The Earliest Arithmetics in English
Chapter 9
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, I wyll 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, I myghte 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: I fynde 3 in the fyrste summe, and 6 in the seconde, whiche make 900, them do I take vp {and} set in the thyrd roume where is one hundred all redy, to whiche I put 900, and it wyll be 1000, therfore I set one 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 you se, 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: I wyll 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, I take 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, I meane Subtraction, for that onely is a sure waye: but consyderynge that Subtraction must be fyrste knowen, I wyl fyrste teache you the arte of Subtraction, and that by this example: I wolde subtracte 2892 out of 8746. These summes must I set downe as I dyd in Addition: but here it is best [*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 are 8) 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: I muste 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, I take vp one thousande, and leue 5000; but then set I downe the 200 unto the 700 y^t are there all redye, and make them 900 thus.
+-------------- | 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, I loke how moche y^t 90 doth dyffer from the next summe aboue it, that is 100 (or elles whiche is all to one effecte, I loke how moch 9 doth dyffer fro{m} 10) {and} I fynd it to be 1, then in the stede of that 90, I do take from the second summe 100: but consyderynge that it is 10 to moche, I set downe 1 in y^e nexte lyne beneth for it, as you se here.
+------------ | 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, I wyll 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: I wyll 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, I fynde 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, 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-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, 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 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, I saye 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: I wold 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, I haue 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, I haue 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 you se.
+---------+-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-----+-------+------+------+------+------+------+-