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Accomptynge by counters.
[Transcriber's Note:
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.]
[*116b]
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 cou{n}ters: 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 co{m}menly. 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 p{er}ceaue your meanynge.
M. For example of the [*117a.] ly[*]nes:
——1-0-0-0-0-0— ——1-0-0-0-0—— -X—1-0-0-0——— ——1-0-0———— ——1-0————— ——1——————
[Sidenote: Numeration.]
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. 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 [*117b] by places, so that you shall not nede longer to stande about Numeration, excepte there be any other difference.
M. Yf you do vndersta{n}de it, then how wyll you set 1543?
S. Thus, as I suppose.
———- -X—1— ——5— ——4— ——3—
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 cou{n}ter round, as you se here, where I haue expressed that same summe.
——————-
-X—o———— 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 [*118a] 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, Iknowe 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 no{m}bers aboue 5. And this is true bothe of digettes and articles. And for example I wyll set downe this su{m}me 287965,
-X—————-
———o-o—— o ———o-o-o— o -X——o-o—— o ——o-o-o-o— o ——o———— o ——————-
which su{m}me yf you marke well, you nede none other exa{m}ples for to lerne the numeration of [*118b] this forme. But this shal you marke, that as you dyd in the other kynde of arithmetike, set a pricke in the places of thousa{n}des, in this worke you shall sette a starre, as you se here.
[Headnote: Addition on the Counting Board.]
[Sidenote: 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 su{m}mes at ones together: how be it you may adde more, as I wyll tell you anone. Therfore when you wyll adde two su{m}mes, 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 su{m}me, so that that lyne may be betwene them, as yf you wolde adde 2659 to 8342, you must set your su{m}mes as you se
- - o -X o-o-o o-o o o-o-o o o o-o-o-o - o o-o o-o-o-o
here. And then yf you lyst, you [*119a] 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, Iwyll showe you fyrst. Therfore wyl I begynne at the vnites, whiche in the fyrst su{m}me is but 2, {and} in y^e second su{m}me 9, that maketh 11, those do I take vp, and for them I set 11 in the new roume, thus,
- - - o -X o-o-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 su{m}me 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 [*119b] ryght lynes as you se here.
- - o -X o-o-o o-o - - o o-o-o o - - o - o-o-o-o-o - o
Where I haue taken awaye 40 fro{m} the fyrste su{m}me, 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, Imyghte better for those 6 cou{n}ters 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 cou{n}ters in any line, for that may be done with 1 cou{n}ter in a hygher place.
S. I iudge that good reaso{n}, for many are vnnedefull, where one wyll serue.
M. Well, then [*120a] wyll I adde forth of hundredes: Ifynde 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 cou{n}ter in the fourth lyne for them all, as you se here.
- - o -X o-o-o o-o o - - - - - - - o -
Then adde I y^e thousandes together, whiche in the fyrst su{m}me are 8000, {and} in y^e second 2000, that maketh 10000: them do I take vp fro{m} those two places, and for them I set one counter in the fyfte lyne, and then appereth as youse, to be 11001, for so many doth amount of the addition of 8342 to 2659.
——o——-
-X—o——-
—————
—————
——o——-
[*120b] S. Syr, this I do perceave: but how shall I set one su{m}me to an other, not chaungynge them to a thyrde place?
M. Marke well how I do it: Iwyll adde together 65436, and 3245, whiche fyrste I set downe thus.
- o - o - o -X o-o-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 su{m}me, {and} the 5 of the seco{n}de, and for them I set 1 in the seco{n}d lyne, [*121a] as you se here.
- o - o - 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 su{m}me {and} seconde lyne (which make 40) and adde them to the 4 counters of the same lyne, in the second su{m}me, and it maketh 80, But as I sayde I maye not conueniently set aboue 4 cou{n}ters in one lyne, therfore to those 4 that I toke vp in the fyrst su{m}me, Itake one also of the seconde su{m}me, 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 - o - o -X o-o-o o-o o-o-o-o - o - o-o-o - - o -
[*121b.] 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 su{m}me, 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 - o - o -X o-o-o o - o - o - o-o-o - - o -
Then I take y^e 3000 in y^e fyrste su{m}me, 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 -o - o -X -o-o-o - o -o - o -o-o-o - -o -
[*122a.] And so you see the hole su{m}me, that amou{n}teth of the addytio{n} of 65436 with 3245 to be 6868[1]. And yf you haue marked these two exa{m}ples well, you nede no farther enstructio{n} in Addition of 2 only summes: but yf you haue more then two summes to adde, you may adde them thus. 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.
- - o -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 - o o o o-o-o-o o
[*122b.] And then I adde the thyrde thereto thus. And so of more yf you haue them.
- - o o -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 o o - o -
[Headnote: Subtraction on the Counting Board.]
[Sidenote: Subtraction.]
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 in the other Addition by the penne, Imeane Subtraction, for that onely is a sure waye: but consyderynge that Subtraction must be fyrste knowen, Iwyl fyrste teache you the arte of Subtraction, and that by this example: Iwolde subtracte 2892 out of 8746. These summes must I set downe as I dyd in Addition: but here it is best [*116a (sic).] to set the lesser no{m}ber fyrste, thus.
- o -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 -
Then shall I begynne to subtracte the greatest nombres fyrste (contrary to the vse of the penne) y^t is the thousandes in this exa{m}ple: therfore I fynd amongest the thousandes 2, for which I withdrawe so many fro{m} the seconde summe (where are8) and so remayneth there 6, as this exa{m}ple showeth.
- o - - o - o o o-o-o o-o - o o-o-o-o o-o-o-o - o o-o o -
Then do I lyke wayes with the hundredes, of whiche in the fyrste summe [*116b] I fynde 8, and is the seconde summe but 7, out of whiche I can not take 8, therfore thus muste I do: Imuste loke how moche my summe dyffereth from 10, whiche I fynde here to be 2, then must I bate for my su{m}me 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 fro{m} the fyrste su{m}me I take that 800, and from the second su{m}me where are 6000, Itake 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.
- o - - o - o-o-o-o - o o-o-o-o o-o-o-o - o o-o o -
Then come I to the articles of te{n}nes where in the fyrste su{m}me I fynde 90, [*117a] and in the seconde su{m}me but only 40: Now consyderyng that 90 can not be bated from 40, Iloke how moche y^t 90 doth dyffer from the next summe aboue it, that is 100 (or elles whiche is all to one effecte, Iloke how moch 9 doth dyffer fro{m} 10) {and} I fynd it to be 1, then in the stede of that 90, Ido take from the second summe 100: but consyderynge that it is 10 to moche, Iset downe 1 in y^e nexte lyne beneth for it, as you se here.
- o - - o - o-o-o - o - o o-o o -
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.
o - o o-o-o - o o-o-o-o -
So y^t yf I subtracte 2892 fro{m} 8746, the remayner wyll be 5854, [*117b] And that this is truely wrought, you maye proue by Addition: for yf you adde to this remayner the same su{m}me that you dyd subtracte, then wyll the formar su{m}me 8746 amount agayne.
S. That wyll I proue: and fyrst I set the su{m}me that was subtracted, which was 2892, {and} the{n} the remayner 5854, thus.
+ o - o-o + o o -o-o-o + o-o-o - o o -o-o-o-o + -o-o + o-o-o-o -
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.
+ o - o-o + o o -o-o-o + o-o-o - o o -o-o-o-o + o + o -
[*118a] 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.
+ o - o-o + o o -o-o-o + o-o-o-o - + o-o-o-o - o + o -
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.
+ o - o-o + o - o + o-o - + o-o-o-o - o + o -
Then is there lefte in the fyrste summe but only 2000, whiche I shall take vp from thence, and set [*118b] in the same lyne in y^e second su{m}me, to y^e one y^t is there all redy: {and} then wyll the hole su{m}me 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. And thus you may se how Subtraction maye be tryed by Addition.
+ o -X + o-o-o - o + o-o - + o-o-o-o - o + o -
S. I perceaue the same order here w^t cou{n}ters, 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 exa{m}ple of Additio{n} where I haue added 2189 to 4988, and the hole su{m}me appereth to be 7177,
- 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 o-o - o o o -o-o-o-o o-o-o o-o -
[*119a] Nowe to trye whether that su{m}me be well added or no, Iwyll subtract one of the fyrst two su{m}mes from the thyrd, and yf I haue well done y^e remayner wyll be lyke that other su{m}me. As for example: Iwyll subtracte the fyrste summe from the thyrde, whiche I set thus in theyr order.
+ o - o-o + o-o - -o + o - o o -o-o-o + o-o - o o -o-o-o-o + o-o -
Then do I subtract 2000 of the fyrste summe fro{m} y^e second su{m}me, and then remayneth there 5000 thus.
-+ o -X -+ - o + o - o o o-o-o + o-o - o o o-o-o-o + o-o -
Then in the thyrd lyne, I subtract y^e 100 of the fyrste summe, fro{m} the second su{m}me, where is onely 100 also, and then in y^e thyrde lyne resteth nothyng. Then in the second lyne with his space ouer hym, Ifynde 80, which I shuld subtract [*119b] from the other su{m}me, 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, Isette downe so many in the seconde roume, whiche with the 70 beynge there all redy do make 4990, &then the summes doth stande thus.
+ - -+ o-o-o-o - o + o-o-o-o - o + o-o-o-o - o o -o-o-o-o + o-o -
Yet remayneth there in the fyrst su{m}me 9, to be bated from the second summe, where in that place of vnities dothe appere only 7, then I muste bate a hygher su{m}me, 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 [*120a] or lowest lyne, as you se here.
-+ - + o-o-o-o - o -+ o-o-o-o - o -+ o-o-o - o -+ o-o-o -
And so haue I ended this worke, {and} the su{m}me appereth to be y^e same, whiche was y^e seconde summe of my addition, and therfore I perceaue, Ihaue 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 su{m}me, 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.
+ - o + o-o - o -o-o-o + o-o-o - o -o-o-o-o + o-o - o o -o + o-o-o -
[*120b] And fyrste they take 6 whiche is in the lower lyne, and his space from 8 in the same roumes, in y^e second su{m}me, 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.
-+ - + o -+ -+ o-o-o - -+ o-o -
So that (as you se) it differeth not greatly whether you begynne subtractio{n} at the hygher lynes, or at [*121a] 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.
[Headnote: Multiplication by Counters.]
[Sidenote: Multiplication.]
But nowe touchynge Multiplicatio{n}: you shall set your no{m}bers 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 no{m}bers 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, Isaye that lyne for the vnites: {and} when you haue so done with the hygheest no{m}ber 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 [*121b] 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 chau{n}ce 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 cou{n}ter 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 exa{m}ple: Iwold multiply 1542 by 365, therfore I set y^e nombers thus.
+ - -+ o - o -o-o-o + o -o + o-o-o-o - o + o-o -
[*122a] 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:
- - - - o-o-o - o - - o - o -X - - [<-] o o-o-o - o o o-o-o-o o - o-o
where for the one counter taken vp from the fourth lyne, Ihaue sette downe other 6, whiche make y^e su{m}me 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 [*122b] of 1000, multiplyed by 365.
M. Well the{n} 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 no{m}ber) which is 182, {and} here I do styll let that fourth place stand, as yf it were y^e fyrst:
- 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-o -
as in this fourme you se, where I haue set this multiplycatio{n} with y^e other: but for the ease of your vndersta{n}dynge, Ihaue set a lytell lyne betwene them: now shulde they both in one su{m}me stand thus.
- 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
[*123a] Howe be it an other fourme to multyplye suche cou{n}ters i{n} space is this: Fyrst to remoue the fynger to the lyne nexte benethe y^e space, {and} then to take vp y^e cou{n}ter, {and} to set downe y^e multiplyer .v. tymes, as here youse.
- --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 o - - -o - +o -+o -o -- o o o o o [->]-X-o-o-o- - - - o -o --o-o-o-o- - - o --o-o - - -
Which su{m}mes yf you do adde together into one su{m}me, you shal p{er}ceaue that it wyll be y^e same y^t appeareth of y^e other worki{n}g before, so that [*123b] bothe sortes are to one entent, but as the other is much shorter, so this is playner to reason, for suche as haue had small exercyse in this arte. Not withstandynge you maye adde them in your mynde before you sette them downe, as in this exa{m}ple, you myghte haue sayde 5 tymes 300 is 1500, {and} 5 tymes 60 is 300, also 5 tymes 5 is 25, whiche all put together do make 1825, which you maye at one tyme set downe yf you lyste. But nowe to go forth, Imust remoue the hand to the nexte counters, whiche are in the second lyne, and there must I take vp those 4 counters, settynge downe for them my multiplyer 4 tymes, whiche thynge other I maye do at 4 tymes seuerally, or elles I may gather that hole summe in my mynde fyrste, and then set it downe: as to saye 4 tymes 300 is 1200: 4 tymes 60 are 240: and 4 tymes 5 make 20: y^t is in all 1460, y^t shall I set downe also: as here youse. o - - - - - o-o-o-o o - o -X - - o-o o-o-o-o - o o o-o-o - - o - o [->] o - - o - o-o -
[*124a] whiche yf I ioyne in one summe with the formar nombers, it wyll appeare thus. o - - o - - o - - - o-o - o-o-o --o o [->] o - o - o-o
Then to ende this multiplycation, I remoue the fynger to the lowest lyne, where are onely 2, them do I take vp, and in theyr stede do I set downe twyse 365, that is 730, for which I set [*124b] one in the space aboue the thyrd lyne for 500, and 2 more in the thyrd lyne with that one that is there all redye, and the reste in theyr order, {and} so haue I ended the hole summe thus. o - - o - - o - - - o-o - o o-o-o - o-o-o - o o - o-o-o - o - -
Wherby you se, that 1542 (which is the nomber of yeares syth Ch[r]ystes incarnation) beyng multyplyed by 365 (which is the nomber of dayes in one yeare) dothe amounte vnto 562830, which declareth y^e no{m}ber of daies sith Chrystes incarnatio{n} vnto the ende of 1542[{1}] yeares. (besyde 385 dayes and 12 houres for lepe yeares).
S. Now wyll I proue by an other exa{m}ple, as this: 40 labourers (after 6d. y^e day for eche man) haue wrought 28 dayes, Iwold [*125a] know what theyr wages doth amou{n}t vnto: In this case muste I worke doublely: fyrst I must multyplye the nomber of the labourers by y^e wages of a man for one day, so wyll y^e charge of one daye amount: then secondarely shall I multyply that charge of one daye, by the hole nomber of dayes, {and} so wyll the hole summe appeare: fyrst therefore I shall set the su{m}mes thus.
+ + + + o-o-o-o - o o -+
Where in the fyrste space is the multyplyer (y^t is one dayes wages for one man) {and} in the second space is set the nomber of the worke men to be multyplyed: the{n} saye I, 6 tymes 4 (reckenynge that second lyne as the lyne of vnites) maketh 24, for whiche summe I shulde set 2 counters in the thyrde lyne, and 4 in the seconde, therfore do I set 2 in the thyrde lyne, and let the 4 stand styll in the seconde lyne, thus.[*125b]
-+ -+ -+ o-o - -+ o-o-o-o - -+
So apwereth the hole dayes wages to be 240d. that is 20s. Then do I multiply agayn the same summe by the no{m}ber of dayes and fyrste I sette the nombers, thus.
-+ -+ -+ o-o - o-o + o-o-o-o - o o-o-o + -
The{n} bycause there are counters in dyuers lynes, Ishall begynne with the hyghest, and take them vp, settynge for them the multyplyer so many tymes, as I toke vp counters, y^t is twyse, then wyll y^e su{m}me stande thus.
-+ o -+ o -+ o - -+ o-o-o-o - -+
Then come I to y^e seconde lyne, and take vp those 4 cou{n}ters, settynge for them the multiplyer foure tymes, so wyll the hole summe appeare thus.[*126a]
-+ o -+ o - o -+ o-o - -+ o-o - -+
So is the hole wages of 40 workeme{n}, for 28 dayes (after 6d. eche daye for a man) 6720d. that is 560s. or 28l'i.
[Headnote: Division on the Counting Board.]
[Sidenote: Diuision.]
M. Now if you wold proue Multiplycatio{n}, the surest way is by Dyuision: therfore wyll I ouer passe it tyll I haue taught you y^e arte of Diuision, whiche you shall worke thus. Fyrste sette downe the Diuisor for feare of forgettynge, and then set the nomber that shalbe deuided, at y^e ryghte syde, so farre from the diuisor, that the quotient may be set betwene them: as for exa{m}ple: Yf 225 shepe cost 45l'i. what dyd euery shepe cost? To knowe this, Ishulde diuide the hole summe, that is 45l'i. by 225, but that can not be, therfore must I fyrste reduce that 45l'i. into a lesser denomination, as into shyllynges: then I multiply 45 by 20, and it is 900, that summe shall I diuide by the no{m}ber of [*126b] shepe, whiche is 225, these two nombers therfore I sette thus.
- - - - o o-o - o-o-o-o - o-o - o - -
Then begynne I at the hyghest lyne of the diuident, and seke how often I may haue the diuisor therin, and that maye I do 4 tymes, then say I, 4 tymes 2 are 8, whyche yf I take from 9, there resteth but 1, thus
- - - - o-o - o - o-o - o - o-o-o-o
And bycause I founde the diuisor 4 tymes in the diuidente, Ihaue set (as you se) 4 in the myddle roume, which [*127a] is the place of the quotient: but now must I take the reste of the diuisor as often out of the remayner: therfore come Ito the seconde lyne of the diuisor, sayeng 2 foure tymes make 8, take 8 from 10, {and} there resteth 2, thus.
- - - - -o-o - -o-o - o-o - o o-o-o-o
Then come I to the lowest nomber, which is 5, and multyply it 4 tymes, so is it 20, that take I from 20, and there remayneth nothynge, so that I se my quotient to be 4, whiche are in valewe shyllynges, for so was the diuident: and therby I knowe, that yf 225 shepe dyd coste 45l'i. euery shepe coste 4s.
S. This can I do, as you shall perceaue by this exa{m}ple: Yf 160 sowldyars do spende euery moneth 68l'i. what spendeth eche man? Fyrst [*127b] bycause I can not diuide the 68 by 160, therfore I wyll turne the pou{n}des into pennes by multiplicacio{n}, so shall there be 16320d. Nowe muste I diuide this su{m}me by the nomber of sowldyars, therfore I set the{m} i{n} order, thus.
- - o - o - - o - -o - - o-o-o - o -o - - o-o - - -
Then begyn I at the hyghest place of the diuidente, sekynge my diuisor there, whiche I fynde ones, Therfore set I 1 in the nether lyne.
M. Not in the nether line of the hole summe, but in the nether lyne of that worke, whiche is the thyrde lyne.
S. So standeth it with reason.
M. Then thus do they stande.[*128a]
- - - - -o - o o-o-o - o -o - - o-o - - -
Then seke I agayne in the reste, how often I may fynde my diuisor, and I se that in the 300 I myghte fynde 100 thre tymes, but then the 60 wyll not be so often founde in 20, therfore I take 2 for my quotient: then take I 100 twyse from 300, and there resteth 100, out of whiche with the 20 (that maketh 120) Imay take 60 also twyse, and then standeth the nombers thus,
- - - - - - -o - o - o -o - - - - o-o -
[*128b] where I haue sette the quotient 2 in the lowest lyne: So is euery sowldyars portion 102d. that is 8s. 6d.
M. But yet bycause you shall perceaue iustly the reason of Diuision, it shall be good that you do set your diuisor styll agaynst those nombres fro{m} whiche you do take it: as by this example I wyll declare. Yf y^e purchace of 200 acres of ground dyd coste 290l'i. what dyd one acre coste? Fyrst wyl I turne the poundes into pennes, so wyll there be 69600d Then in settynge downe these nombers I shall do thus.
- - o o-o - o - o -X - - o-o-o-o - o - - o - - - - -
Fyrst set the diuident on the ryghte hande as it oughte, and then [*129a] the diuisor on the lefte hande agaynst those nombers, fro{m} which I entende to take hym fyrst as here you se, wher I haue set the diuisor two lynes hygher the{n} is theyr owne place.
S. This is lyke the order of diuision by the penne.
M. Truth you say, and nowe must I set y^e quotient of this worke in the thyrde lyne, for that is the lyne of vnities in respecte to the diuisor in this worke. Then I seke howe often the diuisor maye be founde in the diuident, {and} that I fynde 3 tymes, then set I 3 in the thyrde lyne for the quotient, and take awaye that 60000 fro{m} the diuident, and farther I do set the diuisor one line lower, as yow se here.
- o - o-o - o-o-o-o - o o-o-o o - - -
[*129b] And then seke I how often the diuisor wyll be taken from the nomber agaynste it, whiche wyll be 4 tymes and 1 remaynynge.
S. But what yf it chaunce that when the diuisor is so remoued, it can not be ones taken out of the diuident agaynste it?
M. Then must the diuisor be set in an other line lower.
S. So was it in diuision by the penne, and therfore was there a cypher set in the quotient: but howe shall that be noted here?
M. Here nedeth no token, for the lynes do represente the places: onely loke that you set your quotient in that place which standeth for vnities in respecte of the diuisor: but now to returne to the example, Ifynde the diuisor 4 tymes in the diuidente, and 1 remaynynge, for 4 tymes 2 make 8, which I take from 9, and there resteth 1, as this figure sheweth:
- - - o-o - o o o-o-o o o-o-o-o - - -
and in the myddle space for the quotient I set 4 in the seconde lyne, whiche is in this worke the place of vnities.[*130a] Then remoue I y^e diuisor to the next lower line, and seke how often I may haue it in the dyuident, which I may do here 8 tymes iust, and nothynge remayne, as in this fourme,
- - - o-o - - o-o-o - o-o-o-o - o o-o-o -
where you may se that the hole quotient is 348d, that is 29s. wherby I knowe that so moche coste the purchace of one aker.
S. Now resteth the profes of Multiplycatio{n}, and also of Diuisio{n}.
M. Ther best profes are eche [*130b] one by the other, for Multyplication is proued by Diuision, and Diuision by Multiplycation, as in the worke by the penne you learned.
S. Yf that be all, you shall not nede to repete agayne that, y^t was sufficye{n}tly taughte all redye: and excepte you wyll teache me any other feate, here maye you make an ende of this arte I suppose.
M. So wyll I do as touchynge hole nomber, and as for broken nomber, Iwyll not trouble your wytte with it, tyll you haue practised this so well, y^t you be full perfecte, so that you nede not to doubte in any poynte that I haue taught you, and thenne maye I boldly enstructe you in y^e arte of fractions or broken no{m}ber, wherin I wyll also showe you the reasons of all that you haue nowe learned. But yet before I make an ende, Iwyll showe you the order of co{m}men castyng, wher in are bothe pennes, shyllynges, and poundes, procedynge by no grounded reason, but onely by a receaued [*131a] fourme, and that dyuersly of dyuers men: for marchau{n}tes vse one fourme, and auditors an other:
[Headnote: Merchants' Casting Counters.]
[Sidenote: Merchants' casting.]
But fyrste for marchauntes fourme marke this example here,
o o o o o o o o o o o o o o o o o o o o o o
in which I haue expressed this summe 198 l'i.[{2}] 19s. 11d. So that you maye se that the lowest lyne serueth for pe{n}nes, the next aboue for shyllynges, the thyrde for poundes, and the fourth for scores of pou{n}des. And farther you maye se, that the space betwene pennes and shyllynges may receaue but one counter (as all other spaces lyke wayes do) and that one standeth in that place for 6d. Lyke wayes betwene the shyllynges {and} the pou{n}des, one cou{n}ter standeth for 10s. And betwene the poundes and 20l'i. one counter standeth for 10 pou{n}des. But besyde those you maye see at the left syde of shyllynges, that one counter standeth alone, {and} betokeneth 5s. [*131b] So agaynste the poundes, that one cou{n}ter standeth for 5l'i. And agaynst the 20 poundes, the one counter standeth for 5 score pou{n}des, that is 100l'i. so that euery syde counter is 5 tymes so moch as one of them agaynst whiche he standeth.
[Sidenote: Auditors' casting.]
Now for the accompt of auditors take this example.
o o o o o o o o o o o o o o o o o o o o o o
where I haue expressed y^e same su{m}me 198l'i. 19s. 11d. But here you se the pe{n}nes stande toward y^e ryght hande, and the other encreasynge orderly towarde the lefte hande. Agayne you maye se, that auditours wyll make 2 lynes (yea and more) for pennes, shyllynges, {and} all other valewes, yf theyr summes extende therto. Also you se, that they set one counter at the ryght ende of eche rowe, whiche so set there standeth for 5 of that roume: and on [*132a] the lefte corner of the rowe it sta{n}deth for 10, of y^e same row. But now yf you wold adde other subtracte after any of both those sortes, yf you marke y^e order of y^t other feate which I taught you, you may easely do the same here without moch teachynge: for in Additio{n} you must fyrst set downe one su{m}me and to the same set the other orderly, and lyke maner yf you haue many: but in Subtraction you must sette downe fyrst the greatest summe, and from it must you abate that other euery denominatio{n} from his dewe place.
S. I do not doubte but with a lytell practise I shall attayne these bothe: but how shall I multiply and diuide after these fourmes?
M. You can not duely do none of both by these sortes, therfore in suche case, you must resort to your other artes.
S. Syr, yet I se not by these sortes how to expresse hu{n}dreddes, yf they excede one hundred, nother yet thousandes.
M. They that vse such accomptes that it excede 200 [*132b] in one summe, they sette no 5 at the lefte hande of the scores of poundes, but they set all the hundredes in an other farther rowe {and} 500 at the lefte hand therof, and the thousandes they set in a farther rowe yet, {and} at the lefte syde therof they sette the 5000, and in the space ouer they sette the 10000, and in a hygher rowe 20000, whiche all I haue expressed in this exa{m}ple,
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 o o o o o
which is 97869l'i. 12s. 9 d ob. q. for I had not told you before where, nother how you shuld set downe farthynges, which (as you se here) must be set in a voyde space sydelynge beneth the pennes: for q one counter: for ob. 2 counters: for ob. q. 3 counters: {and} more there can not be, for 4 farthynges [*133a] do make 1d. which must be set in his dewe place.
[Headnote: Auditors' Casting Counters.]
And yf you desyre y^e same summe after audytors maner, lo here itis.
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 o o o o o
But in this thyng, you shall take this for suffycyent, and the reste you shall obserue as you maye se by the working of eche sorte: for the dyuers wittes of men haue inuented dyuers and sundry wayes almost vnnumerable. But one feate I shall teache you, whiche not only for the straungenes and secretnes is moche pleasaunt, but also for the good co{m}moditie of it ryghte worthy to be well marked. This feate hath ben vsed aboue 2000 yeares at the leaste, and yet was it neuer come{n}ly knowen, especyally in Englysshe it was neuer taughte yet. This is the arte of nombrynge on the hand, with diuers gestures of the fyngers, expressynge any summe conceaued in the [*133b] mynde. And fyrst to begynne, yf you wyll expresse any summe vnder 100, you shall expresse it with your lefte hande: and from 100 vnto 10000, you shall expresse it with your ryght hande, as here orderly by this table folowynge you may perceaue.
+ Here foloweth the table of the arte of the hande+
The arte of nombrynge by the hande.
[Transcriber's Note:
Footnote 3 reads: "Bracket ([) denotes new paragraph in original." For this e-text, the brackets have been omitted in favor of restoring the paragraph breaks. Changes of speaker (M, S) are also marked by paragraphs, as in the previous selection.
The illustration includes the printed page number 134; there is therefore no sidenote *134a. The sidenote for "4" is missing.]
[Sidenote: 1]
[*134b] In which as you may se 1 is expressed by y^e lyttle fynger of y^e lefte hande closely and harde croked.
[Sidenote: 2]
[{3}]2 is declared by lyke bowynge of the weddynge fynger (whiche is the nexte to the lyttell fynger) together with the lytell fynger.
[Sidenote: 3]
3 is signified by the myddle fynger bowed in lyke maner, with those other two.
4 is declared by the bowyng of the myddle fynger and the rynge fynger, or weddynge fynger, with the other all stretched forth.
[Sidenote: 5, 6]
5 is represented by the myddle fynger onely bowed.
And 6 by the weddynge fynger only crooked: and this you may marke in these a certayne order. But now 7, 8, and 9, are expressed w{i}t{h} the bowynge of the same fyngers as are 1, 2, and 3, but after an other fourme.
[Sidenote: 7]
For 7 is declared by the bowynge of the lytell fynger, as is 1, saue that for 1 the fynger is clasped in, harde {and} [*135a] rounde, but for to expresse 7, you shall bowe the myddle ioynte of the lytell fynger only, and holde the other ioyntes streyght.
S. Yf you wyll geue me leue to expresse it after my rude maner, thus I vnderstand your meanyng: that 1 is expressed by crookynge in the lyttell fynger lyke the head of a bysshoppes bagle: and 7 is declared by the same fynger bowed lyke a gybbet.
M. So I perceaue, you vnderstande it.
[Sidenote: 8]
Then to expresse 8, you shall bowe after the same maner both the lyttell fynger and the rynge fynger.
[Sidenote: 9, 10]
And yf you bowe lyke wayes with them the myddle fynger, then doth it betoken 9.
Now to expresse 10, you shall bowe your fore fynger rounde, and set the ende of it on the hyghest ioynte of the thombe.
[Sidenote: 20]
And for to expresse 20, you must set your fyngers streyght, and the ende of your thombe to the partitio{n} of the [*135b] fore moste and myddle fynger.
[Sidenote: 30]
30 is represented by the ioynynge together of y^e headdes of the foremost fynger and the thombe.
[Sidenote: 40]
40 is declared by settynge of the thombe crossewayes on the foremost fynger.
[Sidenote: 50]
50 is signified by ryght stretchyng forth of the fyngers ioyntly, and applyenge of the thombes ende to the partition of the myddle fynger {and} the rynge fynger, or weddynge fynger.
[Sidenote: 60]
60 is formed by bendynge of the thombe croked and crossynge it with the fore fynger.
[Sidenote: 70]
70 is expressed by the bowynge of the foremost fynger, and settynge the ende of the thombe between the 2 foremost or hyghest ioyntes of it.
[Sidenote: 80]
80 is expressed by settynge of the foremost fynger crossewayes on the thombe, so that 80 dyffereth thus fro{m} 40, that for 80 the forefynger is set crosse on the thombe, and for 40 the thombe is set crosse ouer y^e forefinger.
[Sidenote: 90]
[*136a] 90 is signified, by bendynge the fore fynger, and settyng the ende of it in the innermost ioynte of y^e thombe, that is euen at the foote of it. And thus are all the no{m}bers ended vnder 100.
[Sidenote: 11, 12, 13, 21, 22, 23]
S. In dede these be all the nombers fro{m} 1 to 10, {and} then all the tenthes within 100, but this teacyed me not how to expresse 11, 12, 13, {et}c. 21, 22, 23, {et}c. and such lyke.
M. You can lytell vnderstande, yf you can not do that without teachynge: what is 11? is it not 10 and 1? then expresse 10 as you were taught, and 1 also, and that is 11: and for 12 expresse 10 and 2: for 23 set 20 and 3: and so for 68 you muste make 60 and there to 8: and so of all other sortes.
[Sidenote: 100]
But now yf you wolde represente 100 other any nomber aboue it, you muste do that with the ryghte hande, after this maner. [You must expresse 100 in the ryght hand, with the lytell fynger so bowed as you dyd expresse 1 in the left hand.
[Sidenote: 200]
[*136b] And as you expressed 2 in the lefte hande, the same fasshyon in the ryght hande doth declare 200.
[Sidenote: 300]
The fourme of 3 in the ryght hand standeth for 300.
[Sidenote: 400]
The fourme of 4, for 400.
[Sidenote: 500]
Lykewayes the fourme of 5, for 500.
[Sidenote: 600]
The fourme of 6, for 600. And to be shorte: loke how you did expresse single vnities and tenthes in the lefte hande, so must you expresse vnities {and} tenthes of hundredes, in the ryghte hande.
[Sidenote: 900]
S. I vnderstande you thus: that yf I wold represent 900, Imust so fourme the fyngers of my ryghte hande, as I shuld do in my left hand to expresse 9,
[Sidenote: 1000]
And as in my lefte hand I expressed 10, so in my ryght hande must I expresse 1000.
And so the fourme of euery tenthe in the lefte hande serueth to expresse lyke no{m}ber of thousa{n}des,
[Sidenote: 4000]
so y^e fourme of 40 standeth for 4000.
[Sidenote: 8000]
The fourme of 80 for 8000.
[Sidenote: 9000]
[*137a]
And the fourme of 90 (whiche is the greatest) for 9000, and aboue that I can not expresse any nomber. M. No not with one fynger: how be it, w{i}t{h} dyuers fyngers you maye expresse 9999, and all at one tyme, and that lac keth but 1 of 10000. So that vnder 10000 you may by your fyngers ex- presse any summe. And this shal suf- fyce for Numeration on the fyngers. And as for Addition, Subtraction, Multiplicatio{n}, and Diuision (which yet were neuer taught by any man as farre as I do knowe) I wyll enstruct you after the treatyse of fractions. And now for this tyme fare well, and loke that you cease not to practyse that you haue lear ned. S. Syr, with moste harty mynde I thanke you, bothe for your good learnyng, {and} also your good cou{ns}el, which (god wyllyng) I truste to folow.
Finis.
FOOTNOTES (Accomptynge by counters and The arte of nombrynge by the hande):
[1: 1342 in original.] [2: 168 in original.] [3: Bracket ([) denotes new paragraph in original.]
APPENDIX I.
A Treatise on the Numeration of Algorism.
[From a MS. of the 14th Century.]
To alle suche even nombrys the most have cifrys as to ten. twenty. thirtty. an hundred. an thousand and suche other. but ye schal vnderstonde that a cifre tokeneth nothinge but he maketh other the more significatyf that comith after hym. Also ye schal vnderstonde that in nombrys composyt and in alle other nombrys that ben of diverse figurys ye schal begynne in the ritht syde and to rekene backwarde and so he schal be wryte as thus—1000. the sifre in the ritht side was first wryte and yit he tokeneth nothinge to the secunde no the thridde but thei maken that figure of 1 the more signyficatyf that comith after hem by as moche as he born oute of his first place where he schuld yf he stode ther tokene but one. And there he stondith nowe in the ferye place he tokeneth a thousand as by this rewle. In the first place he tokeneth but hymself. In the secunde place he tokeneth ten times hymself. In the thridde place he tokeneth an hundred tymes himself. In the ferye he tokeneth a thousand tymes himself. In the fyftye place he tokeneth ten thousand tymes himself. In the sexte place he tokeneth an hundred thousand tymes hymself. In the seveth place he tokeneth ten hundred thousand tymes hymself, &c. And ye schal vnderstond that this worde nombre is partyd into thre partyes. Somme is callyd nombre of digitys for alle ben digitys that ben withine ten as ix, viii, vii, vi, v, iv, iii, ii, i. Articules ben alle thei that mow be devyded into nombrys of ten as xx, xxx, xl, and suche other. Composittys be alle nombrys that ben componyd of a digyt and of an articule as fourtene fyftene thrittene and suche other. Fourtene is componyd of four that is a digyt and of ten that is an articule. Fyftene is componyd of fyve that is a digyt and of ten that is an articule and so of others . . . . . . But as to this rewle. In the firste place he tokeneth but himself that is to say he tokeneth but that and no more. If that he stonde in the secunde place he tokeneth ten tymes himself as this figure 2 here 21. this is oon and twenty. This figure 2 stondith in the secunde place and therfor he tokeneth ten tymes himself and ten tymes 2 is twenty and so forye of every figure and he stonde after another toward the lest syde he schal tokene ten tymes as moche more as he schuld token and he stode in that place ther that the figure afore him stondeth: lo an example as thus 9634. This figure of foure that hath this schape 4 tokeneth but himself for he stondeth in the first place. The figure of thre that hath this schape 3 tokeneth ten tyme himself for he stondeth in the secunde place and that is thritti. The figure of sexe that hath this schape 6 tokeneth ten tyme more than he schuld and he stode in the place yer the figure of thre stondeth for ther he schuld tokene but sexty. And now he tokeneth ten tymes that is sexe hundrid. The figure of nyne that hath this schape 9 tokeneth ten tymes more than he schulde and he stode in the place ther the figure of 6 stondeth inne for thanne he schuld tokene but nyne hundryd. And in the place that he stondeth inne nowe he tokeneth nine thousand. Alle the hole nombre of these foure figurys. Nine thousand sexe hundrid and foure and thritti.
APPENDIX II.
Carmen de Algorismo.
[From a B.M. MS., 8 C. iv., with additions from 12 E. 1 & Eg. 2622.]
Hec algorismus ars presens dicitur[{1}]; in qua Talibus Indorum[{2}] fruimur his quinque figuris. 0. 9. 8. 7. 6. 5. 4. 3. 2. 1. Prima significat unum: duo vero secunda: Tercia significat tria: sic procede sinistre 4 Donec ad extremam venies, qua cifra vocatur; [{3}][Que nil significat; dat significare sequenti.] Quelibet illarum si primo limite ponas, Simpliciter se significat: si vero secundo, 8 Se decies: sursum procedas multiplicando.[{4}] [Namque figura sequens quevis signat decies plus, Ipsa locata loco quam significet pereunte: 12 Nam precedentes plus ultima significabit.] [{5}]Post predicta scias quod tres breuiter numerorum Distincte species sunt; nam quidam digiti sunt; Articuli quidam; quidam quoque compositi sunt. 16 [Sunt digiti numeri qui citra denarium sunt; Articuli decupli degitorum; compositi sunt Illi qui constant ex articulis digitisque.] Ergo, proposito numero tibi scribere, primo 20 Respicias quis sit numerus; quia si digitus sit, [{5}][Una figura satis sibi; sed si compositus sit,] Primo scribe loco digitum post articulum fac Articulus si sit, cifram post articulum sit, 24 [Articulum vero reliquenti in scribe figure.] Quolibet in numero, si par sit prima figura, Par erit et totum, quicquid sibi continetur; Impar si fuerit, totum sibi fiet et impar. 28 Septem[{6}] sunt partes, non plures, istius artis; Addere, subtrahere, duplare, dimidiare; Sexta est diuidere, set quinta est multiplicare; Radicem extrahere pars septima dicitur esse. 32 Subtrahis aut addis a dextris vel mediabis; A leua dupla, diuide, multiplicaque; Extrahe radicem semper sub parte sinistra.
[Sidenote: Addition.]
Addere si numero numerum vis, ordine tali 36 Incipe; scribe duas primo series numerorum Prima sub prima recte ponendo figuram, Et sic de reliquis facias, si sint tibi plures. Inde duas adde primas hac condicione; 40 Si digitus crescat ex addicione priorum, Primo scribe loco digitum, quicunque sit ille; Si sit compositus, in limite scribe sequenti Articulum, primo digitum; quia sic iubet ordo. 44 Articulus si sit, in primo limite cifram, Articulum vero reliquis inscribe figuris; Vel per se scribas si nulla figura sequatur. Si tibi cifra superueniens occurrerit, illam 48 Deme suppositam; post illic scribe figuram: Postea procedas reliquas addendo figuras.
[Sidenote: Subtraction.]
A numero numerum si sit tibi demere cura, Scribe figurarum series, vt in addicione; 52 Maiori numero numerum suppone minorem, Siue pari numero supponatur numerus par. Postea si possis a prima subtrahe primam, Scribens quod remanet, cifram si nil remanebit. 56 Set si non possis a prima demere primam; Procedens, vnum de limite deme sequenti; Et demptum pro denario reputabis ab illo, Subtrahe totaliter numerum quem proposuisti. 60 Quo facto, scribe supra quicquit remanebit, Facque novenarios de cifris, cum remanebis, Occurrant si forte cifre, dum demseris vnum; Postea procedas reliquas demendo figuras. 64
[Sidenote: Proof.]
[{7}][Si subtracio sit bene facta probare valebis, Quas subtraxisti primas addendo figuras. Nam, subtractio si bene sit, primas retinebis, Et subtractio facta tibi probat additionem.] 68
[Sidenote: Duplation.]
Si vis duplare numerum, sic incipe; solam Scribe figurarum seriem, quamcumque voles que Postea procedas primam duplando figuram; Inde quod excrescet, scribens, vbi iusserit ordo, 72 Juxta precepta que dantur in addicione. Nam si sit digitus, in primo limite scribe; Articulus si sit, in primo limite cifram, Articulum vero reliquis inscribe figuris; 76 Vel per se scribas, si nulla figura sequatur: Compositus si sit, in limite scribe sequenti Articulum primo, digitum; quia sic jubet ordo: Et sic de reliquis facias, si sint tibi plures. 80 [{8}][Si super extremam nota sit, monadem dat eidem, Quod tibi contingit, si primo dimidiabis.]
[Sidenote: Mediation.]
Incipe sic, si vis aliquem numerum mediare: Scribe figurarum seriem solam, velud ante; 84 Postea procedens medias, et prima figura Si par aut impar videas; quia si fuerit par, Dimidiabis eam, scribens quicquit remanebit; Impar si fuerit, vnum demas, mediare, 88 Nonne presumas, sed quod superest mediabis; Inde super tractum, fac demptum quod notat unum; Si monos, dele; sit ibi cifra post nota supra. Postea procedas hac condicione secunda:[{9}] 92 Impar[{10}] si fuerit hic vnum deme priori, Inscribens quinque, nam denos significabit Monos prdictam: si vero secunda dat vnam, Illa deleta, scribatur cifra; priori 96 Tradendo quinque pro denario mediato; Nec cifra scribatur, nisi inde figura sequatur: Postea procedas reliquas mediando figuras, Quin supra docui, si sint tibi mille figure. 100 [{11}][Si mediatio sit bene facta probare valebis, Duplando numerum quem primo dimidiasti.] Si super extremam nota sit monades dat eidem Quod contingat cum primo dimiabis Atque figura prior nuper fuerit mediando.]
[Sidenote: Multiplication.]
Si tu per numerum numerum vis multiplicare, Scribe duas, quascunque volis, series numerorum; 104 Ordo tamen seruetur vt vltima multiplicandi Ponatur super anteriorem multiplicantis; [{12}][A leua relique sint scripte multiplicantes.] In digitum cures digitum si ducere, major 108 Per quantes distat a denis respice, debes Namque suo decuplo tociens delere minorem; Sicque tibi numerus veniens exinde patebit. Postea procedas postremam multiplicando, 112 Juste multiplicans per cunctas inferiores, Condicione tamen tali; quod multiplicantis Scribas in capite, quicquid processerit inde; Set postquam fuerit hec multiplicata, figure 116 Anteriorentur seriei multiplicantis; Et sic multiplica, velut istam multiplicasti, Qui sequitur numerum scriptum quicunque figuris. Set cum multiplicas, primo sic est operandum, 120 Si dabit articulum tibi multiplicacio solum; Proposita cifra, summam transferre memento. Sin autem digitus excrescerit articulusque, Articulus supraposito digito salit ultra; 124 Si digitus tamen, ponas illum super ipsam, Subdita multiplicans hanc que super incidit illi Delet eam penitus, scribens quod provenit inde; Sed si multiplices illam posite super ipsam, 128 Adiungens numerum quem prebet ductus earum; Si supraimpositam cifra debet multiplicare, Prorsus eam delet, scribi que loco cifra debet, [{12}][Si cifra multiplicat aliam positam super ipsam, 132 Sitque locus supra vacuus super hanc cifra fiet;] Si supra fuerit cifra semper pretereunda est; Si dubites, an sit bene multiplicando secunda, Diuide totalem numerum per multiplicantem, 136 Et reddet numerus emergens inde priorem.
[Sidenote: Mental Multiplication.]
[{13}][Per numerum si vis numerum quoque multiplicare Tantum per normas subtiles absque figuris Has normas poteris per versus scire sequentes. 140 Si tu per digitum digitum quilibet multiplicabis Regula precedens dat qualiter est operandum Articulum si per reliquum vis multiplicare In proprium digitum debebit uterque resolvi 144 Articulus digitos post per se multiplicantes Ex digitis quociens teneret multiplicatum Articuli faciunt tot centum multiplicati. Articulum digito si multiplicamus oportet 148 Articulum digitum sumi quo multiplicare Debemus reliquum quod multiplicaris ab illis Per reliquo decuplum sic omne latere nequibit In numerum mixtum digitum si ducere cures 152 Articulus mixti sumatur deinde resolvas In digitum post hec fac ita de digitis nec Articulusque docet excrescens in detinendo In digitum mixti post ducas multiplicantem 156 De digitis ut norma docet sit juncta secundo Multiplica summam et postea summa patebit Junctus in articulum purum articulumque [{14}][Articulum purum comittes articulum que] 160 Mixti pro digitis post fiat et articulus vt Norma jubet retinendo quod egreditur ab illis Articuli digitum post in digitum mixti duc Regula de digitis ut percipit articulusque 164 Ex quibus excrescens summe tu junge priori Sic manifesta cito fiet tibi summa petita. Compositum numerum mixto sic multiplicabis Vndecies tredecem sic est ex hiis operandum 168 In reliquum primum demum duc post in eundem Unum post deinde duc in tercia deinde per unum Multiplices tercia demum tunc omnia multiplicata In summa duces quam que fuerit te dices 172 Hic ut hic mixtus intentus est operandum Multiplicandorum de normis sufficiunt hec.]
[Sidenote: Division.]
Si vis dividere numerum, sic incipe primo; Scribe duas, quascunque voles, series numerorum; 176 Majori numero numerum suppone minorem, [{15}][Nam docet ut major teneat bis terve minorem;] Et sub supprima supprimam pone figuram, Sic reliquis reliquas a dextra parte locabis; 180 Postea de prima primam sub parte sinistra Subtrahe, si possis, quociens potes adminus istud, Scribens quod remanet sub tali conditione; Ut totiens demas demendas a remanente, 184 Que serie recte ponentur in anteriori, Unica si, tantum sit ibi decet operari; Set si non possis a prima demere primam, Procedas, et eam numero suppone sequenti; 188 Hanc uno retrahendo gradu quo comites retrahantur, Et, quotiens poteris, ab eadem deme priorem, Ut totiens demas demendas a remanenti, Nec plus quam novies quicquam tibi demere debes, 192 Nascitur hinc numerus quociens supraque sequentem Hunc primo scribas, retrahas exinde figuras, Dum fuerit major supra positus inferiori, Et rursum fiat divisio more priori; 196 Et numerum quotiens supra scribas pereunti, Si fiat saliens retrahendo, cifra locetur, Et pereat numero quotiens, proponas eidem Cifram, ne numerum pereat vis, dum locus illic 200 Restat, et expletis divisio non valet ultra: Dum fuerit numerus numerorum inferiore seorsum Illum servabis; hinc multiplicando probabis,
[Sidenote: Proof.]
Si bene fecisti, divisor multiplicetur 204 Per numerum quotiens; cum multiplicaveris, adde Totali summ, quod servatum fuit ante, Reddeturque tibi numerus quem proposuisti; Et si nil remanet, hunc multiplicando reddet, 208
[Sidenote: Square Numbers.]
Cum ducis numerum per se, qui provenit inde Sit tibi quadratus, ductus radix erit hujus, Nec numeros omnes quadratos dicere debes, Est autem omnis numerus radix alicujus. 212 Quando voles numeri radicem querere, scribi Debet; inde notes si sit locus ulterius impar, Estque figura loco talis scribenda sub illo, Que, per se dicta, numerum tibi destruat illum, 216 Vel quantum poterit ex inde delebis eandem; Vel retrahendo duples retrahens duplando sub ista Que primo sequitur, duplicatur per duplacationem, Post per se minuens pro posse quod est minuendum. 220 [{16}]Post his propones digitum, qui, more priori Per precedentes, post per se multiplicatus, Destruat in quantum poterit numerum remanentem, Et sic procedens retrahens duplando figuram, 224 Preponendo novam donec totum peragatur, Subdupla propriis servare docetque duplatis; Si det compositum numerum duplacio, debet Inscribi digitus a parte dextra parte propinqua, 228 Articulusque loco quo non duplicata resessit; Si dabit articulum, sit cifra loco pereunte Articulusque locum tenet unum, de duplicata resessit; Si donet digitum, sub prima pone sequente, 232 Si supraposita fuerit duplicata figura Major proponi debet tantummodo cifra, Has retrahens solito propones more figuram, Usque sub extrema ita fac retrahendo figuras, 236 Si totum deles numerum quem proposuisti, Quadratus fuerit, de dupla quod duplicasti, Sicque tibi radix illius certa patebit, Si de duplatis fit juncta supprima figura; 240 Radicem per se multiplices habeasque Primo propositum, bene te fecisse probasti; Non est quadratus, si quis restat, sed habentur Radix quadrati qui stat major sub eadem; 244 Vel quicquid remanet tabula servare memento; Hoc casu radix per se quoque multiplicetur, Vel sic quadratus sub primo major habetur, Hinc addas remanens, et prius debes haberi; 248 Si locus extremus fuerit par, scribe figuram Sub pereunte loco per quam debes operari, Que quantum poterit supprimas destruat ambas, Vel penitus legem teneas operando priorem, 252 Si suppositum digitus suo fine repertus, Omnino delet illic scribi cifra debet, A leva si qua sit ei sociata figura; Si cifre remanent in fine pares decet harum 256 Radices, numero mediam proponere partem, Tali quesita radix patet arte reperta. Per numerum recte si nosti multiplicare Ejus quadratum, numerus qui pervenit inde 260 Dicetur cubicus; primus radix erit ejus; Nec numeros omnes cubicatos dicere debes, Est autem omnis numerus radix alicujus;
[Sidenote: Cube Root.]
Si curas cubici radicem qurere, primo 264 Inscriptum numerum distinguere per loca debes; Que tibi mille notant a mille notante suprema Initiam, summa operandi parte sinistra, Illic sub scribas digitum, qui multiplicatus 268 In semet cubice suprapositum sibi perdat, Et si quid fuerit adjunctum parte sinistra Si non omnino, quantum poteris minuendo, Hinc triplans retrahe saltum, faciendo sub illa 272 Que manet a digito deleto terna, figuram Illi propones quo sub triplo asocietur, Ut cum subtriplo per eam tripla multiplicatur; Hinc per eam solam productum multiplicabis, 276 Postea totalem numerum, qui provenit inde A suprapositis respectu tolle triplate Addita supprimo cubice tunc multiplicetur, Respectu cujus, numerus qui progredietur 280 Ex cubito ductu, supra omnes adimetur; Tunc ipsam delens triples saltum faciendo, Semper sub ternas, retrahens alias triplicatas Ex hinc triplatis aliam propone figuram, 284 Que per triplatas ducatur more priori; Primo sub triplis sibi junctis, postea perse, In numerum ducta, productum de triplicatis: Utque prius dixi numerus qui provenit inde 288 A suprapositis has respiciendo trahatur, Huic cubice ductum sub primo multiplicabis, Respectumque sui, removebis de remanenti, Et sic procedas retrahendo triplando figuram. 292 Et proponendo nonam, donec totum peragatur, Subtripla sub propriis servare decet triplicatis; Si nil in fine remanet, numerus datus ante Est cubicus; cubicam radicem sub tripla prebent, 296 Cum digito juncto quem supprimo posuisti, Hec cubice ducta, numerum reddant tibi primum. Si quid erit remanens non est cubicus, sed habetur Major sub primo qui stat radix cubicam, 300 Servari debet quicquid radice remansit, Extracto numero, decet hec addi cubicato. Quo facto, numerus reddi debet tibi primus. Nam debes per se radicem multiplicare 304 Ex hinc in numerum duces, qui provenit inde Sub primo cubicus major sic invenietur; Illi jungatur remanens, et primus habetur, Si per triplatum numerum nequeas operari; 308 Cifram propones, nil vero per hanc operare Set retrahens illam cum saltu deinde triplata, Propones illi digitum sub lege priori, Cumque cifram retrahas saliendo, non triplicabis, 312 Namque nihil cifre triplacio dicitur esse; At tu cum cifram protraxeris aut triplicata, Hanc cum subtriplo semper servare memento: Si det compositum, digiti triplacio debet 316 Illius scribi, digitus saliendo sub ipsam; Digito deleto, que terna dicitur esse; Jungitur articulus cum triplata pereunte, Set facit hunc scribi per se triplacio prima, 320 Que si det digitum per se scribi facit illum; Consumpto numero, si sole fuit tibi cifre Triplato, propone cifram saltum faciendo, Cumque cifram retrahe triplam, scribendo figuram, 324 Preponas cifre, sic procedens operare, Si tres vel duo serie in sint, pone sub yma, A dextris digitum servando prius documentum. Si sit continua progressio terminus nuper 328 Per majus medium totalem multiplicato; Si par, per medium tunc multiplicato sequentem. Set si continua non sit progressio finis: Impar, tunc majus medium si multiplicabis, 332 Si par per medium sibi multiplicato propinquum. 333
FOOTNOTES (Appendix II, Carmen de Algorismo):
[1: "Hec prsens ars dicitur algorismus ab Algore rege ejus inventore, vel dicitur ab algos quod est ars, et rodos quod est numerus; qu est ars numerorum vel numerandi, ad quam artem bene sciendum inveniebantur apud Indos bis quinque (id est decem) figur." —Comment. Thom de Novo-Mercatu. MS. Bib. Reg. Mus. Brit. 12 E.1.]
[2: "H necessari figur sunt Indorum characteros." MS. de numeratione. Bib. Sloan. Mus. Brit. 513, fol. 58. "Cum vidissem Yndos constituisse IX literas in universo numero suo propter dispositionem suam quam posuerunt, volui patefacere de opere quod sit per eas aliquidque esset levius discentibus, si Deus voluerit. Si autem Indi hoc voluerunt et intentio illorum nihil novem literis fuit, causa que mihi potuit. Deus direxit me ad hoc. Si vero alia dicam preter eam quam ego exposui, hoc fecerunt per hoc quod ego exposui, eadem tam certissime et absque ulla dubitatione poterit inveniri. Levitasque patebit aspicientibus et discentibus." MS. U.L.C., Ii. vi. 5, f.102.]
[3: From Eg. 2622.]
[4: 8 C. iv. inserts Nullum cipa significat: dat significare sequenti.]
[5: From 12 E. 1.]
[6: En argorisme devon prendre Vii especes . . . . Adision subtracion Doubloison mediacion Monteploie et division Et de radix eustracion A chez vii especes savoir Doit chascun en memoire avoir Letres qui figures sont dites Et qui excellens sont ecrites. —MS. Seld. Arch. B.26.]
[7: From 12 E. 1.]
[8: From 12 E. 1.]
[9: 8 C. iv. inserts Atque figura prior nuper fuerit mediando.]
[10: I.e. figura secundo loco posita.]
[11: So 12 E. 1; 8 C. iv. inserts—
[12: 12 E. 1 inserts.]
[13: 12 E. 1 inserts to l. 174.]
[14: 12 E. 1 omits, Eg. 2622 inserts.]
[15: 12 E. 1 inserts.]
[16: 8 C. iv. inserts— Hinc illam dele duplans sub ei psalliendo Que sequitur retrahens quicquid fuerit duplicatum.]
INDEX OF TECHNICAL TERMS[1*]
[Footnote 1*: This Index has been kindly prepared by Professor J.B. Dale, of King's College, University of London, and the best thanks of the Society are due to him for his valuable contribution.]
[Transcriber's Note: The Technical Terms and Glossary (following) refer to page and line numbers in the printed book. Information in [[double brackets]] has been added by the transcriber to aid in text searching.]
algorisme, 33/12; algorym, augrym, 3/3; the art of computing, using the so-called Arabic numerals. The word in its various forms is derived from the Arabic al-Khowarazmi (i.e. the native of Khwarazm (Khiva)). This was the surname of Ja'far Mohammad ben Musa, who wrote a treatise early in the 9th century (see p.xiv). The form algorithm is also found, being suggested by a supposed derivation from the Greek arithmos (number).
antery, 24/11; to move figures to the right of the position in which they are first written. This operation is performed repeatedly upon the multiplier in multiplication, and upon certain figures which arise in the process of root extraction.
anterioracioun, 50/5; the operation of moving figures to the right. [[written anteriorac{i}o{u}n or anterioracio{u}n]]
article, 34/23; articul, 5/31; articuls, 9/36, 29/7,8; anumber divisible by ten without remainder. [[also articull{e}]]
cast, 8/12; to add one number to another. 'Addition is a casting together of two numbers into one number,' 8/10.
cifre, 4/1; the name of the figure 0. The word is derived from the Arabic sifr = empty, nothing. Hence zero. A cipher is the symbol of the absence of number or of zero quantity. It may be used alone or in conjunction with digits or other ciphers, and in the latter case, according to the position which it occupies relative to the other figures, indicates the absence of units, or tens, or hundreds, etc. The great superiority of the Arabic to all other systems of notation resides in the employment of this symbol. When the cipher is not used, the place value of digits has to be indicated by writing them in assigned rows or columns. Ciphers, however, may be interpolated amongst the significant figures used, and as they sufficiently indicate the positions of the empty rows or columns, the latter need not be indicated in any other way. The practical performance of calculations is thus enormously facilitated (see p.xvi).
componede, 33/24; composyt, 5/35; with reference to numbers, one compounded of a multiple of ten and a digit. [[written componed{e}]]
conuertide = conversely, 46/29, 47/9. [[written co{n}u{er}tid{e} or {con}u{er}tid{e}]]
cubicede, 50/13; to be c., to have its cube root found. [[written cubiced{e}]]
cubike nombre, 47/8; anumber formed by multiplying a given number twice by itself, e.g. 27 = 333. Now called simply a cube. [[written cubik{e} ...]]
decuple, 22/12; the product of a number by ten. Tenfold.
departys = divides, 5/29. [[written dep{ar}tys]]
digit, 5/30; digitalle, 33/24; anumber less than ten, represented by one of the nine Arabic numerals. [[written digitall{e}]]
dimydicion, 7/23; the operation of dividing a number by two. Halving. [[written dimydicio]]
duccioun, multiplication, 43/9. [[written duccio{u}n]]
duplacion, 7/23, 14/15; the operation of multiplying a number by two. Doubling. [[written duplacio or duplacion with fancy "n"]]
i-mediet = halved, 19/23.
intercise = broken, 46/2; intercise Progression is the name given to either of the Progressions 1, 3, 5, 7, etc.; 2, 4, 6, 8, etc., in which the common difference is2. [[written int{er}cise]]
lede into, multiply by, 47/18. [[words always separated, as "lede ... into"]]
lyneal nombre, 46/14; a number such as that which expresses the measure of the length of a line, and therefore is not necessarily the product of two or more numbers (vide Superficial, Solid). This appears to be the meaning of the phrase as used in The Art of Nombryng. It is possible that the numbers so designated are the prime numbers, that is, numbers not divisible by any other number except themselves and unity, but it is not clear that this limitation is intended.
mediacioun, 16/36, 38/16; dividing by two (see also dimydicion). [[written mediacion with fancy "n", generally without "u"]]
medlede nombre, 34/1; anumber formed of a multiple of ten and a digit (vide componede, composyt). [[written medled{e} ...]]
medye, 17/8, to halve; mediete, halved, 17/30; ymedit, 20/9.
naturelle progressioun, 45/22; the series of numbers 1, 2, 3, etc. [[written naturell{e} p{ro}gressio{u}n]]
produccioun, multiplication, 50/11. [[written produccio{u}n]]
quadrat nombre, 46/12; a number formed by multiplying a given number by itself, e.g. 9 = 33, asquare.
rote, 7/25; roote, 47/11; root. The roots of squares and cubes are the numbers from which the squares and cubes are derived by multiplication into themselves.
significatyf, significant, 5/14; The significant figures of a number are, strictly speaking, those other than zero, e.g. in 3 6 5 0 4 0 0, the significant figures are 3, 6, 5, 4. Modern usage, however, regards all figures between the two extreme significant figures as significant, even when some are zero. Thus, in the above example, 3 6 5 0 4 are considered significant.
solide nombre, 46/37; anumber which is the product of three other numbers, e.g. 66 = 1123. [[usually written solid{e}]]
superficial nombre, 46/18; anumber which is the product of two other numbers, e.g. 6 = 23. [[written sup{er}ficial or sup{er}ficiall{e}]]
ternary, consisting of three digits, 51/7. [[written t{er}nary]]
vnder double, a digit which has been doubled, 48/3.
vnder-trebille, a digit which has been trebled, 49/28; vnder-triplat, 49/39. [[written vnder-trebill{e}, vnder-t{r}iplat]]
w, a symbol used to denote half a unit, 17/33 [[printed as superscript^w]]
GLOSSARY
[Transcriber's Note:
Words whose first appearance is earlier than the page cited in the Glossary are identified in double-bracketed notes. To aid in text searching, words written with internal {italics} are also noted, and context is given for common words.]
ablacioun, taking away, 36/21 [[written ablacio{u}n]] addyst, haddest, 10/37 agregacioun, addition, 45/22. (First example in N.E.D., 1547.) [[written ag{r}egacio{u}n]] a-[gh]enenes, against, 23/10 allgate, always, 8/39 als, as, 22/24 and, if, 29/8; &, 4/27; & yf, 20/7 a-nendes, towards, 23/15 aproprede, appropriated, 34/27 [[written ap{ro}pred{e}]] apwereth, appears, 61/8 a-risy[gh]t, arises, 14/24 a-rowe, in a row, 29/10 arsemetrike, arithmetic, 33/1 [[written arsemetrik{e}]] ayene, again, 45/15
bagle, crozier, 67/12 bordure = ordure, row, 43/30 [[written bordur{e}]] borro, inf. borrow, 11/38; imp. s. borowe, 12/20; pp. borwed, 12/15; borred, 12/19 boue, above, 42/34
caputule, chapter, 7/26 [[written caputul{e}]] certayn, assuredly, 18/34 [[written c{er}tayn]] clepede, called, 47/7 [[written cleped{e}]] competently, conveniently, 35/8 compt, count, 47/29 contynes, contains, 21/12; [[written {con}tynes]] pp. contenythe, 38/39 [[written co{n}tenyth{e}]] craft, art, 3/4
distingue, divide, 51/5
egalle, equal, 45/21 [[written egall{e}]] excep, except, 5/16] exclusede, excluded, 34/37 [[written exclused{e}]] excressent, resulting, 35/16 [[written exc{re}ssent]] exeant, resulting, 43/26 expone, expound, 3/23
ferye = fere, fourth, 70/12 figure = figures, 5/1 [[written fig{ure}]] for-by, past, 12/11 fors; no f., no matter, 22/24 forseth, matters, 53/30 forye = fore, forth, 71/8] fyftye = fyfte, fifth, 70/16
grewe, Greek, 33/13
haluendel, half, 16/16; haldel, 19/4; pl. haluedels, 16/16 hayst, hast, 17/3, 32 hast, haste, 22/25 [[in "haue hastto"]] heer, higher, 9/35 here, their, 7/26 [[in "in her{e} caputul{e}"]] here-a-fore, heretofore, 13/7 [[written her{e}-a-for{e}]] heyth, was called, 3/5 hole, whole, 4/39; holle, 17/1; hoole, of three dimensions, 46/15 holdye, holds good, 30/5 how be it that, although, 44/4
lede = lete, let, 8/37 lene, lend, 12/39 lest, least, 43/27 [[in "at the lest"]] lest = left, 71/9 [[in "the lest syde"]] leue, leave, 6/5; pr. 3 s. leues, remains, 11/19; [[first in 10/40]] leus, 11/28; pp. laft, left, 19/24 lewder, more ignorant, 3/3 [[written lewd{er}]] lust, desirest to, 45/13 ly[gh]t, easy, 15/31 lymytes, limits, 34/18; lynes, 34/12; lynees, 34/17; Lat. limes, pl. limites.
maystery, achievement; [[written mayst{er}y]] no m., no achievement, i.e. easy, 19/10 me, indef. pron. one, 42/1 [[first in 34/16]] mo, more, 9/16 moder = more (Lat. majorem), 43/22 most, must, 30/3 [[first in 3/12 and many more]] multipliede, to be m. = multiplying, 40/9 mynvtes, the sixty parts into which a unit is divided, 38/25 [[written mynvt{es}]] myse-wro[gh]t, mis-wrought, 14/11
nether, nor, 34/25 [[in "It was, netheris"]] nex, next, 19/9 no[gh]t, nought, 5/7 [[first in 4/8]] note, not, 30/5
oo, one, 42/20; o, 42/21 [[first in 34/27; 33/22]] omest, uppermost, higher, 35/26; omyst, 35/28 omwhile, sometimes, 45/31 [[first in 39/17]] on, one, 8/29 [[in "on vnder an-o{er}"]] opyne, plain, 47/8 [[written opyn{e}]] or, before, 13/25 [[in "or ou be-gan"]] or = e o{er}, the other, 28/34 [[in "or by-twene"]] ordure, order, 34/9; row, 43/1 [[word form is "order"]] other, or, 33/13, 43/26; [[in "art other craft" on 33/13, "other how oft" on 43/26; note also "one other other" on 35/24]] other ... or, either .. . or, 38/37 [[in "other it is even or od{e}" on 38/37; there are earlier occurrences]] ouerer, upper, 42/15 [[written ou{er}er]] ouer-hippede, passed over, 43/19 [[written ou{er}-hipped{e}]]
recte, directly, 27/20 [[in "stondes not recte"; also on 26/31 in "recte ou{er} his hede"]] remayner, remainder, 56/28 representithe, represented, 39/14 [[written rep{re}sentith{e}]] resteth, remains, 63/29 [[first in 57/29 and others]] rewarde, regard, 48/6 [[written reward{e}]] rew, row, 4/8 rewle, row, 4/20, 7/12; [[in "place of e rewle", "e rewle of fig{ure}s"]] rewele, 4/18; rewles, rules, 5/33
s. = scilicet, 3/8 [[in "s. Algorism{us}"]] sentens, meaning, 14/29 signifye(tyf), 5/13. The last three letters are added above the line, evidently because of the word 'significatyf' in l.14. But the 'Solucio,' which contained the word, has been omitted. sithen, since, 33/8 some, sum, result, 40/17, 32 [[first in 36/21 in "me may see a some", then in "the same some" and "to someof"]] sowne, pronounce, 6/29 singillatim, singly, 7/25 spices, species, kinds, 34/4 [[first in 5/34 and others]] spyl, waste, 14/26 styde, stead, 18/20 subtrahe, subtract, 48/12; pp. subtrayd, 13/21 sythes, times, 21/16
ta[gh]t, taught, 16/36 take, pp. taken; t. fro, starting from, 45/22 [[in "fro oone or tweyn{e} take"]] taward, toward, 23/34 thou[gh]t, though, 5/20 trebille, multiply by three, 49/26 [[written trebill{e}]] twene, two, 8/11 [[first in 4/23]] ow, though, 25/15 [[in "ow {o}u take"]] ow[gh]t, thought; be ., mentally, 28/4 us = is, this, 20/33 [[in "us nombur 214"]]
vny, unite, 45/10
wel, wilt, 14/31 [[in "If {o}u wel"]] wete, wit, 15/16; wyte, know, 8/38; pr. 2 s. wost, 12/38 wex, become, 50/18 where, whether, 29/12 [[written wher{e} in "wher{e} in e secunde,or"]] wher-thurghe, whence, 49/15 [[written Wher-thurgh{e}]] worch, work, 8/19; [[first in 7/35]] wrich, 8/35; wyrch, 6/19; imp. s. worch, 15/9; [[first in 9/6]] pp. y-wroth, 13/24 write, written, 29/19; [[first in 6/37 in "hast write", "be write"]] y-write, 16/1 wryrchynge = wyrchynge, working, 30/4 [[written wryrchyng{e}]] w^t, with, 55/8
y-broth, brought, 21/18 ychon, each one, 29/10 [[written ychon]] ydo, done, added, 9/6 [[first in 8/37 in "haue ydo"; 9/6 in "ydo all to-ged{er}"]] ylke, same, 5/12 y-lyech, alike, 22/23 y-my[gh]t, been able, 12/2 y-now[gh]t, enough, 15/31; ynov[gh]t, 18/34 yove, given, 45/33 y^t, that, 52/8 y-write, v. write. y-wroth, v. worch.
* * * * * * * * * * * * * *
MARGINAL NOTES:
Headnotes have been moved to the beginning of the appropriate paragraph. Headnotes were omitted from the two Appendixes, as sidenotes give the same information.
Line Numbers are cited in the Index and Glossary. They have been omitted from the e-text except in the one verse selection (App. II, Carmen de Algorismo). Instead, the Index and Glossary include supplemental information to help locate each word.
Numbered Notes:
Numbered sidenotes show page or leaf numbers from the original MSS. In the e-text, the page number is shown as [*123b] inline; mid-word page breaks are marked with a supplemental asterisk [*]. Numbers are not used.
Footnotes give textual information such as variant readings. They have been numbered sequentially within each title, with numbers shown as [{1}] to avoid confusion with bracked text—including single numerals—in the original. Editorial notes are shown as [1*]. When a footnote calls for added text, the addition is shown in the body text with [[double brackets]].
Sidenotes giving a running synopsis of the text have been moved to the beginning of each paragraph, where they are shown as a single note.
ERRORS AND ANOMALIES (Noted by Transcriber):
Introduction:
dated Mij^c [In this and the remainder of the paragraph, the letter shown as ^c is printed directly above the preceding j.]
The Crafte of Nombrynge:
sursu{m} {pr}ocedas m{u}ltiplicando [Italicized as shown: error for "p{ro}cedas"?] Sidenote: Our author makes a slip here [Elsewhere in the book, numerical errors are corrected in the body text, with a footnote giving the original form.] ten tymes so mych is e nounb{re} [text unchanged: error for "as"?] 6 tymes 24, [{19}]en take [misplaced footnote anchor in original: belongs with "6 times 24"] Fn. 7: 'Subt{ra}has a{u}t addis a dext{ri}s [open quote missing]
The Art of Nombryng:
oone of the digitis as .10. of 1.. 20. of. 2. [text unchanged: error for "as .10. of .1. 20. of .2."?] sette a-side half of tho m{inutes} [text unchanged: error for "the"?] and. 10. as before is come therof [text unchanged: error for "and .10."?] Sidenote: Where to set the quotiente [spelling (1922) unchanged] Sidenote: Definition of Progression. [f in "of" illegible] Sidenote: ... giving the value of ab.^2 [That is, "a(b^2)."]
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