sequentem4 ,set si continua non sit progressio finis3 ,impar1 tunc majus medium si multiplicabis1 #ccb ,si par per medium sibi multiplicato propinquum4 #ccc ,,footnotes 7,appendix ,,ii1 ,carmen de ,algorismo73 #a3 8,hec prsens ars dicitur algorismus ab ,algore rege ejus inventore1 vel dicitur ab algos quod est ars1 et rodos quod est numerus2 qu est ars numerorum vel numerandi1 ad quam artem bene sciendum inveniebantur apud ,indos bis quinque 7id est decem7 figur40 --,comment4 ,thom de ,novo-,mercatu4 ,,ms4 ,bib4 ,reg4 ,mus4 ,brit4 #ab ,e4 #a4 #b3 8,h necessari figur sunt ,indorum characteros40 ,,ms4 de numeratione4 ,bib4 ,sloan4 ,mus4 ,brit4 #eac1 fol4 #eh4 8,cum vidissem ,yndos constituisse ,,ix literas in universo numero suo propter dispositionem suam quam posuerunt1 volui patefacere de opere quod sit per eas aliquidque esset levius discentibus1 si ,deus voluerit4 ,si autem ,indi hoc voluerunt et intentio illorum nihil novem literis fuit1 causa que mihi potuit4 ,deus direxit me ad hoc4 ,si vero alia dicam preter eam quam ego exposui1 hoc fecerunt per hoc quod ego exposui1 eadem tam certissime et absque ulla dubitatione poterit inveniri4 ,levitasque patebit aspicientibus et discentibus40 ,,ms4 ,u4,l4,c41 ,ii4 vi4 #e1 f4 #ajb4 #c3 ,from ,eg4 #bfbb4 #d3 #h ,c4 iv4 inserts ,nullum cipa significat3 dat significare sequenti4 #e3 ,from #ab ,e4 #a4 #f3 ,en argorisme devon prendre ,vii especes 4 4 4 4 ,adision subtracion ,doubloison mediacion ,monteploie et division ,et de radix eustracion ,a chez vii especes savoir ,doit chascun en memoire avoir ,letres qui figures sont dites ,et qui excellens sont ecrites4 --,,ms4 ,seld4 ,arch4 ,b4 #bf4 #g3 ,from #ab ,e4 #a4 #h3 ,from #ab ,e4 #a4 #i3 #h ,c4 iv4 inserts ,atque figura prior nuper fuerit mediando4 #aj3 ,i4e4 figura secundo loco posita4 #aa3 ,so #ab ,e4 #a2 #h ,c4 iv4 inserts-- #ab3 #ab ,e4 #a inserts4 #ac3 #ab ,e4 #a inserts to l4 #agd4 #ad3 #ab ,e4 #a omits1 ,eg4 #bfbb inserts4 #ae3 #ab ,e4 #a inserts4 #af3 #h ,c4 iv4 inserts-- ,hinc illam dele duplans sub ei psalliendo ,que sequitur retrahens quicquid fuerit duplicatum4 ,,index ,,of ,,technical ,,terms#a99 ,footnote #a993 ,this ,index has been kindly prepared by ,professor ,j4 ,b4 ,dale1 of ,king's ,college1 ,university of ,london1 and the best thanks of the ,society are due to him for his valuable contribution4 ,transcriber's ,note3 ,the ,technical ,terms and ,glossary 7following7 refer to page and line numbers in the printed book4 ,information in double brackets has been added by the transcriber to aid in text searching4 algorisme1 #cc_/ab2 algorym1 augrym1 #c_/c2 the art of computing1 using the so-called ,arabic numerals4 ,the word in its various forms is derived from the ,arabic al-,khowarazmi 7i4e4 the native of ,khwarazm 7,khiva774 ,this was the surname of ,ja'far ,mohammad ben ,musa1 who wrote a treatise early in the #ith century 7see p4 xiv74 ,the form algorithm is also found1 being suggested by a supposed derivation from the ,greek 7number74 antery1 #bd_/aa2 to move figures to the right of the position in which they are first written4 ,this operation is performed repeatedly upon the multiplier in multiplication1 and upon certain figures which arise in the process of root extraction4 anterioracioun1 #ej_/e2 the operation of moving figures to the right4 written anterioracioun or anterioracioun article1 #cd_/bc2 articul1 #e_/ca2 articuls1 #i_/cf1 #bi_/g1h2 a number divisible by ten without remainder4 also articulle cast1 #h_/ab2 to add one number to another4 #8,addition is a casting together of two numbers into one number1' #h_/aj4 cifre1 #d_/a2 the name of the figure #j4 ,the word is derived from the ,arabic sifr empty1 nothing4 ,hence zero4 ,a cipher is the symbol of the absence of number or of zero quantity4 ,it may be used alone or in conjunction with digits or other ciphers1 and in the latter case1 according to the position which it occupies relative to the other figures1 indicates the absence of units1 or tens1 or hundreds1 etc4 ,the great superiority of the ,arabic to all other systems of notation resides in the employment of this symbol4 ,when the cipher is not used1 the place value of digits has to be indicated by writing them in assigned rows or columns4 ,ciphers1 however1 may be interpolated amongst the significant figures used1 and as they sufficiently indicate the positions of the empty rows or columns1 the latter need not be indicated in any other way4 ,the practical performance of calculations is thus enormously facilitated 7see p4 xvi74 componede1 #cc_/bd2 composyt1 #e_/ce2 with reference to numbers1 one compounded of a multiple of ten and a digit4 written componede conuertide conversely1 #df_/bi1 #dg_/i4 written conuertide or conuertide cubicede1 #ej_/ac2 to be c41 to have its cube root found4 written cubicede cubike nombre1 #dg_/h2 a number formed by multiplying a given number twice by itself1 e4g4 #bg #c #c #c4 ,now called simply a cube4 written cubike 444 decuple1 #bb_/ab2 the product of a number by ten4 ,tenfold4 departys divides1 #e_/bi4 written departys digit1 #e_/cj2 digitalle1 #cc_/bd2 a number less than ten1 represented by one of the nine ,arabic numerals4 written digitalle dimydicion1 #g_/bc2 the operation of dividing a number by two4 ,halving4 written dimydicio duccioun1 multiplication1 #dc_/i4 written duccioun duplacion1 #g_/bc1 #ad_/ae2 the operation of multiplying a number by two4 ,doubling4 written duplacio or duplacio with fancy 8n0 i-mediet halved1 #ai_/bc4 intercise broken1 #df_/b2 intercise ,progression is the name given to either of the ,progressions #a1 #c1 #e1 #g1 etc42 #b1 #d1 #f1 #h1 etc41 in which the common difference is #b4 written intercise lede into1 multiply by1 #dg_/ah4 words always separated1 as 8lede 444 into0 lyneal nombre1 #df_/ad2 a number such as that which expresses the measure of the length of a line1 and therefore is not necessarily the product of two or more numbers 7vide ,superficial1 ,solid74 ,this appears to be the meaning of the phrase as used in ,the ,art of ,nombryng4 ,it is possible that the numbers so designated are the prime numbers1 that is1 numbers not divisible by any other number except themselves and unity1 but it is not clear that this limitation is intended4 mediacioun1 #af_/cf1 #ch_/af2 dividing by two 7see also dimydicion74 written mediacio with fancy 8n01 generally without 8u0 medlede nombre1 #cd_/a2 a number formed of a multiple of ten and a digit 7vide componede1 composyt74 written medlede 444 medye1 #ag_/h1 to halve2 mediete1 halved1 #ag_/cj2 ymedit1 #bj_/i4 naturelle progressioun1 #de_/bb2 the series of numbers #a1 #b1 #c1 etc4 written naturelle progressioun produccioun1 multiplication1 #ej_/aa4 written produccioun quadrat nombre1 #df_/ab2 a number formed by multiplying a given number by itself1 e4g4 #i #c #c1 a square4 rote1 #g_/be2 roote1 #dg_/aa2 root4 ,the roots of squares and cubes are the numbers from which the squares and cubes are derived by multiplication into themselves4 significatyf1 significant1 #e_/ad2 ,the significant figures of a number are1 strictly speaking1 those other than zero1 e4g4 in #c #f #e #j #d #j #j1 the significant figures are #c1 #f1 #e1 #d4 ,modern usage1 however1 regards all figures between the two extreme significant figures as significant1 even when some are zero4 ,thus1 in the above example1 #c #f #e #j #d are considered significant4 solide nombre1 #df_/cg2 a number which is the product of three other numbers1 e4g4 #ff #aa #b #c4 usually written solide superficial nombre1 #df_/ah2 a number which is the product of two other numbers1 e4g4 #f #b #c4 written superficial or superficialle ternary1 consisting of three digits1 #ea_/g4 written ternary vnder double1 a digit which has been doubled1 #dh_/c4 vnder-trebille1 a digit which has been trebled1 #di_/bh2 vnder-triplat1 #di_/ci4 written vnder-trebille1 vnder-triplat w1 a symbol used to denote half a unit1 #ag_/cc shown in e-text as superscript ,,glossary ,transcriber's ,note3 ,words whose first appearance is earlier than the page cited in the ,glossary are identified in double-bracketed notes4 ,to aid in text searching1 words written with internal italics are also noted1 and context is given for common words4 ablacioun1 taking away1 #cf_/ba written ablacioun addyst1 haddest1 #aj_/cg agregacioun1 addition1 #de_/bb4 7,first example in ,n4,e4,d41 #aedg47 written agregacioun a-enenes1 against1 #bc_/aj allgate1 always1 #h_/ci als1 as1 #bb_/bd and1 if1 #bi_/h2 1 #d_/bg2 yf1 #bj_/g a-nendes1 towards1 #bc_/ae aproprede1 appropriated1 #cd_/bg written aproprede apwereth1 appears1 #fa_/h a-risyt1 arises1 #ad_/bd a-rowe1 in a row1 #bi_/aj arsemetrike1 arithmetic1 #cc_/a written arsemetrike ayene1 again1 #de_/ae bagle1 crozier1 #fg_/ab bordure ordure1 row1 #dc_/cj written bordure borro1 inf4 borrow1 #aa_/ch2 imp4 s4 borowe1 #ab_/bj2 pp4 borwed1 #ab_/ae2 borred1 #ab_/ai boue1 above1 #db_/cd caputule1 chapter1 #g_/bf written caputule certayn1 assuredly1 #ah_/cd written certay clepede1 called1 #dg_/g written clepede competently1 conveniently1 #ce_/h compt1 count1 #dg_/bi contynes1 contains1 #ba_/ab2 written contynes pp4 contenythe1 #ch_/ci written contenythe craft1 art1 #c_/d distingue1 divide1 #ea_/e egalle1 equal1 #de_/ba written egalle excep1 except1 #e_/af exclusede1 excluded1 #cd_/cg written exclusede excressent1 resulting1 #ce_/af written excressent exeant1 resulting1 #dc_/bf expone1 expound1 #c_/bc ferye fere1 fourth1 #gj_/ab figure figures1 #e_/a written figure for-by1 past1 #ab_/aa fors2 no f41 no matter1 #bb_/bd forseth1 matters1 #ec_/cj forye fore1 forth1 #ga_/h fyftye fyfte1 fifth1 #gj_/af grewe1 ,greek1 #cc_/ac haluendel1 half1 #af_/af2 haldel1 #ai_/d2 pl4 haluedels1 #af_/af hayst1 hast1 #ag_/c1 #cb hast1 haste1 #bb_/be in 8haue hast to0 heer1 higher1 #i_/ce here1 their1 #g_/bf in 8in here caputule0 here-a-fore1 heretofore1 #ac_/g written here-a-fore heyth1 was called1 #c_/e hole1 whole1 #d_/ci2 holle1 #ag_/a2 hoole1 of three dimensions1 #df_/ae holdye1 holds good1 #cj_/e how be it that1 although1 #dd_/d lede lete1 let1 #h_/cg lene1 lend1 #ab_/ci lest1 least1 #dc_/bg in 8at the lest0 lest left1 #ga_/i in 8the lest syde0 leue1 leave1 #f_/e2 pr4 #c s4 leues1 remains1 #aa_/ai2 first in #aj_/dj leus1 #aa_/bh2 pp4 laft1 left1 #ai_/bd lewder1 more ignorant1 #c_/c written lewder lust1 desirest to1 #de_/ac lyt1 easy1 #ae_/ca lymytes1 limits1 #cd_/ah2 lynes1 #cd_/ab2 lynees1 #cd_/ag2 ,lat4 limes1 pl4 limites4 maystery1 achievement2 written maystery no m41 no achievement1 i4e4 easy1 #ai_/aj me1 indef4 pron4 one1 #db_/a first in #cd_/af mo1 more1 #i_/af moder more 7,lat4 majorem71 #dc_/bb most1 must1 #cj_/c first in #c_/ab and many more multipliede1 to be m4 multiplying1 #dj_/i mynvtes1 the sixty parts into which a unit is divided1 #ch_/be written mynvtes myse-wrot1 mis-wrought1 #ad_/aa nether1 nor1 #cd_/be in 8,it was1 nether is0 nex1 next1 #ai_/i not1 nought1 #e_/g first in #d_/h note1 not1 #cj_/e oo1 one1 #db_/bj2 o1 #db_/ba first in #cd_/bg2 #cc_/bb omest1 uppermost1 higher1 #ce_/bf2 omyst1 #ce_/bh omwhile1 sometimes1 #de_/ca first in #ci_/ag on1 one1 #h_/bi in 8on vnder an-oer0 opyne1 plain1 #dg_/h written opyne or1 before1 #ac_/be in 8or ou be-gan0 or e oer1 the other1 #bh_/cd in 8or by-twene0 ordure1 order1 #cd_/i2 row1 #dc_/a word form is 8order0 other1 or1 #cc_/ac1 #dc_/bf2 in 8art other craft0 on #cc_/ac1 8other how oft0 on #dc_/bf2 note also 8one other other0 on #ce_/bd other 4 4 4 or1 either 4 4 4 or1 #ch_/cg in 8other it is even or ode0 on #ch_/cg2 there are earlier occurrences ouerer1 upper1 #db_/ae written ouerer ouer-hippede1 passed over1 #dc_/ai written ouer-hippede recte1 directly1 #bg_/bj in 8stondes not recte02 also on #bf_/ca in 8recte ouer his hede0 remayner1 remainder1 #ef_/bh representithe1 represented1 #ci_/ad written representithe resteth1 remains1 #fc_/bi first in #eg_/bi and others rewarde1 regard1 #dh_/f written rewarde rew1 row1 #d_/h rewle1 row1 #d_/bj1 #g_/ab2 in 8place of e rewle01 8e rewle of figures0 rewele1 #d_/ah2 rewles1 rules1 #e_/cc s4 scilicet1 #c_/h in 8s4 ,algorismus0 sentens1 meaning1 #ad_/bi signifye7tyf71 #e_/ac4 ,the last three letters are added above the line1 evidently because of the word #8significatyf' in l4 #ad4 ,but the #8,solucio1' which contained the word1 has been omitted4 sithen1 since1 #cc_/h some1 sum1 result1 #dj_/ag1 #cb first in #cf_/ba in 8me may see a some01 then in 8the same some0 and 8to some of0 sowne1 pronounce1 #f_/bi singillatim1 singly1 #g_/be spices1 species1 kinds1 #cd_/d first in #e_/cd and others spyl1 waste1 #ad_/bf styde1 stead1 #ah_/bj subtrahe1 subtract1 #dh_/ab2 pp4 subtrayd1 #ac_/ba sythes1 times1 #ba_/af tat1 taught1 #af_/cf take1 pp4 taken2 t4 fro1 starting from1 #de_/bb in 8fro oone or tweyne take0 taward1 toward1 #bc_/cd thout1 though1 #e_/bj trebille1 multiply by three1 #di_/bf written trebille twene1 two1 #h_/aa first in #d_/bc ow1 though1 #be_/ae in 8ow ou take0 owt1 thought2 be 41 mentally1 #bh_/d us is1 this1 #bj_/cc in 8us nombur #bad0 vny1 unite1 #de_/aj wel1 wilt1 #ad_/ca in 8,if ou wel0 wete1 wit1 #ae_/af2 wyte1 know1 #h_/ch2 pr4 #b s4 wost1 #ab_/ch wex1 become1 #ej_/ah where1 whether1 #bi_/ab written where in 8where in e secunde1 or0 wher-thurghe1 whence1 #di_/ae written ,wher-thurghe worch1 work1 #h_/ai2 first in #g_/ce wrich1 #h_/ce2 wyrch1 #f_/ai2 imp4 s4 worch1 #ae_/i2 first in #i_/f pp4 y-wroth1 #ac_/bd write1 written1 #bi_/ai2 first in #f_/cg in 8hast write01 8be write0 y-write1 #af_/a wryrchynge wyrchynge1 working1 #cj_/d written wryrchynge wt1 with1 #ee_/h y-broth1 brought1 #ba_/ah ychon1 each one1 #bi_/aj written ycho ydo1 done1 added1 #i_/f first in #h_/cg in 8haue ydo02 #i_/f in 8ydo all to-geder0 ylke1 same1 #e_/ab y-lyech1 alike1 #bb_/bc y-myt1 been able1 #ab_/b y-nowt1 enough1 #ae_/ca2 ynovt1 #ah_/cd yove1 given1 #de_/cc yt1 that1 #eb_/h y-write1 v4 write4 y-wroth1 v4 worch4 99 99 99 99 99 99 99 99 99 99 99 99 99 99 ,,marginal ,,notes3 ,headnotes have been moved to the beginning of the appropriate paragraph4 ,headnotes were omitted from the two ,appendixes1 as sidenotes give the same information4 ,line ,numbers are cited in the ,index and ,glossary4 ,they have been omitted from the e-text except in the one verse selection 7,app4 ,,ii1 ,carmen de ,algorismo74 ,instead1 the ,index and ,glossary include supplemental information to help locate each word4 ,numbered ,notes3 ,numbered sidenotes show page or leaf numbers from the original ,,mss4 ,in the e-text1 the page number is shown as 99#abc;b inline2 mid-word page breaks are marked with a supplemental asterisk 994 ,numbers are not used4 ,footnotes give textual information such as variant readings4 ,they have been numbered sequentially within each title1 with numbers shown as #a to avoid confusion with bracked text--including single numerals--in the original4 ,editorial notes are shown as #a994 ,when a footnote calls for added text1 the addition is shown in the body text with double brackets4 ,sidenotes giving a running synopsis of the text have been moved to the beginning of each paragraph1 where they are shown as a single note4 ,,errors ,,and ,,anomalies 7,noted by ,transcriber73 ,introduction3 dated ,mijc ,in this and the remainder of the paragraph1 the letter shown as c is printed directly above the preceding j4 ,the ,crafte of ,nombrynge3 sursum procedas multiplicando ,italicized as shown3 error for 8procedas08 ,sidenote3 ,our author makes a slip here ,elsewhere in the book1 numerical errors are corrected in the body text1 with a footnote giving the original form4 ten tymes so mych is e nounbre text unchanged3 error for 8as08 #f tymes #bd1 #ai;en take misplaced footnote anchor in original3 belongs with 8#f times #bd0 ,fn4 #g3 #8,subtrahas aut addis a dextris open quote missing ,the ,art of ,nombryng3 oone of the digitis as 4#aj4 of #a44 #bj4 of4 #b4 text unchanged3 error for 8as 4#aj4 of 4#a4 #bj4 of 4#b408 sette a-side half of tho minutes text unchanged3 error for 8the08 and4 #aj4 as before is come therof text unchanged3 error for 8and 4#aj408 ,sidenote3 ,where to set the quotiente spelling 7#aibb7 unchanged ,sidenote3 ,definition of ,progression4 f in 8of0 illegible ,sidenote3 444 giving the value of ab4#b ,that is1 8a7b#b740 ,accomptynge by counters3 ,for example of the 99#aag;a4 ly99nes final 4 in sidenote missing or invisible also in #aba;b1 #abb;a which in the fyrst summe is #e invisible 8#e0 supplied by transcriber 99#aaf;a 7sic74 ,editor's 8sic03 page numbering jumps back to #aaf instead of the expected #abc1 and continues from #aaf4 99#abc;a 444 set downe ye multiplyer 4v4 tymes1 as here you se ,diagram shown as printed1 with #ceejj for #cfejj in one column1 and apparent misplaced 8thousands0 marker #cfe 7which is the nomber of dayes 444 open 7 missing ,the arte of nombrynge by the hande3 for #a the fynger is clasped in ,in at least one printing of the text1 8clasped0 is misprinted as 8elasped0 but this teacyed me not text unchanged ,appendix ,i3 ,a ,treatise on the ,numeration of ,algorism3 ,see ,introduction and ,glossary for 3f and 3y errors ,appendix ,,ii3 ,carmen de ,algorismo3 ,in this selection1 errors that are not explained in footnotes were assumed to be typographic4 l4 #ii ,postea procedas procdeas l4 #afc ,articuli digitum post in digitum mixti duc post iu