“ If it is a Composite, ¶ Here he puttes þe Thryd case, þe quych is þis, yf of duplacioɳ of a figure come a Composit. þou schalt do away þe figure þat is doublet & set þere a digit of þe Composit, write down the digit, and ‘carry’ the tens. & sett þe articulle ouer þe next figures hede, & after draw hym downe with þe figure ouer whos hede he stondes, & make þere-of an nombre as þou hast done 16 afore, & yf þere come no figure after þat digit þat þou hast y-write, þan set þe articulle next after hym in þe same rewe as þus, 67: double 6 þat wel be 12, do away 6 & write þere þe digit leaf 148 a. ”

“ Then antery the lower number: ¶ Now þou most antery þe nether nounbre, þat is to say, þou most sett þe neþer nounbre more towarde þe ryȝt side, as þus. Take þe neþer 2 toward þe ryȝt side, & sette it eueɳ vnder þe 4 of þe hyer nounbre, & antery alle þe figures þat comes after þat 2, as þus; sette 2 vnder þe 4. þen sett þe figure of 3 þere þat þe figure of 2 stode, þe quych is now vndur þat 4 in þe hier nounbre; þen sett þe oþer figure of 2, þe quych is þe last figure toward þe lyft side of þe neþer nomber þere þe figure of 3 stode. ”

“ It may now be regarded as proved by Bubnov that our present numerals are derived from Greek sources through the so-called Boethian “apices,” which are first found in late tenth century manuscripts. That they were not derived directly from the Arabic seems certain from the different shapes of some of the numerals, especially the 0, which stands for 5 in Arabic. Another Greek form existed, which was introduced into Europe by John of Basingstoke in the thirteenth century, and is figured by Matthew Paris (V. 285) ”

“ Definition of mediation. Mediacioun is the fyndyng of the halfyng of euery nombre, that it may be seyne what and how moche is 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, and Where to begin. begynne at the right, that is to sey, fro the first figure to the right side, so that it be signyficatife other represent vnyte or eny other digitalle nombre. If it be vnyte write in his place a cifre for the If the first figure is unity. ”

“ Specimen sums have indeed been constructed by moderns which show its possibility, but it is absurd to think that men of science, acquainted with Egyptian methods and in possession of the abacus,2 were unable to devise methods for its use. ”

“ But I know not by what reason you set that one counter for 500 betwene two lynes.
M. you shall remember this, that when so euer you nede to set downe 5, 50, or 500, or 5000, or so forth any other nomber, whose numerator 118 a. *is 5, you shall set one counter for it, in the next space aboue the lyne that it hath his denomination of, as in this example of that 500, bycause the numerator is 5, it must be set in a voyd space: and bycause the denominator is hundred, I knowe that his place is the voyde space next aboue hundredes, that is to say, aboue the thyrd lyne. ”

“ Now what is the valewe of euery place or lyne, you may perceaue by the figures whiche I haue set on them, which is accordynge as you learned before in the Numeration of figures by the penne: for the fyrste place is the place of vnities or ones, and euery counter set in that lyne betokeneth but one: and the seconde lyne is the place of 10, for euery counter there, standeth for 10. ”

“ Than me shalle brynge the last of the multipliant in hym to be multipliede, vnder the whiche is the first multipliant. Work as before. And than wolle growe oþer a digit, an article, or a componede nombre. If it be a digit, adde hym even above his hede; If it be an article, transferre hym to the lift side; And if it be a nombre componede, adde a digit to the figure above his hede, and sette to the lift hande the article. ”

“ But now yf you wold adde other subtracte after any of both those sortes, yf you marke ye order of yt other feate which I taught you, you may easely do the same here without moch teachynge: for in Addition you must fyrst set downe one summe and to the same set the other orderly, and lyke maner yf you haue many: but in Subtraction you must sette downe fyrst the greatest summe, and from it must you abate that other euery denomination from his dewe place. ”

“ Cubike is þat nombre þat comythe of ledynge of hym-selfe twyes, or ones in his quadrat. And here-by it is open that o nombre is the roote of a quadrat and of a cubike. Natheles the same nombre is not quadrat and cubike. No number may be both linear and solid. Opyne it is also that alle nombres may be a rote to a quadrat and cubike, but not alle nombre quadrat or cubike. Therfor sithen þe ledynge of vnyte in hym-self ones or twies nought comethe but vnytes, Seithe Boice in Arsemetrike, Unity is not a number. that vnyte potencially is al nombre, and none in act. ”

“ And if it hap þat no digit may be founde, Than sette a cifre vndre a cifre, and cesse not tille thow fynde a digit; and whan thow hast founde it to double it, neþer to sette the doublede forwarde nether the vnder doublede, Special cases. Till thow fynde vndre the first figure a digit, the whiche lade in alle double, settyng away alle that is ouer hym in respect of the doublede: Than lede hym into hym-selfe, and put a-way alle in regarde of hym, other as nyghe as thow maist. ”

“ Therfor yf thow wilt prove yf thow have wele do or no, Multiplie the digit last founde with the vnder-double oþer vnder-doublis, and thow shalt fynde the same figures that thow haddest before; And so that nought be the Fol. 55. *residue. And yf thow have any residue, than with the addicioun þerof that is reseruede with-out in thy table, thow shalt fynde thi first figures as thow haddest them before, etc.
Chapter XI. Extraction of Cube Root.
Definition of a cubic number and a cube root. Heere folowithe the extraccioun of rotis in cubike nombres ”

“ Why says he þat a cifre makys a figure to signifye (tyf) more &c. ¶ I speke for þis worde significatyf, ffor sothe it may happe aftur a cifre schuld come a-noþur cifre, as þus 2φφ. And ȝet þe secunde cifre shuld token neuer þe more excep he schuld kepe þe order of þe place. and a cifre is no figure significatyf.
¶ Quam precedentes plus ultima significabit /
The last figure means more than all the others, since it is of the highest value. ”

“ 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 thaire nombre, and þan worche as it shewith in 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 passide ouer, sette figures of .9. as here it is specifiede:—
And whan me comethe to the nombre wherof me intendithe, there remaynethe alle-wayes .10. ”

“ Ife þe nombur þat þou hast write be an Articul, write first a cifre & aftur þe cifer write an Articulle þus. 2φ. tens, fforthermore þou schalt vndirstonde yf þou haue an Articul, loke how 7 mych he is, yf he be with-ynne an hundryth, þou schalt write bot one cifre, afore, as here .9φ. hundreds, If þe articulle be by hym-silfe & be an hundrid euene, þen schal þou write .1. & 2 cifers afore, þat he may stonde in þe thryd place, for euery figure in þe thryd place schal token a hundrid tymes hym selfe. thousands, &c. ”

“ The word “withdraw” or “withdraw” was inconsistently hyphenated; it was left as printed, and line-end hyphens were retained. All brackets [ ] are in the original.
The diagrams in “Accomptynge by Counters” may not line up perfectly in all browsers, but the contents should still be intelligible. ”

“ Therfore sithen the science of the whiche at this tyme we The name of the art. intendene to write of standithe alle and about nombre: ffirst we most se, what is the propre name therofe, and fro whens the name come: Afterwarde what is nombre, And how manye spices of nombre ther ben. The name is clepede Algorisme, Derivation of Algorism. hade out of Algore, other of Algos, in grewe, That is clepide in englisshe art other craft, And of Rithmus that is callede nombre. So algorisme is clepede the art of nombryng, Another. ”

“ Digit by article. Whan a digit multipliethe an article, thow most bryng the digit into þe digit, of þe whiche the article [has] 13 his name, and euery vnyte 41 shalle stonde for .10., and euery article an .100. (3) Composite by digit. Whan the digit multipliethe a nombre componede, þou most bryng the digit into aiþer part of the nombre componede, so þat digit be had into digit by the first rule, into an article by þe seconde rule ”

“ The seven rules. ¶ Here telles þat þer beɳ .7. spices or partes of þis craft. The first is called addicioñ, þe secunde is called subtraccioñ. The thryd is called duplacioñ. The 4. is called dimydicioñ. The 5. is called multiplicacioñ. The 6 is called diuisioñ. The 7. is called extraccioñ of þe Rote. What all þese spices bene hit schalle be tolde singillatim in here caputule.
¶ Subtrahis aut addis a dextris vel mediabis:
Add, subtract, or halve, from right to left. ”

“ To alle suche even nombrys the most have cifrys as to ten. twenty. thirtty. an hundred. an thousand and suche other. but ye schal vnderstonde that a cifre tokeneth nothinge but he maketh other the more significatyf that comith after hym. Also ye schal vnderstonde that in nombrys composyt and in alle other nombrys that ben of diverse figurys ye schal begynne in the ritht syde and to rekene backwarde and so he schal be wryte as thus—1000. ”