Chapter 6

Therfore sithen the science of the whiche at this tyme we intenden{e} to write of standith{e} all{e} and about nombre: ffirst we most se, what is the p{ro}pre name therof{e}, and fro whens the name come: Afterward{e} what is nombre, And how manye spices of nombre ther ben. The name is cleped{e} Algorisme, had{e} out of Algor{e}, other of Algos, in grewe, That is clepid{e} in englissh{e} art other craft, And of Rithm{us} that is called{e} nombre. So algorisme is cleped{e} the art of nombryng, other it is had of{e} en or in, and gogos that is introduccio{u}n, and Rithm{us} nombre, that is to say Interduccio{u}n of nombre. And thirdly it is had{e} of the name of a kyng that is cleped{e} Algo and Rythm{us}; So called{e} Algorism{us}. Sothely .2. maner{e} of nombres ben notified{e}; Formall{e},[{1}] as nombr{e} i{s} vnitees gadred{e} to-gedres; Materiall{e},[{2}] as nombr{e} is a colleccio{u}n of vnitees. Other nombr{e} is a multitude had{e} out of vnitees, vnitee is that thynge wher-by eu{er}y thynge is called{e} oone, other o thynge. Of nombres, that one is cleped{e} digitall{e}, that other{e} Article, Another a nombre componed{e} oþ{er} myxt. Another digitall{e} is a nombre w{i}t{h}-in .10.; Article is þ{a}t nombre that may be dyvyded{e} in .10. p{ar}ties egally, And that there leve no residue; Componed{e} or medled{e} is that nombre that is come of a digite and of an article. And vndrestand{e} wele that all{e} nombres betwix .2. articles next is a nombr{e} componed{e}. Of this art ben{e} .9. spices, that is forto sey, num{er}acio{u}n, addicio{u}n, Subtraccio{u}n, Mediac{i}o{u}n, Duplacio{u}n, Multipliacio{u}n, Dyvysio{u}n, Progressio{u}n, And of Rootes the extraccio{u}n, and that may be had{e} in .2. maners, that is to sey in nombres quadrat, and in cubic{es}: Amonge the which{e}, ffirst of Num{er}acio{u}n, and aft{er}ward{e} of þe oþ{er}s by ordure, y entende to write.

[Headnote: Chapter I. Numeration.]

[*Fol. 48b]

+For-soth{e} num{er}acio{u}n is of eu{er}y numbre by competent figures an artificiall{e} rep{re}sentacio{u}n.+

[Sidenote: Figures, differences, places, and limits. The 9 figures. The cipher. The numeration of digits, of articles, of composites. The value due to position. Numbers are written from right to left.]

Sothly figure, difference, places, and lynes supposen o thyng other the same, But they ben sette here for dyue{r}s resons. ffigure is cleped{e} for p{ro}traccio{u}n of figuracio{u}n; Difference is called{e} for therby is shewed{e} eu{er}y figure, how it hath{e} difference fro the figures before them: place by cause of space, where-in me writeth{e}: lynees, for that is ordeyned{e} for the p{re}sentacio{u}n of eu{er}y figure. And vnderstonde that ther ben .9. lymytes of figures that rep{re}senten the .9. digit{es} that ben these. 0. 9. 8. 7. 6. 5. 4. 3. 2. 1. The .10. is cleped{e} theta, or a cercle, other a cifre, other a figure of nought for nought it signyfieth{e}. Nathelesse she holdyng that place giveth{e} others for to signyfie; for with{e}-out cifre or cifres a pure article may not be writte. And sithen that by these .9. figures significatif{es} Ioyned{e} w{i}t{h} cifre or w{i}t{h} cifres all{e} nombres ben and may be rep{re}sented{e}, It was, nether is, no nede to fynde any more figures. And note wele that eu{er}y digite shall{e} be writte w{i}t{h} oo figure allone to it ap{ro}pred{e}. And all{e} articles by a cifre, ffor eu{er}y article is named{e} for oone of the digitis as .10. of 1.. 20. of. 2. and so of the others, &c. And all{e} nombres digitall{e} owen to be sette in the first difference: All{e} articles in the seconde. Also all{e} nombres fro .10. til an .100. [which] is excluded{e}, with .2. figures mvst be writte; And yf it be an article, by a cifre first put, and the figure y-writte toward{e} the lift hond{e}, that signifieth{e} the digit of the which{e} the article is named{e}; And yf it be a nombre componed{e}, ffirst write the digit that is a part of that componed{e}, and write to the lift side the article as it is seid{e} be-fore. All{e} nombre that is fro an hundred{e} tille a thousand{e} exclused{e}, owith{e} to be writ by .3. figures; and all{e} nombre that is fro a thousand{e} til .x. Mł. mvst be writ by .4. figures; And so forthe. And vnderstond{e} wele that eu{er}y figure sette in the first place signyfieth{e} his digit; In the second{e} place .10. tymes his digit; In the .3. place an hundred{e} so moche; In the .4. place a thousand{e} so moche; In the .5. place .x. thousand{e} so moch{e}; In the .6. place an hundred{e} thousand{e} so moch{e}; In the .7. place a thousand{e} thousand{e}. And so infynytly mvltiplying by [*Fol. 49.] these .3. 10, 100, 1000. And vnderstand{e} wele that competently me may sette vpon figure in the place of a thousand{e}, a prik{e} to shewe how many thousand{e} the last figure shall{e} rep{re}sent. We writen{e} in this art to the lift side-ward{e}, as arabien{e} writen{e}, that weren fynders of this science, other{e} for this reso{u}n, that for to kepe a custumable ordr{e} in redyng, Sette we all{e}-wey the more nombre before.

[Headnote: Chapter II. Addition.]

[Sidenote: Definition. How the numbers should be written. The method of working. Begin at the right. The Sum is a digit, or an article, or a composite.]

Addicio{u}n is of nombre other of nombres vnto nombre or to nombres aggregacio{u}n, that me may see that that is come therof as exc{re}ssent. In addicio{u}n, 2. ordres of figures and .2. nombres ben necessary, that is to sey, a nombre to be added{e} and the nombre wherto the addic{i}oun shold{e} be made to. The nombre to be added{e} is that þat shold{e} be added{e} therto, and shall{e} be vnderwriten; the nombre vnto the which{e} addicio{u}n shall{e} be made to is that nombre that resceyueth{e} the addicion of þat other, and shall{e} be writen above; and it is convenient that the lesse nombre be vnderwrit, and the more added{e}, than the contrary. But whether it happ{e} one other other, the same comyth{e} of, Therfor, yf þow wilt adde nombre to nombre, write the nombre wherto the addicio{u}n shall{e} 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. That done, adde the first of the lower ordre to the first of the omyst ordre. And of such{e} addicio{u}n, other þ{er}e grow{i}t{h} therof a digit, An article, other a composed{e}. If it be digit{us}, In the place of the omyst shalt thow write the digit excrescyng, as thus:--

+----------------------------+---+ |The resultant | 2 | +----------------------------+---+ |To whom it shal be added{e} | 1 | +----------------------------+---+ |The nombre to be added{e} | 1 | +----------------------------+---+

If the article; in the place of the omyst put a-way by a cifre writte, and the digit transferred{e}, of þe which{e} the article toke his name, toward{e} the lift side, and be it added{e} to the next figure folowyng, yf ther be any figure folowyng; or no, and yf it be not, leve it [in the] void{e}, as thus:--

+---------------------------------+----+ | The resultant | 10 | +---------------------------------+----+ | To whom it shall{e} be added{e} | 7 | +---------------------------------+----+ | The nombre to be added{e} | 3 | +---------------------------------+----+

+----------------------+---+---+---+---+---+ | Resultans | 2 | 7 | 8 | 2 | 7 | +----------------------+---+---+---+---+---+ | Cui d{ebet} addi | 1 | 0 | 0 | 8 | 4 | +----------------------+---+---+---+---+---+ | Num{erus} addend{us} | 1 | 7 | 7 | 4 | 3 | +----------------------+---+---+---+---+---+

And yf it happe that the figure folowyng wherto the addicio{u}n shall{e} be made by [the cifre of] an article, it sette a-side; In his place write the [*Fol. 49b] [digit of the] Article as thus:--

+---------------------------------+----+ | The resultant | 17 | +---------------------------------+----+ | To whom it shall{e} be added{e} | 10 | +---------------------------------+----+ | The nombre to be added{e} | 7 | +---------------------------------+----+

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 cifre {and} write þe article toward{e} þe lift hond{e} as bifore, and thus:--

+---------------------------------+----+ | The resultant | 10 | +---------------------------------+----+ | To whom it shall{e} be added{e} | 9 | +---------------------------------+----+ | The nombre to be added{e} | 1 | +---------------------------------+----+

And yf[{3}] [therefrom grow a] nombre componed,[{4}] [in the place of the nombre] put a-way[{5}][let] the digit [be][{6}]writ þ{a}t is part of þ{a}t co{m}posid{e}, and þan put to þe lift side the article as before, and þus:--

+---------------------------------+----+ | The resultant | 12 | +---------------------------------+----+ | To whom it shall{e} be added{e} | 8 | +---------------------------------+----+ | The nombre to be added{e} | 4 | +---------------------------------+----+

This done, adde the seconde to the second{e}, and write above oþ{er} as before.

[Sidenote: The translator’s note.]

Note wele þ{a}t in addic{i}ons and in all{e} spices folowyng, whan he seith{e} one the other shall{e} be writen aboue, and me most vse eu{er} figure, as that eu{er}y figure were sette by half{e}, and by hym-self{e}.

[Headnote: Chapter III. Subtraction.]

[Sidenote: Definition of Subtraction. How it may be done. What is required. Write the greater number above. Subtract the first figure if possible. If it is not possible ‘borrow ten,’ and then subtract.]

Subtraccio{u}n is of .2. p{ro}posed{e} nombres, the fyndyng of the excesse of the more to the lasse: Other subtraccio{u}n is ablacio{u}n of o nombre fro a-nother, that me may see a some left. The lasse of the more, or even of even, may be w{i}t{h}draw; The more fro the lesse may neu{er} be. And sothly that nombre is more that hath{e} more figures, So that the last be signyficatife{s}: 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. More-ou{er} in w{i}t{h}-drawyng .2. nombres ben necessary; A nombre to be w{i}t{h}draw, And a nombre that me shall{e} w{i}t{h}-draw of. The nombre to be w{i}t{h}-draw shall{e} be writ in the lower ordre by his differences; The nombre fro the which{e} me shall{e} with{e}-draw in the omyst ordre, so that the first be vnder the first, the second{e} vnder the second{e}, And so of all{e} others. With{e}-draw therfor the first of the lower{e} ordre fro the first of the ordre above his hede, and that wolle be other more or lesse, oþ{er} egall{e}.

+---------------------------------+----+ | The remanent | 20 | +---------------------------------+----+ | Wherof me shall{e} w{i}t{h}draw | 22 | +---------------------------------+----+ | The nombre to be w{i}t{h}draw | 2 | +---------------------------------+----+

yf it be egall{e} or even the figure sette beside, put in his place a cifre. And yf it be more put away þ{er}fro als many of vnitees the lower figure conteyneth{e}, and writ the residue as thus

+----------------------------------+---+---+ | The remanent | 2 | 2 | +----------------------------------+---+---+ | Wherof me shall{e} w{i}t{h}-draw | 2 | 8 | +----------------------------------+---+---+ | Þe nombre to be w{i}t{h}draw | | 6 | +----------------------------------+---+---+

[*Fol. 50.]

+--------------------------+---+---+-----+---+---+---+---+---+---+ | Remane{n}s | 2 | 2 | 1 | 8 | 2 | 9 | 9 | 9 | 8 | +--------------------------+---+---+-----+---+---+---+---+---+---+ | A quo sit subtraccio | 8 | 7 | 2 | 4 | 3 | 0 | 0 | 0 | 4 | +--------------------------+---+---+-----+---+---+---+---+---+---+ | Numerus subt{ra}hend{us} | 6 | 5 |[{7}]|[6]| . | . | . | . | 6 | +--------------------------+---+---+-----+---+---+---+---+---+---+

And yf it be lesse, by-cause the more may not be w{i}t{h}-draw ther-fro, borow an vnyte of the next figure that is worth{e} 10. Of that .10. and of the figure that ye wold{e} have w{i}t{h}-draw fro be-fore to-gedre Ioyned{e}, w{i}t{h}-draw þe figure be-nethe, and put the residue in the place of the figure put a-side as þ{us}:--

+----------------------------------+---+---+ | The remanent | 1 | 8 | +----------------------------------+---+---+ | Wherof me shall{e} w{i}t{h}-draw | 2 | 4 | +----------------------------------+---+---+ | The nombre to be w{i}t{h}-draw | 0 | 6 | +----------------------------------+---+---+

[Sidenote: 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 þ{er}of, lest the figures folowing faile of thair{e} nombre, and þan worch{e} as it shew{i}t{h} in this figure here:--

+--------------------------------+---+---+------+ | The remanent | 3 | 0 |9[{8}]| +--------------------------------+---+---+------+ | Wherof me shal w{i}t{h}-draw | 3 | 1 | 2 | +--------------------------------+---+---+------+ | The nombre to be w{i}t{h}-draw | . | . | 3 | +--------------------------------+---+---+------+

[Sidenote: If the second figure is a cipher.]

And yf the vnyte wherof me shal borow be a cifre, go ferther to the figure signyficatif{e}, and ther borow one, and reto{ur}nyng bak{e}, in the place of eu{er}y cifre þ{a}t ye passid{e} ou{er}, sette figures of .9. as here it is specified{e}:--

+----------------------------------+---+---+---+---+---+ | The remenaunt | 2 | 9 | 9 | 9 | 9 | +----------------------------------+---+---+---+---+---+ | Wherof me shall{e} w{i}t{h}-draw | 3 | 0 | 0 | 0 | 3 | +----------------------------------+---+---+---+---+---+ | The nombre to be w{i}t{h}-draw | | | | | 4 | +----------------------------------+---+---+---+---+---+

[Sidenote: A justification of the rule given. Why it is better to work from right to left. How to prove subtraction, and addition.]

And whan me cometh{e} to the nombre wherof me intendith{e}, there remayneth{e} all{e}-wayes .10. ffor þe which{e} .10. &c. The reson why þat for eu{er}y cifre left behynde me setteth figures ther of .9. this it is:--If fro the .3. place me borowed{e} an vnyte, that vnyte by respect of the figure that he came fro rep{re}sentith an .C., In the place of that cifre [passed over] is left .9., [which is worth ninety], and yit it remayneth{e} as .10., And the same reson{e} wold{e} be yf me had{e} borowed{e} an vnyte fro the .4., .5., .6., place, or ony other so vpward{e}. This done, withdraw the second{e} of the lower ordre fro the figure above his hede of þe omyst ordre, and wirch{e} as before. And note wele that in addicion or in subtracc{i}o{u}n me may wele fro the lift side begynne and ryn to the right side, But it wol be more p{ro}fitabler to be do, as it is taught. And yf thow wilt p{ro}ve yf thow have do wele or no, The figures that thow hast withdraw, adde them ayene to the omyst figures, and they wolle accorde w{i}t{h} the first that thow haddest yf thow have labored wele; and in like wise in addicio{u}n, whan thow hast added{e} all{e} thy figures, w{i}t{h}draw them that thow first [*Fol. 50b] addest, and the same wolle reto{ur}ne. The subtraccio{u}n is none other but a p{ro}uff{e} of the addicio{u}n, and the contrarye in like wise.

[Headnote: Chapter IV. Mediation.]

[Sidenote: Definition of mediation. Where to begin. If the first figure is unity. What to do if it is not unity.]

Mediacio{u}n is the fyndyng of the halfyng of eu{er}y nombre, that it may be seyn{e} what and how moch{e} is eu{er}y half{e}. In halfyng ay oo order of figures and oo nombre is necessary, that is to sey the nombre to be halfed{e}. Therfor yf thow wilt half any nombre, write that nombre by his differences, and begynne at the right, that is to sey, fro the first figure to the right side, so that it be signyficatif{e} other rep{re}sent vnyte or eny other digitall{e} nombre. If it be vnyte write in his place a cifre for the figures folowyng, [lest they signify less], and write that vnyte w{i}t{h}out in the table, other resolue it in .60. mynvt{es} and sette a-side half of tho m{inutes} so, and reserve the remen{au}nt w{i}t{h}out in the table, as thus .30.; other sette w{i}t{h}out thus .{dī}: that kepeth{e} none ordre of place, Nathelesse it hath{e} signyficacio{u}n. And yf the other figure signyfie any other digital nombre fro vnyte forth{e}, oþ{er} the nombre is od{e} or even{e}. If it be even, write this half in this wise:--

+-----------------+---+---+ | Halfed{e} | 2 | 2 | +-----------------+----+--+ | to be halfed{e} | 4 | 4 | +-----------------+---+---+

And if it be odde, Take the next even vndre hym conteyned{e}, and put his half in the place of that odde, and of þe vnyte that remayneth{e} to be halfed{e} do thus:--

+-----------------+---+---+ | halfed{e} | 2 | 3 | [di] +-----------------+---+---+ | To be halfed{e} | 4 | 7 | +-----------------+---+---+

[Sidenote: Then halve the second figure. If it is odd, add 5 to the figure before.]

This done, the second{e} is to be halfed{e}, yf it be a cifre put it be-side, and yf it be significatif{e}, other it is even or od{e}: If it be even, write in the place of þe nombres wiped{e} out the half{e}; yf it be od{e}, take the next even vnder it co{n}tenyth{e}, and in the place of the Impar sette a-side put half of the even: The vnyte that remayneth{e} to be halfed{e}, respect had{e} to them before, is worth{e} .10. Dyvide that .10. in .2., 5. is, and sette a-side that one, and adde that other to the next figure p{re}cedent as here:--

+-----------------+---+---+---+ | Halfed{e} | | | | +-----------------+---+---+---+ | to be halfed{e} | | | | +-----------------+---+---+---+

And yf þe addicio{u}n shold{e} be made to a cifre, sette it a-side, and write in his place .5. And vnder this fo{ur}me me shall{e} write and worch{e}, till{e} the totall{e} nombre be halfed{e}.

+------------------+---+---+---+---+---+----+----+---+ | doubled{e} | 2 | 6 | 8 | 9 | 0 | 10 | 17 | 4 | +------------------+---+---+---+---+---+----+----+---+ | to be doubled{e} | 1 | 3 | 4 | 4 | 5 | 5 | 8 | 7 | +------------------+---+---+---+---+---+----+----+---+

[Headnote: Chapter V. Duplation.]

[Sidenote: Definition of Duplation. Where to begin. Why. What to do with the result.]

Duplicacio{u}n is ag{re}gacion of nombre [to itself] þat me may se the nombre growen. In doublyng{e} ay is but one ordre of figures necessarie. And me most be-gynne w{i}t{h} the lift side, other of the more figure, And after the nombre of the more figure rep{re}sentith{e}. [*Fol. 51.] In the other .3. before we begynne all{e} way fro the right side and fro the lasse nombre, In this spice and in all{e} other folowyng we wolle begynne fro the lift side, ffor and me bigon th{e} double fro the first, omwhile me myght double oo thynge twyes. And how be it that me myght double fro the right, that wold{e} 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 doubly{n}g other growith{e} a nombre digital, article, or componed{e}. [If it be a digit, write it in the place of the first digit.] If it be article, write in his place a cifre and transferre the article toward{e} the lift, as thus:--

+------------------+----+ | double | 10 | +------------------+----+ | to be doubled{e} | 5 | +------------------+----+

And yf the nombre be componed{e}, write a digital that is part of his composicio{u}n, and sette the article to the lift hand{e}, as thus:--

+------------------+----+ | doubled{e} | 16 | +------------------+----+ | to be doubled{e} | 8 | +------------------+----+

That done, me most double the last save one, and what groweth{e} þ{er}of me most worche as before. And yf a cifre be, touch{e} it not. But yf any nombre shall{e} be added{e} to the cifre, in þe place of þe figure wiped{e} out me most write the nombre to be added{e}, as thus:--

+------------------+---+---+---+ | doubled{e} | 6 | 0 | 6 | +------------------+---+---+---+ | to be doubled{e} | 3 | 0 | 3 | +------------------+---+---+---+

[Sidenote: How to prove your answer.]

In the same wise me shall{e} wirch{e} of all{e} others. And this p{ro}bacio{u}n: If thow truly double the halfis, and truly half the doubles, the same nombre and figure shall{e} mete, such{e} as thow labo{ur}ed{e} vpon{e} first, And of the contrarie.

+------------------+---+---+---+ | Doubled{e} | 6 | 1 | 8 | +------------------+---+---+---+ | to be doubled{e} | 3 | 0 | 9 | +------------------+---+---+---+

[Headnote: Chapter VI. Multiplication.]

[Sidenote: Definition of Multiplication. Multiplier. Multiplicand. Product.]

Multiplicacio{u}n of nombre by hym-self other by a-nother, w{i}t{h} p{ro}posid{e} .2. nombres, [is] the fyndyng of the third{e}, That so oft conteyneth{e} that other, as ther ben vnytes in the oþ{er}. In multiplicacio{u}n .2. nombres pryncipally ben necessary, that is to sey, the nombre multiplying and the nombre to be multiplied{e}, as here;--twies fyve. [The number multiplying] is designed{e} adu{er}bially. The nombre to be multiplied{e} resceyveth{e} a no{m}i{n}all{e} appellacio{u}n, as twies .5. 5. is the nombre multiplied{e}, and twies is the nombre to be multipliede.

+-----------------+-----+---+---++---+---+---+---+---+---+---+---+---+ | Resultans |[{9}]| 1 | 0 || 1 | 3 | 2 | 6 | 6 | 8 | 0 | 0 | 8 | +-----------------+-----+---+---++---+---+---+---+---+---+---+---+---+ | Multiplicand{us}| . | . | 5 || . | . | 4 | . | 3 | 4 | 0 | 0 | 4 | +-----------------+-----+---+---++---+---+---+---+---+---+---+---+---+ | Multiplicans | . | 2 | 2 || . | 3 | 3 | 2 | 2 | 2 | . | . | . | +-----------------+-----+---+---++---+---+---+---+---+---+---+---+---+

Also me may thervpon{e} to assigne the. 3. nombre, the which{e} is [*Fol. 51b] cleped{e} p{ro}duct or p{ro}venient, of takyng out of one fro another: as twyes .5 is .10., 5. the nombre to be multiplied{e}, 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 multiplied{e}, and of the contrarie, remaynyng eu{er} the same some, and herof{e} cometh{e} the comen speche, that seith{e} all nombre is converted{e} by Multiplying in hym-self{e}.