Chapter 10

Which su{m}mes yf you do adde together into one su{m}me, you shal p{er}ceaue that it wyll be y^e same y^t appeareth of y^e other worki{n}g before, so that [*123b] bothe sortes are to one entent, but as the other is much shorter, so this is playner to reason, for suche as haue had small exercyse in this arte. Not withstandynge you maye adde them in your mynde before you sette them downe, as in this exa{m}ple, you myghte haue sayde 5 tymes 300 is 1500, {and} 5 tymes 60 is 300, also 5 tymes 5 is 25, whiche all put together do make 1825, which you maye at one tyme set downe yf you lyste. But nowe to go forth, 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: y^t is in all 1460, y^t shall I set downe also: as here you se. o -----------+-------+-----------+-------------- | | | -----------+-------+--o-o-o-o--+--o----------- | | o | -X---------+-------+--o-o------+--o-o-o-o----- | | o | o ----o-o-o--+-------+-----------+--o----------- o | | | [->] ----o------+-------+-----------+-------------- o | | | -----------+--o-o--+-----------+--------------

[*124a] whiche yf I ioyne in one summe with the formar nombers, it wyll appeare thus. o ---------+-------+---------- | | o ---------+-------+--o------- | | ---------+-------+--o-o----- | | --o-o-o--+-------+-o-------- o | | [->] --o------+-------+---------- o | | ---------+--o-o--+----------

Then to ende this multiplycation, I remoue the fynger to the lowest lyne, where are onely 2, them do I take vp, and in theyr stede do I set downe twyse 365, that is 730, for which I set [*124b] one in the space aboue the thyrd lyne for 500, and 2 more in the thyrd lyne with that one that is there all redye, and the reste in theyr order, {and} so haue I ended the hole summe thus. o ---------+-----+------------ | | o ---------+-----+--o--------- | | ---------+-----+--o-o------- | | o --o-o-o--+-----+--o-o-o----- o | | --o------+-----+--o-o-o----- o | | ---------+-----+------------

Wherby you se, that 1542 (which is the nomber of yeares syth Ch[r]ystes incarnation) beyng multyplyed by 365 (which is the nomber of dayes in one yeare) dothe amounte vnto 562830, which declareth y^e no{m}ber of daies sith Chrystes incarnatio{n} vnto the ende of 1542[{1}] yeares. (besyde 385 dayes and 12 houres for lepe yeares).

_S._ Now wyll I proue by an other exa{m}ple, as this: 40 labourers (after 6 d. y^e day for eche man) haue wrought 28 dayes, I wold [*125a] know what theyr wages doth amou{n}t vnto: In this case muste I worke doublely: fyrst I must multyplye the nomber of the labourers by y^e wages of a man for one day, so wyll y^e charge of one daye amount: then secondarely shall I multyply that charge of one daye, by the hole nomber of dayes, {and} so wyll the hole summe appeare: fyrst therefore I shall set the su{m}mes thus.

+-------------- | ------+-------------- | ------+-------------- | ------+--o-o-o-o----- o | --o---+--------------

Where in the fyrste space is the multyplyer (y^t is one dayes wages for one man) {and} in the second space is set the nomber of the worke men to be multyplyed: the{n} saye I, 6 tymes 4 (reckenynge that second lyne as the lyne of vnites) maketh 24, for whiche summe I shulde set 2 counters in the thyrde lyne, and 4 in the seconde, therfore do I set 2 in the thyrde lyne, and let the 4 stand styll in the seconde lyne, thus.[*125b]

+-------------- | -----+-------------- | -----+--o-o--------- | -----+--o-o-o-o----- | -----+--------------

So apwereth the hole dayes wages to be 240d’. that is 20 s. Then do I multiply agayn the same summe by the no{m}ber of dayes and fyrste I sette the nombers, thus.

+-------------- | ---------+-------------- | ---------+--o-o--------- | --o-o----+--o-o-o-o----- o | --o-o-o--+-------------

The{n} bycause there are counters in dyuers lynes, I shall begynne with the hyghest, and take them vp, settynge for them the multyplyer so many tymes, as I toke vp counters, y^t is twyse, then wyll y^e su{m}me stande thus.

+-------------- | o -----+-------------- | o -----+--o----------- | -----+--o-o-o-o----- | -----+--------------

Then come I to y^e seconde lyne, and take vp those 4 cou{n}ters, settynge for them the multiplyer foure tymes, so wyll the hole summe appeare thus.[*126a]

+---------- | o -----+--o------- | o -----+--o-o----- | -----+--o-o----- | -----+----------

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

[Headnote: Division on the Counting Board.]

[Sidenote: Diuision.]

_M._ Now if you wold proue Multiplycatio{n}, the surest way is by Dyuision: therfore wyll I ouer passe it tyll I haue taught you y^e arte of Diuision, whiche you shall worke thus. Fyrste sette downe the Diuisor for feare of forgettynge, and then set the nomber that shalbe deuided, at y^e ryghte syde, so farre from the diuisor, that the quotient may be set betwene them: as for exa{m}ple: Yf 225 shepe cost 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 no{m}ber of [*126b] shepe, whiche is 225, these two nombers therfore I sette thus.

+-----+-------------- | | -------+-----+-------------- | | o --o-o--+-----+--o-o-o-o----- | | --o-o--+-----+-------------- o | | -------+-----+--------------

Then begynne I at the hyghest lyne of the diuident, and seke how often I may haue the diuisor therin, and that maye I do 4 tymes, then say I, 4 tymes 2 are 8, whyche yf I take from 9, there resteth but 1, thus

+-----------+-------- | | -------+-----------+-------- | | --o-o--+-----------+--o----- | | --o-o--+-----------+-------- o | | -------+--o-o-o-o--+--------

And bycause I founde the diuisor 4 tymes in the diuidente, I haue set (as you se) 4 in the myddle roume, which [*127a] is the place of the quotient: but now must I take the reste of the diuisor as often out of the remayner: therfore come I to the seconde lyne of the diuisor, sayeng 2 foure tymes make 8, take 8 from 10, {and} there resteth 2, thus.

+-----------+---------- | | -||-------+-----------+---------- | | -----o-o--+-----------+---------- | | -----o-o--+-----------+--o-o----- o | | ----------+--o-o-o-o--+----------

Then come I to the lowest nomber, which is 5, and multyply it 4 tymes, so is it 20, that take I from 20, and there remayneth nothynge, so that I se my quotient to be 4, whiche are in valewe shyllynges, for so was the diuident: and therby I knowe, that yf 225 shepe dyd coste 45 l’i. euery shepe coste 4 s.

_S._ This can I do, as you shall perceaue by this exa{m}ple: Yf 160 sowldyars do spende euery moneth 68 l’i. what spendeth eche man? Fyrst [*127b] bycause I can not diuide the 68 by 160, therfore I wyll turne the pou{n}des into pennes by multiplicacio{n}, so shall there be 16320 d’. Nowe muste I diuide this su{m}me by the nomber of sowldyars, therfore I set the{m} i{n} order, thus.

+-----+--o--------- | | o -||------+-----+--o--------- | | -----o---+-----+--o-o-o----- o | | -----o---+-----+--o-o------- | | ---------+-----+------------

Then begyn I at the hyghest place of the diuidente, sekynge my diuisor there, whiche I fynde ones, Therfore set I 1 in the nether lyne.

_M._ Not in the nether line of the hole summe, but in the nether lyne of that worke, whiche is the thyrde lyne.

_S._ So standeth it with reason.

_M._ Then thus do they stande.[*128a]

+-----+------------ | | -||------+-----+------------ | | -----o---+--o--+--o-o-o----- o | | -----o---+-----+--o-o------- | | ---------+-----+------------

Then seke I agayne in the reste, how often I may fynde my diuisor, and I se that in the 300 I myghte fynde 100 thre tymes, but then the 60 wyll not be so often founde in 20, therfore I take 2 for my quotient: then take I 100 twyse from 300, and there resteth 100, out of whiche with the 20 (that maketh 120) I may take 60 also twyse, and then standeth the nombers thus,

+-------+----- | | -||------+-------+----- | | -----o---+--o----+----- o | | -----o---+-------+----- | | ---------+--o-o--+-----

[*128b] where I haue sette the quotient 2 in the lowest lyne: So is euery sowldyars portion 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 fro{m} whiche you do take it: as by this example I wyll declare. Yf y^e purchace of 200 acres of ground dyd coste 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 ----o-o--+-----+--o----------- | | o -X-------+-----+--o-o-o-o----- | | o ---------+-----+--o----------- | | ---------+-----+-------------- | | ---------+-----+--------------

Fyrst set the diuident on the ryghte hande as it oughte, and then [*129a] the diuisor on the lefte hande agaynst those nombers, fro{m} which I entende to take hym fyrst as here you se, wher I haue set the diuisor two lynes hygher the{n} is theyr owne place.

_S._ This is lyke the order of diuision by the penne.

_M._ Truth you say, and nowe must I set y^e quotient of this worke in the thyrde lyne, for that is the lyne of vnities in respecte to the diuisor in this worke. Then I seke howe often the diuisor maye be founde in the diuident, {and} that I fynde 3 tymes, then set I 3 in the thyrde lyne for the quotient, and take awaye that 60000 fro{m} the diuident, and farther I do set the diuisor one line lower, as yow se here.

+---------+-------------- | | o -||--o-o--+---------+--o-o-o-o----- | | o ----------+--o-o-o--+----o--------- | | ----------+---------+-------------- | | ----------+---------+--------------

[*129b] And then seke I how often the diuisor wyll be taken from the nomber agaynste it, whiche wyll be 4 tymes and 1 remaynynge.

_S._ But what yf it chaunce that when the diuisor is so remoued, it can not be ones taken out of the diuident agaynste it?

_M._ Then must the diuisor be set in an other line lower.

_S._ So was it in diuision by the penne, and therfore was there a cypher set in the quotient: but howe shall that be noted here?

_M._ Here nedeth no token, for the lynes do represente the places: onely loke that you set your quotient in that place which standeth for vnities in respecte of the diuisor: but now to returne to the example, I fynde the diuisor 4 tymes in the diuidente, and 1 remaynynge, for 4 tymes 2 make 8, which I take from 9, and there resteth 1, as this figure sheweth:

+-----------+--------- | | -||--o-o--+-----------+--o------ | | o ----------+--o-o-o----+--o------ | | ----------+--o-o-o-o--+--------- | | ----------+-----------+---------

and in the myddle space for the quotient I set 4 in the seconde lyne, whiche is in this worke the place of vnities.[*130a] Then remoue I y^e diuisor to the next lower line, and seke how often I may haue it in the dyuident, which I may do here 8 tymes iust, and nothynge remayne, as in this fourme,

+-----------+----- | | -||--o-o--+-----------+----- | | ----------+--o-o-o----+----- | | ----------+--o-o-o-o--+----- | o | ----------+--o-o-o----+-----

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

_S._ Now resteth the profes of Multiplycatio{n}, and also of Diuisio{n}.

_M._ Ther best profes are eche [*130b] one by the other, for Multyplication is proued by Diuision, and Diuision by Multiplycation, as in the worke by the penne you learned.

_S._ Yf that be all, you shall not nede to repete agayne that, y^t was sufficye{n}tly taughte all redye: and excepte you wyll teache me any other feate, here maye you make an ende of this arte I suppose.

_M._ So wyll I do as touchynge hole nomber, and as for broken nomber, I wyll not trouble your wytte with it, tyll you haue practised this so well, y^t you be full perfecte, so that you nede not to doubte in any poynte that I haue taught you, and thenne maye I boldly enstructe you in y^e arte of fractions or broken no{m}ber, wherin I wyll also showe you the reasons of all that you haue nowe learned. But yet before I make an ende, I wyll showe you the order of co{m}men castyng, wher in are bothe pennes, shyllynges, and poundes, procedynge by no grounded reason, but onely by a receaued [*131a] fourme, and that dyuersly of dyuers men: for marchau{n}tes vse one fourme, and auditors an other:

[Headnote: Merchants’ Casting Counters.]

[Sidenote: Merchants’ casting.]

But fyrste for marchauntes fourme marke this example here,

o o o o o o o o o o o o o o o o o o o o o o

in which I haue expressed this summe 198 l’i.[{2}] 19 s. 11 d’. So that you maye se that the lowest lyne serueth for pe{n}nes, the next aboue for shyllynges, the thyrde for poundes, and the fourth for scores of pou{n}des. And farther you maye se, that the space betwene pennes and shyllynges may receaue but one counter (as all other spaces lyke wayes do) and that one standeth in that place for 6 d’. Lyke wayes betwene the shyllynges {and} the pou{n}des, one cou{n}ter standeth for 10 s. And betwene the poundes and 20 l’i. one counter standeth for 10 pou{n}des. But besyde those you maye see at the left syde of shyllynges, that one counter standeth alone, {and} betokeneth 5 s. [*131b] So agaynste the poundes, that one cou{n}ter standeth for 5 l’i. And agaynst the 20 poundes, the one counter standeth for 5 score pou{n}des, that is 100 l’i. so that euery syde counter is 5 tymes so moch as one of them agaynst whiche he standeth.

[Sidenote: Auditors’ casting.]

Now for the accompt of auditors take this example.

o o o o o o o o o o o o o o o o o o o o o o

where I haue expressed y^e same su{m}me 198 l’i. 19 s. 11 d’. But here you se the pe{n}nes stande toward y^e ryght hande, and the other encreasynge orderly towarde the lefte hande. Agayne you maye se, that auditours wyll make 2 lynes (yea and more) for pennes, shyllynges, {and} all other valewes, yf theyr summes extende therto. Also you se, that they set one counter at the ryght ende of eche rowe, whiche so set there standeth for 5 of that roume: and on [*132a] the lefte corner of the rowe it sta{n}deth for 10, of y^e same row. But now yf you wold adde other subtracte after any of both those sortes, yf you marke y^e order of y^t other feate which I taught you, you may easely do the same here without moch teachynge: for in Additio{n} you must fyrst set downe one su{m}me and to the same set the other orderly, and lyke maner yf you haue many: but in Subtraction you must sette downe fyrst the greatest summe, and from it must you abate that other euery denominatio{n} from his dewe place.

_S._ I do not doubte but with a lytell practise I shall attayne these bothe: but how shall I multiply and diuide after these fourmes?

_M._ You can not duely do none of both by these sortes, therfore in suche case, you must resort to your other artes.

_S._ Syr, yet I se not by these sortes how to expresse hu{n}dreddes, yf they excede one hundred, nother yet thousandes.

_M._ They that vse such accomptes that it excede 200 [*132b] in one summe, they sette no 5 at the lefte hande of the scores of poundes, but they set all the hundredes in an other farther rowe {and} 500 at the lefte hand therof, and the thousandes they set in a farther rowe yet, {and} at the lefte syde therof they sette the 5000, and in the space ouer they sette the 10000, and in a hygher rowe 20000, whiche all I haue expressed in this exa{m}ple,

o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o

which is 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: {and} more there can not be, for 4 farthynges [*133a] do make 1 d’. which must be set in his dewe place.

[Headnote: Auditors’ Casting Counters.]

And yf you desyre y^e same summe after audytors maner, lo here it is.

o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o

But in this thyng, you shall take this for suffycyent, and the reste you shall obserue as you maye se by the working of eche sorte: for the dyuers wittes of men haue inuented dyuers and sundry wayes almost vnnumerable. But one feate I shall teache you, whiche not only for the straungenes and secretnes is moche pleasaunt, but also for the good co{m}moditie of it ryghte worthy to be well marked. This feate hath ben vsed aboue 2000 yeares at the leaste, and yet was it neuer come{n}ly knowen, especyally in Englysshe it was neuer taughte yet. This is the arte of nombrynge on the hand, with diuers gestures of the fyngers, expressynge any summe conceaued in the [*133b] mynde. And fyrst to begynne, yf you wyll expresse any summe vnder 100, you shall expresse it with your lefte hande: and from 100 vnto 10000, you shall expresse it with your ryght hande, as here orderly by this table folowynge you may perceaue.

+¶ Here foloweth the table of the arte of the hande+

The arte of nombrynge by the hande.

[Transcriber’s Note:

Footnote 3 reads: “Bracket ([) denotes new paragraph in original.” For this e-text, the brackets have been omitted in favor of restoring the paragraph breaks. Changes of speaker (M, S) are also marked by paragraphs, as in the previous selection.

The illustration includes the printed page number 134; there is therefore no sidenote *134a. The sidenote for “4” is missing.]

[Sidenote: 1]

[*134b] In which as you may se 1 is expressed by y^e lyttle fynger of y^e lefte hande closely and harde croked.

[Sidenote: 2]

[{3}]2 is declared by lyke bowynge of the weddynge fynger (whiche is the nexte to the lyttell fynger) together with the lytell fynger.

[Sidenote: 3]

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

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

[Sidenote: 5, 6]

5 is represented by the myddle fynger onely bowed.

And 6 by the weddynge fynger only crooked: and this you may marke in these a certayne order. But now 7, 8, and 9, are expressed w{i}t{h} the bowynge of the same fyngers as are 1, 2, and 3, but after an other fourme.

[Sidenote: 7]

For 7 is declared by the bowynge of the lytell fynger, as is 1, saue that for 1 the fynger is clasped in, harde {and} [*135a] rounde, but for to expresse 7, you shall bowe the myddle ioynte of the lytell fynger only, and holde the other ioyntes streyght.

_S._ Yf you wyll geue me leue to expresse it after my rude maner, thus I vnderstand your meanyng: that 1 is expressed by crookynge in the lyttell fynger lyke the head of a bysshoppes bagle: and 7 is declared by the same fynger bowed lyke a gybbet.

_M._ So I perceaue, you vnderstande it.

[Sidenote: 8]

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

[Sidenote: 9, 10]

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

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

[Sidenote: 20]

And for to expresse 20, you must set your fyngers streyght, and the ende of your thombe to the partitio{n} of the [*135b] fore moste and myddle fynger.

[Sidenote: 30]

30 is represented by the ioynynge together of y^e headdes of the foremost fynger and the thombe.

[Sidenote: 40]

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

[Sidenote: 50]

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

[Sidenote: 60]

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

[Sidenote: 70]

70 is expressed by the bowynge of the foremost fynger, and settynge the ende of the thombe between the 2 foremost or hyghest ioyntes of it.

[Sidenote: 80]

80 is expressed by settynge of the foremost fynger crossewayes on the thombe, so that 80 dyffereth thus fro{m} 40, that for 80 the forefynger is set crosse on the thombe, and for 40 the thombe is set crosse ouer y^e forefinger.

[Sidenote: 90]

[*136a] 90 is signified, by bendynge the fore fynger, and settyng the ende of it in the innermost ioynte of y^e thombe, that is euen at the foote of it. And thus are all the no{m}bers ended vnder 100.

[Sidenote: 11, 12, 13, 21, 22, 23]

_S._ In dede these be all the nombers fro{m} 1 to 10, {and} then all the tenthes within 100, but this teacyed me not how to expresse 11, 12, 13, {et}c. 21, 22, 23, {et}c. and such lyke.

_M._ You can lytell vnderstande, yf you can not do that without teachynge: what is 11? is it not 10 and 1? then expresse 10 as you were taught, and 1 also, and that is 11: and for 12 expresse 10 and 2: for 23 set 20 and 3: and so for 68 you muste make 60 and there to 8: and so of all other sortes.

[Sidenote: 100]

But now yf you wolde represente 100 other any nomber aboue it, you muste do that with the ryghte hande, after this maner. [You must expresse 100 in the ryght hand, with the lytell fynger so bowed as you dyd expresse 1 in the left hand.

[Sidenote: 200]

[*136b] And as you expressed 2 in the lefte hande, the same fasshyon in the ryght hande doth declare 200.

[Sidenote: 300]

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

[Sidenote: 400]

The fourme of 4, for 400.