THE VVell spryng of sciences, whiche teacheth the perfecte woorke and practise of Arithmeticke, bothe in whole nombers and fractions, vvith suche easie and compendious instruction into the said Arte, a [...] hath not heretofore been [...] 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 for the by Humfrey Baker Citezeine of Lōdon.

¶ Imprinted at London, by Ihon Kyng [...]ton, for Iames Rowbothum.

M. D. LXVIII.

# ¶ TO THE RIGHTE worshipfull Maister Ihon Fitz▪ williams Gouernour of the most famous societie of Marchauntes Aduenturers into Flaunders. And to the righte VVorshipfull the Consult, Assistens, and comminaltie of the same societie, Humfrey Baker wisheth health with continuall increase of commoditie by their worthie trauaile.

ALthough heretofore (right worshipfull) diuers and sundrye wel skilled men in y^{•} feate, and science of numbrynge, haue not spared their labour, not onelie in beatifiyng, but also in augmentyng the same with such increase of witte, as hath excedinglye contained the studientes therein, and made them riper, in the exacte knowledge and redinesse of it. Yet because an arte can not haue to muche setting furthe, neither can be to muche commonded, I (amonge, and after so man [...] pregnaunt [Page] wittes) am bolde to putte forthe the fruicte of my blunt vnderstandyng and trauaile, in the like science, not correctyng or reprouyng any mans former edition, but makyng an attempte of my grosse, and meane studie herein, to the entent that thys my enterprise maye (by other mennes doinges (bee either gentlie amended, or frendlie receiued. And because I desire speciallie that it may bee gratfullie accepted, I haue chosen you (worshipful) to be as defendours of this little lucubration, to whom I knowe it shall bee welcome, not so muche for my sake, as for the science loue it selfe, wherin it is wel knowen you haue ripe iudgement and good vnderstandyng, as by youre daylie exercises it maie appeare. And although some folishe heades of fansie ouerthwart, thinke this Arithmetical art peculier onely to a fewe, vsyng the trade of marchaundise, as though it were not so necessarie for other men: yet I am sure, the wisest and best learned [Page] perceiuyng the wonderfull art, the depe diuises, and the cunnyng conclusions that are comprised in it: affirme it (that of right) to bee the beste whetstone, or sharpnyng of the witte of euerie man that euer was inuented, & thinke it most necessary to be taught vnto children, without the whiche, nothinge either priuate or common can bee well ordered. And true it is, that whosoeuer is ignorant of this science of numbring, he stepeth in all the reste and lacketh that promptitude of witt, whiche other haue in callynge and accomptynge of greate summes. Yea, I saie he that hath the exact knowledge of it, hath a speciall gift of God, and carieth about hym a note and token of a good witte: and I doubt whether euer any was counted or estemed simple or folishe, that had this science in his head Nowe if some curiours braine would aske me this question: six you cōmende much your art of numbering, can you tell who was the firste inuentor of it, or [Page] what this worde Arithmetica doth signifie, I would answere hym that it is vncertaine who inuented it firste, but well I knowe, it is one of the seuen liberal sciences: which, if I say god gaue vnto man to adourne his life with all, I saie as it is, & as wee ought to beleue of whom we acknowledge to haue receiued them & not of the Poetes, or heathen gods, as some schole men wil say. Moreouer, I knowe that Abraham taught it first vnto the Egiptians, and no man wil denie but that he receiued all his knowledge of god for he worshipped no straunge gods neither receiued any giftes or sciences of them. And although many yeres after, Pythagoras declared it to the Grekes, yet was not he the firste inuentor of it, but god. Plato also commended it as most necessarie, willinge it to be taught to all youth before all other thinges. And thinkes be to god, it appeareth that his precept hath been very well folowed here in Englande, for I finde not in any [Page] nation (I speake not flateryng my countrey men but commending them) more ready knowledge in this science, then there is in the moste parte of our youth here in Englande, and namely in London, whiche I séeyng it is in them, it is (out of question) in their masters and tutours, whiche by their greate care ouer theim, haue trained theim vp vnto the practise thereof, to whom the greateste parte of the praise redoundeth. And nowe right worshipfull, when I consider how little profite my simple trauaill shall bryng, beyng but as it were an assaye of further attempt, I am cōpelled to craue at your handes onely to accepte my good will with the worke: for will & abilitie are somewhat vnequall in me, but if abilitie were corespōdent to will, I would gladly pleasure you in greater thinges then this is, whiche is but a shadowe incomparison of other mēnes substanciall labour and trauail in this art: yet I truste it is not altogether so vnpolished, [Page] but it hath passed vnder the file of some workemanshippe, wherby some men maye take occasion of profit and furtherance herein, which is my chiefe desire, although I knowe many are able to do better: for England is not so destitute of learned heades, but my doinges may sone be folowed, yea rather amended. But yet forasmuche as I would thei should knowe, that knowe not▪ I thought it not good to hide that which hath been opened vnto me. And so casting with my self the sōme of all, I destre you eftsones, right worshipful (to whome I thought it moste mete to present the fruicte of my studie) thankefullye to receiue it, as your wonted wisedomes doth all thinges well and vertuouslie purposed.

Thus fare you well, destryng God to maintaine you in youre estate, to prospere & further all your traffickes and voyages, encrease you in vertue, and keepe you in good health.

Amen.

# THE DIFFINITION of Nomber.

NOmber is as much to say, as a multitude compound of manye vnityes, as two is compounde of two vnities, three is compounde of three vnities, foure of foure vnities, fyue of fyue vnities ten of ten, fourtene of fourtene, fyftene of fyftene, twentie of twentie vnities, &c.

And therefore an vnitie is no nomber, but the beginninge and originall of number, as yf you doe multiplye or diuide an vnitie by it selfe, it is resolued into it selfe without increase. But it is in nomber otherwise, for there can be no nomber, howe great so euer it bee, but that it maye continuallye be encreased by addynge euermore one vnitie vnto the same.

## Numeration. Cap .i.

[Page] NUumeration is the art wherby to expresse and declare the value of any summe proposed, and is of two kyndes, the one gathereth y^{•} value of a sum proposed, & the other expresseth anye summe conceyued by oue figures and places, for the value is one thinge, and the figures are another thynge: and that cōmeth partlye by the diuersitye of fygures, but chiefelye of the places wherein they be ordeclye sei. And fyrste marke, that there are but ten fygures or characters whiche are vsed in Arithmeticke, wherof nine of them are called signifiynge fygures, and the tenth is called a ciphar, whiche is made lyke an 0. and of it selfe signifieth nothinge, but it beynge ioyned with any of the other figures, encreaseth their value, and these be they.

- 1 one.
- 2 two.
- 3 thre.
- 4 four.
- 5 fyue.
- 6 syre.
- 7 seuen.
- 8 eyght.
- 9. nyne.

[Page 2] Also you shal vnderstande that euery one of these fygures hath two values: One is alwaye certaine and hathe his signification of his owne forme, and the other is vncertayne whiche be taketh of his place.

A place is called the scate or roome that a figure standeth in, and howe many fygures so euer are written in one summe, so many places hathe the whole value thereof. And that is called the firste place (whiche nexte is towarde the ryghte hande) of anye summe, and so recknynge by order towarde the lefte hande, so that, that place is laste whiche is nexte the lefte hande. And contrarywise, when you expresse the value of the fygures in anye summe you muste begynne at the lefte hande, and so recken towarde the ryght hande.

Euerye of these nine fygures, (whiche are called signifiynge fygures) hath hys owne simple valew [Page] when he is founde alone, or in the fyrste place of anye sume. In the seconde place towarde the lefte hande; he betokeneth his one valewe ten times. As. 70. is, seuen times ten: that is to saye seuentie, 80. is viii. times ten: that is to saye eyghtie. In the thirde place euery fygure betokeneth his owne valew a hundreth times. As. 700. in that thirde place betokeneth, a hundreth tymes. 7. that is to saye, seuen hundred: In the fourthe place euerye fygure betokeneth hys owne valew a thousande times. As. 7000. is seuen thousande, and 8000. is eyghte thousande. These foure fyrste places muste be had perfectlye in minde, ye and that by harte, for by the knowledge of them you maye expresse all kinde of nombres howe great so euer they be.

In the fyfte place euery figure by tokeneth his owne valewe tenne thousande times. As 70000. is ten tymes seauen thousande, that is to [Page 3] sayē seauentie thousand: In the sytte place euerye fygure standeth for his owne valew, a hundreth thousande tymes. As 700000, is seauen hundreth thousande. The vii. place a M. M. times, or a million: as 7000000, is vii. M. M. or vii. millions. And the viii. place .x. M. M. times, or ten millions, so that euery place towarde the lefte hande, excedeth the former ten times. But nowe for the easye readynge, and Redye expressynge orderlye of anye summe proposed you shall practise this maner, folowing. And for example I propone this nomber 765432658. in the whiche are nine places. In the fyrste place is 8. and betokeneth, but eyghte, in the seconde place is 5. and betokeneth, x. times fyue that is fyfte, in the thirde place is syxe, and betokeneth a. C. tymes syxe, that is .vi. C. In the forthe place is. 2. and that is twoo. M. And 3. in the fyfte places is x. M. tymes 3. that is xxx. M. So 4. in the syxie [Page] place is a C. thousand times. 4. that is .iiii. C. M. then fyue in the seuenth place is a. M. M. times 5. that is v. M. or rather 5. millyons. And 6. in the viii. place is .vi. times x. millions, that is lx. millions. And laste of all .vii. in the. ix place, is vii. C. millions now foloweth the practise. Fyrste put a prick ouer the fourthe fygure, and so ouer the seuenthe, and lykewyse ouer the tenthe. And also ouer the 13, 16. or 19 if you had so many, and so stil leauinge twoo fygures betwene euerye twoo prickes and those roomes from one pricke to an other are called ternaries, then you muste pronounce euery three fygures from one pricke to an other as though they were written alone from the reste. And at the ende of theyr valewe, adde so manye tymes thousande, as your nomber hathe prickes (that is to saye, yf there be but 1. pricke, it is but one M. yf two prickes a M. M. or els a million yt 3. prickes a. M. M. M. or a M. [Page 4] million, and so consequentlye of all other fygures folowynge.) Then come lykewyse to the nexte three fygures, and sounde them as yf they were aparte from the reste, and adde to their valewe so manye times thousandes as there are pryckes betwene them and the fyrst place of your whole nomber. And so do by they next three fygures folowyng and of all the reste likewise as in exāple. 451234678567. The fyrste pricke is ouer. 8. in the fourth place, whiche is the place of a. M. the Seconde pricke is ouer 4. in the vii. place, whiche is the place of a M. M. or one million, the third pricke is ouer the .x. place whiche is the place of a M. M. M. or of a M. million, as in the former example. Then for the expressynge of this nomber by the valewe of euerye fygure, accordynge to the place wherin they stande [...], you shall fyrste begynne at the laste prycke ouer one, and take yt and the other twoo fygures. 5. and. 4. [Page] whiche do folowe him & valew them alone and they are iiij. C li. MMM. or els CCCC li. M. millions. Then take the other iij. fygures from one to the nerte pricke, and valewe them as if they were a parte from the other, and they are. 234. which are. CCxxxiiij. million, or 234. MM. Then come to the thyrde pricke ouer 8. and take the other ii. fygures behinde it, and recken them lykewyse as yf they were alone, and they are vi C lxxviii. M. And laste of all come to the other twoo fygures whiche remaine, that is. 567. and they are fyue Clxvii. Thus the whole sume of these fygures, is iiii. Cli. M. ii Cxxxiiii. Millions, viC .lxxviii. M. vClxvii, as before.

Note also that whole nomber is deuided [...] of [...] into three kindes, that is to saye, diget nomber, article, and mixte or compounde nomber. The dyget nomber, is all maner of nombres [...] vnder. 10. whiche are these. 9. [Page 5] fygures 123456789. of the whiche I haue spoken before. The Article nomber Article. is anye kinde whiche beginneth wyth a cyphar as this. 0. and they maye euer be deuided Iuste by. 10. without anye remayne as these. 10. 20. 30. 40. 50. 100. and all other suche like. The mixte, or compounde Mixte or compounde. nomber, conteineth diuers and many articles, or at the lest one article, and a diget, as. 11. 12. 16. 19. 22. 38. 108. 1007. and so forthe. And as any article nomber maye be made a compounde, by puttynge therto a diget, euen so lykewise euery compounde nomber, may be made an article nomber by addinge thervnto a 0.

The denominatours. of the places. | The order of the places. | |

M. of Millions. | 4 | Tenth. |

C. of Millions. | 3 | Nyneth. |

X. of Millions. | 2 | Eyghteth. |

Millions. | 1 | Seuenth. |

C. of Thousandes. | 0 | Sixte. |

X. Thousandes. | 1 | Fyfthe. |

Thousandes. | 8 | Fourth. |

Hundrethes. | 3 | Thyrde. |

Tenthes. | 4 | Seconde. |

Vnities. | 5 | Fyrste. |

## Addition in whole nomber Chap. 2.

ADdicion is as muche as to bringe together twoo summes or more into one, as if there were due to anye man 223, li. by seme one body, and 334 .li. by another, and 431. by another, & you woulde knowe howe many pound is due to the same man in all, these iii. summee shall you set downe orderly the one vnder the other, writinge the greatest summe highest and the nexte to the greatest vnder it, and the least sum vnder the laste, in suche sorte that the firste fygure of the one sūme by directly vnder the firste figure of the other, and the seconde vnder the seconde, an so furth in order, when you haue thus done drawe vnder them a straight line, & then wyll they stande thus. [...]

Nowe begynne alwayes at the fyrste places towarde yours ryghte hande, and put together [Page] the .iii. firste figures of these .iii. summes, and looke what commeth of them, write that vnder thē beneath the lyne, as in sayinge. 3. 4. and 1. beinge put together do make 8. write 8. vnder. 3. as thus. [...]

And then go to the second figures and doe likewise: as in sayinge. 2. 3. and 3. maketh 8. write 8. vnder. 2. as here you see. [...]

And lykewise doe wyth the fygures that be in the thirde place, in sayinge. 2. 3. and 4. are. 9. put. 9. vnder them and so wyll your whole sum appeare thus: [...] whereby you maye perceyue that those three summes beyng added together do make 988 li. and this is y^{•} art of addicion accordinge to his simplicitye, if the sum of anye place doe not excede a diget nomber. But in case the sum of anye one place can not be expressed [Page 7] by one figure but by two you shal put the fyrste of those figures vnder the line, and kepe the other in your minde for to adde it vnto the firste figure of the nexte place. And if the same nexte place can not be aualued but by two fygures, you muste in lyke maner put the fyrste of those fygures vnder the lyne, and reserue the second for the other place nexte after, and thus must you do from one place to another vntyll you haue come to the laste place, where in case you doe fynde that the summe be of twoo fygures, you muste set them both downe because it is the ende of that woorke, as in this example. [...] [Page] where the firste figures are, 3. 1. 5. 6. whiche added togither maketh 15. and for that, that 15. is of twoo figures, I do put the firste fygure 5. vnder the line, and kepe the seconde fygure (which is. [...]) in my minde, the whiche I muste adde with the nexte fygures of the seconde place, that is to saye with 2. 9. 4. and. 5. the whiche togither make 21. I writte 1. vnder the line for the seconde fygure of that addition, that is to saye after. 5. & I kepe 2. to be added vnto the third place the whyche wyth the other fygures 1. 8. 3. 4. do make 18. therefore I put. 8. nexte after. 1. in the third place vnder the line, and kepe 1. to be added vnto the fygures of the fourth place, whiche is with 2. 7. 2. 2. the whiche with the 1. that I kepe doe make 14. I set downe 4. for the fourth fygure (vnder the line) that is to saye, after 8. and I kepe 1. to be added vnto the fygures of the fyfte place, the whiche is 7. 6. 3. 8. with the 1. that I kepe [Page 8] maketh. 25. I put. in the fyfte place vnder the line nexte after, 4. and kepe 2. in mynde to be adoed wyth the fygures of the syxie place, that is with 6. 4. 9. 6. and that 2. whiche I kepe, maketh 27. I write downe 7. vnder the line in the syxte place, and I kepe 2. whiche I adde wyth the fygures in the seuenth place, and they make 13. I put downe 3. vnder the lyne in the seuenth place, and adde 1. vnto the fygures in the eyght place and they are 10. I doe put 0. vnder the lyne in the eyghte place, and then I adde 1. vnto the nynethe place, that is to saye with 4. 7. and they make 12. the whiche 12. I writte at length vnder the lyne bycause it is the ende of this addicion, and this is to be done of all suche lyke. And for the easyer vnderstandinge of that whiche we haue spoken of addition, you may e [...]amen these two other examples folowinge, in the whiche the fyrste hath these nombers. 3570. 2763. 579. 28. [Page] whiche beinge added toghther doe make this nomber 6940, and in the seconde example dothe resolute this nomber 51683. by addynge together of these nombers, 47630. 3756, 272, 25, as here vnder written. [...]

## ¶ Of substraction in whole nomber. chap. 3.

SUbstraction teacheth howe you shal abate one lesser number from a greater, and what there doth reremaine after that you shall haue abated the same, I speake not of the abating of one egall nomber, from an other egall vnto it, for the facilitie therof requireth no Rule.

In substraction are founde three nombers, the one is that, from the whiche the substraction is made. The seconde is the nomber, that is to be substracted, and the thirde is the nomber whiche remaineth after the substraction is ended. As when I woulde substract. 25. from 40. The 40. is the nomber from the which the substraction is made, 25. is the nombre to be substracted, and. 15. is the nomber whiche remaineth after you haue done the substraction, here foloweth the practise. You shall put the lesser nōber vnder the greater in such [Page] sorte, that euerye figure of the one numbre maye aunswere vnto euerye fygure of the other orderlye, and then drawe a right lyne vnder those twoo numbers, as you dyd in Addition. Then muste you beginne at the right hande and take the fyrste figure of the vndermoste number, and abate that from the first figure of the vppermost numbre, and that whiche remaineth you muste set vnderneth the line right vnder that figure which you haue substracted. Then afterwarde take likewise the seconde figure of the neither [...] moste snumbre, and abate that also from the seconde fygure of the higher numbre. The thirde from the thirde, and soe fourth of all the reste tyll you come to the ende, puttynge alwayes the remaine of euerye fygure vnder the line in hys order, example. I will subbstracte. 2345. from 9876. [...] after that I haue put them doune accordynge to the maner aforesaied. [Page 10] I take firste. 5. from. 6. and there resteth. 1. the whiche I sette vnder the lyne right againste. 5. Secondlye I abate. 4. from. 7. and there resteth. 3. the whiche I set in the seconde place vnder the lyne, nexte after. 1. Thirdly, I abate. 3. from. 8. and there resteth 15. The whiche I put vnder the line in the thirde place, finallye I doe abate. 2. from. 9. and there resteth. 7. the whiche I putte vnder the lyne in the fourth and laste place, and thus is this substraction ended, by the whiche there resteth. 753.

But when twoo fygures of one likenes doe chaunce to meete, so that the one muste bee abated from the other, as if I shoulde abate seuen from seuen, there remaineth nothyng, and then muste I set a ciphre. 0. vnder the line. But when the figure whiche is to bee abated, dothe excede the figure whiche is ouer hym, so that it can not be taken out of the same figure. Then muste you abate the neather fygure [Page] from, 10. And that whiche dothe remain you shall adde vnto the same figure whiche is vppermost. And the sum whiche commeth therof shall you set vnder the line. But when so euer you do borow any such. 10. of the ouer nomber: you muste adde. 1. vnto the nexte neather fygure folowynge whiche is to bee abated. And there is nothinge elles to be done in substraction. Example I will substract 93576. from. 4037479. after that I haue placed my two nombers, as I ought to do, I do fyrst.

[...] Abate. 6. from 9. and there resteth. 3. then I put the 3. vnder the line righte against. 6. And secondlye, I abate. 7. from 7. And there resteth nothinge. I doe put a cipher. 0. vnder the line right againste. 7. in the seconde place. Then I come to the third place where I finde. 5. whiche I cannot abate from the fygure ouer hym whiche is but [Page 11] 4: therfore I do abate it from 10. as before I taught, and there resteth. 5. the whiche I doe adde wyth the. 4. whiche is ouer hym, and that maketh. 9. I put 9. in thirde place vnder the line for the thirde fygure, fourthlye for the. 10. whiche I borowed I adde one vnto the nerte neather fygure whiche is 3. and they make 4. the whiche I doe abate from the ouer fygure. 7. and there resteth 3. I put. 3. vnder the lyne for the fourth fygure. And then I come to the fyfte place where I do fynde. 9. whiche I can not abate from the fygure ouer hym whiche is but. 3. but I abate 9. from 10. and there resteth. 1. the whiche I do adde with 3. and they make, 4. I put. 4. vnder the lyne for the fyfte fygure. And if it were not, for that I did laste borowe. 10. the substraction shoulde haue bene ended. But for bycause that I muste (for euerye suche tenne that I borowe) alwayes adde. 1. vnto the nexte neather [Page] figure folowyng, I muste therefore proceade vnto the substraction. And for that, that there is noe other figure folowyng in the neather number, it shall suffice to haue kepte the vnitee, and to abate it from the nerte ouer figure. But I find there. 0. and I cannot abate. 1. from. 0. therefore I abate it from. 10. and there resteth nine whiche I doe putte vnder the lyne in the sixte place, fynallie for the tenthe, which I borowed, I kéepe. 1. in mind. The whiche I doe abate from. 4. and there remaineth thre, the whiche I do [...]tte vnder the lyne in the seauenth place after nine. And the operation is thus ended.

### ¶ Another Example.

[...] But if there were many numbers to be substracted from one numbre alone, then muste you firste adde those numbers together according vnto the [Page 12] doctrine of the chapitre goyng before, and then to make your substraction as aboue saied. As if I would abate these thre summes. 123. 234. 456. frō. 98925 first I do adde the. 3 summes into one and thei are 813. The which I do abate from 98925. and there resteth 98112.

## ¶ Of multiplication. Chapiter. 4.

IN multiplication there are thre numbers to bee noted, that is to saie the numbre whiche is to bee multiplied, the whiche wee will call the multiplicande: and the numbre by the whiche we multiplie, we call the multiplier, or multiplicator. And the third numbre is that which commeth of the multiplication of the one by the other, which is called the product. As when I would knowe how much moūteth. 10. multiplied by 9. y^{•} is to saie how much are. 10. tymes 9. I find that thei ar worth 90. thā 10. is the multiplicāo. 9. is the multipter.

[Page] And 90. is called the producte: Then for to multiplye, is none other thinge, but to fynde a nombre whiche conteineth the multiplicande so many tymes, as the multiplyer conteineth vnities. As 10. multiplyed by 9. do make 90. as before said. And 90. conteyneth 10. so manye tymes, as 9. conteineth vnities, that is to saye nyne times.

In multiplycation, it forceth not muche whiche of the twoo nombers by the multiplicande, nor whiche be the multiplier. For 10. multiplyed by 9. maketh as manye as 9. multiplied by. 10. yet neuertheles it shalbe more commodious that the lesser nomber be alwayes the multiplier.

And for that, that the multiplication of fygures the one by the other, is the moste chiefe and necessaryest kynde, whereby to knowe howe to worke in the multiplication of compounde nombers, and that euery man hath it not at the fingers [Page 13] ende: I wyll therfore gyue you heare certein easye wayes of multiplication of diget nombers. When you woulde multiplye twoo simple fygures, or digits the one by the other, abate eche of those diget nombers from 10. Then multiplye the twoo remaines the one by the other. And if the summe doe excede 10. writte onely the firste fygure, and kepe the other to be added to the nexte operation, whiche is thus. Adde your two simple figures togither: And of the which resulteth of that addicion, take onelye the fyrste fygure, vnto the whiche you muste adde the vnitye whiche you kept before. And that shalbe the seconde fygure of the summe which you do seeke. Example, I woulde multiply 7. by 6. I take 7. from 10. and there resteth 3. likewise I abate 6. from 10. & there resteth 4. then I saye thus 3. times 4. make 12. I writte 2. for my first fygure, & I kepe 1. in my minde, then I adde 6. with 7. and they are 13. of the [Page] whiche I caste awaie the seconde figure. 1. and I take onelie the firste fygure. 3. vnto the whiche I adde the vnitie whiche I kepte, and they make 4. whiche I wryte in the seconde place after. 2. And thus I finde. 42. whiche is the valuer of 7. multiplied by. 6.

Otherwise, and all commeth to one effecte, sette doiune youre two dig [...]e numbers the one right ouer the other, and right against euery or them towarde the right hande wryt & hys owne distaunce from. 10. Then multiplie the two differences together, the figure whiche commeth thereof, shall you set doune vnder both the differences. But if there be two fygures sette doune but the fyrste, and keepe the other in youre mynde, afterwardes abate (from one of the two digit numbers) the difference of the other digite number that is to saie, croswise. And vnto the remayne adde the fygure whiche you kepte, and that shall bée [Page 14] the seconde nomber, and thus you shall haue your multiplication. Exā ple of the like figures, that is to saie, of 7. multiplied by. 6. the distaunce of. 7. vnto. 10. is. 3. And the distaunce of. 6. from. 10. is. 4. I sette them doune crossewaies, as you see: [...] And then I saie. 3. tymes. 4. are. 12. I sette doune. 2. and keepe one in my mynde, then I abate. 4. from. 7. or els. 3. from. 6. it forceth not from whiche of theim: and there resteth alwaies. 3. vnto the whiche I adde the vnitie, whiche I kepte in my mynde, and thei are fower, whiche shall bee the seconde figure of the multiplication. And thus I finde that seuen, multiplied by sixe, maketh fowertle and twoo, as in the other operation. This practise hath no place where the twoo digette noumbers (dooe not exceade tenne) by addyng theim together, and then is [Page] multiplication easye ynoughe without any rule.

An other waye to knowe the multiplication of symple numbers, is by this table folowyng: the vse whereof is thus.

Fyrste you shall vnderstande, that the nombers from 1. and so downewardes to 9. set in the left part or hanging margin of this table do betoken the multipliers of all simple nombers. And the elements or fygures beyng put highest in euery square roome drawynge towarde youre ryght hande right against euery of the multipliers, doe signifie, the multiplicands, vnto the multipliers of the hangyng margin. And the lower or inferiour nombers in euerye square roome, do betoken the product of that multiplication, which is made in multiplyinge the vpper nomber ouer it, wyth the figure in the hanging margin, answering directly vnto the sayd square: as by example.

1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |

2 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |

4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | ||

3 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | ||

9 | 12 | 15 | 18 | 21 | 24 | 27 | |||

4 | 4 | 5 | 6 | 7 | 8 | 9 | |||

16 | 20 | 24 | 28 | 32 | 36 | ||||

5 | 5 | 6 | 7 | 8 | 9 | ||||

25 | 30 | 35 | 40 | 45 | |||||

6 | 6 | 7 | 8 | 9 | |||||

36 | 42 | 48 | 54 | ||||||

7 | 7 | 8 | 9 | ||||||

4 | 56 | 63 | |||||||

8 | 8 | 9 | |||||||

64 | 72 | ||||||||

9 | 9 | ||||||||

81 |

[Page] Firste, because one doeth not multiplie, I sette in the vpper margin the figures from. 1. to. 9. bothe in the hier and also in the inferiour rowes, for one in the hangyng margine, multiplied by one the vpper nomber in the firste square bryngeth but one, so likewise twoo beyng the higher nomber in the seconde square of the vpper margine, multiplied by. 1. in the hangyng margine, bryngeth twoo for the lower nō ber in the seconde square of the vpper margine, for one tymes one maketh but one, and one tymes twoo maketh twoo, then one tymes three maketh. 3. and one tymes fower maketh fower, and so continewyng toward the right hande, vntill you come to the figure of 9. whiche is one tymes. 9. maketh. 9. Then after multiplie twoo of the hangyng margine by twoo, the vpper nō ber of the square, nexte towarde the right hande, and that maketh fower, whiche is the producte of twoo, multiplied by twoo, whiche fower is set vnder [Page 16] the twoo, for twoo times twoo are fower, and twoo tymes thred maketh sixe, then twoo tymes fower maketh 8. and twoo tymes. 5. maketh. 10. and so conunewyng vnto twoo tymes. 9. whiche maketh. 18. The like is to bee doen with the thirde rowe, and so likewise of all the residue.

Example, I would knowe what is the producte of. 9. multiplied by. 8. I seke in the hangyng margin the multiplier eight, and emongest the squares directly againste eight, drawyng toward the right hande, I seke the multiplicande. 9. in the higher rowe, and I finde the producte right vnder. 9. to bee. 72. Then. 72. is the nomber which commeth of the multiplication of. 9. by eight, and so is to bee vnderstande of all the reste of the table, whiche table muste bee of all menne learned by harte, or as thei saie without booke, whiche beyng learned, you shall the better attaine to the resle of multiplication. [Page] To come nowe vnto the practise of multiplication, when you woulde multiplye twoo nombers, the one by the other, you muste set them downe after the same maner as you dyd in addicion, and in substraction. That is to saye, the first fygure of the multiplier, vnder the firste figure of the multiplicande, the second vnder the seconde, and the third vnder the third, yf there be so manye, and then drawe a ryghte lyne vnder them, as in the other operations goynge before.

After you shall multiplie all the fygures of the multiplicande by the multiplyer, and set downe the fygures (commynge of any suche multiplication) vnder the lyne euery one in order.

Example, I woulde multiplye 123. by. 3. that is to saye, I woulde knowe howe muche amounteth iij. tymes one hundreth, twentie and three. The twoo nombers beynge placed in suche order as is before [Page 17] saied, you muste beginne towardes the right hande: and say thus three tymes thrée are nine, wryte downe 9. vnder the line right againste 3. for the firste figure: [...] secondlye by the same thrée you muste multiplie the seconde fygure 2. and they make 6: putte downe 6. after the 9. vnder the line: Thirdly by the same 3. you shall multiplie the last figure. 1. and they are but 3. set doune 3 after 6. for the thirde and laste figure. And thus is that worke ended: wherby you shal finde, that 123. being multiplied by 3. maketh 369.

But when that of the multiplication of one figure by an other: The sum whiche commeth thereof shal be of two figures, as it happeneth moste often, then shall you wryte downe the firste figure, and kepe the other figure to bee added vnto the multiplication of the next figure.

Example. Syxe menne haue gained (euery one of them) 345. crownes, [Page] I woulde knowe howe manye Crownes, thei hadde in all. [...]Fyrste I multiplye 6. times fiue are. 30. I wright. 0. vnder the line, and keepe. 3. to bee added to the next multiplication: Secondlye I saie 6. tymes four, are. 24. vnto the whyche I adde 3. whyche I reserued. And thei make 27. I wryt. 7. in the second place vnder the line, and I kéepe 2. to be added to the next multiplication, thirdlie I saie sire tymes. 3. are 18. vnto the whiche I adde the. 2. whiche I kepe, and thei make. 20. the whiche I wright all doune for because that is the laste woorke. And so I fynde that 345. beyng multiplied by. 6. doe make 2070. Whan the multiplier is of manie figures you muste multiplie al the whole multiplycande by euerye one of those figures, & wryte the productes euerie one vnder his owne figure.

Example. I woulde knowe howe manye dayes are paste from the natiuitie [Page 18] of Iesus Christe vntill the yeare 1560. full complete. I haue to multiplie. 1560. by 365. which are the daies of one whole yere. The leape yeares not beyng reckened, which haue euerie one of them. 366. daies.

First by the figure. 5. I multiplie [...]all the higher fygures, saiyng thus. 5. tymes. 0. maketh. 0. I wryte. 0. vnder the lyne for the firste fygure, and because I kéepe nothyng for the next place. I procede and saye. 5. tymes 6. are. 30. I sette 0. vnder the lyne for the seconde fygure, and I keepe. 3. to bee added to the next multiplication, thirdlie I say 5. tymes. 5. are. 25. The whiche with 3. that I keepe are. 28. I sette doune 8. and keepe. 2. to bée added with the next multiplication. Then comming vnto the fourth and last fygure, I saie fiue tymes. 1. are. 5. the whiche with 2. that I reserued are 7. I putte 7. for the laste fygure of thys fyrste operation [Page] by the figure. 5. with the which figure we haue no more to doe. And therfore I cancel thesame. 8. with a little strik through it, to signifie that wée haue finished with that figure. And for as muche that in multiplication there is alwaies as manie simple operations, as the multiplier conteineth figures. There resteth yet twoo operations to be made. I come then vnto the seconde operation, whiche is by the fygure 6. by the whiche I muste againe multiply all the figures of the multiplicand as I did by 5 and the first figure, which shalbe produced, you muste putte one ranke more lower then the figures of the operation euen nowe made by fiue: not right vnder the firste fygure of the multiplier fiue, but vnder sixe: that is to say: one place more forward then the fiue towarde the left hande, and one rancke more lower then the firste operation: and you shall put afterwarde euerie of the other figures whiche commeth of the same multiplication [Page 19] in their order: thirdlye you muste make the multiplication by the thirde figure and that whiche shall come thereof you muste sette in hys rancke, as here vnder you see. And nowe wee néede make no further discourse hereof, because that hee whiche canne doe the firste multiplication by fiue, may as easely doe all the others. It shall therefore suffice to sette here vnder the examples. [...]

Nowe, if you will knowe howe muche the operations thus placed doe amount vnto, which in value are but one number: you must adde those thre numbers together, but not after the same maner as wee haue done in the chapiter of addition, the first figure of the first ranck with the firste figure of the second ranck, and of the thirde: but [Page] you muste adde them in the same sort as you shall finde them situated or placed: that is to saie, the firste figure of the fyrste rancke alone by it selfe the seconde of that ranck with the first of the seconde rancke. The thirde of the firste rancke wyth the seconde fygure of the thyrde rancke: and so of all other as hereafter doth appeare.

[...] And thus the 1560 yeares doe containe fyue hundred sixty and nine thousand four hundred daies not counting herein the dayes of the leape yeares, whiche are heare in numbre 320. then the whole summe of the daies shoulde be 569790.

### ¶Another Example.

[Page 20] [...] The summe of multiplication, when you would multiplie any number by 10. you muste onelye adde one cipher vnto all the number. As 345. multiplied by 10. maketh 3450. yf you will multiplie by. 100. Adde vnthe whole number two cyphers. 00: yf by 1000. adde. 000. And to be briefe, when the laste figure of the multiplier is. 1. and all the rest be ciphers, adde so many ciphers to your multiplicande, as ther shalbe foūd in your multiplier

But yf in multipliyng, the laste fygure were not one, but that there were onelye certayne Cyphers in the beginnynge: and that the other were signifiynge fygures, and [Page] likewise those of that multiplicande, then shall you putte those cyphers a part, and multiplie the signifiyng figures of the one by the signifiyng figures of the other. Than adde vnto the product of that multiplication, all the ciphers whiche you dyd before put a part. As if I would multiply. 46000 by 3500. I put apart the thrée ciphers of the fyrste, and twoo ciphers of the seconde numbers. And then I multiply fourtie sixe by 35. and thereof commeth 2610: vnto the whiche I add the 00000. and then the whole producte will be. 161000000. [...]

## 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 diuisour 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 [Page] vnderneth, in suche sorte, that the first fygure of the diuisour towarde the lefte hande be vnder the first of the diuidend, 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 other, yf there bée anie more, whiche is contrarie to the other three kyndes before specified, but you muste consyder if all the lower fygures of the diuisor, maye be taken out of the higher figures 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 diuisour. And at the ende of them an other croked line, behinde the whiche towarde [Page 22] the right hand, shal be set your quotient. As by this example following 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 herevnder you may see. [...]

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 example 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 diuisor: [...] and vnder the same multiplication.

I muste sette that number whiche [Page] 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 multiplication (of the quotient by the diuisor) from the higher numbre corespondant 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 operations is comprehended the art of diuition. 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 [...] 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. behinde [Page 23] the croked line behinde. 2. afterwarde 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. Thirdlye for that there yet remaineth an other, 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 [...] the crooked iyne for the third and last figure of the quotient. Then by the same 5. I multiplie the diuisour 4. and that maketh 20. the whiche I abate from the higher number, and there resteth nothing. And so is thys diuisyon ended: and I [Page] haue founde that. 860. being diuided by foure, bringeth for the quotient 215. that is to saie, that. 4. is conteined in 860. twoo hundred and fiftene tymes. This is the moste eastest workynge 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 diuidende, you must not place the firste figure 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 tymes the firste figure of the same towarde the leste hande is contained in [Page 24] the higher numbre aunsweringe to the saied numbre, and then to work after y^{•} same maner as is before taught.

Erample. I haue. 316215. crownes 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 deuidende, whiche is. 3. because that. 4. is greater number then. 3. And further, I can not take. 4. out of. 3. wherfore I muste sette the. 4. vnder the seconde figure of the higher number whiche is. 1. and the fygure. 5. of the diuisour next right vnder the 6. as you maie sée.

I muste firste séeke, home many tymes [...] 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 [Page] 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 remaineth 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. [...]

And when I come to remoue the diuisour, and that I muste séeke howe manye tymes it is contained 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 quotient behind the croked line, and if ther remaine anye fygures in the diuidende [Page 25] whiche are not finished, I must remoue t [...] diuisor againe nerer towarde my right hande by one place, for to finde a newe figure in the quociente. As in this our example, for after that I haue remoued the diuisor, I seke how many tymes. 45. is conteined [...] 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 abatyng, I remoue again the diuisor nerer 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. whereas [...] 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.

[Page] 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 (whiche 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 [Page 26] 121. and there remaineth. 31. And then will the somme stande thus. [...]

And here is also to be noted, that the A note. somme whiche remaineth, must be alwaies lesser then the diuisor. Then finally, 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. [...]

[Page] 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 remaine aparte behinde the croked line, after the entier quotiente, and the diuisor, 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. [...]

[Page 27] 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 tymes the diuisor is conteined in the higher nomber: then, by the quotiente (whiche I finde) I must multiplie the diuisor: finally, I must abate the producte of that multiplicacion, from the higher nomber to thesame correspondent, 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 supputacion, and for the placyng of fewer [Page] figures in the quotiente: and consequently it shall saue muche labor vnto thē, whiche as yet haue not muche studied in this art. The practise wherof is thus, as followeth.

I haue to deuide. 7894658. by 643. In the firste place, you shall vnderstande, that although the firste figure 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 onely 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 behinde the line put this figure. 1. Then double your saied diuisour, and right [Page 28] 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, behinde the line putte the figure of 3. And vnto this thirde producte, you muste adde againe your diuisor: and right againste the same producte, behinde 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 producte behinde the line, the noumber whiche signifieth howe muche he is in order. That is to saie, righte againste the fifte producte, you muste putte. 5. right againste the sixte producte, you must put. 6: and so likewise of all the other.

Example of the diuisour proponed, 643. firste, I write doune. 643. and [Page] right against the same behinde the [...] line, I put. 1. secondly, 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 alwaies by encreasyng so muche euery producte, as the diuisor doeth amount vnto, vntill you haue so doen nine tymes, as you see in this present table.

This beyng dooen, you muste sette downe your diuisor vnder the diuidende, after the same maner as is before declared: that is to saie. 643. vnder [Page 29] the three firste figures of the diuidende, 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 operacion ended. Then I sette forewarde the diuisour, one figure nerer to my right hande, and I seeke a newe quotient, as I sought this, where I finde the higher number ouer my diuisour [Page] 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 againste it this digette. 2. I putte. 2. for the seconde figure of the quotient behinde the line, and I dooe abate. 1286. from. 1464. and there remaineth. 178

Thirdly, I remone forward the diuisor, as before, and I finde the higher number to be. 1786. and that the next lesser number to it in my table, is againe. 1286. I putte the refore ones againe. 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 quotiente, for the fowerth figure. And after that I haue abated. 4501. from 5005. there remaineth. 504.

[Page 30] Finally, I remoue forward my diuisor 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 betwene them, in this maner folowing. [...]

### ¶ 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. shalbe the quotiente, and the. 5. shalbe the number that doeth remain. Likewise when you would diuide any number [Page] by. 100. take awaie the twoo laste figures 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 other, when the first figure of the diuisor toward the lefte hande, shalbe onely. 1. and the reste of the same diuisour beyng but ryphers.

### ¶ Here foloweth the proofes of addition, substraction, multiplication, 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 recken them all, one after an other, casting [Page 31] 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 awaie, 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 taken awaie all the nines, out of the totalle somme of thesame addition, as often [...] as you may ther finde any: as in this addition which here you se. There remaineth. 2. for eche part

#### ¶ The proofe of substraction.

ADde the number whiche you doe substracte, with that number which remaineth after the substraction is made: and if the totall somme of that addition, bee like vnto the number from the whiche the substraction [Page] was made, you haue dooen well, otherwise not: as in this example dooeth appeare, [...] 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 produced of the multiplication, by the multiplier: you shal finde the higher number, 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 [Page 32] 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

## ¶ Of progression, the vi. Chapiter.

PRogression Arithmeticall, Progression Arithmeticalle. is a brief and spedie assemblyng, or addyng together of diuers figures or numbers, euery one surmountyng thother continually, by equall difference: as 1. 2. 3. 4. 5. &c. here the diffrence, from the first to the seconde is but of 1. and so do al the other, euery one excede an other by. 1. still to thende. Like waies. Here 2. 4. 6. 8. &c. doe proceade by the difference of. 2. also. 3. 6. 9. 12. &c. doe euery one differ from other by. 3. and so may these nūters continue. Infinitty after this order, in addyng vnto the thirds number, the quantitie wherein the secōde doth differ frō the first: like wates [Page] addyng the same difference vnto the fowerth number, also to the fifte, and so vnto all the other. As. 14. the difference of the seconde to the firste is. 3: adde. 3. vnto. 4: and thei are. 7. for the thirde number: Then adde. 3. vnto. 7: and thei make. 10. for the fowerth nū ber, and so of all other.

Then if you will adde quickly the number of any progression, you shall doe thus, firste tell how many numbers there are, and write their somme doune by it self, as in this example, 2. 5. 8. 11. 14. where the numbers are 5 as you maie see, therefore you muste sette doune. 5. in a place alone, as I 5. haue dooen here in the margent.

Then shall you adde the first number, and the laste together, whiche in this example are. 14. and. 2. and thei make. 16. take halfe thereof, whiche is. 8. and multiplie it by the. 5. whiche I noted in the margente, for the nomber of the places, and the somme whiche amounteth of that multyplication, [Page 33] is the iuste somme of all those figures added together, as in this example: 8. multiplied by. 5. dooe make. 40. and that is the somme of al the figures.

An other example of parcelles that are euen, as thus. 1. 2. 3. 4. 5. 6. in this exāple you must likewaies note doun the number of the places, as before is taught, and thē adde together the last nomber and the first. And the somme whiche commeth of that additiō, shall you multiplie by halfe the nomber of the places, whiche before are noted, and that, whiche resulteth of the same multiplication, is the whole somme of al those figures, as in this former exā ple, where the nomber of the places is 6. I note the 6. apart, and then I adde 6. 6. and 1. together, whiche are the laste and firste nombers, and thei make. 7. the whiche I multiplie by. 3. whiche is halfe the nomber of places, and thei make. 21. and so muche amounteth all those figures added together.

[Page] Progression Geometricall is, when Progression. geometricall. the second nomber containeth the first in any proporcion. 2. 3. or. 4. times and so forthe. And in like proporcion shall the thirde nomber contain the second, and the fowerth, the third, and the first the fowerth. &c. As. 2. 4. 8. 16. 32. 64. here the proportion is double.

Likewaies. 3. 9. 27. 81. 243. are in triple proportion.

And. 2. 8. 32. 128. 512. are in proporcion quadruple.

That is to saye, in the firste example, where the proporcion is double, euery nomber containeth the other. 2. tymes. In the seconde example of triple proporcion, the noumbers exceade eche other thre times. And in the third example, the nombers exceade eche other fower times, and thus you se that progression Arthmeticalle, differeth from Progression Geometricalle for that, that in y^{•} Arithmeticall. The excesse is only in quātitie, but in the Geometricall, the excesse is in proporcion.

[Page 34] Nowe if you will easelie finde the somme of any soche nombers, you shal dooe thus, consider by what noumber thei be multiplied, whether by. 2. 3. 4. 5. or any other, and by the same nomber you must multiply the last somme in the progression. And from the producte of the same multiplication, you shall abate the first nomber of the progression. And that whiche remaineth of the saied multiplicacion, you shall diuide by. 1. lesse then was the nomber, by the which I did multiplie. And the quotient shall shew you the sōme of all the nombers in any Progression. As in this exaumple. 5. 15. 45. 135. 405. whiche are in triple proporcion: now muste you multiplie. 405. by. 3. and thei are. 1215. from the which you shal abate the first nomber of the progression, whiche is. 5. and there resteth 1210. the whiche you shall diuide by the nōber lesse by. 1. then by the which you did multiplie, that is to saie, by. 2. and you shall finde in the quociēt 605. [Page] whiche is the totall somme of the nō bers of that progression. Likewise. 4. 16. 64. 256. 1024. whiche are in proportion quadruple: therfore multiplie 1024. by. 4. and thereof commeth 4096. from the whiche abate the first nomber. 4. and there resteth. 4092: the whiche you must deuide by. 3. and you shall finde in your quotient. 1364 whiche is the total somme of that progression, and this shalbe sufficient for progression.

## ¶ The .vij. Chapiter treateth of the Rule of. 3. called the golden Rule.

THE rule of three is the chiefest, the moste profitable, and the moste excellente rule of all the rules of Arithmetike. For al other rules haue nede of it, and it passeth all the other, for the whiche cause it is saied, that the Philosophers did name it the golden rule. And after others opinion and iudgemente, it is [Page 35] called the rule of proportions of nombers. But now in these daies, by vs it is called the rule of thrée, be cause it requireth three nombers in his operacion. Of the whiche three nombers, the twoo first are set in a certain proportion. And in suche proportion as thei be stablished, this rule serueth to finde out vnto the third nomber, the fourth nomber to hym proportioned, in suche sort as the seeonde is proportioned vnto the firste. Not for that, that the fower noumbers, nor yet the three, are or bee proportionall, or set in one proportion, but suche proportion, as is from the firste to the seconde, ought to bee from the thirde vnto the fowerth, that is to sate, if the seconde noumber dooe contain the firste, twoo tymes or more, so many tymes shall the fourth nomber containe the thirde. And note well that the firste noumber, and the thirde in euery rule of three, oughte and must bee alwaies semblable, and of one condition. And the second nomber, [Page] and the fowerth muste likewise bee of one semblaunce and nature.

And are dissemblaunte, and contrarie to the other twoo noumbers: that is to saie to the firste, and the thirde. And if you dooe multiplie the firste by the fowerth. And the seconde noumber by the thirde. The twoo multiplications will bee egall. Likewise if you diuide the one sembleaunte by the other, that is to saie, the thirde noumber by the firste. And likewise the one dissembleaunt by the other: that is to saie, the fowerth nomber by the second (which are dissembleaunt to the other twoo nombers) your twoo quocientes will be egall

The stile of this rule in thus, you Regul. muste sette doune your three noumbers in a certaine order, as by example here vnder shal appeare. And then multiplie the thirde noumber, by the seconde. And the producte thereof you must diuide by the firste noumber, or otherwise, diuide the firste noumber [Page 36] by the seconde. And the quocient therof shalbee diuisor also vnto the third nōber that is to saie, the thirde nomber shall bee diuided by the quotient of the foresaied diuision, that is of the firste noumber diuided by the seconde. Or otherwise diuide the second noumber by the firste. And that whiche cōmeth into your quotiente, you shall multiplie it by the thirde nomber. And thus shall you haue the fowerth noumber, whiche you seke for.

### ¶ Example.

IF, 8. be worthe. 12. what are 14. worth after the rate, or els if. 8. require. 12. for his proportionall, what will. 14. demaunde? The whiche three noumbers maie conuenientlie bee sette in soche order, as hereafter doeth appere.

If. 8.-12.-14. multiplie the thirde nomber. 14. by the seconde, whiche is. 12. And thereof commeth [Page] (for the whole producte of this multiplication. 168. the whiche (as the rule teacheth) you muste diuide by the first nomber, that is to saie by. 8. and therof commeth. 21. And so muche are the 14. worthe. This is the waie, whiche is moste vsed.

Otherwise diuide. 8. by. 12. whiche you can not doe, for thei are 8/72. wherefore abreuie. 8/12. and thei are 2/9. for your quotient, then diuide the thirde nomber. 14. by the saied ⅖, and you shall haue. 21. as before. Orels diuide the seconde nomber. 12. by the firste nomber. 8. thereof commeth. 1 ½. the whiche 1 ½ you shall multiplie by 14. and therof will come. 21. as is aboue saied, and thus muste you dooe of all other. And although, that the nōbers of this rule maie bee founde in three differences, for sometymes thei are whole nombers and broken together, sometymes broken and broken together, and somtymes all whole nombers, if thei bee whole nombers, you muste doe none [Page 37] otherwise, then you did in the last erā ple. But in case thei be broken noumbers, or broken and whole noumbers together, the maner and waie to dooe theim, recetueth a certaine variacion, and difficultee, according to the varietie of the noumbers, the whiche operation easeiy to do, and vnuariable, this rule teacheth.

The three noumbers beyng sotte doune, according vnto the order of the whole noumbers aforesaied, without any broken nomber, let. 1. bee put alwaies vnderneath euery whole nomber, with a line betwene them fractiō wise, as thus 8/1. and that. 1. is denominatour to euery soche whole nomber. When you haue whole nomber and broken, thei must bee reduced and added with their broken nomber, and if there bée broken nomber without any whole nomber, thesame broken must remain in their estate.

### A Rule.

This beynge deen, you shall multiplie [Page] the denominatour of the firste nomber, by the numeratour of the second, and the producte thereof againe by the numerator of the third nōber. And so shall you haue the diuidende, or nomber whiche muste bee diuided, then multiplie the numeratour of the first nōber, by the denominator of the seconde, and the product thereof by the denominator of the third nomber, and that which cōmeth of this multiplication shalbe your diuisor. Then diuide the nomber, whiche is to bee diuided, by the diuisour, and you shall finde the fowerth nomber that you seke. Of the which maner and fashions of the rule of. 3. are diuers kindes, wherof the first is of. 3. whole nombers, as was the last example, and here foloweth the second

If. 15. poundes doe buy me two clothes, how many clothes wil. 300. poū des buye me of thesame pr [...]e, that the twoo clothes did cost, sette doune your three nombers thus.

Lib: Clothes. Lib.

[Page 38] [...] And thā as you see, multiplie y^{•} thirde nomber, which is. 300. l [...]. by. 2. whiche is the second nōber, and therof cōmeth 600. the which. 600. you must diuide by the first nōber. 15. and you shal find in your quociēt 40. whiche is. 40. clothes, and so many clothes shall I buye for. 300 .l. as appereth by practise here aboue written. And here you muste marke that the first nōber & the thirde in this questiō be of one denominaciō, and like wise the second & the fowerth which you haue found are of one semblaunce: but in case that the first nomber and the third in any question: bee not of like denominacion, you muste in workyng bryng them into one, as in this exāple folowing. If. 12. nobles do gaine me 6. nobles, how many nobles wil. 48. poundes gatne me: Here you se that y^{•} denominaciō of the first [Page] nomber is nobles, and the denominacion of the thirde, is poundes, wherefore, before you doe procede to worke by the rule of three, you muste firste turne the poūdes into nobles in multipliyng. 48. poundes by three nobles and they make. 144. nobles, for that there is in euery pounde of money. 3. nobles, or otherwise if you will, you maie bryng the first nomber beyng. 12 nobles, into poundes, by diuiding thē by. 3. and thus shal your first and third nombers, bee brought into one denominacion. Then shal you sette doune your. 3. nombers in order thus.

If. 12. nobles doe gaine me. 6. nobles, what shall. 144. nobles gaine? the which. 144. are the nobles which are in. 48. [...]. Then multiplie the third nomber. 144. by the seconde nomber 6. and thereof cometh 864. the whiche you muste diuide by. 12. nobles, and thereof commeth. 72. nobles.

But here it maie perchaunce make some menne masse, to see all the three [Page 39] nombers in this rule of three, to bee of one denomination, whiche can not otherwise bee dooen, if you reduce the third nomber, to the denominacion of the firste. But if you will reduce the firste nomber, to the denomination of the thirde, that is to saie the. 12. nobles into poundes, then shall the firste and the third nombers onely agree in one denominacion, and the fowerth nomber which you seke, shalbe of the same denominacion as is the seconde, as in the former example. If. 12. nobles doe yelde me. 6. nobles, what will. 48. poundes yeld me: first you shall diuide 12. nobles by three to bryng theim in poundes, and thei shall bee like to the thirde nomber, whiche is also poundes, then will thei stande thus. [...]

[Page] There is yet a more exacte waie to woorke in this rule of three, whiche is thus. You muste marke if the third & first nombers in the rule of three, maie bee bothe diuided by one like diuisor: the which after you haue diuided thē, you shall write doune eche of the quocientes orderly, in the saied rule of. 3. euery one of theim in his owne place, as though those were twoo of the nō bers of your question, and not chaungyng the middle noumber, that is to saie the seconde, as thus, if. 50. Croudes doe buye me. 44. yardes of clothe, howe manye yardes shall I haue for 120, here you maye see that the thirde and the firste nombers, maie be diuided by. 10. whiche in the thirde nomber is. 12. tymes, and in the firste. 5. times. Wherefore you shall put. 12. for the thirde nomber in the rule of three▪ in stede of 120. and 5. for the first nomber in stede of 50. and let. 44. remain still in the middest for the second nomber, after this sorte as foloweth, and [Page 40] then worke by the rule as before. [...]

Multiplie. 44. by. 12. and thereof commeth. 528. diuide thesame. 528. by 5. and you shall finde in your quocient 105. ⅕. and euē so many yardes should you haue founde, if you had wrought the rule of thre, by the first nōbers proposed. There is yet certaine other varieties, in woorkinge by the rule of three, but for that they require the knowledge of fractions, and because thei are not so easte as this first waie, whiche is common, therefore content your selues with this same, vntil you haue tasted the fractions, the whiche by gods helpe I intende to fet forth in seconde part of this boke, incontinently after that I haue firste taught you [Page] the backer rule of three.

The backer rule of thre is so called: because it requireth a contrary woorkyng to that, whiche doeth the rule of three directe, where of wee haue nowe treated. For in the directe rule of three the greater the thirde noumber is, so moche the greater will the fourth be. But here in this backer rule it is contrariwise, for the greater the third nō ber is, so moche lesser wil the fowerth bee. Then, where as in the rule of. 3. directe, the third noumber is multiplied by the seconde, and the producte thereof diuided by the firste. Here you muste multiplie the seconde noumber by the firste, and diuide the producte of the same by the third, and the nomber whiche commeth in the quotiente, answereth to the questiō. For suche practise commeth often tymes in vse: In suche sorte, that if you woorke thesame by the rule of three directe (not hauyng a regarde vnto the Proposition of the question) you should then [Page 41] committe an euidente and open errour.

### ¶ Example.

If. 15. shillynges worthe of Wine wil serue for the ordinary of. 46 men when the Tonne of Wine is worthe 12. poundes: for howe many menne will the same. 15. shillynges suffice, when the Tonne of Wine is worthe but. 8. poundes? It is certain, that the lower the price is, that the Tonne of Wine doeth coste, and so many more persones will the saied. 15. shillynges in Wine suffice. Therefore set doune your nombers thus, if. 12. poundes suffice. 46. menne, how many will. 8. poundes suffice, you muste multiplie 46. by. 12. and thereof commeth. 552. the whiche. 552. you shall diuide by. 8. and thereof commeth. 69. and vnto 69. menne will the saied. 15. shillynges worthe in Wine suffice, when the Tonne of Wine is worthe but eight poundes, as hereafter dooeth appeare by practise. [Page] [...]

Likewise, a messenger maketh a iourney in. 24. daies, when the daie is but, 12. houres long: how many daies shal he be vpon the same iourney, whē the daie is. 16 houres in length? Here you must perceiue, that the more houres are in a daie, the fewer daies will the messenger bee in goyng his iouruey. Therefore write doune your nō bers thus, as here you maie se. [...]

And then multiplie 24. daies by. 12 [Page 42] houres, and thereof commeth. 288. diuide the same. 288. by the thirde nomber. 16. and you shall finde 18. the whiche is. 18. daies, and in so many daies will the messenger make his iourney when the daie is. 16. houres long.

Likewise, when the Bushell of wheate doeth coste. 3. shillynges, the penie loafe of bread waieth. 4. lib.

I demaunde what the same penie loafe shall weye, when the bushell of wheate is worthe but twoo shillynges: here is to bee considered, that the better cheape the Wheate is, the heauier shall the pennie loafe wiie, and therefore write doune your. 3. nombers, thus. [...]

Then multiplie. 4. lib. whiche is the secōde nomber, by the first nomber 3. and thei make. 12. the which. 12 you [Page] shall diuide by the thirde nōber. 2. and thereof commeth. 6. lib. and so muche muste the penie loafe of breade waye, when the bushell of wheate is worthe but twoo shillynges, as maie appere.

And nowe, accordyng to my former promise, shall followe the seconde parte of Arithmeticke, whiche teacheth the workyng by Fractions.

# ¶ The seconde parte of Arithmeticke, whiche treateth of Fractions or broken nombers.

## ¶ The firste Chapiter treateth of Fractions, or broken nombers, and the difference thereof.

BRoken nomber is as muche as a parte or many partes of one, whereof there are two noumbers with a line betwene them bothe: That is to saie, the one whiche is aboue the line, is cal led the numerator. And the other vnderneath the line, is called the denominator: as by example, three quarters, whiche must bee set doune thus, 4: whereof. 3. whiche is the higher nō ber aboue the line, is called the numerator, and. 4. whiche is vnder the line is called the denominator. And it is al waies conuenient, that the numerator [Page] be lesse in nomber, then the denominator. For if the numerator, and the denominator were egall in value: then should thei represente a whole nomber, thus, as 1/1, 2/2, 3/3, whiche are whole nombers: by reason that the numerators of these, and al suche like maie bee diuided by their denominators, and their quotiētes will alwaies be but. 1. But in case that y^{•} numerator do excede his denominator, then it is more then I whole: as 20/18, is more thā a whole nomber by 2/18, other diffinition dooeth not hereunto appertaine. Furthermore it is to bée vnderstande, that the middest of all broken nombers, is the iuste halfe of. 1. whole, as 5/12, 7/14, 8/16, 9/18 and other like, are the halfes of one whole nomber, wherof doeth growe, and come forthe 2. progressions natural: the one progrediyng by augmentyng, or encreasyng, as these.

½ ⅔ ¾ ⅘ ⅚ 6/7 ⅞ 8/9 9./10 &c.

[Page 44] And thei doe proceade infinitely, and wil neuer reche to make a whole nomber, thus 1/1. And the other progression, dooeth progrede by diminishyng or decreasyng, as thus.

½ ⅓ ¼ ⅕ ⅙ 1/7 ⅛ 1/9 1/10 &c.

And these dooe proceade infinitely, and shall neuer come to make a. ○. whiche signifieth nothyng, but shall euer reiaine some certaine nomber whatsoeuer, whereby it doeth appere that broken nombers are infinite.

## ¶ The seconde Chapiter treateth of the reducyng or bringyng together of twoo nombers, or many broken dissemblyng, vnto one broken semblyng.

REduction, is as muche as to bryng together, or to put in sembleaunce, twoo or many noumbers dissembling one from the other, in reducyng [Page] them vnto a common denominator. For bicause the diuersitie and difference of the broken numbers, doe come of the denominators part, or of diuers denominators, and for the vnderstanding hereof, there is a general rule whose operation is thus. Multiple the Denominators the one by the other, and so you shall haue a new denominator cōmon to al, the whiche denominator diuide by the perticuler denominators, and multiplye euerie quotiēt by his numerator and so you shall haue newe numerators, for the numbers whiche you woulde reduce, as appeareth by thys example followyng.

### ¶ Reduction in common denomination.

IF you wil reduce 3/3 and ⅘ together, you must firste multiplie 1. the. 2. denominators the one by the other, that is to saie 3. by 5. maketh 15. which is your common denominator, [Page 45] that sette vnder the crosse, [...] then deuide. 15. by the denominator 3 and you shal haue 5. whiche multiply by the numerator. 2. & you shall finde 10. sette that ouer the ⅔. and thei are 10/11 for the. ⅔. Afterwardes diuide. 15. by the denominator. 5. and thereof commeth 3. the whiche multiplie by the numerator. 4. and you shall finde 12. whiche sette ouer the heade of the. ⅘. and thei make. 12/15. for the. ⅘: as appeareth more plainer in the margent.

If you will reduce ½, ⅔, ¾, ⅚, together, you must multiplie all the denominators, the one by the other, that is to saie, 2. by. 3. maketh. 6. then. 6. by 4. and mounteth. 24. Laste of all. 24. by. 6. and thereof commeth. 144. for the common denominator. Then, for the firste diuide. 144. by the denominator. 2. and thereof commeth. 72. the whiche multiplie by the numerator 1. [Page] and it is stil 72. set that ouer the ½ and it is 27/144, for the ½: Then deuide 144 by the second denominator 3. and therof commeth 48: the whiche multiplie by the seconde numerator 2. and they are 96. whiche set ouer the ⅔ and they make 96/144, for the ⅔: Then diuide 144. by the thirde denominator 4. & thereof commeth 36. the whiche multiplie by the thirde numerator 3. and thei make 108. whiche sette ouer the ¾. and thei are 108/144 for the 5/4.

Finally diuide. 144. by the last denominator. 6. and thereof commeth 24: the whiche multiplie by the laste numerator 5. and thereof commet 120 Whiche sette ouer the. ⅚. and thei are 120/144, for the ⅚, as appeareth here by pratise.

### ¶ The example.

[Page 46] [...]

### ¶ Reduction of broken nombers of broken.

IF you will reduce the broken 3. of broken together, as thus, the ⅔ of ¼ of ⅘, you must multiplie the numerators the one by the other, to make one broken nō ber, of three broken nombers, that is to saie. 2. by. 1. maketh. 2. and then. 2. by 4. maketh 8. which is your numerator. Then [...] multiplie the Denominators, the one by the other, that is to saie. 3. by 4. maketh. 12. and then [Page] 12. by. 5. maketh. 60. for your denominator, sette. 8. ouer. 60. with a line betwene them, and thei be. 1/60. whiche beyng abreuied, are. 2/15. and so muche are the ⅔. of. 5/4. of. ⅘. as appeareth in the margent.

### ¶ An other example of the same reduction, and of the seconde reduction.

IF you will reduce. ⅔. of. 5/4. of ⅘. the 5/4. of 5/7: and the ½. of the ½. of the. ⅔. of the. ⅓. Firste, it behoueth you of euery partie of the broken nōbers, to make of eche of them one broken, as by the third reduction is taughte: That is to saie, in multipliyng the numerators by numerators, and denominators by denominators: firste, for the firste parte, whiche is ⅔. of ¼. of ⅘, you muste as is said before, multiplie. 2 by. 1. and then by. 4. and you shall haue. 8. for the numerator, likewise multiplie. 3. by. 4. and the producte by. 5. and you shall [Page 47] haue. 60. for the denominator, so thei make. 8/60. whiche beyng abreuied, are 2/15 for the first parte, that is to saie, for the ⅔ of [...] of ⅘, secondly for the ¾ of 5/7 multiplie likewise the numerator. 3. by. 5 maketh. 15. for the numerator, and multiplie. 4. by. 7. maketh. 28. for the denominator, and then thei bee. 15/28. for the seconde parte, that is to saie, for the ¾ of 5/7. Thirdly, for the ½ of ½ of ⅔ of ⅓. multiplie the numerators thone by the other, that is to saie. 1. by. 1. and then by. 2. and laste by. 1. and all maketh but 2 for the numerator, likewise multiplie. 2. by. 2. maketh. 4. and 4. by 3. maketh. 12. and then. 12. by. 3. maketh. 36. for the denominator, and thei are 2/36, whiche beyng abreuied maketh 1/18. for the thirde parte, that is to saie, for ½ of the ½. of ⅔ of ⅓. Last of al, take the 2/15 the 15/28. and the 1/18. and reduce them according to the order of the second reduction, and you shall finde 1000/7 [...]60. for the 2/15. And 4050/ [...]500 for the 15/28. And 426/7560 for 1/18: and thus are broken nombers of broken, [Page] reduced, as appeareth by by practise. [...]

### ¶ Reduction of broken nombers, and the partes of broken together.

IF you will reduce ⅓. and the ½ of ⅓. together, to bryng them into one broken nomber, you muste firste sette doune the ⅓. and ½. as appeareth in the margent with a crosse betwene [...] theim, and then multiply the twoo denominators. The one by the other, that is to saie, 2. by 3. maketh. 6 set that vnder the crosse, then multiplie the firste numerator, one by the laste denominator twoo, and that maketh. 2. vnto the whiche [Page 48] adde the laste numerator one, and thei be three, whiche set aboue your crosse, so you shall finde that the ⅕ and the ½ of [...]/3. doe make 3/6. whiche beyng abreuied doeth make ⅓, which is as much as the. ⅓. and the. ½. of. ⅓. Likewise if you will reduce the ⅔, and the ¼. of ⅓. you must doe as before, set doune the ⅔ and ¼ with a crosse betwene thē, & then multiplie the twoo denominators, the one by the other, that is to saie. 3. by. 4 maketh. 12. whiche set vnder the crosse [...] as you see in the margent, and then multiplie the firste numerator 2 by the last denominator. 4. and thereof commeth. 8. whereunto adde the laste numerator. 1. and that maketh: 9. whiche sette ouer the crosse, so shall you finde that the ⅔. and the ¼ of ⅓ are worthe 9/ [...], whiche abreuied, dooe make ¼, as appeareth by example in the margent.

### ¶ Reduction of whole nombers and broken together into a Fraction.

IF you will reduce whole nō ber 6. with broken, you muste bryng the whole nomber into brokē, as by this example maie appere: reduce 17. ⅝ into a brokē nomber, firste you must multiple the whole nomber. 17. by the denominator of the broken, whiche is. 8. in saiyng. 8. tymes. 17. doe masse. 136. vnto the whiche you muste adde the numerator of. ⅚ whiche is. 5. and al amounteth to. 141. whiche sette ouer. 8. with a line betwene theim, and thei will bee 141/8 so muche is. 17. ⅝. worthe in a fraction, as it appeareth here by practise. [...]

[Page 49] In case you haue whole number and broken to bee reduced, with broken you muste bringe the whole number into his broken, in multiplyinge it by the denominator of the broken number going therwith, and adde thereunto the numerator of the saide broken number, as in the laste example, and then reduce that broken number wyth the other broken, as here appereth by this example. Reduce 10. ⅔ & 4/7 togither, first bring 10⅔ al into thirds, as by the syxt reduction, and you shal finde [...], then reduce the 32/3 and 4/7 togither, by the fyrste reduction, and you shall finde 224/21 for the 32/3: and [...] for [...] as appereth here by practise. [...]

[Page] Also in case you haue in bothe partes of your Reduction, as well whole nū ber as broken, you must alwayes put the whole into their broken (as by the syxte reduction) of either part.

### ¶ Example.

If you wyll reduce 12. ¼ wyth 14. ⅔ to bringe them into one denomination, first bringe the 12. ¼ all into fourthes, and you shall fynde 49/4: then likewise reduce 14. ⅔ all into thirdes, and you shall haue 4 [...]/3, for the 14. ⅔, then reduce [...] and 44/3 togither, by the order of the firste Reduction, and you shall fynde [...] for the [...]. And [...] for the 14. ⅔ as here by practise dothe plainlye appere. [...]

## ¶ The thirde Chapter treateth of abbreuiation of one greate broken number into a lesser broken.

ABbreuiation is asmuch as to set downe, or to write a brokē nūber by figures of lesse signification, & not diminishing y^{•} value thereof. The whiche to doe, there is a rule whose operation is thus, diuide the numerator and likewise the denominator, by one whole number, the greatest y^{t} you maye fynde in the same broken number, and of the quotient of that numerator, make it the numerator, and likewise of that of the denominator, make it your denominator, as by example.

1. If you wyl abbreuiat 54/8 [...], you shal vnderstande that the greatest whole number that you maye take, by the which you maye diuide the numerator & denominator is 27, which is the half of y^{•} numerator, & that is a whole number, for you can not take a whole [Page] number oute of the denominator. 81. but that there will bee either more or lesse than a whole number, therefore if you diuide 54. by 27. you shall [...] finde 2. for the numerator, likewise if you diuide 81. by 27. you shall finde 3. for the denominator. then put. 2. ouer the 3. with a line betwene thē, and you shall find ⅔ and thus by this rule the 53/81 are abreuted vnto ⅔, as appeareth in the margent, and so is to be vnderstande all other.

¶ The forme & maner how to finde oute the greater number, by the which you mai wholy diuide, y^{•} numerator & denominator (to thende y^{t} you may abreuiat them) is thus.

[Page 51] First, diuide the denominator by hys numerator, and if anye number doe remaine, let your diuisor be diuided by the same number, and so you must continue vntill you haue so diuided y^{•} there maye nothing remaine, then is it to be vnderstande, that your last diuisor (wherat you did ende, and that o did remaine after your last diuision) is the greatest number, by the whiche you must abreuiat, as you did in the laste example, but in case that your last diuisor be 1. it is a token that the same nūber can not bee abreuied. Example, of 54/81 diuide 81. (whiche is the denomination) by 54. which is his numerator, and there resteth 27. then diuide 54. by 27. and there remaineth nothinge, wherefore your last diuisor 27. is the number, by the whiche you must abreuiat 54/81 as in the laste example is specifyed.

### ¶ An other stile of abbreuiation.

[Page] 2. Mediate the numerator, and also the denominator of your fraction in case the noumbers be euen, that is to saye, take alwayes the halfe of the numerator, and likewise of the denominator, and of the mediatiō or halfe of the numerator, make the numerator, also of ½ the denominator, make your denominator, and so continue as often as you may in takinge alwayes the ½ of the numerator, and semblablie of the denominator, or els see if you may abbreuiate the numbers which doe remaine, by 3. by 4. by 5. 6. 7. 8. 9. or by 10. for you must abbreuiate them as often as you can by any of the saide numbers, and it is to bée noted, that with whatsoeuer number of these, you doe abbreuiate the Numerator of your Fraction, by the same you muste abbreuiate likewyse the Denominator, so continuynge vntil they can no more bee abbreuied. And it is to bee vnderstande that if the Numerator and the Denominator [Page 52] be euen numbers, as you maye know when the fyrste fygure is an euen nūber, or a [...], thē maye you perceaue if both the Numerator and the Denominator may be abbreuied by 10. by 8 by 4. or by 2. although y^{t} some times they maye bee abbreuied by three.

And if they be odde numbers, then muste you consider if they maye bee abbreuied by 9. by 7 by. 5. or by 3: but when the first number, as well of the Numerator, as of the Denominator are euen numbers, then may you well knowe that suche numbers maye bee abbreuied be 2. as is aforesaide. And if you adde the fygures of the Numerator togither, in suche manner as you doe in makynge the proofe by nyne in whole Numbers: that is, if you fynde 9. [...] appeareth that you maye abb [...] that number by 9. And lykewise by 3. and sometimes by 6. if you fynde 6. it maye bee abbreuied by 6. and [Page] alwaies by 3. if you finde 3. it is a signe that you may abreuiate by 3. And by whatsoeuer nomber that you doe abreuiate the numerator, by the same must you abreuiate likewise the denominator, and if the first figures of the same nomber be. 5. or 0. you maye abreuiate them by 5. but if the firste fygures be both 0. they may be abreuied by 10. in cutting awaye the twoo Cyphers thus, as [...] whiche maketh 2/ [...], & sometimes by 100. thus, as [...] in cutting away the foure ciphers after this sorte, [...] and then the 100/200 doe make ½, and after this maner haue I set here diuers examples, althoughe that all numbers cannot be abreuied by this rule, that is to saye, all those whiche maye bee well abreuyed by the fyrste rule aforesayde. [Page 53] [...]

3. Furthermore you shall vnderstande that sometimes it happeneth, that all the fygures of the numerator are egall vnto them of the denominator, which when it so happeneth, you maye thē take one of them of the numerator, and also one of them of the denominator, and it shall bee abreuyed as 555/888, beinge abreuiated after this maner commeth to ⅝. And yet it happeneth sometimes, that two, or manye fygures of the numerator are proportioned vnto two, or many fyguree [Page] of their denominators and the other fygures of the same number doe beholde the one the other in thys proportion? Then may you take twoo or many fygures, as well of the numerator as of the denominator, and by this maner the same number shall bee abbreuied, as 4747/ [...] whiche beinge abbreuied by this rule, do come to [...].

4. Also it happeneth somtimes that you woulde abbreuiate one number vnto the semblaunce or likenesse of another. And for to knowe if the same maye by abbreuied, and also by what number it maye bee abbreuied, you must diuide the numerator of the one number, by the numerator of the other, and likewise the denominator of the one, by the denominator of the other, for in case that after euery diuision there doe remaine 0. and that the twoo quotiens be [...] all, then is one of them the number by the whiche the saide fraction must be abbreuied, as by exāple of 11 [...]/ [...]. I woulde knowe if [Page 54] they maye be abbreuied vnto 5/9, and for to doe this, you must diuide 115. by 5. and you must diuide 207. by 9. and there will come into bothe the quotients 23. by the which it appeareth that this number may be abbreuied by 23. [...]

## ¶ The 4. Chapter treateth of the assembling of two or many broken numbers togither, as by example.

FOr to adde broken numbers togither, there is a generall rule, which is thus, if the nū bers be vnlike y^{t} one to the other, you must reduce thē into a cōmō denomination, whiche after you haue reduced thē, you must then adde both y^{•} numerators togither, & set y^{•} product of the saide addition ouer the crosse, & diuide the same by the common denominator, as by this exāple folowinge.

[Page] 1. If you wyll adde ⅔ wyth ¾, you muste fyrst reduce the twoo fractions bothe into one denomination, accordinge to the introduction of the fyrste reduction, that is to saye, in multiplyinge the denominator of the firste fraction whiche is 3, by the denominator of the other fraction whiche is foure, and they make 12. for your common denominator, [...] the which 12. set vnder the crosse, thē multiplie y^{t} fyrst numerator 2. by the last denominator 4. and thereof commeth 8. whiche sett ouer the ⅔, and then multiply y^{•} last numerator 3. by the fyrste denominator 3. and therof commeth 9. whiche you must set ouer the ¾, then adde the numerator 8. with the numerator 9. & they make 17. which set ouer the crosse, and then your fraction wyll be 17/12 whiche is the addition of the ⅔ wyth ¾. And bycause [Page 55] your numerator 17. is greater thā his denominator 12. therfore you must diuide 17. by 12. and thereof will come 1. and 5. remaining, which 5. are worth [...]/12, and so muche are the ⅔ added with ¾ as doth appere.

### ¶ Addition in broken. numbers.

2. Also if you will adde ½, ⅔, ¾, ⅘, togither, you muste fyrste adde the [...] and ⅔ together, accordynge to the doctrine of the laste rule, and you shall finde 7/6: then adde ¾ and ⅘ togither by the saide last chapter, and they make [...]. Then finally adde the 7/6 (whiche came of the ½ and ⅔ added together) with 31/20, & you shall fynde by the foresaide addition that they amounte vnto 326/120, Wherfore diuide 326. by 120. & therof cōmeth 2. and 86. remaineth whiche is 86/120 of one whole, & thei being abreuied do make 43/60: & thus the ½, ⅔, ¾, ⅘, added together doe amount to 2. 4 [...]/ [...], [Page] as here vnder doth appere. [...]

### ¶ Addition of broken numbers of broken.

3. Furthermore, if you will adde the broken numbers of broken togither, as to adde the ⅔ of ¾ of ⅘ with the ⅚ of ½ of [...]: first you must reduce the nū bers accordyng to the order of y^{•} fourth reduction, in multiplying the numerator of the fyrst 3. fractions, the one by the other, and of the producte make your numerator, & likewise you must multiplie y^{•} de nominators of the foresaid thre fractions, the one by the other [Page 56] and of the product make your denominator, and you shal finde 24/60 for the first three broken numbers, which being abbreuied do make ⅖, then reduce the other 3. fractiōs, by the said fourth reduction, in multiplyinge the numerators by numerators, & denominators, by denominators, as you did by the first 3. broken numbers, & you shal finde 25/96 then must you adde y^{•} ⅖ whiche came of the fyrste 3. broken numbers, & 25/56 whiche are of the last 3. fractions, both togither, by the instruction of y^{•} first additiō & you shall find 317/480 whiche cannot be abreuied, but is the product of y^{•} addition: so muche are the ⅔ of ¾ of ⅘ added with the ⅚ of [...] of ⅝ as hereafter by practise doth euidently appere. [...]

### ¶ Addition of broken number the partes of broken together.

4. Likewise if you will addde the ½, and the ½ of ⅓ with the ⅘ and [...]/4 of [...], you must reduce the 11/32 by the fyft reduction and therof cōmeth 3/6 for the [...] & [...], of one of the saide thirdes, then reduce the ⅘ and ¼ by the saide fift reduction, and thereof commeth [...].

Last of all adde the ⅚ and 17/20 togither according to the firste rule of addition, and you shall fynde 20 [...]/ [...] which beinge diuided bringeth 1. & [...] part remaining, whiche abrouied maketh 41/60 and thus you doe perceaue that the 2/1 & [...] added with the ⅘ and ¼ doe amounte vnto 1. [...] as hereafter by practise doth plainly appere. [...]

### ¶ Addition of whole nomber and broken together.

5. Also if you will adde. 12. ⅘. with 20. ⅚, you maie (if you will) adde. 1 [...]. and. 20. together, and thei make. 32. and then adde the twoo broken nombers together, that is to saie ⅘, and ⅚, by the order of the firste addition, and thei make 49/ [...]: therefore diuide 49. by [...], and thereof commeth 1. and [...] partes remaine, whiche. 1. you must adde vnto the. [...]2 and the whole additiō wil be. 33 [...], or otherwise, you maie reducte 12. ⅘. into the likenesse of a fraction by the sixt reduction, and thei will bee [...], and likewise by thesame reduction, reduce. 2 [...], and thei be [...], then adde [...] with the [...], by the firste addition, and you shall finde. [...]. Therefore diuide [...]09. by. [...]0, and thereof commeth [...]. [Page] as before, and as by practise of thesame bothe the waies, doeth here vnder appeare. [...]

## ¶ The .v. Chapiter treateth of Substraction in broken nombers.

IF you will substracte ⅔| from 2/4 you must firste reduce bothe the fractions into a common denomination by the fyrst reduction, and you shall finde 8/12 for the ⅔, and 9/12 for the ¾. Therefore abate the numerator 8. from the numerator 9. and there remaineth 1/12 as may appere here by practise. [Page 59] [...]

2. But if you haue a broken nōber, to bee substracted from a whole nomber, you must borow one of the whole nomber, and resolue it into a fraction of like denomination, as is the fraction, which you would abate from the same whole nomber, and then abate the saied fraction there from, and you shall finde what doeth remaine, as by this example. If you abate ⅘ from. 8. you must borowe out of the said. 8. and resolue it into fiftes like vnto the fraction, because it is 4. fiftes, that. 1. will bee 5. fiftes thus 5/5. therefore abate ⅘. from 5/5. and there will remaine ⅘, and substract that. 1. whiche you borowed from 8. and there doeth remain. 7. and also the ⅕. Thus the ⅘ being substracted frō. 8. doeth leaue. 7. ⅕, as by practise [Page] doeth plainly appere [...]

3. If you will substract broken nomber, from whole nomber, and broken beyng together: thus, as if you would substract ¾. from. 6. ⅚, you maie by the first substraction, abate ¾. from ⅚, and there will remaine 1/12, and the 6. doeth still remaine whole, because that 5/4. beyng abated from. 6. ⅚. leaueth. 6. 1/12. as appeareth by practise. [...]

Likewise if you will abate ⅔, from 14. ⅖, you muste firste reduce. 14. ⅖. all into fiftes by the. 6. reduction, and thei [Page 59] bee [...], then reduce ⅔. into a common denomination, by the firste reduction, and you shall finde 10/15. for the ⅔: and 21 [...]/15 for the 70/5: then substracte the numerator. 10. of the firste fraction, from the numerator. 216. of the seconde fractiō, and there remaineth 206/15. Therfore diuide 206. by. 15. and thereof commeth 13.11/15, and so muche remain of this substraction, as maie appeare. [...]

4. If you will substracte whole nō ber and broken, from whole and broken, [Page] as thus, if you will substract. 9¼. from. 20. ½. you must reduce. 9. ¼. into fowerthes, and likewise the. 20 ½. into halfes by the sixt reductiō: and you shall find 37/4 for the. 9. ¼. And 42/2. for the 20. [...]/2. Then reduce 57/4 and 31/2 into one denominatiō, accordyng vnto the first reduction and you shall finde 74/8 for the 37/4, and [...]64/ [...] for the 41/2 thā abate the numerator of [...]65/8 and there remaineth 90/8 then diuide 90. by 8. and thereof commeth 11. ½ whiche is the remaine of this substraction. [...]

### ¶ Substraction of broken nombers of broken.

[Page 60] 5. If you will substracte, the ½ of ⅔ of ⅗. from the ⅚. of ¾. of ⅞, you muste firste bryng the ½. of ⅔. of ⅗. into one fraction by the. 3. reduction, and the ⅚ of ¾ of ⅞ likewise into one fraction by the same reduction, and you shal finde 6/30. for the firste. 3. broken nombers, whiche beyng abreuied dooe make. ⅕: and for the other 3. broken nombers, you shall finde 105/192: whiche beyng likewise abreuted dooe make 35/64. then you shall substracte ⅕. from. 35/64. by the instruction of the firste substraction, in reducyng bothe the fractions into a common denomination, as before is dooen, and you shall finde remainyng [...]/3 [...]0, as maie appeare by example. [...]

## ¶ The sixt Chapiter is of multiplication in broken nombers.

FIrste, for to multiplie in broken nomber, there is a rule, whiche is thus, multiplie the numerator of the one fraction, by the numerator of the other. And then diuide that fraction if you maie, or els abreuiate it, and you haue do [...]en: but if there be whole nomber and broken together, you muste reduce the whole nombers into broken, and adde thervnto the numerator of his brokē, and then multiplie, as is before saied, as also hereafter by examples shal more plainly appeare.

1. If you will multiplie ⅔ by ¾, you muste multiplie the numerator. 2. by the numerator. 3. and therof commeth 6. for the numerator. Likewise multiplie the denominators thone by the other, that is to saie. 3. by. 4. and thereof [Page 61] commeth. 12. for the denominator, so that this multiplication commeth to 6/12, whiche beyng abreuied doe make ½. and so muche amounteth the multiplicatiō of the ⅔. by ¾ as by practise. [...]

2. Likewise, if you will multiplie a broken nomber by whole nomber, or whole nomber by broken, whiche is all one, as ⅘. by. 18 or els. 18. by ⅘, you must set. 1. vnder. 18. thus. 18/1: and then multiplie. 18. by the numerator 4. and therof commeth. 72. the whiche diuide by the denominator. 5. and thereof cō meth. 14. ⅖. for the whole multiplication, or otherwise abate from. 18. his [...]/5. parte, whithe is. 3. ⅗, and there remaineth. 14. ⅖. as aboue. [...]

[Page] 3. Also if you wil multiplie a whole nomber, by whole nomber & brokē, or els whole nomber & brokē by a whole nōber, whiche is all one. As by example. If you will multiplie. 15. by. 16. ¾. or els. 16. ¾ by. 15 First reduce. 16 ¾ all into fourthes, in multipliyng 16. by y^{e} deneminator of ¾ whiche is 4. & therof commeth 64 whereunto adde the numerator. 3. & it maketh 67/4, whiche multiplie by 15/1, accordyng vnto thinstruction of the laste example, and you shall find y^{e} product of this multiplicatiō to be 251 ¼ as by practise doth here appere [...]

[Page 62] 4. And if you will multiple a broken nomber, by whole nomber and broken, or els whole nomber and broken by a broken. As by example, if you will multiplie ¼. by. 18. ⅔, or els 18 ⅔. by ¼, whiche is all one: you muste reduce the whole noumber into his broken by the sixt reduction. And you shall finde, 56/3, whiche you shall multiplie by the ¼, after the doctrine of the first multiplication, that is to saie: in multipliyng the numerator. 56. by the Numerator of ¼, which is 1. And it is still 56. because 1. doth neither multiplie nor deuide. And likewise you muste multiplie the Denominator. 3. by the Denominatour. 4. and it maketh. 12. then diuide. 56. by. 12. and thereof commeth. 4. ⅔. And so muche [Page] amounnteth the multiplication of the [...]. ⅔. multiplied by ¼, as by example. [...]

5. If you will multiplie whole nō ber and broken, with whole and broken, you muste firste put either whole nomber into his broken, accordyng to the instruction of the sixte reduction, and [...] multiplie the one numera [...] the other, and of the producte make your numerator. And likewise multiplie the denominators, the one by the other, and thereof make the denominator, then diuide the numerator, by the denominator, and the quotient shalbe the encrease of this multiplication.

### ¶ Example.

If you would multiplie. 12. ⅘. b [...]. ¾: firste by the sixte reduction, the. 12. ⅘ [Page 63] will make 64/5, and the. 6. ¾. will make 27/4 then multiplie the numerator. 64. by the numerator. 27. and thereof commeth. 1728. for the numerator. And then you must multiplie the denominator. 5. by the denominator. 4. and thei doe make. 20. then diuide. 1728. by. 20. and thereof commeth. 86. ⅖. for the whole multiplicatiō, as by exāple. [...]

6. If you will multiplie one broken nomber by many broken nombers, thus: As to multiplie ⅔. by 5/7. and by 4/9: you muste multiplie the numerators of all the fractions, the one by [Page] the other, and of the product make the numerator, that is to saie: 2. by. 5. and thei be. 10. then. 10. by. 4. and thei bee 40. for the numerator. Likewise you must multiplie the denominators, the one by the other, that is to saie. 3. by. 7. maketh. 21. then. 21. by. 9. maketh. 189 for the denominator: then sette. 40. ouer the. 189. with a line betwene thē, and thei make 40/189. And so muche amounteth the whole multiplication of the ⅔, multiplied by 3/7 and 4/9, as by example folowyng. And thus is to be vnderstande of all suche like. [...]

## ¶ The .vii. Chapiter treateth of diuision in broken nombers.

NOte, that in diuision of broken nombers, you muste sette [Page 64] your diuisor downe firste, nexte vnto the lefte hande, and the diuidende or nomber, whiche is to bee diuided alwaies towarde the right hande.

And then multiplie crosse wise, that is to saie, the numeratour of youre Diuisour by the Denominator of the Diuidende, and the producte shalbee the Denominatour, whiche afterwarde shall bee your Diuisour. And likewise you muste multiplie the Denominatour of your firste noumber, that is to saie of your Diuisour: By the Numeratour of the Diuidende, whiche afterwarde shall bee the Diuidende, and that muste bee sette ouer the Crosse, and the Denominatour vnder the Crosse, then shall you diuide the Numeratour by the Denominatour, if any maie bee diuided, if not, you muste abreuiate theim, as hereafter by examples shall more plainly appeare.

1. If you will diuide ¾. by ⅔, you muste sette the Diuisour (whiche is [Page] ⅖) nexte to the lefte hande, and the diuidende ¾. towarde your right hande, with a crosse betwene them: as maie appere by this example in the margente. Then [...] you shall multiplie the numer at or of the ⅔, whiche is. 2. by the denominator of the ¾. which is 4. and thereof commeth. 8. which shalbe your newe diuison: set that. 8. vnder the [...]osse, as the denominator, then multiplie the numerator of the diuidende, that is to saie, of the ¾ whiche is 3. by the denominator of the diuisour, that is to wit, of the. ⅔. whiche is. 3. set that ouer the crosse, and it is. 9. for the numerator, whiche shalbe now the diuidende, or nōber to be diuided. Then finally, you shall diuide. 9. by. 8. and thereof commeth into the quotient. 1. 2/8 and so often times is ⅔. conteined in ¾, as dooeth appeare before in the margente. But in case you would diuide ⅔. by. ¾, you muste likewise sette your [Page 65] diuisor ¾ nexte to your left hande, as is before said. And then procede, as is aboue declared, & you shall finde that ⅔ diuided by ¾ bringeth into y^{e} quotiēt 2/9, whiche can not be diuided nor abbreuied, wherfore it appereth that ⅔ diuided by ¾ bringeth but 2/9 of one vnitie into the quotient as doth appere. [...]

2. Likewise if you wil diuide a broken number by a whole nomber, or els a whole number by a broken, as to diuide ¾ by 13. you shall put 1. vnder 13. and it wil be 13/1 which is your diuisor, set y^{t} toward [...] your left hande, and then multiply 13. by 4. accordīg to the first diuision, & therof commeth 52. for the denominator, [Page] set that vnder y^{e} crosse & multiply 3. by 1. which is 3. for the numerator, that set ouer the crosse, and it is 3/52 as appeareth in the margent. But if you will diuide 13. by ¾ then set the ¾ nexte your left hand and put one vnder 13. as in the last example, & it is [...]/3 set y^{t} toward your right hande thus, as appeareth in the margent [...], and then worke according to y^{e} doctrine of the first diuision, & you shall finde that 13. being diuided by ¾ bringeth into y^{e} quotient 52/4, then diuide 52. by 3. and therof commeth 17. [...]/3, and so oftentimes is ¾ conteined in 13. as doth appere. [...]

3. And if you wil diuide whole nū ber by whole number and broken, or els whole nūber and broken by whole number, as to diuide 20. by 5. ⅚, you shall reduce 5. ⅚ into his broken by y^{e} sixt reduction, & it maketh 35/ [...] for your [Page 66] diuisor, then put 1. vnder 20. And it wyll bee 20/1, then shall you multiply [...] 35. by 1. and 20. by 6. as is taught in the other diuisions, and you shall finde 220/3: then diuide 12. by 35. and you shall fynde in your quotient 3. 3/ [...] & so many tymes is 5. [...]/6 conteined in 20. as in the margent doth appere.

But if you will diuide 5. 3/6 by 20. you must diuide 35. by 120. which you can not, wherefore you shall abbreuiate 35/120, and thereof commeth. 7/24.

4. If you will diuide a broken number by whole number and broken, or els a whole number and broken, by a broken number. As to diuide ¾ by 13. ⅔, you muste reduce 13. ⅔, into hys broken, by the syxte reduction [Page] And they be 41/3 for your diuisor, then multiplye 41. by 4. & they make 164. for your denominator, likewise multiplye 3. by 3. and they make 9. for the numerator, and then will your sūme be 9/104. But yf you will diuide 13. ⅔ by ¾ then you must diuide 164. by 9. and you shall fynde 18. 2/9.

5. If you will diuide whole number and broken, by whole number & broken, as to diuide 7. ¾ by 13. ⅔ you must reduce the whole numbers into their broken, by the doctrine of the sixt reduction, & you shall fynde 31/4 for the 7. ¾, & 41/3 for the 13. ⅔. Then sett downe 41/3 towarde [...] the left hande bycause it is your diuisor, and the 31/4 towarde the right hande, and multiplie 41. by 4. for your denominator, and thereof commeth 164. Likewise multiply 31. by 3. for your numerator, and it amounteth to 93. the whiche diuision will bee thus [...] as before [Page 67] doth appere.

But if you will diuide 13⅔. by 7. ¾ you muste contrariwyse to the other example, diuide 164. by 93. and you shall fynde in the quotient 1. 71/93.

6. The broken numbers of broken, muste be diuided in suche maner as broken numbers are, & there is no difference, sauing onely that of many broken numbers you must make but two broken numbers, that is to saye y^{e} diuisor, and the diuidend, or number that is to be diuided, example. If you will diuide the ¾ of ⅗ of ½, by the 2/2 of 4/7. For the fyrst, the ¾ of ⅗ of ½ are 9/4. by the thirde reduction: and the ⅔ of 4/7 are by the same Reductiō 8/21, then haue you 8/21 for your [...] diuisor, & 9/40 for your nū ber to bee diuided, then multiply 8. by 40. which maketh 320. set that vnder the crosse and multiply 9. by 21: & thereof cōmeth 189. which set ouer the crosse for the numerator, and they [Page] make 189/120 for this diuision as doth appere.

But if you woulde diuide 1/21 by 9/200. you must worke contrary to the laste example, that is to saye, you must diuide 320. by 189. And therof commeth in the quotient 1. 131/189.

## ¶ The eyght Chapter treateth of duplation, triplation, and quadruplation of all broken numbers.

IF you wyll double any broken number, you shall diuide y^{e} same by½: likewise if you wyll triple any fraction you muste diuide it by ⅔. And for to quadruple any broken nū ber, you shall diuide it by ¼, and so is to be vnderstande of all other.

### Example of duplation.

IF you will double ⅜ you shal diuide 3/ [...] by ½, and thereof commeth 6/8, which being abbreuied [...] are ¾: as by example.

[Page 68] Or otherwise, in case the denominator of any fraction bee an euen number, you may take halfe the sayde denominator, without anye other operation, and the numerator to abyde still y^{e} numerator, vnto the said halfe of the denominator of the Fractiō, as by the other exāple before rehearsed: that is to say of [...], take ½ of 8. which is 4. and that is the denominator, and 3. remaineth stil numerator to 4. and it maketh ¾ and so of all other. But in case the denominator bee an odde nomber, that is to say, not euen, then you may multiply the numerator by 2. or els double y^{e} numerator, whiche is al one thing; and that fraction shall bee doubled. Example, if you will double [...]/5 you must onely multiply the numerator 3. by 2. & they be 6. whiche maketh that fraction to be [...], the which 6. being diuided by 5. bringeth 1. ⅕ and so much is the double of 3/ [...].

### Example of Triplation.

[Page] If you wil triple ⅗ you muste diuide [...]/5 by [...]/3 and thereof commeth 9/5 whiche beinge diuided bringeth 1⅘, or otherwise, bicause the denominator is an odde number you maye multiplie the numerator 3. by 3. and therfore commeth 9. which maketh [...] as before.

### Example of quadruplacion.

If you will quadruple ⅘, you shall diuide ⅘ by ¼ and thereof commeth ⅘ which 16. being diuided by 5. bringeth 3 [...]/5, or otherwise, bicause the denominator of the fraction is an odde nūber, you shall multiplie the numerator of the [...] that is to say 4. by 4. and therof commeth 16. the whiche diuide by 5. and you shall finde 3. [...]/5 as before, and this sufficeth for duplacion, tripiacion and quadruplacion.

## ¶The 9. Chapter treateth of the proofes of broken numbers. And first of Reduction.

[Page 69] IF you doe abbreuiate y^{•} broken nū bers whiche bee reduced, you shall retourne them into their firste estate: as by example, if you reduce [...]/5 wyth ⅘ you shall fynde 10/ [...] and 12/ [...], then abbreuiate 10/15 and you shall fynde ⅔, abbreuiate likewise [...] and thereof commeth ⅘ as before.

### The proofe of Abbreuiation.

IF you doe multiplie that number whiche you haue abbreuied by that or those numbers, by the whiche you haue abbreuied them, you shall returne them againe into their firste estate. Example, if you wyll abbreuiate 32/48 by 16. in takyng y^{•} [...] part both of the numerator, and also of the denominator, you shall finde ⅔, y^{•} proofe is thus, you must multiplye bothe the numerator & denominator of 2/ [...] by 16. that is to say, three by 16. maketh 48. for the denominator, & 2. by 16. maketh 32. for the numerator, then set the numerator 32. ouer the denominator 48 [Page] and they be 32/48 as before.

### The proofe of Addition.

If you doe substract one of the nū bers, or manye of them (which you haue added) from the totall summe, there shal remaine the other, or others; Example: if you do adde 1/ [...] with ¼ you shall fynde 7/12. The proofe is, is if you substract ⅓ from 7/12 you shall fynde remaining the other number whiche is 2/4, or els yf you doe substract ¼ frō [...] there will remains the other nomber, which is 1/ [...].

### The proofe of Substraction.

If you do adds that number whiche remaineth, with the number whyche you did substract, you shall fynde the totall summe, oute of the which you made y^{•} abateinēt: or otherwise, if you adde the twoo lesser nōbers togither, you shall finde the greater. Examplet [Page 70] if you doe abate or substracte ¼ from ⅓ there will remaine [...]/12. The profe is thus: you muste adde 1/12 & ¼ togither, and you shall fynde [...]/3, whiche is the greatest nomber.

### The proofe of multiplication.

If you diuide the producte of the whole multiplication, by the multiplicator, you shal fynde in your quotient, the multiplicande or nomber by the which you haue multiplyed: or els if you diuide the totall sōme which is come of the multiplication, by the multiplicande: you shall finde in the quotiēt the multiplicator. Example, if you multiply [...] by ⅘, the product of this multiplication will bee 8/15. The proofe is thus: you shal diuide 8/15 by y^{•} multiplicator ⅘, and therof cōmeth ⅔. Or els diuide 8/15 by ⅔ & you shall finde the ⅘ which is the multiplicator.

### The proofe of Diuision.

[Page] If you doe multiplye the quotient by the diuisor, you shall finde the number which you did diuide, y^{t} is to say, your diuidende. Example: if you diuide ⅔ by ¾, your quotient will bee 2/9 y^{•} proofe is thus, you must multiply 8/9 by ¾, and thereof commeth 24/36 whiche being abbreuiated art ⅔ whiche is your diuidende, & by this maner all whole numbers haue their proofes as well as broken numbers.

## ¶ The tenth Chapter treateth of certaine questions done by broken numbers. And first by Reduction.

FInde twoo numbers, where of the 2/ [...] of the one number may bee egal vnto the [...] of the other. Aunswere: you shall reduce 2/7 & ⅜ crossewise, and you shall finde 16. ouer the ⅔ and 21. ouer the ⅜, which are the two numbers that you seeke: for the ⅜ of 16. are 6. and so are the 2/ [...] of 21. lykewise 6. wherefore you may perceiue that the [Page 71] the ⅜ of 16. which are 6. are egall vnto the 2/7 of 21. whiche is also 6.

2. Finde twoo numbers, wherof y^{•} 2/ [...] of the one may be double to the ¼ of the other. Aunswere: double ¼ & you shal haue 2/4, which being abbreuiated is ½: thē reduce ⅔ & ½ crossewise, & you shall finde 4. ouer the ⅔ & three ouer the ½ which are the two numbers that you seeke. For the 2/1 of 3. which is 2. is double vnto the ¼ of 4. which is but 1.

3. Finde two numbers whereof the ⅔ and the ¼ of the one, maye bee egall vnto the ¼ & ⅕ of the other. Aunswere: Adde the ⅓ and ¼ togither, and they make 7/12 then adde ¼ and [...]/5 togither, & they are 6/20, then reduce 7/17 & 9/20 crossewise, & you shall haue 140. ouer the 7/12 & 108. ouer the 9/20, whiche are the two nūbers that you seke. For 63. whiche are the 7/12 of 108. are also the 9/20 of 140.

4. Finde two numbers, wherof the [...]/2 the and the ¼ of the one of them, maye by egall vnto the ⅕ the ⅙ and 1/7 of the other number. Aunswere: first you [Page] must adde ½, ⅓, and ¼ togither, & they make 13/12: then adde ⅕, ⅙ and 1/7 togither, & they make 207/210. Then reduce 13/12 and 107/210 crossewise, as by the fyrste question of reduction, and you shall finde 2730. ouer the 13/12 and 1284. ouer the 107/210, whiche are the twoo nombers that you seeke: for 1391 which is the ½ the ⅓ and ¼ of 1284. is lyke to the 11/56 & ¼ of 2730, which is also 1391.

5. Finde three nombers, whereof the ⅖ of the first, the 4/7 of the seconde, & the 4/9 of the thirde, maye be egall the one to the other. Aunswere: set downe the 23/57 and [...], and then multiplie the Denominator of the ⅖ that is to saye 5. by the Numerators of the other twoo Fractions, that is to saye, by the Numerator of 3/7, and by the Numerator of 4/9, whiche is 3. and 4. And thereof commeth 60. for your fyrste nomber, then shall you multiplye the Denominator of the 3/7 whiche is 7. by the Numerators of ⅕ and 4/9, that is to say by 2. and 4. and thereof commeth 56. [Page 72] for the seconde number: Then multiplie the Denominator of 4/9, that is 9. by the Numerators of ⅖ and [...]/7 that is by 2. and by 3. and thereof commeth 54. for the third nomber

And thus the ⅖ of 60. which is 24. is likewise the 3/7 of 56. whiche is the second nomber and the 4/9 of 54. whiche is the thirde nomber.

6. Finde three nombers, of whiche the fyrste and the seconde maye bee in suche proporcion as ½ and ⅓, and the seconde and thirde in suche proportion as ¼ and ⅕. Aunswere: reduce ½ and ⅓ crossewise, and you shall haue 3. ouer the ½ and 2. ouer the ⅓, then reduce ¼ and ⅕ in lyke maner, and you shall fynde 5. ouer the ¼ and 4. ouer the 3/ [...]. Then say by the Rule of three, [...] 5. do gyue me 4. what shall twoo gyue me, whiche is the seconde proportionall, multiply the seconde nomber 4. by the thyrde nomber twoo, and therof commeth eyght, the whiche diuide [Page] by the first number 5. and therof commeth 1. ⅗ for the third proportionall, and you shal fynde that 3. 2. 1. ⅗ are the three nombers proportionall that I demaunde, or els 15. 10. & 8. in whole numbers.

1. What number is that, vnto the whiche if you doe adde 13. the whole amounteth to 31. Aunswere: rebate 13. from 31. and there wyll remaine 18. which is the number that you seeke.

2. What number is that, vnto the which if you adde ⅖ the addition wyll be ⅚. Answere: abate ⅖ from ⅚, and there will remaine 13/30, whiche is the number that you desire.

3. What number is that, whereunto if you adde 7. ⅔, the whole additiō will be 12. ¼. Aunswere: abate 7. ⅔ frō 12. ¼. & the remaine will be 4 7/12 which is the number y^{t} you desire to know.

4. What number is that, whereinto if you adde the ¾ of it selfe, that is to say, of the nūber that your séeke, the whole addition may be ⅚.

[Page 57] Aunswere: Here followeth a generall rule for all suche like questions. First, of. 3. whiche is the numeratour of. ¾. make still the numeratour, and likewise of. 3. and. 4 together, whiche is bothe the numerator, and the denominator of the. ¾. make your denominatour, so you shall finde. 3/7, then take the 3/7 of ⅚, whiche is 15/42, or 5/14; and substracte theim from ⅚, and there will remaine 10/21, whiche is the nomber that you seke.

5. What nomber is that, vnto the whiche if you adde his owne ⅔. that is to saie ⅔. of it self, the whole addition shall bee. 20. Aunswere: do [...]e as in the laste question: of the numeratour of ⅔ that is to saie, of. 2. make stil your numerator. And likewise of the numerator. 2. and the denominator. 3. of the ⅔, make of them bothe, your denominator, and you shall finde ⅖, then take the ⅖. of. 20. whiche are. 8. And abate them from. 20. and there will remain 12. whiche is the nomber that you desire, [Page] and so is to bee dooen of all suche like reasons.

1. What nomber is that, from the whiche if you dooe abate. 17. the reste maie bee. 19. Aunswere: adde. 17. and 19. together, and you shall finde. 36. whiche is the nomber that you seke.

2. What nomber is that, from the whiche if you abate ⅗, the rest maie bée ⅛. Aunswere: adde ⅗ and ⅛ together, and you shall finde 29/40, whiche is the nomber that you demaunde.

3. What nomber is that, from the whiche if you deduct. 13. ½ the rest maie be. 5. 5/7. Aunswere: adde. 13. ½ and. 5. 5/7. together, and there of commeth. 19. 1¾, whiche is the nomber that you seke.

4. What nomber is that, from the whiche if you substracte his ⅖, the reste maie bee. 12. Aunswere: and a rule for A rule. suche like reasons, that is to saie, from the denominator of 5/2. whiche is. 5. abate. 2. whiche is his numerator, and there resteth. 3. for the denominator, and thus of ⅖. you haue you made ⅖, [Page 74] then take the [...] of. 12. whiche are 8: and adde them vnto. 12. and therof cōmeth 20. for the nomber whiche you desire.

5. What nomber is that, from the whiche if you doe abate his ¾, the reste maie bee 8/ [...]. Aunswere: from the denominator of ¾, whiche is. 4. substracte his numerator. 3. and there resteth. 1. Thus of ¾. you haue made. 3/1. Then multiplie 5/ [...] by 8/9, and therof commeth 2. ⅔, the whiche adde vnto 8/9, and you shall haue. 3. 5/9, whiche is the nomber that you seke.

6. What nomber is that, from the whiche if ye abate his ⅘, the rest maie be. 12. ⅔. Aunswere: Doe as you did in the laste question, and you shall finde that the ⅘. will bee 4/1. And therefore multiplie. 12. ⅔ by 4/1, and thereof commeth. 50 ⅔, the whiche adde vnto. 12 ⅔, and you shall finde. 93. [...], for the nomber that you demaunde. And thus of all like questions.

1. What nomber is that, which beyng multiplied by 13. the whole. Mul [...]iplication [Page] shal m [...]nte to. 221. Aunswere: [...]de. 221. by. 13. and thereof comnieth. 17. whiche is the nomber that you seeke.

2. What nomber is that, whiche beyng multiplied by. 15. the whole multiplication wil amount to ¾. Aunswere: diuide ¼. by 15/1. and thereof commeth 1/20. whiche is the noumber, that you seeke.

3. What nomber is that, whiche beeyng multiplied by. 21. the whole multiplication will bee: 16. ⅘. Aunswere: diuide. 16. ⅘. by. 21/5, and you shall finde ⅘, whiche is the nomber that you demaund [...].

4. What nomber is that, whiche beyng multiplied by ¾. the multipliplication will amounte to. 18. Aunswere: diuide 18/1 by ¾, and there of commeth. 24. whiche is the nomber that you desire to knowe.

5. What nomber is that, whiche if it bee multiplied by ⅔. the whole multiplication will bee. [...]/4. Aunswere: diuide [Page 75] ¼ by ⅔. and the quotient will bee ⅜, whiche is the nomber that you require to knowe.

6. What noumber is that, whiche beyng multiplied by ⅝, the product of that multiplication will bee. 16. ⅔. Aunswere: diuide. 16. by ⅔. by [...]. and thereof commeth. 26. ⅔, whiche is the nomber that you seeke.

### ¶ Here ensueth other necessarie questions, whiche are wrought by multiplication in broken nombers.

I Demaunde howe muche the ⅝ of. 20. shillynges are worth or what are the ⅝. of. 20. shillynges. Aunswere: you muste multiplie ⅝ by 1 [...]/5, and the product will bee 100/8, therefore diuide. 100. by. 8. and thereof commeth. 12. ½, whiche is to saie. 12. shillynges. 6. pence, and so muche are the 5/ [...]. of. 20. shillynges worthe.

[Page] 2. I demaunde what the ¼ of ⅚ of [...] pounde of money are worthe, that is to sa [...]e, of. 20. shillynges. Aunswere: multiplie. ¾. by ⅚. And thereof commeth 5/2. Then take the 5/ [...] of 20. shillinges, as in the last questiō goyng before and you shall finde. 12. shillynges. 6. pence, and so muche are the ¾ of ⅚, of 20 shillynges worthe.

3. I demaunde what the ⅔ of. 8. pence [...]/2 are worth. Aunswere: multiplie. 8. [...]/2 by ⅔, or els ⅔ by. 8. & [...]/2 whiche is all one, and you shall finde 34/6. Then diuide 34 by. 6. and your quotient will bee fiue pence ⅔, and so muche are the. [...]. of. 8. pence [...] [...]the.

4. What are the ¾. of. 14. pence [...], Aunswers multiplie. 14. ⅗ by ¾, and therof commeth 219/20. Therefore diuide 219. by. 20. and your quotiente will bee. 10. pence 19/20. and so muche are the [...]/4 of. 14. [...]/5.

5. How many [...] of sworth [...] [...]contain [...]r in. 7. ⅔. Aunswer multiplie, 7. ⅔ by 4/ [...] (because one whole [Page 76] containeth. 4. quarters) and thereof commeth, 30. ⅔, and so many quarters are in the. 7. ⅔. that is to saie. 30. quarters, and ⅔ of a quarter.

6. Howe many thirdes are in ¾ and [...]/2, that is to saie in. 3, quarters, and [...]/2 of one quarter, whiche are ⅞ by the [...] reduction. Aunswere: multiplie 7/ [...] by [...] (for because that in one whol [...] are cō tained. 3. thirdes) and thereof wil come ⅔ and ⅝ of a third, and so many thirdes are in ¾ and ½ or in ⅞, whiche is al one

### ¶ Question doen by diuision in broken nomber.

WHat nomber is that, which beyng diuided by. 17. the quotiente will bee. 13. Aunswere: multiplie. 17. by. 13. And thereof commeth. 221. whiche is the nomber that you seke.

2. What nomber is that, which beyng diuided by ¾, the quotient will be 21. Aunswere: multiplie 23/1. by [...]/4 and therof cōmeth 63/4. Then diuide. [...] by 4 [Page] and thereof commeth. 15. ¾, whiche is the nomber that you seke.

3. What nomber is that, which beyng diuided by. ⅛, the quotiente will bee ⅔. Aunswere: multiplie 2/1. by ⅛. and thereof commeth 2/24, whiche beeyng abreuiated are 1/12. for the nomber, whiche you require.

4. What nomber is that, whiche beyng diuided by ⅘. the quotient will be. 16. ⅔? Aunswere: multiplie. 16. ⅔, by ⅘, and thereof commeth 200/15. Therefore diuide. 200. by. 15. and thereof cō meth. 13. [...]/3, whiche is the nomber that you desire to finde.

5. What nomber is that, which beyng diuided by. 13. ⅓, the quotient will bee. 20. Aunswere: multiplie [...], by. 13 [...] and thereof commeth 900/3, then diuide 800. by. 3. and thereof commeth. 266 [...] for the nomber, whiche you seeke,

6. What nomber is that, whiche it it hee diuided, by. 12. ½, the quotiente wil be 7/ [...]. Answere: multiplie 7/ [...] by. 12 ½ and thereof commeth 175/ [...], then diuide [Page 77] 175 by. 16. and thereof commeth. 10 15/ [...] for the nomber whiche you desire.

### ¶ Other necessarie questions dooen by diuision in broken nomber.

I Demaunde what parte. 30. is of. 70. Aunswer: diuide. 30. by 70. whiche you can not, for thei are 30/7 [...], but abreuiate them and thei are 3/7. Thus. 30. are the 3/7. of. 70.

2. I demaunde what parte. 10. is of 16 ⅔. Aunswere: diuide. 10/ [...]. by. 16 ⅔, and thereof commeth 30/50, whiche beyng abreuiated are [...]. And thus 10. is found to bee ⅗ of. 16. 2/ [...].

3. More, what parte is. 25. of. 5/ [...]. Aunswere: diuide 5/ [...]. by. 25/10, and thereof commeth 5/20 [...], whiche beyng abreniated is. 1/40. And thus. 5/ [...]. is but the 1/40. of. 25.

4. More, ⅚ what parte are thei of ⅞ Aunswere: diuide ⅚ by ⅞, and you shal [Page] finde 4 [...]/42 whiche abreuiated are 20/21.

5. More, ⅘ what parte are thei of. 13. [...]. Aunswer: diuide ⅘ by. 13. ⅓. and you shall finde 12/200, whiche beyng abbreuiated are [...]/50. And thus 4/1 are the 3/50. of 13. ⅓.

6. More. 12. ½. what parte are thei of. Aunswere: diuide. 12. ½. by 30/ [...], and you shall finde. [...]5/60, whiche beyng abreuiated are [...]/12, and thus. 12. ½, are the 5/12 of. 30.

7. More. 16. [...]/3, what parte are thei of. 57. 1/7. Aunswere: diuide. 16. ⅔ by. 57 [...]/7. and therof commeth 550/1200, whiche beyng abreuiated are 7/24, and thus. 16. ⅔. are the. 7/24 of. 57. 1/7.

8. More, [...]/4 and ⅔ of [...]/4, or three quarters, and ⅖ of one quarter, what part are thei of. 1. Aunswere: reduce 5/4, and the ⅔ of [...]/4, into one broken by the first reduction, and you shall finde 11/12. And thus the [...]/4, and [...]/3 of ¼ are the 11/12 of one whole.

9. More, of what nomber are. 9. the [...]. Aunswere: diuide. 9. by 3/3, and thereof [Page 78] commeth. 13. [...]/2, whiche is the nomber whereof. 9. are the [...].

10. More, of what nomber are ⅘ the [...]/4. Aunswere: diuide ⅖ by ¾, and thereof commeth 8/15, whiche is the nōber wherof 2/ [...] are the ¾ of the same nomber.

11. More, of what nomber are. 5. ¾. the [...]/7. Aunswere: diuide. 15. ¾ by 3/ [...], and you shall finde. 13. 5/12, whiche is the nō ber whereof. 5, [...]/4 are the 3/7.

12. More. 9. [...] what parte are thei of. 33. Aunswere: diuide. 9. 2/ [...] by 33. ½. And thereof commeth 18/20 [...]: and thus 9. [...] are the 58/201 of 33. ½ as appereth.

# ¶The thirde parte treateth of certaine brief rules, called rules of practise, with diuerse necessarie questions, profitable for Marchauntes.

## ¶The first Chapiter.

SOme there be, which doe cal these rules of practise br [...] rules, for that by thē many questions maie be doen with quicker expeditiō, then by the rule of thre. There bée others, whiche call them the small multiplication, for because that the producte, is alwaies lesse in quantitie, then the nomber whiche is to be multiplied. This practise commeth not in vse, but onely emong small kindes of nombers, whiche haue ouer theim, other nombers that are greater. And this beyng well considered, is no other thyng, but to conuerte lesser and perticuler kindes of nōber, into greater, the whiche maie bee dooen by the [Page 79] meanes of diuision, in taking the half, the third, the fowerth, the fift, or suche other partes of the somme, whiche is to bee multiplied, as the multiplier is part of his greater kinde, & that which commeth thereof is worthe as muche (not in quantitie, but in his owne forme) as if you did multiplie simplie the twoo sommes, the one by thother, and for the better vnderstandyng of suche conuersions, you must haue respecte to one of these twoo considerations. The first is, when one would demaunde this question. At 6. pence the yarde of Cotton, what are. 18. yardes worth by the price? It is manifest that thei are worthe. 18. peeces of. 6. pence the pece, or. 18. halfe shillynges, which must be tourned into shillynges, in takyng the halfe of. 18. shillynges, and thei make. 9. shillinges. Or otherwise you muste consider, that at. 1 shillyng the yarde, the. 18. yardes are worth. 18. shillynges, wherefore at. 6. pence thei shalbe but halfe so muche, for. 6. pence [Page] is but the ½ of. 1. shiliyng. Therefore you must take ½ of. 18. and thei make 9 shillinges, whiche are worth as much as. 108. pence, that is to saie, as. 18. times. 6. pence.

2. Firste, if you will multiplie any nomber, after this maner by pence, whereof the nomber of thesame pence doe not extende vnto. 12. and therof to bryng shillinges into the product: you must knowe the certaine partes of. 12 whiche are these: that is to saie, 6, 4. 3. 2. and. 1. For. 6. is the ½ of. 12. and. 4. is the [...]. of. 12: 3. is the ¼: 2 is the ⅙: and 1. is the. [...]/12. Then for. 6. pence, whiche is the halfe of. 1. shillyng, you must take the ½ of all the nomber, whiche is to be multiplied. And that which commeth thereof, shalbe shillynges, if there do remaine. 1. it is. 6. pence.

For fower pence you must take the [...]/4 of all the nomber, that is to bée multiplied: and if any vnities doe remaine thei shalbe thirdes of a shillyng, euery one beyng in value. 4. pence.

[Page 80] For. 3. pence you muste take the, ¼. of all the somme: if any vnities dooe remaine, thei shall bee fowerthes of a shillyng, euery one beyng worthe three pence.

For 2 pēce you must take the ⅙ of all the the somme, and if any vnities doe remaine, thei shall bee sixe partes of a shillyng, beyng euery one of theim worthe twoo pence.

For. 1. penie, take the 1/12 of the whole somme, if any vnities remaine, thei are. 12. partes of a shillyng, eche of them beyng in value. 1. penie, as by these examples folowing doeth plainly appeare. [...]

[Page] Here you maie se in the first exāple, that. 59. yardes, at. 6. pence the yarde is worthe. 29. shillynges. 6. in taking the ½ of. 59. And in the seconde example, the. 82. yardes at. 4. pēce the yarde is worthe. 27. shillynges. 4. pence, in takyng the ⅓ of. 82.

Likewise, in the third example. 97 yardes, at three pence the yarde, bringeeh. 24. shillynges. 3. pence, in takyng the ¼ of. 97. Also in the [...]owerth example. 346. yardes, a [...]. 2. pence the yarde, maketh. 57. shillynges eighte pēce, in taking the 1/ [...] of. 346. And final- [Page 81] in the fyft example. 343. yardes, at I. d. the yarde, amount to 28. shill. 7. d. in taking the 1/12 of 343. And so is to be done of all suche lyke, when the nomber of the pence, is any of the certaine partes of 12.

But if the nomber of the pence be not a certain parte of 12. you muste reduce them into some certaine partes of 12. and after the foresayd maner you shall make two or three productes as neede shall require, and adde them togither into one summe as 5. d. may be reduced into 4. & I. or els into 3. & 2. wherfore if you wil work by 4. & by I: you muste for 4. d. take fyrst the ⅓. of y^{e} nomber, that is to bée multiplied, and for I. d. take the 1/12, or rather for I. d. you may take the ¼ of the producte whiche did come of the 4. d. bycause that I. d. is the ¼ of 4. d. But if you wyll worke by. 3. and you shal take for 3. d. the ¼ of the nomber whiche is to bee multiplied: and likewise for 2. d. the ⅙ of the same nomber, adding togyther both [Page] the productes. The totall summe of those two nombers shall be the solution to the question. And in like maner is to be done of all other. As by these former folowyng may appeare. [...]

[Page 82] Here in this same example where it is demaunded (at 5 pence y^{e} yard) howe much are nine and fourty yardes worth? Firste for four pence, [Page] I take y^{e} [...]/3 of 49. s. and thereof cōmeth 16. s. 4. d. thē for 1. d. I take the ¼ of the same product, that is to say, of 16. s. 4. d. and that bringeth. 4. shill. 1. d. these. twoo sūmes added togither, do make 20. s. 5. d. And so much are the 49. yardes worth at 5. d. the yarde.

For 7. d. take the ⅓ and the ¼ of the whole summe whiche is to be multiplied, and adde them togither, that is to say, for 4. d. the [...] and for 3. d. the ¼: bycause 4. d. is the [...] of 12. d. and 3. d. is the ¼ as in the second example before doth appeare: Where the question is thus, at 7. d. the yarde what are 54. yardes worth? Firste for 4. d. I take y^{e} [...] of 54: and they make 18. s. Likewise for 3. [...]. I take the [...] of 54. and they are 13. s. 6. d. Then I adde 18. s. and 13. s. 6 d. togither, so both amount to [...]1. s. 6. [...] and so muche are the 54 yardes worth at 5. d. the yarde.

Otherwise for 7. d. take first the ½, of the whole sūme for 6. d. Then for 1. d. take the ⅙ of the same product, and [Page 83] adde them togither, so shall you haue the like summe as before.

For eight pence you must first take [...]/3 of the whole sūme for 4. pence, and another [...]/ [...] for other 4. d. and adde thē togither as in the. 3. example doth euidently appeare. Where the question is thus, at 8. d. the yarde, what are 40 yardes worth? Firste for 4. d. I take the ⅓ of 40. which is 13. s. 4. d. Againe, I take another ⅓ for the other 4 pence which is also 13. shillings & 4. pence. These twoo summes being added togither, do make 26. shillings 8. pence, and so muche are the 40. yards worth at 8. pence the yard, as in the third example abouesayd doth appeare.

Otherwayes, for eyght pence you may take first the ½ of the whole sūme for 6. d. Then for 2. d. you shal take the [...]/3 of the product, which did come of the said ½, and adde them togither, so shall you haue likewise the solution to the question. As in the same third exāple of 40. yardes, I take first the ½ of 40. [Page] for 6. d. and thereof commeth 20. shil. then for 2. d. I take [...]/3 of the saide product, that is to say of 20. s. which bringeth 6. [...]. 8. d. these two summes (20. s and 6. s. 8. d.) I adde togither and they make 26. s. 8. d. as before.

For 9. d. you must take the [...]/2 & the ¼ of the whole sūme, and adde them togither: or els for 6. d. take fyrst ½ of the whole summe, then for 3. d. take y^{e} [...]/2 of the same product, bicause 3. d. is y^{e} halfe of 6. d: And 6. d. added with 3. d. bringeth 9. d. as by the fourth example, where it is demaunded after this sort: at 9. d. the yarde, what are 73. yardes worthe. First for 6. d. I take the ½ of 73. and therof commeth 36. s. 6. d. then for 3. d. I take ½ of y^{e} same 36. shil. 6. d. which is 18. s. 3. d. these twoo summes doe I adde togither, & they make 54. shil. 9. d. as in y^{e} said fourth example is euident.

For 10. d. take first the ½, then y^{e} ⅓ of the whole summe, & adde thē togither

For 11. [...]take fyrst ⅓ for 4. pence, secondely, [Page 84] another ⅓ for other 4. d. and thirdly ¼ for 3. d. of all the whole sūme. and adde them togither.

Or els for 11. d take first the ½ then then ⅓ of the whole summe, and finallye the ¼ of the laste producte, addinge them togither.

3. Lykewise by the same reason, when you wil multiply (by shillings) anye nomber that is vnder xx. s. you shall haue in the product poundes, if you knowe the certaine partes of 20: which are these. 10. 5. 4. 2. &. 1. For 10. is the ½ of 20. 5. is the ¼ part: 4 is the ½. 2. is the 1/10: and 1. is the 1/20.

Then for 10. s. which is the ½ of a pounde: you muste take the [...]/2 of the nomber, whiche is to bee multiplied, and you shal haue poundes in the product. If there doe remaine 1, it shalbe worth ten shillings.

For 5. shillinges you muste take the ¼ of the nomber whiche is to bee multiplied, & if there do remaine any vnities, they shall be foure partes of [Page] a poūd, euery one being in value 5. s.

For 4. [...]. you must take the [...]/5 of the number whiche is to bee multiplyed. And if there do remaine any vnities, they shall be fift parts of a pound, euery one being worth foure shillings. [...]

For 2. shillings you must take the 1/10 of the nōber that is to be multiplied. Wherefore, if you wyll take the 1/10 of any nomber, you muste separate the last figure of the same number whiche is nerest your righte hande, from all [Page 85] the other fygures. For all the other figures whiche doe remaine towarde your lefte hande, from the same fygure, which is separated, shall bee the sayde 1/1 [...] of a pounde: and that separated fygure, towarde your right hand shall be so many peeces of 2. shillings the peece: the whiche fygure must be doubled, to make therof shillings, as by these examples appeareth. [...]

Herevpō doth depend an other exact way for to multiply by shillings (if y^{e} nomber of shillings be euen) which is thus: you shall take [...] the nōber of the same shillings, and conuert them into peeces of 2. shillings. Then by the [Page] nomber of this halfe, you must firste multiply the last figure towarde your right hande, of the nōber which is to be multiplied: And if ther be any tennes in the same product, those must you reserue in your minde: But if (wyth the same of els without the same) you doe finde any diget nomber, the same diget nomber shall you double, & put it in the place of shillings: Then must you proceede to the multiplication of the other figures, adding vnto y^{e} product the tennes whiche you before reserued: and therof shall come pounds.

Nowe, for your better vnderstanding of this which hath bene said and by the way of example, I will propone vnto you this question.

At 8. shillings the grosse, what are [...]7. grosse worth after the rate?

Firste in this example I take halfe the nomber of Shillinges, as before is taught, that is to say of eighte shillings, which is foure shillinges, this 4. shil. I put apart, behinde a crooked [Page 86] line righte againste 97. towardes the left hand, as here you may see and as here after dooeth appeare by diuers examples. [...]

Nowe in the first example, where it is demaunded, at 8. [...]. the grosse, what are 97 grosse? First the ½ of 8. s. whiche is 4. s. being set apart behind the crooked line, as before is said: thē I multiplye y^{e} 97 by 4. saying first, 4. times 7. is 28. I double y^{e} diget nūber 8. and [Page] that maketh 16, the which 16, I do put vnder the line, in the place of shillings & I kepe y^{•} tennes in my mind, which here are 2. For 20. are two times ten: Then secondly, I multiply 9. by the sayd 4, and thereof cōmeth 36: whervnto I adde the 2. tennes, whiche before I reserued, and they make 38. Therfore I put 38, vnder the lyne in the place of poundes, and the whole summe will be 38 .li. 16. s. Thus much are the 97. grosse worth, at eight shillings the grosse: the like is to be done of all other. As of 12. shillings in multipliyng by 6. Likewise of 6. shillings if you multiply by 3. also of 14. if you multiply by 7. And so of all euen nombers after the same maner.

For 1. Shilling you must take the ½ of the 1/10 parte of any nomber that is to be multiplied. And if any thyng [...] do remaine, they are shil. Thus by thys maner shil. [Page 87] are conuerted into poundes: for it is euen like, as if you did diuide thē by 20. s. as by this exāple in the margent doth appeare. Wher it is demaunded at 1. s. the yard, the peece, or any other thing, what are 350. worth?

First I separate the laste fygure of 359. nexte to my ryght hand, which is the 0 with a line betwene it and the figure 5. Then I make a line vnder the 3 |0, and I take the ½ of 35, after this maner: saiyng the ½ of 3. is I. and I remaineth, whiche remaine signifieth 10. in that secōd place. Then I put I. vnder the line against 3, & I proceede to the rest, saiyng: the halfe of 15. is 7. (which 15. came of the I. that remayned, and of the 5. in y^{•} first place) I put 7. vnder the line right against 5, and they make 17 li. The I. which did last remaine, is 10. s. Therfore I put 10. s aparte vnder the line, and the whole summe is 17 .li. 19. s. so much are 350. worth at I. s. the peece.

But when the nomber of shillings [Page] is not some certaine parte of 20. shil. you must then conuert the same nomber of shillings, into the certain parts of 20. and make twoo or thrée products, as nede shall require, the whiche must bee added togither after this maner following.

For 3. shillings you must firste take for 2. shil. the 1/10 of the nomber that is to be multiplied, thē for 1. shilling you must take the ½ of the producte whiche did come of the same 1/10 part: and adde those twoo sūmes togither, as appeareth by this example following.

At 3. s. the peece of any thing, what shall 684 peeces coste mee after the rate. First, for 2 shillings I take the 1/10 of 684, which is 68: in separating the last [...] figure 4, whiche 4 I must double, & they be 8. I set eight shillings aparte from the place of poundes, and then I haue 68. poū des 8. s. for the 1/10 parte, that is to say, [Page 88] for the 2. s. secondlye, for 1. shil. I take the ½ of the product, that is to saye: of 68 .li. 8. s. whiche is 34 .li. 4. s. and I put the same vnder the 68 .li. 8. shil. Then finally, I adde those two summes together, that is to saie, 68 .li. 8. s. and 34 .li. 4. shil. so they make 102 .li. 12. s. and so much are the 684. peeces worth at 3. shillings the peece, as may appere in the margent.

For 6. shll. take 3/ [...] of the nomber whiche is to be multiplied: that is to say, first 1/10, then double the product of the same 1/10 and adde them together. Or otherwise for 4. s. take first y^{e} ⅕ of y^{e} nōber that is to be multiplied, then take the ½ of the product which is for two. s. and adde them togither.

Or els take for 5. shill. the ¼ of the whole summe, then for 1. shil. the 1/ [...] of the product and adde them togither.

Likewise for 7. shil. take firste for 5. shil. the ¼ then for 2. shillings take the 1/10 of the nomber whiche is to be multiplied, and adde them togither. [Page] For eyght shillings take the ⅖ at two sundry times, that is to say, first ⅖ for 4. shil. and then as muche more for other 4. shil. and adde them togyther.

For 9. shil. take first the ¼ and lykewise the ⅓ of the nomber that is to be multiplied, and adde them togither.

For 11. shil. take first ½ for 10. s. Thē for 1. shil. take the 1/10 of the producte, & adde them togither.

For 12. shil. take first the ½ for 10. shil then for 2. s. take the ⅖ part of the product, and adde them togither,

For 13. shil. take the ¼ then the ⅕, and againe another ⅕ of the nomber which is to be multiplied. And adde the productes togither, that is to saye: fyrste for 5. shil. take the ¼, then for 4. shil. take the ⅕. And againe, another ⅕ for the other 4 shil. and assemble the three productes, the like is to be done of all others, when the price of the thynge which is valued, is onely of shillings. And as by these examples followyng doth plainly appeare. [Page 89] [...]

[Page] 4. Likewise in multipliyng by pēce you shall haue (at the firste instaunte) poundes in the producte, in case you knowe the certaine partes of the 1/10 of a pounde, or of. 24. pence, whiche are these, 12. pence. 6. 4. 3. and. 2. For. 12. is the ½ of. 24:8. is the ⅕:6. is the ¼. 4. is the ⅙:3. is the ⅛:2. the 1/12: but for. 12. pence, whiche is 1. shillyng: wee haue before made mention thereof.

For 8 pence you muste take the ⅓ of the 1/10, and the reste whiche are the peces of 8 pence, muste bee doubled to make of them peeces of. 4. pence. And of the same nomber beyng doubled, you muste take the ⅓, whiche will bee [Page 90] shillynges, and if there doe yet remain any thing, thei are thirds of a shilling being in value 4 pence the pece.

For 6 pence take the ¼ of the 1/12, and of that whiche remaineth, you muste take the ½, whiche shall be shillynges, if there doe yet remaine. 1. it shall bee in value. 6. pence.

For 4 pence you must take the ¼ of the 1/10, and of that whiche resteth, take the ⅓ to make therof shillynges, if any thyng dooe yet remaine, thei are thirdes of a shillyng, beyng in value. 4. pence the pece.

For 3 pence take the ⅕ of the 1/10, and of that whiche remaineth, take the ¼, to make of theim shillynges: if any thyng doe yet remaine, thei are fourethes of a shillyng, euery one of theim beyng worthe 3. pence.

For 2 pence take the 1/12 of the 1/10: and of that whiche resteth take the ⅙, the whiche are shi. if there do still remain any thing, thei shalbe sixte partes of a shilling, euery one being in value 2 d.

[Page] For 1, penie it is impossible with ease, to bryng of pence, poundes (into the producte) vpon the totall somme: But firste you must bryng them into shillinges by the order of the seconde rule of this chapiter, and then afterwarde you shall conuerte them into pounds, if nede so require. As by these examples followyng may appere. [...]

[Page 91] But if the number of pence, be not a certaine parte of 24. pence. Then must you bring them into the certain partes of 24. and make thereof diuers productes, which must be added together, as shall hereafter appeare.

For. 5. pence you shall first take for 3. pence, then for. 2. pence, and adde thē together, accordyng to the instruction of the laste rule. Or els firste take for 4. pence, and then for. 1. penie.

For. 7. pence, first take for. 4. pence then for. 3. d, and adde them together.

For 9 pence, firste take for 6 pence, then for 3 pence adoyng thē together.

For. 10. pence, firste take for 6. d. then for. 4. d, and adde them together.

For. 11. pence take firste for. 8. pence then for. 3. pence, and adde them together: as by these examples followyng doeth appeare. [Page] [...]

5. If you will multiplie any nōber by shillynges and pence, beyng bothe [Page 92] together, you must take first for the. s. accordyng to the instructiō of the third rule of this chapiter, then take for the pence, after the order of the [...]owerth rule before mentioned: but if there be any certaine partes of. 1. pounde, containyng bothe shillynges and pence, then for suche partes you shal take the like parte of the nomber that is to bee multiplied, as the nomber is part of 1 .li. the whiche certain partes are these, 6. s. 8. d: 3. s. 4. d: 2. s. 6. d: and. 1. s. 8. d: For 6. s. 8. d. is the ⅓ of a li. 3. s. 4. d. is the ⅙ of a li. 2. s. 6. d. is the ⅛: and 1. s. 8. d. is the 1/12: then for 6. s. 8. d. you muste take the ⅓ of the nomber that is to bee multiplied: and if any thyng dooe remain, thei are thirdes of a pound, euery one being worthe 6 s. 8. pence. For 3. s. 4 d. you must take y^{e} ⅙ if any thing doe remain, thei are 6 partes of a li. euery one beyng in valor 3. s. 4. d. For 2. s. 6. pēce, you must take the ⅛▪ if any thing be remainyng, the [...] are 8 partes of a li. euery one being worth 2. [...]. 6. d.

[Page] For. 1. shillyng. 8. pence, you shall take the [...]/12, if there dooe any thyng remaine, thei are twelfth partes of a .li. euery one beyng valued at. 1. s. 8. d. [...]

6. Here shall you accustome your self, to multiplie by all sortes of sommes, beyng compounde of shillynges, and pence, whiche maie come to practise. As thus, for. 1. shillyng. 1. penie: for. 1. shillyng. 2. pence: for. 1. shillyng 3. pence: for. 1. shillyng 4. pence. Likewise for 2. shillynges 1. penie: 2. shilly. 2. pence: 2. shillynges. 3. pence. 2. s. 4. d And so of all other: consideryng moreouer, [Page 93] many subtile abreuiations, whiche happen oftentymes, that are easie to bee conceiued. As thus: at 11. s. 3. d. after that I haue taken firste the ½ for 10. shillynges. Then for 1. shillyng. 3. pence. I take the ⅛ of the product, because 1. shillyng. 3. pence is the ⅛ of 10 shillynges, in takyng thesaied ⅛ of the producte. And by this meanes, when ye haue taken one producte, ye maie oftentymes vpon the same, take an other more briefly, then vpon the sōme that is to be multiplied, whiche thyng you must foresée. [...]

[Page] 7. But if you will multiplie, by poundes, shillinges & pence being together. Firste you muste wholly multiplie by poundes. Then take for the shillyngs & pēce, as in the. 5. rule of this chapiter is plainly declared. And as by these examples folowyng maie appeare. [...]

[Page 94] 8. So these rules do serue, both to buy & sel, at suche a price the ell, the yard, y^{•} pece, the li. waight or any other thing: how much such a thing. Likewise thei are very necessary to conuert all peces of gold & siluer into li. for I may aswel saie, at. 4. s. 8. d. the Frenche croune, what are 135 crounes worth? As to say at. 4. s. 8. d. the yarde of clothe, what are 135. yardes worthe?

9. When any 1 of the sōmes (which is to be multiplied) is cōpounde of many denominations: & thother is of one figure alone: then shall ye multiplie al the denominations of thother somme, by the same one figure, beginning first with y^{•} somme, which is least in value towards your right hand, & bryng the product of those pence into shi. and the product of the shillinges into poundes as by this example doeth appeare. [...]

[Page] 10. But ( [...] of the nombers whiche are to be multiplied) there bee with it a broken nomber, you muste (accordyng to his denominator) take one or many partes of the nomber, as nede doeth require: and sette the nomber whiche commeth thereof, vnder the productes, addyng the same together. As thus: At 5. pounde 7. shillynges 8. pence the grosse, what shall. 34. grosse ½ cost? First you shall multiplie [...] 5. pound. 7. s. 8. pence by. 34. grosse, saiyng. 5. tymes. 34. dooe make 170. pound then for. 6. shilly. 8. pence, take the [...]/3 of. 34. whiche is. 11. pounde. 6. shillynges, 8. pence. Thirdly, for. 1. shillyng, take. 34. shillynges, whiche is 1 pounde 14. shillynges. 0.

Lastly, for the ½ grosse, you muste take ½ of the. 5. pounde 7. shillynges 8 [Page 95] pence, whiche is. 2. pounde. 13. s. 10. d. And then adde thē all together, so you shall finde that the. 34. grosse ½ at. 5 .li. 7. shillynges. 8 pence is worthe. 185. pounde. 14. shillynges. 6. pence, as appeareth in the margent.

And as in this last example, you did take the halfe of the money, (whiche one grosse was worthe) for the ½ grosse Because that one grosse beyng worth 5. pounde. 7. shillynges. 8, pence, the ½ grosse must be worthe halfe so muche So likewise, if you haue ⅓ of a grosse or of any other thyng, you muste take the. ⅓. of the price, that one grosse is worthe. Semblable, for the. ¼. of any thyng you shall take the ¼ of the price, that one is worth, and of all other fractions, as by these examples folowing doth appeare. [...]

[Page] 11. If you will make the proofe of these rules aforsaid, you must first abate the somme of money (which the fractiō of the multiplicand doeth importe) from the total somme. And diuide the rest of the poundes of the said totall sōme, by the whole multiplicande, the fraction onely accepted. And if any thing dooe remain after the diuision is made, that remain shalbe multiplied by 20 & vnto [Page 96] the producte of that multiplication, you shal adde the shi. which remained of the rest of the totall sōme. Again, if anything do remain after the same diuision, you must multiply the same by 12 & vnto y^{e} product adde the pence of y^{e} total sōme that remained if any be left And thus if ye haue truely wrought, you shall finde again the higher sōme of your questiō, that is to saie, the price that one grosse, or any other thyng is worthe, whereof you demaunde.

Or otherwise, reduce the remaine of the totall somme (the value of the money that the fraction is worthe, beyng firste deducted) all into pence, in multipliyng the poundes by. 20. and the shillynges by. 12. addyng thereunto the shillynges and pence, which are i [...]igned with the remaine of the saied totall somme, if any suche be, then diuide those pence by the foresaied nomber that is to be multiplied, the fraction of the same nomber being also abated. So shall you finde the price that [Page] one pece, I grosse, or any other thyng is valued at. As in the firste example goyng before, where the totall sōme is. 201. pounde. 10. shillynges, from the whiche I dooe first abate the price of the halfe grosse, whiche is. 2. pound 3. shillynges. 4. pence, the reste is 199 .li. 6. shillynges 8. pence, whiche being reduced into pence, bringeth 47840. d. I diuide thesame by 46. and thereof commeth 1040. d. Then I diuide that 1040. d. by 12, and thei bryng 86. s. 8. d. that is to saie 4 .li. 6. s. 8. pēce, which is the price that one grosse, or any other thyng did coste, as in that firste example doeth eppeare.

12. The like is to bee doen of any maner of thyng, that is solde by the hundred, or by Kintall. As thus: at 12. pounde seuen shillynges six pence the hundreth pounde waight: what shall. 374. pounde waight coste. You shall first multiplie twelue pound seuen shillinges, sixe pence by thre: that is to saie, by three hundreth. Then for [Page 97] 50 .li. waight, you [...] shall take the ½ of 12 .li. 7. s. 6. d. bicause 50 .li. is the ½ of 100 .li. Likewise for 20. pound waight, which is the ⅕ of 100 .l. take the ⅕ of 12 .li. 7. shil. 6. d. lastly for 4 .li. waight take the ⅕ of the laste product. This done, you muste adde all these productes into one summe, whiche will make the summe of 64 .li. 5. s. 7. d. ⅘, as by this exāply aboue written doth appeare.

The proofe is made by reducinge the totall summe into pence. And to diuide the product by the nomber y^{•} is to be multiplyed, that is to saie by 374 likewise diuide the quotient produced of that first diuision by 12. so shall you finde againe the higher summe 12 .li. 7 shil. 6. d. whiche is the price of 100 .li. wayght, as before.

13. Also the like maye be done of [Page] our vsuall waight here in Englande (whiche is 112 .li. for euerye hundred pounde waight). in case you knowe the certaine parts of a hundred, that is to say, of 112 .li. waight, whiche are these 56 .li. 28 .li. 14 .li. 7 .li. For 56 .li. is the ½ of 112. 28 .li. is the ¼ of 112 .li. 14 .li. is the ⅛, and 7 .li. is the 1/16.

Therfore, for 56 .li. take the ½ of the summe of money, that the 112. pound waight is worth.

For 28 .li. take the ¼ of the summe of money that the 112 .li. is worth.

For 14 li. take the ⅛ of the summe that the C. is worth.

For 7 .li. take the 1/16 of the summe of money that the C, is worth.

As thus: at 3 .li. 6. s. 8. d. the hundreth pounds waight, that is to saye, the 112, li. What shall 24. C. 3. quar. 21, li. cost after the ra [...]e?

Fyrst, you shall multiply 24, hundreth by 3. whiche is the 3 .li. & thereof cōmeth 72 .li. then for 6. s, 8, d. whiche is the ⅓ of 20. s. you shall take y^{•} ⅓ of 24 [Page 98] which is 8, li. for [...] 24. nobles maketh 8, li. afterwarde, for the 3. quarters of the C. you shall first for the 56 .li. take [...]the ½ of 3 .li. 6. s. 8, d. bicause 56. li is the ½ of the C. & thereof cōmeth 1 .li. 13. shil. 4. d. then for 28 .li. (whiche is the quar. of a C.) you shall take the ¼ of 3 .li. 6. s. 8. d. or els the ½ of the product, whiche came of 56 .li. which is 16, s. 8, d likewise for 14 .li. take the ⅛ of 3 .li. 6, s. 8. d. whiche is 8, d. 4, d. or els the ½ of the producte of 28 .li. which is all one: lastly for 7 .li. take the 1/16 of 3, li. 6. s. 8, d. or els the ½ of the product, that came of 14 li. and therof cōmeth 4, s. 2, d. Then adde al these products togither: & the totall summe wil be 83 .li. 2. s. 6. d. so muche are y^{•} 24 .c. 3. quar. 21 .l. waight worth after 3, li. 6. s. 8. d. y^{•} C. as appereth in [Page] the margent.

The proofe hereof is made, lyke to to the other proofes aforesaide, saning that where in those proofes, you aba [...]e the price of the money, that the fraction was worthe, from the totall summe: here in this example (and in suche other like) you muste abate the price of money, that the odde waight amounteth vnto (ouer and aboue the iust hundrethes) from the saide totall summe, the rest thereof shall you conuert into pence, diuidinge the product of y^{e} multiplication by the iuste nūber of the hundrethes, so shall you finde the pence y^{e} one hundreth is worthe, whiche you shall bringe into poundes by the order of diuision, & so all other.

## ¶The second Chapter treateth of the rule of three compounde, which are foure in nomber.

THere belongeth to the fyrst & seconde partes of the rule of thre compound alwaies fyue numbers: whereof (in the first [Page 99] part of the rule of three compo [...]nde the seconde nomber and the [...], are alwayes of one semblaunce, and [...]ke denomination: whose rule is thus, multiply y^{e} first nōber by the seconde, & that shalbe your diuisor: then multiplie the other three nōbers the oneby the other to be your diuidende. Exā ple, of this first part: if 100. crowns in 12. monthes, do gaue 15 .li. what will 60. crownes gaine. in 8. monthes? Answere, first multiplie 100. crownes by 12. monthes, & therof cōmeth 1200. for your diuisor: then multiply 15 .li. by 60. crownes, & by 8. monthes & you shal haue 7200. diuide 7200. by 1200, & therof cōmeth 6 .li. so many li. wyl 6. crownes gaine in 8. mōthes: this questiō may be done by y^{e} double rule of 3. y^{t} is to say by y^{e} rule of 3. at 2. times, but yet this rule of 3 cōpoūd is more brief. [...]

[Page] 2. In the seconde part of the rule of third compound, the 3. nomber is like vnto the fift, wherof the rule is thus: multiplie the 3. nomber by the 4. the product shalbe your diuisor: then multiply the first nomber by y^{e} seconde, & the product therof by the fift, y^{t} whiche nomber shalbe your diuidend, or nō ber y^{t} is to be diuided: as by example,

When 60. crownes in 8 monthes do gaine 6 .li. in howe many monthes wil: 100. crownes gain. 15 .li. Aunswer Multiply the thirde nomber 6. by the fourth nōber 100, & ther of cōmeth 600 then multiplye the first nōber 60. by the secōd nōber 8. & by the fift nōber 15 thereof will come 7200. then diuide 7200. by 600. & y^{e} quotiēt wilbe 12: in so many monthes will 100. crownes gaine 15 .li. This question may likewise be done by the double rule of 3. [...]

[Page 100] 3. In the thirde part of the rule of 3. compound, there may be 5. nombers or more: & in this rule the first nōber & the last alwayes dissemblaunt the one to thother: & the questiō is from the last nōber vnto the first, wherof y^{e} rule is thus: multiply that nomber which you woulde know by those nō bers which do giue the value, & diuide the product of the same, by y^{e} multiplication of the nōbers which are already valued, as by exāple. If 4. deniers Parisis, be worth 5. deniers arnols, & 10. deniers tournois, be world 12. deniers of Sauoy, I demaūd how many deniers Parisis are 8. deniers of Sauoy worth▪ Aunswere: Multiply 8. deniers of Sauoy (which is the nomber y^{t} you would know) by 4. deniers parisis, & by 10 deniers t [...]is whiche are y^{e} nōbers that giue y^{e} value, & they make 320: then multiplie 5. denters tournois, by 12 deniers of sauoy (which are the nombers alredye valued) and they make 60: lastly diuide 320. by 60 [Page] and you shall finde 5. deniers ⅓ parisis, so muche are the denies of Sauoye worth. [...]

4. In the fourthe parte of the rule of three compounde: the first nomber and the last are always semblant and of one denomination, and the questiō of this rule, is alwayes from the last nōber to the last sauing one. Whereof there is a rule which is thus. You must multiplye that nomber whiche you woulde knowe, by the nombers that are alreadye valued, and diuide the product of the same, by the multiplication which commeth of the nombers that giue the value, as by exāple

If 4: deniers Parisis, bee worth 5. Deniers Lournois, and 10. Deniers Lournois, be worthe 12. Deniers of Sauoy, I demaunde how many Deniers [Page 101] of Sauye. are 15. Deniers Parisis worth. Aunswere: Multiplye 15. Deniers Parisis that you woulde knowe, by 5. Deniers Lournois, & by 12. Deniers of Sauoye, which are the nombers alreadye valued, and they make 900 Diuide the same by 4 times 10. which are the nombers that doe giue the value, and you shal finde 22. Deniers ½ of Sauoye, so much are the 15. Deniers Parisis worth. [...]

## The thirde Chapter treateth of questions of the trade of Marchaundise.

IF 31. Deuonsh. dosēs do cost me 100 .li. 15. shil. What shal 4. dosens cost? Aunswere: first bring the 100 .li. 15. shill. all into shillings, [Page] in multiplying y^{e} 100 .li. by 20. adding to the product the 15. shill. and thereof commeth 2015. shill. then multiplye 2015. by the thirde number 4. and diuide the product by 31. and the quetietit wilbe 260. s. The which diuide aagaine by 20. & therof commeth 13 .li. [...]

If foure Dosens be worth 13. pound What are 31. Dosens worthe by the price? Aunswere: Multiply 31. by 13. & therof cōmeth 403. The whiche you shall diuide by 4. and thereof commeth [Page 102] 100 .li. ¾, which ¾ are 15. s. and so much are 31. Dosens worth as before. [...]

If 49 elles be worth 2, li. 4, s. 11, d. what are 18 elles worthe by the price? First you must bring 2 .li. 4, s. 11, d. all into pence, in multipliyng 2 .li. by 20. maketh 40. adde thereto 4, shil. they make 44. s. y^{•} which multiply by 12. d. & they make 528, d. wherūto adde 11. d all is 539, d. the whiche 539, d. muste be your second nōber in y^{•} rule of 3. then multiply 539 by 18, & therof commeth 9792. diuide y^{•} same by 49, & you shall haue in your quotient 198, d. y^{•} whiche diuide by 12. & you shall finde 16 s. 6 d. so much are the 18. elles worth. [Page] [...]

IF 18. elles be worth 16 s. 6. d. what are 49. elles worth by the price? Auns. bring 19. s. 6. d. into pence, in multiplying 19. by 12. and thereof commeth 198. d. with the 6. d. adde dto it, then multiplye 198 by 49. the product will be 2702. The which diuide by 18. elles and therof commeth 539. d. Then diuide 539. d. by 12. and the product therof by 20. So shall you haue 2 .li. 4. sh. [Page 105] 11. d. so muche are the 49. elles worth. [...]

If a yarde of Ueluet cost 19. s. what shalt ¾ of a yarde cost? Aunswere: sette down your nombers thue. If 1/1 | 19/1 | 3/ [...], Then multiplye 1. times 16. by 3. and therof cōmeth 57, for your diuidēde, or nomber to be diuided. The whiche 57. you shall diuide by 1, times 1, foure times; which are 4, and your quotient wil be 14 s. ¼, which ¼ is worth 3. d. so [Page] muche are the ¼ of a yarde worth after 19. shil. the yarde, as by practise followeth. [...]

Or otherwise by the rules of practise first for 2/4 of a yarde whiche is ½ of a yarde, you muste take the ½ of 19 s. which is 9, s. 6, d. then for ¼, take the ½ of the product, that is to saye, of 9, s. 6, d. and therof cōmeth 4, s. 9. d. adde these nōbers togither, & [...] you shall haue 14, s. 3, d. as aboue is sayd, and as appeareth here in the margent.

IF ¾ of a yarde of Ueluet do cost 14. shil. 3. d. What shall 1. yarde coste, set your nombers downe thus: if ¾ | 14 ¼ | 1/1. Reduce 14, ¼ into a traction, and they wil be 57/4 thē multiply 57. by 1. 4, times, & thereof cōmeth 228. for your diuidend. Likewise multiply 1. times [Page 104] 4, 3, times, & therof cōmeth 12, for your diuisor: then diuide 228 by 12. & your quotient will be 19. shil. so much is the yarde of veluet worth. [...]

Or otherwise by y^{•} rule of practise: you shall take the ⅓ parte of 14, sh. 3. d. and adde it with the same 14, sh. 3, d. and you shall haue 19, shill. as before. [...]

If one elle of Hollonde clothe be worth 5, s. what are ⅔ worth after the rate? Aunswere: say thus if 1/1 | 5/1 | ⅔. Then multiply 2 times 5, one time, and therof commeth 10, for your diuidende: likewise multiply three times 1 one time, they make 3, for your diuisor, then diuide 10, by 3. & thereof commeth 30. s. ⅓ which [...]/3 is worth 4. pēce, & [Page] so much are the [...] of an ell worth. [...]

Or otherwise, by the rule of practise: take first the ⅔ of 5. s. for the ⅔ of an ell, which is 1. s. 8. d. Likewise, for the other ⅔ of an ell take againe another ⅔ of 5. s. which is also 1. sh. 8, d. and adde them together, and so shall you haue 3. s. 4. d. as before. [...]

If of an ell of Hollande cloth doe cost me 3. s. 4. d. what shal the el cost? Aunswere: set down your sūme thus, if [...]|3 [...]| [...]. Firste reduce 31/ [...] all into thirds, and it will be 10/3. Then multiply 1. times 10. 3, times, and thereof cōmeth 30. for your diuidēde. Likewise multiplie 1. times 3. 2 times, your quotiēt wil be 6. then diuide 30, by 6. & you shall haue 5, s. so much is the ell [Page 105] of Hollande clothe worthe. [...]

Or otherwise by practise, take the ½ of. 3. s. 4. whiche is. 1. s. 8. d. and adde it to the same 3. s. 4. d. and thereof will come. 5. as before. For the ⅓ of. 5. s. is as muche as the ½ of 3. s. 4. pēce, whiche was the [...] price that the ⅔ of an elle did coste, as appereth.

If one elle coste me. 17. s. what shal 15. elles ⅔ part cost? Whiche ⅛ is halfe a quarter of an elle. Aunswere: safe of [...]| [...]|15. ½. First reduce 15. 1/ [...] into. 8. partes, and thei make 121/8, then multiplie 121. by. 17. 1. tyme, and thereof cōmeth 2057. for your diuidende. Likewise multiplie. 8. times. 1. 1. time, and your quotiente will bee. 8. for your diuisor, then diuide. 2057. by. 8. and you shall find. 257. shillynges ⅛, which is. 12. [...]. 17. shillynges. 1. penie ½, and so muche are the. 15. elles ⅛ worthe, as by practise doeth appere. [Page] [...]

Or otherwise, for 10. s. take the ½ of 15, which is 7 li. 10. s. then for 5 s. take the ½ of 7 li. 10s: which is 3 li. 15 s: thirdly, for 2: s: take ⅕ of 7 li: 10: s: because the ½ of 10. s: is 2: s: Fourthly, for the ⅛ of [...] the ell, you shal take the [...] of 17: s: whiche is 2: s: 1: d: ½: Lastlie, adde all these sōmes together, and then shall you finde 12 .li. 17: s: 1: d: ½ as before, and as appereth more plainly in the margent.

If 25 elles bee worthe 2, li: 3: s: 4: d: what are 18 elles ¾ worth by the price? Answer: first put 3. s. 4. d. into the part of a li. and you shall haue ⅙, then saie, if 25/1 geue me 2 li. ⅙ what shall 18¾ geue: put the whole nomber into his brokē, and then multiplie 1 tymes 13 by 75: the product will be 975 the which you shall diuide by 25 tymes 6: 4 tymes, [Page 106] which maketh 600. Then diuide 975 by 600, and your quotient will be 1 li. and 375. remaineth, the which 375. you shal multiply by 20. thereof commeth 7500. diuide the same by 600. youre quotient wil be 12. s. & 300. remaineth y^{t} which abreuiated bringeth ½ whiche is 6. d. thus the 18. elles ¾ are worth 1 .li. 12. s. 6. d. as by practise appeareth. [...]

Or otherwise by the rules of practise: for because that 12. elles ½ is the ½ of 25. elles, therefore take the ½ of 2 .li. 3. s. 4. d. which is. 1 .li. 1. s. 8. d. then for 9. elles ¼ take the ¼ of 2 .li. 3. s. 4. d. or else the ½ of the last product (that is to safe of 1 .li. 1. s. 8. d.) whiche is all one, and adde them together, so shall you haue 1 .li. 12. s. 6. d: as before. [...]

[Page] If. 15. yardes be worthe. 32. s. what are half a yard, or half a quarter, or els [...] of a yarde worthe. Aunswere: saie, if [...] geue 32/ [...] what will ⅝ geue? Multiplie [...] times 32. by 5, and diuide the product by. 15. tymes. [...]. s. and 4, remaineth, whiche is ⅓ of a shi. that is to saie 4. d and so muche are 5/2 of a yarde worthe. [...]

Otherwise, see what the yarde is worthe, after the maner aforesaied in the other examples, and you shall find that the yarde is worthe, 2. s. 1. d. ⅗ of the whiche nomber take first the ½ for 4/6, whiche is 1. s. 0. d. ⅘, of the whiche nomber, take the ¼ for the other [...], whiche is 3. d. ⅕, adde these. 2. nombers together, and you shall finde the ⅝ to be worthe 1. s. 4. d. as before is saied. [...]

If 13 els [...], be worthe 27. s. what are 10 elles ⅔ worthe by the price? Aunswere: [Page 107] saie if 13⅚ geue 27/ [...] which shal 10 ⅔ geue: putte the whole nombers into their broken, and you shall finde 83/6, 27/1, end 32/3. Then multiplie 6 tymes 27 by 32 and thereof cōmeth 5184 the which nomber you shall diuide by 83 times [...] thre times, and you shall finde 20. s. [...] whiche is worth 9. d. [...]/85 part of a peny. [...]

If two yardes ½ be worthe 4. s. 8. d. what are 8 yardes ¼ worthe? Aunswer put the 8. d. into the part of a shillyng, which wilbe ⅔, then reduce the whole nombers into their broken, and thei will stand thus, 5/2, 14/3, 33/4, then multiplie twoo tymes 14 by 33, and diuide the producte by. 5. tymes 3. 4. tymes, and you shall finde 15. s. 4. d. ⅘, so muche are the eight yardes ¼ worthe. [...]

If one Kersey bee worthe 2 .li. 6. s. 8. d. how many Kerseis shal I buy for [Page] 36 .li. 3. s. 4 d. after y^{e} rate: Answere: put 6. s. 8. d. into the parte of a li. and you shall haue 2 li. ⅓ for the first nomber in the rule of 3. and 1 elle for the seconde nomber: then put 3 s. 4 d: into the part of a li. and you shal finde 36 li. 3/6 for the 3 nomber, then will your 3 nombers in the rule of 3 stand thus. 2⅓ | 1/1 | 36⅙. Therefore reduce the whole nombers into their broken, and you shall haue 7/3|1/1|217/6. Then multiplie 1 by 217, and thereof will come 651 for your diuidende. Likewise, multiplie 7 tymes 1 by 6, and the producte thereof will be 42. Then diuide 651 by 42. and you shall finde 15 ½. So many Kerseis of 2 pounde, 6 shillinges 8 pence the pece shall you haue for. 36. pounde, 3. shillynges 4 pence. [...]

## ¶The fowerth Chapiter treateth of losses and gaines, in the trade of marchaundise.

[Page 108] IF 13 yardes ⅓ be worth 22 li. 10 shillynges, how shall I sell the yard to gaine ⅓, or to make of 3. 4? whiche is all one? Answere: saie by the rule of 3, if 3 be come of 4. or if 3. yelde 4. what will 22. ½: multiplie and diuide, and you shal find 30 .li. Then saie again by the rule of 3. if 13 yardes 1/ [...] doe geue 30 .li. aswell of principall as of gain: what will 1 yard be worthe by the price? Multiplie and diuide, and you shall finde 2. pounde 5 shil. and for that price must the yard be sold to gaine the ⅓, or to make of 3. 4. [...]

Or otherwise, take the ⅓ part of 22. [...]i. 10. s. which is 7 .l. 10. s. that shal you adde with 22 pounde 10 shillynges, & you shall haue 30 pound as before. Then diuide [...] 30. by 13. ⅓. and you shall finde 2 pounde 5 shillynges, as aboue is saied.

[Page] If one yarde bee worthe 27. s. 6. d. for how muche shall 16 yardes ⅔ bee solde, to gaine 2. s vpon the pounde of money, that is to saie: open 20. s. Answere: adde 2 vnto 20 & you shall haue 22, then saie: if 20. s. of principall, doe geue 22. s. aswel of principal as gain: howe muche will 27. s. 6. d. principall yeld. Multiplie and diuide & you shall find 30. s. ¼: then saie again by the rule of 3, if 1. yard do geue me 30. s ¼ (which is aswell the principall as the gaine) what shal 16 yardes ⅔ geue me? Multiplie and diuide, and you shall finde 25 .li. 4. s. 2. d. For the same price shall the 16 yardes ⅔ be solde to gaine after the rate of 2. s vpon the pound of money, or in 20. s. whiche is all one. [...]

If 10 yardes ⅔ be worthe 25 .li. 10. s For how muche shall 2 yards ¼ be sold to gaine after 10 .li. vpon the 100 .li. of money? Answere: saie if 100 of principall [Page 109] yeld 110 aswell principall as gain how muche will 25 .li. 10. s. yelde me? Multiplie and diuide, and you shall finde 28 .li. 1. s. Then saie if 10 yardes ⅔ doe yeld me 28 .li. 1. s. aswell of principall as of gaine, how muche shall. 2. yardes ¼ yelde me? Multiplie and diuide, and you shall finde 5 .li. 18. s. 4. d. 1/ [...]2, for so muche shall the twoo yardes be solde, to gaine after 10 .li. vpon the 100 .li. of money. [...]

And although that in these questions of gaine and losse, sometymes the firste nōbers is not like vnto the third nomber, that is to saie, of the same denomination: as one would saie: if 20. shillynges gaine. 2. shillynges, what shall. 50 .li. gaine? Or 25 .li. &c. Or if 20 .li. doe gaine 2 .li. What shall 25. s. gaine me, or what shall. 27. s. gaine? Yet neuerthelesse, the rule is not therfore false. For if 20. s. dooe gaine 2. s: [Page] 20 .li. shall gaine 2 .li. and 20. d. shall gaine 2. d. likewise 20 crounes shall gaine 2 crounes, and so of all other: therefore it is to bee vnderstande, that the first nōber in these reasons, is presupposed to be semblable to the thirde.

When one marchaunt selleth wares to an other, and he geueth to the buyer 2 vpon 15: how muche shall the buyer gain vpō the 100 after the rate? Answere: saie if 15 geue 17, what shall 100 geue? Multiplie and diuide, and you shall finde 113⅓, so the buyer getteth after the rate of 13⅓ vpon the 100. [...]

If 1. Northe dosen cost me 3 li. 5 s. & I sel y^{e} same again for 3 .li. 12. s. 6 d. how muche do I gain vpō the li. of money after the rate. Answere saie if 3 .li. ¼ do geue 3. li ⅝, what shall 20/8 geue, put the whole nōber into their broken, & you shall haue 13/4, 29/8, 20/1, then multiplie 4 tymes 29, by 20, & therof commeth 2320 for your nomber that is to be diuided, likewise multiplie 13 times 8. 1 time, & [Page 110] therof cōmeth 104. Then diuide 2320 by 104, and you shall finde 22 shillinges 4/13. So I shall get 2 shillynges 4/13 vpon 20 shill. or vpon the li. of money. [...]

If a yarde of clothe coste me 7 shillynges 8 pence, and afterward I deliuer out 13 yardes ¼, for 4 pounde 13. s. 4 pence. I would know whether I do winne or lose, and how muche vpon the 100 pounde of money? Answere: se first at 7 shillynges 8 pence the yarde, what the 13 yardes ¼ shall cost, and you shall finde 5 pound 1 shil. 7 pence. And I solde them but for 4 pounde 13 shil. 4 pence, so that I doe lose vpon the 13 yardes ¼ the somme of 8 shilly. 3 pence. Then for to knowe how muche is lost vpon the 100: saie by the rule of 3, [...] 5. pound 1 shil. 7. pence doe lose 8 shil. 3. d What will 100 lose? First put 1 shill. 7 pence into the part of a pounde, and it will be 19/240. Likewise put 8 shill. 3. d. into the part of a li. and it is 33/80. Then [Page] will your nomber stande thus 5. 19/240, 33/80, 100/ [...], put the whole into his broken, and then multiplie and diuide, so you shall finde 8 .li. 1184/9752, whiche is worthe 2. shillynges. 5. d. 169/1219 and so muche is lost vpon the 100 .li. of money. [...]

More, if 12 yardes ½ of Scarlet bes solde for 30 pounde 15 shillynges, vpō the whiche is gained after the rate of [...] 1/9 vpon the 100. I demaunde what the yarde did coste at the firste. Aunswere: from 30 pounde 15 shillynges, substracte his 1/10 part, whiche is 3 .li. 1. s 6. d. and there resteth 27 .li. 13. s. 6. d. the whiche number multiplied by. 2. bryngeth 55 .li. 7. s. of the which is 2 pounde, one shillyng and [...]ower pēce. Then take againe the ⅕ of thesaied [...] pounde 1 shillyng 4 pence, whiche is 2 pounde [...]ower shillynges three pence, 9/25. And so muche did the elle cost at the firste penie. [Page 111] [...]

More, if 15 yardes ¾ of Scarlette doe cost me 32 pound 13. s. 4. pence. And I sell the yarde again for 2 pound, whether dooe I winne or lose, and howe muche vpon the pounde of money.

Answere: loke what the 15 yards ¾ are worthe at 2 .li. the yarde, and you shall find that thei are worth 31 .l. 10. s. But thei did cost 32 .li. 13. s. 4. d. so that there is lost vpō the whole 1 .li. 3, s 4. d. Thē to knowe how muche is lost vpon the li. saie by the rule of 3, if 32 li. ⅛ doe lose 1 li. ⅙: what will ⅓ lose? That is to saie: what will 1 li. lose? Reduce the whole nombers into their broken, and then multiplie and diuide, so shal you finde 21/ [...]8 [...] parte of a li. Then multiplie 21 by [Page] 240. because so many pence are in a li. and deuide the producte by 588. so shall you finde 8. d. ⅗ ⅜ ⅝ which beyng abreuiated doe make 4/7, and thus you se that 8. d. 4/7 is lost vpō the li. of money. [...]

If 1. yarde of cloth of tissue be solde for 3 .li. 15. s. whervpon is lost after the rate of 10. s. vpon the 100. I demaund what 12. yardes ½ of the same tissue did cost? Aunswere: adde vnto 3 .li. 15. hys owne 1/10 part, whiche is 7. s. 6. d. and al amounteth to 4 .li. 2. s. 6. d. then looke what the 12. yardes ½ wil amount vnto, after 4 .li. 2. s. 6. d. & you shall finde that thei will come to 51 .li. 11. s. 3. d. so muche did the 12. yardes ½ coste. [...]

[Page 112] More, if I sell 1 Wilshire white for 6 li. 12 s. whereupon I doe gain after the rate of 2 shil. vpon the li. of money, that is to saie, vpon 20 s. I demaunde what 11 peeces of the same whites did cost me? Answere: abate from 6 pound 12 s. (whiche is 132 s) his 1/ [...] parte, and thereof cōmeth 12 s. and there remaineth 120 s. or. 6 li. Then se at 6 .li. the clothe, what the 11 clothes are worthe and you shall finde them to be worthe 66 li. so muche did the 11 clothes coste. [...]

If I sel 10 elles ½ of Hollād for 12 s. 6 d. wherupon I doe lose after the rate of 2 s. vpon the li. of money. I demaūd what the ell did cost me? Answere: saie by the rule of: 3, if 18 geue 20: s. what will 20 s: 6: d: geue? Multiplie and diuide, and you shall find 25 s: Then dinide 25 s. by 10 ½, and thereof commeth 2 shil. 4. d. 4/7. So muche did the ell cost [...]

[Page] If I sell 1 clothe for 5 li. wherupon I do lose 10 vpō the 100, I demaunde how muche I should lose or gaine vppon the 100, in case I had sold the same for 5 li. 10. s. Answere: saie, if 90 yelde 100, how muche will 5 li. geue. Multiplie and diuide, and you shall finde 5 li. 5/9: then saie again by the rule of 3, if 5/9 come to 5 ½, what will 100 come vnto? Multiplie and diuide, and you shal finde 99 li. whiche beyng abated from 100. there will remain 1 pounde, and so muche is loste vpon the 100. [...]

## ¶The fifte Chapiter treateth of lengthes & breadthes of tapistrie, and other clothes.

IF a peece of Tapistri [...]e bee 5 elles ¾ longe, and 4 elles ⅔ in bredth, how many ells square doeth the same pece containe? Answere: multiplie the length by the breadth, that is to saie 5 ¾ by 4 ⅔, and therof cōmeth 26 elles ⅚, so many elles [Page 113] square doth the same peece conteine.

More, if a peece of Tapistrie doe conteine 32. elles square, and the same being in length 6 elles ¼. I demaūde howe many elles in breadth the same peece doth conteine. Auns. diuide 32. elles by 6 ¼ and thereof commeth 5. 3/28: So manye elles dothe the same péece conteine in breadth.

More, a péece of clothe beynge 13. yardes ⅓ in length, and 5, quarters ½ in breadth, how many yardes of ⅔ and ½ broade will the same peece make? Aunswere: see what parte of a yarde, the 5/4 and ½ be, and you shall finde that they make 1 yarde ⅜. Then multiplye 13 yardes [...] by 1 yarde ⅜ and you shall haue 18. yardes ⅓ in square the whiche you must diuide by ⅔ & ½ that is to saye by [...], (bicause the ⅔, ½ beinge brought into 1 fraction maketh 5/ [...]) and you shall finde 22. yardes: So many yardes of [...] & [...] large doth the same peece conteine.

[Page] More, a marchant hath bought 4. yardes ⅔ of cloth beinge syre quarters [...] broade to make hym a gowne the whiche he will line thoro [...]out, wyth black Say of three quarters of a yarde broad, I demaund how much Say he must bye? Aunswere: Multiplye the length of the cloth, by the breadth, that is to say 4 ⅔ by 1 ⅝, (which is the syx quarters ½) and thereof commeth 7. yardes 7/12, the which diuide by ¾ and you shal finde ten yardes 1/9. So many yardes of Say must he haue to line the same 4 yardes ⅔ of clothe of 6. quart. [...] broad.

More, at 6, s. 8, d. the elle square, what shall a peece of tapistrie cost me, which is fiue elles ½ lōg and 4, elles ¼ broade? Aunswere, multiply 5, ½ by 4, [...] and therof cōmeth 23 elles ⅜ square: then say by the rule of three, if one elle square cost me 6, s. 8, d. what shal 23 ⅜ cost? Multiplye and diuide, and you shall finde 7 li. 15. s. 10 d. so muche the said peece of tapistrie did cost.

[Page 114] Or otherwise, by the rules of practise, take the ⅓ of 23. ⅜: and you shall finde 7 .li. 15. s. 10. d. as aboue is sayde.

More, a peece of Hollande clothe conteining 42. elles ⅔ Flemish, howe many elles Englishe doe they make? Here must you fyrst note that 100. els Flemish, do make but 60. elles Englishe, and so consequentlye fiue elles Flemishe do make but 3. els English. Therfore say by the rule of 3. if 5. elles flemish doe make three elles English, how many elles English will 42, els ⅔ Flemish make. Multiply and diuide so shall you finde 25. elles ⅗ Englishe, and so many elles Englishe doth 42. ⅔ Flemish conteine, the lyke is to bee done of all others.

More, I haue boughte a peece of Tapistrie, being 5. elles ¾ longe, and 4. elles ⅔ broade measure of Flaunders, I demaunde howe manye elles square it maketh Englishe measure? Aunswere: [Page] First, forasmuch as three elles english are worth 5 elles flemishe, therefore put 3 elles english into hys square, in multipliyng 3. by himself whiche maketh 9: likewise multiplye 5. in hymselfe squarely, and it wilbe 25. Then multiplye 5¾ whiche is the length of the peece, by 4⅔ which is the breadth, & therof cōmeth 26 elles ⅚ square, then say by y^{•} rule of thrée, if 25 elles square of flemishe measure, be worth 9 elles square of englishe measure, what are 26 elles flemish ⅚ worth? multiply and diuide, and you shall finde that they are worthe nine elles 33/50 square of english measure.

More at 3. s. 6. d: the ell flemish what is the englishe ell worth after y^{•} rate. Answere: saye if 5. elles flemishe bee worth three ells english, what is 1 ell flemishe worth? multiply and diuide, & you sh [...]ll fiynde 3/ [...] of an englishe ell. Then saye by the rule of 3, if [...] of an englishe ell, be worth 3 s. 6 d. what is 1. englishe ell worth? multiplye and [Page 115] diuide, and you shall finde 5 s. 10. d. so much shall the englishe ell be worth.

More at 6 s. 8 d. the flemishe elle square, what is the english ell worth. Aunswere: say by the aforesaid reasō, if 25 elles flemishe square, be worth 9. elles square englishe, what is one ell square flemishe worth? multiply and diuide, & you shall finde 9/25 of a square englishe elle: Then saye, if 9/25 of an englishe elle be worth 6 s. 8. d. what is one square elle englishe worth? multiplie and diuide, and you shall fynde 18. s. 6. d. [...]/9, so muche shal one englishe elle square be worth.

## ¶ The sixt Chapter treateth of the reducing of the paumes of Genes into englishe yardes, wherof foure Paumes maketh one english yarde.

I Haue bought 97. paumes [...] of Genes veluet, & I would know howe manye yardes they will [Page] make? Aunswere, Diuide 97. [...] by 4. and you shall haue 24. yardes [...]. So manye yardes doe the 97. paumes [...]/2 conteine.

Or otherwise, take some other nō ber at your pleasure, as 20, paumes, which doe make fiue yardes, and then say by the rule of three, if 20/1 paumes, giue 5/1 yardes, what will 97. ½ giue? Multiplye and diuide, and you shall finde 24. yardes ⅜ as before.

More, at two shillings 7, d. y^{•} pau [...] of Genes, what wil the English yarde be worth after the rate? Aunswere, say by the rule of three, if ¼ of an English yarde bee worthe twoo shillings 7/12. What is [...] yarde worth? Multiplie & diuide, and you shall finde ten shillings 4. d. So muche is the Englishe yarde worth.

Or otherwise, multiply 4. paumes (which is one yarde) by two shillings 7, pence, and you shal finde 10, s. 4. d. as before.

[Page 116] If 257. Paumes ½ bee worth 20 .li. 16. s. 8. d. What is one yarde worthe after the rate? Annswere, saye: by the rule of 3, if 257, ½ paumes be worthe 20. 5/ [...], what are 4/ [...] paumes worth. Multiply and diuide, and you shall fynde 100/309 part of a pounde, which is worthe 6 s. 5. pence, [...]: so much is one yarde worthe.

## ¶ The vii. Chapter treateth of marchaundise solde by waight.

AT 9. d. ½ the ounce, what is the li. waight worth? Answere, say if 3/ [...] giue 9. ½ what will [...] gyue multiply and diuide, & you shal finde 12. s. 8 d. so much is the yarde worth?

Or otherwise, by the rules of practise for syre pence, take the ½ of 16. which is 8. s. then for 3. d. take the ½ of 16. s. whiche is 4. s. Finally, for the halpenye, take 16. ob. which are 8. d. adde all these nombers togither and you shal finde 12. s. 8. d. as before.

[Page] More, at 10 d. ½ the ounce, what are 112 .li. waight worth after the rate? Aunswere reduce. 112 .li. into ounces: in multipliyng. 112 .li. by 16. ounces & you shall haue 1792. ounces, then say by the rule of 3. if 1/ [...]|10½|1722/1: Multiplie and diuide, and you shall finde 18816 d. whiche do make 78 .li. 8. s. and so much are the 112 .li. worth after 10. d. ½ the ounce.

At 12. s. 8. d. the li. waight, what is the ounce worth: Aunswere: put 12. s. 8 d. into pence, and you shall haue 152. pence: then say by the rule of 3. if 16. ounces cost 152. d. what shall 1. ounce coste, multiplye and diuide, and you shall finde 9. d. 1/1, so much is the ounce worth.

Or otherwise, take the ¼ of 12 s. 8. d. for 4 ounces, and thereof commeth 3. s. 2. d. then for one ounce, take the ¼ of 3. s. 2 d. and you shall haue 9. d. ½ as before.

At 32 .li. 10. s. the quintall, that is to [Page 117] saye, the 100 .li. waight what is 1 .li. waight worthe after the same rate? Aunswere: Put 32 .li. 12. s. al into shillings and you shal haue 650. s.

Then say, by the rule of three, if [...] 650 [...]. multiply and diuide, and you shall finde, 6 s. 6. d. so muche is the 11. worthe.

If one pound waight of saffron do cost me 18. s. 8 d. what shall 355 .li. 10. ounces cost me by the same price? Aunswere, saye by the rule of 3. if ½|18 ⅔|355⅝. Multiply and diuide, and you shall finde 331 .li. 18. s. 4. d. so much are the 355 .li. ten ounces worth.

### Brief rules of waight.

WHo that multiplieth the pēce that 1 .li. waight is worth by [...], & diuideth the product therof by. 12. hee shall finde howe manye pounds in mony the quintal is worth that is to saye, howe much the 100 .li. waight is worth.

And contrariwise he that multiplieth [Page] the pounds of money that the 100 waight is worth by 12. and deuide the product by 5. and you shall fynde howe many pēce the poūd waight is worth.

### ¶ Example.

AT seuentene pence the pounde waight, what is the 100 pounde waight worth? Aunswere, Multiplie 17. by 5, and thereof cōmeth 85. diuide the same by 12. and you shall finde 7. pound 1/12, whiche 1/12 is worth one shilling and eight pence. So much is the 100 pounde waight worth.

More, at 13, li. the 100 li. waight, what is one pounde waight worth? Aunswere, Multiplie 13. by 12. amounteth to 156. the whiche diuide by 5. and you shal finde 31. d. [...]/5 which is 2. s. 7 d. 1/ [...] and so muche is one pounde waight worth.

The lyke is to be done of yardes, elles, or of any other measure, when we recken but fyue score to the hundred.

### Briefe Rules for measure.

[Page 118] Who that multiplieth the pence that one ell is worthe, by 6. And diuideth the product by 12. he shall fynde how many poundes in money y^{e} 120. elles are worth, which 120 elles wee count but for a C.

And contrariwise, be that multiplieth the poundes in money that the 120 elles are worth by 12, and diuideth the multiplicotion by 6, shall fynde howe many pence the ell is worthe.

### ¶ Example.

At ten pence the ell, what are 120. elles worth: Answere, multiplie 10. d. by 6. and thereof commeth 60: The which diuide by 12. and you shall find fyue pounde, so many pounds in money are 120. ells worth at 10. d. the ell.

More. at 9. pounde, the 120, elles, what is one elle worthe? Aunswere, Multiplie nine pound by twelue, and therof commeth 108, the which diuide by 6. and you shal finde 18. d. so much is one elle worth.

The like is to be done of alll maner [Page] of wares, which are sold after 120. for the hundred.

### ¶ Briefe Rules for [...]ure hundreth waight here at London, whiche is after 112 .li. for the C.

WHo that multiplieth y^{e} pence that one pounde waight is worthe by 28. and diuideth the product by 60. shall finde howe many pounds in money the 112 .li. waight is worth.

ANd contrariwise, he that multiplied y^{•} poundes in money that the 112, li. is worth by 6 [...], and diuideth the product by 28. shal finde howe manye pence one li. waight is worth.

### Example.

AT nine pence the pounde waight, what is the 112 li. waight worth [...] Aunswere: multiplie 6. [...]. by 28, and thereof cōmeth 2, 2, the whiche diuide by 60, and you shal finde 4 .li. 12/60 which beinge abbreuiated is [...] of a pounde, whiche is worthe 4, s. And thus the [...] .li. 2. is worth 4. pound 4, shil.

[Page 119] At 8 li. the 112, li. waight, what is 1 .li. waight worth? Aunswere, Multiplie 8 li. by 60, and thereof commeth 480, the whiche diuide by 28. and you shall finde 17. d. [...]: so muche is 1, li. waight worth.

## ¶ The .viij. Chapter treateth of tares and allowances of marchaundise solde by waight:

AT 12 li. the 100. suttell, what shall 987 li. suttell be worth? in giuing 4 li. waight vppon euery 100 for tret? Answere, adde 4 li. vnto 100. & you shall haue 104. Then say by the rule of thre, if 104. be worth 1 [...] .li. what are 987 li. waight worthe? multiply & diuide, & you shal finde 113, li. [...] which is worthe 17. s. 8, d. 4/ [...]. So much shal y^{e} 987 .li. waight be worth. [...]

At 6, s. 8. d. the pound waight what shall 345, li. ½ be worth in giuing 4, li. waight vpon euery 100, for the tret. Aunswere, see first by the rule of three, [Page] what the 100 pound is worth saying, if [...]|6⅓|100/1 Multiplie and diuide, and you shall finde 33 .li. ⅓ then adde 4 .li. vnto 100, & they are 104. then say again by the rule of 3. if a 104 .li. be sold for 33, li. ⅓ for how muche shall 345 .li. ½ be sole? multiply & diuide, and you shal finde 110 .l. 14. s. 8. d. 12/13 So much shal the 345 .li. [...]/2 be worth, at 6. s. 8. d. the pound, in giuing 4. vpon the 100.

More, if 100 bee worth 36. s. 8 d. what shall 780 .li. bee worth in rebating 4 li. vpon euery 100, for Tare & Close? Answere, Multiply 780. by 4. and therof commeth 3120. The which diuide by 100. and you shal haue 31, li. ⅕ abate 31. ⅕ from 780. and there wyll remaine 748⅘. Then say by the rule of three, if 100/1 do cost 36. ⅔, what will 748. ⅘ cost after the rate? Multiplie & diuide so shall you finde 274. s. 6. d. [...]8/25, and so much shall the 780 .li. cost, in rebatinge 4 .li. vpon euery 100. for Tare and Cloffe.

[Page 120] More, whether doth he lose more that giueth 5 li. vpō the 100. or he that rebateth 5 .li. vpō the 100. for care and cloffe? Answere. First, note that hee which giueth 5 .li. vpō the 100. giueth 105. for 100: and he which rebateth 5 .li. vpon the 100. giueth the 100. for 95. Therefore saye by the rule of 3. if 105. be giuen for 100: for how much shal y^{•} 100. be giuen? Multiply and diuide & you shal finde 65. 5/2 [...]: and be whiche rebateth 5. vpō the 100. maketh but 95. of 100: so that he loseth 5. vpon the 100 & the other which giueth 5. vpon y^{•} 100 loseth but 4. 16/21 vpon the 100. Thus he that rabateth 5. vpon the 100. loseth more by 5/21 vppon the 100. than the other whiche gaue 5. vppon the 100. for tare and clo [...]fe.

If 100. of Allam do [...] me. 26. s. 8. d. how shall I sell the li. waight to gaine after the rate of 10. vpon y^{•} 100. Answere, put 26. s. 8. d. al into pence. & you shal haue 320. d. Thē say by y^{•} rule at 3. if 100. giue 110 what shal 320 giue [Page] multiplie and diuide, and you shall finde 352. d. Thē say, if 100 li. be worth 352 d. what is [...], li. multiplye & diuide, and you shall haue 3, d. 26/50 whiche 26/50 is worth ½, and 1/25 of ½. That is to saye, the pounde waight shalbe worth 3. d. ½, 1/25 of a halfe pennye, in gaining 10. vpon the 100.

If one pound waight doe cost me, 6 s. 10, d. and I sell the same for 7, s. 2 d. I demaund howe much I should gaine vpon the 100 li, of money after the rate? Answere, say by the rule of 3. if 6, ⅚ yelde 7. 2/6 what will 105/1 yelde? Put the whole nombers into theyr broken, the [...]ul [...]iplie and diuide, & you shall finde 10436/41 from the whiche substract 100, and there resteth 4 .li. 36/41 so muche is gained vpon the hundred pounde of money after the rate.

More, if one pound do cost me 5, s. 4. d. and I sell the same againe for 4, s. 9 d. I demaunde how much I shal lose vpon the 100. pounde of money? saye, if 5, ⅓ doe giue but 4, ¾, what [Page 121] shall 100/1 geue? Putte the whole nomber into their broken. Then multiplie and diuide, and you shall finde 89 1/10. the whiche you must substract from 100. and there will remain. 10. pound 15/ [...]6, so muche is lost vpō the. 100. pound of money.

More, if the pounde waight doe cost me. 3. shillynges. 2. pence: and I sell it againe for. 3. shillynges. 4. pence, how muche shall I gaine vpon. 20. shillin. Aunswere: saie if 3⅙ geue 4⅓, what shall 20/1 geue. Multiplie and diuide and you shall finde 27. shillinges 7/19: out of the whiche abate. 20. shillynges, and there will remaine. 7. shillynges 7/19, whiche is. 4. pence 4/19: and so muche is gained vpon the pound of money, that is to saie vpon 20. shillinges.

If the pounde waight doe coste me 4. shillinges. 4. pēce, and sell it again for. 3. shillinges. 2. pence. I demaunde how muche I shall lose vpō the pound of money? That is to saie, vpon. 20. shillynges. Answere: saie, if, 4. ⅓ geue [Page] but 3⅙, what will 20/1 geue, multiplie and diuide, and you shall finde. 14. s. [...], the whiche you must abate from. 20 shillynges, and there will remaine. 5. shillinges 5/13, whiche [...], is worthe. 4. d. [...] of a penie, and so muche is lost vpon the pounde of money.

## ¶ The ninth Chapter treateth of certain questions, doen by the double rule, and also by the rule of three compounde.

WHen the quarter of Wheate doeth cost 6. s. 8. d the loafe of breade waiyng. 20. ounces is solde for a ob. I demaunde that if the quarter of Wheate did cost 10 shillinges, for how muche shall the loafe of bread be sold, that waieth. 16. ounces? Answere: by the first part of the rule of 3. compounde, whiche is mentioned in the third parte of this boke, and in the seconde Chaptier of the same. Therefore saie by thesame firste parte of the rule of. 3. compounde, if 6⅔|20/1|½|10/1|16/1.

[Page 122] Then multiplie the firste nomber by the seconde, and the producte thereof shalbe your diuisor. Likewise multiplie the other three nombers, the one by the other, and the producte thereof shalbe your diuidende: As thus, firste multiplie. 6⅔ by. 20. and thereof commeth 400/ [...] for your diuisor, then multiplie ½ by 10/1, and the producte thereof by 16/1, so you shall haue [...] for youre nomber that is to bee diuided, then diuide 150/ [...] by 400/ [...], and thereof commeth 480/800. the whiche beeyng abbreuiated, bryngeth [...]. of a penie: and for that price muste the loafe of breade bee solde, whiche waieth sixtene ounces, and the quarter of Wheate beeyng worthe tenne shillynges.

Or otherwise by the rule of three, at twoo tymes. Firste saie if [...]0/1 ounces geue ½, what will 16/1 ounces geue? Multiplie and diuide, and you shall finde 2/ [...], of a pennie. Then saie againe, if. 6. [...] dooe geue me ⅖, what will 10/2 geue? Multiplie and diuide, [Page] and you shall finde ⅗ of a penie, as afore is saied.

When the cariage of one hundreth waighte of Marchaundise. 50. miles doeth cost. 5. shillinges, what shall the cariage of. 500. waight cost me for. 16. mile? Aunswere. By the firste parte of the rule of. 3. compounde, saiyng, if 100|50|5|500|16 Multiplie. 100. by 50. the produ [...]e will be. 5000. whiche shalbee your diuisor. Then multiplie 5. tymes. 500. by. 16. and thereof commeth. 40000. for your diuidende. Therefore diuide. 40000. by. 5000. and you shall finde. 8. shillynges, so muche shall coste the cariage of. 500. waight. 16. miles.

Or otherwise by the double rule of three, that is to saie, by the rule of three at twoo tymes: firste safe if. 50. miles dooe paie. 5. shillinges, what shall. 16. miles paie? Multiplie and diuide, and you shall finde. 1. shilling ⅗. Then saie againe, if. 100. waight doe coste me. 1: shilling 3/ [...], what shall. 500. waighte [Page 123] coste? Multiplie and diuide, and you shall finde. 8. shillinges, as before.

When the cariage of. 800. pounde waight of Marchaundise. 84. miles, doeth cost me. 6. shillinges, how many miles maie I haue. 64. pound waight carted for. 3. shillinges. 4 pence. Aunswere, by the second part of the rule of 3. compounde: saie if [...] | 84/1 | 6/1 | 64/1 | 3 ⅓.

Then multiplie the fowerth nomber 64/1 by the third nomber ⅙, and therof commeth [...] for your diuisor. Likewise multiplie. 3⅓ by 100/1, and by 14/1, and you shal haue in the product 24000/ [...]. then diuide [...]000/3 by 384/1 and you shall find. 72. miles 11/12 of a mile. So many miles shal the. 64. pounde waight be caried, for three shillinges. 4. pence.

Otherwise by the rule of. 3. at twoo tymes: first saie, if. 100. waight do cost me. 6. shillinges, what shall. 64. p [...]ūd waight cost? Multiplie and diuide. and you shall finde. 3. shillinges 21/25. Then saie, if 3 21/25. hee paied for. 84. miles carriage: for how many miles shal [...]. shillynges [Page] ⅓ bee paied? Multiplie and diuide, and you shall finde 72 miles 1 [...]/12.

If 100 horses in 100. daies do spend 180. quarters of otes: how many quarters of Otes will. 300. horses spende in. 150. daies? Aunswere: by the firste parte of the rule of three compounde: multiplie. 180. tymes. 350. by. 150. and diuide the product by. 100. tymes. 100 and you shall finde. 945. quarters. So many quarters of Otes will 350. horses spende in. 150. daies.

Or otherwise by the rule of three at twoo tymes: first saie, if 100. daies doe yelde me. 180. quarters of Otes: what shall. 150. daies yelde: multiplie and diuide, and you shall finde. 270. quarters: then saie againe, if. 100. horses doe spende. 270. quarters of otes, how many quarters of otes wil. 350. horses spende? Multiplie and diuide, as you shall finde. 945. quarters as before.

## ¶ The tenth Chapiter treateth of the rule of fellowship, with out any time limited.

[Page 124] THE rule of Fellowship is thus: you must sette doune eche mannes somme of money that he laieth into companie, euery one directly vnder the other, the whiche you shall adde altogether, and the totall somme of all their whole stocke beyng thus assembled, shalbe your commō diuisor, to the findyng out of euery mannes part of the gaine. Then shall you multiplie the gaine, or els the losse, by eche mannes portion of money that be laied in, and diuide the productes by the saied diuisor: so shall you haue in your quotient euery mannes parte of the gaine, or els of the losse, if any thyng be lost.

### ¶ Example.

1. Twoo Marchauntes haue made companie together, the firste laied in fiue hundred pounde. The second put in three hundred pounde, and with occupiyng thei haue gained. 64. pound: I demaunde how muche eche mā shal haue of thesame gaines, accordyng to [Page] the money that he laied in. Aunswere: Adde. 500. and. 300. bothe together, which are the percelles that thei laied in, and thereof commeth. 800. for your diuisor: then saie by the rule of three, if 300 .li. (whiche is their stocke) do gain 64 .li. what shall. 500 .li. gain? (which is the first mānes money that he laied in) multiplie and deuide, and you shal finde. 40 .li. for the first mans parte of the gaine: then saye if. 800. giue. 64. what will. 300. giue? Multiply and diuide, and you shall finde 24 .li. for the seconde mannes parte of the gaine. [...]

Or otherwise, put 500 .li. which is the first mans money that he laied in, ouer the 800 li. whiche is the whole, stocke, and you shal haue 509/800 which being abbreuiated, do make ⅝, and suche part of the gain shal the first mā take, that is to say ⅝ of 64 .li. which is 40 .li. [Page 125] And consequentlie, by the same manner, the seconde shal take the ⅜ of 64. which is 24. pound for his part of the gaine as before. [...]

2. Twoo Marchauntes haue compainied together, the first put in 640 .li. and he taketh ⅝ partes of the gaine. I demaunde what the seconde Marchaunte layed in? Aunswere. Seeyng that the first Marchaunt taketh ⅝ of the gaine, it followeth that the second must haue ⅜ which is the rest, and ther fore saie by the rule of thrée, if ⅝ of the gaine, whiche the first manne taketh, did saie into the stocke [...]. How muche shall the ⅜ of the gaine laie in, whiche is the seconde mannes gaine? Multitiplie and diuide, and you shall finde 384. pounde, so muche ought the seconde man to laie into companie.

3. Twoo Marchauntes haue companied together, the first manne laied in. 640. pounde, and the seconde hath [Page] laied in so muche, that he muste haue 60. pounde for his part of. 100. pounde whiche thei haue gained. I demaunde howe muche the seconde man did laie into companie? Answere: seyng that the seconde man taketh. 60. pounde of the gaine: it followeth that the firste must haue but. 40. pounde. Therfore saie by the rule of. 3. if. 40. pounde doe laie in. 640. pounde, what shall. 60. pounde laie in? Multiplie and diuide, and you shall finde. 260 .li. so muche did the seconde Marchaunt laie in.

4. Twoo Marchauntes haue companied together, the firste laied in. 83. pounde 6. shillinges. 8: pence, the seconde put in. 170. duckettes: and thei haue gained 100. li of y^{t} which the first man must haue 60 pounde. I demaūd what the ducket was worth? Answer seyng that the first man must haue 60 pounde, it followeth that the seconde muste haue. 40. pounde, therefore saie by the rule of three, if. 60. pounde of gaine that the first man taketh did lay [Page 126] in. 83 .li. 6 s. 8. pence of principall, how muche shall. 40 .li. of gain put it, mulplie and diuide, and you shall finde 55 .li. 5/9: so muche are the 170. duckets worth. Then put 55 .li. 5/9 into shillings and you shall haue 1111. s. 5/9 then to knowe what the ducket is worth, saie by the rule of thrée, if a [...] giue 1111. [...], what will [...] giue? Multiplie and deuide, and you shall finde 6. s. 6. d. 12/51, so muche is the ducket worth.

5. Twoo Marchauntes haue companied together, the second man laied in more by 30. poundes then did the firste man: and they gained 120. poundes of the whiche the firste man ought to haue 50. pounds. I demaund what eche of thē did laie in. Aunsware, from 120. pound abate 50. pounde and ther resteth 70. pound for the second mans parte: so that by thys meanes the seconde manne (because he laide in 30. pound more then the first man did) taketh 20 .l. more of the gain: & therfore say by the rule of 3. if 20 .li. of gaine [Page] did laie in. 30. pounde of principall, how muche shall. 50. pounde laie in Multiplie and diuide, and you shall finde. 75. pounde, so muche did the first man laie in, and consequently the seconde laied in. 105. pounde.

6. Twoo Marchauntes haue companied together, the second hath laied in twise so muche as the first man did, and. 10. pounde more: and thei gained 100. pounde, of the whiche, the firste ought to haue. 32. pounde for his part: I demaunde how muche eche of them did laie into companie? Answere. If it were not for the. 10. pound that the seconde manne laied in more: he should haue had but. 64. pounde of the gaine, which is the double of the first mānes parte. But because he laied in. 10 .li. more, he hath [...]ower pounde more of the gaine, and therfore saie by the rule of. 3. if. 4. pound of gaine did laie in 10 pound of principall (whiche was ouer and aboue the double of the first mannes laiyng in) what shall 32 .li. of gaines [Page 127] laie in? whiche is the first mannes part of the gaines that he taketh. Multiplie and diuide, and you shall finde 80 .li. for the firste mannes laiyng in: and consequently. 170. pounde for the second mannes portion that he laid in.

4. Twoo Marchauntes haue companied together, and thei haue gained 100. pound of the which the first must haue after the rate of 10 vpon the 100 pound, and the second must haue after the rate of 15 pound vpon the 100 .li. I demaunde howe muche eche of theim ought to haue? Answere. But 10 .li. for the first mannes laiyng in, and. 15 li. for the second mannes laiyng in Adde 10 .li. and. 15. pound together, and thei make 25. pound. Then put 10. ouer 25 and it is 10/25, whiche beyng abreuiated are ⅖. Therefore he that taketh 10 li. vpon the 100. it must haue the 2/ [...] of the gaine, whiche is 40 .li. Then put. 1 [...]. ouer 25. and it is 15/25, which being abreuiated are 3/ [...]. Therfore the second must haue 3/ [...] of the 100 .li. which is. 60 .li.

[Page] 8. Twoo Marchautes haue companied together, the first laied in 46 .li. 18. shilling. and the seconde laied in 33. pounde 2. s. so thei haue gained 30. pound. I demaunde how much euerie man shal haue for his part of the gain Answere: Adde 46. pound 18. shilling and 33. pound 2. shilling both togither and you shall finde 80. pound for your common diuisor: then saie if 80. pound which is all their stocke do gain 30 .li. what wil 46. 6/10 gaine, whiche is the firste mans laiyng in: Multiplie and deuide, and you shall finde 17. pounde 11. shillyng 9. pence for the firste mans part of the gaine. Then saie again, if 80. pound do gain 30. pound what wil 33. pound 1/10 gain, whiche was the seconde mans laiyng in: multiplie & deuide, and you shall finde 12 .li. 8. s. 3. d. for the seconde mans part of the gain.

And after the same maner shal you doe, in case there were three or foure Marchauntes that woulde companie together: Addyng al their summes of [Page 128] money (which thei lay into the stocke) into one totall summe: whiche shal be your common deuisor: and then work with the reste, as is taught in the former questions of the rule of company.

9. Three Marchautes haue companied togither, the first laide in I know not how muche: the seconde did put in 20. peeces of clothe, and the thirde hath laide 500. pounde. So at the ende of their companie, their gaines amounted vnto a thousande pounde, wherof the firste manne ought to haue 350. pounde, and the seconde muste haue foure hundred pounde.

Now I demaunde how muche the first man did laye in, and for how muche the twentie péeces of clothe were put into companie?

Aunswere.

Seeyng that the fyrste and the seconde Marchautes must haue 750 .li. for their partes of the gain. Then the thirde manne muste haue the reste of the thousande pound which is 250 .li. [Page] And theerfore saie by the rule of. 3. [...] 250. of gaine, be come of 500. pound of principall: of how muche shall come 350 pounde of gaine? whiche the firste man taketh, multiplie and diuide, and you shall finde 700. pound. So muche did the first mā laie in then saie if. 250 pound of gaine be come of. 500. pound of principall, of how muche will come 400. pounde, whiche is the gain that the second man taketh. Multiplie and diuide, and you shal finde. 800. pound For so muche were the. 20. peeces of clothe laied into companie.

10. Three Marchauntes haue gained. 100 .li. the firste must haue the ½, the seconde must haue ⅓: and the third must haue ¼. I demaunde how muche euery manne must haue of the gaine? Answere: reduce ½, ⅓, ¼, into a commō denomination, after the order of the second reduction in fractions, and you shall finde 12/24, for the ½, 8/24, for the ½: and 6/ [...], for the ¼: Then take twelue for the firste mannes laiyng in, 8. for the second [Page 129] mannes laying in: and 6 for the thirde mannes laying in. The whiche three nombers beinge added together shall be your common diuisor, whiche do make 25. Then multiply 100 by 12 for the firste man, by 8, for the second man, and by 6 for the third man. And diuide euery multiplication by 26. So shall you fynde 46 li. 2/13 for the fyrste mannes part of the gaine. 30 .li. 10/13 for the second mannes parte: and 23 li. 1/1 [...]0 for the third mannes parte.

11. Two marchauntes haue gained 100 li. the firste muste haue ½ and 5 li. more: the second must haue ⅓ and 4 li. more: I demaunde howe muche ethe of them shall haue? Aunswere, From 100 abate 5 and 4. so ther wil remain 91. then take the ½ of 100 l. which is 50 li. for the first mans laying in: Likewise, take ⅓ of 100 li. for the seconde mans laying in, which is 33 li. ⅓. Then adde 50 li. and 33 li. ⅓ togither, and you shall haue 83 li. ⅓ for youre common [Page] diuisor, then multiply 91. pound by 50. and diuide by 83. ⅓: and thereof cōmeth 54 pound, ⅗ vnto the whiche nomber adde 5, and all is 59 .li. ⅗ for y^{e} first mans part. Likewise multiplye. 91. by 33. ⅓: and diuide by 83. ⅓: and you shal finde 36 li. ⅖ vnto the which adde 4: and you shal haue fourty pound, [...] for the seconde mans part.

12. Two Marchauntes haue gayned a hundred pound, the firste muste haue the ½ lesse by 4 pound, the second must haue ⅓: lesse by 2 pounde. I demaund how much eche of them shall haue? Answere, Adde 4 & 2. with 100. & they make 106. Thē take as before is saide [...]0 pounde, for the first man, & 33. ⅓ for the seconde, adde them bothe togither, & they be 83. ⅓ whiche shalbe your diuisor. Then multiplie 106. by 50. and diuide the product by 83 ⅓, so thereof commeth 63 .li. ⅗. From the whiche abate the foure pounde lesse y^{t} the fyrste mā taketh, and then is there remaining 59. pounde, ⅗ for his parte. [Page 130] Likewise multiplie 106 by 33. ⅓ and diuide by 83⅓ & you shall finde 42 .li. ⅖: from the which abate 2 .li. lesse and there remaineth 40. pounde, ⅖ for the seconde mans part.

### ¶ The rule of Felowship with time.

THe money that euery mā layeth in, muste be multiplied by the time that it remaineth in company: and of that whiche cōmeth thereof you shall make their newe layings in for eche of them: and then multiplye the gaines by euery one of them seuerally, the whiche you shall diuide by all their new layings in adding togither, and you shall haue proportionally eche mannes parte of the gaine according to his laying in.

### ¶ Example.

1. Two Marchauntes haue companied togither, the first hath put in the fyrst of Ianuary 450. pounde, the second did lay in y^{e} 2. of May. 750. pound [Page] And at the yeres ende, they had gayned 100 li. I demaunde howe muche eche of them shall haue of the gayne? Answere: forasmuche as the firste did put 450 .li. the fyrste of January: his money remained in company 12. monethes, and therefore multiply 450. by 12 monethes, and therof commeth 5400. for his newe laying in. And the seconde layed in his 7, 0 li. but at the first daye of Maye: so that his money remained in companye but 8 monethes. Therefore multiplye his 750 li. by 8. and therof commeth 6000 for his new laying in: Thē [...]dde 5400. with 6000. and they make 11400 for your common diuisor: Then multiply 100 li. which is the gaynes by 5400, and diuide the product by 11400. and therof commeth 48 li. 7/19 for the first mannes part of the gayne. Likewise multiplye 100. by 6000, and diuide the producte by 11400, and you shal finde 52. 12/19 and so muche must the second man haue for his parte of the gayne.

[Page 131] 2. Two marchaunts haue companied togither, the first hath put in the first of Ianuary 640 .li. The seconde can lay in nothing vntil the first of Aprill. I demaunde how much he shall then laye in, to the ende that he maye take halfe the gaines? Aunswere, Multiply 640 li. by 12. monethes that his money abideth in the companye, and therof cōmeth 7680 li. for his layinge in. And so muche oughte the seconde mannes layinge in to be, for because he taketh ½ of the gaine: But for that, that be putteth in nothinge vntill the first of Aprill, his money can be in cō pany no lōger than 9 monethes. And therefore diuide 8680 by 9, and therof commeth 753 li. ⅓ So much oughte the seconde marchaunt to laye in the first of Aprill, to the ende that he maye take the one moy [...]y of the gaynes.

3. Three Marchayntes haue companied togither. the firste layed in the firste of Marche 100 li. The seconde [Page] laide in y^{e} first of Iune so muche money, that of the gaine, he muste haue the ⅓ parte: and the thirde laide in y^{e} first of Nouember so muche money, that of the gaines he muste haue likewise ⅓ and thei continued in company, vntil y^{e} next Marche folowing. I demaunde howe muche the seconde and the thirde Marchauntes did laye in? Answere, Multiply 100. which the firste man did lay in, by 12. monethes that his money continued in companie, and therof commeth 1200. for his laying in: and so muche ought the seconde and the thirde marchaunt eche of them to lay in: Bicause they parte the gaynes by thirdes. But for that, that the seconde Marchaunt putteth in nothinge tyll the first of Iune, hys money can bee in companye but nine monethes. Therefore diuide 1200. by nine monethes, and therof commeth 133. ⅓. And so muche ought the seconde Marchaunt to laye in: Then, forasmuch as the thirde Marchaunt, dyd [Page 132] laye in nothing vntil the first of Nouember: His money abideth in companye but the space of foure monethes. Therefore diuide 1200. by 4. and thereof commeth three hundred pounde. And so much ought the thirde marchaunt to lay into companye.

4. Three Marchauntes haue companied togither, the fyrste layde in the fyrst of Ianuary a hundred Duckettes. The seconde hath layed in fyftye pounde, the fyrste of Marche: And the thirde putte in a Iewell the fyrste of Iulye: And at the yeares ende, they had gained foure hundred crownes: of the whiche, the fyrste marchaunt must haue fifty crownes, and the seconde muste haue 80. I demaunde what the Duc [...]et was worth and at what price the Iewell was valued, whiche the thirde Marchaunte layde in?

[Page] Aunswere: the firste mannes money is 1200 as afore is sayde, and he taketh 50 crownes of the gayne: therefore say, if fifty crownes of gayne be come of 1200, whice was his stock, of how much shall come 80. crownes of gaine that the seconde man taketh? multiplye and diuide, and you shall finde 1920. for the second marchaunts laying in. Then say again, if 50 crownes be come of 1200. stocke: of howe much shal come 270. crownes, which the thirde man taketh of the gayne? Multiply and diuide, & you shall finde 6480. for the third marchauntes laying in. Then diuide 1920, whiche is the seconde mannes laying in, by 10. monethes that his money did continue in company, and you shall fynde 192 Duckets, which are worth 50 .li. bicause he layed in 50 li. Then diuide 192 Duckets by the sayde 50, li. (being reduced into shillinges) and thereof commeth 5. shillings 2. pence, ½. So muche was the Ducket worth: Finallye, [Page 133] diuide 6480. (whiche is the third mannes layinge in) by 6. monethes that his Iewel remained in companye, and you shall finde 1080 Duckets: and for that price was y^{e} Iewell put in company.

5. Three Marchauntes haue companied togither: the first layed in the first of Ianuary 100 li. and the firste of April he hath taken backe againe 20 .li. The second hath layed in the firste of Marche 60 .li. and afterward he dyd put in more 100 li. the first of August. The third layd in the first of Iuly 150 .li. And the first of October he did take backe againe 50 li. And at the yeres end, they found that they had gained 160 li. I demaunde how muche euery man shall haue? Aunswere, Multiply 100 li. which the first man layed, by 12 monethes, and therof commeth 1200. li from that nomber abate 9 times 20 which are 180. and there wil remaine 1020, for the first mans laying in. Thē [Page] multiplie 60. which the seconde man layde in, by ten & you shall haue 600 vnto the which adde 5. times one hundred, which are 500. so all amounteth to 1100. for the second mans laying in Afterwardes, multiplie 150. pounde, which the thyrde man hath layed in, by 6. monethes, and therof commeth 900. from the which nōber abate three times 50. and they are 150: so there resteth 750. for the thyrde mans laying in. Then procede wyth the reste, as in the firste Question of the rule of felowship with tyme, in adding 1020, 1100. and 750. altogither, whiche shall be your Diuisor: Then multiplie 160 by 1020. by 1100. and by 750. & diuide at euery time by your Diuisor, which is by all theyr layinges in added togither, and they make 2870, so you shal fynde 56. 241/287: for the fyrste man, 61. 93/ [...]87 for the seconde, and 41. 233/287 for the thirde man.

6. Two Marchaunts haue companied [Page 134] togither, the fyrste hath put in 960. pounde, for the space of 12. monthes, and he ought to haue 8. pounde vpon the hundred pound of the gaine. The second hath layed in 1120 .li. for the space of eight monthes, & he ought to haue after 12. pounde vpon the 100. pound of the gaine.

And at the yeares ende, they haue gained eyght hundred pounde. I demaunde how much eche of them shall haue of the gaine. Answere, multiplie 960. that the first mā did lay in, by 12. monethes, and the product thereof, multiplie againe by 8. and you shall haue 62160. for the fyrst mans laying in: then multiplye the 1120. that the seconde hath layed in by eyght monethes, and that whiche commeth thereof you shall multiplie againe by 12. and you shal finde 10 [...]520. for the seconde mans layinge in, then proceede with the rest, as in the first Question of the Rule of Felowshippe, and as in the laste Exaumple, and you shall [Page] finde 3993/13 for the first man: and 430 .li. 10/13 for the second man.

### ¶ The rule of company, betwene Marchauntes and their Factours.

7. The estimation of the bodye or persone of a Factour, is in suche proportion to the stocke, whiche the Marchaunt layeth in: as the gaine of the sayd Factour is vnto the gayne of the sayd Marchaunt. As thus: if a Marchaunt do put into the handes of hys Factour 200 li. to employe, and he to haue halfe the profise, the persone of the sayd Factour shal be esteemed 200 li. And if the Factour do take but the [...]/3 of the gaine, he shoulde haue but ½ so much of the gaine as the marchaunt taketh, which should take ⅔ wherfore the persone of the Factour is estemed but the ½ of that which the Marchaunt layeth in, that is to say 100 li.

And if the Factour did take the ⅖ of [Page 135] the gaine, then the Marchaunt shall take the residue, which are ⅗ of y^{e} gaine wherefore the gayne of the Mayster vnto that of the Factoure is in suche proportion as 3 vnto 2. Then if you will knowe the estimation of the persone of the Factour, say if 3 giue me 2 what wil 200 giue? Multiply 200 by 2 and diuide be 3 so you shall finde 133 [...]/3 Otherwise, consider that the Factour taketh the ⅗ of that whiche the Marchaunte taketh. And therfore take the ⅔ of 200, and you shall fynde 133. ⅓ as before: and so muche is the persone of the Factoure esteemed to bee worthe.

8. And if the Marchaunt should deliuer vnto his Factoure 200 .li. and the Factour would laye in 40 .li. and his person, to the ende he might haue the halfe of the gaine: I demaund for how much shall his person be esteemed Ans. abate 40 li. from 200 li. and ther will remaine 160 .li. And at so much [Page] shall his person be estemed.

And if the factour woulde take the ⅔ of the gaine, his person with his 40 pounde, shall bee esteemed twise as muche as the stock that the marchant layeth in, which shoulde haue but ⅔ of the gaine: for ⅔ vnto ⅓, is in double proportion. Therefore double twoo hundred pounde, therof cōmeth 400 .li. from the which abate 40 .li. & there will remaine 360 .li. And if the Factour would take but the ⅓ of the gaine, that shall bee but the ½ of ⅔ whiche the marchaunt taketh: then the estima [...]ion of his person, with his layinge in should be esteemed but the halfe of the whiche the marchaunt layeth in: take therfore the [...]/2 of 200 .li. whiche is 100 .li. from the which abate fourty pound and the rest whiche is 60 .li. is the estimation of his person.

9. If it so chaunce that for to make traffick of 240 .li. y^{e} person of the factour should be so estemed, y^{t} he shoulde [Page 136] haue but the ¼ of the gaine, and yet he would haue the ⅔, I demaunde howe much he shoulde put in of readye money, besides his person? Aunswere, seing that his person gaineth the ¼, all the whole layinge in, shall gaine the rest that is to saye the ¾: nowe for bycause ¼ is the ⅓ of ¾ therfore his person shalbe estemed the ⅓ of all the layinge in. Take then the ⅓ of 240. and you shall haue 80. for the estimation of his person, and for that, that he wyll haue the halfe of y^{e} gaine, you shal adde 80. with 240 .li. and therof commeth 320. of the which take the halfe, which is 160. and from the same you shal abate the 80. and there wyll remaine other 80. which he ought to lay in of readye money, and the marchaunt muste lay in the ouerplus, whiche amounteth to 160 .li.

10. A marchaunt hath deliuered to his Factour 1200 .li. to gouerne them in the trade of Marchandise vpon such condition that hee for hys seruyce [Page] shall haue the ⅓ of y^{e} gaine if any thing be gayned, or of the losse if any thing be lost: I demaund for how much hys person was estemed? Answere, seeing that the Factoure taketh the ⅓ of the gaine, hys persone ought to bee esteemed as muche as ½ of the stock whiche the Marchaunte layeth in, that is to say y^{e} ½ of 1200 li. which is 900 l. The reason is, bycause the ⅓ of the gayne that the Factoure taketh, is the [...]/2 of the ⅔ of the gaine that the Marchaunt taketh.

11. A Marchaunt that deliuereth vnto his Factour 1200 li. and y^{e} Factour layeth in 500 li. and his person: Now, bicause he laieth in 500 li. and his persone, it is agreed betwene them that be shal take the ⅖ of y^{•} gain: I demaūde for how much his persone was esteemed? Aunswere, Forasmuche as the Factour taketh the ⅖ of the gaine, he saketh the ⅔ of that which y^{•} March [...]ūt taketh, for ⅖ are the ⅔ of ⅗: and therefore [Page 137] the Factours laiyng in ought to be 800. pound, which is the ⅔ of 1200. pounde, that the marchaunt laied in: Then abate 500. pounde, whiche the Factour did lay in, from 800. pounde, whiche should be his whole stock and there remaineth thrée hundred pound for the estimation of his persone.

12. More, a marchaunt hath deliuered vnto hys factoure a thousande pounde vpon suche condition, that the factour for his paines and seruice, shal haue the gaines of 2. hundred pound, as though he had laide so muche in of redy money: I demand what the portion of the gaine, the saide Factour shall take? Aunswere: Sée what parte the two hundreth pounde (whiche the Factour layed in) is of one thousande twoo hundreth whiche is the whole stocke of their companie, and you shal finde that it is the ⅙, and such parte of the gaine shall the Factour take.

But in case that in making the couenauntes, it were agreed that the [Page] Factour should haue the gaine of two boundred pounde of the stocke, whiche the marchaunte layeth in, that is to saie of the thousande pounde. Then shoulde the Factour take the ⅕ part of the gaine. For twoo hundred pounde is the ⅕ of a thousande pounde.

## ¶ The 11 Chapiter treatteth of the Rules of barter.

TWoo Marchaunts will chaūge their marchandise, the one with the other. The one of them hath clothe of 7. s. 1. d. the yarde to selle for readie money, but in barter he wil sel it for 8. s. 4. d. The other hath Sinamon of 4. s. 7. d. the li. to sell for readie money. I demaunde how he shall sell it in barter to the ende he be no loser? Aunswere, saie, if 7. 1/12 (whiche is the price that the yarde of clothe is worth in readie money (be solde in barter for 8. ⅓ for what shal 4. 7/12 be solde in barter which 4. 7/12 is the price that the l. of Synamon is worth in readie money, reduce the whole numbers in to their [Page 138] broken, and then multiply and diuide, and you shall finde 5. s. 4. d. [...]2/1 [...] parts, of a penie, and for so much shall he sell the pounde of Synamon in barter.

2. Two Marchaunts will chaunge their marchaundise the one with the other, the one of them hath Chamlets of two pounde 18. s. 4. pence the péece to sell for ready money, and in barter he will sell the péece for 4 .li. 3, s. 4. d. the other hath fine cappes of 35. s. 10. d. the dossen to sell in barter. I demaund what the dosen of caps did cost in redy money? Answere: saie if 4 .li. 3. s. 4. d. whiche is the ouerprice of the péece of Chamlet, become of 2 .li. 18. s. 4. pence whiche was the iust price of the same, of what shal come 35. s. 10. d. whiche is the ouerprice of the dossen of cappes? Multiplie and deuide, and you shall finde 25. shil. 1. d. and so muche are the dossen of caps worthe in redie money.

3. Two Mharchants will chaunge their marchaundise the one with the other: the one of them hath Fustians [Page] of 18. s. 4. d. the peece to sell for readye money, and in barter hee will sell the pece for 26. s. 8. pence. The other hath tapistrie of 15. d. the ell to sel for readie money, and in barter he will sell it for 20. pence the ell: I demaunde whiche of them gaineth, and how muche vppon the hundred pounde of money?

Aunswere: saye if 18. s. ⅓ (which is the iuste price of the peece of F [...]itian) bee solde in barter for 16. s. ⅔: for howe muche shall 1. s. ¼ (whiche is the iuste price of the ell of Tapistry) be solde in barter? Multiplie and diuide, and you shall finde 21. d. 9/11. And he doeth ouersel it but for 20. d. so that of 21. d. 9/11: bée maketh but 20. d. And therefore saye by the rule of three, yf the seconde marchaunte, of 21 9/11, do make but 20/1 how muche shall he lose vpon the 100/1? Multiplie and deuide, and you shall find 21. ⅔, the which being abated from a hundred there wil remaine 8. ⅓. And after the rate of 8. ⅓. doeth the seconde marchant lose vpon the 100. And consèquently [Page 139] the firste marchant, of 20. d. maketh 21. d. 9/11: and therfore say again by the rule of thre, if the first marchāt of 10/11, doe make 21. 9/11 how much shall he gaine vpon 100/1? Multiplie and diuide, and you shall finde 102 .li. 1/11. Thus the firste gaineth after the rate of nine pounde. 1/11 vpon the hundred pounde of money.

For your better vnderstandyng of these questions, you muste note that when one Marchante gaineth of an other after the rate of ten pound vpon the hundred pounde hee gaineth the [...] of his owne principall, and the other whiche loseth after the rate of 9. 1/11 vppon the hundred he loseth the 1/11 of his principal. And it may be proued thus: When one marchaunt will sell hys wares vnto an other, whiche wares stand him but in 100 .li. and he wil sel thē for 110 .li. he of his 100 .li. maketh 110 .li. wherfore he gaineth after 10. li vpon the 100. whiche is the 1/1 [...] of his principall, and the other whiche byeth [Page] wares for 110 .li. that coste but 110 .li. of the 110. poūd he maketh but 100 .li. And therfore say by the rule of three, if 110. become of 100. of howe muche shall come 100? Multiplie and deuide, and you shall finde 90. 10/11, the whiche abate from 100: and there resteth 9. 1/11 whiche is the 1/11 of his principall that y^{•} second leseth vpon the 100. as afo [...] is saied. And therefore, who so that will know what one marchant gaineth of another, either after the rate of tenne vpon the hundred, whiche is the 1/11 of hys principall, or else after the rate of twentie vpon the hundred which is the ⅕, or of anye other parte, and that he would likewise knowe what parts the other loseth of his principall: hee muste take for the numeratour of the broken number of hym that loseth, as much as for him that gaineth, then adde the numerator and the denominator (of the broken number of hym that gaineth) both togither, and make therof the denominator of the broken [Page 140] number of hym that loseth, and then shall you haue the parte of hym that loseth, as by example, of him that gaineth after ten pound vpon the 100 .li. which is the 1/10 of his principall: take the numerator whiche is 1. and make that the numeratoure of the broken number of hym that loseth, then adde 1. whiche is the numerator of the fraction of hym that gaineth with tenne, whiche is his denominator, and you shall haue 11. for the denominatour of the fraction of hym that loseth. Then put one ouer the 11. and you shal haue 1/11. Thus it appeareth when one marchaunt gaineth of an other after ten vpon the hundred, he gaineth the 1/10 of his principall, and the other loseth 9. 1/11 whiche is the 1/11 of hys principall. And if he woulde gaine after twentie vppon the hundred whiche is the ⅕ of hys principall, the other shoulde lose sixtine ⅖ whiche is the ⅙ of hys principall, and so is to bee vnderstande of al other fractions.

[Page] 4. Two marchaunts wil chaunge their marchaundise the one with the other, the one of them hath Seies of 20. s. & 10. d. the peece to sel for readie moneye, and in barter he will sell the peece for 23. s. 4 d. and yet he wil gain moreouer after ten pounde vppon the hundred pounde. The other hath wol of 50. s. the hundred to sell for readye money. I demaunde how he shall sell the C. of woll in barter; Answere: say if 20. s. 10. d. whiche is the iuste price of the peece of Seye, be solde in barter for 23. s. 4. d. for how much shall 50. s. (which is the iust price of y^{•} C. of woll) be solde in barter? Multipliye & diuide, and you shal finde 56. s. Then for because the first marchant gaineth after 10. li vpō the C .li. he maketh of his c .li. 110 .li. and consequently the second marchant maketh of 110 .li. but 100 .l. And therefore saie, if the seconde marchaunt of 110. doe make but 100 how much shal he mak of 56: Multiply and diuide & you shal finde 50. s. 20. d. 10/12 of [Page 141] a penie, and for so muche shall he selle the hundred of woll in barter.

5. More, two Marchauntes wyll chaunge their marchaundise, the one with the other, the one of theim hath Taffeta, of 16 crownes the peece to sell for readie money, and in barter he will sel the peece for twenty crownes, and yet he wyll gaine moreouer after ten pounde, vpon the hundred pound. The other hath ginger of 3. s 9. d. the pounde waight, to sell in barter. I demaunde what the pounde did coste in readie moneye? Answere: saye if twenty crownes which is the surprice of the peece of Laffeta, become of 16. crownes the iuste price, of how muche shall come. 3. s. 9. d. whiche is the price of the ouerselling the pounde of Ginger? Multiply and diuide and you shal finde 3. s. Then, for because that the Marchaunt of Laffeta will gaine after the rate of ten vppon the hundred▪ say if 100. doe giue 100. what shal 3. s. giue? Multiplie and deuide and you [Page] shall finde thre shillinges thre pence [...] and so much did the pound of Ginger coste in readite money.

6. More twoo marchauntes will thaunge their marchaundise the one with the other, the one of theim hath Worsteds of 25 shil. the péece to sel for ready money, and in barter he wil sell the péece for 33. shill. 4. pence and yet he loseth after ten vpon the hundred: the other hath waxe of 3 pound 6. shil. 8. pence the hundred to sell for readye money. I would know how he should sell his ware in barter? Aunswere: say if 25. shil. whiche is the iuste price of the peece of Worsted be solde in barter for three and thirtie shill. 4. pence, for how much shall thre pound 6. shil. 8. d. be solde, whiche is the iust price of the hundred of waxe. Multiply and diuide, and you shall find foure pound 4/9 whiche is eight shillinges ten pente, ⅔ then for because that the Marchaunt of Worsteds, loseth after ten vppon the hundred: Of a hundred he maketh [Page 142] but sixtie. And thererefore, saie: If ninetie geue a hundred what giueth foure pounde. 4/9? Multiply and diuide, and you shall finde foure 76/81 whiche is worth eightine shillinges nine pence 5/27, and for so much shall he sell the one hundreth of Waxe in barter.

7. More, twoo Marchauntes wyll chaunge their marchaundise the one with the other, the one of them hath Worsteds of 5. pounde six shillinges, eight pence the peece to sell for readie moneye, and in barter he will sell the peece for 6. pounde, 13. shillinges. 4. d. and yet he loseth after tenne vpon the hundred, and the other hath Muske of two shillings. nine pence ⅓ the pound waight, to sell in barter? I demaunde what the pound did cost in readie money? Answere: say if 6 pound. ⅔ whiche is the ouerprice of the peece of Worsted, become of 5. pound, 1/9 whiche is the iust price of the same, of how much shall come twoo shillinges 9. pence. ⅓. [Page] Multiplye and deuide and you shall finde 2. 2/9. whiche is twoo pence ⅔ then for because that the marchaunte of Worsteds loseth after tenne vpon the hundred, of a hundred he maketh but 90. and therfore saie if a hundred giue but 90. how muche shall 2. s. 2/9 giue? Multiply and diuide and you shal find twoo shillynges and so muche coste the pound of Muske in readie money.

### Other Rules of Barter, wherein is giuen some parte in readie money.

WHen a Marchaunt ouerselleth hys marchaundise and he will giue also some part of hys ouerprice in readie money as in the ½ the ⅓ or the ¼ &c. He must substract the same part of money from the iuste price, and also from the ouerprice of his marchandise: and the two numbers that remaine after the substraction is made, shalbe that twoo firste numbers in the rule of three and the iuste price of the second marchant [Page 143] shalbe y^{•} thirde, to know how much he shal ouersel the part of his marchādise

8: Two Marchaunts wil chaunge their marchaundise the one with the other the one of theim hath fine woll at fiue pounde the hundreth, to sell for readie moneye, and in barter hee will sell it for sixe pounde, and yet hee will haue the [...]/3 in readye money. The other hath clothe of 13. shillinges four pence to sel for ready money. I would knowe howe he shall sell the same in barter? Aunswere: take the ⅓ of 6 .li. whiche is the ouerprice of the 100. of wolle, and you shall haue 2. pound the whiche abate from 5 .li. whiche is the iust price of the 100. of woll and from 6 .li. which is the ouerprice, and there shal rest 3 .l. and 4 .l. for the 2 first numbers in the rule of thrée, then take 1 [...]. s. 4. d. whiche is the iust price of a yarde of clothe for the thirde number: Then multiplie and deuide and you shall finde 17. shill. 9. d. ⅓. for so muche shall the seconde selle his clothe in barter.

[Page] 9. More, twoo Marchauntes will chaunge their marchaundise the one with the other, the one of them hath waxe of thrée pound. 6. s. 8. d the C. to sell for readie money, and in barter he will sell the same for 4 .li. 3. s. 4. d. and yet he will haue the ¼ in ready money, and the other hath fine Crimson sattine of 15. s. the yarde to sell in barter. I demaund what it is worth in ready money. Aunswere. Take the ¼ of 4 .li. 3. s. 4. d. and abate it from 4 .li. 3. s. 4. d. and from thrée pounde 6. s. 8. pencs, and there resteth 3 .li. 2. s. 6. d. and 2 .li. 5. shillinges 10. pence for the two first numbers in the rule of thrée, and 15. s for the thirde number whiche is the ouerprice of the yard of sattine. Then multiply and diuide, and you shal find 11. shillinges. And soe muche did the yarde of sattine cost in readie money.

10. Two Marchants will chaunge their marchandise the one with the other, the one of them hath tinne of 50. shillings the hundred to sell for ready [Page 144] money, and in barter he wil sell it for three pounde 6. s. 8. d and he wil gaine after ten vppon the hundred, and yet he wil haue the one half in readie money: and the other hath leade of 3. halfpence the li. to sell for readie money. I demaund how he shall sell the pounde in barter? Aunswere: See first at tenne vpon the hundred, what the thre poūd ⅓ will come vnto, and you shall finde that they will come to 3 .li. ⅔, which is 13. s. 4. d. of the which, the halfe which he demaundeth in readie money, is 36 shillinges and 8. pence the whiche beyng abated from fiftie shillinges, and also from three pounde 13. shillinges four pence, there shal rest 13. shillings foure pence, and one pound sixtine. s. eight pence, for the twoo first numbers in the rule of three, whiche you muste put al into halfepence, and thre halfepence for the thirde number, and then multiplye and diuide, and you shall finde 4. pence ⅛, and for so muche shall hée sell the pounde of leade in barter.

[Page] 11. More twoo marchauntes will chaunge their marchaundise the one with the other, the one of theim hath steele of 16. s. 8. d. the hundred waight to sell for readie money, and in barter he wil sell it for 25. s. and yet he loseth after tenne vpon the hundred, but hee will haue the ½ in readie moneye, the other hath yron of 6. shill. 8. pence the dundred to sell in barter, I demaunde what it did coste in readye money?

Aunswere: saie if a hundred come but to 90. how muche shall 25. s. come to? Multiplie and deuide, and you shall finde 22. s. 6. d. of the whiche number, take ½ which is 11. s. 3. d. and substract it from 22. s. 6. d. and from 16. s. 8. d. and there shall rest 11 s. 3. d. and 5. s. 5. pence for the two first numbers in the rule of thrée, and 6. shil. 8. pence which is the ouerprice of a hundred of yron for the thirde number, then multiplie and deuide, and you shall finde. 3. s. 2. pence, ½4/7: and so muche did the hunof yron coste in readie moneye.

[Page 145] 12. More, twoo marchauntes wyll chaunge their marchaundise, the one with the other, the one of them hath sayes of 200 shill. 10 d. the pece to sell for ready money, and in barter he wil sell the pece for 21 shillings, and he wyll haue the ¼ in readie money: The other hath cappes of 35 shill. the dosen to tell for readye money: but he wyll gayne after 10 vppon the 100. I demaunde howe he shall sell the same cappes in barter? Aunswere, saye, if 100 be worthe 110: What shall 35. shill. be worthe, which is the iust price of the dosen of cappes? Multiplye and diuide, and you shall fynd 38 shill. 6. d. Then take the ¼ of 25. whiche is 6 sh. 3. d. and substracte it from 20 shill. 10. d. and from 25. sh. and there shall rest 14 sh. 7. d. & 18 shill. 9. d. for the two firste numbers in the rule of three, and 38 shill. 6. d. whiche is the iuste price with his gaine of the dosen of cappes, for the thyrd nomber: then multiplie and diuide, and you shall finde 49. [Page] shill. 6. d. and for so muche shall he sell the dosen of cappes in barter.

## ¶ The 12 Chapter treateth exchaunging of money from one place to another.

FIrste, you muste note, that at Andwerpe they vse to make their accomptes by Deniers de gros, that is to saye by pence Flemishe, whereof 12 doe make 1 shill. Flemishe, and 20 shil. Flemish do make 1 li. de gros.

1. If I deliuer in Flaunders, 500 li. Flemishe, at 19 shil. 6. de gros that is to saye, at 19s. wil. 6. d. Flemish, to receiue 20 shil at London: I demaund howe much I shall receyue sterlynge at London for the sayde 500 li. Flemishe? Aunswere: Saye, yf 19 ½ geue 20/1 [...] what wyll 50 [...]/1 geue? Multiplie and diuide, and you shal finde 512 li. 16. sh. 4. d. 12/13 of pennye. And so muche sterling shall I receyue in London for my 500 li. Flemishe.

[Page 146] 2. If I deliuer in London 375 li. sterlyng, to receyue in Andwerpe 21 shill. 9. d. de gros, that is to saye Flemishe, for euery pounde sterlinge. I demaunde howe many poundes Flemishe I shall receiue in Andwerpe, for the sayd 375 pound sterling?

Answere, say if 23/ [...] giue 21 ¾: what will 375/ [...] giue? Multiplie and diuide, and you shall finde 407 .li. So many poundes Flemishe shall I receyue for the sayde 375 .li. sterlinge in Andwerpe.

3. If I take vp monie at Andwerpe after 19 sh. 6 d. Flemishe to pay for y^{e} same at London 20 s. ster. & when the daye of paiment is come, I am forced to rechaunge the same, and to take vp money againe here in London to repay the same, so that for 20 shillings, which I take vp here, I must repaye, 19 shill. 9. at Andwerpe. I demaunde whether I do winne or lose, and how muche vpon the 100 pound of money? Aunswere, Saye, if 19 [...]/4 giue 19 [...]/ [...], what wyll 100/ [...] giue?

[Page] Multiplie and diuide, and you shall finde 9858/79, the whiche beinge abated from 100 there will remayne 121/79. And so muche doe I lose vpon the 100 li. of money.

4. If I take vp at London 20. shill. ster. to paye at Andwerpe 21. shill. 8. d. Flemishe, and when the day of payment is come. I am constrained to take vp money againe at Andwerpe wherewith to repay the foresayde summe: and there I doe receyue 22. shill. Flemishe to paye 20. shill. at London. Nowe I demaund whether I do winne or losse and howe much vpon the 100 pounde of money after the rate.

Answere: say, if 212/2 giue 22/1. What wil 100/1 giue [...] Multiplie and diuide, and you shall finde 1017/13, from the whiche abate 100, and there will remaine 1 [...] and so muche shall I gaine vpon the 100 li. of money.

The exchaunce from London into Fraunce, is not lyke as it is into [Page 147] Flaunders but is deliuered by the Frenche crowne, whiche is worth 50 souse Tournois the pece.

And here muste you note that in Fraunce they make theyr accompte by Deniers Tournoys, whereof 12 maketh one souse Tournois, and 20. souse. Tournois maketh 1. pound Tournois, whiche thei call a Liure, and the Frenche Crowne is currant amonge Marchauntes for 51 souse Tournois, but by excaunge it is otherwyse, for they wyll deliuer but 50 souse. Tournois, whiche is 2 li. 10 souse. Tournois for a crowne, or at suche price as the takers vp of money can agree with the deliuerer. As by Example.

5. If I deliuer 340 pound ster. here in London after 6. shill. 4. d. ster. the crowne, to receyue at Roan or at Paris 50 souse Tournois for euerye crowne, I woulde knowe howe manye [Page] liures Tournoys I shall receiue there for my 340 li. ster. Aunswere, say if 6 sh. ⅓ do giue me 2 li. Tournois what will 6100/1 sh. giue (whiche is the 340 li. reduced into shillings) multiplie and diuide, & you shall finde 2684 Liores 4/19 whiche is worthe 4. shill. 4/19 Tournois, and so much shal I receiue in Roan or Paris for my 340 li. sterl.

6. If I deliuer in Paris or Roan, or els where in Fraunce 1250 liures Tournyis, at 50 souse tournois the crowne, to recyue for euerye suche crowne 6 shillings 3. d. sterling at London. I demaunde howe much sterling money I shall receyue at London for my 1250 .li. Tournois. Aunswere: saye, if twoo li. ½ doe giue me 6 shillinges ¼, what wyll 1250/1 giue? Multiplie and diuide, and you shal finde 3125 shyllynges sterlynge, whiche maketh 156, pounde, fyue shillyngs sterlynge. And so many poundes shal I receyue at London, for the sayde [Page 148] 1250 pounde Tournois, after 6 shillinges 3 pence for euery crowne.

## ¶ The 13 Chapter treateth of the Rule of Alligation.

THe Rule of Alligation is so named, for that it teacheth to alligate or binde togither diuers percelles of sundrie prices, and to knowe howe muche you muste take of euerye percell, accordynge to the nombers of the Question.

### Example.

1 A Goldsmythe hath three sortes of Golde. The fyrste is worth thirtye Crownes the pounde weyghte: The secende is worthe 36 Crownes. And the thyrde is worthe 45. crownes, and of these three sortes he will make a Scepter of sixe pound weight, whiche shalbe worthe 40, Crownes the pounde. I demaunde howe muche he muste take of euery sorte? [Page] Answere: first you must set downe the nōbers whereof you wyl make the Alligation (which are 30. 36. 38, and 45) orderly the one vnder the other, as if you shoulde make of them an addiciō: and the common nomber whereunto you will reduce them, shall you set on the left hand, which common nomber in this example is 40. Then marke what summes bee lesser, then that common nomber, and whiche bee greater, and wyth a draught of your penne, euermore lynke two nombers togither, so that the one be lesser then that common nomber, and the other greater then he. For 2 greater nor twoo smaller nombers maye not bee linked together, for they wyll eyther be lesser, or els greater then the common nomber: but one greater nomber, and one smaller maye be so mixed, that they will make the common nomber. And two greater or twoo smaller nombers, can neuer make the common nomber in dewe order, [Page 149] as here after shall appeare.

After that you haue thus lynked them, then make howe muche eche of the lesser nombers is smaller then the common nomber, and that difference shall you set against the greater nombers, whiche bee linked with those smaller, eche of them with his matche styll on the righte hande. And lykewise you muste set the excesse of the greater nombers againste the lesser which be combyned with them. Then shall you adde all those differences into one summe, whiche shalbe the first nomber in the rule of three, and the second nomber shalbe the whole massye pece that you will haue of all the perticulers, the thirde summe shalbe eche difference by it selfe, and by them shall you finde out the fourth nomber declarynge the luste portion of euerye perticuler in that mixture, as nowe by the former example, I wyll make it plaine. [Page] [...]

Here in this former example, you see that I haue set downe the seuerall prices, whiche be 30, 36. 42. 45, and haue linked together 30, with 45. & 36. with 42. The cōmon price 40, I haue set on the lefte syde, and the difference of it from euerye seuerall price, I haue set on the ryght hande, agaynste that summe wyth the whiche it is lynked. So the difference of 30 from [Page 150] 40 is 10, whiche I set against 45, that he is lynked wythall, and the difference of 45, aboue 40 is 5. whiche I haue set against 30.

So lykewayes, the differences of 42 aboue 40, is 2, that I haue sette againste 36. And the difference betwen 36 and 40 (whiche is 4) I haue sette againste 42. Then I adde all those differences togither and they make 21, whiche I make the fyrste nomber in the Rule of three, and 6 the seconde nomber, which is the weyght of the Scepter of Golde, and the thyrde nomber shall bee euerye perticuler difference. Then I worke by the Rule of three: sayinge yf 21. (whiche is the differences added together) doe geue me 6 pounde, whiche is the weyghte of the Scepter, what shall 5 gyue, whiche is the first difference?

Multiplie and diuide, and you shall fynde one pounde 1/7: so muche muste I haue of the fyrste price. [Page] Then doe likewise with the reste and you shall finde 4/7 of the seconde price, 1 li. 1/7 of the thirde price, and 2 .li. 6/7 of the fourth, the whiche 4 summes beinge added together, do make 6 li. whiche is the totall that I would haue. And now to proue if the prices do agre, you shall do thus: Firste multiplie this totall summe 6 by the common price 40 and it will make 240 Crownes, whiche you shall kepe be it selfe. And afterward multiplie euery seuerall summe of weight by the price belongynge to the same weyght, and if that sume do agree with the first that you kepte by it selfe, then is youre worke well done, as here 1 li. 3/7 is the weyght of that sorte of golde whiche is of 30 Crownes price. Then multiplie 30. by 1 li. 3/ [...], and it maketh 42. crownes, 6/7, which you shall set downe. Then multiplie 4/7 (whiche is the weyght of the seconde sorte of golde) by 36, which is the price of the same, & therof cōmeth 20 crownes 4/2: so again

[Page 151] 1 .li. ⅓ multiplyed by 42, doth make 48 crownes. And laste of all 2 pounde. [...] multiplied by 45. maketh 128 crownes 4/7. Al these added together dothe make 240 crownes, agreable to the former summe of 40 multiplied by 6: And thus I maye affirme that this worke is well done.

2. A Tauerner hathe foure sortes of wine, of foure seuerall prices, the firste of 8 pence the Gallonde, the seconde of tenne pence the Gallonde, the third of 15 pence, and the fourht of 18 pence. And he will mingle one pū chen with all these sortes, so that the Gallonde shalbe worthe but twelfe pence. I demaunde howe many Gallondes he muste take of euery sorte? Aunswere: Firste suppose the punchen to holde some certayne measure, as to conteyne 84 Gallondes and then the (forme wyll bee after this sorte, as you see here after folowynge. [Page] [...]

## ¶ The 14 chapter treateth of the rule of falshode, or false positions.

THe Rule of falsehod is so named not for that it teacheth any deceyte of falsehode, but that by fayned nombers taken at all aduentures, it teacheth to finde out the true nōber that is demaunded. And this (of all the vulgare Rules whiche are in practise) is the most excellent: this rule hath two parts, the one is of one false position alone: the other is of two positions as here after shall appeare.

[Page 152] Those questions whiche are done by false positions, haue theyr operations, in a maner like vnto that of the rule of three, but onely that in the rule of three, we haue three nombers knowen, and here in this rule wee haue but one (I meane that commeth in operation) vnto the likenes whereof we muste diuise two other, the one multiplying, and the other diuidyng, as by example.

1. I haue deliuered to a banker a certein summe of pounds in money, to haue of him by the yeare 6 li. vpon the 100 li. And at the ende of 10 yeres he payed me 500 pounde for all bothe principall an gaine. I demaund howe muche was the principall summe that I deliuered at the fyrste. Here you see that there are diuers termes: but the chiefe to worke with all is 500 li. whiche commeth of the other nombers, that is to saye, of 10, and 100 for of them is compounde the tenour [Page] of the question, the practise whereof is thus.

Let vs faine a nomber at pleasure, and with the same let vs make oure discourse, euen as though it were the principalll summe that we seke for. As by Example. Suppose that I deliuered hym at the firste 200 pounde, the whiche were worthe to me in 10 yeares 120 pounde after the rate of 6 vpon the hundred pound. Then 120 pounde added with 200 pounde. Do make but 320 li. and I must haue 500 li. Thus you see that I haue three termes for the rule of three: the one which shall conteine the Question the other two, which I haue 200, & 320: in such sorte, that 320, ought ta haue suthe proporcion to 200, as 500 hat vnto the nomber that I seeke: that is to say, vnto the true principal summe, then must I haue recourse vnto the rule of three, after this sorte, saying. [Page 153] If 320. pound become of 200. pounde of howe muche shall come 500. pound Multiply 500: by 200. and thei are 100000. the whiche you shall diuide by 320. pounde and thereof commeth 312. pounde. ½ whiche is the summe that I deliuered at the firste, and thus, this rule hath some congruence with the double rule of three.

2. I haue a Cesterne with 3. vnegal cockes conteining 60. pipes of water: And if the greatest cocke bée opened, the water will auoyde cleane in one houre, at the seconde yt will anoyde in twoo houres, and at the thirde it will require three houres. Nowe I demande in what space wil it auoide, all the cocks beyng set open. Suppose that it wyll auoide in halfe an houre, that is to saie, in 30. minutes. Then muste there auoide at the firste cocke, the ½, whiche is thirtie pipes, and by the seconde cocke the ¼ whiche is 15, pipes, and by the thirde cocke the ⅙, that is tenne pypes, all the whiche [Page] summes beeyng added together doeo make fiftie fiue pipes, but it shoulde be sixtie pipes. Therefore I saie by the rule of thrée, if fiftie fiue pipes doe voide in thirtie minutes: in how manye minutes will sixtie pipes voide? Multiplie and diuide, and you shall finde thirtie twoo minutes 40/55. And in that space will the water auoide if all the cockes be set open.

### ¶ Of the rule of two false positions.

THe summe of this rule of two false positions is thus, when any question is proponed appertaining to this rule. First imagine any nūber at your pleasure, which you shal name the first position, and with the same shall you worke in stede of the true number, as the question doth import, and if you sée that you haue missed. Then is the last number of y^{e} worke either to great or to litle, y^{t} shal you note for to be the first errour, [Page 154] in the which you haue missed with the signe of more, or lesse, whiche signes shalbe noted with these figures, 4 [...]:—This figure 4: betokeneth more, and this plaine line—signifieth lesse, that is to saie the one signifieth to muche & the other to little: then begin again, & take an other number, whiche shal be the second position, and worke by the question as before, if you haue missed againe, note the excesse or want, for that is the seconde errour. Then shall you multiplie the first position by the second errour crossewise, and againe the seconde position by the first errour (and this must alwaies be obserued) & keepe the two productes: then if the signes be both like, that is to saie, either both to muche, or both to little, abate the lesser producte frō the greater and likewise, you shall substract the lesser error from y^{e} greater, & by the remaine of those errours, you shal diuide the residue of the products: the quotient shal be the true number that you séeke for. [Page] But if the twoo signes be vnlyke, that is to say, the one to muche and the other to lyttle, then shall you adde those productes together so shall you also adde bothe the errours together, and by the summe of those errours, diuide the totall summe of both the products: the quotient shall be likewise the true number that the question seketh, and this is the wholerule, as by example. [...]3. A manlying at the point of death, saied that he had in a certain Coffer a hundred duckets, the which he bequea ched to 3. of his frends by him named, after this sorte. The first must haue a certain portion, the 2. must haue twise so manie as the first abatyng 8. Duckets & the third muste haue 3. times so many as the first, lesse by 15 Duckets. Nowe, I demaunde howe many euerie of them must haue. Answere: First I doe imagine that the firste man had thirtie Duckets, then by the order of the question, the seconde shoulde haue fiftie twoo: and the third seuentie fiue. [Page 155] These three summes beyng added together doe make 157: and I shoulde haue but a hundred soe that this fiirste errour is to much by fiftie seuen, then I note a parte the first position thirtie, with hys errour fiftie seuen, to muche after this sorte thirtie. 457.

Therefore I prosecute my worke and I suppose that the firste hadde twentie foure, then by the orderder of the question, the seconde should haue 40. and the third, fiftie seuen, these three summes beyng added together, doe ma [...] 121. and I must haue but a hundred, so the seconde errour, is to muche by 21. Therefore I note 24. 421. vnder the thirtie. 457. as you maye see in the margent of the nexte side following. Then I multiplie crossewaies, thirty (which is the firste position) by twelue which is the seconde errour, and therof commeth 630. likewise I multiplie twentie and four (which is the second position) by fiftie seuen, whiche is the firste errour, and I finde 1368: then [Page] because the signes of the errours are [...] bothe like: that is to saye muche, I must therfore substracte 630. from 1368. and ther wil remain 731 which is the diuidende: again I must substract the lesser error from the greater, that is to wit, 21. out of 57. and there will remain 36. whiche shalbee my diuisor. Thys done I diuide 738 by 36 and the quotient will bee 20. [...]. The which 20. ½ is the iust number of the duckets that the first man had for his parte, so consequently the seconde manne had thirtie three Duckets, and the thirde fourty six ½, as by proofe in the margent maie appeare.

[Page 156] The like number wil also appeare. in case the errours wer both to little, as in makyng the twoo positions by 18. and 20. where you shal finde the [...] two errours both to little, the firste will bée to lyttle by 15. and the seconde to little by 3. as by perusing this worke in the margent you shal well perceiue.

Againe if one of the errours were to muche, and the other to little, yet shall I haue the true number, as before: As if the twoo positions were 24. and 20. I shall finde that the firste errour will be 21. to muche, and the seconde wil be three to little: Therfore I multiplie twentie foure by thirtie, crossewaies, thereof commeth 72. [Page] Likewise I multiplie twentie by 21. [...] the product will be 420: These twoo summes 72. and 420, I adde both together because y^{e} signes of the errours bee vnlyke, and thei mak. 492. the whiche shalbee my diuidende, and againe, I addde the lesser errour three, with the greater errour 21. and they mak 24. for my diuisor, then diuiding foure hundred ninetie two by twenty foure, the quotiet wilbe twentie ½: as in the margent doth plainlie appeare.

And now because you shall not forget this parte of the rule, learne thys briefe remembraunce followyng.

[Page 157] The meanyng whereof is thus yf both the errours haue lyke sygnes, then muste the diuidende and the diuisor bee made by substraction, as is taught before, and if those sygnes bee vnlike, then muste I by addition gather the diuidende, and the diuisor, as I haue done in this laste example.

4. A man hath twoo siluer cups of vnegall waight, hauynge to them bothe, but one couer, the waight whereof is fiue ounces, if the couer be putte to the lesser cuppe, it wyll bee in double proportion vnto the waight of the greater, and the couer béeyng put to the greater cuppe, will bee in triple proportion, vnto the waight of the lesser. I demaunde what was the waight of euery cuppe.

Suppose that the lesser cuppe dyd wayghe seauen ounces, then wyth the couer it muste waigh twelue, and this waight shoulde be in double proportion vnto the greater, therefore [Page] the greater muste waigh six ounces [...] adde vnto it 5 ounces for y^{e} couer, all wil be 11. ounces, but it should be 21 for to haue it in triple proportion, vnto 7. which represēteth y^{e} weght of the lesser cuppe So that this first errour is to little by 10. which you shall note after seuen in this sorte. 7.-10.

After you shall suppose some other number, as 9. and make the like work as before, so shal you finde 15. to little, for the second errour, which you shall put behind nine, and then worke with the reste as aboue is sayed, and you shall finde that the lesser cuppe wayed three ounces, and consequentlye the greater foure ounces.

5. One man demaunded of another [Page 158] in a mornynge what a clocke it was, the other made hym thys aunswere, if you doe [...] (saieth hee) the ½ of the houres whiche he passte synce mindnight, with the ⅓ of the houres whiche are to come vntill noone, you shall haue the iuste houre, that is to saie, you shall knowe what a clocke it was: Suppose that it was 4 [...] clocke in the mornynge, so shoulde there remaine 8. vntill none: then I take the [...]/4 of 4. whiche is 1. and the ⅔ of 8. whiche is 5. ⅓, and I adde them together, so I finde 6. ⅓ and I supposed but 4. therfore thys first errour is to muche by 2. [...], which I note after my position thus 4. † 2. [...]/3: then againe I suppose an other number, that is to saye nine, soe shoulde remaine but 3. houres vntill noone., I take the ¼ of 9. and the ⅔ of 3. whiche is 2. ¼ and 2. these I adde together and they make 4. ¼: but I supposed that it was 9. therfore the second errour is 4. ¾ to little whiche I note behinde my position thus. 9. † 4. 2/4. [Page] And then I multiplie crossewyse, as before [...] is taught, and bicause th [...] sygnes of the errours are vnlike, that is to saye, the one to muche, and the other to lyttle, therefore in thys woorke I muste adde the products, and they will bee fourtie. Likewise I adde the errours, and thei be seuen, [...]. Then I diuide fourtie by seuen 1/12, and thereof commeth fiue houres 11/17, and that houre it was in the mornyng.

## ¶ The fiueth Chapiter treateth of sportes, and pastime, done by number.

IF you would know the number that any man doth thinke or imagine in hys minde, as though you coulde deuine.

Bydde hym triple the same number, then of the product let hym take the [...] [Page] if the number be euen, or els the greater halfe, if the same bee odde, then bid him triple againe the saied ½: after say to him that he put awaie, if he can 36. 27. 18. or. 9. from the laste number beyng tripled: that is to saie, cause hym subtellye to put awaie nine as many times as is possible and kepe the number secretly: and when ye can no more take away 9. then to know if that yet there remaine anye nomber, bid hym abate three and twoo or one, if he can: this doone, see how manie tymes nine you haue caused hym to abate, for the which keepe you in mind so manie tymes 2. and if that you knowe that hee had any thing remaining beside the nines, y^{e} same shall also note vnto you [...].

Suppose that he thought 6. whiche beyng tripled is 18. wherof the [...]/2 is 9. the triple of that is 27: now cause hym to abate 18, or 9. or 27. and againe 9: but then he will saie vnto you that he cannot, bid hym then abate 3. or 2. or 1. [...]e wyl saie also that he can not: wherfore [Page] consideryng that you haue made hym to abate 3. tymes nine iustly, you shall tell him that he thought 6. for 3. tymes 2. maketh 6. If he had thought 5. the triple therof is fiftene, wherfore the greater ½ is 8. the triple of that maketh 24. which conteineth 2. times 9. thei ar worth 4. and the remain signifieth 1. the which added togither make 5. which is the nōber that he thought.

2. If in any companie, one of them hath a ringe vpon his finger, and you woulde know by maner of deuining, who hath the same and vpon what finger and what ioint: cause the persons to sit doune in order, & keepe likewise an order of their fingers: then separate your self from them in some certaine place, and saie vnto one of the lookers on, that hee double the number (markyng the order) of hym that hath the ring: and vnto the double bid hym ad 5. and then cause him to multiply this addition by 5 and vnto the product bid hym adde the number of the finger of [Page] the person whiche hath the ring: bee if that the same last sum did amounte to 89. then afterward say to him y^{t} he put after the same laste number towarde his right hand a figure signifiyng vppon whiche of the iointes hee hath the ring. As if it be vpon the third iointe, lette hym put 3. after 89. and it will bee 893: this doen, you shal aske him what number hee keepeth, from the whiche you shal abate 250. and you shal haue thrée figures remainyng at the least. The fyrste towarde youre lefte hande shall signifie the number of the person whiche hath the ring. The second or middle figure shall represente the number of the finger. And the last figure towarde your right hande, shall be token the number of the ioynte. As if the number whiche hée dyd keepe were 893. from that you shall abate 250. and there will remaine 64 [...] Which do note vnto you, that the sixt person hath the ringe vpon the fourth finger, and vpon his thirde iointe.

[Page] But note that when you haue made your substraction, if there do remaine a cipher in the place of tennes, that is to saie in the seconde place, you muste then abate from that figure whiche is in the place of hundreds, that is to saie from the figure which is next your left hand, & that shalbe worth ten tenthes, signifiyng the tenth finger: as if there should remain 703. you must saye that the sixte person) vpon his tenth finger and vpon his third ioint) hath the ring

3. And after the same maner, if a man do cast thrée dice, you maie know the poincts of euery one of them, for if you do cause him to doube the poincts of one die, & vnto that double to ad 5. and the same sum to multiplye by 5. & vnto the product ad the poincts of one of the other dice, and behind the number towarde the right hand, to put the figure which signifieth the poinctes of y^{e} last die, & then shal you aske him what nūber he kepeth [...] y^{e} which abat 250 [...] ther wil rema [...] figures, which do [Page 161] note vnto you the pointes of euery die.

4. Likewise if 3. of your companions, to saie Peter, Iames, and Ihon, that would (in your absence) giue them selfe euerie one a contrarie name: as for example: Peter woulde be called a king, Iames a duke, and Ihon a countie: And I woulde deuine whiche of them is called the kinge, whiche the duke, and which the county. Take 24. stones, or other peces whatsoeuer, and geue vnto Peter. 1. vnto Iames 2. and vnto Ihon. 3. or otherwise. But marke well vnto whiche of them you haue 1. vnto which 2. and vnto whom 3. Then leuing the 18. stones (before them) that are ramainyng, you shall absent your selfe from their sighte, or elles tourne your face from them, saiyng thus vnto them, whosoeuer nameth himselfe a kinge: for euerie stone that I gaue him lette hym take one of the residue, and he that nameth hymselfe a duke foreuerie stone that I gaue hym lette hym take twoo of them that remain, and he [Page] that calleth hym selfe a countie, for euerie stone that I gaue hym let, hym take foure, this beyng done approache nere them, and marke towe man [...]e stones are remainyng: and know this that there can not remaine any other, but one of these sire, 1, 2, 3, 5, 6, 7, for the whiche sire numbers we haue chosen to euerie of them a seuerall name, whiche are these: Angeli, Beati, qualiter, Messias, Israel, p [...]etas: eche of them containyng three [...]oweiles, a, e, i, whiche doe shewe the names by order: that is to say, a, sheweth whiche is the king, e, telleth which is the duke, i, sheweeth which is the coū [...] folewynge the order how, and to whom you haue geuen 1. [...] [Page 162] stone, to which [...]. and to which [...]. Then there doe remaine but 1 stone, the first name Augeli [...] by these three vowels a, e. i, sheweth that Peter is the kinge, Iames the doke, and Ihon the county. And if there doe remaine 2 stones, the second name Bea [...], shall shewe you by these three vowels, e, a, i, that Peter is the duke. Iames the king, and Ihon the countie. And so of the other as by this Table dooeth plainly appeare.

# ¶ Here beginneth the table of this Booke.

- THe diffinition of number. fol. 1
- Numeration in whole number. Fol. eodem Chap. 1.
- Addition in whole number. ca. 1. fo. 6.
- Substraction. capt. 3. fol. 9.
- Multiplication. capi. 4. fol. 12.
- Diuision. capi. 5. fol. 21.
- Progression Arithmeticall and Geometricall. capi. 6. fol. 32.
- The Rule of three called the Golden Rule. And the backer Rule, vnto all these is added their profes. ca. 7. fo. 34

- The first chapter sheweth what a fraction or a broken number is. fol. 43.
- Reduction of fractions. capi. 2. fol. 44.
- Abbreutation. capi. 3. fol. 50.
- Addition in fract. capi. 4. fol. 54.
- Substraction. capi. 5. fol. 58.
- Multiplication in fract. capi. 6. fol. 61
- [...]sion in broken num. cap. 7. fo. 64
- [Page] Duplation, Triplation, and [...]uadruplation. ca. 8. fol. 68
- All the profes of fractions. ca. 9. fo 69
- Questions done by Reduction, by Addition, Substraction, Multiplication, and by Diuision in broken numbers. Chapter. 10. fol.. 71.

- Rules of practise called briefe rules. capi. 1. fol. 79.
- Rules of three compounde, beyng. 4. in number. capi. 2. fol. 99.
- Questions of the trade of Marchaundise. capi. 3. fol. 101.
- Questions of losse and gaine in the trade of marchaundise. cap. 4. fol. 108.
- Questions of diuers breadthes and lengthes of tapistries. cap. 5. fol. 113.
- Questions of the reducyng of paumes of Genes into yardes. ca. 6. fol. 115.
- Questions of Marchaundise solde by weight, with briefe rules for the same. Chapter. 7. fol. 116.
- Questions of Tares and allowaunces. cap 8. fol. 119.
- Questions done by the double rule of [Page] thrée, cop. 9. fol. 122.
- Questions of the rule of felowshippe. Chapter. 10. fol. 124.
- Questions of bartering. cap. 11. fo. 138.
- Questions of the exchaunges. Chapter. 12. Fol. 146.
- Questions of the rule of Alligation. Chapter. 13. fol. 148.
- Questions of the rule of falsehode. Chapter 14. fol. 152.
- Questions of sporte and pastime do [...]n by number.