Chapter 5

[Ashmole MS. 396, fol. 48.]Boys seying in the begynnyng of hisArsemetrike:—AlleFol. 48.thynges that benefro the first begynnyng of thynges have procedede, and come forthe, And by resoun of nombre ben formede; And in wise as they bene, So owethethey to be knowene; wherfor in vniuersalleknowlechyng of thynges the Art of nombrynge is best, and most operatyfe.Therforesithenthe science of the whiche at this tyme weThe name of the art.intendeneto write of standithealleand about nombre: ffirst we most se, what is the propre name therofe, and fro whens the name come: Afterwardewhat is nombre, And how manye spices of nombre ther ben. The name is clepedeAlgorisme,Derivation of Algorism.hadeout of Algore, other of Algos, ingrewe, That is clepidein englissheartothercraft, And of Rithmusthat is calledenombre. Soalgorismeis clepedethe art of nombryng,Another.other it is had ofeen or in, and gogos that is introduccioun, and Rithmusnombre, that is to say Interduccioun of nombre.Another.And thirdly it is hadeof the name of a kyng that is clepedeAlgo and Rythmus; So calledeAlgorismus.Kinds of numbers.Sothely .2. manereof nombres ben notifiede; Formalle,1as nombreisvnitees gadredeto-gedres; Materialle,2as nombreis a colleccioun of vnitees. Other nombreis a multitude hadeout of vnitees, vnitee is that thynge wher-by euery thynge is calledeoone, otherothynge. Of nombres, that one is clepededigitalle, that othereArticle, Another a nombrecomponedeoþermyxt. Another digitalleis a nombre with-in .10.; Article is þat nombre that may be dyvydedein .10. parties egally, And that thereleve no residue; Componedeormedledeis that nombre that is come of a digite and of anarticle. And vndrestandewele that allenombres betwix .2. articles next is a nombrecomponede.The 9 rules of the Art.Of this art bene.9.spices, that is forto sey, numeracioun, addicioun, Subtraccioun, Mediacioun, Duplacioun, Multipliacioun, Dyvysioun, Progressioun, And of Rootes the extraccioun, and that may be hadein .2. maners, that is to sey in nombres quadrat, and in cubices: Amonge the whiche, ffirst of Numeracioun, and afterwardeof þe oþers byordure, y entende to write.Chapter I. Numeration.Fol. 48b.*For-sothenumeracioun is of euery numbre by competent figures an artificiallerepresentacioun.Figures, differences, places, and limits.Sothly figure, difference, places, andlynessupposen o thyng other the same, But they ben sette here for dyuers resons. ffigure is clepedefor protraccioun of figuracioun; Difference is calledefor therby is shewedeeuery figure, how it hathedifference fro the figures before them: place by cause of space, where-inmewritethe:lynees, for that is ordeynedefor the presentacioun of euery figure.The 9 figures.And vnderstonde that ther ben .9.lymytesof figures that representen the .9. digitesthat ben these. 0. 9. 8. 7. 6. 5. 4. 3. 2. 1.The cipher.The .10. is clepedetheta, or a cercle, other a cifre, other a figure of nought for nought it signyfiethe. Nathelesse she holdyng that place givetheothers for to signyfie; for withe-out cifre or cifres a pure article may not be writte.The numerationAnd sithen that by these .9. figures significatifesIoynedewithcifre or withcifres allenombres ben and may be representede, It was,netheris, no nede to fynde any more figures.of digits,And note wele that euery digite shallebe writte withoofigure allone to itaproprede.of articles,And allearticles by a cifre, ffor euery article is namedefor oone of the digitis as.10. of 1.. 20. of. 2.and so of the others, &c. And allenombres digitalleowen to be sette in the first difference: Allearticles in the seconde. Also allenombres fro .10. til an .100. [which] is excludede, with .2. figures mvst be writte; And yf it be an article, by a cifre first put, and the figure y-writte towardethe lift honde, that signifiethethe digit of the whichethe article is namede;of composites.And yf it be a nombre componede, ffirst write the digit that is a part of that componede, and write to the lift side the article as it is seidebe-fore. Allenombre that is fro an hundredetille a thousandeexclusede, owitheto be writ by .3. figures; and allenombre that is fro a thousandetil .x. Mł. mvst be writ by .4. figures; And so forthe.The value due to position.And vnderstondewele that euery figure sette in the first place signyfiethehis digit; In the secondeplace .10. tymes his digit; In the .3. place an hundredeso moche; In the .4. place a thousandeso moche; In the .5. place .x. thousandeso moche; In the .6. place an hundredethousandeso moche; In the .7. place a thousandethousande. And so infynytly mvltiplying byFol. 49.*these .3. 10, 100, 1000. And vnderstandewele thatcompetentlyme may sette vpon figure in the place of a thousande, a priketo shewe how many thousandethe last figure shallerepresent.Numbers are written from right to left.We writenein this art to the lift side-warde, as arabienewritene, that weren fynders of this science, otherefor this resoun, that for to kepe a custumable ordrein redyng, Sette we alle-wey the more nombre before.Chapter II. Addition.Definition.Addicioun is of nombre other of nombres vnto nombre or to nombres aggregacioun, that me may see that that is come therof asexcressent. In addicioun, 2. ordres of figures and .2. nombres ben necessary, that is to sey, a nombre to be addedeand the nombre wherto the addicioun sholdebe made to. The nombre to be addedeis that þat sholdebe addedetherto, and shallebe vnderwriten; the nombre vnto the whicheaddicioun shallebe made to is that nombre that resceyuethethe addicion of þat other, and shallebe writen above;How the numbers should be written.and it is convenient that the lesse nombre be vnderwrit, and the more addede, than the contrary. But whether it happeoneotherother, the same comytheof, Therfor, yf þow wilt adde nombre to nombre, write the nombre wherto the addicioun shallebe made in theomestordre by his differences, so that the first of the lower ordre be vndre the first of theomystordre, and so of others.The method of working.That done, adde the first of the lower ordre to the first of the omyst ordre. And of sucheaddicioun, other þere growiththerof a digit, An article, other a composede.Begin at the right.If it be digitus, In the place of the omyst shalt thow write the digit excrescyng, as thus:—The resultant2To whom it shal be addede1The nombre to be addede1The Sum is a digit,If the article; in the place of the omyst put a-way by a cifre writte, and the digit transferrede, of þe whichethe article toke his name, towardethe lift side, and be it addedeto the next figure folowyng, yf ther be any figure folowyng; or no, and yf it be not, leve it [in the] voide, as thus:—or an article,The resultant10To whom it shallebe addede7The nombre to be addede3Resultans27827Cui debetaddi10084Numerusaddendus17743And yf it happe that the figure folowyng wherto the addicioun shallebe made by [the cifre of] an article, it sette a-side;The resultant17To whom it shallebe addede10The nombre to be addede7In his place write theFol. 49b.*[digit of the] Article as thus:—And yf it happe that a figure of .9. by the figure that me mvst adde [one] to,The resultant10To whom it shallebe addede9The nombre to be addede1In the place of that 9. put a cifreandwrite þe article towardeþe lift hondeas bifore, and thus:—or a composite.And yf3[therefrom grow a] nombre componed,4[in the place of the nombre] put a-way5The resultant12To whom it shallebe addede8The nombre to be addede4[let] the digit [be]6writ þat is part of þat composide, and þan put to þe lift side the article as before, and þus:—The translator’s note.This done, adde the seconde to the seconde, and write above oþeras before. Note wele þat in addicions and in allespices folowyng, whan he seitheone the other shallebe writen aboue, and me most vse euerfigure, as that euery figure were sette by halfe, and by hym-selfe.Chapter III. Subtraction.Definition of Subtraction.Subtraccioun is of .2. proposedenombres, the fyndyng of the excesse of the more to the lasse: Other subtraccioun isablaciounof o nombre fro a-nother, that me may see asomeleft. The lasse of the more, or even of even, may be withdraw; The more fro the lesse may neuerbe.How it may be done.And sothly that nombre is more that hathemore figures, So that the last be signyficatifes: And yf ther ben as many in that one as in that other, me most deme it by the last, other by the next last.What is required.More-ouerin with-drawyng .2. nombres ben necessary; A nombre to be withdraw, And a nombre that me shallewith-draw of. The nombre to be with-draw shallebe writ in the lower ordre by his differences;Write the greater number above.Thenombre fro the whicheme shallewithe-draw in the omyst ordre, so that the first be vnder the first, the secondevnder the seconde, And so of alleothers.Subtract the first figure if possible.Withe-draw therfor the first of the lowereordre fro the first of the ordre above his hede, and that wolle beothermore or lesse, oþeregalle.The remanent20Wherof me shallewithdraw22The nombre to be withdraw2The remanent22Wherof me shallewith-draw28Þe nombre to be withdraw6yf it be egalleor even the figure sette beside, put in his place a cifre. And yf it be more put away þerfro als many of vnitees the lower figure conteynethe, and writ the residue as thusFol. 50.*Remanens221829998A quo sit subtraccio872430004Numerus subtrahendus657[6]....6If it is not possible ‘borrow ten,’And yf it be lesse, by-cause the more may not be with-draw ther-fro, borow an vnyte of the next figure that is worthe10. Of that .10. and of the figure that ye woldehave with-draw froand then subtract.be-fore to-gedre Ioynede,The remanent18Wherof me shallewith-draw24The nombre to be with-draw06with-draw þe figure be-nethe, and put the residue in the place of the figure put a-side as þus:—If the second figure is one.And yf the figure wherof me shal borow the vnyte be one, put it a-side, and write a cifre in the place þerof, lest the figures folowing faile of thairenombre, and þan worcheas it shewithin this figure here:—The remanent3098Wherof me shal with-draw312The nombre to be with-draw..3If the second figure is a cipher.And yf the vnyte wherof me shal borow be a cifre, go ferther to the figure signyficatife, and ther borow one, and retournyng bake, in the place of euery cifre þat ye passideouer, sette figures of .9. as here it is specifiede:—The remenaunt29999Wherof me shallewith-draw30003The nombre to be with-draw4And whan me cometheto the nombre wherof me intendithe, there remaynethealle-wayes .10. ffor þe whiche.10. &c.A justification of the rule given.The reson why þat for euery cifre left behynde me setteth figures ther of .9. this it is:—If fro the .3. place me borowedean vnyte, that vnyte by respect of the figure that he came fro representith an .C., In theplace of that cifre [passed over] is left .9., [which is worth ninety], and yit it remaynetheas .10., And the same resonewoldebe yf me hadeborowedean vnyte fro the .4., .5., .6., place, or ony other so vpwarde. This done, withdraw the secondeof the lower ordre fro the figure above his hede of þe omyst ordre, and wircheas before.Why it is better to work from right to left.And note wele that in addicion or in subtraccioun me may wele fro the lift side begynne and ryn to the right side, But it wol be more profitabler to be do, as it is taught.How to prove subtraction,And yf thow wilt prove yf thow have do wele or no, The figures that thow hast withdraw, adde themayeneto the omyst figures, and they wolle accorde withthe first that thow haddest yf thow have labored wele;and addition.and in like wise in addicioun, whan thow hast addedeallethy figures, withdraw them that thow firstFol. 50b.*addest, and the same wolle retourne. The subtraccioun is none other but a prouffeof the addicioun, and the contrarye in like wise.Chapter IV. Mediation.Definition of mediation.Mediaciounis the fyndyng of the halfyng of euery nombre, that it may be seynewhat and how mocheis euery halfe. In halfyng ay oo order of figures and oo nombre is necessary, that is to sey the nombre to be halfede. Therfor yf thow wilt half any nombre, write that nombre by his differences, andWhere to begin.begynne at the right, that is to sey, fro the first figure to the right side, so that it be signyficatifeother represent vnyte or eny other digitallenombre. If it be vnyte write in his place a cifre for theIf the first figure is unity.figures folowyng, [lest they signify less], and write that vnyte without in the table, other resolue it in .60.mynvtesand sette a-side half ofthominutesso, and reserve the remenaunt without in the table, as thus .30.; other sette without thus .dī: that kepethenone ordre of place, Nathelesse it hathesignyficacioun. And yf the other figure signyfie any other digital nombre fro vnyte forthe, oþerthe nombre is odeor evene.Halfede22to be halfede44halfede23[di]To be halfede47What to do if it is not unity.If it be even, write this half in this wise:—And if it be odde, Take the next even vndre hym conteynede, and put his half in the place of that odde, and of þe vnyte that remaynetheto be halfededo thus:—Then halve the second figure.This done, the secondeis to be halfede, yf it be a cifre put it be-side, and yf it be significatife,other it is even or ode: If it be even, write in the place of þe nombres wipedeout the halfe; yf it be ode, take the next even vnder itcontenythe, and in the place of the Impar sette a-side put half of the even: Thevnyte that remaynetheto be halfede, respect hadeto them before, is worthe.10.Halfedeto be halfedeIf it is odd, add 5 to the figure before.Dyvide that .10. in .2., 5. is, and sette a-side that one, and adde that other to the next figure precedent as here:—And yf þe addicioun sholdebe made to a cifre, sette it a-side, and write in his place .5.doublede2689010174to be doublede13445587And vnder this fourme me shallewrite and worche, tillethe totallenombre be halfede.Chapter V. Duplation.Definition of Duplation.Duplicacioun is agregacion of nombre [to itself] þat me may se the nombre growen. In doublyngeay is but one ordre of figures necessarie. And me most be-gynne withthe lift side, other of the more figure, And after the nombre of the more figurerepresentithe.Fol. 51.*In the other .3. before we begynne alleway fro the right side and fro the lasse nombre,Where to begin.In this spice and in alleother folowyng we wolle begynne fro the lift side, ffor and me bigon thedouble fro the first,omwhileme myght double oo thynge twyes.Why.And how be it that me myght double fro the right, that woldebe harder in techyng and in workyng. Therfor yf thow wolt double any nombre, write that nombre by his differences, and double the last. And of that doublyng other growithea nombre digital, article, or componede. [If it be a digit, write it in the place of the first digit.]double10to be doublede5What to do with the result.If it be article, write in his place a cifre and transferre the article towardethe lift, as thus:—And yf the nombre be componede,doublede16to be doublede8write a digital that is part of his composicioun, and sette the article to the lift hande, as thus:—That done, me most double the last save one, and what growetheþerof me most worche as before. And yf a cifre be, toucheit not. But yf any nombre shallebe addedeto the cifre,doublede606to be doublede303in þe place of þe figure wipedeout me most write the nombre to be addede, as thus:—In the same wise me shallewircheof alleothers.How to prove your answer.And this probacioun:Doublede618to be doublede309If thow truly double the halfis, and truly half the doubles, the same nombre and figure shallemete, sucheas thow labouredevponefirst, And of the contrarie.Chapter VI. Multiplication.Definition of Multiplication.Multiplicacioun of nombre by hym-self other by a-nother, withproposide.2. nombres, [is] the fyndyng of the thirde, That so oft conteynethethat other, as ther ben vnytes in the oþer. In multiplicacioun .2. nombres pryncipally ben necessary, that is to sey, the nombre multiplying and the nombre to be multipliede, as here;—twies fyve.Multiplier.[The number multiplying] is designedeaduerbially.Multiplicand.The nombre to be multipliederesceyvethea nominalleappellacioun, as twies .5. 5. is the nombre multipliede, and twies is the nombre to bemultipliede.Resultans910132668008Multiplicandus..5..4.34004Multiplicans.22.33222...Product.Also me may thervponeto assigne the. 3. nombre, the whicheisFol. 51b.*clepedeproduct or provenient, of takyng out of one fro another: as twyes .5 is .10., 5. the nombre to be multipliede, and .2. the multipliant,and. 10.as before is come therof. And vnderstonde wele, that of the multipliant may be made the nombre to be multipliede, and of the contrarie, remaynyng euerthe samesome, and herofecomethethe comen speche, that seitheall nombre is convertedeby Multiplying in hym-selfe.The Cases of Multiplication.There are 6 rules of Multiplication.123456789102468101010141618203691215182124273048121620242832364051015202530354045506121824303642485660714212835424956637081624324048566472809182736455463728190102030405060708090100And ther ben .6 rules of Multiplicacioun;(1) Digit by digit.ffirst, yf a digit multiplie a digit, considrehow many of vnytees ben betwix the digit by multiplying and his .10. betheto-gedre accomptede, and so oft with-draw the digit multiplying, vnder the article of his denominacioun. Example of grace. If thow wolt wete how mocheis .4. tymes .8.,11se how many vnytees ben betwix .8.12and .10. to-geder rekenede, and it shewiththat .2.: withdraw ther-for the quaternary, of the article of his denominacion twies, of .40., And ther remaynethe.32., that is, tosomeof allethe multiplicacioun.See the table above.Wher-vpon for more evidence and declaracion the seidetable is made.(2) Digit by article.Whan a digit multipliethean article, thow most bryng the digit into þe digit, of þe whichethe article [has]13his name, and euery vnyteshallestondefor .10., and euery article an .100.(3) Composite by digit.Whan the digit multipliethea nombre componede, þou most bryng the digit into aiþerpart of the nombre componede, so þat digit be had into digit by the first rule, into an article by þe seconderule; and afterwardeIoyne the produccioun, and þere wol be the some totalle.Resultans1267361201208Multiplicandus23264Multiplicans632320302(4) Article by article.Whan an article multipliethean article, the digit wherof he is namedeis to be brought Into the digit wherof the oþeris namede, and euery vnyte wol be wortheFol. 52.*an .100., and euery article. a .1000.(5) Composite by article.Whan an article multipliethea nombre componede, thow most bryng the digit of the article into aither part of the nombre componede; and Ioyne the produccioun, and euery article wol be worthe.100., and euery vnyte .10., and so wollethe some be opene.(6) Composite by composite.Whan a nombre componedemultipliethea nombre componede, euery part of the nombre multiplying is to be hadeinto euery part of the nombre to be multipliede, and so shallethe digit be hadetwies, onys in the digit, that other in the article. The article also twies, ones in the digit, that other in the article. Therfor yf thow wilt any nombre by hym-self other by any other multiplie, write the nombre to be multipliedein the ouerordre by his differences,How to set down your numbers.The nombre multiplying in the lower ordre by his differences, so that the first of the lower ordre be vnder the last of the ouerordre. This done, of the multiplying, the last is to be hadeinto the last of the nombre to be multipliede. Wherof than wolle grow a digit, an article, other a nombre componede.If the result is a digit,The resultant6To be multipliede3Þe nombre multipliyng2If it be a digit, even above the figure multiplying is hede write his digit that come of, as it apperethehere:—an article,And yf an article had be writ ouerthe figure multiplying his hede, put a cifre þerand transferre the article towardethe lift hande, as thus:—The resultant10to be multipliede5þe nombre multipliyng2or a composite.And yf a nombre componedebe writ ouerthe figure multyplying is hede, write the digit in the nombre componedeis place, and sette the article to the lift hande, as thus:—Resultant12to be multipliede4the nombre multipliyng3Multiply next by the last but one, and so on.This done,memost bryng the last save one of the multipliyng into the last of þe nombre to be multipliede, and se what comythetherof as before, and so do withalle, tille me come to the first of the nombre multiplying, that must be brought into the last of the nombre to be multipliede, wherof growitheoþera digit, an article,Fol. 52b.*other a nombre componede.Resultant66to be multipliede3the nombre multipliyng22The resultant110to be multipliede5þe nombre multiplying22The resultant13152to be multipliede4þe nombremultipliant33If it be a digit, In the place of theouerer, sette a-side, as here:If an article happe, there put a cifre in his place, and put hym to the lift hande, as here:If it be a nombre componede, in the place of the ouerer sette a-side, write a digit that14is a part of the componede, and sette on the left hondethe article, as here:Then antery the multiplier one place.That done, sette forwardethe figures of the nombre multiplying byoodifference, so that the first of the multipliant be vnder the last save one of the nombre to be multipliede, the other byoplace sette forwarde. Than me shallebryngethe last of the multipliant in hym to be multipliede, vnder the whicheis the first multipliant.Work as before.And than wolle growe oþera digit, an article, or a componedenombre. If it be a digit, adde hym even above his hede; If it be an article, transferre hym to the lift side; And if it be a nombre componede, adde a digit to the figure above his hede, and sette to the lift handethe article. And alle-wayes euery figure of the nombre multipliant is to be brought to the last save one nombre to be multipliede, til me come to the first of the multipliant, where me shallewirche as it is seidebefore of the first, and afterwardeto put forwardethe figures by o difference and one tillethey allebe multipliede.How to deal with ciphers.And yf it happe that the first figure of þe multipliant be a cifre, andboueit is sette the figure signyficatife, write a cifre in the place of the figuresette a-side, as thus,etc.:The resultant120to be multipliede6the multipliant20How to deal with ciphers.And yf a cifre happe in the lowerorderbe-twix the first and the last, and even above be sette the figure signyficatif,The resultant22644To be multipliede222The multipliant102leve it vntouchede, as here:—And yf the space above sette be voide, in that place write thow a cifre. And yf the cifre happe betwix þe first and the last to be multipliede, me most sette forwardethe ordre of the figures by thairedifferences, for oft ofducciounof figures in cifres nought is the resultant, as here,Resultant8008to be multipliede4004the multipliant2...Fol. 53.*wherof it is evident and open, yf that the first figure of the nombre be to be multipliedebe a cifre, vndir it shallebe none sette as here:—Resultant32016To be multipliede80The multipliant4Leave room between the rows of figures.Vnder[stand] also that in multiplicacioun, divisioun, and of rootis the extraccioun, competently me may leve a mydel space betwix .2. ordres of figures, that me may write there what is come of addyng other withe-drawyng, lest any thynge sholdebeouer-hippedeand sette out of mynde.Chapter VII. Division.Definition of division.For to dyvyde oo nombre by a-nother, it is of .2. nombres proposede, It is forto depart themodernombre into as many partis as ben of vnytees in the lasse nombre. And note wele that in makyngeof dyvysioun ther ben .3. nombres necessary:Dividend, Divisor, Quotient.that is to sey, the nombre to be dyvydede; the nombre dyvydyng and the nombreexeant,otherhow oft, or quocient. Ay shallethe nombre that is to be dyvydedebe more, other at thelestevenewiththe nombre the dyvysere, yf the nombre shallebe madeby hole nombres.How to set down your Sum.Therfor yf thow wolt any nombre dyvyde, write the nombre to be dyvydedein þe ouererbordureby his differences, the dyviserein the lower ordureby his differences, so that the last of the dyviser be vnder the last of the nombre to be dyvyde, the next last vnder the next last, and so of the others, yf it may competently be done;An example.as here:—The residue27The quotient5To be dyvydede342The dyvyser63Examples.Residuum82726Quociens212259Diuidendus68066342332Diuiser3236334When the last of the divisor must not be set below the last of the dividend.And ther ben .2. causes whan the last figure may not be sette vnder the last, other that the last of the lower nombre may not be with-draw of the last of the ouerer nombre for it is lasse than the lower, otherhow be it, thatit myght be with-draw as for hym-self fro the ouerer the remenaunt may not so oft of them above, other yf þe last of the lower be even to the figure above his hede, and þe next last oþerthe figure be-fore þat be more þan the figure above sette.Fol. 532.*These so ordeynede, me most wirchefrom the last figure of þe nombre of the dyvyser, and se how oft it may be with-draw ofHow to begin.and fro the figure aboue his hede, namly so that the remenaunt may be take of so oft, and to se the residue as here:—The residue26The quocient9To be dyvydede332The dyvyser34An example.And note wele that me may not withe-draw more than .9. tymes nether lasse than ones. Therfor se how oft þe figures of the lower ordre may be with-draw fro the figures of the ouerer, and the nombre that shewithþe quocient most be writ ouerthe hede of þat figure, vnder the whichethe first figure is, of the dyviser;Where to set thequotienteAnd by that figure me most withe-draw alleoþerfigures of the lower ordir and that of the figures aboue thairehedis. This so done, me most sette forwardeþe figures of the diuiser by o difference towardesthe right hondeand worcheas before; and thus:—Examples.Residuum.12quociens6542004Diuidendus3551228863704Diuisor5434423The quocient654To be dyvydede355122The dyvyser543A special case.And yf it happeafter þe settyng forwardeof the figures þat þe last of the divisor may not so oft be withdraw of the figure above his hede, above þat figure vnder the whichethe first of the diuiser is writ me most sette a cifre in ordre of the nombre quocient, and sette the figures forwardeas be-fore be o difference alone, and so me shalledo in allenombres to be dyvidede, for where the dyviser maynot be with-draw me most sette there a cifre, and sette forwardethe figures; as here:—The residue12The quocient2004To be dyvydede8863704The dyvyser4423Another example.And me shallenot cesse fro suchesettyng of figures forwarde, nether of settyngeof þe quocient into the dyviser, neþerof subtraccioun of the dyvyser, tillethe first of the dyvyser be with-draw fro þe first to be dividede. The whichedone, or ought,17oþernought shalleremayne: and yf it be ought,17kepe it in the tables, And euervnyit to þe diviser. And yf þou wilt wete how many vnytees of þe divisiounFol. 533.*wol growe to the nombre of the divisere,What the quotient shows.the nombre quocient wol shewe it: and whan suchedivisioun is made, and þoulustprove yf thow have wele done orHow to prove your division,no, Multiplie the quocient by the diviser, And the same figures wolle come ayene that thow haddest bifore and none other. And yf ought be residue, than withaddicioun therof shallecome the same figures: And so multiplicacioun provithedivisioun, and dyvisioun multiplicacioun:or multiplication.as thus, yf multiplicacioun be made, divide it by the multipliant, and the nombre quocient wol shewe the nombre that was to be multipliede,etc.Chapter VIII. Progression.DefinitionofProgression.Progressioun is of nombre afteregalleexcesse fro oone or tweynetakeagregacioun. of progressioun one isnaturelleor contynuelle, þat oþerbroken and discontynuelle.Natural Progression.Naturelleit is, whan me begynnethewithone, and kepetheordure ouerlepyng one; as .1. 2. 3. 4. 5. 6.,etc., so þat the nombre folowyngepassithethe other be-fore in one.Broken Progression.Broken it is, whan me lepithefro o nombre tilleanother, and kepithenot the contynuel ordire; as 1. 3. 5. 7. 9,etc. Ay me may begynne with.2., as þus; .2. 4. 6. 8.,etc., and the nombre folowyng passethethe others by-fore by .2. And note wele, that naturelleprogressioun ay begynnethewithone, and Intercise or broken progressioun,omwhilebegynnythewith one, omwhile withtwayne. Of progressioun naturell .2. rules ther beyove, of the whichethe first is this;The 1st rule for Natural Progression.whan the progressioun naturelleendithein even nombre, by the half therof multiplie þe next totalleouererenombre; Example of grace: .1. 2. 3. 4. Multiplie .5. by .2. and so .10. cometheof, that is the totallenombre þerof.The second rule.The seconderule is suche, whan the progressioun naturelleendithein nombre ode. Take the more porcioun of the oddes, and multiplie therby the totallenombre. Example of grace 1. 2. 3. 4. 5., multiplie.5. by .3, and thryes .5. shallebe resultant. so the nombre totalleis .15.The first rule of Broken Progression.Of progresiounintercise, ther ben also .2.18rules; and þe first is þis: Whan the Intercise progression endithein even nombre by half therof multiplie the next nombre to þat halfeas .2.184. 6. Multiplie .4. by .3. so þat is thryes .4., and .12. the nombre of allethe progressioun, wollefolow.The second rule.The seconderule is this: whan the progressioun interscise endithein ode, take þe more porcioun of alleþe nombre,Fol. 534.*and multiplie by hym-selfe; as .1. 3. 5. Multiplie .3. by hym-selfe, and þe some of allewolle be .9.,etc.Chapter IX. Extraction of Roots.The preamble of the extraction of roots.Here folowithethe extraccioun of rotis, and first in nombre quadrates. Wherfor me shallese what is anombre quadrat, and what is the rote of a nombre quadrat, and what it is to draw out the rote of a nombre. And before other note this divisioun:Linear, superficial, and solid numbers.Of nombres one islyneal, anoþersuperficialle, anoþerquadrat, anoþercubikeorhoole. lyneal is that þat is considredeafter the processe, havyngeno respect to the direccioun of nombre in nombre, As a lyne hathebut one dymensioun that is to sey after the lengthe.Superficial numbers.Nombre superficial is þat cometheof ledyngeof oo nombre into a-nother, wherfor it is calledesuperficial, for it hathe.2. nombres notyng or mesuryngehym, as a superficiallethyngehathe.2. dimensions, þat is to sey lengtheand brede.Square numbers.And for bycause a nombre may be hadein a-nother by .2. maners, þat is to sey other in hym-selfe, oþerin anoþer, Vnderstondeyf it be had in hym-self, It is a quadrat. ffor dyvisioun write by vnytes, hathe.4. sides even as a quadrangille. and yf the nombre be hadein a-noþer, the nombre is superficiel and not quadrat, as .2. hadein .3. makethe.6. that is þe first nombre superficielle; wherfor it is open þat allenombre quadrat is superficiel, and notconuertide.The root of a square number.The rote of a nombre quadrat is þat nombre that is had of hym-self, as twies .2. makithe4. and .4. is the first nombre quadrat, and 2. is his rote. 9. 8. 7. 6. 5. 4. 3. 2. 1. / The rote of the more quadrat .3. 1. 4. 2. 6.Notes of some examples of square roots here interpolated.The most nombre quadrat 9. 8. 7. 5. 9. 3. 4. 7. 6. / the remenent ouerthe quadrat .6. 0. 8. 4. 5. / The first caas of nombre quadrat .5. 4. 7. 5. 6. The rote .2. 3. 4. The secondecaas .3. 8. 4. 5. The rote .6. 2. The thirdecaas .2. 8. 1. 9. The rote .5. 3. The .4. caas .3. 2. 1. The rote .1. 7. / The 5. caas .9. 1. 2. 0. 4. / The rote 3. 0. 2.Solid numbers.Thesolidenombreor cubikeis þat þat comytħe of double ledyng of nombre in nombre;Three dimensions of solids.And it is clepedea solidebody that hatheþer-in .3[dimensions] þat is to sey, lengthe, brede, and thiknesse. so þat nombre hathe.3. nombres to be brought forthein hym. But nombre may be hadetwies in nombre, for other it is hadein hym-selfe, oþerin a-noþer.Cubic numbers.If a nombre be hadetwies in hym-self, oþerones in his quadrat, þat is the same, þat acubikeFol. 54.*is, And is the same that is solide. And yf a nombre twies be hadein a-noþer, the nombre isclepedesolide and not cubike, as twies .3. and þat .2. makithe.12.All cubics are solid numbers.Wherfor it isopynethat allecubikenombre is solide, and notconuertide. Cubikeis þat nombre þat comytheof ledyngeof hym-selfetwyes, or ones in his quadrat. And here-by it is open that o nombre is therooteof a quadrat and of a cubike. Natheles the same nombre is not quadrat and cubike.No number may be both linear and solid.Opyneit is also that allenombres may be a rote to a quadrat and cubike, but not allenombre quadrat or cubike. Therfor sithen þe ledyngeof vnyte in hym-self ones or twies nought comethebut vnytes, SeitheBoice in Arsemetrike,Unity is not a number.that vnyte potencially is al nombre, and none in act. And vndirstondewele also that betwix euery .2. quadratesther is a meene proporcionalle,Examples of square roots.That is openedethus;lede the rote of o quadrat intothe rote of the oþerquadrat, and þan wolle þe meene shew.Residuum0400Quadrande43563029174241936Duplum121026[8]19Subduplum665513244A note on mean proportionals.Also betwix the next .2. cubikis, me may fynde a double meene, that is to sey a more meene and a lesse. The more meene thus, as to bryngethe rote of the lesse into a quadrat of the more. The lesse thus, If the rote of the more be brought Into the quadrat of the lesse.Chapter X. Extraction of Square Root.To20draw a rote of the nombre quadrat it is What-euernombre be proposedeto fynde his rote and to se yf it be quadrat.To find a square root.And yf it be not quadrat the rote of the most quadrat fynde out, vnder the nombre proposede. Therfor yf thow wilt the rote of any quadrat nombre draw out, write the nombre by his differences, andcomptthe nombre of the figures, and wete yf it be odeor even. And yfBegin with the last odd place.it be even, than most thow begynne worche vnder the last save one. And yf it be odewiththe last; and forto sey it shortly, al-weyes fro the last odeme shallebegynne. Therfor vnder the last in an od place sette,Find the nearest square root of that number, subtract,me most fyndea digit, the whicheladein hym-selfeit puttitheaway that, þat is ouerhis hede, oþeras neigheas memay: suche a digit foundeand withdraw fro his ouerer, me most double that digit and sette the double vnder the next figure towardethe right honde, and hisvnder doublevnder hym.double it,That done, than me most fyndea-noþerdigit vnder the next figure bifore the doublede,and set the double one to the right.the whicheFol. 54b.*brought in double settethea-way allethat is ouerhis hede as torewardeof the doublede: Than brought into hym-self settitheall away in respect of hym-self,Find the second figure by division.Other do it as nye as it may be do: other me may with-draw the digit21[last] founde, and lede hym in double or double hym, and after in hym-selfe;Multiply the double by the second figure, and add after it the square of the second figure, and subtract.Than Ioyne to-geder the produccioneof them bothe, So that the first figure of the last product be addedebefore the first of the first productes, the secondeof the first,etc. and so forthe,subtrahefro the totallenombre in respect of þe digit.Examples.The residue5432To be quadrede41209151399005432The double4024600The vnder double203123[3][0][0]0

[Ashmole MS. 396, fol. 48.]

Boys seying in the begynnyng of hisArsemetrike:—AlleFol. 48.thynges that benefro the first begynnyng of thynges have procedede, and come forthe, And by resoun of nombre ben formede; And in wise as they bene, So owethethey to be knowene; wherfor in vniuersalleknowlechyng of thynges the Art of nombrynge is best, and most operatyfe.

Therforesithenthe science of the whiche at this tyme weThe name of the art.intendeneto write of standithealleand about nombre: ffirst we most se, what is the propre name therofe, and fro whens the name come: Afterwardewhat is nombre, And how manye spices of nombre ther ben. The name is clepedeAlgorisme,Derivation of Algorism.hadeout of Algore, other of Algos, ingrewe, That is clepidein englissheartothercraft, And of Rithmusthat is calledenombre. Soalgorismeis clepedethe art of nombryng,Another.other it is had ofeen or in, and gogos that is introduccioun, and Rithmusnombre, that is to say Interduccioun of nombre.Another.And thirdly it is hadeof the name of a kyng that is clepedeAlgo and Rythmus; So calledeAlgorismus.Kinds of numbers.Sothely .2. manereof nombres ben notifiede; Formalle,1as nombreisvnitees gadredeto-gedres; Materialle,2as nombreis a colleccioun of vnitees. Other nombreis a multitude hadeout of vnitees, vnitee is that thynge wher-by euery thynge is calledeoone, otherothynge. Of nombres, that one is clepededigitalle, that othereArticle, Another a nombrecomponedeoþermyxt. Another digitalleis a nombre with-in .10.; Article is þat nombre that may be dyvydedein .10. parties egally, And that thereleve no residue; Componedeormedledeis that nombre that is come of a digite and of anarticle. And vndrestandewele that allenombres betwix .2. articles next is a nombrecomponede.The 9 rules of the Art.Of this art bene.9.spices, that is forto sey, numeracioun, addicioun, Subtraccioun, Mediacioun, Duplacioun, Multipliacioun, Dyvysioun, Progressioun, And of Rootes the extraccioun, and that may be hadein .2. maners, that is to sey in nombres quadrat, and in cubices: Amonge the whiche, ffirst of Numeracioun, and afterwardeof þe oþers byordure, y entende to write.

Chapter I. Numeration.

Figures, differences, places, and limits.Sothly figure, difference, places, andlynessupposen o thyng other the same, But they ben sette here for dyuers resons. ffigure is clepedefor protraccioun of figuracioun; Difference is calledefor therby is shewedeeuery figure, how it hathedifference fro the figures before them: place by cause of space, where-inmewritethe:lynees, for that is ordeynedefor the presentacioun of euery figure.The 9 figures.And vnderstonde that ther ben .9.lymytesof figures that representen the .9. digitesthat ben these. 0. 9. 8. 7. 6. 5. 4. 3. 2. 1.The cipher.The .10. is clepedetheta, or a cercle, other a cifre, other a figure of nought for nought it signyfiethe. Nathelesse she holdyng that place givetheothers for to signyfie; for withe-out cifre or cifres a pure article may not be writte.The numerationAnd sithen that by these .9. figures significatifesIoynedewithcifre or withcifres allenombres ben and may be representede, It was,netheris, no nede to fynde any more figures.of digits,And note wele that euery digite shallebe writte withoofigure allone to itaproprede.of articles,And allearticles by a cifre, ffor euery article is namedefor oone of the digitis as.10. of 1.. 20. of. 2.and so of the others, &c. And allenombres digitalleowen to be sette in the first difference: Allearticles in the seconde. Also allenombres fro .10. til an .100. [which] is excludede, with .2. figures mvst be writte; And yf it be an article, by a cifre first put, and the figure y-writte towardethe lift honde, that signifiethethe digit of the whichethe article is namede;of composites.And yf it be a nombre componede, ffirst write the digit that is a part of that componede, and write to the lift side the article as it is seidebe-fore. Allenombre that is fro an hundredetille a thousandeexclusede, owitheto be writ by .3. figures; and allenombre that is fro a thousandetil .x. Mł. mvst be writ by .4. figures; And so forthe.The value due to position.And vnderstondewele that euery figure sette in the first place signyfiethehis digit; In the secondeplace .10. tymes his digit; In the .3. place an hundredeso moche; In the .4. place a thousandeso moche; In the .5. place .x. thousandeso moche; In the .6. place an hundredethousandeso moche; In the .7. place a thousandethousande. And so infynytly mvltiplying byFol. 49.*these .3. 10, 100, 1000. And vnderstandewele thatcompetentlyme may sette vpon figure in the place of a thousande, a priketo shewe how many thousandethe last figure shallerepresent.Numbers are written from right to left.We writenein this art to the lift side-warde, as arabienewritene, that weren fynders of this science, otherefor this resoun, that for to kepe a custumable ordrein redyng, Sette we alle-wey the more nombre before.

Chapter II. Addition.

Definition.Addicioun is of nombre other of nombres vnto nombre or to nombres aggregacioun, that me may see that that is come therof asexcressent. In addicioun, 2. ordres of figures and .2. nombres ben necessary, that is to sey, a nombre to be addedeand the nombre wherto the addicioun sholdebe made to. The nombre to be addedeis that þat sholdebe addedetherto, and shallebe vnderwriten; the nombre vnto the whicheaddicioun shallebe made to is that nombre that resceyuethethe addicion of þat other, and shallebe writen above;How the numbers should be written.and it is convenient that the lesse nombre be vnderwrit, and the more addede, than the contrary. But whether it happeoneotherother, the same comytheof, Therfor, yf þow wilt adde nombre to nombre, write the nombre wherto the addicioun shallebe made in theomestordre by his differences, so that the first of the lower ordre be vndre the first of theomystordre, and so of others.The method of working.That done, adde the first of the lower ordre to the first of the omyst ordre. And of sucheaddicioun, other þere growiththerof a digit, An article, other a composede.Begin at the right.If it be digitus, In the place of the omyst shalt thow write the digit excrescyng, as thus:—

The Sum is a digit,If the article; in the place of the omyst put a-way by a cifre writte, and the digit transferrede, of þe whichethe article toke his name, towardethe lift side, and be it addedeto the next figure folowyng, yf ther be any figure folowyng; or no, and yf it be not, leve it [in the] voide, as thus:—

And yf it happe that the figure folowyng wherto the addicioun shallebe made by [the cifre of] an article, it sette a-side;

In his place write theFol. 49b.*[digit of the] Article as thus:—

And yf it happe that a figure of .9. by the figure that me mvst adde [one] to,

In the place of that 9. put a cifreandwrite þe article towardeþe lift hondeas bifore, and thus:—

or a composite.And yf3[therefrom grow a] nombre componed,4[in the place of the nombre] put a-way5

[let] the digit [be]6writ þat is part of þat composide, and þan put to þe lift side the article as before, and þus:—

The translator’s note.This done, adde the seconde to the seconde, and write above oþeras before. Note wele þat in addicions and in allespices folowyng, whan he seitheone the other shallebe writen aboue, and me most vse euerfigure, as that euery figure were sette by halfe, and by hym-selfe.

Chapter III. Subtraction.

Definition of Subtraction.Subtraccioun is of .2. proposedenombres, the fyndyng of the excesse of the more to the lasse: Other subtraccioun isablaciounof o nombre fro a-nother, that me may see asomeleft. The lasse of the more, or even of even, may be withdraw; The more fro the lesse may neuerbe.How it may be done.And sothly that nombre is more that hathemore figures, So that the last be signyficatifes: And yf ther ben as many in that one as in that other, me most deme it by the last, other by the next last.What is required.More-ouerin with-drawyng .2. nombres ben necessary; A nombre to be withdraw, And a nombre that me shallewith-draw of. The nombre to be with-draw shallebe writ in the lower ordre by his differences;Write the greater number above.Thenombre fro the whicheme shallewithe-draw in the omyst ordre, so that the first be vnder the first, the secondevnder the seconde, And so of alleothers.Subtract the first figure if possible.Withe-draw therfor the first of the lowereordre fro the first of the ordre above his hede, and that wolle beothermore or lesse, oþeregalle.

yf it be egalleor even the figure sette beside, put in his place a cifre. And yf it be more put away þerfro als many of vnitees the lower figure conteynethe, and writ the residue as thus

A quo sit subtraccio

Numerus subtrahendus

If it is not possible ‘borrow ten,’And yf it be lesse, by-cause the more may not be with-draw ther-fro, borow an vnyte of the next figure that is worthe10. Of that .10. and of the figure that ye woldehave with-draw froand then subtract.be-fore to-gedre Ioynede,

with-draw þe figure be-nethe, and put the residue in the place of the figure put a-side as þus:—

If the second figure is one.And yf the figure wherof me shal borow the vnyte be one, put it a-side, and write a cifre in the place þerof, lest the figures folowing faile of thairenombre, and þan worcheas it shewithin this figure here:—

If the second figure is a cipher.And yf the vnyte wherof me shal borow be a cifre, go ferther to the figure signyficatife, and ther borow one, and retournyng bake, in the place of euery cifre þat ye passideouer, sette figures of .9. as here it is specifiede:—

And whan me cometheto the nombre wherof me intendithe, there remaynethealle-wayes .10. ffor þe whiche.10. &c.A justification of the rule given.The reson why þat for euery cifre left behynde me setteth figures ther of .9. this it is:—If fro the .3. place me borowedean vnyte, that vnyte by respect of the figure that he came fro representith an .C., In theplace of that cifre [passed over] is left .9., [which is worth ninety], and yit it remaynetheas .10., And the same resonewoldebe yf me hadeborowedean vnyte fro the .4., .5., .6., place, or ony other so vpwarde. This done, withdraw the secondeof the lower ordre fro the figure above his hede of þe omyst ordre, and wircheas before.Why it is better to work from right to left.And note wele that in addicion or in subtraccioun me may wele fro the lift side begynne and ryn to the right side, But it wol be more profitabler to be do, as it is taught.How to prove subtraction,And yf thow wilt prove yf thow have do wele or no, The figures that thow hast withdraw, adde themayeneto the omyst figures, and they wolle accorde withthe first that thow haddest yf thow have labored wele;and addition.and in like wise in addicioun, whan thow hast addedeallethy figures, withdraw them that thow firstFol. 50b.*addest, and the same wolle retourne. The subtraccioun is none other but a prouffeof the addicioun, and the contrarye in like wise.

Chapter IV. Mediation.

Definition of mediation.Mediaciounis the fyndyng of the halfyng of euery nombre, that it may be seynewhat and how mocheis euery halfe. In halfyng ay oo order of figures and oo nombre is necessary, that is to sey the nombre to be halfede. Therfor yf thow wilt half any nombre, write that nombre by his differences, andWhere to begin.begynne at the right, that is to sey, fro the first figure to the right side, so that it be signyficatifeother represent vnyte or eny other digitallenombre. If it be vnyte write in his place a cifre for theIf the first figure is unity.figures folowyng, [lest they signify less], and write that vnyte without in the table, other resolue it in .60.mynvtesand sette a-side half ofthominutesso, and reserve the remenaunt without in the table, as thus .30.; other sette without thus .dī: that kepethenone ordre of place, Nathelesse it hathesignyficacioun. And yf the other figure signyfie any other digital nombre fro vnyte forthe, oþerthe nombre is odeor evene.

What to do if it is not unity.If it be even, write this half in this wise:—

And if it be odde, Take the next even vndre hym conteynede, and put his half in the place of that odde, and of þe vnyte that remaynetheto be halfededo thus:—

Then halve the second figure.This done, the secondeis to be halfede, yf it be a cifre put it be-side, and yf it be significatife,other it is even or ode: If it be even, write in the place of þe nombres wipedeout the halfe; yf it be ode, take the next even vnder itcontenythe, and in the place of the Impar sette a-side put half of the even: Thevnyte that remaynetheto be halfede, respect hadeto them before, is worthe.10.

If it is odd, add 5 to the figure before.Dyvide that .10. in .2., 5. is, and sette a-side that one, and adde that other to the next figure precedent as here:—

And yf þe addicioun sholdebe made to a cifre, sette it a-side, and write in his place .5.

And vnder this fourme me shallewrite and worche, tillethe totallenombre be halfede.

Chapter V. Duplation.

Definition of Duplation.Duplicacioun is agregacion of nombre [to itself] þat me may se the nombre growen. In doublyngeay is but one ordre of figures necessarie. And me most be-gynne withthe lift side, other of the more figure, And after the nombre of the more figurerepresentithe.Fol. 51.*In the other .3. before we begynne alleway fro the right side and fro the lasse nombre,Where to begin.In this spice and in alleother folowyng we wolle begynne fro the lift side, ffor and me bigon thedouble fro the first,omwhileme myght double oo thynge twyes.Why.And how be it that me myght double fro the right, that woldebe harder in techyng and in workyng. Therfor yf thow wolt double any nombre, write that nombre by his differences, and double the last. And of that doublyng other growithea nombre digital, article, or componede. [If it be a digit, write it in the place of the first digit.]

What to do with the result.If it be article, write in his place a cifre and transferre the article towardethe lift, as thus:—

And yf the nombre be componede,

write a digital that is part of his composicioun, and sette the article to the lift hande, as thus:—

That done, me most double the last save one, and what growetheþerof me most worche as before. And yf a cifre be, toucheit not. But yf any nombre shallebe addedeto the cifre,

in þe place of þe figure wipedeout me most write the nombre to be addede, as thus:—

In the same wise me shallewircheof alleothers.How to prove your answer.And this probacioun:

If thow truly double the halfis, and truly half the doubles, the same nombre and figure shallemete, sucheas thow labouredevponefirst, And of the contrarie.

Chapter VI. Multiplication.

Definition of Multiplication.Multiplicacioun of nombre by hym-self other by a-nother, withproposide.2. nombres, [is] the fyndyng of the thirde, That so oft conteynethethat other, as ther ben vnytes in the oþer. In multiplicacioun .2. nombres pryncipally ben necessary, that is to sey, the nombre multiplying and the nombre to be multipliede, as here;—twies fyve.Multiplier.[The number multiplying] is designedeaduerbially.Multiplicand.The nombre to be multipliederesceyvethea nominalleappellacioun, as twies .5. 5. is the nombre multipliede, and twies is the nombre to bemultipliede.

Product.Also me may thervponeto assigne the. 3. nombre, the whicheisFol. 51b.*clepedeproduct or provenient, of takyng out of one fro another: as twyes .5 is .10., 5. the nombre to be multipliede, and .2. the multipliant,and. 10.as before is come therof. And vnderstonde wele, that of the multipliant may be made the nombre to be multipliede, and of the contrarie, remaynyng euerthe samesome, and herofecomethethe comen speche, that seitheall nombre is convertedeby Multiplying in hym-selfe.The Cases of Multiplication.

And ther ben .6 rules of Multiplicacioun;(1) Digit by digit.ffirst, yf a digit multiplie a digit, considrehow many of vnytees ben betwix the digit by multiplying and his .10. betheto-gedre accomptede, and so oft with-draw the digit multiplying, vnder the article of his denominacioun. Example of grace. If thow wolt wete how mocheis .4. tymes .8.,11se how many vnytees ben betwix .8.12and .10. to-geder rekenede, and it shewiththat .2.: withdraw ther-for the quaternary, of the article of his denominacion twies, of .40., And ther remaynethe.32., that is, tosomeof allethe multiplicacioun.See the table above.Wher-vpon for more evidence and declaracion the seidetable is made.(2) Digit by article.Whan a digit multipliethean article, thow most bryng the digit into þe digit, of þe whichethe article [has]13his name, and euery vnyteshallestondefor .10., and euery article an .100.(3) Composite by digit.Whan the digit multipliethea nombre componede, þou most bryng the digit into aiþerpart of the nombre componede, so þat digit be had into digit by the first rule, into an article by þe seconderule; and afterwardeIoyne the produccioun, and þere wol be the some totalle.

(4) Article by article.Whan an article multipliethean article, the digit wherof he is namedeis to be brought Into the digit wherof the oþeris namede, and euery vnyte wol be wortheFol. 52.*an .100., and euery article. a .1000.(5) Composite by article.Whan an article multipliethea nombre componede, thow most bryng the digit of the article into aither part of the nombre componede; and Ioyne the produccioun, and euery article wol be worthe.100., and euery vnyte .10., and so wollethe some be opene.(6) Composite by composite.Whan a nombre componedemultipliethea nombre componede, euery part of the nombre multiplying is to be hadeinto euery part of the nombre to be multipliede, and so shallethe digit be hadetwies, onys in the digit, that other in the article. The article also twies, ones in the digit, that other in the article. Therfor yf thow wilt any nombre by hym-self other by any other multiplie, write the nombre to be multipliedein the ouerordre by his differences,How to set down your numbers.The nombre multiplying in the lower ordre by his differences, so that the first of the lower ordre be vnder the last of the ouerordre. This done, of the multiplying, the last is to be hadeinto the last of the nombre to be multipliede. Wherof than wolle grow a digit, an article, other a nombre componede.If the result is a digit,

If it be a digit, even above the figure multiplying is hede write his digit that come of, as it apperethehere:—

an article,And yf an article had be writ ouerthe figure multiplying his hede, put a cifre þerand transferre the article towardethe lift hande, as thus:—

or a composite.And yf a nombre componedebe writ ouerthe figure multyplying is hede, write the digit in the nombre componedeis place, and sette the article to the lift hande, as thus:—

Multiply next by the last but one, and so on.This done,memost bryng the last save one of the multipliyng into the last of þe nombre to be multipliede, and se what comythetherof as before, and so do withalle, tille me come to the first of the nombre multiplying, that must be brought into the last of the nombre to be multipliede, wherof growitheoþera digit, an article,Fol. 52b.*other a nombre componede.

If it be a digit, In the place of theouerer, sette a-side, as here:

If an article happe, there put a cifre in his place, and put hym to the lift hande, as here:

If it be a nombre componede, in the place of the ouerer sette a-side, write a digit that14is a part of the componede, and sette on the left hondethe article, as here:

Then antery the multiplier one place.That done, sette forwardethe figures of the nombre multiplying byoodifference, so that the first of the multipliant be vnder the last save one of the nombre to be multipliede, the other byoplace sette forwarde. Than me shallebryngethe last of the multipliant in hym to be multipliede, vnder the whicheis the first multipliant.Work as before.And than wolle growe oþera digit, an article, or a componedenombre. If it be a digit, adde hym even above his hede; If it be an article, transferre hym to the lift side; And if it be a nombre componede, adde a digit to the figure above his hede, and sette to the lift handethe article. And alle-wayes euery figure of the nombre multipliant is to be brought to the last save one nombre to be multipliede, til me come to the first of the multipliant, where me shallewirche as it is seidebefore of the first, and afterwardeto put forwardethe figures by o difference and one tillethey allebe multipliede.How to deal with ciphers.And yf it happe that the first figure of þe multipliant be a cifre, andboueit is sette the figure signyficatife, write a cifre in the place of the figuresette a-side, as thus,etc.:

How to deal with ciphers.And yf a cifre happe in the lowerorderbe-twix the first and the last, and even above be sette the figure signyficatif,

leve it vntouchede, as here:—

And yf the space above sette be voide, in that place write thow a cifre. And yf the cifre happe betwix þe first and the last to be multipliede, me most sette forwardethe ordre of the figures by thairedifferences, for oft ofducciounof figures in cifres nought is the resultant, as here,

Fol. 53.*wherof it is evident and open, yf that the first figure of the nombre be to be multipliedebe a cifre, vndir it shallebe none sette as here:—

Leave room between the rows of figures.Vnder[stand] also that in multiplicacioun, divisioun, and of rootis the extraccioun, competently me may leve a mydel space betwix .2. ordres of figures, that me may write there what is come of addyng other withe-drawyng, lest any thynge sholdebeouer-hippedeand sette out of mynde.

Chapter VII. Division.

Definition of division.For to dyvyde oo nombre by a-nother, it is of .2. nombres proposede, It is forto depart themodernombre into as many partis as ben of vnytees in the lasse nombre. And note wele that in makyngeof dyvysioun ther ben .3. nombres necessary:Dividend, Divisor, Quotient.that is to sey, the nombre to be dyvydede; the nombre dyvydyng and the nombreexeant,otherhow oft, or quocient. Ay shallethe nombre that is to be dyvydedebe more, other at thelestevenewiththe nombre the dyvysere, yf the nombre shallebe madeby hole nombres.How to set down your Sum.Therfor yf thow wolt any nombre dyvyde, write the nombre to be dyvydedein þe ouererbordureby his differences, the dyviserein the lower ordureby his differences, so that the last of the dyviser be vnder the last of the nombre to be dyvyde, the next last vnder the next last, and so of the others, yf it may competently be done;An example.as here:—

When the last of the divisor must not be set below the last of the dividend.And ther ben .2. causes whan the last figure may not be sette vnder the last, other that the last of the lower nombre may not be with-draw of the last of the ouerer nombre for it is lasse than the lower, otherhow be it, thatit myght be with-draw as for hym-self fro the ouerer the remenaunt may not so oft of them above, other yf þe last of the lower be even to the figure above his hede, and þe next last oþerthe figure be-fore þat be more þan the figure above sette.Fol. 532.*These so ordeynede, me most wirchefrom the last figure of þe nombre of the dyvyser, and se how oft it may be with-draw ofHow to begin.and fro the figure aboue his hede, namly so that the remenaunt may be take of so oft, and to se the residue as here:—

An example.And note wele that me may not withe-draw more than .9. tymes nether lasse than ones. Therfor se how oft þe figures of the lower ordre may be with-draw fro the figures of the ouerer, and the nombre that shewithþe quocient most be writ ouerthe hede of þat figure, vnder the whichethe first figure is, of the dyviser;Where to set thequotienteAnd by that figure me most withe-draw alleoþerfigures of the lower ordir and that of the figures aboue thairehedis. This so done, me most sette forwardeþe figures of the diuiser by o difference towardesthe right hondeand worcheas before; and thus:—Examples.

A special case.And yf it happeafter þe settyng forwardeof the figures þat þe last of the divisor may not so oft be withdraw of the figure above his hede, above þat figure vnder the whichethe first of the diuiser is writ me most sette a cifre in ordre of the nombre quocient, and sette the figures forwardeas be-fore be o difference alone, and so me shalledo in allenombres to be dyvidede, for where the dyviser maynot be with-draw me most sette there a cifre, and sette forwardethe figures; as here:—

Another example.And me shallenot cesse fro suchesettyng of figures forwarde, nether of settyngeof þe quocient into the dyviser, neþerof subtraccioun of the dyvyser, tillethe first of the dyvyser be with-draw fro þe first to be dividede. The whichedone, or ought,17oþernought shalleremayne: and yf it be ought,17kepe it in the tables, And euervnyit to þe diviser. And yf þou wilt wete how many vnytees of þe divisiounFol. 533.*wol growe to the nombre of the divisere,What the quotient shows.the nombre quocient wol shewe it: and whan suchedivisioun is made, and þoulustprove yf thow have wele done orHow to prove your division,no, Multiplie the quocient by the diviser, And the same figures wolle come ayene that thow haddest bifore and none other. And yf ought be residue, than withaddicioun therof shallecome the same figures: And so multiplicacioun provithedivisioun, and dyvisioun multiplicacioun:or multiplication.as thus, yf multiplicacioun be made, divide it by the multipliant, and the nombre quocient wol shewe the nombre that was to be multipliede,etc.

Chapter VIII. Progression.

DefinitionofProgression.Progressioun is of nombre afteregalleexcesse fro oone or tweynetakeagregacioun. of progressioun one isnaturelleor contynuelle, þat oþerbroken and discontynuelle.Natural Progression.Naturelleit is, whan me begynnethewithone, and kepetheordure ouerlepyng one; as .1. 2. 3. 4. 5. 6.,etc., so þat the nombre folowyngepassithethe other be-fore in one.Broken Progression.Broken it is, whan me lepithefro o nombre tilleanother, and kepithenot the contynuel ordire; as 1. 3. 5. 7. 9,etc. Ay me may begynne with.2., as þus; .2. 4. 6. 8.,etc., and the nombre folowyng passethethe others by-fore by .2. And note wele, that naturelleprogressioun ay begynnethewithone, and Intercise or broken progressioun,omwhilebegynnythewith one, omwhile withtwayne. Of progressioun naturell .2. rules ther beyove, of the whichethe first is this;The 1st rule for Natural Progression.whan the progressioun naturelleendithein even nombre, by the half therof multiplie þe next totalleouererenombre; Example of grace: .1. 2. 3. 4. Multiplie .5. by .2. and so .10. cometheof, that is the totallenombre þerof.The second rule.The seconderule is suche, whan the progressioun naturelleendithein nombre ode. Take the more porcioun of the oddes, and multiplie therby the totallenombre. Example of grace 1. 2. 3. 4. 5., multiplie.5. by .3, and thryes .5. shallebe resultant. so the nombre totalleis .15.The first rule of Broken Progression.Of progresiounintercise, ther ben also .2.18rules; and þe first is þis: Whan the Intercise progression endithein even nombre by half therof multiplie the next nombre to þat halfeas .2.184. 6. Multiplie .4. by .3. so þat is thryes .4., and .12. the nombre of allethe progressioun, wollefolow.The second rule.The seconderule is this: whan the progressioun interscise endithein ode, take þe more porcioun of alleþe nombre,Fol. 534.*and multiplie by hym-selfe; as .1. 3. 5. Multiplie .3. by hym-selfe, and þe some of allewolle be .9.,etc.

Chapter IX. Extraction of Roots.

The preamble of the extraction of roots.Here folowithethe extraccioun of rotis, and first in nombre quadrates. Wherfor me shallese what is anombre quadrat, and what is the rote of a nombre quadrat, and what it is to draw out the rote of a nombre. And before other note this divisioun:Linear, superficial, and solid numbers.Of nombres one islyneal, anoþersuperficialle, anoþerquadrat, anoþercubikeorhoole. lyneal is that þat is considredeafter the processe, havyngeno respect to the direccioun of nombre in nombre, As a lyne hathebut one dymensioun that is to sey after the lengthe.Superficial numbers.Nombre superficial is þat cometheof ledyngeof oo nombre into a-nother, wherfor it is calledesuperficial, for it hathe.2. nombres notyng or mesuryngehym, as a superficiallethyngehathe.2. dimensions, þat is to sey lengtheand brede.Square numbers.And for bycause a nombre may be hadein a-nother by .2. maners, þat is to sey other in hym-selfe, oþerin anoþer, Vnderstondeyf it be had in hym-self, It is a quadrat. ffor dyvisioun write by vnytes, hathe.4. sides even as a quadrangille. and yf the nombre be hadein a-noþer, the nombre is superficiel and not quadrat, as .2. hadein .3. makethe.6. that is þe first nombre superficielle; wherfor it is open þat allenombre quadrat is superficiel, and notconuertide.The root of a square number.The rote of a nombre quadrat is þat nombre that is had of hym-self, as twies .2. makithe4. and .4. is the first nombre quadrat, and 2. is his rote. 9. 8. 7. 6. 5. 4. 3. 2. 1. / The rote of the more quadrat .3. 1. 4. 2. 6.Notes of some examples of square roots here interpolated.The most nombre quadrat 9. 8. 7. 5. 9. 3. 4. 7. 6. / the remenent ouerthe quadrat .6. 0. 8. 4. 5. / The first caas of nombre quadrat .5. 4. 7. 5. 6. The rote .2. 3. 4. The secondecaas .3. 8. 4. 5. The rote .6. 2. The thirdecaas .2. 8. 1. 9. The rote .5. 3. The .4. caas .3. 2. 1. The rote .1. 7. / The 5. caas .9. 1. 2. 0. 4. / The rote 3. 0. 2.Solid numbers.Thesolidenombreor cubikeis þat þat comytħe of double ledyng of nombre in nombre;Three dimensions of solids.And it is clepedea solidebody that hatheþer-in .3[dimensions] þat is to sey, lengthe, brede, and thiknesse. so þat nombre hathe.3. nombres to be brought forthein hym. But nombre may be hadetwies in nombre, for other it is hadein hym-selfe, oþerin a-noþer.Cubic numbers.If a nombre be hadetwies in hym-self, oþerones in his quadrat, þat is the same, þat acubikeFol. 54.*is, And is the same that is solide. And yf a nombre twies be hadein a-noþer, the nombre isclepedesolide and not cubike, as twies .3. and þat .2. makithe.12.All cubics are solid numbers.Wherfor it isopynethat allecubikenombre is solide, and notconuertide. Cubikeis þat nombre þat comytheof ledyngeof hym-selfetwyes, or ones in his quadrat. And here-by it is open that o nombre is therooteof a quadrat and of a cubike. Natheles the same nombre is not quadrat and cubike.No number may be both linear and solid.Opyneit is also that allenombres may be a rote to a quadrat and cubike, but not allenombre quadrat or cubike. Therfor sithen þe ledyngeof vnyte in hym-self ones or twies nought comethebut vnytes, SeitheBoice in Arsemetrike,Unity is not a number.that vnyte potencially is al nombre, and none in act. And vndirstondewele also that betwix euery .2. quadratesther is a meene proporcionalle,Examples of square roots.That is openedethus;lede the rote of o quadrat intothe rote of the oþerquadrat, and þan wolle þe meene shew.

A note on mean proportionals.Also betwix the next .2. cubikis, me may fynde a double meene, that is to sey a more meene and a lesse. The more meene thus, as to bryngethe rote of the lesse into a quadrat of the more. The lesse thus, If the rote of the more be brought Into the quadrat of the lesse.

Chapter X. Extraction of Square Root.

To20draw a rote of the nombre quadrat it is What-euernombre be proposedeto fynde his rote and to se yf it be quadrat.To find a square root.And yf it be not quadrat the rote of the most quadrat fynde out, vnder the nombre proposede. Therfor yf thow wilt the rote of any quadrat nombre draw out, write the nombre by his differences, andcomptthe nombre of the figures, and wete yf it be odeor even. And yfBegin with the last odd place.it be even, than most thow begynne worche vnder the last save one. And yf it be odewiththe last; and forto sey it shortly, al-weyes fro the last odeme shallebegynne. Therfor vnder the last in an od place sette,Find the nearest square root of that number, subtract,me most fyndea digit, the whicheladein hym-selfeit puttitheaway that, þat is ouerhis hede, oþeras neigheas memay: suche a digit foundeand withdraw fro his ouerer, me most double that digit and sette the double vnder the next figure towardethe right honde, and hisvnder doublevnder hym.double it,That done, than me most fyndea-noþerdigit vnder the next figure bifore the doublede,and set the double one to the right.the whicheFol. 54b.*brought in double settethea-way allethat is ouerhis hede as torewardeof the doublede: Than brought into hym-self settitheall away in respect of hym-self,Find the second figure by division.Other do it as nye as it may be do: other me may with-draw the digit21[last] founde, and lede hym in double or double hym, and after in hym-selfe;Multiply the double by the second figure, and add after it the square of the second figure, and subtract.Than Ioyne to-geder the produccioneof them bothe, So that the first figure of the last product be addedebefore the first of the first productes, the secondeof the first,etc. and so forthe,subtrahefro the totallenombre in respect of þe digit.

To be quadrede

The vnder double

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