AMEN CORNER, E.C. 4. 1922. (continued 2)

The Cases of this Craft.

leaf 158

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What to do if the first multiplication results in an article. ¶ Here he puttes þe fyrst case of þis craft, þe quych is þis: yf þere come an articulle of þe multiplicacioɳ ysette before the articulle in þe lyft side as þus 51 23. multiplye 5 by 2, þat wol be 10; sette ouer þe hede of þat 2 a 0, & sett þat on, þat is þe articul, in þe lyft side, þat is next hym, þen þou schalt haue þis nounbre 1051. 23 ¶ And þen worch forth as þou diddist afore. And þou schalt vnderstonde þat þou schalt write no 0. but whan þat place where þou schal write þat 0 has no figure afore hym noþer after. versus.

What to do if the result is a composite number. ¶ Here is þe secunde case, þe quych is þis: yf hit happe þat þere come a composyt, þou schalt write þe digitte ouer þe hede of þe neþer figure by þe quych þou multipliest þe hier figure; and sett þe articulle next hym toward þe lyft side, as þou diddyst afore, as þus 83. 83 Multiply 8 by 8, þat wol be 64. Write þe 4 ouer 8, þat is to say, ouer þe hede of þe neþer 8; & set 6, þe quych leaf 159 a. *is an articul, next after. And þen þou schalt haue such a nounbre as is here, 648313, 83 And þen worch forth.

What if it be a digit. ¶ Here is þe thryde case, þe quych is þis: yf hit happe þat of þi multiplicaciouɳ come a digit, þou schalt write þe digit ouer þe hede of þe neþer figure, by the quych þou multipliest þe hiere figure, for þis nedes no Ensampul.

The fourth case of the craft. ¶ Here is þe 4 case, þe quych is: yf hit be happe þat þe neþer figure schal multiplye þat figure, þe quych stondes ouer þat figures hede, þou schal do away þe hier figure & sett þere þat þat comys of þat multiplicacioɳ. As yf þere come of þat multiplicacioɳ an articuls þou schalt write þere þe hier figure stode a 0. ¶ And write þe articuls in þe lyft side, yf þat hit be a digit write þere a digit. yf þat hit be a composit, write þe digit of þe composit. And þe articul in þe lyft side. al þis is lyȝt y-nowȝt, þere-fore þer nedes no Ensampul.

leaf 159

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The fifth case of the craft. ¶ Here is þe 5 case, þe quych is þis: yf *þe neþer figure schul multiplie þe hier, and þat hier figure is not recte ouer his hede. And þat neþer figure hase oþer figures, or on figure ouer his hede by multiplicacioɳ, þat hase be afore, þou schalt write þat nounbre, þe quych comes of þat, ouer alle þe ylke figures hedes, as þus here: 236 234 Multiply 2 by 2, þat wol be 4; set 4 ouer þe hede of þat 2. þen14 multiplies þe hier 2 by þe neþer 3, þat wol be 6. set ouer his hede 6, multiplie þe hier 2 by þe neþer 4, þat wol be 8. do away þe hier 2, þe quych stondes ouer þe hede of þe figure of 4, and set þere 8. And þou schalt haue þis nounbre here 46836 234 And antery þi figures, þat is to say, set þi neþer 4 vnder þe hier 3, and set þi 2 other figures nere hym, so þat þe neþer 2 stonde vndur þe hier 6, þe quych 6 stondes in þe lyft side. And þat 3 þat stondes vndur 8, as þus aftur ȝe may se, 46836 234 Now worch forthermore, And multiplye þat hier 3 by 2, þat wol be 6, set þat 6 þe quych stondes ouer þe hede of þat 2, And þen worch as I taȝt þe afore.

leaf 160

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The sixth case of the craft. ¶ Here is þe 6 case, þe quych is þis: yf hit happe þat þe figure by þe quych þou schal multiplye þe hier figure, þe quych stondes ryght ouer hym by a 0, þou schalt do away þat figure, þe quych ouer þat cifre hede. ¶ And write þere þat nounbre þat comes of þe multiplicacioɳ as þus, 23. do away 2 and sett þere a 0. vnde versus.

The seventh case of the craft. ¶ Here is þe 7 case, þe quych is þis: yf a 0 schal multiply a figure, þe quych stondes not recte ouer hym, And ouer þat 0 stonde no thyng, þou schalt write ouer þat 0 anoþer 0 as þus: 24 03 multiplye 2 be a 0, it wol be nothynge. write þere a 0 ouer þe hede of þe neþer 0, And þen worch forth til þou come to þe ende.

The eighth case of the craft. ¶ Here is þe 8 case, þe quych is þis: yf þere be a 0 or mony cifers in þe hier rewe, þou schalt not multiplie hem, bot let hem stonde. And antery þe figures beneþe to þe next figure sygnificatyf as þus: 00032. 22 Ouer-lepe alle þese cifers & sett þat leaf 160 b. *neþer 2 þat stondes toward þe ryght side, and sett hym vndur þe 3, and sett þe oþer nether 2 nere hym, so þat he stonde vndur þe thrydde 0, þe quych stondes next 3. And þan worch. vnde versus.

How to prove the multiplication. ¶ Here he teches how þou schalt know wheþer þou hase wel I-do or no. And he says þat þou schalt deuide alle þe nounbre þat comes of þe multiplicacioɳ by þe neþer figures. And þen þou schalt haue þe same nounbur þat þou hadyst in þe begynnynge. but ȝet þou hast not þe craft of dyuisioɳ, but þou schalt haue hit afterwarde.

Mental multiplication. ¶ Here he teches þe to multiplie be þowȝt figures in þi mynde. And þe sentence of þis verse is þis: yf þou wel multiplie on nounbre by anoþer in þi mynde, þou schal haue þereto rewles in þe verses þat schal come after.

Digit by digit is easy. ¶ Here he teches a rewle as þou hast afore to multiplie a digit be anoþer, as yf þou wolde wete qwat is sex tymes 6. þou leaf 161 a. *schalt wete by þe rewle þat I taȝt þe before, yf þou haue mynde þerof.

The first case of the craft. ¶ Here he teches þe furst rewle, þe quych is þis: yf þou wel multiplie an articul be anoþer, so þat both þe articuls bene with-Inne an hundreth, þus þou schalt do. Article by article; take þe digit of bothe the articuls, for euery articul hase a digit, þen multiplye þat on digit by þat oþer, and loke how mony vnytes ben in þe nounbre þat comes of þe multiplicacioɳ of þe 2 digittes, & so mony hundrythes ben in þe nounbre þat schal come of þe multiplicacioɳ of þe ylke 2 articuls as þus. an example: yf þou wold wete qwat is ten tymes ten. take þe digit of ten, þe quych is 1; take þe digit of þat oþer ten, þe quych is on. ¶ Also multiplie 1 be 1, as on tyme on þat is but 1. In on is but on vnite as þou wost welle, þerefore ten tymes ten is but a hundryth. another example: ¶ Also yf þou wold wete what is twenty tymes 30. take þe digit of twenty, þat is 2; & take þe digitt of thrytty, þat is 3. multiplie 3 be 2, þat is 6. Now in 6 ben 6 vnites, ¶ And so mony hundrythes ben in 20 tymes 30*, leaf 161 b. þerefore 20 tymes 30 is 6 hundryth eueɳ. loke & se. ¶ But yf it be so þat one articul be with-Inne an hundryth, or by-twene an hundryth and a thowsande, so þat it be not a þowsande fully. þen loke how mony vnytes ben in þe nounbur þat comys of þe multiplicacioɳ 16And so mony tymes16 of 2 digittes of ylke articuls, so mony thowsant ben in þe nounbre, the qwych comes of þe multiplicacioɳ. And so mony tymes ten thowsand schal be in þe nounbre þat comes of þe multiplicacion of 2 articuls, as yf þou wold wete qwat is 4 hundryth tymes [two hundryth]. Multiply 4 be 2,17 þat wol be 8. in 8 ben 8 vnites. How to work subtly without Figures. Mental multiplication. ¶ And so mony tymes ten thousand be in 4 hundryth tymes [2]17 hundryth, þat is 80 thousand. Take hede, I schall telle þe a Another example. generalle rewle whan þou hast 2 articuls, And þou wold wete qwat comes of þe multiplicacioɳ of hem 2. multiplie þe digit of þat on articuls, and kepe þat nounbre, þen loke how mony cifers schuld go before þat on articuls, and he were write. Als mony cifers schuld go before þat other, & he were write of cifers. And haue alle þe ylke cifers togedur in þi mynde, leaf 162 a. *a-rowe ychoɳ aftur other, and in þe last plase set þe nounbre þat comes of þe multiplicacioɳ of þe 2 digittes. And loke in þi mynde in what place he stondes, where in þe secunde, or in þe thryd, or in þe 4, or where ellis, and loke qwat þe figures by-token in þat place; & so mych is þe nounbre þat Another example. comes of þe 2 articuls y-multiplied to-gedur as þus: yf þou wold wete what is 20 thousant tymes 3 þowsande. multiply þe digit of þat articulle þe quych is 2 by þe digitte of þat oþer articul þe quych is 3, þat wol be 6. þen loke how mony cifers schal go to 20 thousant as hit schuld be write in a tabul. certainly 4 cifers schuld go to 20 þowsant. ffor þis figure 2 in þe fyrst place betokenes twene. Notation. ¶ In þe secunde place hit betokenes twenty. ¶ In þe 3. place hit betokenes 2 hundryth. .¶. In þe 4 place 2 thousant. ¶ In þe 5 place hit betokenes twenty þousant. þerefore he most haue 4 cifers a-fore hym þat he may stonde in þe 5 place. kepe þese 4 cifers in thy mynde, þen loke how mony cifers goɳ to 3 thousant. Certayn to 3 thousante leaf 162 b. *goɳ 3 cifers afore. Now cast ylke 4 cifers þat schuld go to twenty thousant, And thes 3 cifers þat schuld go afore 3 thousant, & sette hem in rewe ychoɳ after oþer in þi mynde, as þai schuld stonde in a tabulle. And þen schal þou haue 7 cifers; þen sett þat 6 þe quych comes of þe multiplicacioɳ of þe 2 digittes aftur þe ylke cifers in þe 8 place as yf þat hit stode in a tabul. And loke qwat a figure of 6 schuld betoken in þe 8 place. yf hit were in a tabul & so mych it is. & yf þat figure of 6 stonde in þe fyrst place he schuld betoken but 6. ¶ In þe 2 place he schuld betoken sexty. ¶ In the 3 place he schuld betokeɳ sex hundryth. Notation again. ¶ In þe 4 place sex thousant. ¶ In þe 5 place sexty þowsant. ¶ In þe sext place sex hundryth þowsant. ¶ In þe 7 place sex þowsant thousantes. ¶ In þe 8 place sexty þowsant thousantes. þerfore sett 6 in octauo loco, And he schal betoken sexty þowsant thousantes. Mental multiplication. And so mych is twenty þowsant tymes 3 thousant, ¶ And þis rewle is generalle for alle maner of articuls, Whethir þai be hundryth or þowsant; but þou most know well þe craft of þe wryrchynge in þe tabulle leaf 163 a. *or þou know to do þus in þi mynde aftur þis rewle. Thou most þat þis rewle holdyþe note but where þere ben 2 articuls and no mo of þe quych ayther of hem hase but on figure significatyf. As twenty tymes 3 thousant or 3 hundryth, and such oþur.

The third case of the craft; ¶ Here he puttes þe thryde rewle, þe quych is þis. yf þou wel multiply in þi mynde, And þe Articul be a digitte, þou schalt loke þat þe digitt be with-Inne an hundryth, þen þou schalt multiply the digitt of þe Articulle by þe oþer digitte. And euery vnite in þe nounbre þat schalle come þere-of schal betoken ten. As þus: an example. yf þat þou wold wete qwat is twyes 40. multiplie þe digitte of 40, þe quych is 4, by þe oþer diget, þe quych is 2. And þat wolle be 8. And in þe nombre of 8 ben 8 vnites, & euery of þe ylke vnites schuld stonde for 10. þere-fore þere schal be 8 tymes 10, þat wol be 4 score. And so mony is twyes 40. ¶ If þe articul be a hundryth or be 2 hundryth And a þowsant, so þat hit be notte a thousant, leaf 163 b. *worch as þou dyddyst afore, saue þou schalt rekene euery vnite for a hundryth.

The fourth case of the craft: Here he puttes þe 4 rewle, þe quych is þis: yf þou multipliy on composit be a digit as 6 tymes 24, 19þen take þe diget of þat composit, & multiply þat digitt by þat oþer diget, and kepe þe nombur þat comes þere-of. þen take þe digit of þat composit, & multiply þat digit by anoþer diget, by þe quych þou hast multiplyed þe diget of þe articul, and loke qwat comes þere-of. Composite by digit. þen take þou þat nounbur, & cast hit to þat other nounbur þat þou secheste as þus yf þou wel wete qwat comes of 6 tymes 4 & twenty. Mental multiplication. multiply þat articulle of þe composit by þe digit, þe quych is 6, as yn þe thryd rewle þou was tauȝt, And þat schal be 6 score. þen multiply þe diget of þe composit, leaf 164 a. *þe quych is 4, and multiply þat by þat other diget, þe quych is 6, as þou wast tauȝt in þe first rewle, yf þou haue mynde þerof, & þat wol be 4 & twenty. cast all ylke nounburs to-gedir, & hit schal be 144. And so mych is 6 tymes 4 & twenty.

How to multiply without Figures.

The fifth case of the craft: ¶ Here he puttes þe 5 rewle, þe quych is þis: yf þou wel multiply an Articul be a composit, multiplie þat Articul by þe articul of þe composit, and worch as þou wos tauȝt in þe secunde rewle, of þe quych rewle þe verse begynnes þus. Article by Composite. ¶ Articulum si per Relicum vis multiplicare. þen multiply þe diget of þe composit by þat oþir articul aftir þe doctrine of þe 3 rewle. take þerof gode hede, I pray þe as þus. Yf þou wel wete what is 24 tymes ten. An example. Multiplie ten by 20, þat wel be 2 hundryth. þen multiply þe diget of þe 10, þe quych is 1, by þe diget of þe composit, þe quych is 4, & þat leaf 164 b. *wol be 4. þen reken euery vnite þat is in 4 for 10, & þat schal be 40. Cast 40 to 2 hundryth, & þat wol be 2 hundryth & 40. And so mych is 24 tymes ten.

How to work without Figures.

The sixth case of the craft: ¶ Here he puttes þe 6 rewle, & þe last of alle multiplicacioɳ, þe quych is þis: yf þou wel multiplye a composit by a-noþer composit, þou schalt do þus. Composite by Composite. multiplie þat on composit, qwych þou welt of the twene, by þe articul of þe toþer composit, as þou were tauȝt in þe 5 rewle, þen multiplie þat same composit, þe quych þou hast multiplied by þe oþer articul, by þe digit of þe oþer composit, Mental multiplication. as þou was tauȝt in þe 4 rewle. An example As þus, yf þou wold wete what is 11 tymes 13, as þou was tauȝt in þe 5 rewle, & þat schal be an hundryth & ten, afterwarde multiply þat same composit þat þou hast multiplied, þe quych is a .11. And multiplye hit be þe digit of þe oþer composit, þe quych is 3, for 3 is þe digit of 13, And þat wel be 30. þen take þe digit of þat composit, þe quych composit þou multiplied by þe digit of þat oþer composit, leaf 165 a. *þe quych is a 11. of the sixth case of the craft. ¶ Also of the quych 11 on is þe digit. multiplie þat digitt by þe digett of þat other composit, þe quych diget is 3, as þou was tauȝt in þe first rewle in þe begynnynge of þis craft. þe quych rewle begynnes “In digitum cures.” And of alle þe multiplicacioɳ of þe 2 digitt comys thre, for onys 3 is but 3. Now cast alle þese nounbers togedur, the quych is þis, a hundryth & ten & 30 & 3. And al þat wel be 143. Write 3 first in þe ryght side. And cast 10 to 30, þat wol be 40. set 40 next aftur towarde þe lyft side, And set aftur a hundryth as here an Ensampulle, 143.

(Cetera desunt.)

[Ashmole MS. 396, fol. 48.]

Boys seying in the begynnyng of his Arsemetrike:—Alle Fol. 48. thynges that bene fro the first begynnyng of thynges have procedede, and come forthe, And by resoun of nombre ben formede; And in wise as they bene, So owethe they to be knowene; wherfor in vniuersalle knowlechyng of thynges the Art of nombrynge is best, and most operatyfe.

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. other it is had ofe en or in, and gogos that is introduccioun, and Rithmus nombre, that is to say Interduccioun of nombre. Another. And thirdly it is hade of the name of a kyng that is clepede Algo and Rythmus; So callede Algorismus. Kinds of numbers. Sothely .2. manere of nombres ben notifiede; Formalle,1 as nombre is vnitees gadrede to-gedres; Materialle,2 as nombre is a colleccioun of vnitees. Other nombre is a multitude hade out of vnitees, vnitee is that thynge wher-by euery thynge is callede oone, other o thynge. Of nombres, that one is clepede digitalle, that othere Article, Another a nombre componede oþer myxt. Another digitalle is a nombre with-in .10.; Article is þat nombre that may be dyvydede in .10. parties egally, And that there leve no residue; Componede or medlede is that nombre that is come of a digite and of an article. And vndrestande wele that alle nombres betwix .2. articles next is a nombre componede. 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 hade in .2. maners, that is to sey in nombres quadrat, and in cubices: Amonge the whiche, ffirst of Numeracioun, and afterwarde of þe oþers by ordure, y entende to write.

Chapter I. Numeration.

Fol. 48

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*For-sothe numeracioun is of euery numbre by competent figures an artificialle representacioun.

Figures, differences, places, and limits. Sothly figure, difference, places, and lynes supposen o thyng other the same, But they ben sette here for dyuers resons. ffigure is clepede for protraccioun of figuracioun; Difference is callede for therby is shewede euery figure, how it hathe difference fro the figures before them: place by cause of space, where-in me writethe: lynees, for that is ordeynede for the presentacioun of euery figure. The 9 figures. And vnderstonde that ther ben .9. lymytes of figures that representen the .9. digites that ben these. 0. 9. 8. 7. 6. 5. 4. 3. 2. 1. The cipher. The .10. is clepede theta, or a cercle, other a cifre, other a figure of nought for nought it signyfiethe. Nathelesse she holdyng that place givethe others for to signyfie; for withe-out cifre or cifres a pure article may not be writte. The numeration And sithen that by these .9. figures significatifes Ioynede with cifre or with cifres alle nombres ben and may be representede, It was, nether is, no nede to fynde any more figures. of digits, And note wele that euery digite shalle be writte with oo figure allone to it aproprede. of articles, And alle articles by a cifre, ffor euery article is namede for oone of the digitis as .10. of 1.. 20. of. 2. and so of the others, &c. And alle nombres digitalle owen to be sette in the first difference: Alle articles in the seconde. Also alle nombres 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 towarde the lift honde, that signifiethe the digit of the whiche the 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 seide be-fore. Alle nombre that is fro an hundrede tille a thousande exclusede, owithe to be writ by .3. figures; and alle nombre that is fro a thousande til .x. Mł. mvst be writ by .4. figures; And so forthe. The value due to position. And vnderstonde wele that euery figure sette in the first place signyfiethe his digit; In the seconde place .10. tymes his digit; In the .3. place an hundrede so moche; In the .4. place a thousande so moche; In the .5. place .x. thousande so moche; In the .6. place an hundrede thousande so moche; In the .7. place a thousande thousande. And so infynytly mvltiplying by Fol. 49. *these .3. 10, 100, 1000. And vnderstande wele that competently me may sette vpon figure in the place of a thousande, a prike to shewe how many thousande the last figure shalle represent. Numbers are written from right to left. We writene in this art to the lift side-warde, as arabiene writene, that weren fynders of this science, othere for this resoun, that for to kepe a custumable ordre in 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 as excressent. In addicioun, 2. ordres of figures and .2. nombres ben necessary, that is to sey, a nombre to be addede and the nombre wherto the addicioun sholde be made to. The nombre to be addede is that þat sholde be addede therto, and shalle be vnderwriten; the nombre vnto the whiche addicioun shalle be made to is that nombre that resceyuethe the addicion of þat other, and shalle be 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 happe one other other, the same comythe of, Therfor, yf þow wilt adde nombre to nombre, write the nombre wherto the addicioun shalle be made in the omest ordre by his differences, so that the first of the lower ordre be vndre the first of the omyst ordre, 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 suche addicioun, other þere growith therof 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 resultant 2

To whom it shal be addede 1

The nombre to be addede 1

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 whiche the article toke his name, towarde the lift side, and be it addede to 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 resultant 10

To whom it shalle be addede 7

The nombre to be addede 3

Resultans 2 7 8 2 7

Cui debet addi 1 0 0 8 4

Numerus addendus 1 7 7 4 3

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

The resultant 17

To whom it shalle be addede 10

The nombre to be addede 7

In his place write the Fol. 49 b. *[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 resultant 10

To whom it shalle be addede 9

The nombre to be addede 1

In the place of that 9. put a cifre and write þe article towarde þe lift honde as bifore, and thus:—

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

The resultant 12

To whom it shalle be addede 8

The nombre to be addede 4

[let] the digit [be]6 writ 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 oþer as before. Note wele þat in addicions and in alle spices folowyng, whan he seithe one the other shalle be writen aboue, and me most vse euer figure, as that euery figure were sette by halfe, and by hym-selfe.

Chapter III. Subtraction.

Definition of Subtraction. Subtraccioun is of .2. proposede nombres, the fyndyng of the excesse of the more to the lasse: Other subtraccioun is ablacioun of o nombre fro a-nother, that me may see a some left. The lasse of the more, or even of even, may be withdraw; The more fro the lesse may neuer be. How it may be done. And sothly that nombre is more that hathe more 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-ouer in with-drawyng .2. nombres ben necessary; A nombre to be withdraw, And a nombre that me shalle with-draw of. The nombre to be with-draw shalle be writ in the lower ordre by his differences; Write the greater number above. The nombre fro the whiche me shalle withe-draw in the omyst ordre, so that the first be vnder the first, the seconde vnder the seconde, And so of alle others. Subtract the first figure if possible. Withe-draw therfor the first of the lowere ordre fro the first of the ordre above his hede, and that wolle be other more or lesse, oþer egalle.

The remanent 20

Wherof me shalle withdraw 22

The nombre to be withdraw 2

The remanent 2 2

Wherof me shalle with-draw 2 8

Þe nombre to be withdraw 6

yf it be egalle or 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

Fol. 50.

*Remanens 2 2 1 8 2 9 9 9 8

A quo sit subtraccio 8 7 2 4 3 0 0 0 4

Numerus subtrahendus 6 5 7 [6] . . . . 6

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 worthe 10. Of that .10. and of the figure that ye wolde have with-draw fro and then subtract. be-fore to-gedre Ioynede,

The remanent 1 8

Wherof me shalle with-draw 2 4

The nombre to be with-draw 0 6

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 thaire nombre, and þan worche as it shewith in this figure here:—

The remanent 3 0 98

Wherof me shal with-draw 3 1 2

The nombre to be with-draw . . 3

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:—

The remenaunt 2 9 9 9 9

Wherof me shalle with-draw 3 0 0 0 3

The nombre to be with-draw 4

And whan me comethe to the nombre wherof me intendithe, there remaynethe alle-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 borowede an vnyte, that vnyte by respect of the figure that he came fro representith an .C., In the place of that cifre [passed over] is left .9., [which is worth ninety], and yit it remaynethe as .10., And the same resone wolde be yf me hade borowede an vnyte fro the .4., .5., .6., place, or ony other so vpwarde. This done, withdraw the seconde of the lower ordre fro the figure above his hede of þe omyst ordre, and wirche as 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 them ayene to the omyst figures, and they wolle accorde with the first that thow haddest yf thow have labored wele; and addition. and in like wise in addicioun, whan thow hast addede alle thy figures, withdraw them that thow first Fol. 50 b. *addest, and the same wolle retourne. The subtraccioun is none other but a prouffe of the addicioun, and the contrarye in like wise.

Chapter IV. Mediation.

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. figures folowyng, [lest they signify less], and write that vnyte without in the table, other resolue it in .60. mynvtes and sette a-side half of tho minutes so, and reserve the remenaunt without in the table, as thus .30.; other sette without thus .dī: that kepethe none ordre of place, Nathelesse it hathe signyficacioun. And yf the other figure signyfie any other digital nombre fro vnyte forthe, oþer the nombre is ode or evene.

Halfede 2 2

to be halfede 4 4

halfede 2 3 [di]

To be halfede 4 7

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 remaynethe to be halfede do thus:—

Then halve the second figure. This done, the seconde is 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 wipede out the halfe; yf it be ode, take the next even vnder it contenythe, and in the place of the Impar sette a-side put half of the even: The vnyte that remaynethe to be halfede, respect hade to them before, is worthe .10.

Halfede

to be halfede

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 sholde be made to a cifre, sette it a-side, and write in his place .5.

doublede 2 6 8 9 0 10 17 4

to be doublede 1 3 4 4 5 5 8 7

And vnder this fourme me shalle write and worche, tille the totalle nombre be halfede.

Chapter V. Duplation.

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

double 10

to be doublede 5

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

And yf the nombre be componede,

doublede 16

to be doublede 8

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 growethe þerof me most worche as before. And yf a cifre be, touche it not. But yf any nombre shalle be addede to the cifre,

doublede 6 0 6

to be doublede 3 0 3

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

In the same wise me shalle wirche of alle others. How to prove your answer. And this probacioun:

Doublede 6 1 8

to be doublede 3 0 9

If thow truly double the halfis, and truly half the doubles, the same nombre and figure shalle mete, suche as thow labourede vpone first, And of the contrarie.

Chapter VI. Multiplication.

Definition of Multiplication. Multiplicacioun of nombre by hym-self other by a-nother, with proposide .2. nombres, [is] the fyndyng of the thirde, That so oft conteynethe that 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 designede aduerbially. Multiplicand. The nombre to be multipliede resceyvethe a nominalle appellacioun, as twies .5. 5. is the nombre multipliede, and twies is the nombre to be multipliede.

Resultans 9 1 0 1 3 2 6 6 8 0 0 8

Multiplicandus . . 5 . . 4 . 3 4 0 0 4

Multiplicans . 2 2 . 3 3 2 2 2 . . .

Product. Also me may thervpone to assigne the. 3. nombre, the whiche is Fol. 51 b. *clepede product 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 euer the same some, and herofe comethe the comen speche, that seithe all nombre is convertede by Multiplying in hym-selfe. The Cases of Multiplication.

There are 6 rules of Multiplication.

1 2 3 4 5 6 7 8 9 10

2 4 6 8 10 1010 14 16 18 20

3 6 9 12 15 18 21 24 27 30

4 8 12 16 20 24 28 32 36 40

5 10 15 20 25 30 35 40 45 50

6 12 18 24 30 36 42 48 56 60

7 14 21 28 35 42 49 56 63 70

8 16 24 32 40 48 56 64 72 80

9 18 27 36 45 54 63 72 81 90

10 20 30 40 50 60 70 80 90 100

And ther ben .6 rules of Multiplicacioun; (1) Digit by digit. ffirst, yf a digit multiplie a digit, considre how many of vnytees ben betwix the digit by multiplying and his .10. bethe to-gedre accomptede, and so oft with-draw the digit multiplying, vnder the article of his denominacioun. Example of grace. If thow wolt wete how moche is .4. tymes .8., 11se how many vnytees ben betwix .8.12 and .10. to-geder rekenede, and it shewith that .2.: withdraw ther-for the quaternary, of the article of his denominacion twies, of .40., And ther remaynethe .32., that is, to some of alle the multiplicacioun. See the table above. Wher-vpon for more evidence and declaracion the seide table is made. (2) 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 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; and afterwarde Ioyne the produccioun, and þere wol be the some totalle.

Resultans 1 2 6 7 3 6 1 2 0 1 2 0 8

Multiplicandus 2 3 2 6 4

Multiplicans 6 3 2 3 2 0 3 0 2

(4) Article by article. Whan an article multipliethe an article, the digit wherof he is namede is to be brought Into the digit wherof the oþer is namede, and euery vnyte wol be worthe Fol. 52. *an .100., and euery article. a .1000. (5) Composite by article. Whan an article multipliethe a 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 wolle the some be opene. (6) Composite by composite. Whan a nombre componede multipliethe a nombre componede, euery part of the nombre multiplying is to be hade into euery part of the nombre to be multipliede, and so shalle the digit be hade twies, onys in the digit, that other in the article. The article also twies, ones in the digit, that other in the article. Therfor yf thow wilt any nombre by hym-self other by any other multiplie, write the nombre to be multipliede in the ouer ordre 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 ouer ordre. This done, of the multiplying, the last is to be hade into 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 resultant 6

To be multipliede 3

Þe nombre multipliyng 2

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

an article, And yf an article had be writ ouer the figure multiplying his hede, put a cifre þer and transferre the article towarde the lift hande, as thus:—

The resultant 1 0

to be multipliede 5

þe nombre multipliyng 2

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

Resultant 1 2

to be multipliede 4

the nombre multipliyng 3

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

Resultant 6 6

to be multipliede 3

the nombre multipliyng 2 2

The resultant 1 1 0

to be multipliede 5

þe nombre multiplying 2 2

The resultant 1 315 2

to be multipliede 4

þe nombre multipliant 3 3

If it be a digit, In the place of the ouerer, 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 that14 is a part of the componede, and sette on the left honde the article, as here:

Then antery the multiplier one place. That done, sette forwarde the figures of the nombre multiplying by oo difference, so that the first of the multipliant be vnder the last save one of the nombre to be multipliede, the other by o place sette forwarde. 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. 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 shalle wirche as it is seide before of the first, and afterwarde to put forwarde the figures by o difference and one tille they alle be multipliede. How to deal with ciphers. And yf it happe that the first figure of þe multipliant be a cifre, and boue it is sette the figure signyficatife, write a cifre in the place of the figure sette a-side, as thus, etc.:

The resultant 1 2 0

to be multipliede 6

the multipliant 2 0

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

The resultant 2 2 6 4 4

To be multipliede 2 2 2

The multipliant 1 0 2

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 forwarde the ordre of the figures by thaire differences, for oft of duccioun of figures in cifres nought is the resultant, as here,

Resultant 8 0 0 8

to be multipliede 4 0 0 4

the multipliant 2 . . .

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

Resultant 3 2 016

To be multipliede 8 0

The multipliant 4

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 sholde be ouer-hippede and 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 the moder nombre into as many partis as ben of vnytees in the lasse nombre. And note wele that in makynge of dyvysioun ther ben .3. nombres necessary: Dividend, Divisor, Quotient. that is to sey, the nombre to be dyvydede; the nombre dyvydyng and the nombre exeant, other how oft, or quocient. Ay shalle the nombre that is to be dyvydede be more, other at the lest evene with the nombre the dyvysere, yf the nombre shalle be made by hole nombres. How to set down your Sum. Therfor yf thow wolt any nombre dyvyde, write the nombre to be dyvydede in þe ouerer bordure by his differences, the dyvisere in the lower ordure by his differences, so that the last of the dyviser be vnder the last of the nombre to be dyvyde, the next last vnder the next last, and so of the others, yf it may competently be done; An example. as here:—

The residue 2 7

The quotient 5

To be dyvydede 3 4 2

The dyvyser 6 3

Examples.

Residuum 8 2 7 2 6

Quociens 2 1 2 2 5 9

Diuidendus 6 8 0 6 6 3 4 2 3 3 2

Diuiser 3 2 3 6 3 3 4

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, other how be it, that it 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 oþer the figure be-fore þat be more þan the figure above sette. Fol. 53². *These so ordeynede, me most wirche from the last figure of þe nombre of the dyvyser, and se how oft it may be with-draw of How 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 residue 2 6

The quocient 9

To be dyvydede 3 3 2

The dyvyser 3 4

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 shewith þe quocient most be writ ouer the hede of þat figure, vnder the whiche the first figure is, of the dyviser; Where to set the quotiente And by that figure me most withe-draw alle oþer figures of the lower ordir and that of the figures aboue thaire hedis. This so done, me most sette forwarde þe figures of the diuiser by o difference towardes the right honde and worche as before; and thus:— Examples.

Residuum . 1 2

quociens 6 5 4 2 0 0 4

Diuidendus 3 5 5 1 2 2 8 8 6 3 7 0 4

Diuisor 5 4 3 4 4 2 3

The quocient 6 5 4

To be dyvydede 3 5 5 1 2 2

The dyvyser 5 4 3

A special case. And yf it happe after þe settyng forwarde of 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 whiche the first of the diuiser is writ me most sette a cifre in ordre of the nombre quocient, and sette the figures forwarde as be-fore be o difference alone, and so me shalle do in alle nombres to be dyvidede, for where the dyviser may not be with-draw me most sette there a cifre, and sette forwarde the figures; as here:—

The residue 1 2

The quocient 2 0 0 4

To be dyvydede 8 8 6 3 7 0 4

The dyvyser 4 4 2 3

Another example. And me shalle not cesse fro suche settyng of figures forwarde, nether of settynge of þe quocient into the dyviser, neþer of subtraccioun of the dyvyser, tille the first of the dyvyser be with-draw fro þe first to be dividede. The whiche done, or ought,17 oþer nought shalle remayne: and yf it be ought,17 kepe it in the tables, And euer vny it to þe diviser. And yf þou wilt wete how many vnytees of þe divisioun Fol. 53³. *wol growe to the nombre of the divisere, What the quotient shows. the nombre quocient wol shewe it: and whan suche divisioun is made, and þou lust prove yf thow have wele done or How 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 with addicioun therof shalle come the same figures: And so multiplicacioun provithe divisioun, 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.

Definition of Progression. Progressioun is of nombre after egalle excesse fro oone or tweyne take agregacioun. of progressioun one is naturelle or contynuelle, þat oþer broken and discontynuelle. Natural Progression. Naturelle it is, whan me begynnethe with one, and kepethe ordure ouerlepyng one; as .1. 2. 3. 4. 5. 6., etc., so þat the nombre folowynge passithe the other be-fore in one. Broken Progression. Broken it is, whan me lepithe fro o nombre tille another, and kepithe not 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 passethe the others by-fore by .2. And note wele, that naturelle progressioun ay begynnethe with one, and Intercise or broken progressioun, omwhile begynnythe with one, omwhile with twayne. Of progressioun naturell .2. rules ther be yove, of the whiche the first is this; The 1st rule for Natural Progression. whan the progressioun naturelle endithe in even nombre, by the half therof multiplie þe next totalle ouerere nombre; Example of grace: .1. 2. 3. 4. Multiplie .5. by .2. and so .10. comethe of, that is the totalle nombre þerof. The second rule. The seconde rule is suche, whan the progressioun naturelle endithe in nombre ode. Take the more porcioun of the oddes, and multiplie therby the totalle nombre. Example of grace 1. 2. 3. 4. 5., multiplie .5. by .3, and thryes .5. shalle be resultant. so the nombre totalle is .15. The first rule of Broken Progression. Of progresioun intercise, ther ben also .2.18 rules; and þe first is þis: Whan the Intercise progression endithe in even nombre by half therof multiplie the next nombre to þat halfe as .2.18 4. 6. Multiplie .4. by .3. so þat is thryes .4., and .12. the nombre of alle the progressioun, wolle folow. The second rule. The seconde rule is this: whan the progressioun interscise endithe in ode, take þe more porcioun of alle þe nombre, Fol. 53⁴. *and multiplie by hym-selfe; as .1. 3. 5. Multiplie .3. by hym-selfe, and þe some of alle wolle be .9., etc.

Chapter IX. Extraction of Roots.

The preamble of the extraction of roots. Here folowithe the extraccioun of rotis, and first in nombre quadrates. Wherfor me shalle se what is a nombre 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 is lyneal, anoþer superficialle, anoþer quadrat, anoþer cubike or hoole. lyneal is that þat is considrede after the processe, havynge no respect to the direccioun of nombre in nombre, As a lyne hathe but one dymensioun that is to sey after the lengthe. Superficial numbers. Nombre superficial is þat comethe of ledynge of oo nombre into a-nother, wherfor it is callede superficial, for it hathe .2. nombres notyng or mesurynge hym, as a superficialle thynge hathe .2. dimensions, þat is to sey lengthe and brede. Square numbers. And for bycause a nombre may be hade in a-nother by .2. maners, þat is to sey other in hym-selfe, oþer in anoþer, Vnderstonde yf 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 hade in a-noþer, the nombre is superficiel and not quadrat, as .2. hade in .3. makethe .6. that is þe first nombre superficielle; wherfor it is open þat alle nombre quadrat is superficiel, and not conuertide. The root of a square number. The rote of a nombre quadrat is þat nombre that is had of hym-self, as twies .2. makithe 4. and .4. is the first nombre quadrat, and 2. is his rote. 9. 8. 7. 6. 5. 4. 3. 2. 1. / The rote of the more quadrat .3. 1. 4. 2. 6. Notes of some examples of square roots here interpolated. The most nombre quadrat 9. 8. 7. 5. 9. 3. 4. 7. 6. / the remenent ouer the quadrat .6. 0. 8. 4. 5. / The first caas of nombre quadrat .5. 4. 7. 5. 6. The rote .2. 3. 4. The seconde caas .3. 8. 4. 5. The rote .6. 2. The thirde caas .2. 8. 1. 9. The rote .5. 3. The .4. caas .3. 2. 1. The rote .1. 7. / The 5. caas .9. 1. 2. 0. 4. / The rote 3. 0. 2. Solid numbers. The solide nombre or cubike is þat þat comytħe of double ledyng of nombre in nombre; Three dimensions of solids. And it is clepede a solide body that hathe þer-in .3 [dimensions] þat is to sey, lengthe, brede, and thiknesse. so þat nombre hathe .3. nombres to be brought forthe in hym. But nombre may be hade twies in nombre, for other it is hade in hym-selfe, oþer in a-noþer. Cubic numbers. If a nombre be hade twies in hym-self, oþer ones in his quadrat, þat is the same, þat a cubike Fol. 54. *is, And is the same that is solide. And yf a nombre twies be hade in a-noþer, the nombre is clepede solide and not cubike, as twies .3. and þat .2. makithe .12. All cubics are solid numbers. Wherfor it is opyne that alle cubike nombre is solide, and not conuertide. 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 vndirstonde wele also that betwix euery .2. quadrates ther is a meene proporcionalle, Examples of square roots. That is openede thus; lede the rote of o quadrat into the rote of the oþer quadrat, and þan wolle þe meene shew.

Residuum 0 4 0 0

Quadrande 4 3 5 6 3 0 2 9 1 7 4 2 4 1 9 3 6

Duplum 1 2 1 0 2 6 [8] 19

Subduplum 6 6 5 5 1 3 2 4 4

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 brynge the rote of the lesse into a quadrat of the more. The lesse thus, If the rote of the more be brought Into the quadrat of the lesse.

Chapter X. Extraction of Square Root.

To20 draw a rote of the nombre quadrat it is What-euer nombre be proposede to 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, and compt the nombre of the figures, and wete yf it be ode or even. And yf Begin with the last odd place. it be even, than most thow begynne worche vnder the last save one. And yf it be ode with the last; and forto sey it shortly, al-weyes fro the last ode me shalle begynne. Therfor vnder the last in an od place sette, Find the nearest square root of that number, subtract, me most fynde a digit, the whiche lade in hym-selfe it puttithe away that, þat is ouer his hede, oþer as neighe as me may: suche a digit founde and withdraw fro his ouerer, me most double that digit and sette the double vnder the next figure towarde the right honde, and his vnder double vnder hym. double it, That done, than me most fynde a-noþer digit vnder the next figure bifore the doublede, and set the double one to the right. the whiche Fol. 54 b. *brought in double settethe a-way alle that is ouer his hede as to rewarde of the doublede: Than brought into hym-self settithe all 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 digit 21[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 produccione of them bothe, So that the first figure of the last product be addede before the first of the first productes, the seconde of the first, etc. and so forthe, subtrahe fro the totalle nombre in respect of þe digit.

Examples.

The residue 5 4 3 2

To be quadrede 4 1 2 0 9 1 5 1 3 9 9 0 0 5 4 3 2

The double 4 0 2 4 6 0 0

The vnder double 2 0 3 1 2 3 [3] [0] [0] 0

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. The residue. That done, other ought or nought wolle be the residue. If nought, than it shewithe that a nombre componede was the quadrat, and his rote a digit last founde with vndere-double other vndirdoubles, so that it be sette be-fore: And yf ought22 remayne, that shewith that the nombre proposede was not quadrat,23 but a digit [last found with the subduple or subduples is]

This table is constructed for use in cube root sums, giving the value of

ab.

2

1 2 3 4 5 6 7 8 9

2 8 12 16 20 24 28 32 36

3 18 27 36 45 54 63 72 81

4 32 48 64 80 96 11224 128 144

5 50 75 100 125 150 175 200 225

6 72 108 144 180 216 252 288 324

7 98 147 196 245 294 343 393 441

8 128 192 256 320 384 448 512 576

9 168 243 324 405 486 567 648 72925

The rote of the most quadrat conteynede vndre the nombre proposede. How to prove the square root without or with a remainder. 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; wher-for me most se what is a nombre cubike, and what is his roote, And what is the extraccioun of a rote. A nombre cubike it is, as it is before declarede, that comethe of ledyng of any nombre twies in hym-selfe, other ones in his quadrat. The rote of a nombre cubike is the nombre that is twies hade in hym-selfe, or ones in his quadrat. Wher-thurghe it is open, that euery nombre quadrat or cubike have the same rote, as it is seide before. And forto draw out the rote of a cubike, It is first to fynde þe nombre proposede yf it be a cubike; And yf it be not, than thow most make extraccioun of his rote of the most cubike vndre the nombre proposide his rote founde. Therfor proposede some nombre, whos cubical rote þou woldest draw out; Mark off the places in threes. First thow most compt the figures by fourthes, that is to sey in the place of thousandes; Find the first digit; And vnder the last thousande place, thow most fynde a digit, the whiche lade in hym-self cubikly puttithe a-way that þat is ouer his hede as in respect of hym, other as nyghe as thow maist. treble it and place it under the next but one, and multiply by the digit. That done, thow most trebille the digit, and that triplat is to be put vnder the .3. next figure towarde the right honde, And the vnder-trebille vnder the trebille; Then find the second digit. Than me most fynde a digit vndre the next figure bifore the triplat, the whiche with his vnder-trebille had into a trebille, afterwarde other vnder[trebille]26 had in his produccioun, puttethe a-way alle that is ouer it in regarde of27 [the triplat. Then lade in hymself puttithe away that þat is over his hede as in respect of hym, other as nyghe as thou maist:] Multiply the first triplate and the second digit, twice by this digit. That done, thow most trebille the digit ayene, and the triplat is to be sette vnder the next .3. figure as before, And the vnder-trebille vnder the trebille: and than most thow sette forwarde the first triplat with his vndre-trebille by .2. differences. And than most thow fynde a digit vnder the next figure before the triplat, the whiche withe his vnder-triplat had in his triplat afterwarde, Subtract. other vnder-treblis lad in product Fol. 55 b. *It sittethe a-way ałł that is ouer his hede in respect of the triplat than had in hym-self cubikly,28 or as nyghe as ye may.

Examples.

Residuum 5 4 1 0 1 9

Cubicandus 8 3 6 5 4 3 2 3 0 0 7 6 7 1 1 6 6 7

Triplum 6 0 1 8 4

Subtriplum 2 0 [3] 6 7 2 2

Continue this process till the first figure is reached. Nother me shalle not cesse of the fyndynge of that digit, neither of his triplacioun, neþer of the triplat-is 29anterioracioun, that is to sey, settyng forwarde by .2. differences, Ne therof the vndre-triple to be put vndre the triple, Nether of the multiplicacioun þerof, Neither of the subtraccioun, tille it come to the first figure, vnder the whiche is a digitalle nombre to be founde, the whiche withe his vndre-treblis most be hade in tribles, After-warde without vnder-treblis to be hade into produccioun, settyng away alle that is ouer the hede of the triplat nombre, After had into hymselfe cubikly, Examples. and sette alle-way that is ouer hym.

To be cubicede 1 7 2 8 3 2 7 6 8

The triple 3 2 9

The vnder triple 1 2 [3] 3 3

Also note wele that the produccion comynge of the ledyng of a digite founde30 me may adde to, and also with-draw fro of the totalle nombre sette above that digit so founde.31 The residue. That done ought or nought most be the residue. If it be nought, It is open that the nombre proposede was a cubike nombre, And his rote a digit founde last with the vnder-triples: If the rote therof wex bade in hym-selfe, and afterwarde product they shalle make the first figures. And yf ought be in residue, kepe that without in the table; and it is opene that the nombre was not a cubike. but a digit last founde with the vndirtriplis is rote of the most cubike vndre the nombre proposede conteynede, the whiche rote yf it be hade in hym-selfe, Special cases. And afterwarde in a product of that shalle growe the most cubike vndre the nombre proposede conteynede, And yf that be addede to a cubike the residue reseruede in the table, wolle make the same figures that ye hade first. Special case. Fol. 56. *And yf no digit after the anterioracioun32 may not be founde, than put there a cifre vndre a cifre vndir the thirde figure, And put forwarde þe figures. Note also wele that yf in the nombre proposede ther ben no place of thowsandes, me most begynne vnder the first figure in the extraccioun of the rote. some vsen forto distingue the nombre by threes, and ay begynne forto wirche vndre the first of the last ternary other uncomplete nombre, the whiche maner of operacioun accordethe with that before.

Examples.

The residue 0 1 1

The cubicandus 8 0 0 0 0 0 0 8 2 4 2 4 1 9

The triple 33 0 0 6

The vndertriple [2] 0 0 2 6 2

And this at this tyme suffisethe in extraccioun of nombres quadrat or cubikes etc.

Table of Numbers, &c.

A table of numbers; probably from the Abacus.

1 2 3 4 5

one. x. an. hundrede / a thowsande / x. thowsande /

6 7

An hundrede thowsande / A thowsande tymes a thowsande /

x. thousande tymes a thousande / An hundrede thousande tymes a thousande A thousande thousande tymes a thousande / this is the x place etc.

[Ende.]

The original text was printed as a single continuous paragraph, with no break between speakers; all examples were shown inline. It has been broken up for this e-text.

116

b

.

* ¶ The seconde dialoge of accomptynge by counters.

Mayster.

Nowe that you haue learned the commen kyndes of Arithmetyke with the penne, you shall se the same art in counters: whiche feate doth not only serue for them that can not write and rede, but also for them that can do bothe, but haue not at some tymes theyr penne or tables redye with them. This sorte is in two fourmes commenly. The one by lynes, and the other without lynes: in that y^t hath lynes, the lynes do stande for the order of places: and in y^t that hath no lynes, there must be sette in theyr stede so many counters as shall nede, for eche lyne one, and they shall supplye the stede of the lynes.

S. By examples I shuld better perceaue your meanynge.

M. For example of the 117 a ly*nes:

1 0 0 0 0 0

1 0 0 0 0

X 1 0 0 0

1 0 0

1 0

1

Lo here you se .vi. lynes whiche stande for syxe places so that the nethermost standeth for y^e fyrst place, and the next aboue it, for the second: and so vpward tyll you come to the hyghest, which is the syxte lyne, and standeth for the syxte place. Numeration. 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. The thyrd lyne the place of hundredes: the fourth of thousandes: and so forth.

S. Syr I do perceaue that the same order is here of lynes, as was in the other figures 117 b. *by places, so that you shall not nede longer to stande about Numeration, excepte there be any other difference.

M. Yf you do vnderstande it, then how wyll you set 1543?

X 1

5

4

3

S. Thus, as I suppose.

M. You haue set y^e places truely, but your figures be not mete for this vse: for the metest figure in this behalfe, is the figure of a counter round, as you se here, where I haue expressed that same summe.

× o

°

o o o o

o o o

S. So that you haue not one figure for 2, nor 3, nor 4, and so forth, but as many digettes as you haue, you set in the lowest lyne: and for euery 10 you set one in the second line: and so of other. But I know not by what reason you set that one counter for 500 betwene two lynes.

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. And farther you shall marke, that in all workynge by this sorte, yf you shall sette downe any summe betwene 4 and 10, for the fyrste parte of that nomber you shall set downe 5, & then so many counters more, as there reste nombers aboue 5. And this is true bothe of digettes and articles. And for example I wyll set downe this summe 287965,

X

o o

o o°o

X o o°

o o°o o

o°

which summe yf you marke well, you nede none other examples for to lerne the numeration of 118 b. *this forme. But this shal you marke, that as you dyd in the other kynde of arithmetike, set a pricke in the places of thousandes, in this worke you shall sette a starre, as you se here.

Addition on the Counting Board.

Addition. S. Then I perceave numeration, but I praye you, howe shall I do in this arte to adde two summes or more together?

M. The easyest way in this arte is, to adde but 2 summes at ones together: how be it you may adde more, as I wyll tell you anone. Therfore when you wyll adde two summes, you shall fyrst set downe one of them, it forseth not whiche, and then by it drawe a lyne crosse the other lynes. And afterward set downe the other summe, so that that lyne may be betwene them, as yf you wolde adde 2659 to 8342, you must set your summes as you se here.

X o°o o o o

o o o o°

o o o o °

o o o°o o o

And then yf you lyst, you 119 a. *may adde the one to the other in the same place, or els you may adde them both together in a newe place: which waye, bycause it is moste playnest, I wyll showe you fyrst. Therfore wyl I begynne at the vnites, whiche in the fyrst summe is but 2, and in y^e second summe 9, that maketh 11, those do I take vp, and for them I set 11 in the new roume, thus,

X o°o o o o

o o o o°

o o o o ° o

o

Then do I take vp all y^e articles vnder a hundred, which in the fyrst summe are 40, and in the second summe 50, that maketh 90: or you may saye better, that in the fyrste summe there are 4 articles of 10, and in the seconde summe 5, which make 9, but then take hede that you sette them in theyr 119 b. *ryght lynes as you se here.

X o°o o o o

o o o o°

o°o o o o

o

Where I haue taken awaye 40 from the fyrste summe, and 50 from y^e second, and in theyr stede I haue set 90 in the thyrde, whiche I haue set playnely y^t you myght well perceaue it: how be it seynge that 90 with the 10 that was in y^e thyrd roume all redy, doth make 100, I myghte better for those 6 counters set 1 in the thyrde lyne, thus:

X

o

o

For it is all one summe as you may se, but it is beste, neuer to set 5 counters in any line, for that may be done with 1 counter in a hygher place.

S. I iudge that good reason, for many are vnnedefull, where one wyll serue.

M. Well, then 120 a. *wyll I adde forth of hundredes: I fynde 3 in the fyrste summe, and 6 in the seconde, whiche make 900, them do I take vp and set in the thyrd roume where is one hundred all redy, to whiche I put 900, and it wyll be 1000, therfore I set one counter in the fourth lyne for them all, as you se here.

X o°o o o o o

o

Then adde I y^e thousandes together, whiche in the fyrst summe are 8000, and in y^e second 2000, that maketh 10000: them do I take vp from those two places, and for them I set one counter in the fyfte lyne, and then appereth as you se,

o

X o

o

to be 11001, for so many doth amount of the addition of 8342 to 2659.

120 b. *S. Syr, this I do perceave: but how shall I set one summe to an other, not chaungynge them to a thyrde place?

M. Marke well how I do it: I wyll adde together 65436, and 3245, whiche fyrste I set downe thus.

o °

X o o o °

o o o o o o

o o o o o o o

° o °

Then do I begynne with the smalest, which in the fyrst summe is 5, that do I take vp, and wold put to the other 5 in the seconde summe, sauynge that two counters can not be set in a voyd place of 5, but for them bothe I must set 1 in the seconde lyne, which is the place of 10, therfore I take vp the 5 of the fyrst summe, and the 5 of the seconde, and for them I set 1 in the second lyne, 121 a. *as you se here.

o°

X o o o °

o o o o o o

o o o o o o o o

o

Then do I lyke wayes take vp the 4 counters of the fyrste summe and seconde lyne (which make 40) and adde them to the 4 counters of the same lyne, in the second summe, and it maketh 80, But as I sayde I maye not conueniently set aboue 4 counters in one lyne, therfore to those 4 that I toke vp in the fyrst summe, I take one also of the seconde summe, and then haue I taken vp 50, for whiche 5 counters I sette downe one in the space ouer y^e second lyne, as here doth appere.

o°

X o o o °

o o o o o o

o°o o

o

121 b *and then is there 80, as well w^t those 4 counters, as yf I had set downe y^e other 4 also. Now do I take the 200 in the fyrste summe, and adde them to the 400 in the seconde summe, and it maketh 600, therfore I take vp the 2 counters in the fyrste summe, and 3 of them in the seconde summe, and for them 5 I set 1 in y^e space aboue, thus.

o°

X o o o °

o°

o°o o

o

Then I take y^e 3000 in y^e fyrste summe, vnto whiche there are none in the second summe agreynge, therfore I do onely remoue those 3 counters from the fyrste summe into the seconde, as here doth appere.

o°

X o°o o

o °

o°o o

o

122 a *And so you see the hole summe, that amounteth of the addytion of 65436 with 3245 to be 6868[1]. And yf you haue marked these two examples well, you nede no farther enstruction in Addition of 2 only summes: but yf you haue more then two summes to adde, you may adde them thus.

X o o o o o o o°

o° o o o°o o o

o°o o°o o o°

o°o o o o° °

Fyrst adde two of them, and then adde the thyrde, and y^e fourth, or more yf there be so many: as yf I wolde adde 2679 with 4286 and 1391. Fyrste I adde the two fyrste summes thus. 122 b. *And then I adde the thyrde thereto thus.

X o o° o o o°

o o o o°o o o o o o

o°o o o °o °

o ° o°

And so of more yf you haue them.

Subtraction on the Counting Board.

S. Nowe I thynke beste that you passe forth to Subtraction, except there be any wayes to examyn this maner of Addition, then I thynke that were good to be knowen nexte.

M. There is the same profe here that is Subtraction. in the other Addition by the penne, I meane Subtraction, for that onely is a sure waye: but consyderynge that Subtraction must be fyrste knowen, I wyl fyrste teache you the arte of Subtraction, and that by this example: I wolde subtracte 2892 out of 8746. These summes must I set downe as I dyd in Addition: but here it is best 116 a (sic). *to set the lesser nomber fyrste, thus.

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Then shall I begynne to subtracte the greatest nombres fyrste (contrary to the vse of the penne) y^t is the thousandes in this example: therfore I fynd amongest the thousandes 2, for which I withdrawe so many from the seconde summe (where are 8) and so remayneth there 6, as this example showeth.

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Then do I lyke wayes with the hundredes, of whiche in the fyrste summe 116 b. *I fynde 8, and is the seconde summe but 7, out of whiche I can not take 8, therfore thus muste I do: I muste loke how moche my summe dyffereth from 10, whiche I fynde here to be 2, then must I bate for my summe of 800, one thousande, and set downe the excesse of hundredes, that is to saye 2, for so moche 100[0] is more then I shuld take vp. Therfore from the fyrste summe I take that 800, and from the second summe where are 6000, I take vp one thousande, and leue 5000; but then set I downe the 200 unto the 700 y^t are there all redye, and make them 900 thus.

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Then come I to the articles of tennes where in the fyrste summe I fynde 90, 117 a. *and in the seconde summe but only 40: Now consyderyng that 90 can not be bated from 40, I loke how moche y^t 90 doth dyffer from the next summe aboue it, that is 100 (or elles whiche is all to one effecte, I loke how moch 9 doth dyffer from 10) and I fynd it to be 1, then in the stede of that 90, I do take from the second summe 100: but consyderynge that it is 10 to moche, I set downe 1 in y^e nexte lyne beneth for it, as you se here.

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Sauynge that here I haue set one counter in y^e space in stede of 5 in y^e nexte lyne. And thus haue I subtracted all saue two, which I must bate from the 6 in the second summe, and there wyll remayne 4, thus.

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So y^t yf I subtracte 2892 from 8746, the remayner wyll be 5854, 117 b. *And that this is truely wrought, you maye proue by Addition: for yf you adde to this remayner the same summe that you dyd subtracte, then wyll the formar summe 8746 amount agayne.

S. That wyll I proue: and fyrst I set the summe that was subtracted, which was 2892, and then the remayner 5854, thus.

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Then do I adde fyrst y^e 2 to 4, whiche maketh 6, so take I vp 5 of those counters, and in theyr stede I sette 1 in the space, as here appereth.

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118 a. *Then do I adde the 90 nexte aboue to the 50, and it maketh 140, therfore I take vp those 6 counters, and for them I sette 1 to the hundredes in y^e thyrde lyne, and 4 in y^e second lyne, thus.

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Then do I come to the hundredes, of whiche I fynde 8 in the fyrst summe, and 9 in y^e second, that maketh 1700, therfore I take vp those 9 counters, and in theyr stede I sette 1 in the .iiii. lyne, and 1 in the space nexte beneth, and 2 in the thyrde lyne, as you se here.

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Then is there lefte in the fyrste summe but only 2000, whiche I shall take vp from thence, and set 118 b. *in the same lyne in y^e second summe, to y^e one y^t is there all redy: and then wyll the hole summe appere (as you may wel se) to be 8746, which was y^e fyrst grosse summe, and therfore I do perceaue, that I hadde well subtracted before.

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And thus you may se how Subtraction maye be tryed by Addition.

S. I perceaue the same order here w^t counters, y^t I lerned before in figures.

M. Then let me se howe can you trye Addition by Subtraction.

S. Fyrste I wyl set forth this example of Addition where I haue added 2189 to 4988, and the hole summe appereth to be 7177,

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119 a. *Nowe to trye whether that summe be well added or no, I wyll subtract one of the fyrst two summes from the thyrd, and yf I haue well done y^e remayner wyll be lyke that other summe. As for example: I wyll subtracte the fyrste summe from the thyrde, whiche I set thus in theyr order.

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Then do I subtract 2000 of the fyrste summe from y^e second summe, and then remayneth there 5000 thus.

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Then in the thyrd lyne, I subtract y^e 100 of the fyrste summe, from the second summe, where is onely 100 also, and then in y^e thyrde lyne resteth nothyng. Then in the second lyne with his space ouer hym, I fynde 80, which I shuld subtract 119 b. *from the other summe, then seyng there are but only 70 I must take it out of some hygher summe, which is here only 5000, therfore I take vp 5000, and seyng that it is to moch by 4920, I sette downe so many in the seconde roume, whiche with the 70 beynge there all redy do make 4990, & then the summes doth stande thus.

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Yet remayneth there in the fyrst summe 9, to be bated from the second summe, where in that place of vnities dothe appere only 7, then I muste bate a hygher summe, that is to saye 10, but seynge that 10 is more then 9 (which I shulde abate) by 1, therfore shall I take vp one counter from the seconde lyne, and set downe the same in the fyrst 120 a. *or lowest lyne, as you se here.

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And so haue I ended this worke, and the summe appereth to be y^e same, whiche was y^e seconde summe of my addition, and therfore I perceaue, I haue wel done.

M. To stande longer about this, it is but folye: excepte that this you maye also vnderstande, that many do begynne to subtracte with counters, not at the hyghest summe, as I haue taught you, but at the nethermoste, as they do vse to adde: and when the summe to be abatyd, in any lyne appeareth greater then the other, then do they borowe one of the next hygher roume, as for example: yf they shuld abate 1846 from 2378, they set y^e summes thus.

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120 b. *And fyrste they take 6 whiche is in the lower lyne, and his space from 8 in the same roumes, in y^e second summe, and yet there remayneth 2 counters in the lowest lyne. Then in the second lyne must 4 be subtracte from 7, and so remayneth there 3. Then 8 in the thyrde lyne and his space, from 3 of the second summe can not be, therfore do they bate it from a hygher roume, that is, from 1000, and bycause that 1000 is to moch by 200, therfore must I sette downe 200 in the thyrde lyne, after I haue taken vp 1000 from the fourth lyne: then is there yet 1000 in the fourth lyne of the fyrst summe, whiche yf I withdrawe from the seconde summe, then doth all y^e figures stande in this order.

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So that (as you se) it differeth not greatly whether you begynne subtraction at the hygher lynes, or at 121 a. *the lower. How be it, as some menne lyke the one waye beste, so some lyke the other: therfore you now knowyng bothe, may vse whiche you lyst.

Multiplication by Counters.

Multiplication. But nowe touchynge Multiplication: you shall set your nombers in two roumes, as you dyd in those two other kyndes, but so that the multiplier be set in the fyrste roume. Then shall you begyn with the hyghest nombers of y^e seconde roume, and multiply them fyrst after this sort. Take that ouermost lyne in your fyrst workynge, as yf it were the lowest lyne, setting on it some mouable marke, as you lyste, and loke how many counters be in hym, take them vp, and for them set downe the hole multyplyer, so many tymes as you toke vp counters, reckenyng, I saye that lyne for the vnites: and when you haue so done with the hygheest nomber then come to the nexte lyne beneth, and do euen so with it, and so with y^e next, tyll you haue done all. And yf there be any nomber in a space, then for it 121 b. *shall you take y^e multiplyer 5 tymes, and then must you recken that lyne for the vnites whiche is nexte beneth that space: or els after a shorter way, you shall take only halfe the multyplyer, but then shall you take the lyne nexte aboue that space, for the lyne of vnites: but in suche workynge, yf chaunce your multyplyer be an odde nomber, so that you can not take the halfe of it iustly, then muste you take the greater halfe, and set downe that, as if that it were the iuste halfe, and farther you shall set one counter in the space beneth that line, which you recken for the lyne of vnities, or els only remoue forward the same that is to be multyplyed.

S. Yf you set forth an example hereto I thynke I shal perceaue you.

M. Take this example: I wold multiply 1542 by 365, therfore I set y^e nombers thus.

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122 a. *Then fyrste I begynne at the 1000 in y^e hyghest roume, as yf it were y^e fyrst place, & I take it vp, settynge downe for it so often (that is ones) the multyplyer, which is 365, thus, as you se here:

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where for the one counter taken vp from the fourth lyne, I haue sette downe other 6, whiche make y^e summe of the multyplyer, reckenynge that fourth lyne, as yf it were the fyrste: whiche thyng I haue marked by the hand set at the begynnyng of y^e same,

S. I perceaue this well: for in dede, this summe that you haue set downe is 365000, for so moche doth amount 122 b. *of 1000, multiplyed by 365.

M. Well then to go forth, in the nexte space I fynde one counter which I remoue forward but take not vp, but do (as in such case I must) set downe the greater halfe of my multiplier (seyng it is an odde nomber) which is 182, and here I do styll let that fourth place stand, as yf it were y^e fyrst:

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as in this fourme you se, where I haue set this multiplycation with y^e other: but for the ease of your vnderstandynge, I haue set a lytell lyne betwene them: now shulde they both in one summe stand thus.

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123 a. *Howe be it an other fourme to multyplye suche counters in space is this: Fyrst to remoue the fynger to the lyne nexte benethe y^e space, and then to take vp y^e counter, and to set downe y^e multiplyer .v. tymes, as here you se.