The welspring of sciences, which teacheth the perfecte worke and practise of arithmeticke both in vvhole numbers & fractions, with such easie and compendious instruction into the saide art, as hath not heretofore been by any set out nor laboured, : Beautified vvith most necessary rules and questions, not onely profitable for marchauntes, but also for all artificers, as in the table doth plainely appere..

Of Diuision the fift Chap.

DIuision or partition is, to seeke how many times one number doth conteine an other for in this operation are firste required twoo numbers for the fynding out of the thirde. The firste number is called the diuidende or number which is to be diuided & that muste be the greater number, the other number is called the diuisor, & that is the lesser. And the third number which we seke is called the quotiēt. As if I would diuide 36. by 9. the di∣uidend shall be .36. and the diuisour is 9. And for bicause that 9. is cōtei∣ned in 36. foure times, that is to say y^t 4. times 9. do make 36. The quo∣tient shall be 4. as in marking how many times 9. is conteined in .36.

¶ The practise.

Wryte downe fyrste the deui∣dende in the higher number, and the diuisour vnderneth, in suche

sorte, that the fyrst fygure of the di∣uisour towarde the lefte hande be vnder the fyrst of the diuidend and euery figure of the same diuisor vn∣der his like, that is to saye, the fyrst vnder the fyrst, the seconde vnder y^e seconde, the thirde vnder the third, and so consequentlye of the other, if there be any more, which is contra∣rye to the other three kindes before specifyed, but you muste consider if all the lower figures of the diuisor, maye be taken out of the higher fy∣gures of the diuidend, by the order of substraction. The which if you can not doe, then muste you set the fyrst fygure of the Diuisor (toward the lefte hande) vnder the seconde fygure of y^e diuidend, and so conse∣quently the reast, if any be to be set down euery one of them vnder his like as before is said. And thē draw a lyne betweene the diuidende and the diuisor. And at the ende of them an other crooked lyne, behinde the

which toward the right hand, shall be set your quotient. As by this ex∣ample folowing where the diuisor is but of one fygure.

If you woulde diuide 860. by 4. you must set downe 4. vnder the 8. with a line betwene them as here vnder you may see.

The diuidende	860
Diuisor.	4

And then you muste seeke howe many times the diuisor is cōteined in y^e higher nūber, or diuidend aun∣swering to him, as in this our exā∣ple I must seke how many times 4 is conteined in 8. in y^e which I finde 2. times, then I write downe 2. apart behinde the crooked line, as you se, which shalbe the first figure of the quotient to come, secondly by this fygure (being thus

860

4  (2

8

put apart) I must mul∣tiple the diuisor: and vn∣der the same multiplica∣tion. I must set that number which

commeth of the same multiplicati∣on as 2. times 4. doe make 8. which 8. I do set vnder the 4. which is the diuisor. Thirdly, I do substract the product of the sayde multiplication (of the quotiēt by the diuisor) from the higher number correspondant to the same, as if I abate 8. from 8. there remaineth nothing, and then I cansell or strike out that whych is done as you sée. In these three operations is comprehēded the arte of diuision. The which are to bee obserued frō point to point for ther is no diuersitie in the finishing of the same which is thus.

I must remoue my diuisor one place nerer toward my right hand: as in proceding with our

2

260 (21

4

exāple I remoue my de∣uisor 4. which was vn∣der 8. and I set it vnder 6. then I seeke how many times 4. is conteined in 6. where I fynde but one time thē I set 1. behinde y^e croo∣ked

line behinde 2. afterwarde by this last & new figure 1. I multiplie the diuisor 4. & that maketh but 4. (for an vnity which is but 1. encrea∣seth nothing) I abate 4. from the higher figure 6. and there resteth 2. y^e which 2. I set ouer the 6. & I can∣cell the 6. for so must you doe when there resteth anye thing after you haue made y^e substraction. Thirdly for y^t there yet remaineth another fygure in the diuidend, I remoue again the diuisor, and I set it vnder the cipher 0. Then I seeke how ma∣ny times 4. is in the higher nūber which is 20. where I

2

860 (215

4

20

fynde 5. times, I put 5. behinde y^e crooked line for the thirde and laste figure of the quotient. Then by the same 5. I multiply y^e diuisor 4. and y^t maketh 20. the which I abate frō y^e higher number, and there resteth nothing. And so is this diuision en∣ded: & I haue founde y^t. 860. being

diuided by 4. bringeth for the quo∣tient 215. that is to say, that 4. is cō∣teyned in 860. twoo hundreth & fyf∣tene tymes. Thys is the most easi∣est working that is in diuision, but that which foloweth, apperteyneth to the whole and perfect vnderstan∣ding of the same. When the fyrste fygure of your diuisor toward your left hande is greater than the fyrst of y^e diuidende, you must not place the fyrste fygure of your diuisour right vnderneth the fyrst of the di∣uidende, but vnder the seconde fy∣gure of the same diuidende, neerer to your right hāde, as before is said. Whē y^e diuisor is of many figures, and y^t you haue to seeke how many times it is conteined in the higher number (for the more easyer wor∣king) you muste not seeke to abate the diuisor all at one time, but you muste see and marke howe many times the fyrst fygure of the same towarde the lefte hande is con∣teyned

in the hygher number aun∣swering to the said number, & then to worke after the same maner as is before taught.

Exāple, I haue 316215. crownes to be deuided amonge 45. men for to make my diuision I muste not put the fyrste fygure of the diuisor which is 4. vnder the first of the di∣uidende, which is 3. bicause that 4. is greater number than 3. And fur∣ther, I cannot take 4. out of three, wherefore I must set the 4. vnder the seconde fygure of the hygher number which is 1. and the fygure 5. of the diuisor next right vnder the 6. as you may see.

316215

45

I must first seeke, how many times 45. is cō∣teyned in 316. which is but parte of the diui∣dende, wherefore for the more easie working I neede but to seeke howe many times 4. is conteined in 31. & bicause I may haue it seauē tymes

I put 7. behinde the crooked line, as is aforesaide, then by 7. I multiply all the diuisor 45. and they are 315: the which I set vnder y^e same diui∣sor, the fyrst fygure vnder the fyrst: And the other in order towarde the lefte hande. Then I substract. 315. from the higher number 316. and of this fyrst working there remaineth but 1. the which I set o∣uer

1

316 215

45 (7

315

the 6. and I cancell the. 315. & the other fy∣gures 3, 1, 6, and also the diuisor: and then it wil stande thus.

And when I come to remoue the diuisor, and that I must seeke how many times it is conteined in the higher number, if I see that I can∣not fynde it there, that is to say that if the higher number be lesser than the diuisor, as it is in this example, then must I put a cipher in y^e quo∣tient behinde the crooked line, & if there remain any figures in y^e diui∣dende

whiche are not finished, I must remoue the diuisor agayne nerer to∣warde my right hande by one place, for to finde a newe fygure in the quo∣tient. As in this our example, for af∣ter that I haue remoued the diuisor,

1

52 6215

45 (70

I séeke howe many times. 45. is contei∣ned in. 12. and be∣cause I cannot haue 45. in 12. I put a 0. behinde the croked line after 7. then without multipliyng or abatyng, I remoue againe the diuisor nerer to∣warde my ryght hande, and I seeke howe many tymes 4. (whiche is the first figure of the diuisor) is in the hy∣gher

1

316215

45 (703

135

numbre, that is to say, in 12. whereas I finde it 3. tymes: I put 3. behinde the crooked lyne, for the thyrde fygure of the quotient: then by 3. I multiplie the diuisor. 45. and therof commeth. 135.

But here is to bee noted, that if it happen that the fygure beynge laste founde whiche is put in the quotient, doe produce or bringe foorthe a grea∣ter noumbre (in multipliyng all the dyuisor by the same) then that whiche is ouer the saied diuisor: you muste then make the same figure of youre quotient (whiche you doe put downe) lesser by one: and after that you haue cancelled the firste multiplication, you must make a newe. And the same must bee so doone as often times: as (in decreasing the same) it produceth a lesser noumbre, or at the least, a noumbre egall to that whiche is ouer it. As in the laste woorke: for because that the diuisor, being multiplied by 3. bringeth foorthe. 135. whiche a∣mounteth more then. 121. the same producte must be cancelled. And like∣wyse the figure. 3. whiche I did put in the quotient, must bee chaunged into a figure of 2. Then by the saied 2. I must multiplie the diuisor. 45. and

thereof commeth 90, the whiche I a∣bate from. 121. and there remaineth. 31. And then wil y^e somme stand thus.

13

316215

45 2

135 (703

90

And here is also to bee noted that the somme whiche remaineth must be alwayes lesser then the diuisor. Thē finally I remoue the diuisor to the 2. last figures towarde the right hande, and I seeke howe many times 4. is in 31. And for because I finde it 7. times, I put 7. in the quotient: by the whiche I multiplie the diuisor, and thereof commeth 315. the whiche I abate frō the hygher noumbre of the diuidend, and there remaineth nothing as here you may see.

13

516215

45 (7027

515

But in case that after the diuision is ended, there doe remaine any thing in the diuidende, as most often times there dothe: I must them sette that re∣maine aparte behinde the croked line after the entier quotient, and the di∣uisor right vnder the same remayne, with a lyne betwene them bothe, as in this diuision followyng, where there remayneth 3. in the last woorke of the same. And we shal sée what the same doth signifie, whē we shal treate of fractions or broken numbres.

〈 math 〉〈 math 〉

In summe, all the whole practise of diuision may bée kepte in remem∣braunce by thrée lettres, that is to say: S. M. A. whiche thrée letters doe sig∣nifie to seeke, to multiplie, to abate.

First, I must séeke howe manye times the diuisor is conteined in the higher numbre: then, by the quotient (whiche I finde) I must multiplie the diuisor: finally, I must abate the pro∣duct of that multiplication, from the higher numbre to the same correspon∣dent, that is to saye: out of the diui∣dende, aunswering to the diuisor.

And further, besides this kynde of woorkyng in diuision. The whiche is reguler and commune: I wyll here put an other manner of woorkynge very easye. The whiche shall serue for soche diuisions as are difficyll to bée wrought. That is to wytte, when the number to bee diuided is verye great, and the diuisor great also, and it shall serue againe for to auoyde er∣rour in supputacion, and for the pla∣cing

of fewer figures in the quotient: and consequently it shall saue muche labour vnto them whiche as yet haue muche studied in this arte. The prac∣tise whereof is thus, as foloweth.

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

And thus you shall at no time putte any noumbre in your quotient which exceadeth the fygure of. 9. that is to saie any noumbre being greater then 9. for to come then vnto our practise, wryte downe your dyuisor one time: and behynde it towarde your ryghte hande, drawe a lyne downe straighte, and right against the same diuisor be∣hinde the lyne put this fygure 1. Thē

double your saied diuisor, and righte against the same (beyng doubled) put behinde the lyne the fygure of 2. Af∣ter, adde vnto the same noumbre (whiche you doubled) your saied di∣uisor and right against the same pro∣duct, behinde the line put the fygure of. 3. And vnto this thyrde producte, you muste adde againe your diuisor: and ryght against the same producte behinde the lyne sette the fygure. 4. And this muste you dooe, vntill you come to the fygure of. 9: in soche sorte that euerye of the productes dooe sur∣mounte so muche his former noum∣bre, as all the diuysor dothe amounte vnto: placing at the right syde of eue∣rye producte behynde the lyne, the noumbre whiche signifieth howe muche he is in order. That is to saye, right against the fifte producte, you must put. 5. right against the sixte pro¦ducte, you muste put. 6: And so like∣wyse of all the other.

Example of the diuisor proponed,

643. first, I wryte downe 643. and

643	1
1286	2
1929	3
2572	4
3215	5
3858	6
4501	7
5144	8
5787	9

right against the same behind the lyne I put .1. se∣condely, I dou∣ble 643. and thei make 1286 & right againste him behinde the lyne I put .2. Thirdlye, vnto that same 1286. I adde the diuisor 643. and they are 1929. and right against the same I sette 3. Fourthely, vnto the sayed 1929. I adde the diuisor 643. and they are 2572. and right against the same I put .4. And thus must you doe alwayes by encreasing so muche euery product, as the diuisor dothe a∣mounte vnto, vntyll you haue so done nyne tymes, as you sée in this pre∣sent table.

This being done, you must sette downe your diuisor vnder the diui∣dent

after the same maner as is be∣fore declared: that is to saye, 643. vnder the thrée first figures of the di∣uidende towarde your right hande, which are .789. Then must you seke howe many times .643. are contey∣ned in .789. And for to knowe the same, I looke in my foresaied table if I may there finde the same numbres, 789. the whiche is not there: There∣fore I must take a lesser noumbre the nerest to it in quantitie that I can finde in the table, the whiche is .643. whiche noumbre hathe against it on the ryght hande of the lyne this diget. 1. Then I take the sayed 1. and I put it behynde the croked lyne, for the first fygure of the quotient.

Then I dooe abate .643. from 789. and there remayneth .146. the whiche I put ouer the .789. and I cancell the 789. and thus is the fyrst operation ended. Then I sette for∣warde the deuisour one fygure ne∣rer to my ryghte hande, and I séeke

a newe quotient as I soughte this, where I finde the higher noumber o∣uer my diuisour to bee 1464. The whiche I doe séeke in the table, and because I can not fynde it there, I take a lesser noumbre, the nighest to it that I can finde, and that is 1286: whiche noumbre hathe against it this digette .2. I put .2. for the seconde fi∣gure of the quotient behinde the line, and I doe abate 1286. from .1464. and there remaineth .178. Thirdly, I remoue forewarde the diuisour, as before, and I finde the higher noum∣bre to bee .1786. and that the nexte lesser noumbre to it in my table, is a∣gaine .1286. I put therefore ones againe .2. in the quotient for the third figure: and I abate .1286. from, 1786. and there remaineth .500. Fourthly, I set forward the diuisour, and the higher noumbre ouer it, is 5005. and the next lesser noumbre to it in my table is .4501. right against the whiche noumbre is .7. I put my .7

in the quotient, for the fourth figure. And after that I haue abated .4501. from .5005. there remayneth .504. Finally, I remoue forwarde my diui∣sor vnto the last place: and I finde the higher noumbre to bee .5048. And the nexte lesser noumbre to it in my table, is. 4501. I set .7. againe in the quotient, for the fifte and last fygure. Then I take .4501. from .5048. and there remaineth .547. whiche must bee put at the ende of the whole quotiēt with the diuisor vnder it, and a lyne betwene them in this maner folowing.

(1227 547/043

¶ The summe of diuision.

WHen you would diuide any nū∣bre by .10. you must take away y^e last figure next towardes your right hand & the rest shalbe y^e quotient. As if you would diuide .46845. by .10. take away the .5. and then .4684. shal bee the quotient, and the .5, shalbe the nombre that doth remaine. Likewise

when you woulde diuide any num∣bre by 100. take awaye the twoo last figures towardes your right hande, and if you woulde diuide by .1000. take away thrée figures, if by .10000. take away foure figures. And so of all other, when the first figure of the dy∣uisor towarde the lefte hande shalbe onely 1. and the rest of the same diuisor being but cyphers.

¶Here foloweth the proofes of addicion, substraction, mul∣tiplication, and diui∣sion.

¶The profe of Addition.

WHen you would proue whe¦ther your addicion be well made, consider the figures of the noumber whiche bée added, euery one in his simple value: not hauinge any regarde to the place where he standeth, but to recken him as though he were alone by himselfe

and then recken them all, one after an other, casting away from them the noumber of 9. as ofte as you maye.

And after your discourse made, kepe in minde the same figure which remaineth after the nynes bee taken away, or set the same in a voyde place at the vpper ende of a line. For if your addicion be well made, the like figure will remaine, after that you haue ta∣ken away all the nines, out of the to∣tall summe of the

24567	2
5329
431
30377	2

same addicion, as of ten as you mayther finde any: as in this addiciō whiche here you se. Ther remai¦neth .2. for echepart.

¶The profe of substraction.

ADde the noumbre whiche you doe substract with that numbre which remaineth after the substraction, is made: and if the totall somme of that addicion, be like vnto the nombre frō the whiche the substraction was made

you haue done well, o∣therwyse

5463

3584

1879

5463

not: as in this example doth appeare, where you sée the nom∣bre whiche is to be sub∣stracted is, 3584. and the noumbre whiche doth remayne, is .1879. the whiche twoo summes being added together, dooe make .5463. whiche is like to the hi∣gher noumbre, out of the whiche the substraction was made, as before is saied.

¶The profe of multiplication.

THe profe of multiplication is made by the helpe of diuision, for if you di¦uide the nombre produced of the mul∣tiplicatiō, by the multiplier: you shall finde the higher noumbre, whiche is the multiplicande.

¶The profe of diuision.

TO knowe if your diuision be well made: you must multiplie all the quotient by your diuisor, and if any thinge remained after your diuision

was made. Thesame shall you adde vnto the producte whiche commeth of the multiplication: and you shal finde the like nombre vnto your diuidend if you haue wel diuided: otherwise not.

Notes

• A note.
• A note.