The vvell spryng of sciences whiche teacheth the perfecte woorke and practise of arithmeticke, bothe in whole nombers and fractions, with suche easie and compendious instruction into the said arte, as hath not heretofore been by any sette out nor laboured. Beautified with moste necessary rules and questions, not onely profitable for marchauntes, but also for all artificers, as in the table doeth partlie appere: set forthe by Humfrey Baker citezeine of Lo[n]don.

Of diuition the 5. chapiter.

DIuition or partition is, to seeke how many times one numbre dothe containe an other for in thys operation at first required two numbers for the findyng out of the thirde. The firste numbre is called the diuidende or numbre whiche is to bee deuided, and that muste bee the greater numbre, the other numbre is called the diuisor, and that is the lesser. And the thirde numbre which we seeke is called the quotient. As if I woulde deuide 36. by 9. the diuidend shal be 36. and the diui¦sour is 9. And for because that nine is conteined in 36. foure times, that is to saie, that 4: times 9. do make 36. The quotient shalbe 4. as in marking how many tymes 9. is conteined in 36.

The practise.

Write doune firste the diuidende in the hygher number, and the diuisor

vnderneth, in suche sorte, that the first fygure of the diuisour towarde the lefte hande be vnder the first of the di∣uidend, and euerie figure of the same diuisour vnder hys lyke, that is to say, the fyrste vnder the firste, the seconde vnder the seconde, the third vnder the thyrde, and so consequentlye of the o∣ther, yf there bée anie more, whiche is contrarie to the other three kyndes be∣fore specified, but you muste consyder if all the lower fygures of the diuisor, maye be taken out of the higher figu∣res of the diuidence, by the order of substraction. The whiche if you canne not doe, then muste you sette the fyrste fygure of the Dyuysour (towarde the lefte hande) vnder the seconde figure of the diuidende, and soe consequently the reste, if anie bée to be sette doune euerie one of them vnder his like as before is saied. And than drawe a line betwene the diuidence and the diui∣sour. And at the ende of them an other croked line, behinde the whiche to∣warde

the right hand, shal be set your quotient. As by this example follow∣ing wher the diuisor is but of 1. figure

If you woulde diuide 860. by. 4. you muste sette doune. 4. vnder the. 8. with a line betwene them as herevn∣der you may see. 〈 math 〉〈 math 〉

And then you muste seeke howe many times the diuisour in contained in the higher numbre, or diuidende aunswering to him, as in this our ex∣ample I muste seke how many tymes 4. is contained in. 8. in the whiche I finde 2. tymes, then I wryte doune. 2. aparte behinde the crooked lyne, as you se, whiche shall be the firste figure of the quotient to come, secondlye by this figure (beeynge thus putte aparte) I must multiply the di∣uisor: 〈 math 〉〈 math 〉 and vnder the same multiplication.

I muste sette that number whiche

commcth of the same multiplication, as two tymes foure doe make eight, whiche eight I doe set vnder the four, whiche is the diuisour. Thirdlie I doe substract the producte of the saied mul∣tiplication (of the quotient by the diui∣sor) from the higher numbre corespon∣dant to the same, as if I abate 8. from 8. there rematneth nothyng, and then I cancell or stryke out that whiche is doen as you see. In these three operati∣ons is comprehended the art of diuiti∣on. The which ar to be obseraed from point to point, for there is no diuersity in the finishyng of the same whiche is thus.

I muste remoue mye diuisor one place nerer towarde my right hande, as in procedyng with. oure exāple, I remoue 〈 math 〉〈 math 〉 my diuisour 4. whiche was vnder. 8. and I set it vnder. 6. then I seeke howe manye tymes. 4. is conteined in. 6. where I finde but one tyme, then I sette. 1. be∣hinde

the croked line behinde. 2. after∣warde by this laste and new figure. 1. I multiplie the diuisour. 4 and that maketh but. 4 (for an vninitie whiche is but. 1. encreaseth nothyng) I abate 4. from the higher figure. 6. and there resteth. 2. the whiche 2. I sette ouer the 6. and I cancell the. 6. for so must you do when there resteth any thing after you haue made the substractiō. Third∣lye for that there yet remaineth an o∣ther, figure in the diuidend, I remoue againe the diuisour, and I sett vnder the cipher. 0. Then I seeke how many tymes foure is in the higher number whiche is. 10. where I fynde 5. times, I put. 5. behinde 〈 math 〉〈 math 〉 the crooked iyne for the third and last figure of the quotient. Then by the same 5. I mul∣tiplie the diuisour 4. and that maketh 20. the whiche I abate from the high∣er number, and there resteth nothing. And so is thys diuisyon ended: and I

haue founde that. 860. being diuided by foure, bringeth for the quotient 215. that is to saie, that. 4. is contei∣ned in 860. twoo hundred and fiftene tymes. This is the moste eastest wor∣kynge that is in diuision, but that whiche foloweth, appertayneth to the whole and perfecte vnderstandyng of the same. When the firste figure of your diuisor towarde your lefte hande is greater then the firste of diui∣dende, you must not place the firste fi∣gure of your diuisor right vnderneth the first of your diuidend, but vnder y^• 2. figure of the same diuidende, nerer to your right hand, as before is saied.

When the diuisour is of manye figures, and that you haue to séeke howe manye tymes it is contained in the higher numbre (for the more easier workyng) you must uot seeke to abate the diuisour all at one tyme, but you muste see and marke howe maye ty∣mes the firste figure of the same to∣warde the leste hande is contained in

the higher numbre aunsweringe to the saied numbre, and then to work af¦ter y^• same maner as is before taught.

Erample. I haue. 316215. crow∣nes to be deuided amonge. 45. menne for to make my diuision, I muste not putte the firste figure of the diuisour whiche is. 4. vnder the firste of the de∣uidende, whiche is. 3. because that. 4. is greater number then. 3. And fur∣ther, I can not take. 4. out of. 3. wher∣fore I muste sette the. 4. vnder the se∣conde figure of the higher number whiche is. 1. and the fygure. 5. of the di¦uisour next right vnder the 6. as you maie sée.

I muste firste séeke, home many tymes 〈 math 〉〈 math 〉 45. ie contayned in 316. whiche is but parte of the diuidende, wherefore for the more casie workyng I nede but to seke how many tymes 4. is conteined in 31. & because I may haue it 7. tymes I put. 7. behinde the croked line, as is

afore saied, then by. 7. I multiply all the diuisour. 45. and they are 315: the whiche I set vnder the same diuisour, the fyrste figure vnder the fyrste. And the other in order towarde the lefte hande. Then I substract thre hundred fiftine, from the higher number. 316. and of this fyrst working there remai∣neth but. 1. the whiche I sette ouer the 6 and I cancell the 315, and the other figures 3, 1, 6, also the diuisour: and then it will stande thus. 〈 math 〉〈 math 〉

And when I come to re∣moue the diuisour, and that I muste séeke howe manye tymes it is contai∣ned in the higher numbre, if I se that I can not finde it there, that is to saie that if the higher numbre be lesser thā the diuisonr, as it is in this example, then must I put a cipher in the quoti∣ent behind the croked line, and if ther remaine anye fygures in the diui∣dende

whiche are not finished, I must remoue t〈…〉〈…〉 diuisor againe nerer to∣warde my right hande by one place, for to finde a newe figure in the quo∣ciente. As in this our example, for af∣ter that I haue remoued the diuisor, I seke how many tymes. 45. is con∣teined 〈 math 〉〈 math 〉 in. 12. and because I can not haue. 45. in. 12. I put a 0. behinde the croked line after 7. then without multipliyng or aba∣tyng, I remoue again the diuisor ne∣rer toward my right hand, and I seke how many tymes 4 (which is the first figure of the diuisour) is in the higher nomber, that is to saie, in. 12. where∣as 〈 math 〉〈 math 〉 I find it 3. times I putte. 3. behinde the croked line, for the thirde figure of the quocient: then by. 3. I multiply the diuisor. 45. and thereof commeth 135.

But here is to bee noted, that if it happen that the figure beyng laste founde, whiche is put in the quotient, doe produce or bryng for the a greater number (in multipliyng al the diuisor by the same) then that whiche is ouer the saied diuisor: you must then make the same figure of your quotient (whi∣che you dooe put doune) lesser by one. and after that you haue cancelled the firste multiplication, you muste make a newe. And the same must be so doen as often tymes: as (in decreasyng the same) it produceth a lesser number, or at the leaste, a number egall to that, whiche is ouer it. As in the laste work for because that the diuisour, beyng multiplied by. 3. bryngeth for the 135. whiche amounteth more then 121. the same producte must be cancelled. And likewise the figure. 3. whiche I did put in the quotiēt, must be chaunged into a figure of. 2. Then by the saied. 2. I must multiplie the diuisor. 45. & therof commeth 90. the whiche I abate from

121. and there remaineth. 31. And then will the somme stande thus. 〈 math 〉〈 math 〉

And here is also to be noted, that the * 1.1 somme whiche remaineth, must be al∣waies lesser then the diuisor. Then fi∣nally, I remoue the diuisor to the. 2. last figures towarde the right hande, and I seeke how many tymes 4. is in 31. And for because I finde it. 7. times I put. 7. in the quotient: by the which I multiplie the diuisoure, and thereof commeth. 315 the whiche I abate frō the higher number of the diuidende, and there remaineth nothyng, as here you maie see. 〈 math 〉〈 math 〉

But in case that after the diuision is ended, there doe remaine any thyng in the diuidende, as moste often times there doeth: I must then sette that re∣maine aparte behinde the croked line, after the entier quotiente, and the di∣uisor, right vnder thesame remaine, with a line betwene them bothe, as in this diuision followyng, where there remaineth. 3. in the last woorke of the same. And we shall see what the same doeth signifie, when we shall treate of fractions, or broken nombers. 〈 math 〉〈 math 〉

In somme, all the whole practise of diuision, maie be kept in remēbraūce by three letters, that is to saie: S. M. A whiche three letters dooe signifie to seeke, to multiplie, to abate.

Firste, I must seeke how many ty∣mes the diuisor is conteined in the hi∣gher nomber: then, by the quotiente (whiche I finde) I must multiplie the diuisor: finally, I must abate the pro∣ducte of that multiplicacion, from the higher nomber to thesame correspon∣dent, that is to saie: out of the diuidēde aunsweryng to the diuisor.

And further, besides this kinde of woorkyng in diuision. The whiche is reguler and common: I will here put an other maner of woorkyng verie easie. The whiche shall serue for suche diuisiōs as are difficil to be wrought. That is to witte, when the nomber to bee diuided is verie greate, and the diuisor greate also, and it shall serue againe for to auoide errour in suppu∣tacion, and for the placyng of fewer

figures in the quotiente: and conse∣quently it shall saue muche labor vn∣to thē, whiche as yet haue not muche studied in this art. The practise wher∣of is thus, as followeth.

I haue to deuide. 7894658. by 643. In the firste place, you shall vn∣derstande, that although the firste fi∣gure of the diuisor towarde your lefte hande, maie bee founde many tymes in the higher number, as. 10. tymes, 12. tymes or more: yet is it so, that you must neuer putte but one figure one∣ly at a tyme in your quotient.

And thus you shall at no tyme put any number in your quotient, whiche exceadeth the figure of. 9. that is to saie, any number beyng greater then 9. for to come then vnto your practise, write donne your diuisour one tyme: and behinde it towarde youre righte hande, drawe a line doune straighte, and right against thesame diuisor be∣hinde the line put this figure. 1. Then double your saied diuisour, and right

againste the same (beyng doubled) put behinde the line the figure of. 2. After adde vnto the same number (whiche you doubled) your saied diuisour, and right againste the same producte, be∣hinde the line putte the figure of 3. And vnto this thirde producte, you muste adde againe your diuisor: and right againste the same producte, be∣hinde the line sette the figure. 4. And this muste you dooe, vntill you come to the figure of. 9: in suche sorte that euery of the productes doe surmounte so muche his former noumber, as all the diuisoure dooeth amounte vnto: placyng at the right side of euery pro∣ducte behinde the line, the noumber whiche signifieth howe muche he is in order. That is to saie, righte a∣gainste the fifte producte, you muste putte. 5. right againste the sixte pro∣ducte, you must put. 6: and so likewise of all the other.

Example of the diuisour proponed, 643. firste, I write doune. 643. and

right against the same behinde the 〈 math 〉〈 math 〉 line, I put. 1. se∣condly, I double 643. and they make. 1286. and righte againste hym behinde the line I put. 2.

Thirdly, vnto that same. 1286. I adde the diuisor. 643. and thei are 1929. and right againste the same I set. 3. Fourthly, vnto the saied 1929. I adde the diuisor. 643. and thei are 2572. and right againste the same I putte. 4. And thus must you dooe al∣waies by encreasyng so muche euery producte, as the diuisor doeth amount vnto, vntill you haue so doen nine ty∣mes, as you see in this present table.

This beyng dooen, you muste sette downe your diuisor vnder the diui∣dende, after the same maner as is be∣fore declared: that is to saie. 643. vn∣der

the three firste figures of the diui∣dende, towarde your righte hande, which are. 789: Then must you seke how many times. 643. are conteined in. 789: And for to knowe the same, I looke in my foresaied table, if I maie there finde the same nombers. 789. the whiche is not there: The refore I must take a lesser nomber, the nereste to it in quantitie, that I can finde in the table, the whiche is. 643. whiche nomber hath againste it on the righte hande of the line this diget. 1. Then I take the said. 1. and I put it behind the croked line, for the firste figure of the quotiente.

Then I doe abate. 643. from. 789 and there remaineth. 146. the which I putte ouer the. 789. and I cancell the. 789. and thus is the firste opera∣cion ended. Then I sette forewarde the diuisour, one figure nerer to my right hande, and I seeke a newe quo∣tient, as I sought this, where I finde the higher number ouer my diuisour

to bée. 1464. The whiche I doe seeke in the table, and because I can not finde it there, I take a lesser number, the nigh••st to it that I can finde, and that is. 1286: whiche number hath a∣gainste it this digette. 2. I putte. 2. for the seconde figure of the quotient be∣hinde the line, and I dooe abate. 1286. from. 1464. and there remaineth. 178

Thirdly, I remone forward the di∣uisor, as before, and I finde the higher number to be. 1786. and that the next lesser number to it in my table, is a∣gaine. 1286. I putte the refore ones a∣gaine. 2. in the quotient for the thirde figure: and I abate. 1286. from. 1786. and there remaineth. 500.

Fourthly, I set forward the diuisor and the higher nūber ouer it, is. 5005 and the next lesser number to it in my table is. 4501. right against the which noumber is. 7. I putte. 7. in the quo∣tiente, for the fowerth figure. And af∣ter that I haue abated. 4501. from 5005. there remaineth. 504.

Finally, I remoue forward my di∣uisor vnto the laste place: and I finde the higher number to bee. 5048. And the nexte lesser noumber to it in my table, is. 4501. I sette. 7. againe in the quotient, for the fifte and laste figure. Then I take. 4501. from. 5048. and there remaineth. 547. whiche must be put at the ende of the whole quotieute with the diuisor vnder it, and a line be¦twene them, in this maner folowing. 〈 math 〉〈 math 〉

¶ The somme of deuision.

WHen you would deuide any number by. 10. you muste take awaie the laste figure nexte towardes your right hand, and the rest shalbe the quotient. As if you would diuide. 46845 by 10. take awaie the. 5. & then. 4684. shal∣be the quotiente, and the. 5. shalbe the number that doeth remain. Likewise when you would diuide any number

by. 100. take awaie the twoo laste fi∣gures towardes your righte hande, and if you would diuide by. 1000 take awaie thre figures, if by. 10000. take awaie fower figures. And so of all o∣ther, when the first figure of the diui∣sor toward the lefte hande, shalbe one∣ly. 1. and the reste of the same diuisour beyng but ryphers.

¶ Here foloweth the proofes of addition, substraction, mul∣tiplication, and diuision.

¶ The proofe of addition.

WHen you would proue, whether your additiō boe well made, consider the figures of the numbers, whiche bee added, euery one in his simple value: not hauyng any regarde to the place where he stā∣deth, but to recken hym as though he were alone by hymself, and then rec∣ken them all, one after an other, ca∣sting

awaie fram them the number of 9. as ofte as you maie.

And after youre discourse made, kepe in minde thesame figure whiche remaineth after the nines be taken a∣waie, or sette thesame in a voide place at the vpper ende of a line. For if your addition be well made, the like figure will remaine, after that you haue ta∣ken awaie all the nines, out of the to∣talle somme of the∣same addition, as of∣ten 〈 math 〉〈 math 〉 as you may ther finde any: as in this addition which here you se. There remai∣neth. 2. for eche part

¶ The proofe of substraction.

ADde the number whiche you doe substracte, with that num∣ber which remaineth after the substraction is made: and if the totall somme of that addition, bee like vnto the number from the whiche the sub∣straction

was made, you haue dooen well, otherwise not: as in this example dooeth ap∣peare, 〈 math 〉〈 math 〉 where you see the number whiche is to bee substracted, is. 3584. and the number whiche doeth remaine, is 1879. the whiche twoo sommes beyng added together, doe make 5463. which is like to the higher number, out of the which the substraction was made, as before is sated.

¶ The proofe of multiplication.

THe proofe of multiplication, is made by the help of diuision, for if you diuide the number produ∣ced of the multiplication, by the mul∣tiplier: you shal finde the higher num∣ber, whiche is the multiplicande.

¶ The proofe of diuision.

TO knowe if your diuision be wel made: you must multiplie all the quotiente by your diuisor, and if any thyng remained after youre diuision

was made. The same shall you adde vnto the producte, whiche commeth of the multiplication: and you shal finde the like number vnto your diuidende if you haue wel diuided: otherwise not

Notes

• * 1.1

  A note.