Chapter 7

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 counters: whiche feate doth not only serue for them that can not write and rede, but also for them that can do bothe, but haue not at some tymes theyr penne or tables redye with them. This sorte is in two fourmes commenly. The one by lynes, and the other without lynes: in thatythath lynes, the lynes do stande for the order of places: and in ytthat hath no lynes, there must be sette in theyr stede so many counters as shall nede, for eche lyne one, and they shall supplye the stede of the lynes.S.By examples I shuld better perceaue your meanynge.M.For example of the117aly*nes:10 0 0 0 010 0 0 0X10 0 010 0101Lo here you se .vi. lynes whiche stande for syxe places so that the nethermost standeth for yefyrst place, and the next aboue it, for the second: and so vpward tyll you come to the hyghest, which is the syxte lyne, and standeth for the syxte place.Numeration.Now what is the valewe of euery place or lyne, you may perceaue by the figures whiche I haue set on them, which is accordynge as you learned before in the Numeration of figures by the penne: for the fyrste place is the place of vnities or ones, and euery counter set in that lyne betokeneth but one:andthe 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:andso forth.S.Syr I do perceaue that the same order is here of lynes, as was in the other figures117b.*by places, so that you shall not nede longer to stande about Numeration, excepte there be any other difference.M.Yf you do vnderstande it, then how wyll you set 1543?X1543S.Thus, as I suppose.M.You haue set yeplaces truely, but your figures be not mete for this vse:for the metest figure in this behalfe, is the figure of a counter round, as you se here, where I haue expressed that same summe.×o°o o o oo o oS.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 numerator118a.*is 5, you shall set one counter for it, in the next space aboue the lyne that it hath his denomination of, as in this example of that 500, bycause the numerator is 5, it must be set in a voyd space: and bycause the denominator is hundred, I knowe that his place is the voyde space next aboue hundredes, that is to say, aboue the thyrd lyne. And farther you shall marke, that in all workynge by this sorte, yf you shall sette downe any summe betwene 4 and 10, for the fyrste parte of that nomber you shall set downe 5, & then so many counters more, as there reste nombers aboue 5. And this is true bothe of digettes and articles. And for example I wyll set downe this summe 287965,Xo oo o°oXo o°o o°o oo°which summe yf you marke well, you nede none other examples for to lerne the numeration of118b.*this forme. But this shal you marke, that as you dyd in the other kynde of arithmetike, set a pricke in the places of thousandes, in this worke you shall sette a starre, as you se here.Addition on the Counting Board.Addition.S.Then I perceave numeration, but I praye you, howe shall I do in this arte to adde two summes or more together?M.The easyest way in this arte is, to adde but 2 summes at ones together: how be it you may adde more, as I wyll tell you anone. Therfore when you wyll adde two summes, you shall fyrst set downe one of them, itforsethnot whiche,andthen by it drawe a lyne crosse the other lynes. And afterward set downe the other summe, so that that lyne may be betwene them, as yf you wolde adde 2659 to 8342, you must set your summes as you se here.Xo°o oo oo o oo°o o o o°o oo°o o oAnd then yf you lyst, you119a.*may adde the one to the other in the same place, or els you may adde them both together in a newe place: which waye, bycause it is moste playnest, I wyll showe you fyrst. Therfore wyl I begynne at the vnites, whiche in the fyrst summe is but 2,andin yesecond summe 9, that maketh 11, those do I take vp, and for them I set 11 in the new roume, thus,Xo°o oo oo o oo°o o o o°ooThen do I take vp all yearticles vnder a hundred, which in the fyrst summe are 40, and in the second summe 50, that maketh 90: or you may saye better, that in the fyrste summe there are 4 articles of 10, and in the seconde summe 5, which make 9, but then take hede that you sette them in theyr119b.*ryght lynes as you se here.Xo°o oo oo o oo°o°o o o ooWhere I haue taken awaye 40 fromthe fyrste summe, and 50 from yesecond, and in theyr stede I haue set 90 in the thyrde, whiche I haue set playnely ytyou myght well perceaue it: how be it seynge that 90 with the 10 that was in yethyrd roume all redy, doth make 100, I myghte better for those 6 counters set 1 in the thyrde lyne, thus:XooFor it is all one summe as you may se, but it is beste, neuer to set 5 counters in any line, for that may be done with 1 counter in a hygher place.S.I iudge that good reason, for many are vnnedefull, where one wyll serue.M.Well, then120a.*wyll I adde forth of hundredes: I fynde 3 in the fyrste summe, and 6 in the seconde, whiche make 900, them do I take vpandset in the thyrd roume where is one hundred all redy, to whiche I put 900, and it wyll be 1000, therfore I set one counter in the fourth lyne for them all, as you se here.Xo°o oo oooThen adde I yethousandes together, whiche in the fyrst summe are 8000,andin yesecond 2000, that maketh 10000: them do I take vp fromthose two places, and for them I set one counter in the fyfte lyne, and then appereth as you se,oXooto be 11001, for so many doth amount of the addition of 8342 to 2659.120b.*S.Syr, this I do perceave: but how shall I set one summe to an other, not chaungynge them to a thyrde place?M.Marke well how I do it: I wyll adde together 65436, and 3245, whiche fyrste I set downe thus.o°Xo o o°o oo o o oo o o oo o o°o°Then do I begynne with the smalest, which in the fyrst summe is5, that do I take vp, and wold put to the other 5 in the seconde summe, sauynge that two counters can not be set in a voyd place of 5, but for them bothe I must set 1 in the seconde lyne, which is the place of 10, therfore I take vp the 5 of the fyrst summe,andthe 5 of the seconde, and for them I set 1 in the second lyne,121a.*as you se here.o°Xo o o°o oo o o oo o o oo o o ooThen do I lyke wayes take vp the 4 counters of the fyrste summeandseconde lyne (which make 40) and adde them to the 4 counters of the same lyne, in the second summe, and it maketh 80, But as I sayde I maye not conueniently set aboue 4 counters in one lyne, therfore to those 4 that I toke vp in the fyrst summe, I take one also of the seconde summe, and then haue I taken vp 50, for whiche 5 counters I sette downe one in the space ouer yesecond lyne, as here doth appere.o°Xo o o°o oo o o oo°o oo121b*and then is there 80, as wellwtthose 4 counters, as yf I had set downe yeother 4 also. Now do I take the 200 in the fyrste summe, and adde them to the 400 in the seconde summe, and it maketh 600, therfore I take vp the 2 counters in the fyrste summe, and 3 of them in the seconde summe, and for them 5 I set 1 in yespace aboue, thus.o°Xo o o°o°o°o ooThen I take ye3000 in yefyrste summe, vnto whiche there are none in the second summe agreynge, therfore I do onely remoue those 3 counters from the fyrste summe into the seconde, as here doth appere.o°Xo°o oo°o°o oo122a*And so you see the hole summe, that amounteth of the addytionof 65436 with 3245 to be 6868[1]. And yf you haue marked these two examples well, you nede no farther enstructionin Addition of 2 only summes: but yf you haue more then two summes to adde, you may adde them thus.Xo oo o o oo°o°o oo°o o oo°oo°o oo°o°o o oo°°Fyrst adde two of them, and then adde the thyrde, and yefourth, 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.122b.*And then I adde the thyrde thereto thus.Xoo°o o o°o o oo°o o oo o oo°o o o°o°o°o°And so of more yf you haue them.Subtraction on the Counting Board.S.Nowe I thynke beste that you passe forth to Subtraction, except there be any wayes to examyn this maner of Addition, then I thynke that were good to be knowen nexte.M.There is the same profe here that isSubtraction.in the other Addition by the penne, I meane Subtraction, for that onely is a sure waye: but consyderynge that Subtraction must be fyrste knowen, I wyl fyrste teache you the arte of Subtraction, and that by this example: I wolde subtracte 2892 out of 8746. These summes must I set downe as I dyd in Addition: but here it is best116a(sic).*to set the lesser nomber fyrste, thus.Xo oo o°oo°o oo°oo°o o oo o o oo oo°Then shall I begynne to subtracte the greatest nombres fyrste (contrary to the vse of the penne)ytis the thousandes in this example: therfore I fynd amongest the thousandes 2, for which I withdrawe so many fromthe seconde summe (where are 8) and so remayneth there 6, as this example showeth.o°+o°o oo°oo°o o oo o o oo oo°Then do I lyke wayes with the hundredes, of whiche in the fyrste summe116b.*I fynde 8, and is the seconde summe but 7, out of whiche I can not take 8, therfore thus muste I do: I muste loke how moche my summe dyffereth from 10, whiche I fynde here to be 2, then must I bate for my summe of 800, one thousande, and set downe the excesse of hundredes, that is to saye 2, for so moche 100[0] is more then I shuld take vp. Therfore fromthe fyrste summe I take that 800, and from the second summe where are 6000, I take vp one thousande, and leue 5000; but then set I downe the 200 unto the 700 ytare there all redye, and make them 900 thus.+°o°o o oo o°o oo o o oo oo°Then come I to the articles of tennes where in the fyrste summe I fynde 90,117a.*and in the seconde summe but only 40: Now consyderyng that 90 can not be bated from 40, I loke how moche yt90 doth dyffer from the next summe aboue it, that is 100 (or elles whiche is all to one effecte, I loke how moch 9 doth dyffer from10)andI fynd it to be 1, then in the stede of that 90, I do take from the second summe 100: but consyderynge that it is 10 to moche, I set downe 1 in yenexte lyne beneth for it, as you se here.+°o°o o°o oo°Sauynge that here I haue set one counter in yespace in stede of 5 in yenexte 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 oSo ytyf I subtracte 2892 from8746, theremaynerwyll be 5854,117b.*And that this is truely wrought, you maye proue by Addition: for yf you adde to this remayner the same summe that you dyd subtracte, then wyll the formar summe 8746 amount agayne.S.That wyll I proue: and fyrst I set the summe that was subtracted, which was 2892,andthenthe remayner 5854, thus.||o o°o°o oo o°oo o°o o°o oo o o oThen do I adde fyrst ye2 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 oo°o oo°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 yethyrde lyne,and4 in yesecond lyne, thus.||o o°o°o oo°o o oo o o o°oThen do I come to the hundredes, of whiche I fynde 8 in the fyrst summe, and 9 in yesecond, 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 oo°o°oo o o oo°Then is there lefte in the fyrste summe but only 2000, whiche I shall take vp from thence, and set118b.*in the same lyne in yesecond summe, to yeone ytis there all redy:andthen wyll the hole summe appere (as you may wel se) to be 8746, which was yefyrst grosse summe,andtherfore I do perceaue, that I hadde well subtracted before.Xo o°oo o°o o o o°oAnd thus you may se how Subtraction maye be tryed by Addition.S.I perceaue the same order here wtcounters, ytI lerned before in figures.M.Then let me se howe can you trye Addition by Subtraction.S.Fyrste I wyl set forth this example of Additionwhere I haue added 2189 to 4988, and the hole summe appereth to be 7177,||o oo o o oo°ooo°o o ooo°o oo°o oo°oo°o o oo°o oo°o119a.*Nowe to trye whether that summe be well added or no, I wyll subtract one of the fyrst two summes from the thyrd, and yf I haue well done yeremayner wyll be lyke that other summe. As for example: I wyll subtracte the fyrste summe from the thyrde, whiche I set thus in theyr order.||o oo°oooo°o oo°oo°o o oo°oThen do I subtract 2000 of the fyrste summe fromyesecond summe, and then remayneth there 5000 thus.X°ooo°o oo°oo°o o oo°oThen in the thyrd lyne, I subtract ye100 of the fyrste summe, fromthe second summe, where is onely 100 also, and then in yethyrde lynerestethnothyng. Then in the second lyne with his space ouer hym, I fynde 80, which I shuld subtract119b.*from the other summe, then seyng there are but only 70 I must take it out of some hygher summe, which is here only 5000, therfore I take vp 5000, and seyng that it is to moch by 4920, I sette downe so many in the seconde roume, whiche with the 70 beynge there all redy do make 4990, & then the summes doth stande thus.||o o o oo°o o oo°o o oo°o o oo°oYet remayneth there in the fyrst summe 9, to be bated from the second summe, where in that place of vnities dothe appere only 7, then I muste bate a hygher summe, that is to saye 10, but seynge that 10 is more then 9 (which I shulde abate) by 1, therfore shall I take vp one counter from the seconde lyne,andset downe the same in the fyrst120a.*orlowest lyne, as you se here.||o o o oo°o o oo°o oo°o oAnd so haue I ended this worke,andthe summe appereth to be yesame, whiche was yeseconde summe of my addition, and therfore I perceaue, I haue wel done.M.To stande longer about this, it is but folye: excepte that this you maye also vnderstande, that many do begynne to subtracte with counters, not at the hyghest summe, as I haue taught you, but at the nethermoste, as they do vse to adde: and when the summe to be abatyd, in any lyne appeareth greater then the other, then do they borowe one of the next hygher roume, as for example: yf they shuld abate 1846 from 2378, they set yesummes thus.||oo oo°o oo o oo o o oo°oo°o°o o120b.*And fyrste they take 6 whiche is in the lower lyne, and his space from 8 in the same roumes, in yesecond summe, and yet there remayneth 2 counters in the lowest lyne. Then in the second lyne must 4 be subtracte from 7, and so remayneth there 3. Then 8 in the thyrde lyne and his space, from 3 of the second summe can not be, therfore do they bate it from a hygher roume, that is, from 1000, and bycause that 1000 is to moch by 200, therfore must I sette downe 200 in the thyrde lyne, after I haue taken vp 1000 from the fourth lyne: then is there yet 1000 in the fourth lyne of the fyrst summe, whiche yf I withdrawe from the seconde summe, then doth all yefigures stande in this order.||°o o oo oSo that (as you se) it differeth not greatly whether you begynne subtractionat the hygher lynes, or at121a.*the lower. How be it, as some menne lyke the one waye beste, so some lyke the other: therfore you now knowyng bothe, may vse whiche you lyst.Multiplication by Counters.Multiplication.But nowe touchynge Multiplication: you shall set your nombers in two roumes, as you dyd in those two other kyndes, but so that the multiplier be set in the fyrste roume. Then shall you begyn with the hyghest nombers of yeseconde roume, and multiply them fyrst after this sort. Take that ouermost lyne in your fyrst workynge, as yf it were the lowest lyne, setting on it some mouable marke, as you lyste, and loke how many counters be in hym, take them vp, and for them set downe the hole multyplyer, so many tymes as you toke vp counters, reckenyng, I saye that lyne for the vnites:andwhen you haue so done with the hygheest nomber then come to the nexte lyne beneth,anddo euen so with it, and so with yenext, tyll you haue done all. And yf there be any nomber in a space, then for it121b.*shall you take yemultiplyer 5 tymes, and then must you recken that lyne for the vnites whiche is nexte beneth that space: or elsafter a shorter way, you shall take only halfe the multyplyer, but then shall you take the lyne nexte aboue that space, for the lyne of vnites: but in suche workynge, yf chaunce your multyplyer be an odde nomber, so that you can not take the halfe of it iustly, then muste you take the greater halfe, and set downe that, as if that it were the iuste halfe, and farther you shall set one counter in the space beneth that line, which you recken for the lyne of vnities, or els only remoue forward the same that is to be multyplyed.S.Yf you set forth an example hereto I thynke I shal perceaue you.M.Take this example: I wold multiply 1542 by 365, therfore I set yenombers thus.||oo o o°o°o o o o°o o122a.*Then fyrste I begynne at the 1000 in yehyghest roume, as yf it were yefyrst 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 oo°X°<--o o o°o°o o o o°o owhere for the one counter taken vp from the fourth lyne, I haue sette downe other 6, whiche make yesumme 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 yesame,S.I perceaue this well: for in dede, this summe that you haue set downe is 365000, for so moche doth amount122b.*of 1000, multiplyed by 365.M.Well thento go forth, in the nexte space I fynde one counter which I remoue forward but take not vp, but do (as in such case I must) set downe the greater halfe of my multiplier (seyng it is an odde nomber) which is 182,andhere I do styll let that fourth place stand, as yf it were yefyrst:o o ooo°o°o o||°o°o<--o o o°o°o o o o°o oas in this fourme you se, where I haue set this multiplycationwith yeother: but for the ease of your vnderstandynge, I haue set a lytell lyne betwene them: now shulde they both in one summe stand thus.o o o o oo o o o||o°o<--o o o°o°o o o o°o o123a.*Howe be it an other fourme to multyplye suche counters inspace is this: Fyrst to remoue the fynger to the lyne nexte benethe yespace,andthen to take vp yecounter,andto set downe yemultiplyer .v. tymes, as here you se.o o oo°o o oo o oo o oo o oo o o°o°°o°o°o°-->Xo o o°°°°°o°o o o o°o oWhich summes yf you do adde together into one summe, you shal perceaue that it wyll be yesame ytappeareth of yeother working before, so that123b.*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 example, you myghte haue sayde 5 tymes 300 is 1500,and5 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, I must 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: ytis in all 1460, ytshall I set downe also: as here you se.°o o o ooXo°oo o o oo o o°o°o°<--°o o124a.*whiche yf I ioyne in one summe with the formar nombers, it wyll appeare thus.°o°o o<--o o ooo°°o oThen 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 set124b.*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,andso haue I ended the hole summe thus.°o°o oo o oo°o oo°o o o°Wherby you se, that 1542 (which is the nomber of yeares syth Ch[r]ystes incarnation) beyng multyplyed by 365(whichis the nomber of dayes in one yeare) dothe amounte vnto 562830, which declareth yenomber of daies sith Chrystes incarnationvnto the ende of 15421yeares. (besyde 385 dayes and 12 houres for lepe yeares).S.Now wyll I proue by an other example, as this: 40 labourers (after 6 d. yeday for eche man) haue wrought 28 dayes, I wold125a.*know what theyr wages doth amount vnto: In this case muste I worke doublely: fyrst I must multyplye the nomber of the labourers by yewages of a man for one day, so wyll yecharge of one daye amount: then secondarely shall I multyply that charge of one daye, by the hole nomber of dayes,andso wyll the hole summe appeare: fyrst therefore I shall set the summes thus.o o o oo°Where in the fyrste space is the multyplyer (ytis one dayes wages for one man)andin the second space is set the nomber of the worke men to be multyplyed: thensaye 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 oo o o oSoapwereththe hole dayes wages to be 240d’. that is 20 s. Then do I multiply agayn the same summe by the nomber of dayes and fyrste I sette the nombers, thus.o oo oo o o oo°o oThenbycause there are counters in dyuers lynes, I shall begynne with the hyghest, and take them vp, settynge for them the multyplyer so many tymes, as I toke vp counters, ytis twyse, then wyll yesumme stande thus.°o°o o o oThen come I to yeseconde lyne, and take vp those 4 counters, settynge for them the multiplyer foure tymes, so wyll the hole summe appeare thus.*126a.o°o°oo oSo is the hole wages of 40 workemen, for 28 dayes (after 6d’. eche daye for a man) 6720d’. that is 560 s. or 28 l’i.Division on the Counting Board.Diuision.M.Now if you wold proue Multiplycation, the surest way is by Dyuision: therfore wyll I ouer passe it tyll I haue taught you yearte 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 yeryghte syde, so farre from the diuisor, that the quotient may be set betwene them: as for example: Yf 225 shepe cost 45 l’i. what dyd euery shepe cost? To knowe this, I shulde diuide the hole summe, that is 45 l’i. by 225, but that can not be, therfore must I fyrste reduce that 45 l’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 nomber of126b.*shepe, whiche is 225, these two nombers therfore I sette thus.o oo°o o oo 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, thuso ooo o°o o o oAnd bycause I founde the diuisor 4 tymes in the diuidente, I haue set (as you se) 4 in the myddle roume, which127a.*is the place of the quotient: but now must I take the reste of the diuisor as often out of the remayner: therfore comeI to the seconde lyne of the diuisor, sayeng 2 foure tymes make 8, take 8 from 10,andthere resteth 2, thus.||o oo oo o°o o o oThen 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 45 l’i. euery shepe coste 4 s.S.This can I do, as you shall perceaue by this example: Yf 160 sowldyars do spende euery moneth 68 l’i. what spendeth eche man? Fyrst127b.*bycause I can not diuide the 68 by 160, therfore I wyll turne the poundes into pennes by multiplicacion, so shall there be 16320 d’. Nowe muste I diuide this summe by the nomber of sowldyars, therfore I set theminorder, thus.o||o°oo o oo°o oThen 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.||ooo o oo°o oThen 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) I may take 60 also twyse, and then standeth the nombers thus,||ooo°o o128b.*where I haue sette the quotient 2 in the lowest lyne: So is euery sowldyars portion 102 d’. that is 8 s. 6 d’.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 fromwhiche you do take it: as by this example I wyll declare. Yf yepurchace of 200 acres of ground dyd coste 290 l’i. what dyd one acre coste? Fyrst wyl I turne the poundes into pennes, so wyll there be 69600 d’· Then in settynge downe these nombers I shall do thus.o oo°Xo°o o oo°Fyrst set the diuident on the ryghte hande as it oughte, and then129a.*the diuisor on the lefte hande agaynst those nombers, fromwhich I entende to take hym fyrst as here you se, wher I haue set the diuisor two lynes hygher thenis theyr owne place.S.This is lyke the order of diuision by the penne.M.Truth you say, and nowe must I set yequotient 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,andthat I fynde 3 tymes, then set I 3 in the thyrde lyne for the quotient, and take awaye that 60000 fromthe diuident, and farther I do set the diuisor one line lower, as yow se here.||o oo°o o oo o o°o129b.*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, I fynde the diuisor 4 tymes in the diuidente, and 1 remaynynge, for 4 tymes 2 make 8, which I take from 9, and thereresteth1, as this figure sheweth:||o ooo o oo°o o o oand 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 yediuisor 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 oo o oo o o oo°o owhere you may se that the hole quotient is 348 d’, that is 29 s. wherby I knowe that so moche coste the purchace of one aker.S.Now resteth the profes of Multiplycation, and also of Diuision.M.Ther best profes are eche130b.*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, ytwas sufficyently 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, I wyll not trouble your wytte with it, tyll you haue practised this so well, ytyou 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 yearte of fractions or broken nomber, wherin Iwyll also showe you the reasons of all that you haue nowe learned. But yet before I make an ende, I wyll showe you the order of commen castyng, wher in are bothe pennes, shyllynges, and poundes, procedynge by no grounded reason, but onely by a receaued131a.*fourme, and that dyuersly of dyuers men: for marchauntes vse one fourme, and auditors an other:Merchants’ Casting Counters.Merchants’ casting.But fyrste for marchauntes fourme marke this example here,oo o o oooo o oooo o o ooo o o o oin which I haue expressed this summe 198 l’i.219 s. 11 d’. So that you maye se that the lowest lyne serueth for pennes, the next aboue for shyllynges, the thyrde for poundes, and the fourth for scores of poundes. 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 6 d’. Lyke wayes betwene the shyllyngesandthe poundes, one counter standeth for 10 s. And betwene the poundes and 20 l’i. one counter standeth for 10 poundes. But besyde those you maye see at the left syde of shyllynges, that one counter standeth alone,andbetokeneth 5 s.131b.*So agaynste the poundes, that one counter standeth for 5 l’i. And agaynst the 20 poundes, the one counter standeth for 5 score poundes, that is 100 l’i. so that euery syde counter is 5 tymes so moch as one of them agaynst whiche he standeth.Auditors’ casting.Now for the accompt of auditors take this example.oo oo ooo o oo o oo o oo o oooo owhere I haue expressed yesame summe 198 l’i. 19 s. 11 d’. But here you se the pennes stande toward yeryght 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,andall 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 on132a.*the lefte corner of the rowe it standeth for 10, of yesame row. But now yf you wold adde other subtracte after any of both those sortes, yf you marke yeorder of ytother feate which I taught you, you may easely do the same here without moch teachynge: for in Additionyou must fyrst set downe one summe and to the same set the other orderly, and lyke maner yf you haue many: but in Subtraction you must sette downe fyrst the greatest summe, and from it must you abate that other euery denominationfrom his dewe place.S.I do not doubte but with alytell 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 hundreddes, yf they excede one hundred, nother yet thousandes.M.They that vse such accomptes that it excede 200132b.*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 roweand500 at the lefte hand therof, and the thousandes they set in a farther rowe yet,andat 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 example,o o o oooo ooo o oo o ooo o o ooo ooo o oo oowhich is 97869 l’i. 12 s. 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:andmore there can not be, for 4 farthynges133a.*do make 1 d’. which must be set in his dewe place.Auditors’ Casting Counters.And yf you desyre yesame summe after audytors maner, lo here it is.o oooooo o oo oo o oo o oo o oo oo o oooo ooBut 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 commoditie of it ryghte worthy to be well marked. This feate hath ben vsed aboue 2000 yeares at the leaste, and yet was it neuer comenly 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 the133b.*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 tableof the arte of thehandepage number ‘134’ from original illustrationhand numbering as described in text1134b.*In which as you may se 1 is expressed by yelyttle fynger of yelefte hande closely and harde croked.232 is declared by lyke bowynge of the weddynge fynger (whiche is the nexte to the lyttell fynger) together with the lytell fynger.33 is signified by the myddle fynger bowed in lyke maner, with those other two.44 is declared by the bowyng of the myddle fynger and the ryngefynger, or weddynge fynger, with the other all stretched forth.55 is represented by the myddle fynger onely bowed.6And 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 withthe bowynge of the same fyngers as are 1, 2, and 3, but after an other fourme.7For 7 is declared by the bowynge of the lytell fynger, as is 1, saue that for 1 the fynger isclaspedin, hardeand135a.*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 bysshoppesbagle: and 7 is declared by the same fynger bowed lyke a gybbet.M.So I perceaue, you vnderstande it.8Then to expresse 8, you shall bowe after the same maner both the lyttell fynger and the rynge fynger.9And yf you bowe lyke wayes with them the myddle fynger, then doth it betoken 9.10Now to expresse 10, you shall bowe your fore fynger rounde, and set the ende of it on the hyghest ioynte of the thombe.20And for to expresse 20, you must set your fyngers streyght, and the ende of your thombe to the partitionof the135b.*fore moste and myddle fynger.3030 is represented by the ioynynge together of yeheaddes of the foremost fynger and the thombe.4040 is declared by settynge of the thombe crossewayes on the foremost fynger.5050 is signified by ryght stretchyng forth of the fyngers ioyntly, and applyenge of the thombes ende to the partition of the myddle fyngerandthe rynge fynger, or weddynge fynger.6060 is formed by bendynge of the thombe croked and crossynge it with the fore fynger.7070 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.8080 is expressed by settynge of the foremost fynger crossewayes on the thombe, so that 80 dyffereth thus from40, that for 80 the forefynger is set crosse on the thombe, and for 40 the thombe is set crosse ouer yeforefinger.90136a.*90 is signified, by bendynge the fore fynger, and settyng the ende of it in the innermost ioynte of yethombe, that is euen at the foote of it. And thus are all the nombers ended vnder 100.S.In dede these be all the nombers from1 to 10,andthen all the tenthes within 100,11, 12, 13,21, 22, 23but thisteacyedme not how to expresse 11, 12, 13,etc. 21, 22, 23,etc. and such lyke.M.You can lytell vnderstande, yf you can not do that without teachynge: what is11? 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.100But 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.200136b.*And as you expressed 2 in the lefte hande, the same fasshyon in the ryght hande doth declare 200.300The fourme of 3 in the ryght hand standeth for 300.400The fourme of 4, for 400.500Lykewayes the fourme of 5, for 500.600The 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 vnitiesandtenthes of hundredes, in the ryghte hande.900S.I vnderstande you thus: that yf I wold represent 900, I must so fourme the fyngers of my ryghte hande, as I shuld do in my left hand to expresse 9,1000And 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 nomber of thousandes,4000so yefourme of 40 standeth for 4000.8000The fourme of 80 for 8000.9000137a.*And the fourme of 90 (whiche isthe greatest) for 9000, and aboue thatI can not expresse any nomber.M.No not with one fynger: how be it,withdyuers fyngers you maye expresse9999, and all at one tyme, and that lacketh but 1 of 10000. So that vnder10000 you may by your fyngers ex-presse any summe. And this shal suf-fyce for Numeration on the fyngers.And as for Addition, Subtraction,Multiplication, and Diuision (whichyet were neuer taught by any man asfarre as I do knowe) I wyll enstructyou after the treatyse of fractions.And now for this tyme fare well,and loke that you cease not topractyse that you haue learned.S.Syr, with mosteharty mynde I thankeyou, bothe for yourgood learnyng,andalso your goodcounsel, which(god wyllyng) I truste to folow.Finis.1.1342 in original.2.168 in original.3.Bracket ([) denotes new paragraph in original.For this e-text, the brackets have been omitted in favor of restoring the paragraph breaks. Numbers 200 and up were printed as separate paragraphs and are unchanged. Sidenote 4 was missing and has been supplied by the transcriber; the pairs 5, 6 and 9, 10 (originally on one line) have been separated.

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.

Mayster.

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

S.By examples I shuld better perceaue your meanynge.

M.For example of the117aly*nes:

Lo here you se .vi. lynes whiche stande for syxe places so that the nethermost standeth for yefyrst place, and the next aboue it, for the second: and so vpward tyll you come to the hyghest, which is the syxte lyne, and standeth for the syxte place.Numeration.Now what is the valewe of euery place or lyne, you may perceaue by the figures whiche I haue set on them, which is accordynge as you learned before in the Numeration of figures by the penne: for the fyrste place is the place of vnities or ones, and euery counter set in that lyne betokeneth but one:andthe 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:andso forth.

S.Syr I do perceaue that the same order is here of lynes, as was in the other figures117b.*by places, so that you shall not nede longer to stande about Numeration, excepte there be any other difference.

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

S.Thus, as I suppose.

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

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 numerator118a.*is 5, you shall set one counter for it, in the next space aboue the lyne that it hath his denomination of, as in this example of that 500, bycause the numerator is 5, it must be set in a voyd space: and bycause the denominator is hundred, I knowe that his place is the voyde space next aboue hundredes, that is to say, aboue the thyrd lyne. And farther you shall marke, that in all workynge by this sorte, yf you shall sette downe any summe betwene 4 and 10, for the fyrste parte of that nomber you shall set downe 5, & then so many counters more, as there reste nombers aboue 5. And this is true bothe of digettes and articles. And for example I wyll set downe this summe 287965,

which summe yf you marke well, you nede none other examples for to lerne the numeration of118b.*this forme. But this shal you marke, that as you dyd in the other kynde of arithmetike, set a pricke in the places of thousandes, in this worke you shall sette a starre, as you se here.

Addition on the Counting Board.

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

M.The easyest way in this arte is, to adde but 2 summes at ones together: how be it you may adde more, as I wyll tell you anone. Therfore when you wyll adde two summes, you shall fyrst set downe one of them, itforsethnot whiche,andthen by it drawe a lyne crosse the other lynes. And afterward set downe the other summe, so that that lyne may be betwene them, as yf you wolde adde 2659 to 8342, you must set your summes as you se here.

And then yf you lyst, you119a.*may adde the one to the other in the same place, or els you may adde them both together in a newe place: which waye, bycause it is moste playnest, I wyll showe you fyrst. Therfore wyl I begynne at the vnites, whiche in the fyrst summe is but 2,andin yesecond summe 9, that maketh 11, those do I take vp, and for them I set 11 in the new roume, thus,

Then do I take vp all yearticles vnder a hundred, which in the fyrst summe are 40, and in the second summe 50, that maketh 90: or you may saye better, that in the fyrste summe there are 4 articles of 10, and in the seconde summe 5, which make 9, but then take hede that you sette them in theyr119b.*ryght lynes as you se here.

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

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

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

M.Well, then120a.*wyll I adde forth of hundredes: I fynde 3 in the fyrste summe, and 6 in the seconde, whiche make 900, them do I take vpandset in the thyrd roume where is one hundred all redy, to whiche I put 900, and it wyll be 1000, therfore I set one counter in the fourth lyne for them all, as you se here.

Then adde I yethousandes together, whiche in the fyrst summe are 8000,andin yesecond 2000, that maketh 10000: them do I take vp fromthose two places, and for them I set one counter in the fyfte lyne, and then appereth as you se,

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

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

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

Then do I begynne with the smalest, which in the fyrst summe is5, that do I take vp, and wold put to the other 5 in the seconde summe, sauynge that two counters can not be set in a voyd place of 5, but for them bothe I must set 1 in the seconde lyne, which is the place of 10, therfore I take vp the 5 of the fyrst summe,andthe 5 of the seconde, and for them I set 1 in the second lyne,121a.*as you se here.

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

121b*and then is there 80, as wellwtthose 4 counters, as yf I had set downe yeother 4 also. Now do I take the 200 in the fyrste summe, and adde them to the 400 in the seconde summe, and it maketh 600, therfore I take vp the 2 counters in the fyrste summe, and 3 of them in the seconde summe, and for them 5 I set 1 in yespace aboue, thus.

Then I take ye3000 in yefyrste summe, vnto whiche there are none in the second summe agreynge, therfore I do onely remoue those 3 counters from the fyrste summe into the seconde, as here doth appere.

122a*And so you see the hole summe, that amounteth of the addytionof 65436 with 3245 to be 6868[1]. And yf you haue marked these two examples well, you nede no farther enstructionin 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 yefourth, 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.122b.*And then I adde the thyrde thereto thus.

And so of more yf you haue them.

Subtraction on the Counting Board.

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

M.There is the same profe here that isSubtraction.in the other Addition by the penne, I meane Subtraction, for that onely is a sure waye: but consyderynge that Subtraction must be fyrste knowen, I wyl fyrste teache you the arte of Subtraction, and that by this example: I wolde subtracte 2892 out of 8746. These summes must I set downe as I dyd in Addition: but here it is best116a(sic).*to set the lesser nomber fyrste, thus.

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

Then do I lyke wayes with the hundredes, of whiche in the fyrste summe116b.*I fynde 8, and is the seconde summe but 7, out of whiche I can not take 8, therfore thus muste I do: I muste loke how moche my summe dyffereth from 10, whiche I fynde here to be 2, then must I bate for my summe of 800, one thousande, and set downe the excesse of hundredes, that is to saye 2, for so moche 100[0] is more then I shuld take vp. Therfore fromthe fyrste summe I take that 800, and from the second summe where are 6000, I take vp one thousande, and leue 5000; but then set I downe the 200 unto the 700 ytare there all redye, and make them 900 thus.

Then come I to the articles of tennes where in the fyrste summe I fynde 90,117a.*and in the seconde summe but only 40: Now consyderyng that 90 can not be bated from 40, I loke how moche yt90 doth dyffer from the next summe aboue it, that is 100 (or elles whiche is all to one effecte, I loke how moch 9 doth dyffer from10)andI fynd it to be 1, then in the stede of that 90, I do take from the second summe 100: but consyderynge that it is 10 to moche, I set downe 1 in yenexte lyne beneth for it, as you se here.

Sauynge that here I haue set one counter in yespace in stede of 5 in yenexte 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.

So ytyf I subtracte 2892 from8746, theremaynerwyll be 5854,117b.*And that this is truely wrought, you maye proue by Addition: for yf you adde to this remayner the same summe that you dyd subtracte, then wyll the formar summe 8746 amount agayne.

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

Then do I adde fyrst ye2 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.

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 yethyrde lyne,and4 in yesecond lyne, thus.

Then do I come to the hundredes, of whiche I fynde 8 in the fyrst summe, and 9 in yesecond, 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.

Then is there lefte in the fyrste summe but only 2000, whiche I shall take vp from thence, and set118b.*in the same lyne in yesecond summe, to yeone ytis there all redy:andthen wyll the hole summe appere (as you may wel se) to be 8746, which was yefyrst grosse summe,andtherfore I do perceaue, that I hadde well subtracted before.

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

S.I perceaue the same order here wtcounters, ytI lerned before in figures.

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

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

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

Then do I subtract 2000 of the fyrste summe fromyesecond summe, and then remayneth there 5000 thus.

Then in the thyrd lyne, I subtract ye100 of the fyrste summe, fromthe second summe, where is onely 100 also, and then in yethyrde lynerestethnothyng. Then in the second lyne with his space ouer hym, I fynde 80, which I shuld subtract119b.*from the other summe, then seyng there are but only 70 I must take it out of some hygher summe, which is here only 5000, therfore I take vp 5000, and seyng that it is to moch by 4920, I sette downe so many in the seconde roume, whiche with the 70 beynge there all redy do make 4990, & then the summes doth stande thus.

Yet remayneth there in the fyrst summe 9, to be bated from the second summe, where in that place of vnities dothe appere only 7, then I muste bate a hygher summe, that is to saye 10, but seynge that 10 is more then 9 (which I shulde abate) by 1, therfore shall I take vp one counter from the seconde lyne,andset downe the same in the fyrst120a.*orlowest lyne, as you se here.

And so haue I ended this worke,andthe summe appereth to be yesame, whiche was yeseconde summe of my addition, and therfore I perceaue, I haue wel done.

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

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

So that (as you se) it differeth not greatly whether you begynne subtractionat the hygher lynes, or at121a.*the lower. How be it, as some menne lyke the one waye beste, so some lyke the other: therfore you now knowyng bothe, may vse whiche you lyst.

Multiplication by Counters.

Multiplication.But nowe touchynge Multiplication: you shall set your nombers in two roumes, as you dyd in those two other kyndes, but so that the multiplier be set in the fyrste roume. Then shall you begyn with the hyghest nombers of yeseconde roume, and multiply them fyrst after this sort. Take that ouermost lyne in your fyrst workynge, as yf it were the lowest lyne, setting on it some mouable marke, as you lyste, and loke how many counters be in hym, take them vp, and for them set downe the hole multyplyer, so many tymes as you toke vp counters, reckenyng, I saye that lyne for the vnites:andwhen you haue so done with the hygheest nomber then come to the nexte lyne beneth,anddo euen so with it, and so with yenext, tyll you haue done all. And yf there be any nomber in a space, then for it121b.*shall you take yemultiplyer 5 tymes, and then must you recken that lyne for the vnites whiche is nexte beneth that space: or elsafter a shorter way, you shall take only halfe the multyplyer, but then shall you take the lyne nexte aboue that space, for the lyne of vnites: but in suche workynge, yf chaunce your multyplyer be an odde nomber, so that you can not take the halfe of it iustly, then muste you take the greater halfe, and set downe that, as if that it were the iuste halfe, and farther you shall set one counter in the space beneth that line, which you recken for the lyne of vnities, or els only remoue forward the same that is to be multyplyed.

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

M.Take this example: I wold multiply 1542 by 365, therfore I set yenombers thus.

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

where for the one counter taken vp from the fourth lyne, I haue sette downe other 6, whiche make yesumme 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 yesame,

S.I perceaue this well: for in dede, this summe that you haue set downe is 365000, for so moche doth amount122b.*of 1000, multiplyed by 365.

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

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

123a.*Howe be it an other fourme to multyplye suche counters inspace is this: Fyrst to remoue the fynger to the lyne nexte benethe yespace,andthen to take vp yecounter,andto set downe yemultiplyer .v. tymes, as here you se.

Which summes yf you do adde together into one summe, you shal perceaue that it wyll be yesame ytappeareth of yeother working before, so that123b.*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 example, you myghte haue sayde 5 tymes 300 is 1500,and5 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, I must 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: ytis in all 1460, ytshall I set downe also: as here you se.

124a.*whiche yf I ioyne in one summe with the formar nombers, it wyll appeare thus.

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 set124b.*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,andso haue I ended the hole summe thus.

Wherby you se, that 1542 (which is the nomber of yeares syth Ch[r]ystes incarnation) beyng multyplyed by 365(whichis the nomber of dayes in one yeare) dothe amounte vnto 562830, which declareth yenomber of daies sith Chrystes incarnationvnto the ende of 15421yeares. (besyde 385 dayes and 12 houres for lepe yeares).

S.Now wyll I proue by an other example, as this: 40 labourers (after 6 d. yeday for eche man) haue wrought 28 dayes, I wold125a.*know what theyr wages doth amount vnto: In this case muste I worke doublely: fyrst I must multyplye the nomber of the labourers by yewages of a man for one day, so wyll yecharge of one daye amount: then secondarely shall I multyply that charge of one daye, by the hole nomber of dayes,andso wyll the hole summe appeare: fyrst therefore I shall set the summes thus.

Where in the fyrste space is the multyplyer (ytis one dayes wages for one man)andin the second space is set the nomber of the worke men to be multyplyed: thensaye 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.

Soapwereththe hole dayes wages to be 240d’. that is 20 s. Then do I multiply agayn the same summe by the nomber of dayes and fyrste I sette the nombers, thus.

Thenbycause there are counters in dyuers lynes, I shall begynne with the hyghest, and take them vp, settynge for them the multyplyer so many tymes, as I toke vp counters, ytis twyse, then wyll yesumme stande thus.

Then come I to yeseconde lyne, and take vp those 4 counters, settynge for them the multiplyer foure tymes, so wyll the hole summe appeare thus.*126a.

So is the hole wages of 40 workemen, for 28 dayes (after 6d’. eche daye for a man) 6720d’. that is 560 s. or 28 l’i.

Division on the Counting Board.

Diuision.M.Now if you wold proue Multiplycation, the surest way is by Dyuision: therfore wyll I ouer passe it tyll I haue taught you yearte 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 yeryghte syde, so farre from the diuisor, that the quotient may be set betwene them: as for example: Yf 225 shepe cost 45 l’i. what dyd euery shepe cost? To knowe this, I shulde diuide the hole summe, that is 45 l’i. by 225, but that can not be, therfore must I fyrste reduce that 45 l’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 nomber of126b.*shepe, whiche is 225, these two nombers therfore I sette thus.

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

And bycause I founde the diuisor 4 tymes in the diuidente, I haue set (as you se) 4 in the myddle roume, which127a.*is the place of the quotient: but now must I take the reste of the diuisor as often out of the remayner: therfore comeI to the seconde lyne of the diuisor, sayeng 2 foure tymes make 8, take 8 from 10,andthere resteth 2, thus.

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 45 l’i. euery shepe coste 4 s.

S.This can I do, as you shall perceaue by this example: Yf 160 sowldyars do spende euery moneth 68 l’i. what spendeth eche man? Fyrst127b.*bycause I can not diuide the 68 by 160, therfore I wyll turne the poundes into pennes by multiplicacion, so shall there be 16320 d’. Nowe muste I diuide this summe by the nomber of sowldyars, therfore I set theminorder, thus.

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.

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) I may take 60 also twyse, and then standeth the nombers thus,

128b.*where I haue sette the quotient 2 in the lowest lyne: So is euery sowldyars portion 102 d’. that is 8 s. 6 d’.

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 fromwhiche you do take it: as by this example I wyll declare. Yf yepurchace of 200 acres of ground dyd coste 290 l’i. what dyd one acre coste? Fyrst wyl I turne the poundes into pennes, so wyll there be 69600 d’· Then in settynge downe these nombers I shall do thus.

Fyrst set the diuident on the ryghte hande as it oughte, and then129a.*the diuisor on the lefte hande agaynst those nombers, fromwhich I entende to take hym fyrst as here you se, wher I haue set the diuisor two lynes hygher thenis theyr owne place.

S.This is lyke the order of diuision by the penne.

M.Truth you say, and nowe must I set yequotient 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,andthat I fynde 3 tymes, then set I 3 in the thyrde lyne for the quotient, and take awaye that 60000 fromthe diuident, and farther I do set the diuisor one line lower, as yow se here.

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, I fynde the diuisor 4 tymes in the diuidente, and 1 remaynynge, for 4 tymes 2 make 8, which I take from 9, and thereresteth1, as this figure sheweth:

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 yediuisor 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,

where you may se that the hole quotient is 348 d’, that is 29 s. wherby I knowe that so moche coste the purchace of one aker.

S.Now resteth the profes of Multiplycation, and also of Diuision.

M.Ther best profes are eche130b.*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, ytwas sufficyently 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, I wyll not trouble your wytte with it, tyll you haue practised this so well, ytyou 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 yearte of fractions or broken nomber, wherin Iwyll also showe you the reasons of all that you haue nowe learned. But yet before I make an ende, I wyll showe you the order of commen castyng, wher in are bothe pennes, shyllynges, and poundes, procedynge by no grounded reason, but onely by a receaued131a.*fourme, and that dyuersly of dyuers men: for marchauntes vse one fourme, and auditors an other:

Merchants’ Casting Counters.

Merchants’ casting.But fyrste for marchauntes fourme marke this example here,

in which I haue expressed this summe 198 l’i.219 s. 11 d’. So that you maye se that the lowest lyne serueth for pennes, the next aboue for shyllynges, the thyrde for poundes, and the fourth for scores of poundes. 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 6 d’. Lyke wayes betwene the shyllyngesandthe poundes, one counter standeth for 10 s. And betwene the poundes and 20 l’i. one counter standeth for 10 poundes. But besyde those you maye see at the left syde of shyllynges, that one counter standeth alone,andbetokeneth 5 s.131b.*So agaynste the poundes, that one counter standeth for 5 l’i. And agaynst the 20 poundes, the one counter standeth for 5 score poundes, that is 100 l’i. so that euery syde counter is 5 tymes so moch as one of them agaynst whiche he standeth.Auditors’ casting.Now for the accompt of auditors take this example.

where I haue expressed yesame summe 198 l’i. 19 s. 11 d’. But here you se the pennes stande toward yeryght 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,andall 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 on132a.*the lefte corner of the rowe it standeth for 10, of yesame row. But now yf you wold adde other subtracte after any of both those sortes, yf you marke yeorder of ytother feate which I taught you, you may easely do the same here without moch teachynge: for in Additionyou must fyrst set downe one summe and to the same set the other orderly, and lyke maner yf you haue many: but in Subtraction you must sette downe fyrst the greatest summe, and from it must you abate that other euery denominationfrom his dewe place.

S.I do not doubte but with alytell 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 hundreddes, yf they excede one hundred, nother yet thousandes.

M.They that vse such accomptes that it excede 200132b.*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 roweand500 at the lefte hand therof, and the thousandes they set in a farther rowe yet,andat 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 example,

which is 97869 l’i. 12 s. 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:andmore there can not be, for 4 farthynges133a.*do make 1 d’. which must be set in his dewe place.

Auditors’ Casting Counters.

And yf you desyre yesame summe after audytors maner, lo here it is.

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 commoditie of it ryghte worthy to be well marked. This feate hath ben vsed aboue 2000 yeares at the leaste, and yet was it neuer comenly 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 the133b.*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.

hand numbering as described in text

1134b.*In which as you may se 1 is expressed by yelyttle fynger of yelefte hande closely and harde croked.

232 is declared by lyke bowynge of the weddynge fynger (whiche is the nexte to the lyttell fynger) together with the lytell fynger.

33 is signified by the myddle fynger bowed in lyke maner, with those other two.

44 is declared by the bowyng of the myddle fynger and the ryngefynger, or weddynge fynger, with the other all stretched forth.

55 is represented by the myddle fynger onely bowed.

6And 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 withthe bowynge of the same fyngers as are 1, 2, and 3, but after an other fourme.

7For 7 is declared by the bowynge of the lytell fynger, as is 1, saue that for 1 the fynger isclaspedin, hardeand135a.*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 bysshoppesbagle: and 7 is declared by the same fynger bowed lyke a gybbet.

M.So I perceaue, you vnderstande it.

8Then to expresse 8, you shall bowe after the same maner both the lyttell fynger and the rynge fynger.

9And yf you bowe lyke wayes with them the myddle fynger, then doth it betoken 9.

10Now to expresse 10, you shall bowe your fore fynger rounde, and set the ende of it on the hyghest ioynte of the thombe.

20And for to expresse 20, you must set your fyngers streyght, and the ende of your thombe to the partitionof the135b.*fore moste and myddle fynger.

3030 is represented by the ioynynge together of yeheaddes of the foremost fynger and the thombe.

4040 is declared by settynge of the thombe crossewayes on the foremost fynger.

5050 is signified by ryght stretchyng forth of the fyngers ioyntly, and applyenge of the thombes ende to the partition of the myddle fyngerandthe rynge fynger, or weddynge fynger.

6060 is formed by bendynge of the thombe croked and crossynge it with the fore fynger.

7070 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.

8080 is expressed by settynge of the foremost fynger crossewayes on the thombe, so that 80 dyffereth thus from40, that for 80 the forefynger is set crosse on the thombe, and for 40 the thombe is set crosse ouer yeforefinger.

90136a.*90 is signified, by bendynge the fore fynger, and settyng the ende of it in the innermost ioynte of yethombe, that is euen at the foote of it. And thus are all the nombers ended vnder 100.

S.In dede these be all the nombers from1 to 10,andthen all the tenthes within 100,11, 12, 13,21, 22, 23but thisteacyedme not how to expresse 11, 12, 13,etc. 21, 22, 23,etc. and such lyke.

M.You can lytell vnderstande, yf you can not do that without teachynge: what is11? 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.

100But 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.

200136b.*And as you expressed 2 in the lefte hande, the same fasshyon in the ryght hande doth declare 200.

300The fourme of 3 in the ryght hand standeth for 300.

400The fourme of 4, for 400.

500Lykewayes the fourme of 5, for 500.

600The 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 vnitiesandtenthes of hundredes, in the ryghte hande.

900S.I vnderstande you thus: that yf I wold represent 900, I must so fourme the fyngers of my ryghte hande, as I shuld do in my left hand to expresse 9,1000And 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 nomber of thousandes,4000so yefourme of 40 standeth for 4000.

8000The fourme of 80 for 8000.

*And the fourme of 90 (whiche isthe greatest) for 9000, and aboue thatI can not expresse any nomber.M.No not with one fynger: how be it,withdyuers fyngers you maye expresse9999, and all at one tyme, and that lacketh but 1 of 10000. So that vnder10000 you may by your fyngers ex-presse any summe. And this shal suf-fyce for Numeration on the fyngers.And as for Addition, Subtraction,Multiplication, and Diuision (whichyet were neuer taught by any man asfarre as I do knowe) I wyll enstructyou after the treatyse of fractions.And now for this tyme fare well,and loke that you cease not topractyse that you haue learned.S.Syr, with mosteharty mynde I thankeyou, bothe for yourgood learnyng,andalso your goodcounsel, which(god wyllyng) I truste to folow.

Finis.

1.1342 in original.2.168 in original.3.Bracket ([) denotes new paragraph in original.For this e-text, the brackets have been omitted in favor of restoring the paragraph breaks. Numbers 200 and up were printed as separate paragraphs and are unchanged. Sidenote 4 was missing and has been supplied by the transcriber; the pairs 5, 6 and 9, 10 (originally on one line) have been separated.

1.1342 in original.

2.168 in original.

3.Bracket ([) denotes new paragraph in original.

For this e-text, the brackets have been omitted in favor of restoring the paragraph breaks. Numbers 200 and up were printed as separate paragraphs and are unchanged. Sidenote 4 was missing and has been supplied by the transcriber; the pairs 5, 6 and 9, 10 (originally on one line) have been separated.

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