Appendix Ii.

Carmen de Algorismo.

Hec algorismus ars presens dicitur[{1}]; in qua Talibus Indorum[{2}] fruimur his quinque figuris. 0. 9. 8. 7. 6. 5. 4. 3. 2. 1. Prima significat unum: duo vero secunda: Tercia significat tria: sic procede sinistre 4 Donec ad extremam venies, qua cifra vocatur; [{3}][Que nil significat; dat significare sequenti.] Quelibet illarum si primo limite ponas, Simpliciter se significat: si vero secundo, 8 Se decies: sursum procedas multiplicando.[{4}] [Namque figura sequens quevis signat decies plus, Ipsa locata loco quam significet pereunte: 12 Nam precedentes plus ultima significabit.] [{5}]Post predicta scias quod tres breuiter numerorum Distincte species sunt; nam quidam digiti sunt; Articuli quidam; quidam quoque compositi sunt. 16 [Sunt digiti numeri qui citra denarium sunt; Articuli decupli degitorum; compositi sunt Illi qui constant ex articulis digitisque.] Ergo, proposito numero tibi scribere, primo 20 Respicias quis sit numerus; quia si digitus sit, [{5}][Una figura satis sibi; sed si compositus sit,] Primo scribe loco digitum post articulum fac Articulus si sit, cifram post articulum sit, 24 [Articulum vero reliquenti in scribe figure.] Quolibet in numero, si par sit prima figura, Par erit et totum, quicquid sibi continetur; Impar si fuerit, totum sibi fiet et impar. 28 Septem[{6}] sunt partes, non plures, istius artis; Addere, subtrahere, duplare, dimidiare; Sexta est diuidere, set quinta est multiplicare; Radicem extrahere pars septima dicitur esse. 32 Subtrahis aut addis a dextris vel mediabis; A leua dupla, diuide, multiplicaque; Extrahe radicem semper sub parte sinistra.

Addere si numero numerum vis, ordine tali 36 Incipe; scribe duas primo series numerorum Prima sub prima recte ponendo figuram, Et sic de reliquis facias, si sint tibi plures. Inde duas adde primas hac condicione; 40 Si digitus crescat ex addicione priorum, Primo scribe loco digitum, quicunque sit ille; Si sit compositus, in limite scribe sequenti Articulum, primo digitum; quia sic iubet ordo. 44 Articulus si sit, in primo limite cifram, Articulum vero reliquis inscribe figuris; Vel per se scribas si nulla figura sequatur. Si tibi cifra superueniens occurrerit, illam 48 Deme suppositam; post illic scribe figuram: Postea procedas reliquas addendo figuras.

A numero numerum si sit tibi demere cura, Scribe figurarum series, vt in addicione; 52 Maiori numero numerum suppone minorem, Siue pari numero supponatur numerus par. Postea si possis a prima subtrahe primam, Scribens quod remanet, cifram si nil remanebit. 56 Set si non possis a prima demere primam; Procedens, vnum de limite deme sequenti; Et demptum pro denario reputabis ab illo, Subtrahe totaliter numerum quem proposuisti. 60 Quo facto, scribe supra quicquit remanebit, Facque novenarios de cifris, cum remanebis, Occurrant si forte cifre, dum demseris vnum; Postea procedas reliquas demendo figuras. 64

Si vis duplare numerum, sic incipe; solam Scribe figurarum seriem, quamcumque voles que Postea procedas primam duplando figuram; Inde quod excrescet, scribens, vbi iusserit ordo, 72 Juxta precepta que dantur in addicione. Nam si sit digitus, in primo limite scribe; Articulus si sit, in primo limite cifram, Articulum vero reliquis inscribe figuris; 76 Vel per se scribas, si nulla figura sequatur: Compositus si sit, in limite scribe sequenti Articulum primo, digitum; quia sic jubet ordo: Et sic de reliquis facias, si sint tibi plures. 80 [{8}][Si super extremam nota sit, monadem dat eidem, Quod tibi contingit, si primo dimidiabis.]

Incipe sic, si vis aliquem numerum mediare: Scribe figurarum seriem solam, velud ante; 84 Postea procedens medias, et prima figura Si par aut impar videas; quia si fuerit par, Dimidiabis eam, scribens quicquit remanebit; Impar si fuerit, vnum demas, mediare, 88 Nonne presumas, sed quod superest mediabis; Inde super tractum, fac demptum quod notat unum; Si monos, dele; sit ibi cifra post nota supra. Postea procedas hac condicione secunda:[{9}] 92 Impar[{10}] si fuerit hic vnum deme priori, Inscribens quinque, nam denos significabit Monos prædictam: si vero secunda dat vnam, Illa deleta, scribatur cifra; priori 96 Tradendo quinque pro denario mediato; Nec cifra scribatur, nisi inde figura sequatur: Postea procedas reliquas mediando figuras, Quin supra docui, si sint tibi mille figure. 100 [{11}][Si mediatio sit bene facta probare valebis, Duplando numerum quem primo dimidiasti.] Si super extremam nota sit monades dat eidem Quod contingat cum primo dimiabis Atque figura prior nuper fuerit mediando.]

Si tu per numerum numerum vis multiplicare, Scribe duas, quascunque volis, series numerorum; 104 Ordo tamen seruetur vt vltima multiplicandi Ponatur super anteriorem multiplicantis; [{12}][A leua relique sint scripte multiplicantes.] In digitum cures digitum si ducere, major 108 Per quantes distat a denis respice, debes Namque suo decuplo tociens delere minorem; Sicque tibi numerus veniens exinde patebit. Postea procedas postremam multiplicando, 112 Juste multiplicans per cunctas inferiores, Condicione tamen tali; quod multiplicantis Scribas in capite, quicquid processerit inde; Set postquam fuerit hec multiplicata, figure 116 Anteriorentur seriei multiplicantis; Et sic multiplica, velut istam multiplicasti, Qui sequitur numerum scriptum quicunque figuris. Set cum multiplicas, primo sic est operandum, 120 Si dabit articulum tibi multiplicacio solum; Proposita cifra, summam transferre memento. Sin autem digitus excrescerit articulusque, Articulus supraposito digito salit ultra; 124 Si digitus tamen, ponas illum super ipsam, Subdita multiplicans hanc que super incidit illi Delet eam penitus, scribens quod provenit inde; Sed si multiplices illam posite super ipsam, 128 Adiungens numerum quem prebet ductus earum; Si supraimpositam cifra debet multiplicare, Prorsus eam delet, scribi que loco cifra debet, [{12}][Si cifra multiplicat aliam positam super ipsam, 132 Sitque locus supra vacuus super hanc cifra fiet;] Si supra fuerit cifra semper pretereunda est; Si dubites, an sit bene multiplicando secunda, Diuide totalem numerum per multiplicantem, 136 Et reddet numerus emergens inde priorem.

Si vis dividere numerum, sic incipe primo; Scribe duas, quascunque voles, series numerorum; 176 Majori numero numerum suppone minorem, [{15}][Nam docet ut major teneat bis terve minorem;] Et sub supprima supprimam pone figuram, Sic reliquis reliquas a dextra parte locabis; 180 Postea de prima primam sub parte sinistra Subtrahe, si possis, quociens potes adminus istud, Scribens quod remanet sub tali conditione; Ut totiens demas demendas a remanente, 184 Que serie recte ponentur in anteriori, Unica si, tantum sit ibi decet operari; Set si non possis a prima demere primam, Procedas, et eam numero suppone sequenti; 188 Hanc uno retrahendo gradu quo comites retrahantur, Et, quotiens poteris, ab eadem deme priorem, Ut totiens demas demendas a remanenti, Nec plus quam novies quicquam tibi demere debes, 192 Nascitur hinc numerus quociens supraque sequentem Hunc primo scribas, retrahas exinde figuras, Dum fuerit major supra positus inferiori, Et rursum fiat divisio more priori; 196 Et numerum quotiens supra scribas pereunti, Si fiat saliens retrahendo, cifra locetur, Et pereat numero quotiens, proponas eidem Cifram, ne numerum pereat vis, dum locus illic 200 Restat, et expletis divisio non valet ultra: Dum fuerit numerus numerorum inferiore seorsum Illum servabis; hinc multiplicando probabis,

Si bene fecisti, divisor multiplicetur 204 Per numerum quotiens; cum multiplicaveris, adde Totali summæ, quod servatum fuit ante, Reddeturque tibi numerus quem proposuisti; Et si nil remanet, hunc multiplicando reddet, 208

Cum ducis numerum per se, qui provenit inde Sit tibi quadratus, ductus radix erit hujus, Nec numeros omnes quadratos dicere debes, Est autem omnis numerus radix alicujus. 212 Quando voles numeri radicem querere, scribi Debet; inde notes si sit locus ulterius impar, Estque figura loco talis scribenda sub illo, Que, per se dicta, numerum tibi destruat illum, 216 Vel quantum poterit ex inde delebis eandem; Vel retrahendo duples retrahens duplando sub ista Que primo sequitur, duplicatur per duplacationem, Post per se minuens pro posse quod est minuendum. 220 [{16}]Post his propones digitum, qui, more priori Per precedentes, post per se multiplicatus, Destruat in quantum poterit numerum remanentem, Et sic procedens retrahens duplando figuram, 224 Preponendo novam donec totum peragatur, Subdupla propriis servare docetque duplatis; Si det compositum numerum duplacio, debet Inscribi digitus a parte dextra parte propinqua, 228 Articulusque loco quo non duplicata resessit; Si dabit articulum, sit cifra loco pereunte Articulusque locum tenet unum, de duplicata resessit; Si donet digitum, sub prima pone sequente, 232 Si supraposita fuerit duplicata figura Major proponi debet tantummodo cifra, Has retrahens solito propones more figuram, Usque sub extrema ita fac retrahendo figuras, 236 Si totum deles numerum quem proposuisti, Quadratus fuerit, de dupla quod duplicasti, Sicque tibi radix illius certa patebit, Si de duplatis fit juncta supprima figura; 240 Radicem per se multiplices habeasque Primo propositum, bene te fecisse probasti; Non est quadratus, si quis restat, sed habentur Radix quadrati qui stat major sub eadem; 244 Vel quicquid remanet tabula servare memento; Hoc casu radix per se quoque multiplicetur, Vel sic quadratus sub primo major habetur, Hinc addas remanens, et prius debes haberi; 248 Si locus extremus fuerit par, scribe figuram Sub pereunte loco per quam debes operari, Que quantum poterit supprimas destruat ambas, Vel penitus legem teneas operando priorem, 252 Si suppositum digitus suo fine repertus, Omnino delet illic scribi cifra debet, A leva si qua sit ei sociata figura; Si cifre remanent in fine pares decet harum 256 Radices, numero mediam proponere partem, Tali quesita radix patet arte reperta. Per numerum recte si nosti multiplicare Ejus quadratum, numerus qui pervenit inde 260 Dicetur cubicus; primus radix erit ejus; Nec numeros omnes cubicatos dicere debes, Est autem omnis numerus radix alicujus;

Si curas cubici radicem quærere, primo 264 Inscriptum numerum distinguere per loca debes; Que tibi mille notant a mille notante suprema Initiam, summa operandi parte sinistra, Illic sub scribas digitum, qui multiplicatus 268 In semet cubice suprapositum sibi perdat, Et si quid fuerit adjunctum parte sinistra Si non omnino, quantum poteris minuendo, Hinc triplans retrahe saltum, faciendo sub illa 272 Que manet a digito deleto terna, figuram Illi propones quo sub triplo asocietur, Ut cum subtriplo per eam tripla multiplicatur; Hinc per eam solam productum multiplicabis, 276 Postea totalem numerum, qui provenit inde A suprapositis respectu tolle triplate Addita supprimo cubice tunc multiplicetur, Respectu cujus, numerus qui progredietur 280 Ex cubito ductu, supra omnes adimetur; Tunc ipsam delens triples saltum faciendo, Semper sub ternas, retrahens alias triplicatas Ex hinc triplatis aliam propone figuram, 284 Que per triplatas ducatur more priori; Primo sub triplis sibi junctis, postea per se, In numerum ducta, productum de triplicatis: Utque prius dixi numerus qui provenit inde 288 A suprapositis has respiciendo trahatur, Huic cubice ductum sub primo multiplicabis, Respectumque sui, removebis de remanenti, Et sic procedas retrahendo triplando figuram. 292 Et proponendo nonam, donec totum peragatur, Subtripla sub propriis servare decet triplicatis; Si nil in fine remanet, numerus datus ante Est cubicus; cubicam radicem sub tripla prebent, 296 Cum digito juncto quem supprimo posuisti, Hec cubice ducta, numerum reddant tibi primum. Si quid erit remanens non est cubicus, sed habetur Major sub primo qui stat radix cubicam, 300 Servari debet quicquid radice remansit, Extracto numero, decet hec addi cubicato. Quo facto, numerus reddi debet tibi primus. Nam debes per se radicem multiplicare 304 Ex hinc in numerum duces, qui provenit inde Sub primo cubicus major sic invenietur; Illi jungatur remanens, et primus habetur, Si per triplatum numerum nequeas operari; 308 Cifram propones, nil vero per hanc operare Set retrahens illam cum saltu deinde triplata, Propones illi digitum sub lege priori, Cumque cifram retrahas saliendo, non triplicabis, 312 Namque nihil cifre triplacio dicitur esse; At tu cum cifram protraxeris aut triplicata, Hanc cum subtriplo semper servare memento: Si det compositum, digiti triplacio debet 316 Illius scribi, digitus saliendo sub ipsam; Digito deleto, que terna dicitur esse; Jungitur articulus cum triplata pereunte, Set facit hunc scribi per se triplacio prima, 320 Que si det digitum per se scribi facit illum; Consumpto numero, si sole fuit tibi cifre Triplato, propone cifram saltum faciendo, Cumque cifram retrahe triplam, scribendo figuram, 324 Preponas cifre, sic procedens operare, Si tres vel duo serie in sint, pone sub yma, A dextris digitum servando prius documentum. Si sit continua progressio terminus nuper 328 Per majus medium totalem multiplicato; Si par, per medium tunc multiplicato sequentem. Set si continua non sit progressio finis: Impar, tunc majus medium si multiplicabis, 332 Si par per medium sibi multiplicato propinquum. 333

FOOTNOTES (Appendix II, Carmen de Algorismo):

INDEX OF TECHNICAL TERMS[1*]

+algorisme+, 33/12; +algorym+, +augrym+, 3/3; the art of computing, using the so-called Arabic numerals. The word in its various forms is derived from the Arabic al-Khowarazmi (i.e. the native of Khwarazm (Khiva)). This was the surname of Ja’far Mohammad ben Musa, who wrote a treatise early in the 9th century (see p. xiv). The form algorithm is also found, being suggested by a supposed derivation from the Greek ἀριθμός (number).

+antery+, 24/11; to move figures to the right of the position in which they are first written. This operation is performed repeatedly upon the multiplier in multiplication, and upon certain figures which arise in the process of root extraction.

+anterioracioun+, 50/5; the operation of moving figures to the right. [[written anteriorac{i}o{u}n or anterioracio{u}n]]

+article+, 34/23; +articul+, 5/31; +articuls+, 9/36, 29/7,8; a number divisible by ten without remainder. [[also articull{e}]]

+cast+, 8/12; to add one number to another. ‘Addition is a casting together of two numbers into one number,’ 8/10.

+cifre+, 4/1; the name of the figure 0. The word is derived from the Arabic sifr = empty, nothing. Hence zero. A cipher is the symbol of the absence of number or of zero quantity. It may be used alone or in conjunction with digits or other ciphers, and in the latter case, according to the position which it occupies relative to the other figures, indicates the absence of units, or tens, or hundreds, etc. The great superiority of the Arabic to all other systems of notation resides in the employment of this symbol. When the cipher is not used, the place value of digits has to be indicated by writing them in assigned rows or columns. Ciphers, however, may be interpolated amongst the significant figures used, and as they sufficiently indicate the positions of the empty rows or columns, the latter need not be indicated in any other way. The practical performance of calculations is thus enormously facilitated (see p. xvi).

+componede+, 33/24; +composyt+, 5/35; with reference to numbers, one compounded of a multiple of ten and a digit. [[written componed{e}]]

+conuertide+ = conversely, 46/29, 47/9. [[written co{n}u{er}tid{e} or {con}u{er}tid{e}]]

+cubicede+, 50/13; +to be c.+, to have its cube root found. [[written cubiced{e}]]

+cubike nombre+, 47/8; a number formed by multiplying a given number twice by itself, e.g. 27 = 3 × 3 × 3. Now called simply a cube. [[written cubik{e} ...]]

+decuple+, 22/12; the product of a number by ten. Tenfold.

+departys+ = divides, 5/29. [[written dep{ar}tys]]

+digit+, 5/30; +digitalle+, 33/24; a number less than ten, represented by one of the nine Arabic numerals. [[written digitall{e}]]

+dimydicion+, 7/23; the operation of dividing a number by two. Halving. [[written dimydicioñ]]

+duccioun+, multiplication, 43/9. [[written duccio{u}n]]

+duplacion+, 7/23, 14/15; the operation of multiplying a number by two. Doubling. [[written duplacioñ or duplacioɳ with fancy “n”]]

+i-mediet+ = halved, 19/23.

+intercise+ = broken, 46/2; intercise Progression is the name given to either of the Progressions 1, 3, 5, 7, etc.; 2, 4, 6, 8, etc., in which the common difference is 2. [[written int{er}cise]]

+lede into+, multiply by, 47/18. [[words always separated, as “lede ... into”]]

+lyneal nombre+, 46/14; a number such as that which expresses the measure of the length of a line, and therefore is not necessarily the product of two or more numbers (vide Superficial, Solid). This appears to be the meaning of the phrase as used in The Art of Nombryng. It is possible that the numbers so designated are the prime numbers, that is, numbers not divisible by any other number except themselves and unity, but it is not clear that this limitation is intended.

+mediacioun+, 16/36, 38/16; dividing by two (see also +dimydicion+). [[written mediacioɳ with fancy “n”, generally without “u”]]

+medlede nombre+, 34/1; a number formed of a multiple of ten and a digit (vide componede, composyt). [[written medled{e} ...]]

+medye+, 17/8, to halve; +mediete+, halved, 17/30; +ymedit+, 20/9.

+naturelle progressioun+, 45/22; the series of numbers 1, 2, 3, etc. [[written naturell{e} p{ro}gressio{u}n]]

+produccioun+, multiplication, 50/11. [[written produccio{u}n]]

+quadrat nombre+, 46/12; a number formed by multiplying a given number by itself, e.g. 9 = 3 × 3, a square.

+rote+, 7/25; +roote+, 47/11; root. The roots of squares and cubes are the numbers from which the squares and cubes are derived by multiplication into themselves.

+significatyf+, significant, 5/14; The significant figures of a number are, strictly speaking, those other than zero, e.g. in 3 6 5 0 4 0 0, the significant figures are 3, 6, 5, 4. Modern usage, however, regards all figures between the two extreme significant figures as significant, even when some are zero. Thus, in the above example, 3 6 5 0 4 are considered significant.

+solide nombre+, 46/37; a number which is the product of three other numbers, e.g. 66 = 11 × 2 × 3. [[usually written solid{e}]]

+superficial nombre+, 46/18; a number which is the product of two other numbers, e.g. 6 = 2 × 3. [[written sup{er}ficial or sup{er}ficiall{e}]]

+ternary+, consisting of three digits, 51/7. [[written t{er}nary]]

+vnder double+, a digit which has been doubled, 48/3.

+vnder-trebille+, a digit which has been trebled, 49/28; +vnder-triplat+, 49/39. [[written vnder-trebill{e}, vnder-t{r}iplat]]

+w+, a symbol used to denote half a unit, 17/33 [[shown in e-text as superscript ʷ]]

GLOSSARY

Words whose first appearance is earlier than the page cited in the Glossary are identified in double-bracketed notes. To aid in text searching, words written with internal {italics} are also noted, and context is given for common words.]

+ablacioun+, taking away, 36/21 [[written ablacio{u}n]] +addyst+, haddest, 10/37 +agregacioun+, addition, 45/22. (First example in N.E.D., 1547.) [[written ag{r}egacio{u}n]] +a-ȝenenes+, against, 23/10 +allgate+, always, 8/39 +als+, as, 22/24 +and+, if, 29/8; +&+, 4/27; +& yf+, 20/7 +a-nendes+, towards, 23/15 +aproprede+, appropriated, 34/27 [[written ap{ro}pred{e}]] +apwereth+, appears, 61/8 +a-risyȝt+, arises, 14/24 +a-rowe+, in a row, 29/10 +arsemetrike+, arithmetic, 33/1 [[written arsemetrik{e}]] +ayene+, again, 45/15

+bagle+, crozier, 67/12 +bordure+ = ordure, row, 43/30 [[written bordur{e}]] +borro+, inf. borrow, 11/38; imp. s. +borowe+, 12/20; pp. +borwed+, 12/15; +borred+, 12/19 +boue+, above, 42/34

+caputule+, chapter, 7/26 [[written caputul{e}]] +certayn+, assuredly, 18/34 [[written c{er}tayɳ]] +clepede+, called, 47/7 [[written cleped{e}]] +competently+, conveniently, 35/8 +compt+, count, 47/29 +contynes+, contains, 21/12; [[written {con}tynes]] pp. +contenythe+, 38/39 [[written co{n}tenyth{e}]] +craft+, art, 3/4

+distingue+, divide, 51/5

+egalle+, equal, 45/21 [[written egall{e}]] +excep+, except, 5/16] +exclusede+, excluded, 34/37 [[written exclused{e}]] +excressent+, resulting, 35/16 [[written exc{re}ssent]] +exeant+, resulting, 43/26 +expone+, expound, 3/23

+ferye+ = ferþe, fourth, 70/12 +figure+ = figures, 5/1 [[written fig{ure}]] +for-by+, past, 12/11 +fors; no f.+, no matter, 22/24 +forseth+, matters, 53/30 +forye+ = forþe, forth, 71/8] +fyftye+ = fyftþe, fifth, 70/16

+grewe+, Greek, 33/13

+haluendel+, half, 16/16; +haldel+, 19/4; pl. +haluedels+, 16/16 +hayst+, hast, 17/3, 32 +hast+, haste, 22/25 [[in “haue hast to”]] +heer+, higher, 9/35 +here+, their, 7/26 [[in “in her{e} caputul{e}”]] +here-a-fore+, heretofore, 13/7 [[written her{e}-a-for{e}]] +heyth+, was called, 3/5 +hole+, whole, 4/39; +holle+, 17/1; +hoole+, of three dimensions, 46/15 +holdyþe+, holds good, 30/5 +how be it that+, although, 44/4

+lede+ = lete, let, 8/37 +lene+, lend, 12/39 +lest+, least, 43/27 [[in “at the lest”]] +lest+ = left, 71/9 [[in “the lest syde”]] +leue+, leave, 6/5; pr. 3 s. +leues+, remains, 11/19; [[first in 10/40]] +leus+, 11/28; pp. +laft+, left, 19/24 +lewder+, more ignorant, 3/3 [[written lewd{er}]] +lust+, desirest to, 45/13 +lyȝt+, easy, 15/31 +lymytes+, limits, 34/18; +lynes+, 34/12; +lynees+, 34/17; Lat. limes, pl. limites.

+maystery+, achievement; [[written mayst{er}y]] +no m.+, no achievement, i.e. easy, 19/10 +me+, indef. pron. one, 42/1 [[first in 34/16]] +mo+, more, 9/16 +moder+ = more (Lat. majorem), 43/22 +most+, must, 30/3 [[first in 3/12 and many more]] +multipliede+, +to be m.+ = multiplying, 40/9 +mynvtes+, the sixty parts into which a unit is divided, 38/25 [[written mynvt{es}]] +myse-wroȝt+, mis-wrought, 14/11

+nether+, nor, 34/25 [[in “It was, nether is”]] +nex+, next, 19/9 +noȝt+, nought, 5/7 [[first in 4/8]] +note+, not, 30/5

+oo+, one, 42/20; +o+, 42/21 [[first in 34/27; 33/22]] +omest+, uppermost, higher, 35/26; +omyst+, 35/28 +omwhile+, sometimes, 45/31 [[first in 39/17]] +on+, one, 8/29 [[in “on vnder an-oþ{er}”]] +opyne+, plain, 47/8 [[written opyn{e}]] +or+, before, 13/25 [[in “or þou be-gan”]] +or+ = þe oþ{er}, the other, 28/34 [[in “or by-twene”]] +ordure+, order, 34/9; row, 43/1 [[word form is “order”]] +other+, or, 33/13, 43/26; [[in “art other craft” on 33/13, “other how oft” on 43/26; note also “one other other” on 35/24]] +other . . . or+, either . . . or, 38/37 [[in “other it is even or od{e}” on 38/37; there are earlier occurrences]] +ouerer+, upper, 42/15 [[written ou{er}er]] +ouer-hippede+, passed over, 43/19 [[written ou{er}-hipped{e}]]

+recte+, directly, 27/20 [[in “stondes not recte”; also on 26/31 in “recte ou{er} his hede”]] +remayner+, remainder, 56/28 +representithe+, represented, 39/14 [[written rep{re}sentith{e}]] +resteth+, remains, 63/29 [[first in 57/29 and others]] +rewarde+, regard, 48/6 [[written reward{e}]] +rew+, row, 4/8 +rewle+, row, 4/20, 7/12; [[in “place of þe rewle”, “þe rewle of fig{ure}s”]] +rewele+, 4/18; +rewles+, rules, 5/33

+s.+ = scilicet, 3/8 [[in “s. Algorism{us}”]] +sentens+, meaning, 14/29 +signifye(tyf)+, 5/13. The last three letters are added above the line, evidently because of the word ‘significatyf’ in l. 14. But the ‘Solucio,’ which contained the word, has been omitted. +sithen+, since, 33/8 +some+, sum, result, 40/17, 32 [[first in 36/21 in “me may see a some”, then in “the same some” and “to some of”]] +sowne+, pronounce, 6/29 +singillatim+, singly, 7/25 +spices+, species, kinds, 34/4 [[first in 5/34 and others]] +spyl+, waste, 14/26 +styde+, stead, 18/20 +subtrahe+, subtract, 48/12; pp. +subtrayd+, 13/21 +sythes+, times, 21/16

+taȝt+, taught, 16/36 +take+, pp. taken; +t. fro+, starting from, 45/22 [[in “fro oone or tweyn{e} take”]] +taward+, toward, 23/34 +thouȝt+, though, 5/20 +trebille+, multiply by three, 49/26 [[written trebill{e}]] +twene+, two, 8/11 [[first in 4/23]] +þow+, though, 25/15 [[in “þow þ{o}u take”]] +þowȝt+, thought; +be þ.+, mentally, 28/4 +þus+ = þis, this, 20/33 [[in “þus nombur 214”]]

+vny+, unite, 45/10

+wel+, wilt, 14/31 [[in “If þ{o}u wel”]] +wete+, wit, 15/16; +wyte+, know, 8/38; pr. 2 s. +wost+, 12/38 +wex+, become, 50/18 +where+, whether, 29/12 [[written wher{e} in “wher{e} in þe secunde, or”]] +wher-thurghe+, whence, 49/15 [[written Wher-thurgh{e}]] +worch+, work, 8/19; [[first in 7/35]] +wrich+, 8/35; +wyrch+, 6/19; imp. s. +worch+, 15/9; [[first in 9/6]] pp. +y-wroth+, 13/24 +write+, written, 29/19; [[first in 6/37 in “hast write”, “be write”]] +y-write+, 16/1 +wryrchynge+ = wyrchynge, working, 30/4 [[written wryrchyng{e}]] +w^t+, with, 55/8

+y-broth+, brought, 21/18 +ychon+, each one, 29/10 [[written ychoɳ]] +ydo+, done, added, 9/6 [[first in 8/37 in “haue ydo”; 9/6 in “ydo all to-ged{er}”]] +ylke+, same, 5/12 +y-lyech+, alike, 22/23 +y-myȝt+, been able, 12/2 +y-nowȝt+, enough, 15/31; +ynovȝt+, 18/34 +yove+, given, 45/33 +y^t+, that, 52/8 +y-write+, v. +write.+ +y-wroth+, v. +worch.+

* * * * * * * * * * * * * *

MARGINAL NOTES:

+Headnotes+ have been moved to the beginning of the appropriate paragraph. Headnotes were omitted from the two Appendixes, as sidenotes give the same information.

+Line Numbers+ are cited in the Index and Glossary. They have been omitted from the e-text except in the one verse selection (App. II, Carmen de Algorismo). Instead, the Index and Glossary include supplemental information to help locate each word.

+Numbered Notes+:

Numbered sidenotes show page or leaf numbers from the original MSS. In the e-text, the page number is shown as [*123b] inline; mid-word page breaks are marked with a supplemental asterisk [*]. Numbers are not used.

Footnotes give textual information such as variant readings. They have been numbered sequentially within each title, with numbers shown as [{1}] to avoid confusion with bracked text--including single numerals--in the original. Editorial notes are shown as [1*]. When a footnote calls for added text, the addition is shown in the body text with [[double brackets]].

+Sidenotes+ giving a running synopsis of the text have been moved to the beginning of each paragraph, where they are shown as a single note.

ERRORS AND ANOMALIES (Noted by Transcriber):

Introduction:

dated Mij^c [In this and the remainder of the paragraph, the letter shown as ^c is printed directly above the preceding j.]

The Crafte of Nombrynge:

sursu{m} {pr}ocedas m{u}ltiplicando [Italicized as shown: error for “p{ro}cedas”?] Sidenote: Our author makes a slip here [Elsewhere in the book, numerical errors are corrected in the body text, with a footnote giving the original form.] ten tymes so mych is þe nounb{re} [text unchanged: error for “as”?] 6 tymes 24, [{19}]þen take [misplaced footnote anchor in original: belongs with “6 times 24”] Fn. 7: ‘Subt{ra}has a{u}t addis a dext{ri}s [open quote missing]

The Art of Nombryng:

oone of the digitis as .10. of 1.. 20. of. 2. [text unchanged: error for “as .10. of .1. 20. of .2.”?] sette a-side half of tho m{inutes} [text unchanged: error for “the”?] and. 10. as before is come therof [text unchanged: error for “and .10.”?] Sidenote: Where to set the quotiente [spelling (1922) unchanged] Sidenote: Definition of Progression. [f in “of” illegible] Sidenote: ... giving the value of ab.^2 [That is, “a(b^2).”]

Accomptynge by counters:

For example of the [*117a.] ly[*]nes [final . in sidenote missing or invisible] [also in 121b, 122a] which in the fyrst summe is 5 [invisible “5” supplied by transcriber] [*116a (sic).] [Editor’s “sic”: page numbering jumps back to 116 instead of the expected 123, and continues from 116.] [*123a] ... set downe y^e multiplyer .v. tymes, as here you se [Diagram shown as printed, with 35500 for 36500 in one column, and apparent misplaced “thousands” marker] 365 (which is the nomber of dayes ... [open ( missing_]

The arte of nombrynge by the hande:

for 1 the fynger is clasped in [In at least one printing of the text, “clasped” is misprinted as “elasped”] but this teacyed me not [text unchanged]